{"id":566,"date":"2026-08-20T20:43:16","date_gmt":"2026-08-20T12:43:16","guid":{"rendered":"https:\/\/www.imageproc.cn\/?p=566"},"modified":"2026-08-20T20:43:16","modified_gmt":"2026-08-20T12:43:16","slug":"566","status":"publish","type":"post","link":"https:\/\/www.imageproc.cn\/index.php\/2026\/08\/20\/566\/","title":{"rendered":""},"content":{"rendered":"

# \u4fe1\u53f7\u4e0e\u7cfb\u7edf\u4e2d\u7684\u76f8\u5173\u3001\u4e92\u76f8\u5173\u3001\u5377\u79ef\u3001\u76f8\u5e72\u6027\u4e0e\u8c31\u5206\u6790<\/p>\n

## \u76ee\u5f55<\/p>\n

1. [\u4e00\u3001\u6982\u5ff5\u603b\u89c8\u4e0e\u4fe1\u53f7\u57fa\u672c\u64cd\u4f5c](#\u4e00\u6982\u5ff5\u603b\u89c8\u4e0e\u4fe1\u53f7\u57fa\u672c\u64cd\u4f5c)
\n2. [\u4e8c\u3001\u5377\u79ef\uff1a\u5b9a\u4e49\u3001\u8ba1\u7b97\u4e0e LTI \u7cfb\u7edf](#\u4e8c\u5377\u79ef\u5b9a\u4e49\u8ba1\u7b97\u4e0e-lti-\u7cfb\u7edf)
\n3. [\u4e09\u3001\u76f8\u5173\u4e0e\u81ea\u76f8\u5173\uff1a\u5b9a\u4e49\u3001\u6027\u8d28\u4e0e\u8ba1\u7b97](#\u4e09\u76f8\u5173\u4e0e\u81ea\u76f8\u5173\u5b9a\u4e49\u6027\u8d28\u4e0e\u8ba1\u7b97)
\n4. [\u56db\u3001\u5377\u79ef\u3001\u76f8\u5173\u4e0e Fourier\/DFT \u5bf9\u5076\u5173\u7cfb](#\u56db\u5377\u79ef\u76f8\u5173\u4e0e-fourierdft-\u5bf9\u5076\u5173\u7cfb)
\n5. [\u4e94\u3001\u80fd\u91cf\u8c31\u3001\u529f\u7387\u8c31\u4e0e\u4e92\u8c31](#\u4e94\u80fd\u91cf\u8c31\u529f\u7387\u8c31\u4e0e\u4e92\u8c31)
\n6. [\u516d\u3001\u76f8\u5173\u6027\u4e0e\u7edf\u8ba1\u4fe1\u53f7\u5206\u6790](#\u516d\u76f8\u5173\u6027\u4e0e\u7edf\u8ba1\u4fe1\u53f7\u5206\u6790)
\n7. [\u4e03\u3001\u76f8\u5e72\u6027\u53ca\u5176\u6269\u5c55](#\u4e03\u76f8\u5e72\u6027\u53ca\u5176\u6269\u5c55)
\n8. [\u516b\u3001\u8c31\u4e0e\u76f8\u5e72\u6027\u7684\u4f30\u8ba1\u65b9\u6cd5](#\u516b\u8c31\u4e0e\u76f8\u5e72\u6027\u7684\u4f30\u8ba1\u65b9\u6cd5)
\n9. [\u4e5d\u3001\u901a\u4fe1\u3001\u68c0\u6d4b\u4e0e\u7cfb\u7edf\u8fa8\u8bc6\u5e94\u7528](#\u4e5d\u901a\u4fe1\u68c0\u6d4b\u4e0e\u7cfb\u7edf\u8fa8\u8bc6\u5e94\u7528)
\n10. [\u5341\u3001\u5178\u578b\u4f8b\u9898\u3001\u5e38\u89c1\u8bef\u533a\u4e0e\u68c0\u67e5\u6e05\u5355](#\u5341\u5178\u578b\u4f8b\u9898\u5e38\u89c1\u8bef\u533a\u4e0e\u68c0\u67e5\u6e05\u5355)<\/p>\n

—<\/p>\n

## \u4e00\u3001\u6982\u5ff5\u603b\u89c8\u4e0e\u4fe1\u53f7\u57fa\u672c\u64cd\u4f5c<\/p>\n

\u5728\u4fe1\u53f7\u5904\u7406\u4e2d\uff0c\u7ecf\u5e38\u4f1a\u770b\u5230\u7c7b\u4f3c\u7684\u4e58\u52a0\u8868\u8fbe\u5f0f\uff1a<\/p>\n

$$
\n\\sum_n x[n]h[n-k],
\n\\qquad
\n\\sum_n x[n]y[n+k],
\n\\qquad
\n\\sum_n x[n]y^*[n-k].
\n$$<\/p>\n

\u5b83\u4eec\u7684\u5f62\u5f0f\u5f88\u63a5\u8fd1\uff0c\u4f46\u542b\u4e49\u5e76\u4e0d\u76f8\u540c\u3002\u6700\u5bb9\u6613\u6df7\u6dc6\u7684\u51e0\u4e2a\u8fd0\u7b97\u662f\uff1a<\/p>\n

| \u8fd0\u7b97 | \u5178\u578b\u8868\u8fbe\u5f0f | \u4e3b\u8981\u7528\u9014 |
\n|—|—|—|
\n| \u7ebf\u6027\u5377\u79ef | $y(t)=\\int x(\\tau)h(t-\\tau)\\,d\\tau$ | \u6c42 LTI \u7cfb\u7edf\u8f93\u51fa\u3001\u63cf\u8ff0\u6ee4\u6ce2 |
\n| \u79bb\u6563\u7ebf\u6027\u5377\u79ef | $y[n]=\\sum_k x[k]h[n-k]$ | \u6570\u5b57\u6ee4\u6ce2\u3001\u7cfb\u7edf\u54cd\u5e94 |
\n| \u81ea\u76f8\u5173 | $R_{xx}(\\tau)=\\int x(t)x^*(t-\\tau)\\,dt$ | \u6d4b\u91cf\u4fe1\u53f7\u4e0e\u81ea\u8eab\u7684\u76f8\u4f3c\u6027 |
\n| \u4e92\u76f8\u5173 | $R_{xy}(\\tau)=\\int x(t)y^*(t-\\tau)\\,dt$ | \u6d4b\u91cf\u4e24\u4e2a\u4fe1\u53f7\u7684\u76f8\u4f3c\u6027\u548c\u76f8\u5bf9\u5ef6\u8fdf |
\n| \u5185\u79ef | $\\langle x,y\\rangle=\\int x(t)y^*(t)\\,dt$ | \u67d0\u4e00\u4e2a\u56fa\u5b9a\u5bf9\u9f50\u4f4d\u7f6e\u7684\u76f8\u4f3c\u6027 |<\/p>\n

\u53ef\u4ee5\u5148\u8bb0\u4f4f\u4ee5\u4e0b\u6838\u5fc3\u533a\u522b\uff1a<\/p>\n

> **\u5377\u79ef\u5305\u542b\u201c\u53cd\u8f6c\u540e\u5e73\u79fb\u201d\uff0c\u76f8\u5173\u5305\u542b\u201c\u5171\u8f6d\u3001\u53cd\u8f6c\u540e\u5e73\u79fb\u201d\u6216\u7b49\u4ef7\u7684\u76f8\u5173\u6392\u5217\u3002**<\/p>\n

\u5bf9\u4e8e\u5b9e\u503c\u4fe1\u53f7\uff0c\u5171\u8f6d\u4e0d\u8d77\u4f5c\u7528\uff0c\u6240\u4ee5\u5377\u79ef\u548c\u76f8\u5173\u5728\u56fe\u5f62\u64cd\u4f5c\u4e0a\u770b\u8d77\u6765\u66f4\u52a0\u76f8\u4f3c\uff1b\u5bf9\u4e8e\u590d\u503c\u4fe1\u53f7\uff0c\u5171\u8f6d\u662f\u76f8\u5173\u5b9a\u4e49\u7684\u91cd\u8981\u7ec4\u6210\u90e8\u5206\u3002<\/p>\n

\u53e6\u4e00\u4e2a\u5173\u952e\u533a\u522b\u662f\uff1a<\/p>\n

– \u5377\u79ef\u63cf\u8ff0\u4e00\u4e2a\u4fe1\u53f7\u7ecf\u8fc7\u7cfb\u7edf\u51b2\u6fc0\u54cd\u5e94\u540e\u7684\u7d2f\u79ef\u4f5c\u7528\uff1b
\n– \u76f8\u5173\u63cf\u8ff0\u4e24\u4e2a\u4fe1\u53f7\u5728\u4e0d\u540c\u76f8\u5bf9\u4f4d\u79fb\u4e0b\u7684\u5339\u914d\u7a0b\u5ea6\u3002<\/p>\n

\u4ece Fourier\/DFT \u5bf9\u5076\u5173\u7cfb\u770b\uff0c\u8fd8\u5e94\u7279\u522b\u533a\u5206\u201c\u76f8\u4e58\u201d\u7684\u5177\u4f53\u5bf9\u8c61\uff1a<\/p>\n

| \u65f6\u57df\u76ee\u6807 | \u9891\u57df\u5bf9\u5e94\u5173\u7cfb | \u7ed3\u679c\u7c7b\u578b |
\n|—|—|—|
\n| \u70b9\u5bf9\u70b9\u4e58\u6cd5 $z[n]=x[n]y[n]$ | $Z[k]=\\frac1N(X\\circledast_NY)[k]$ | \u9891\u57df\u5faa\u73af\u5377\u79ef |
\n| \u7ebf\u6027\u5377\u79ef $x*h$ | $XH$ | \u9891\u57df\u666e\u901a\u9010\u70b9\u4e58\u6cd5 |
\n| \u5faa\u73af\u5377\u79ef $x\\circledast_Nh$ | $X[k]H[k]$ | $N$ \u70b9\u9891\u57df\u666e\u901a\u4e58\u6cd5 |
\n| \u4e92\u76f8\u5173 $R_{xy}[m]$ | $X[k]Y^*[k]$ \u540e IDFT | \u6309 lag \u5c55\u5f00\u7684\u76f8\u5173\u5e8f\u5217 |
\n| \u96f6\u5ef6\u8fdf\u5185\u79ef $\\sum_nx[n]y^*[n]$ | $\\frac1N\\sum_kX[k]Y^*[k]$ | \u4e00\u4e2a\u6807\u91cf |
\n| \u81ea\u529f\u7387\u8c31 | $XX^*=|X|^2$ | \u975e\u8d1f\u5b9e\u6570 |
\n| Hermitian \u68af\u5ea6 $\\boldsymbol X^H\\boldsymbol e$ | $X^*E$ \u540e IDFT | \u81ea\u9002\u5e94\u6ee4\u6ce2\u62bd\u5934\u66f4\u65b0\u65b9\u5411 |<\/p>\n

\u56e0\u6b64\uff0c\u201c\u65f6\u57df\u4fe1\u53f7\u76f8\u4e58\u53ef\u7528\u9891\u57df\u590d\u5171\u8f6d\u76f8\u4e58\u66ff\u4ee3\u201d\u4e0d\u662f\u666e\u904d\u89c4\u5f8b\u3002\u65f6\u57df\u70b9\u4e58\u5bf9\u5e94\u9891\u57df\u5377\u79ef\uff1b\u9891\u57df\u4e00\u65b9\u53d6\u5171\u8f6d\u540e\u9010\u70b9\u76f8\u4e58\uff0c\u901a\u5e38\u5bf9\u5e94\u76f8\u5173\u3001\u590d\u5185\u79ef\u3001\u4e92\u8c31\u6216 Hermitian \u68af\u5ea6\u3002\u770b\u5230 $XE^*$ \u6216 $X^*E$ \u65f6\uff0c\u5fc5\u987b\u7ed3\u5408\u76f8\u5173\u65b9\u5411\u3001\u4fe1\u53f7\u89d2\u8272\u548c\u540e\u7eed\u662f\u6c42\u548c\u8fd8\u662f IFFT \u5224\u65ad\u5176\u542b\u4e49\u3002<\/p>\n

—
\n### \u9884\u5907\u77e5\u8bc6\uff1a\u4fe1\u53f7\u7684\u5e73\u79fb\u3001\u53cd\u8f6c\u4e0e\u5171\u8f6d<\/p>\n

\u7406\u89e3\u5377\u79ef\u548c\u76f8\u5173\u4e4b\u524d\uff0c\u5fc5\u987b\u719f\u6089\u4e09\u4e2a\u57fa\u672c\u64cd\u4f5c\u3002<\/p>\n

#### \u65f6\u95f4\u5e73\u79fb<\/p>\n

\u5bf9\u4fe1\u53f7 $x(t)$\uff1a<\/p>\n

$$
\n x(t-t_0)
\n$$<\/p>\n

\u8868\u793a\u5411\u53f3\u5ef6\u8fdf $t_0$\uff1b<\/p>\n

$$
\n x(t+t_0)
\n$$<\/p>\n

\u8868\u793a\u5411\u5de6\u79fb\u52a8 $t_0$\u3002<\/p>\n

\u79bb\u6563\u4fe1\u53f7\u4e2d\uff1a<\/p>\n

$$
\n x[n-n_0]
\n$$<\/p>\n

\u8868\u793a\u5ef6\u8fdf $n_0$ \u4e2a\u91c7\u6837\u70b9\u3002<\/p>\n

#### \u65f6\u95f4\u53cd\u8f6c<\/p>\n

\u65f6\u95f4\u53cd\u8f6c\u4e3a<\/p>\n

$$
\n x(-t)
\n$$<\/p>\n

\u79bb\u6563\u5f62\u5f0f\u4e3a<\/p>\n

$$
\n x[-n].
\n$$<\/p>\n

\u51e0\u4f55\u4e0a\uff0c\u8fd9\u76f8\u5f53\u4e8e\u628a\u4fe1\u53f7\u5173\u4e8e $t=0$ \u6216 $n=0$ \u955c\u50cf\u7ffb\u8f6c\u3002<\/p>\n

#### \u590d\u5171\u8f6d<\/p>\n

\u590d\u4fe1\u53f7<\/p>\n

$$
\n x(t)=x_R(t)+jx_I(t)
\n$$<\/p>\n

\u7684\u5171\u8f6d\u4e3a<\/p>\n

$$
\n x^*(t)=x_R(t)-jx_I(t).
\n$$<\/p>\n

\u590d\u5185\u79ef\u901a\u5e38\u5b9a\u4e49\u4e3a<\/p>\n

$$
\n\\langle x,y\\rangle=\\int x(t)y^*(t)\\,dt
\n$$<\/p>\n

\u6216\u79bb\u6563\u5f62\u5f0f<\/p>\n

$$
\n\\langle x,y\\rangle=\\sum_n x[n]y^*[n].
\n$$<\/p>\n

\u8fd9\u6837\u5b9a\u4e49\u53ef\u4ee5\u4fdd\u8bc1<\/p>\n

$$
\n\\langle x,x\\rangle=\\int |x(t)|^2dt\\ge 0,
\n$$<\/p>\n

\u56e0\u6b64\u5b83\u80fd\u591f\u8868\u793a\u80fd\u91cf\u6216\u5e73\u65b9\u8303\u6570\u3002<\/p>\n

#### \u4e09\u4e2a\u64cd\u4f5c\u7684\u7ec4\u5408<\/p>\n

\u5377\u79ef\u4e2d\u7684\u88ab\u5377\u79ef\u4fe1\u53f7\u901a\u5e38\u7ecf\u5386\uff1a<\/p>\n

1. \u53cd\u8f6c\uff1a$h(\\tau)\\to h(-\\tau)$\uff1b
\n2. \u5e73\u79fb\uff1a$h(-\\tau)\\to h(t-\\tau)$\uff1b
\n3. \u4e0e $x(\\tau)$ \u76f8\u4e58\u5e76\u79ef\u5206\u3002<\/p>\n

\u76f8\u5173\u4e2d\u7684\u4fe1\u53f7\u901a\u5e38\u7ecf\u5386\uff1a<\/p>\n

1. \u5171\u8f6d\uff1a$y(\\tau)\\to y^*(\\tau)$\uff1b
\n2. \u53cd\u8f6c\u548c\u5e73\u79fb\uff1a$y^*(\\tau)\\to y^*(\\tau-t)$\uff1b
\n3. \u4e0e $x(\\tau)$ \u76f8\u4e58\u5e76\u79ef\u5206\u3002<\/p>\n

—
\n### \u80fd\u91cf\u4fe1\u53f7\u4e0e\u529f\u7387\u4fe1\u53f7<\/p>\n

\u8ba8\u8bba\u80fd\u91cf\u8c31\u548c\u529f\u7387\u8c31\u524d\uff0c\u5fc5\u987b\u5148\u533a\u5206\u80fd\u91cf\u4fe1\u53f7\u4e0e\u529f\u7387\u4fe1\u53f7\u3002\u8fd9\u4e2a\u5206\u7c7b\u51b3\u5b9a\u4e86\u5e94\u4f7f\u7528 Fourier \u53d8\u6362\u8fd8\u662f\u529f\u7387\u8c31\u5bc6\u5ea6\u6765\u63cf\u8ff0\u9891\u7387\u5206\u5e03\u3002<\/p>\n

#### \u8fde\u7eed\u65f6\u95f4\u4fe1\u53f7\u7684\u80fd\u91cf\u548c\u5e73\u5747\u529f\u7387<\/p>\n

\u5bf9\u4fe1\u53f7 $x(t)$\uff0c\u603b\u80fd\u91cf\u5b9a\u4e49\u4e3a<\/p>\n

$$
\n\\boxed{
\nE_x=\\int_{-\\infty}^{\\infty}|x(t)|^2dt
\n}.
\n$$<\/p>\n

\u5e73\u5747\u529f\u7387\u5b9a\u4e49\u4e3a\u957f\u671f\u65f6\u95f4\u5e73\u5747\uff1a<\/p>\n

$$
\n\\boxed{
\nP_x=\\lim_{T\\to\\infty}\\frac{1}{2T}
\n\\int_{-T}^{T}|x(t)|^2dt
\n}.
\n$$<\/p>\n

#### \u79bb\u6563\u65f6\u95f4\u4fe1\u53f7\u7684\u80fd\u91cf\u548c\u5e73\u5747\u529f\u7387<\/p>\n

\u79bb\u6563\u65f6\u95f4\u4fe1\u53f7\u7684\u80fd\u91cf\u4e3a<\/p>\n

$$
\n\\boxed{
\nE_x=\\sum_{n=-\\infty}^{\\infty}|x[n]|^2
\n}.
\n$$<\/p>\n

\u5e73\u5747\u529f\u7387\u4e3a<\/p>\n

$$
\n\\boxed{
\nP_x=\\lim_{N\\to\\infty}\\frac{1}{2N+1}
\n\\sum_{n=-N}^{N}|x[n]|^2
\n}.
\n$$<\/p>\n

\u82e5\u65f6\u95f4\u8d77\u70b9\u4e0d\u91cd\u8981\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528 $0$ \u5230 $N-1$ \u7684\u957f\u7a97\u53e3\u5b9a\u4e49\u76f8\u540c\u7684\u6781\u9650\u3002<\/p>\n

#### \u80fd\u91cf\u4fe1\u53f7<\/p>\n

\u82e5<\/p>\n

$$
\n00$\u3001$b>0$ \u4e14 $a\\ne b$\u3002\u5bf9\u4e8e $t\\ge0$\uff1a<\/p>\n

$$
\n\\begin{aligned}
\ny(t)
\n&=\\int_0^t e^{-a\\tau}e^{-b(t-\\tau)}d\\tau\\\\
\n&=e^{-bt}\\int_0^t e^{-(a-b)\\tau}d\\tau\\\\
\n&=\\frac{e^{-bt}-e^{-at}}{a-b}u(t).
\n\\end{aligned}
\n$$<\/p>\n

\u5377\u79ef\u7ed3\u679c\u7684\u8d77\u59cb\u65f6\u95f4\u7531\u4e24\u4e2a\u4fe1\u53f7\u7684\u8d77\u59cb\u65f6\u95f4\u5171\u540c\u51b3\u5b9a\uff0c\u7cfb\u7edf\u54cd\u5e94\u7684\u8870\u51cf\u5f62\u5f0f\u7531\u8f93\u5165\u548c\u7cfb\u7edf\u6781\u70b9\u5171\u540c\u51b3\u5b9a\u3002<\/p>\n

#### \u8fde\u7eed\u65f6\u95f4\u5377\u79ef\u7684\u91cd\u53e0\u89e3\u91ca<\/p>\n

\u5bf9\u6709\u9650\u65f6\u957f\u4fe1\u53f7\uff0c\u4e92\u76f8\u5173<\/p>\n

$$
\nR_{xy}(\\tau)=\\int x(t)y^*(t-\\tau)dt
\n$$<\/p>\n

\u540c\u6837\u53ef\u4ee5\u7406\u89e3\u4e3a\u4e24\u4e2a\u4fe1\u53f7\u5728\u76f8\u5bf9\u5e73\u79fb\u540e\u91cd\u53e0\u90e8\u5206\u7684\u52a0\u6743\u9762\u79ef\u3002\u4e24\u4e2a\u6ce2\u5f62\u8d8a\u5339\u914d\uff0c\u91cd\u53e0\u4e58\u79ef\u7684\u79ef\u5206\u8d8a\u5927\u3002<\/p>\n

—
\n### \u5377\u79ef\u7684\u91cd\u8981\u6027\u8d28\u4e0e\u5e38\u7528\u7ed3\u8bba<\/p>\n

#### \u5377\u79ef\u7684\u4ea4\u6362\u5f8b<\/p>\n

$$
\n x*h=h*x.
\n$$<\/p>\n

\u8fd9\u8bf4\u660e\u5377\u79ef\u4e2d\u4e24\u4e2a\u4fe1\u53f7\u7684\u89d2\u8272\u53ef\u4ee5\u4ea4\u6362\u3002<\/p>\n

#### \u5377\u79ef\u7684\u7ed3\u5408\u5f8b<\/p>\n

$$
\n (x*h)*g=x*(h*g).
\n$$<\/p>\n

\u5728\u4e32\u8054\u7cfb\u7edf\u4e2d\uff0c\u53ef\u4ee5\u5148\u5408\u5e76\u7cfb\u7edf\u7684\u51b2\u6fc0\u54cd\u5e94\uff0c\u518d\u4e0e\u8f93\u5165\u5377\u79ef\u3002<\/p>\n

#### \u5377\u79ef\u7684\u5206\u914d\u5f8b<\/p>\n

$$
\n x*(h_1+h_2)=x*h_1+x*h_2.
\n$$<\/p>\n

\u8fd9\u5bf9\u5e94\u7ebf\u6027\u7cfb\u7edf\u7684\u53e0\u52a0\u6027\u8d28\u3002<\/p>\n

#### \u5fae\u5206\u4e0e\u5377\u79ef<\/p>\n

\u5728\u9002\u5f53\u8fb9\u754c\u6761\u4ef6\u4e0b\uff1a<\/p>\n

$$
\n\\frac{d}{dt}(x*h)
\n=\\frac{dx}{dt}*h
\n=x*\\frac{dh}{dt}.
\n$$<\/p>\n

\u8fd9\u4f7f\u5f97\u53ef\u4ee5\u5148\u5fae\u5206\u8f93\u5165\u6216\u7cfb\u7edf\u54cd\u5e94\uff0c\u518d\u8fdb\u884c\u5377\u79ef\u3002<\/p>\n

#### \u56e0\u679c\u6027\u4e0e\u5377\u79ef\u652f\u6491\u96c6<\/p>\n

\u82e5 $x(t)$ \u548c $h(t)$ \u90fd\u662f\u56e0\u679c\u4fe1\u53f7\uff0c\u5373 $t<0$ \u65f6\u4e3a\u96f6\uff0c\u5219\n\n$$\n(x*h)(t)=0,\\qquad t<0.\n$$\n\n\u66f4\u4e00\u822c\u5730\uff0c\u4e24\u4e2a\u4fe1\u53f7\u7684\u975e\u96f6\u533a\u95f4\u5377\u79ef\u540e\uff0c\u5176\u652f\u6491\u96c6\u662f\u539f\u4e24\u4e2a\u652f\u6491\u96c6\u7684 Minkowski \u548c\u3002\u6709\u9650\u79bb\u6563\u5e8f\u5217\u957f\u5ea6\u6ee1\u8db3 $N+M-1$\uff0c\u5c31\u662f\u8fd9\u4e00\u7ed3\u8bba\u7684\u79bb\u6563\u4f53\u73b0\u3002\n\n#### \u590d\u6570\u4fe1\u53f7\u7684\u5377\u79ef\uff1a\u4e0d\u4f7f\u7528\u5171\u8f6d\u4e58\u6cd5\n\n\u5f53\u4fe1\u53f7\u4e3a\u590d\u503c\u65f6\uff0c\u7ebf\u6027\u5377\u79ef\u7684\u5b9a\u4e49\u4e0d\u53d8\uff1a\n\n$$\n\\boxed{\ny(t)=\\int_{-\\infty}^{\\infty}x(\\tau)\\,h(t-\\tau)\\,d\\tau,\n\\qquad\ny[n]=\\sum_{k=-\\infty}^{\\infty}x[k]\\,h[n-k]\n}.\n$$\n\n\u88ab\u79ef\u51fd\u6570\u4e2d\u7684 $x(\\tau)\\,h(t-\\tau)$\uff08\u6216 $x[k]\\,h[n-k]$\uff09\u662f\u666e\u901a\u7684\u590d\u6570\u4e58\u6cd5\uff0c\u4e0d\u542b\u4efb\u4f55\u5171\u8f6d\u3002\u8fd9\u4e00\u70b9\u5728\u4fe1\u53f7\u4ece\u5b9e\u503c\u63a8\u5e7f\u5230\u590d\u503c\u65f6\u4fdd\u6301\u4e0d\u53d8\u3002\n\n\u5171\u8f6d\u4e58\u6cd5\u51fa\u73b0\u5728\u53e6\u4e00\u7ec4\u8fd0\u7b97\u4e2d\u2014\u2014\u4e92\u76f8\u5173\u3001\u81ea\u76f8\u5173\u548c\u590d\u5185\u79ef\uff1a\n\n| \u8fd0\u7b97 | \u65f6\u57df\u4e58\u6cd5\u5f62\u5f0f | \u9891\u57df\u5bf9\u5e94 |\n|---|---|---|\n| \u7ebf\u6027\u5377\u79ef $(x*h)(t)$ | $x(\\tau)\\,h(t-\\tau)$\uff08\u666e\u901a\u4e58\u6cd5\uff09 | $X(\\omega)H(\\omega)$ |\n| \u4e92\u76f8\u5173 $R_{xy}(\\tau)$ | $x(t)\\,y^*(t-\\tau)$\uff08\u5171\u8f6d\u4e58\u6cd5\uff09 | $X(\\omega)Y^*(\\omega)$ |\n| \u590d\u5185\u79ef $\\langle x,y\\rangle$ | $x(t)\\,y^*(t)$\uff08\u5171\u8f6d\u4e58\u6cd5\uff09 | $\\frac{1}{2\\pi}\\int X(\\omega)Y^*(\\omega)d\\omega$ |\n\n\u9891\u57df\u540c\u6837\u5982\u6b64\uff1a\u5377\u79ef\u5b9a\u7406\u7ed9\u51fa $Y=XH$\uff0c\u4e0d\u542b\u5171\u8f6d\uff1b\u76f8\u5173\u5b9a\u7406\u7ed9\u51fa $XY^*$\uff0c\u542b\u5171\u8f6d\u3002\n\n**\u590d\u6307\u6570\u9a8c\u8bc1\u3002** \u53d6 $x[n]=e^{j\\omega_0 n}$\u3001$h[n]=e^{j\\omega_1 n}$\uff08\u5747\u4e3a\u590d\u4fe1\u53f7\uff09\u3002\u5377\u79ef\u4e58\u79ef\u4e3a\n\n$$\nx[k]\\,h[n-k]=e^{j\\omega_0 k}\\cdot e^{j\\omega_1(n-k)}=e^{j\\omega_1 n}\\cdot e^{j(\\omega_0-\\omega_1)k}.\n$$\n\n\u5bf9 $k$ \u6c42\u548c\u540e\uff0c\u53ea\u6709 $\\omega_0=\\omega_1$ \u65f6\u4e0d\u62b5\u6d88\uff0c\u8f93\u51fa\u9891\u7387\u4e3a $\\omega_0+\\omega_1$ \u65b9\u5411\u7684\u8d21\u732e\u2014\u2014\u9891\u7387\u76f8\u52a0\u3002\n\n\u800c\u4e92\u76f8\u5173\u4e58\u79ef\u4e3a\n\n$$\nx[n]\\,h^*[n-m]=e^{j\\omega_0 n}\\cdot e^{-j\\omega_1(n-m)}=e^{j(\\omega_0-\\omega_1)n}\\cdot e^{j\\omega_1 m}.\n$$\n\n$\\omega_0=\\omega_1$ \u65f6\u4e58\u79ef\u4e0d\u518d\u65cb\u8f6c\uff0c\u76f8\u5173\u5cf0\u6700\u5927\u2014\u2014\u5171\u8f6d\u628a\u9891\u7387\u6bd4\u8f83\u53d8\u6210\u4e86\u51cf\u6cd5\u3002\n\n\u56e0\u6b64\uff0c\u201c\u590d\u6570\u4fe1\u53f7\u9700\u8981\u5171\u8f6d\u4e58\u6cd5\u201d\u662f\u76f8\u5173\u7684\u6027\u8d28\uff0c\u4e0d\u662f\u5377\u79ef\u56e0\u4fe1\u53f7\u53d8\u4e3a\u590d\u6570\u540e\u7684\u81ea\u52a8\u89c4\u5219\u3002\u5173\u4e8e\u5171\u8f6d\u5728\u76f8\u5173\u4e2d\u7684\u4f5c\u7528\u548c\u5fc5\u8981\u6027\uff0c\u8be6\u89c1\u7b2c\u56db\u7ae0\u201c\u76f8\u5173\u662f\u5e26\u5171\u8f6d\u7684\u5377\u79ef\u53d8\u4f53\u201d\u548c\u201c\u4e3a\u4ec0\u4e48\u76f8\u5173\u4e2d\u6709\u5171\u8f6d\u201d\u3002\n\n---\n### \u7cfb\u7edf\u5206\u6790\u4e2d\u7684\u5377\u79ef\n\n#### LTI \u7cfb\u7edf\u7684\u51b2\u6fc0\u54cd\u5e94\n\n\u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf\uff08LTI\uff09\u7684\u6838\u5fc3\u6027\u8d28\u662f\uff1a\u7cfb\u7edf\u5b8c\u5168\u7531\u51b2\u6fc0\u54cd\u5e94 $h(t)$ \u6216 $h[n]$ \u63cf\u8ff0\u3002\n\n\u8fde\u7eed\u65f6\u95f4\u7cfb\u7edf\u8f93\u5165\u4e3a $x(t)$ \u65f6\uff1a\n\n$$\n\\boxed{\n y(t)=x(t)*h(t)\n}.\n$$\n\n\u79bb\u6563\u65f6\u95f4\u7cfb\u7edf\u8f93\u5165\u4e3a $x[n]$ \u65f6\uff1a\n\n$$\n\\boxed{\n y[n]=x[n]*h[n]\n}.\n$$\n\n#### \u4ece\u51b2\u6fc0\u5206\u89e3\u63a8\u5bfc\u5377\u79ef\n\n\u5229\u7528\u8fde\u7eed\u65f6\u95f4\u51b2\u6fc0\u5206\u89e3\uff1a\n\n$$\n x(t)=\\int_{-\\infty}^{\\infty}x(\\tau)\\delta(t-\\tau)d\\tau.\n$$\n\n\u7531\u4e8e\u7cfb\u7edf\u5bf9\u4f4d\u4e8e $\\tau$ \u7684\u51b2\u6fc0\u54cd\u5e94\u4e3a $h(t-\\tau)$\uff0c\u6839\u636e\u7ebf\u6027\u53e0\u52a0\uff1a\n\n$$\n y(t)=\\int_{-\\infty}^{\\infty}x(\\tau)h(t-\\tau)d\\tau.\n$$\n\n\u79bb\u6563\u65f6\u95f4\u4e2d\uff1a\n\n$$\n x[n]=\\sum_kx[k]\\delta[n-k],\n$$\n\n\u6240\u4ee5\n\n$$\n y[n]=\\sum_kx[k]h[n-k].\n$$\n\n#### \u51b2\u6fc0\u5206\u89e3\u7684\u4e25\u683c\u6b65\u9aa4\n\n\u4e0a\u8ff0\u201c\u5206\u89e3\u2014\u54cd\u5e94\u2014\u53e0\u52a0\u201d\u7684\u601d\u60f3\u53ef\u4ee5\u62c6\u6210\u56db\u4e2a\u53ef\u4ee5\u9010\u6761\u68c0\u9a8c\u7684\u6b65\u9aa4\uff0c\u65b9\u4fbf\u7406\u89e3\u6bcf\u4e00\u6b65\u7528\u5230\u7684\u5047\u8bbe\uff1a\n\n1. **\u91c7\u6837\u8868\u793a**\uff1a\u628a\u8f93\u5165\u5199\u6210\u52a0\u6743\u51b2\u6fc0\u7684\u53e0\u52a0\n $$\n x[n]=\\sum_{k\\in\\mathbb{Z}}x[k]\\,\\delta[n-k].\n $$\n \u8fd9\u662f\u6052\u7b49\u5f0f\uff0c\u4e0d\u6d89\u53ca\u7cfb\u7edf\u6027\u8d28\u3002\n2. **\u65f6\u4e0d\u53d8\u6027**\uff1a\u8bbe\u7cfb\u7edf\u5bf9 $\\delta[n]$ \u7684\u54cd\u5e94\u4e3a $h[n]$\uff0c\u5219\u5bf9 $\\delta[n-k]$ \u7684\u54cd\u5e94\u4e3a $h[n-k]$\u3002\n3. **\u9f50\u6b21\u6027**\uff1a\u5bf9\u7cfb\u6570 $x[k]$ \u7684\u8f93\u5165 $x[k]\\delta[n-k]$\uff0c\u7cfb\u7edf\u8f93\u51fa\u4e3a $x[k]h[n-k]$\u3002\n4. **\u53ef\u52a0\u6027**\uff1a\u628a\u6240\u6709 $k$ \u4e0a\u7684\u54cd\u5e94\u76f8\u52a0\uff0c\u5f97\u5230\n $$\n y[n]=\\sum_{k\\in\\mathbb{Z}}x[k]h[n-k]=(x*h)[n].\n $$\n\n\u5176\u4e2d\u6b65\u9aa4 2 \u7528\u5230\u65f6\u4e0d\u53d8\u6027\uff0c\u6b65\u9aa4 3\u30014 \u5408\u8d77\u6765\u5bf9\u5e94\u7ebf\u6027\u6027\u8d28\u3002\u7f3a\u5931\u5176\u4e2d\u4efb\u4f55\u4e00\u6761\uff0c\u5377\u79ef\u516c\u5f0f\u90fd\u4e0d\u6210\u7acb\u3002\u8fde\u7eed\u65f6\u95f4\u7684\u63a8\u5bfc\u53ea\u9700\u628a\u6c42\u548c\u6362\u6210\u79ef\u5206\uff0c\u628a $\\delta[n-k]$ \u6362\u6210 $\\delta(t-\\tau)$\uff0c\u5e76\u4fdd\u8bc1\u5728\u5408\u9002\u7684\u51fd\u6570\u7a7a\u95f4\u4e2d\u5141\u8bb8\u4ea4\u6362\u79ef\u5206\u4e0e\u7ebf\u6027\u7b97\u5b50\u7684\u987a\u5e8f\u3002\n\n#### \u6709\u9650\u5e8f\u5217\u5377\u79ef\u7684 Toeplitz \u77e9\u9635\u5f62\u5f0f\n\n\u5bf9\u6709\u9650\u957f\u5ea6\u5e8f\u5217 $x=[x_0,\\dots,x_{N-1}]^{\\mathsf T}$\u3001$h=[h_0,\\dots,h_{M-1}]^{\\mathsf T}$\uff0c\u7ebf\u6027\u5377\u79ef $y=x*h$ \u957f\u5ea6\u4e3a $N+M-1$\uff0c\u53ef\u4ee5\u5199\u6210\u77e9\u9635\u2013\u5411\u91cf\u4e58\u79ef\uff1a\n\n$$\n y=H\\,x,\n$$\n\n\u5176\u4e2d $H$ \u662f\u7531 $h$ \u6784\u9020\u7684 $(N+M-1)\\times N$ **Toeplitz \u77e9\u9635**\uff0c\u7b2c $k$ \u5217\u662f\u628a $h$ \u5411\u4e0b\u5e73\u79fb $k$ \u4f4d\u3001\u5176\u4f59\u5143\u7d20\u8865\u96f6\u5f97\u5230\u7684\u5411\u91cf\u3002\u4ee5 $N=3$\u3001$M=3$ \u4e3a\u4f8b\uff1a\n\n$$\nH=\n\\begin{bmatrix}\nh_0 & 0 & 0 \\\\\nh_1 & h_0 & 0 \\\\\nh_2 & h_1 & h_0 \\\\\n0 & h_2 & h_1 \\\\\n0 & 0 & h_2\n\\end{bmatrix},\n\\qquad\n y=H\\,[x_0,x_1,x_2]^{\\mathsf T}.\n$$\n\n\u7531\u4ea4\u6362\u5f8b\uff0c\u4e5f\u53ef\u4ee5\u5199\u6210 $y=X\\,h$\uff0c\u5176\u4e2d $X$ \u662f\u7531 $x$ \u6784\u9020\u7684 $(N+M-1)\\times M$ Toeplitz \u77e9\u9635\u3002\u8fd9\u4e2a\u89c2\u5bdf\u5e26\u6765\u51e0\u4e2a\u76f4\u63a5\u7684\u7528\u5904\uff1a\n\n- **\u6570\u503c\u5b9e\u73b0**\uff1a$H$ \u6bcf\u4e00\u6761\u5bf9\u89d2\u7ebf\u7684\u503c\u76f8\u540c\uff0c\u65e0\u9700\u663e\u5f0f\u5b58\u50a8\u6574\u4e2a\u77e9\u9635\uff1bMATLAB \u4e2d\u7684 `convmtx`\u3001Python \u4e2d\u7684 `scipy.linalg.toeplitz` \u5c31\u662f\u6839\u636e\u8fd9\u4e00\u7ed3\u6784\u6784\u9020\u7684\u3002\n- **\u53cd\u5377\u79ef**\uff1a\u4f30\u8ba1 $x$ \u65f6\u53ef\u4ee5\u6c42\u89e3 $\\min_x\\|Hx-y\\|^2$ \u6216 Tikhonov \u6b63\u5219\u5316\u95ee\u9898\uff0c\u6bd4\u65f6\u57df\u9012\u63a8\u66f4\u7a33\u3002\n- **\u5faa\u73af\u5377\u79ef\u7684\u7c7b\u6bd4**\uff1a\u628a Toeplitz \u7ed3\u6784\u201c\u56de\u7ed5\u201d\u6210 $N\\times N$ \u7684 **\u5faa\u73af\u77e9\u9635**\uff08circulant\uff09\uff0c\u5c31\u662f $N$ \u70b9\u5faa\u73af\u5377\u79ef\uff1b\u5faa\u73af\u77e9\u9635\u603b\u662f\u7531 DFT \u5bf9\u89d2\u5316\uff0c\u8fd9\u6b63\u662f\u7b2c\u56db\u7ae0 FFT \u5feb\u901f\u5377\u79ef\u7684\u4ee3\u6570\u6839\u6e90\u3002\n\n#### \u56e0\u679c\u6027\u4e0e\u5377\u79ef\u652f\u6491\u96c6\u7684\u66f4\u7cbe\u7ec6\u63cf\u8ff0\n\n\u5bf9\u8fde\u7eed\u65f6\u95f4\u4fe1\u53f7\uff0c\u5b9a\u4e49\u652f\u6491\u96c6 $\\operatorname{supp}(x)=\\overline{\\{t:x(t)\\ne 0\\}}$\uff1b\u79bb\u6563\u60c5\u5f62\u7c7b\u4f3c\u3002\u5219\u7ebf\u6027\u5377\u79ef\u6ee1\u8db3\n\n$$\n\\operatorname{supp}(x*h)\\subseteq \\operatorname{supp}(x)+\\operatorname{supp}(h),\n$$\n\n\u5176\u4e2d\u53f3\u4fa7\u662f Minkowski \u548c $\\{a+b:a\\in A,b\\in B\\}$\u3002\u7531\u6b64\u63a8\u51fa\uff1a\n\n- \u82e5 $x$ \u4e0e $h$ \u5747\u56e0\u679c\uff08\u652f\u6491\u96c6\u5728 $[0,\\infty)$ \u4e0a\uff09\uff0c\u5219 $x*h$ \u4e5f\u56e0\u679c\uff1b\n- \u82e5 $\\operatorname{supp}(x)\\subseteq[a_1,b_1]$\uff0c$\\operatorname{supp}(h)\\subseteq[a_2,b_2]$\uff0c\u5219 $\\operatorname{supp}(x*h)\\subseteq[a_1+a_2,b_1+b_2]$\uff1b\n- \u53cd\u56e0\u679c\u3001\u53cc\u8fb9\u4fe1\u53f7\u4e5f\u53ef\u4ee5\u6309\u540c\u6837\u65b9\u5f0f\u5224\u5b9a\u5176\u5377\u79ef\u8f93\u51fa\u7684\u652f\u6491\u533a\u95f4\u3002\n\n\u56e0\u679c LTI \u7cfb\u7edf\u7684\u7b49\u4ef7\u523b\u753b\u662f $h(t)=0,\\ t<0$\u3002\u5bf9\u53ef\u7528\u8f93\u5165 $x(t)$ \u7684\u54cd\u5e94\n\n$$\ny(t)=\\int_{0}^{\\infty}h(\\tau)x(t-\\tau)d\\tau\n$$\n\n\u53ea\u4f7f\u7528\u8fc7\u53bb\u548c\u73b0\u5728\u7684\u8f93\u5165\u3002\u5de5\u7a0b\u4e0a\u5224\u65ad\u7cfb\u7edf\u662f\u5426\u56e0\u679c\uff0c\u6700\u5feb\u7684\u529e\u6cd5\u5c31\u662f\u753b\u51fa $h(t)$ \u6216 $h[n]$\uff0c\u770b\u662f\u5426\u542b\u6709 $t<0$ \u6216 $n<0$ \u7684\u6837\u672c\u3002\n\n#### \u9891\u57df\u7cfb\u7edf\u51fd\u6570\n\n\u5bf9 LTI \u7cfb\u7edf\u505a Fourier \u53d8\u6362\uff1a\n\n$$\nY(\\omega)=X(\\omega)H(\\omega).\n$$\n\n\u56e0\u6b64\n\n$$\nH(\\omega)=\\frac{Y(\\omega)}{X(\\omega)}\n$$\n\n\u8868\u793a\u7cfb\u7edf\u5bf9\u5404\u9891\u7387\u5206\u91cf\u7684\u5e45\u5ea6\u548c\u76f8\u4f4d\u54cd\u5e94\u3002\n\n\u82e5\u8f93\u5165\u4e3a\u590d\u6307\u6570\n\n$$\n x(t)=e^{j\\omega t},\n$$\n\n\u5219\u8f93\u51fa\u4e3a\n\n$$\n y(t)=H(\\omega)e^{j\\omega t}.\n$$\n\n\u8fd9\u8bf4\u660e\u590d\u6307\u6570\u662f LTI \u7cfb\u7edf\u7684\u7279\u5f81\u51fd\u6570\uff0c\u5377\u79ef\u5728\u9891\u57df\u4e2d\u53d8\u6210\u4e58\u6cd5\u3002\n\n#### \u7ea7\u8054\u548c\u5e76\u8054\u7cfb\u7edf\n\n\u4e24\u4e2a LTI \u7cfb\u7edf\u7ea7\u8054\u65f6\uff1a\n\n$$\n h_{\\mathrm{eq}}=h_1*h_2,\n$$\n\n\u9891\u57df\u4e3a\n\n$$\n H_{\\mathrm{eq}}(\\omega)=H_1(\\omega)H_2(\\omega).\n$$\n\n\u4e24\u4e2a\u7cfb\u7edf\u5e76\u8054\u65f6\uff1a\n\n$$\n h_{\\mathrm{eq}}=h_1+h_2,\n$$\n\n\u9891\u57df\u4e3a\n\n$$\n H_{\\mathrm{eq}}(\\omega)=H_1(\\omega)+H_2(\\omega).\n$$\n\n#### \u5377\u79ef\u4e0e\u7cfb\u7edf\u7a33\u5b9a\u6027\n\n\u79bb\u6563\u65f6\u95f4 BIBO \u7a33\u5b9a\u7684 LTI \u7cfb\u7edf\u9700\u8981\u6ee1\u8db3\n\n$$\n\\sum_n|h[n]|<\\infty.\n$$\n\n\u8fde\u7eed\u65f6\u95f4\u5bf9\u5e94\u6761\u4ef6\u4e3a\n\n$$\n\\int_{-\\infty}^{\\infty}|h(t)|dt<\\infty.\n$$\n\n\u8fd9\u662f\u56e0\u4e3a\u5bf9\u4e8e\u6709\u754c\u8f93\u5165 $|x(t)|\\le M$\uff1a\n\n$$\n|y(t)|\n\\le M\\int|h(\\tau)|d\\tau.\n$$\n\n#### BIBO \u7a33\u5b9a\u6027\u7684\u5145\u8981\u6027\u4e0e\u5e38\u89c1\u5224\u636e\n\n\u4e0a\u5f0f\u7ed9\u51fa\u7684\u662f\u5145\u5206\u6027\u3002\u5fc5\u8981\u6027\u53ef\u4ee5\u901a\u8fc7\u6784\u9020\u6700\u574f\u8f93\u5165\u8bc1\u660e\uff1a\u53d6\n\n$$\nx[n]=\\operatorname{sgn}(h^*[-n]),\\qquad |x[n]|\\le 1,\n$$\n\n\u5219 $n=0$ \u5904\u7684\u8f93\u51fa\n\n$$\ny[0]=\\sum_k x[k]h[-k]=\\sum_k |h[-k]|=\\sum_k|h[k]|.\n$$\n\n\u82e5 $\\sum|h[n]|=\\infty$\uff0c\u5c31\u5b58\u5728\u6709\u754c\u8f93\u5165\u4f7f\u8f93\u51fa\u65e0\u754c\uff1b\u56e0\u6b64 $h\\in\\ell^1$ \u662f\u79bb\u6563\u65f6\u95f4 BIBO \u7a33\u5b9a\u7684\u5145\u8981\u6761\u4ef6\u3002\u8fde\u7eed\u65f6\u95f4\u7684\u8bc1\u660e\u601d\u8def\u4e00\u81f4\uff0c\u53ea\u9700\u4fdd\u8bc1\u9009\u53d6\u7684 $x(t)$ \u53ef\u6d4b\u4e14\u6709\u754c\u3002\n\n\u5224\u65ad\u7a33\u5b9a\u6027\u65f6\uff0c\u5e38\u7528\u51e0\u79cd\u89d2\u5ea6\u5bf9\u7167\uff1a\n\n- **\u65f6\u57df**\uff1a\u76f4\u63a5\u8ba1\u7b97\u6216\u4f30\u8ba1 $\\sum|h[n]|$\u3001$\\int|h(t)|dt$\uff0c\u4f8b\u5982\u5224\u65ad $h[n]=a^nu[n]$ \u662f\u5426\u7a33\u5b9a\uff0c\u53ea\u9700 $|a|<1$\uff1b\n- **\u4f20\u9012\u51fd\u6570**\uff1a\u8fde\u7eed\u7cfb\u7edf\u7684\u6240\u6709\u6781\u70b9\u4f4d\u4e8e\u5de6\u534a\u5e73\u9762\uff08\u4e25\u683c $\\operatorname{Re}\\{s_i\\}<0$\uff09\u7b49\u4ef7\u4e8e\u56e0\u679c LTI \u7cfb\u7edf BIBO \u7a33\u5b9a\uff1b\u79bb\u6563\u7cfb\u7edf\u5bf9\u5e94\u6240\u6709\u6781\u70b9\u4f4d\u4e8e\u5355\u4f4d\u5706\u5185\uff08$|z_i|<1$\uff09\uff1b\n- **\u9891\u7387\u54cd\u5e94\u5b58\u5728\u6027**\uff1a$h\\in\\ell^1$ \u4fdd\u8bc1 DTFT $H(e^{j\\hat\\omega})$ \u4f5c\u4e3a\u666e\u901a\u8fde\u7eed\u51fd\u6570\u5b58\u5728\uff0c\u5de5\u7a0b\u4e0a\u5e38\u53cd\u8fc7\u6765\u7528\u201c$H$ \u5e73\u6ed1\u65e0\u5947\u70b9\u201d\u4f5c\u4e3a\u5feb\u901f\u7b5b\u67e5\u4f9d\u636e\uff1b\n- **\u96f6\u8f93\u5165\u54cd\u5e94**\uff1aLTI \u7cfb\u7edf\u7684\u96f6\u8f93\u5165\u54cd\u5e94\u6307\u6570\u8870\u51cf\u5230 0\uff0c\u662f\u7a33\u5b9a\u6027\u7684\u7b49\u4ef7\u4f53\u73b0\u3002\n\n\u56e0\u679c\u6027\u4e0e\u7a33\u5b9a\u6027\u662f\u5f7c\u6b64\u72ec\u7acb\u7684\u4e24\u4e2a\u6027\u8d28\uff1a\u53ef\u4ee5\u6709\u7a33\u5b9a\u4f46\u975e\u56e0\u679c\uff08\u5982\u7406\u60f3\u4f4e\u901a\u7684\u5bf9\u79f0 sinc\uff09\u3001\u4e5f\u53ef\u4ee5\u6709\u56e0\u679c\u4f46\u4e0d\u7a33\u5b9a\uff08\u5982 $h[n]=2^nu[n]$\uff09\u7684\u7cfb\u7edf\uff1b\u53ea\u6709\u4e24\u8005\u540c\u65f6\u6ee1\u8db3\uff0c\u624d\u80fd\u5b9e\u65f6\u4e14\u53ef\u9760\u5730\u5b9e\u73b0\u3002\n\n---\n\n---\n\n## \u4e09\u3001\u76f8\u5173\u4e0e\u81ea\u76f8\u5173\uff1a\u5b9a\u4e49\u3001\u6027\u8d28\u4e0e\u8ba1\u7b97\n\n### \u76f8\u5173\u7684\u57fa\u672c\u601d\u60f3\n\n\u76f8\u5173\u662f\u4e00\u79cd\u76f8\u4f3c\u6027\u5ea6\u91cf\u3002\u5c06\u4e00\u4e2a\u4fe1\u53f7\u76f8\u5bf9\u53e6\u4e00\u4e2a\u4fe1\u53f7\u79fb\u52a8\uff0c\u5728\u6bcf\u4e2a\u76f8\u5bf9\u4f4d\u79fb\u5904\u8ba1\u7b97\u4e58\u79ef\u5e76\u7d2f\u52a0\uff1a\n\n- \u82e5\u4e24\u8005\u5728\u5f53\u524d\u4f4d\u79fb\u4e0b\u5f62\u72b6\u76f8\u4f3c\uff0c\u4e58\u79ef\u5927\u591a\u540c\u53f7\u6216\u76f8\u4f4d\u63a5\u8fd1\uff0c\u7d2f\u52a0\u503c\u8f83\u5927\uff1b\n- \u82e5\u4e24\u8005\u4e0d\u5339\u914d\uff0c\u4e58\u79ef\u4f1a\u53d1\u751f\u6b63\u8d1f\u62b5\u6d88\u6216\u76f8\u4f4d\u62b5\u6d88\uff0c\u7d2f\u52a0\u503c\u8f83\u5c0f\u3002\n\n\u56e0\u6b64\uff0c\u76f8\u5173\u51fd\u6570\u7684\u5cf0\u503c\u901a\u5e38\u8868\u793a\u6700\u4f73\u5339\u914d\u4f4d\u7f6e\u3002\n\n### \u8fde\u7eed\u65f6\u95f4\u4e92\u76f8\u5173\n\n\u4e00\u79cd\u5e38\u7528\u5b9a\u4e49\u662f\n\n$$\n\\boxed{\n R_{xy}(\\tau)=\\int_{-\\infty}^{\\infty}x(t)y^*(t-\\tau)\\,dt\n}.\n$$\n\n\u8fd9\u91cc $x(t)$ \u662f\u53c2\u8003\u4fe1\u53f7\uff0c$y(t)$ \u88ab\u5171\u8f6d\u3001\u53cd\u8f6c\u5e76\u5e73\u79fb\u3002\n\n\u4e5f\u6709\u6587\u732e\u4f7f\u7528\n\n$$\n R_{xy}(\\tau)=\\int_{-\\infty}^{\\infty}x^*(t)y(t+\\tau)\\,dt.\n$$\n\n\u8fd9\u4e24\u4e2a\u5b9a\u4e49\u5728\u6539\u53d8\u76f8\u5173\u65b9\u5411\u6216\u53d6\u5171\u8f6d\u540e\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u3002\u4f7f\u7528\u516c\u5f0f\u524d\uff0c\u5fc5\u987b\u660e\u786e\u6240\u91c7\u7528\u7684\u5b9a\u4e49\u3002\n\n### \u79bb\u6563\u65f6\u95f4\u4e92\u76f8\u5173\n\n\u5bf9\u5e94\u7684\u79bb\u6563\u65f6\u95f4\u5b9a\u4e49\u4e3a\n\n$$\n\\boxed{\n R_{xy}[m]=\\sum_{n=-\\infty}^{\\infty}x[n]y^*[n-m]\n}.\n$$\n\n\u53e6\u4e00\u79cd\u5e38\u89c1\u5199\u6cd5\u662f\n\n$$\n R_{xy}[m]=\\sum_n x^*[n]y[n+m].\n$$\n\n\u53ea\u8981\u524d\u540e\u4e00\u81f4\uff0c\u4e8c\u8005\u90fd\u53ef\u4ee5\u4f7f\u7528\uff1b\u4f46\u76f8\u5173\u5cf0\u7684\u6b63\u8d1f\u5ef6\u8fdf\u65b9\u5411\u53ef\u80fd\u4e0d\u540c\u3002\n\n### \u4e92\u76f8\u5173\u7684\u51e0\u4f55\u610f\u4e49\n\n\u628a\u4fe1\u53f7\u770b\u4f5c\u5411\u91cf\u3002\u5bf9\u6bcf\u4e2a\u5ef6\u8fdf $m$\uff0c\u5b9a\u4e49\u5e73\u79fb\u540e\u7684\u4fe1\u53f7\n\n$$\n y_m[n]=y[n-m].\n$$\n\n\u5219\n\n$$\n R_{xy}[m]=\\langle x,y_m\\rangle.\n$$\n\n\u56e0\u6b64\uff0c\u4e92\u76f8\u5173\u5b9e\u9645\u4e0a\u662f\u5728\u626b\u63cf\u4e0d\u540c\u5e73\u79fb\u91cf\u4e0b\u7684\u5411\u91cf\u5185\u79ef\u3002\u5185\u79ef\u5e45\u5ea6\u8d8a\u5927\uff0c\u8bf4\u660e\u4e24\u4e2a\u4fe1\u53f7\u8d8a\u76f8\u4f3c\u3002\n\n\u5bf9\u4e8e\u590d\u4fe1\u53f7\uff0c$R_{xy}[m]$ \u4e00\u822c\u662f\u590d\u6570\uff1a\n\n- \u5e45\u5ea6\u8868\u793a\u5339\u914d\u7a0b\u5ea6\uff1b\n- \u76f8\u4f4d\u8868\u793a\u76f8\u5bf9\u76f8\u4f4d\u4fe1\u606f\uff1b\n- \u5cf0\u503c\u4f4d\u7f6e\u8868\u793a\u76f8\u5bf9\u65f6\u5ef6\u3002\n\n### \u5f52\u4e00\u5316\u4e92\u76f8\u5173\n\n\u4e0d\u540c\u4fe1\u53f7\u7684\u5e45\u5ea6\u53ef\u80fd\u4e0d\u540c\uff0c\u4ec5\u6bd4\u8f83\u672a\u5f52\u4e00\u5316\u76f8\u5173\u503c\u4f1a\u53d7\u5230\u80fd\u91cf\u5f71\u54cd\u3002\u5e38\u89c1\u7684\u5f52\u4e00\u5316\u4e92\u76f8\u5173\u4e3a\n\n$$\n\\rho_{xy}[m]\n=\\frac{R_{xy}[m]}\n{\\sqrt{R_{xx}[0]R_{yy}[0]}}.\n$$\n\n\u5176\u4e2d\n\n$$\nR_{xx}[0]=\\sum_n|x[n]|^2,\n\\qquad\nR_{yy}[0]=\\sum_n|y[n]|^2.\n$$\n\n\u7531 Cauchy\u2013Schwarz \u4e0d\u7b49\u5f0f\uff0c\u6709\n\n$$\n|\\rho_{xy}[m]|\\le 1.\n$$\n\n\u5f52\u4e00\u5316\u540e\u53ef\u4ee5\u66f4\u516c\u5e73\u5730\u6bd4\u8f83\u4e0d\u540c\u80fd\u91cf\u4fe1\u53f7\u7684\u5339\u914d\u7a0b\u5ea6\u3002\n\n### \u4e92\u76f8\u5173\u4e0e\u5185\u79ef\u7684\u5173\u7cfb\n\n\u5185\u79ef\u662f\u96f6\u5ef6\u8fdf\u4e0b\u7684\u76f8\u5173\uff1a\n\n$$\n\\langle x,y\\rangle=R_{xy}[0]\n$$\n\n\u6216\u5728\u53e6\u4e00\u79cd\u7ea6\u5b9a\u4e0b\u7b49\u4e8e $R_{yx}[0]$\u3002\u76f8\u5173\u5219\u628a\u4e00\u4e2a\u6807\u91cf\u5185\u79ef\u6269\u5c55\u4e3a\u5173\u4e8e\u5ef6\u8fdf\u7684\u51fd\u6570\u3002\n\n---\n### \u81ea\u76f8\u5173\n\n#### \u5b9a\u4e49\n\n\u4fe1\u53f7\u4e0e\u81ea\u8eab\u7684\u4e92\u76f8\u5173\u79f0\u4e3a\u81ea\u76f8\u5173\u3002\n\n\u8fde\u7eed\u65f6\u95f4\uff1a\n\n$$\n\\boxed{\n R_{xx}(\\tau)=\\int_{-\\infty}^{\\infty}x(t)x^*(t-\\tau)\\,dt\n}.\n$$\n\n\u79bb\u6563\u65f6\u95f4\uff1a\n\n$$\n\\boxed{\n R_{xx}[m]=\\sum_n x[n]x^*[n-m]\n}.\n$$\n\n#### \u96f6\u5ef6\u8fdf\u81ea\u76f8\u5173\u7b49\u4e8e\u80fd\u91cf\n\n\u5728\u80fd\u91cf\u6709\u9650\u7684\u60c5\u51b5\u4e0b\uff1a\n\n$$\n R_{xx}(0)=\\int|x(t)|^2dt=E_x\n$$\n\n\u6216\n\n$$\n R_{xx}[0]=\\sum_n|x[n]|^2=E_x.\n$$\n\n\u56e0\u6b64\uff0c\u81ea\u76f8\u5173\u5728\u96f6\u5ef6\u8fdf\u5904\u901a\u5e38\u5177\u6709\u6700\u5927\u5cf0\u503c\u6216\u81f3\u5c11\u6ee1\u8db3\u6700\u5927\u5e45\u5ea6\u4e0d\u8d85\u8fc7\u96f6\u5ef6\u8fdf\u503c\u3002\n\n#### \u81ea\u76f8\u5173\u7684\u5171\u8f6d\u5bf9\u79f0\u6027\n\n\u5bf9\u4e8e\u590d\u4fe1\u53f7\uff1a\n\n$$\n\\boxed{\n R_{xx}(-\\tau)=R_{xx}^*(\\tau)\n}.\n$$\n\n\u79bb\u6563\u5f62\u5f0f\u4e3a\n\n$$\n R_{xx}[-m]=R_{xx}^*[m].\n$$\n\n\u56e0\u6b64\uff1a\n\n- \u5b9e\u4fe1\u53f7\u7684\u81ea\u76f8\u5173\u662f\u5076\u51fd\u6570\uff1b\n- \u590d\u4fe1\u53f7\u7684\u81ea\u76f8\u5173\u4e0d\u4e00\u5b9a\u662f\u5b9e\u6570\uff0c\u4f46\u5177\u6709\u5171\u8f6d\u5bf9\u79f0\u6027\uff1b\n- $R_{xx}(0)$ \u5fc5\u5b9a\u662f\u975e\u8d1f\u5b9e\u6570\u3002\n\n#### \u5468\u671f\u4fe1\u53f7\u7684\u81ea\u76f8\u5173\n\n\u5bf9\u4e8e\u5468\u671f\u4fe1\u53f7\uff0c\u901a\u5e38\u4f7f\u7528\u65f6\u95f4\u5e73\u5747\u5f62\u5f0f\u3002\u4f8b\u5982\u5468\u671f\u4e3a $T_0$ \u7684\u4fe1\u53f7\uff1a\n\n$$\n R_{xx}(\\tau)\n=\\frac{1}{T_0}\\int_{t_0}^{t_0+T_0}x(t)x^*(t-\\tau)\\,dt.\n$$\n\n\u5468\u671f\u4fe1\u53f7\u7684\u81ea\u76f8\u5173\u4e5f\u662f\u5468\u671f\u51fd\u6570\uff0c\u5468\u671f\u901a\u5e38\u4e0e\u539f\u4fe1\u53f7\u76f8\u540c\u3002\u5468\u671f\u7ed3\u6784\u8d8a\u660e\u663e\uff0c\u81ea\u76f8\u5173\u4e2d\u7684\u5468\u671f\u5cf0\u8d8a\u660e\u663e\u3002\n\n#### \u81ea\u76f8\u5173\u7684\u5e94\u7528\n\n\u81ea\u76f8\u5173\u5e38\u7528\u4e8e\uff1a\n\n- \u68c0\u6d4b\u5468\u671f\u548c\u57fa\u9891\uff1b\n- \u6d4b\u91cf\u4fe1\u53f7\u6301\u7eed\u65f6\u95f4\uff1b\n- \u4f30\u8ba1\u566a\u58f0\u767d\u5316\u7a0b\u5ea6\uff1b\n- \u4f30\u8ba1\u968f\u673a\u8fc7\u7a0b\u7684\u7edf\u8ba1\u7ed3\u6784\uff1b\n- \u8bed\u97f3\u57fa\u97f3\u68c0\u6d4b\uff1b\n- \u96f7\u8fbe\u548c\u58f0\u7eb3\u56de\u6ce2\u5206\u6790\u3002\n\n---\n\n\n### \u76f8\u5173\u7684\u5e38\u7528\u6027\u8d28\n\n#### \u76f8\u5173\u7684\u5171\u8f6d\u5bf9\u79f0\u5173\u7cfb\n\n\u4e92\u76f8\u5173\u6ee1\u8db3\n\n$$\n\\boxed{\nR_{yx}(\\tau)=R_{xy}^*(-\\tau)\n}.\n$$\n\n\u79bb\u6563\u5f62\u5f0f\u4e3a\n\n$$\nR_{yx}[m]=R_{xy}^*[-m].\n$$\n\n\u5bf9\u4e8e\u5b9e\u4fe1\u53f7\uff1a\n\n$$\nR_{yx}(\\tau)=R_{xy}(-\\tau).\n$$\n\n#### Cauchy\u2013Schwarz \u4e0d\u7b49\u5f0f\n\n\u76f8\u5173\u5e45\u5ea6\u6ee1\u8db3\n\n$$\n|R_{xy}(\\tau)|^2\n\\le R_{xx}(0)R_{yy}(0).\n$$\n\n\u56e0\u6b64\uff0c\u76f8\u5173\u503c\u7684\u5e45\u5ea6\u4e0d\u4f1a\u8d85\u8fc7\u4e24\u4e2a\u4fe1\u53f7\u80fd\u91cf\u5e73\u65b9\u6839\u7684\u4e58\u79ef\u3002\n\n#### \u76f8\u5173\u7684\u5e73\u79fb\u6027\u8d28\n\n\u82e5\n\n$$\n y(t)=x(t-t_0),\n$$\n\n\u5219\u4e92\u76f8\u5173\u4f1a\u53d1\u751f\u76f8\u5e94\u5e73\u79fb\u3002\u4ee5\n\n$$\nR_{xy}(\\tau)=\\int x(t)y^*(t-\\tau)dt\n$$\n\n\u4e3a\u5b9a\u4e49\u65f6\uff1a\n\n$$\nR_{xy}(\\tau)=R_{xx}(\\tau+t_0).\n$$\n\n\u56e0\u4e3a $y(t)=x(t-t_0)$ \u65f6\uff0c$y^*(t-\\tau)=x^*(t-\\tau-t_0)$\uff0c\u6240\u4ee5 $R_{xy}$ \u7684\u5cf0\u4f4d\u4e3a $\\tau=-t_0$\u3002\u82e5\u91c7\u7528 $R_{yx}$ \u6216\u53e6\u4e00\u79cd\u76f8\u5173\u65b9\u5411\uff0c\u5cf0\u4f4d\u4f1a\u53cd\u53f7\u3002\u7ed3\u8bba\u662f\uff1a\n\n> \u4e00\u4e2a\u4fe1\u53f7\u76f8\u5bf9\u53e6\u4e00\u4e2a\u4fe1\u53f7\u5ef6\u8fdf\uff0c\u4f1a\u4f7f\u76f8\u5173\u5cf0\u79fb\u52a8\uff1b\u5cf0\u79fb\u52a8\u7684\u6b63\u8d1f\u65b9\u5411\u53d6\u51b3\u4e8e\u76f8\u5173\u5b9a\u4e49\u3002<\/p>\n

### \u79bb\u6563\u5377\u79ef\u4e0e\u76f8\u5173\u7684\u8ba1\u7b97\u65b9\u6cd5<\/p>\n

#### \u76f4\u63a5\u8ba1\u7b97\u79bb\u6563\u7ebf\u6027\u5377\u79ef<\/p>\n

\u82e5 $x[n]$ \u957f\u5ea6\u4e3a $N$\uff0c$h[n]$ \u957f\u5ea6\u4e3a $M$\uff0c\u5219\u7ebf\u6027\u5377\u79ef\u957f\u5ea6\u4e3a<\/p>\n

$$
\nN+M-1.
\n$$<\/p>\n

\u8ba1\u7b97\u516c\u5f0f\u4e3a<\/p>\n

$$
\n y[n]=\\sum_{k=0}^{N-1}x[k]h[n-k],
\n$$<\/p>\n

\u5176\u4e2d\u8d85\u51fa $h$ \u6709\u6548\u4e0b\u6807\u7684\u9879\u53d6\u96f6\u3002<\/p>\n

\u5bf9\u4e8e\u6709\u9650\u5e8f\u5217\uff0c\u901a\u5e38\u5efa\u7acb\u4e0b\u6807\u8303\u56f4\uff1a<\/p>\n

$$
\n0\\le n\\le N+M-2.
\n$$<\/p>\n

#### \u5377\u79ef\u8868\u683c\u6cd5<\/p>\n

\u8ba1\u7b97\u79bb\u6563\u5377\u79ef\u65f6\uff0c\u53ef\u4ee5\uff1a<\/p>\n

1. \u5199\u51fa $x[k]$\uff1b
\n2. \u5199\u51fa $h[n-k]$\uff1b
\n3. \u5bf9 $h$ \u8fdb\u884c\u53cd\u8f6c\u5e76\u968f $n$ \u5e73\u79fb\uff1b
\n4. \u627e\u51fa\u4e24\u4e2a\u5e8f\u5217\u7684\u91cd\u53e0\u533a\u95f4\uff1b
\n5. \u5c06\u91cd\u53e0\u6837\u672c\u76f8\u4e58\u5e76\u6c42\u548c\u3002<\/p>\n

#### \u76f4\u63a5\u8ba1\u7b97\u79bb\u6563\u4e92\u76f8\u5173<\/p>\n

\u5bf9\u4e8e<\/p>\n

$$
\nR_{xy}[m]=\\sum_n x[n]y^*[n-m],
\n$$<\/p>\n

\u8ba1\u7b97\u6d41\u7a0b\u662f\uff1a<\/p>\n

1. \u5bf9 $y[n]$ \u53d6\u5171\u8f6d\uff1b
\n2. \u5bf9\u5171\u8f6d\u540e\u7684\u5e8f\u5217\u505a\u65f6\u95f4\u53cd\u8f6c\uff1b
\n3. \u5c06\u5176\u5e73\u79fb\uff1b
\n4. \u4e0e $x[n]$ \u91cd\u53e0\u76f8\u4e58\u5e76\u6c42\u548c\u3002<\/p>\n

\u4e5f\u53ef\u4ee5\u76f4\u63a5\u6839\u636e\u516c\u5f0f\u9010\u4e2a $m$ \u8ba1\u7b97\uff0c\u4e0d\u5fc5\u8fdb\u884c\u56fe\u5f62\u53cd\u8f6c\u3002<\/p>\n

#### \u4f7f\u7528 FFT \u52a0\u901f\u5377\u79ef<\/p>\n

\u76f4\u63a5\u8ba1\u7b97\u957f\u5ea6\u4e3a $N$ \u548c $M$ \u5e8f\u5217\u7684\u5377\u79ef\uff0c\u590d\u6742\u5ea6\u7ea6\u4e3a $O(NM)$\u3002\u5229\u7528 FFT \u53ef\u4ee5\u5c06\u590d\u6742\u5ea6\u964d\u4f4e\u5230\u8fd1\u4f3c<\/p>\n

$$
\nO(L\\log L),
\n$$<\/p>\n

\u5176\u4e2d $L$ \u662f\u8db3\u591f\u5927\u7684 FFT \u957f\u5ea6\u3002<\/p>\n

\u7ebf\u6027\u5377\u79ef\u7684 FFT \u5b9e\u73b0\uff1a<\/p>\n

1. \u53d6 $L\\ge N+M-1$\uff1b
\n2. \u5c06 $x$\u3001$h$ \u8865\u96f6\u5230\u957f\u5ea6 $L$\uff1b
\n3. \u8ba1\u7b97 $X=\\operatorname{FFT}(x)$ \u548c $H=\\operatorname{FFT}(h)$\uff1b
\n4. \u8ba1\u7b97 $Y=XH$\uff1b
\n5. \u8ba1\u7b97 $y=\\operatorname{IFFT}(Y)$\uff1b
\n6. \u4fdd\u7559\u524d $N+M-1$ \u4e2a\u6709\u6548\u6837\u672c\u3002<\/p>\n

\u4e92\u76f8\u5173\u7684 FFT \u5b9e\u73b0\uff1a<\/p>\n

$$
\nR_{xy,\\mathrm{circ}}
\n=\\operatorname{IFFT}\\left\\{X[k]Y^*[k]\\right\\}.
\n$$<\/p>\n

\u5b9e\u9645\u4f7f\u7528\u65f6\u5fc5\u987b\u6839\u636e\u76f8\u5173\u5b9a\u4e49\u5bf9\u7ed3\u679c\u8fdb\u884c\u5faa\u73af\u79fb\u4f4d\u6216\u91cd\u65b0\u6392\u5217\u3002<\/p>\n

#### \u76f8\u5173\u7cfb\u6570\u4e0e\u6ed1\u52a8\u7a97\u53e3\u76f8\u5173<\/p>\n

\u5728\u5b9e\u9645\u68c0\u6d4b\u4e2d\uff0c\u5e38\u5e38\u4e0d\u662f\u5bf9\u4e24\u4e2a\u5b8c\u6574\u5e8f\u5217\u505a\u76f8\u5173\uff0c\u800c\u662f\u5c06\u6a21\u677f $s[n]$ \u5728\u89c2\u6d4b\u4fe1\u53f7 $r[n]$ \u4e0a\u6ed1\u52a8\uff1a<\/p>\n

$$
\nC[m]=\\sum_n r[n]s^*[n-m].
\n$$<\/p>\n

\u4e3a\u907f\u514d\u6a21\u677f\u80fd\u91cf\u548c\u5c40\u90e8\u4fe1\u53f7\u80fd\u91cf\u5f71\u54cd\uff0c\u53ef\u4ee5\u91c7\u7528\u5c40\u90e8\u5f52\u4e00\u5316\uff1a<\/p>\n

$$
\nC_{\\mathrm{norm}}[m]
\n=\\frac{\\sum_n r[n]s^*[n-m]}
\n{\\sqrt{\\sum_n|r[n]|^2\\sum_n|s[n-m]|^2}}.
\n$$<\/p>\n

\u8fd9\u7c7b\u6307\u6807\u5e38\u7528\u4e8e\u6a21\u677f\u5339\u914d\u548c\u540c\u6b65\u68c0\u6d4b\u3002<\/p>\n

#### \u590d\u6709\u9650\u5e8f\u5217\u4e92\u76f8\u5173\u7684\u5b8c\u6574\u4f8b\u9898<\/p>\n

\u8bbe\u4e24\u4e2a\u957f\u5ea6\u4e3a 3 \u7684\u590d\u5e8f\u5217<\/p>\n

$$
\nx[n]=\\{1,\\ j,\\ -1\\}_{n=0,1,2},
\n\\qquad
\ny[n]=\\{1,\\ 1+j,\\ 0\\}_{n=0,1,2}.
\n$$<\/p>\n

\u6309\u672c\u6587\u9ed8\u8ba4\u5b9a\u4e49 $R_{xy}[m]=\\sum_n x[n]y^*[n-m]$\uff0c$m$ \u7684\u6709\u6548\u8303\u56f4\u4e3a $-(N_y-1)\\le m\\le N_x-1$\uff0c\u5373 $-2\\le m\\le 2$\u3002\u4e3a\u4fbf\u4e8e\u8ba1\u7b97\uff0c\u5148\u5199\u51fa $y^*[n]=\\{1,\\ 1-j,\\ 0\\}$\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a $m$ \u627e\u51fa\u4f7f $n$ \u4e0e $n-m$ \u540c\u65f6\u843d\u5728\u6709\u6548\u533a\u95f4\u7684\u9879\uff1a<\/p>\n

– $m=-2$\uff1a\u4ec5 $n=0$ \u6ee1\u8db3\uff0c
\n $$
\n R_{xy}[-2]=x[0]y^*[2]=1\\cdot 0=0.
\n $$
\n– $m=-1$\uff1a$n=0,1$\uff0c
\n $$
\n R_{xy}[-1]=x[0]y^*[1]+x[1]y^*[2]=(1)(1-j)+(j)(0)=1-j.
\n $$
\n– $m=0$\uff1a$n=0,1,2$\uff0c
\n $$
\n R_{xy}[0]=x[0]y^*[0]+x[1]y^*[1]+x[2]y^*[2]=1+(j)(1-j)+0=1+(j+1)=2+j.
\n $$
\n– $m=1$\uff1a$n=1,2$\uff0c
\n $$
\n R_{xy}[1]=x[1]y^*[0]+x[2]y^*[1]=j+(-1)(1-j)=j-1+j=-1+2j.
\n $$
\n– $m=2$\uff1a$n=2$\uff0c
\n $$
\n R_{xy}[2]=x[2]y^*[0]=-1.
\n $$<\/p>\n

\u4e8e\u662f<\/p>\n

$$
\nR_{xy}[m]=\\{0,\\ 1-j,\\ 2+j,\\ -1+2j,\\ -1\\}_{m=-2,\\dots,2}.
\n$$<\/p>\n

\u53d6\u6a21\u5e73\u65b9 $|R_{xy}[m]|^2=\\{0,2,5,5,1\\}$\uff0c\u6700\u5927\u5339\u914d\u4f4d\u7f6e\u5728 $m=0$ \u4e0e $m=1$ \u4e4b\u95f4\u5e76\u5217\uff0c\u53cd\u6620\u51fa $y$ \u4e2d\u7684 $1+j$ \u4e0e $x$ \u4e2d $\\{1,j\\}$ \u7684\u4e24\u79cd\u5bf9\u9f50\u65b9\u5f0f\u76f8\u4f3c\u5ea6\u63a5\u8fd1\u3002\u82e5\u540c\u65f6\u8ba1\u7b97 $R_{yx}[m]$\uff0c\u53ef\u4ee5\u9a8c\u8bc1 $R_{yx}[m]=R_{xy}^*[-m]$\uff0c\u4f8b\u5982 $R_{yx}[-1]=R_{xy}^*[1]=-1-2j$\uff0c\u4e0e\u9010\u9879\u76f4\u63a5\u7b97\u51fa\u7684\u7ed3\u679c\u4e00\u81f4\u3002<\/p>\n

\u8fd9\u4e2a\u4f8b\u5b50\u8bf4\u660e\uff1a\u5728\u590d\u4fe1\u53f7\u4e2d\uff0c$R_{xy}[m]$ \u6bcf\u4e2a\u5206\u91cf\u7684\u76f8\u4f4d\u643a\u5e26\u4e86\u5bf9\u9f50\u540e\u4e24\u6bb5\u4fe1\u53f7\u7684\u76f8\u4f4d\u5dee\u4fe1\u606f\uff1b\u53ea\u770b\u5e45\u5ea6\u4f1a\u4e22\u6389\u76f8\u5bf9\u76f8\u4f4d\uff0c\u56e0\u800c\u4e0d\u80fd\u66ff\u4ee3\u590d\u6570\u7ed3\u679c\u672c\u8eab\u3002<\/p>\n

#### lag \u65b9\u5411\u4e0e\u5b9e\u73b0\u7ea6\u5b9a\u7684\u5bf9\u7167<\/p>\n

\u4e0d\u540c\u6559\u6750\u3001\u4e0d\u540c\u6570\u503c\u5e93\u5bf9\u201clag \u7684\u6b63\u65b9\u5411\u201d\u7ea6\u5b9a\u4e0d\u540c\uff0c\u8fd9\u662f\u521d\u5b66\u8005\u6700\u5bb9\u6613\u51fa\u9519\u7684\u5730\u65b9\u3002\u5bf9\u672c\u6587\u9ed8\u8ba4\u5b9a\u4e49 $R_{xy}[m]=\\sum_n x[n]y^*[n-m]$\uff1a<\/p>\n

| \u7ea6\u5b9a | \u5cf0\u4f4d\u542b\u4e49 | \u6570\u503c\u793a\u4f8b |
\n|—|—|—|
\n| $R_{xy}[m]=\\sum_n x[n]y^*[n-m]$\uff08\u672c\u6587\uff09 | \u82e5 $y[n]=x[n-m_0]$\uff08$y$ \u53f3\u79fb\u3001\u76f8\u5bf9 $x$ \u5ef6\u8fdf $m_0>0$\uff09\uff0c\u5cf0\u5728 $m=-m_0$\uff1b\u6b63 $m$ \u8868\u793a\u6309\u672c\u6587\u516c\u5f0f\u626b\u63cf\u5230\u76f8\u53cd\u65b9\u5411 | \u7528\u5355\u4f4d\u8109\u51b2\u6216\u5df2\u77e5\u79fb\u4f4d\u5e8f\u5217\u9a8c\u8bc1\uff0c\u4e0d\u51ed\u201c\u6b63 lag\u201d\u5b57\u9762\u5224\u65ad |
\n| $R_{xy}[m]=\\sum_n x^*[n]y[n+m]$ | \u4e0e\u4e0a\u5f0f\u7b49\u4ef7\u5730\u63cf\u8ff0\u53e6\u4e00\u65b9\u5411\uff0c\u5cf0\u4f4d\u7b26\u53f7\u968f\u7d22\u5f15\u5b9a\u4e49\u6539\u53d8 | \u5148\u56fa\u5b9a\u516c\u5f0f\uff0c\u518d\u89e3\u91ca\u6b63\u8d1f |
\n| NumPy `np.correlate(x, y, mode=’full’)` | \u8fd4\u56de $\\sum_n x[n+m]\\,\\overline{y[n]}$\uff0c\u6b63 lag \u5bf9\u5e94 $x$ \u76f8\u5bf9 $y$ \u8d85\u524d | \u7ed3\u679c\u7d22\u5f15\u4ece $-(N_y-1)$ \u5230 $N_x-1$ \u9700\u81ea\u884c\u6362\u7b97 |
\n| MATLAB `xcorr(x, y)` | \u8fd4\u56de $\\sum_n x[n+m]\\,y^*[n]$\uff0c\u6b63 lag \u5bf9\u5e94 $x$ \u8d85\u524d $y$ | \u7d22\u5f15\u4ee5\u4e2d\u5fc3\u4e3a\u96f6 lag |<\/p>\n

\u5de5\u7a0b\u4e0a\u907f\u514d\u51fa\u9519\u7684\u4e09\u6b65\u505a\u6cd5\uff1a<\/p>\n

1. \u5728\u62a5\u544a\u6216\u4ee3\u7801\u4e2d\u5199\u6e05\u695a\u201c\u76f8\u5173\u7684\u5b9a\u4e49\u5f0f\u4e0e lag \u7684\u7269\u7406\u542b\u4e49\u201d\uff1b
\n2. \u7528\u4e00\u4e2a\u5df2\u77e5\u5ef6\u8fdf\u7684\u5408\u6210\u4fe1\u53f7\uff08\u5982 $y=x$ \u5e73\u79fb $k$ \u4e2a\u91c7\u6837\uff09\u8dd1\u4e00\u904d\uff0c\u9a8c\u8bc1\u5cf0\u4f4d\u7b26\u53f7\uff1b
\n3. \u82e5\u4e0d\u540c\u6765\u6e90\u7684\u7ed3\u679c\u7b26\u53f7\u76f8\u53cd\uff0c\u76f4\u63a5\u5bf9\u6574\u6bb5\u7ed3\u679c\u505a $m\\to -m$ \u6216\u989d\u5916\u53d6\u5171\u8f6d\u5373\u53ef\u6362\u7b97\u3002<\/p>\n

#### \u5f52\u4e00\u5316\u4f30\u8ba1\u4e0e\u504f\/\u65e0\u504f\u4f30\u8ba1<\/p>\n

\u5bf9\u5bbd\u5e73\u7a33\u8fc7\u7a0b $x[n]$\uff0c\u7528\u6709\u9650\u89c2\u6d4b $\\{x[0],\\dots,x[N-1]\\}$ \u4f30\u8ba1\u81ea\u76f8\u5173\u65f6\u6709\u4e24\u79cd\u5e38\u89c1\u5f62\u5f0f\uff1a<\/p>\n

– **\u6709\u504f\u4f30\u8ba1**\uff08NumPy\/`scipy.signal.correlate` \u9ed8\u8ba4\uff09
\n $$
\n \\hat R^{(b)}_{xx}[m]=\\frac{1}{N}\\sum_{n=0}^{N-1-|m|}x[n+|m|]\\,x^*[n],\\quad |m|\\le N-1.
\n $$
\n \u5206\u6bcd\u56fa\u5b9a\u4e3a $N$\uff0c$|m|$ \u8d8a\u5927\u53c2\u4e0e\u6c42\u548c\u7684\u6837\u672c\u8d8a\u5c11\uff0c\u4f30\u8ba1\u503c\u7cfb\u7edf\u6027\u504f\u4f4e\uff0c\u4f46\u6574\u4e2a\u5e8f\u5217\u4f5c\u4e3a\u6b63\u5b9a\u6838\u66f4\u7a33\u5b9a\uff0c\u5bf9\u5e94\u7684\u529f\u7387\u8c31\u4f30\u8ba1\u975e\u8d1f\u3002
\n– **\u65e0\u504f\u4f30\u8ba1**
\n $$
\n \\hat R^{(u)}_{xx}[m]=\\frac{1}{N-|m|}\\sum_{n=0}^{N-1-|m|}x[n+|m|]\\,x^*[n].
\n $$
\n \u671f\u671b\u503c\u7b49\u4e8e\u771f\u5b9e\u81ea\u76f8\u5173\uff0c\u4f46\u5927 $|m|$ \u65f6\u65b9\u5dee\u6025\u5267\u589e\u5927\uff0c\u7528\u5b83\u505a Fourier \u53d8\u6362\u5f97\u5230\u7684\u529f\u7387\u8c31\u53ef\u80fd\u51fa\u73b0\u8d1f\u503c\u3002<\/p>\n

\u5de5\u7a0b\u5b9e\u8df5\u4e2d\uff1a<\/p>\n

– \u82e5\u53ea\u5173\u5fc3\u5cf0\u4f4d\u6216\u6ce2\u5f62\u5339\u914d\uff0c\u4e24\u79cd\u4f30\u8ba1\u5dee\u522b\u4e0d\u5927\uff0c\u7528\u6709\u504f\u4f30\u8ba1\u66f4\u5b89\u5168\uff1b
\n– \u82e5\u8981\u8fdb\u4e00\u6b65\u505a PSD\u3001Wiener\u2013Khinchin \u6216\u53c2\u6570\u5316\u8c31\u4f30\u8ba1\uff0c\u52a1\u5fc5\u4f7f\u7528\u6709\u504f\u4f30\u8ba1\u6216\u52a0\u7a97\u540e\u7684 Blackman\u2013Tukey \u65b9\u6cd5\uff0c\u4ee5\u4fdd\u8bc1\u6b63\u534a\u5b9a\u6027\uff1b
\n– \u5f53 $N$ \u6709\u9650\u3001\u4e14\u8981\u770b\u5230\u8f83\u5927 $|m|$ \u7684\u81ea\u76f8\u5173\u65f6\uff0c\u53ef\u4ee5\u53ea\u4fe1\u4efb $|m|\\ll N$ \u7684\u90e8\u5206\uff08\u4f8b\u5982 $|m|\\le N\/4$\uff09\uff0c\u5e76\u5728\u62a5\u544a\u4e2d\u6807\u6ce8\u4f30\u8ba1\u5e26\u5bbd\u548c\u6709\u6548\u6837\u672c\u6570\u3002<\/p>\n

\u5bf9\u4e92\u76f8\u5173\u4e5f\u6709\u5b8c\u5168\u7c7b\u4f3c\u7684\u5b9a\u4e49\uff1b\u6b64\u65f6 $N$ \u6362\u6210\u4e24\u6bb5\u4fe1\u53f7\u53ef\u7528\u7684\u91cd\u53e0\u6837\u672c\u6570 $N-|m|$\uff0c\u540c\u6837\u5b58\u5728\u504f\u3001\u65e0\u504f\u4e24\u79cd\u5f52\u4e00\u5316\u65b9\u5f0f\u3002<\/p>\n

—<\/p>\n

—<\/p>\n

## \u56db\u3001\u5377\u79ef\u3001\u76f8\u5173\u4e0e Fourier\/DFT \u5bf9\u5076\u5173\u7cfb<\/p>\n

### \u76f8\u5173\u662f\u5e26\u5171\u8f6d\u7684\u5377\u79ef\u53d8\u4f53<\/p>\n

\u5b9a\u4e49<\/p>\n

$$
\n\\tilde y(t)=y^*(-t).
\n$$<\/p>\n

\u5219<\/p>\n

$$
\n\\begin{aligned}
\n(x*\\tilde y)(\\tau)
\n&=\\int x(t)\\tilde y(\\tau-t)\\,dt\\\\
\n&=\\int x(t)y^*(t-\\tau)\\,dt\\\\
\n&=R_{xy}(\\tau).
\n\\end{aligned}
\n$$<\/p>\n

\u56e0\u6b64<\/p>\n

$$
\n\\boxed{
\nR_{xy}(\\tau)=x(\\tau)*y^*(-\\tau)
\n}.
\n$$<\/p>\n

\u79bb\u6563\u5f62\u5f0f\u4e3a<\/p>\n

$$
\n\\boxed{
\nR_{xy}[m]=x[m]*y^*[-m]
\n}.
\n$$<\/p>\n

\u8fd9\u91cc\u7b49\u5f0f\u4e2d\u7684\u5377\u79ef\u53d8\u91cf\u548c\u7ed3\u679c\u7d22\u5f15\u9700\u8981\u6309\u7167\u5177\u4f53\u5b9a\u4e49\u5bf9\u9f50\uff0c\u4f46\u672c\u8d28\u5173\u7cfb\u662f\uff1a<\/p>\n

> **\u4e92\u76f8\u5173\u7b49\u4e8e\u4e00\u4e2a\u4fe1\u53f7\u4e0e\u53e6\u4e00\u4e2a\u4fe1\u53f7\u7684\u201c\u5171\u8f6d\u65f6\u95f4\u53cd\u8f6c\u7248\u672c\u201d\u505a\u5377\u79ef\u3002**<\/p>\n

### \u4e3a\u4ec0\u4e48\u76f8\u5173\u4e2d\u6709\u5171\u8f6d<\/p>\n

\u5bf9\u4e8e\u590d\u6307\u6570\u4fe1\u53f7\uff1a<\/p>\n

$$
\n x[n]=e^{j\\omega_0n},
\n\\qquad
\n y[n]=e^{j\\omega_1n},
\n$$<\/p>\n

\u9010\u70b9\u5171\u8f6d\u76f8\u4e58\u5f97\u5230<\/p>\n

$$
\n x^*[n]y[n]=e^{j(\\omega_1-\\omega_0)n}.
\n$$<\/p>\n

\u5f53 $\\omega_1=\\omega_0$ \u65f6\uff0c\u7ed3\u679c\u4e3a\u5e38\u6570\uff0c\u7d2f\u52a0\u4e0d\u4f1a\u76f8\u4e92\u62b5\u6d88\uff1b\u5f53\u9891\u7387\u4e0d\u540c\u65f6\uff0c\u7ed3\u679c\u7ee7\u7eed\u65cb\u8f6c\uff0c\u7d2f\u52a0\u4f1a\u90e8\u5206\u62b5\u6d88\u3002\u56e0\u6b64\uff0c\u5171\u8f6d\u76f8\u4e58\u6d4b\u91cf\u7684\u662f\u76f8\u5bf9\u76f8\u4f4d\u548c\u9891\u7387\u5dee\u3002<\/p>\n

\u82e5\u4f7f\u7528\u666e\u901a\u4e58\u6cd5\uff1a<\/p>\n

$$
\n x[n]y[n]=e^{j(\\omega_1+\\omega_0)n},
\n$$<\/p>\n

\u5f97\u5230\u7684\u662f\u9891\u7387\u76f8\u52a0\uff0c\u800c\u4e0d\u662f\u6bd4\u8f83\u4e24\u8005\u7684\u76f8\u5bf9\u9891\u7387\u3002<\/p>\n

### \u4ec0\u4e48\u65f6\u5019\u5377\u79ef\u548c\u76f8\u5173\u5f62\u5f0f\u76f8\u540c<\/p>\n

\u5f53\u4fe1\u53f7\u4e3a\u5b9e\u503c\u4e14\u6ee1\u8db3\u5076\u5bf9\u79f0\u65f6\uff1a<\/p>\n

$$
\n y(t)=y(-t),
\n$$<\/p>\n

\u5171\u8f6d\u548c\u53cd\u8f6c\u90fd\u4e0d\u4f1a\u6539\u53d8\u4fe1\u53f7\uff0c\u6b64\u65f6\u76f8\u5173\u4e0e\u5377\u79ef\u53ef\u80fd\u5177\u6709\u76f8\u540c\u7684\u8868\u8fbe\u5f62\u5f0f\u3002<\/p>\n

\u4f46\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u5c24\u5176\u662f\u4fe1\u53f7\u4e0d\u5bf9\u79f0\u6216\u4e3a\u590d\u4fe1\u53f7\u65f6\uff0c\u4e0d\u80fd\u628a\u76f8\u5173\u76f4\u63a5\u5f53\u6210\u5377\u79ef\u3002<\/p>\n

—
\n### \u5377\u79ef\u548c\u76f8\u5173\u7684 Fourier \u53d8\u6362\u5173\u7cfb<\/p>\n

\u91c7\u7528\u8fde\u7eed\u65f6\u95f4 Fourier \u53d8\u6362\u7ea6\u5b9a\uff1a<\/p>\n

$$
\nX(\\omega)=\\int_{-\\infty}^{\\infty}x(t)e^{-j\\omega t}\\,dt,
\n$$<\/p>\n

$$
\nx(t)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}X(\\omega)e^{j\\omega t}\\,d\\omega.
\n$$<\/p>\n

#### \u5377\u79ef\u5b9a\u7406<\/p>\n

\u82e5<\/p>\n

$$
\n y(t)=x(t)*h(t),
\n$$<\/p>\n

\u5219<\/p>\n

$$
\n\\boxed{
\nY(\\omega)=X(\\omega)H(\\omega)
\n}.
\n$$<\/p>\n

\u53cd\u8fc7\u6765\uff1a<\/p>\n

$$
\n\\boxed{
\n\\mathcal{F}\\{x(t)h(t)\\}
\n=\\frac{1}{2\\pi}X(\\omega)*H(\\omega)
\n}.
\n$$<\/p>\n

\u56e0\u6b64\uff1a<\/p>\n

– \u65f6\u57df\u5377\u79ef\u5bf9\u5e94\u9891\u57df\u9010\u70b9\u76f8\u4e58\uff1b
\n– \u65f6\u57df\u9010\u70b9\u76f8\u4e58\u5bf9\u5e94\u9891\u57df\u5377\u79ef\u3002<\/p>\n

#### \u76f8\u5173\u5b9a\u7406<\/p>\n

\u5bf9\u4e8e\u5b9a\u4e49<\/p>\n

$$
\nR_{xy}(\\tau)=\\int x(t)y^*(t-\\tau)\\,dt,
\n$$<\/p>\n

\u6709<\/p>\n

$$
\n\\boxed{
\n\\mathcal{F}\\{R_{xy}(\\tau)\\}=X(\\omega)Y^*(\\omega)
\n}.
\n$$<\/p>\n

\u6839\u636e\u76f8\u5173\u65b9\u5411\uff0c\u4e5f\u53ef\u80fd\u5199\u4e3a<\/p>\n

$$
\n\\mathcal{F}\\{R_{yx}(\\tau)\\}=Y(\\omega)X^*(\\omega).
\n$$<\/p>\n

\u8fd9\u4e24\u4e2a\u8868\u8fbe\u5f0f\u4e0d\u662f\u7b80\u5355\u7684\u201c\u4e00\u4e2a\u6b63\u786e\u3001\u4e00\u4e2a\u9519\u8bef\u201d\uff0c\u800c\u662f\u5bf9\u5e94\u4e0d\u540c\u7684\u76f8\u5173\u5b9a\u4e49\u548c\u65b9\u5411\u3002<\/p>\n

#### \u76f8\u5173\u5b9a\u7406\u7684\u4e24\u79cd\u7b49\u4ef7\u63a8\u5bfc<\/p>\n

**\u63a8\u5bfc\u4e00\uff1a\u76f4\u63a5\u5c55\u5f00\u5b9a\u4e49\u3002** \u8bb0 $Z(\\omega)=\\mathcal{F}\\{R_{xy}(\\tau)\\}$\uff0c\u5219<\/p>\n

$$
\n\\begin{aligned}
\nZ(\\omega)
\n&=\\int_{-\\infty}^{\\infty}\\left[\\int_{-\\infty}^{\\infty}x(t)y^*(t-\\tau)dt\\right]e^{-j\\omega\\tau}d\\tau\\\\
\n&=\\int_{-\\infty}^{\\infty}x(t)\\int_{-\\infty}^{\\infty}y^*(t-\\tau)e^{-j\\omega\\tau}d\\tau\\,dt.
\n\\end{aligned}
\n$$<\/p>\n

\u5728\u5185\u5c42\u79ef\u5206\u4e2d\u4ee4 $u=t-\\tau$\uff0c$\\tau=t-u$\uff0c$d\\tau=-du$\uff0c\u6539\u53d8\u79ef\u5206\u9650\uff1a<\/p>\n

$$
\n\\int_{-\\infty}^{\\infty}y^*(u)e^{-j\\omega(t-u)}du
\n=e^{-j\\omega t}\\int y^*(u)e^{j\\omega u}du
\n=e^{-j\\omega t}\\,Y^*(\\omega).
\n$$<\/p>\n

\u4ee3\u56de\u5916\u5c42\u79ef\u5206\uff1a<\/p>\n

$$
\nZ(\\omega)=Y^*(\\omega)\\int x(t)e^{-j\\omega t}dt=X(\\omega)Y^*(\\omega).
\n$$<\/p>\n

**\u63a8\u5bfc\u4e8c\uff1a\u501f\u52a9\u201c\u5171\u8f6d\u65f6\u95f4\u53cd\u8f6c\u201d\u4e0e\u5377\u79ef\u5b9a\u7406\u3002** \u8bb0 $\\tilde y(t)=y^*(-t)$\uff0c\u5219\u7531\u57fa\u672c\u53d8\u6362\u516c\u5f0f<\/p>\n

$$
\n\\mathcal{F}\\{y^*(-t)\\}=Y^*(\\omega).
\n$$<\/p>\n

\u518d\u5229\u7528\u7b2c\u4e09\u7ae0\u5f97\u5230\u7684 $R_{xy}(\\tau)=(x*\\tilde y)(\\tau)$ \u4e0e\u5377\u79ef\u5b9a\u7406<\/p>\n

$$
\n\\mathcal{F}\\{x*\\tilde y\\}=X(\\omega)\\cdot Y^*(\\omega),
\n$$<\/p>\n

\u540c\u6837\u5f97\u5230\u76f8\u5173\u5b9a\u7406\u3002\u7b2c\u4e8c\u6761\u63a8\u5bfc\u51f8\u663e\u4e86\u201c\u76f8\u5173\u5c31\u662f\u4e0e\u5171\u8f6d\u53cd\u8f6c\u7248\u672c\u505a\u5377\u79ef\u201d\u8fd9\u4e00\u7ed3\u6784\uff0c\u628a\u65b0\u5b9a\u7406\u6302\u5230\u5df2\u6709\u7ed3\u8bba\u4e0a\uff0c\u53ea\u9700\u4e00\u6b65\u3002<\/p>\n

\u5bf9\u79bb\u6563 DTFT \u6709\u5b8c\u5168\u5bf9\u5e94\u7684\u5f62\u5f0f\uff1a<\/p>\n

$$
\n\\mathcal{F}_{\\mathrm{DTFT}}\\{R_{xy}[m]\\}=X(e^{j\\hat\\omega})\\,Y^*(e^{j\\hat\\omega}).
\n$$<\/p>\n

DFT \u7248\u672c\u5219\u6d89\u53ca\u5faa\u73af\u76f8\u5173\uff0c\u89c1\u4e0b\u6587\u3002<\/p>\n

#### Parseval \u5b9a\u7406\u4e0e\u96f6\u5ef6\u8fdf\u76f8\u5173<\/p>\n

\u5728\u4e0a\u8ff0 Fourier \u7ea6\u5b9a\u4e0b\uff1a<\/p>\n

$$
\nR_{xy}(0)=\\int x(t)y^*(t)\\,dt
\n$$<\/p>\n

\u6ee1\u8db3<\/p>\n

$$
\n\\boxed{
\nR_{xy}(0)=\\frac{1}{2\\pi}
\n\\int X(\\omega)Y^*(\\omega)\\,d\\omega
\n}.
\n$$<\/p>\n

\u79bb\u6563 DFT \u4e0b\uff0c\u82e5<\/p>\n

$$
\nX[k]=\\sum_{n=0}^{N-1}x[n]e^{-j2\\pi kn\/N},
\n$$<\/p>\n

\u5219<\/p>\n

$$
\n\\sum_{n=0}^{N-1}x[n]y^*[n]
\n=\\frac1N\\sum_{k=0}^{N-1}X[k]Y^*[k].
\n$$<\/p>\n

### \u65f6\u57df\u70b9\u4e58\u4e0e\u9891\u57df\u5faa\u73af\u5377\u79ef<\/p>\n

\u9891\u57df\u666e\u901a\u9010\u70b9\u76f8\u4e58\u5bf9\u5e94\u65f6\u57df\u5377\u79ef\uff0c\u800c\u4e0d\u662f\u65f6\u57df\u70b9\u5bf9\u70b9\u4e58\u6cd5\u3002\u5bf9\u957f\u5ea6\u4e3a $N$ \u7684 DFT\uff0c\u82e5<\/p>\n

$$
\nz[n]=x[n]y[n],
\n$$<\/p>\n

\u5c06\u4e24\u4e2a\u9006 DFT \u4ee3\u5165\u5e76\u6309\u9891\u7387\u7d22\u5f15\u6536\u96c6\uff0c\u53ef\u5f97<\/p>\n

$$
\nZ[k]=\\frac{1}{N}\\sum_{r=0}^{N-1}X[r]Y[(k-r)\\bmod N].
\n$$<\/p>\n

\u4e5f\u5c31\u662f\u8bf4\uff0c\u65f6\u57df\u70b9\u4e58\u5bf9\u5e94\u9891\u57df\u5faa\u73af\u5377\u79ef\uff1a<\/p>\n

$$
\n\\boxed{\\mathcal F_N\\{x[n]y[n]\\}=\\frac1N(X\\circledast_NY)[k]}.
\n$$<\/p>\n

\u8fd9\u4e0e<\/p>\n

$$
\n\\operatorname{IDFT}\\{X[k]Y[k]\\}
\n$$<\/p>\n

\u5bf9\u5e94\u7684\u5faa\u73af\u5377\u79ef\u662f\u4e24\u79cd\u4e0d\u540c\u7684\u5bf9\u5076\u5173\u7cfb\u3002\u4e00\u4e2a\u5feb\u901f\u5224\u65ad\u65b9\u6cd5\u662f\u770b\u7ed3\u679c\u7ef4\u5ea6\uff1a\u70b9\u4e58\u4ecd\u662f\u4e0e\u8f93\u5165\u7b49\u957f\u7684\u9010\u6837\u672c\u5411\u91cf\uff0c\u96f6\u5ef6\u8fdf\u5185\u79ef\u662f\u6807\u91cf\uff0c\u76f8\u5173\u5219\u662f\u6bcf\u4e2a lag \u90fd\u6709\u4e00\u4e2a\u503c\u7684\u5e8f\u5217\u3002<\/p>\n

### \u5171\u8f6d\u65b9\u5411\u3001\u4e92\u8c31\u4e0e\u9891\u57df\u68af\u5ea6<\/p>\n

\u672c\u6587\u9ed8\u8ba4\u7684\u76f8\u5173\u5b9a\u4e49\u4e3a<\/p>\n

$$
\nR_{xy}[m]=\\sum_nx[n]y^*[n-m].
\n$$<\/p>\n

\u56e0\u6b64<\/p>\n

$$
\nR_{xy}[m]=\\operatorname{IDFT}\\{X[k]Y^*[k]\\}[m].
\n$$<\/p>\n

\u5982\u679c\u4ea4\u6362\u4fe1\u53f7\u89d2\u8272\uff0c\u6216\u8005\u9700\u8981\u77e9\u9635\u68af\u5ea6\u4e2d\u7684<\/p>\n

$$
\ng[m]=\\sum_nx^*[n-m]e[n],
\n$$<\/p>\n

\u5219\u9891\u57df\u5f62\u5f0f\u4e3a<\/p>\n

$$
\nG[k]=X^*[k]E[k],
\n\\qquad
\n g[m]=\\operatorname{IDFT}\\{X^*[k]E[k]\\}[m].
\n$$<\/p>\n

\u5e38\u89c1\u5f62\u5f0f\u53ef\u4ee5\u5bf9\u7167\u5982\u4e0b\uff1a<\/p>\n

| \u65f6\u57df\u8868\u8fbe\u5f0f | \u9891\u57df\u4e58\u79ef | \u5e38\u89c1\u542b\u4e49 |
\n|—|—|—|
\n| $\\sum_nx[n]y^*[n-m]$ | $XY^*$ | $R_{xy}[m]$ \u6216\u4e92\u529f\u7387\u8c31 |
\n| $\\sum_ny[n]x^*[n-m]$ | $YX^*$ | $R_{yx}[m]$ |
\n| $\\sum_nx^*[n-m]e[n]$ | $X^*E$ | $X^He$ \u578b\u9891\u57df\u68af\u5ea6 |
\n| $\\sum_nx[n]y^*[n]$ | $\\frac1N\\sum_kXY^*$ | \u96f6\u5ef6\u8fdf\u590d\u5185\u79ef |
\n| $x[n]y[n]$ | $\\frac1N X\\circledast_NY$ | \u65f6\u57df\u70b9\u4e58 |<\/p>\n

\u5bf9\u4e8e\u5b9e\u4fe1\u53f7\uff0c\u9891\u8c31\u5171\u8f6d\u5bf9\u79f0\u53ef\u80fd\u4f7f\u65b9\u5411\u5dee\u5f02\u6682\u65f6\u4e0d\u660e\u663e\uff1b\u5bf9\u4e8e\u590d\u4fe1\u53f7\uff0c$XY^*$ \u4e0e $X^*Y$ \u901a\u5e38\u4e92\u4e3a\u5171\u8f6d\u5e76\u4f34\u968f lag \u65b9\u5411\u53d8\u5316\uff0c\u4e0d\u80fd\u4ec5\u51ed\u8bb0\u5fc6\u4e92\u6362\u3002<\/p>\n

### \u76f8\u5bf9\u76f8\u4f4d\u4e0e\u5355\u9891\u9a8c\u8bc1<\/p>\n

\u8bbe<\/p>\n

$$
\nx[n]=e^{j\\omega_0n},
\n\\qquad
\n e[n]=Ae^{j(\\omega_1n+\\phi)}.
\n$$<\/p>\n

\u5171\u8f6d\u4e58\u79ef\u4e3a<\/p>\n

$$
\nx^*[n]e[n]=Ae^{j((\\omega_1-\\omega_0)n+\\phi)}.
\n$$<\/p>\n

\u5f53 $\\omega_1=\\omega_0$ \u65f6\uff0c\u4e58\u79ef\u76f8\u4f4d\u4e0d\u518d\u968f\u65f6\u95f4\u65cb\u8f6c\uff0c\u7d2f\u52a0\u5e45\u5ea6\u6700\u5927\uff1b\u666e\u901a\u4e58\u79ef $x[n]e[n]$ \u5219\u5305\u542b\u9891\u7387\u548c $\\omega_0+\\omega_1$\u3002\u56e0\u6b64\u5171\u8f6d\u7684\u4f5c\u7528\u662f\u6bd4\u8f83\u76f8\u5bf9\u76f8\u4f4d\u548c\u9891\u7387\u5dee\uff0c\u800c\u4e0d\u662f\u628a\u70b9\u4e58\u201c\u6539\u5199\u201d\u4e3a\u9891\u57df\u9010\u70b9\u4e58\u6cd5\u3002
\n#### \u529f\u7387\u8c31\u4e0e Wiener\u2013Khinchin \u5173\u7cfb<\/p>\n

\u81ea\u76f8\u5173\u7684 Fourier \u53d8\u6362\u4e3a\u529f\u7387\u8c31\uff1a<\/p>\n

$$
\n\\boxed{
\nS_{xx}(\\omega)=\\mathcal{F}\\{R_{xx}(\\tau)\\}
\n=X(\\omega)X^*(\\omega)=|X(\\omega)|^2
\n}.
\n$$<\/p>\n

\u5bf9\u4e8e\u968f\u673a\u8fc7\u7a0b\uff0c\u8fd9\u4e00\u7ed3\u8bba\u79f0\u4e3a Wiener\u2013Khinchin \u5b9a\u7406\uff1a\u529f\u7387\u8c31\u5bc6\u5ea6\u662f\u81ea\u76f8\u5173\u51fd\u6570\u7684 Fourier \u53d8\u6362\u3002<\/p>\n

\u4e92\u76f8\u5173\u7684 Fourier \u53d8\u6362\u5219\u662f\u4e92\u529f\u7387\u8c31\uff1a<\/p>\n

$$
\nS_{xy}(\\omega)=X(\\omega)Y^*(\\omega).
\n$$<\/p>\n

\u81ea\u529f\u7387\u8c31\u901a\u5e38\u4e3a\u975e\u8d1f\u5b9e\u6570\uff0c\u800c\u4e92\u529f\u7387\u8c31\u4e00\u822c\u662f\u590d\u6570\uff0c\u5176\u76f8\u4f4d\u5305\u542b\u76f8\u5bf9\u5ef6\u8fdf\u4fe1\u606f\u3002<\/p>\n

\u81ea\u529f\u7387\u8c31\u548c\u4e92\u529f\u7387\u8c31\u7684\u5b8c\u6574\u63a8\u5bfc\u3001\u6838\u5fc3\u6027\u8d28\u548c\u6570\u503c\u7b97\u4f8b\u89c1\u7b2c\u4e94\u7ae0\u3002<\/p>\n

—
\n### \u7ebf\u6027\u5377\u79ef\u4e0e\u5faa\u73af\u5377\u79ef<\/p>\n

#### \u7ebf\u6027\u5377\u79ef<\/p>\n

\u7ebf\u6027\u5377\u79ef\u5047\u8bbe\u4fe1\u53f7\u5728\u6709\u6548\u533a\u95f4\u4e4b\u5916\u4e3a\u96f6\u3002\u5bf9\u4e8e\u6709\u9650\u5e8f\u5217 $x[n]$ \u548c $h[n]$\uff0c\u8f93\u51fa\u957f\u5ea6\u4e3a<\/p>\n

$$
\nN+M-1.
\n$$<\/p>\n

\u7ebf\u6027\u5377\u79ef\u6ca1\u6709\u9996\u5c3e\u76f8\u63a5\u7684\u56de\u7ed5\u3002<\/p>\n

#### \u5faa\u73af\u5377\u79ef<\/p>\n

\u957f\u5ea6\u4e3a $N$ \u7684\u5faa\u73af\u5377\u79ef\u5b9a\u4e49\u4e3a<\/p>\n

$$
\n y_N[n]=\\sum_{k=0}^{N-1}x[k]h[(n-k)\\bmod N].
\n$$<\/p>\n

\u5b83\u628a\u5e8f\u5217\u770b\u6210\u957f\u5ea6\u4e3a $N$ \u7684\u5468\u671f\u5e8f\u5217\uff0c\u56e0\u6b64\u4e00\u4e2a\u5e8f\u5217\u672b\u5c3e\u79fb\u51fa\u540e\u4f1a\u4ece\u5f00\u5934\u91cd\u65b0\u51fa\u73b0\u3002<\/p>\n

#### DFT \u4e0e\u5faa\u73af\u5377\u79ef<\/p>\n

\u957f\u5ea6\u4e3a $N$ \u7684 DFT \u6ee1\u8db3<\/p>\n

$$
\n\\operatorname{DFT}\\{x\\circledast_N h\\}
\n=X[k]H[k].
\n$$<\/p>\n

\u6240\u4ee5\u76f4\u63a5\u5bf9\u4e24\u4e2a\u957f\u5ea6\u4e3a $N$ \u7684\u5e8f\u5217\u505a FFT\u3001\u9891\u57df\u76f8\u4e58\u518d IFFT\uff0c\u5f97\u5230\u7684\u662f $N$ \u70b9\u5faa\u73af\u5377\u79ef\uff0c\u800c\u4e0d\u4e00\u5b9a\u662f\u7ebf\u6027\u5377\u79ef\u3002<\/p>\n

#### \u901a\u8fc7\u8865\u96f6\u5f97\u5230\u7ebf\u6027\u5377\u79ef<\/p>\n

\u82e5 $x$ \u957f\u5ea6\u4e3a $N$\u3001$h$ \u957f\u5ea6\u4e3a $M$\uff0c\u9009\u62e9<\/p>\n

$$
\nL\\ge N+M-1
\n$$<\/p>\n

\u5e76\u5c06\u4e8c\u8005\u8865\u96f6\u81f3\u957f\u5ea6 $L$\uff0c\u5219 $L$ \u70b9\u5faa\u73af\u5377\u79ef\u7684\u524d $N+M-1$ \u4e2a\u6837\u672c\u7b49\u4e8e\u7ebf\u6027\u5377\u79ef\u3002<\/p>\n

#### \u5faa\u73af\u76f8\u5173<\/p>\n

\u540c\u7406\uff0c\u76f4\u63a5\u8ba1\u7b97<\/p>\n

$$
\n\\operatorname{IFFT}\\{X[k]Y^*[k]\\}
\n$$<\/p>\n

\u5f97\u5230\u7684\u662f\u5faa\u73af\u76f8\u5173\u3002\u82e5\u9700\u8981\u7ebf\u6027\u76f8\u5173\uff0c\u4e5f\u5fc5\u987b\u8db3\u591f\u8865\u96f6\uff0c\u5e76\u6839\u636e\u76f8\u5173\u5b9a\u4e49\u622a\u53d6\u6709\u6548\u533a\u95f4\u3002<\/p>\n

#### \u5faa\u73af\u5377\u79ef\u4e0e\u7ebf\u6027\u5377\u79ef\u7684\u5b9a\u91cf\u5173\u7cfb<\/p>\n

\u8bbe $x[n]$\u3001$h[n]$ \u90fd\u5728 $[0,N-1]$ \u4e4b\u5916\u4e3a\u96f6\uff0c\u957f\u5ea6\u5747\u4e0d\u5927\u4e8e $N$\u3002\u7ebf\u6027\u5377\u79ef $y_L[n]=(x*h)[n]$ \u5728 $0\\le n\\le 2N-2$ \u4e0a\u975e\u96f6\uff1b$N$ \u70b9\u5faa\u73af\u5377\u79ef $y_C[n]=(x\\circledast_N h)[n]$ \u53ea\u8986\u76d6 $0\\le n\\le N-1$\u3002\u4e24\u8005\u7684\u5173\u7cfb\u4e3a<\/p>\n

$$
\n\\boxed{
\n y_C[n]=\\sum_{r\\in\\mathbb{Z}}y_L[n+rN],\\qquad 0\\le n\\le N-1
\n}.
\n$$<\/p>\n

\u5373\uff1a**\u5faa\u73af\u5377\u79ef\u7b49\u4e8e\u628a\u7ebf\u6027\u5377\u79ef\u6309\u5468\u671f $N$ \u6298\u53e0\u56de\u4e3b\u533a\u95f4\u3002** \u82e5\u7ebf\u6027\u5377\u79ef\u957f\u5ea6 $L_{\\lin}=N_x+N_h-1\\le N$\uff0c\u6298\u53e0\u4e0d\u4f1a\u4ea7\u751f\u91cd\u53e0\uff0c$y_C$ \u4e0e $y_L$ \u5728 $[0,N-1]$ \u4e0a\u5b8c\u5168\u4e00\u81f4\uff1b\u82e5 $L_{\\lin}>N$\uff0c\u672b\u7aef\u4f1a\u201c\u7ed5\u56de\u201d\u524d\u7aef\uff0c\u5f62\u6210\u65f6\u57df\u6df7\u53e0\uff08time-domain aliasing\uff09\u3002\u56e0\u6b64\u201c\u4f7f\u7528 FFT \u6c42\u7ebf\u6027\u5377\u79ef\u201d\u7684\u9ec4\u91d1\u89c4\u5219\u662f\uff1a<\/p>\n

$$
\nN\\ge N_x+N_h-1,\\qquad \u901a\u5e38\u53d6\\ N=2^{\\lceil\\log_2 L_{\\lin}\\rceil}.
\n$$<\/p>\n

\u5bf9\u4e8e\u76f8\u5173\uff0c\u540c\u6837\u6709<\/p>\n

$$
\nR^{(C)}_{xy}[m]=\\sum_r R^{(L)}_{xy}[m+rN],
\n$$<\/p>\n

\u5176\u4e2d $R^{(L)}$ \u662f\u7ebf\u6027\u76f8\u5173\u3002\u4f7f\u7528 FFT \u6c42\u7ebf\u6027\u76f8\u5173\u65f6\u4e5f\u8981\u6309 $\\ge N_x+N_y-1$ \u8865\u96f6\uff0c\u5e76\u628a IFFT \u7ed3\u679c\u91cd\u6392\u5230\u4e2d\u5fc3 lag \u5904\uff1a\u8d1f lag \u4f4d\u4e8e $[N-\\lfloor(N_y-1)\\rfloor,N-1]$\uff0c\u6b63 lag \u4f4d\u4e8e $[0,N_x-1]$\uff0c\u53ef\u4ee5\u7528 `np.fft.fftshift` \u6216\u624b\u5de5\u5207\u7247\u3002<\/p>\n

#### \u6700\u5c0f NumPy \u793a\u4f8b\uff1a\u5377\u79ef\u4e0e\u76f8\u5173\u7684\u4e09\u79cd\u7b97\u6cd5\u4e92\u8bc1<\/p>\n

\u4e0b\u9762\u7ed9\u51fa\u4e00\u4e2a\u53ef\u4ee5\u76f4\u63a5\u8dd1\u901a\u7684\u6700\u5c0f\u793a\u4f8b\uff0c\u6bd4\u5bf9\u201c\u65f6\u57df\u76f4\u63a5\u7b97\u201d\u201cFFT \u52a0\u96f6\u7b97\u201d\u201c\u5e93\u51fd\u6570\u201d\u4e09\u79cd\u5b9e\u73b0\uff0c\u5de5\u7a0b\u4e0a\u7528\u4f5c\u5355\u5143\u6d4b\u8bd5\u6a21\u677f\uff1a<\/p>\n

“`python
\nimport numpy as np<\/p>\n

rng = np.random.default_rng(0)
\nx = rng.standard_normal(8) + 1j * rng.standard_normal(8)
\nh = rng.standard_normal(5) + 1j * rng.standard_normal(5)
\nN_lin = len(x) + len(h) – 1<\/p>\n

# 1) \u65f6\u57df\u76f4\u63a5\u7ebf\u6027\u5377\u79ef
\ny_direct = np.convolve(x, h, mode=”full”)<\/p>\n

# 2) FFT \u8865\u96f6\u5230 N >= N_lin
\nN = 1 << (N_lin - 1).bit_length()\nX = np.fft.fft(x, N)\nH = np.fft.fft(h, N)\ny_fft = np.fft.ifft(X * H)[:N_lin]\n\n# 3) \u76f8\u5173\uff1aR_xy[m] = sum_n x[n] * conj(y[n-m])\ny_seq = h.copy() # \u76f4\u63a5\u628a h \u5f53\u4f5c\u7b2c\u4e8c\u4e2a\u4fe1\u53f7\nR_direct = np.correlate(x, y_seq, mode=\"full\") # NumPy \u4f7f\u7528 sum x[n+m] * conj(y[n])\nR_fft = np.fft.ifft(np.fft.fft(x, N) * np.conj(np.fft.fft(y_seq, N)))\nR_fft = np.concatenate([R_fft[-(len(y_seq) - 1):], R_fft[: len(x)]])\n\nprint(np.allclose(y_direct, y_fft)) # True\nprint(np.allclose(R_direct, R_fft)) # True\n```\n\n\u5173\u952e\u70b9\uff1a\n\n- FFT \u957f\u5ea6 $N$ \u5fc5\u987b\u5927\u4e8e\u7b49\u4e8e $N_x+N_h-1$\uff0c\u5426\u5219\u6700\u540e\u4e00\u6b65\u7684\u5faa\u73af\u5377\u79ef\u4f1a\u4e0e\u7ebf\u6027\u5377\u79ef\u4e0d\u4e00\u81f4\uff1b\n- \u76f8\u5173\u7684 FFT \u7248\u672c\u4f7f\u7528 $X\\cdot Y^*$\uff0c\u6700\u540e\u8981\u6309\u5b9a\u4e49\u628a IFFT \u7ed3\u679c\u505a `fftshift` \u6216\u624b\u5de5\u5207\u5206\uff1b\n- \u7528 `np.allclose` \u505a\u6570\u503c\u7b49\u4ef7\u6821\u9a8c\uff0c\u5bb9\u5fcd\u6d6e\u70b9\u8bef\u5dee\uff0c\u662f\u81ea\u67e5\u4ee3\u7801\u6b63\u786e\u6027\u7684\u6700\u7b80\u5355\u529e\u6cd5\u3002\n\n#### \u5206\u5757\u5377\u79ef\uff1aOverlap-Add \u4e0e Overlap-Save \u7b80\u8868\n\n\u5de5\u7a0b\u4e0a\u9047\u5230\u957f\u4fe1\u53f7 $x$\uff08\u957f\u5ea6 $L\\gg 1$\uff09\u4e0e\u56fa\u5b9a\u77ed\u6ee4\u6ce2\u5668 $h$\uff08\u957f\u5ea6 $M$\uff09\u65f6\uff0c\u76f4\u63a5\u4e00\u6b21\u505a\u957f FFT \u4f1a\u5360\u7528\u5927\u91cf\u5185\u5b58\u5e76\u5f15\u5165\u5ef6\u8fdf\u3002\u5206\u5757\u5377\u79ef\u628a $x$ \u5207\u6210\u957f\u5ea6 $L_{\\text{blk}}$ \u7684\u6bb5\uff0c\u6bcf\u6bb5\u505a\u957f\u5ea6 $N=L_{\\text{blk}}+M-1$\uff08\u6216 $L_{\\text{blk}}\\ge M$\uff09\u7684 FFT\uff0c\u5c06\u4e24\u79cd\u4e3b\u6d41\u505a\u6cd5\u5217\u8868\u5bf9\u7167\uff1a\n\n| \u7279\u6027 | Overlap-Add (OLA) | Overlap-Save (OLS) |\n|---|---|---|\n| \u5206\u5757\u65b9\u5f0f | \u628a $x$ \u5207\u6210\u4e0d\u91cd\u53e0\u7684\u957f\u5ea6 $L_{\\text{blk}}$ \u6bb5 | \u76f8\u90bb\u6bb5\u6709 $M-1$ \u4e2a\u91cd\u53e0\u6837\u672c |\n| FFT \u957f\u5ea6 | $N=L_{\\text{blk}}+M-1$\uff0c\u5404\u6bb5\u8865\u96f6\u540e FFT | $N=L_{\\text{blk}}$\uff0c$N\\ge M$\uff0c\u5404\u6bb5\u6574\u4f53 FFT |\n| \u8f93\u51fa\u62fc\u63a5 | \u76f8\u90bb\u5757\u5c3e\u90e8 $M-1$ \u4e0e\u4e0b\u4e00\u5757\u5934\u90e8\u76f8\u52a0 | \u4e22\u5f03\u6bcf\u5757\u524d $M-1$ \u6837\u672c\uff0c\u5269\u4f59\u76f4\u63a5\u62fc\u63a5 |\n| \u65f6\u57df\u6df7\u53e0\u5904\u7406 | \u901a\u8fc7\u8865\u96f6\u6d88\u9664 | \u4f9d\u8d56\u201c\u4e22\u5f03\u5934\u90e8\u201d\u628a\u6df7\u53e0\u90e8\u5206\u88c1\u6389 |\n| \u8ba1\u7b97\u91cf\u503e\u5411 | \u6bcf\u5757\u5904\u7406\u91cf\u7565\u9ad8\uff0c\u5b9e\u73b0\u76f4\u89c2 | \u6bcf\u5757\u5c11\u4e00\u6b21\u8865\u96f6\u64cd\u4f5c\uff0c\u5e38\u7528\u9ad8\u6027\u80fd\u5b9e\u73b0 |\n| \u5e38\u89c1\u5e94\u7528 | \u4e00\u822c\u6570\u5b57\u6ee4\u6ce2\u3001\u6559\u5b66\u793a\u4f8b | \u9891\u57df\u81ea\u9002\u5e94\u6ee4\u6ce2\u3001GPU\/DSP \u9ad8\u541e\u5410\u6d41\u6c34\u7ebf |\n\n\u65e0\u8bba OLA \u8fd8\u662f OLS\uff0c\u5176\u6838\u5fc3\u90fd\u662f\u5faa\u73af\u5377\u79ef\u7684**\u6298\u53e0\u516c\u5f0f** $y_C[n]=\\sum_r y_L[n+rN]$\uff1aOLA \u901a\u8fc7\u8865\u96f6\u8ba9\u6298\u53e0\u4e0d\u4ea7\u751f\u91cd\u53e0\uff0cOLS \u901a\u8fc7\u4e22\u5f03\u524d $M-1$ \u6837\u672c\u5254\u9664\u53d7\u6298\u53e0\u6c61\u67d3\u7684\u90e8\u5206\u3002\u7406\u89e3\u4e86\u8fd9\u4e2a\u5173\u7cfb\uff0c\u5c31\u80fd\u5728\u5b9e\u73b0\u4e2d\u968f\u610f\u5207\u6362\u4e24\u79cd\u7b56\u7565\u800c\u4e0d\u51fa\u9519\u3002\n\n---\n\n---\n\n## \u4e94\u3001\u80fd\u91cf\u8c31\u3001\u529f\u7387\u8c31\u4e0e\u4e92\u8c31\n\n### \u80fd\u91cf\u8c31\u7684\u5b9a\u4e49\n\n\u5bf9\u4e8e\u80fd\u91cf\u4fe1\u53f7\uff0c\u4ee4 Fourier \u53d8\u6362\u4e3a\n\n$$\nX(f)=\\int_{-\\infty}^{\\infty}x(t)e^{-j2\\pi ft}dt.\n$$\n\n\u80fd\u91cf\u8c31\u5bc6\u5ea6\uff08energy spectral density\uff0cESD\uff09\u5b9a\u4e49\u4e3a\n\n$$\n\\boxed{\n\\Psi_x(f)=|X(f)|^2=X(f)X^*(f)\n}.\n$$\n\n\u5b83\u8868\u793a\u4fe1\u53f7\u80fd\u91cf\u5728\u9891\u7387\u8f74\u4e0a\u7684\u5206\u5e03\u3002\n\n\u5982\u679c\u4f7f\u7528\u89d2\u9891\u7387 $\\omega$\uff1a\n\n$$\nX(\\omega)=\\int x(t)e^{-j\\omega t}dt,\n\\qquad\n\\Psi_x(\\omega)=|X(\\omega)|^2.\n$$\n\n\u5fc5\u987b\u6ce8\u610f $f$ \u548c $\\omega$ \u7684\u53d8\u91cf\u4e0d\u540c\uff0c\u79ef\u5206\u5173\u7cfb\u4e2d\u4f1a\u51fa\u73b0 $d\\omega=2\\pi df$ \u7684\u6bd4\u4f8b\u56e0\u5b50\u3002\n\n### Parseval \u5b9a\u7406\u4e0e\u603b\u80fd\u91cf\n\n\u91c7\u7528 $f$ \u4e3a\u9891\u7387\u7684 Fourier \u53d8\u6362\u7ea6\u5b9a\uff1a\n\n$$\n\\boxed{\nE_x=\\int_{-\\infty}^{\\infty}|x(t)|^2dt\n=\\int_{-\\infty}^{\\infty}|X(f)|^2df\n=\\int_{-\\infty}^{\\infty}\\Psi_x(f)df\n}.\n$$\n\n\u56e0\u6b64\uff0c\u80fd\u91cf\u8c31\u5bc6\u5ea6\u66f2\u7ebf\u4e0b\u9762\u79ef\u7b49\u4e8e\u4fe1\u53f7\u603b\u80fd\u91cf\u3002\n\n\u82e5\u91c7\u7528\u89d2\u9891\u7387\u7ea6\u5b9a\uff0c\u5219\n\n$$\nE_x=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}|X(\\omega)|^2d\\omega.\n$$\n\n$1\/(2\\pi)$ \u4e0d\u662f\u989d\u5916\u7684\u7269\u7406\u4fee\u6b63\uff0c\u800c\u662f Fourier \u6b63\u9006\u53d8\u6362\u5f52\u4e00\u5316\u7ea6\u5b9a\u5e26\u6765\u7684\u56e0\u5b50\u3002\n\n### \u80fd\u91cf\u8c31\u5bc6\u5ea6\u4e0e\u81ea\u76f8\u5173\n\n\u5bf9\u80fd\u91cf\u4fe1\u53f7\uff0c\u81ea\u76f8\u5173\u5b9a\u4e49\u4e3a\n\n$$\nR_{xx}(\\tau)=\\int x(t)x^*(t-\\tau)dt.\n$$\n\n\u76f8\u5173\u5b9a\u7406\u7ed9\u51fa\n\n$$\n\\boxed{\n\\mathcal F\\{R_{xx}(\\tau)\\}=|X(f)|^2=\\Psi_x(f)\n}.\n$$\n\n\u4e0a\u8ff0\u7ed3\u8bba\u53ef\u4ee5\u4ece\u76f8\u5173\u5b9a\u7406\u76f4\u63a5\u63a8\u51fa\u3002\u81ea\u76f8\u5173\u662f\u4e92\u76f8\u5173 $R_{xy}$ \u5728 $y=x$ \u65f6\u7684\u7279\u6b8a\u60c5\u5f62\uff1a\n\n$$\nR_{xx}(\\tau)=\\int x(t)x^*(t-\\tau)\\,dt.\n$$\n\n\u4ee3\u5165\u76f8\u5173\u5b9a\u7406 $\\mathcal F\\{R_{xy}\\}=X(f)Y^*(f)$\uff0c\u4ee4 $Y=X$\uff1a\n\n$$\n\\mathcal F\\{R_{xx}(\\tau)\\}=X(f)X^*(f)=|X(f)|^2.\n$$\n\n\u81ea\u529f\u7387\u8c31\u4e4b\u6240\u4ee5\u5fc5\u7136\u662f\u5b9e\u975e\u8d1f\u7684\uff0c\u6b63\u662f\u56e0\u4e3a $X\\cdot X^*=|X|^2\\ge 0$\u3002\u8fd9\u91cc\u7684\u5171\u8f6d $X^*$ \u6765\u81ea\u81ea\u76f8\u5173\u5b9a\u4e49\u4e2d\u5bf9\u7b2c\u4e8c\u4e2a\u4fe1\u53f7\u53d6\u5171\u8f6d $x^*(t-\\tau)$\uff0c\u800c\u4e0d\u662f\u6765\u81ea\u5377\u79ef\u2014\u2014\u5377\u79ef\u7684\u9891\u57df\u5bf9\u5e94\u4e3a $X(f)H(f)$\uff0c\u4e0d\u542b\u5171\u8f6d\u3002\n\n\u56e0\u6b64\uff0c\u80fd\u91cf\u8c31\u5bc6\u5ea6\u662f\u80fd\u91cf\u4fe1\u53f7\u81ea\u76f8\u5173\u51fd\u6570\u7684 Fourier \u53d8\u6362\u3002\u53cd\u53d8\u6362\u5173\u7cfb\u4e3a\n\n$$\nR_{xx}(\\tau)=\\int_{-\\infty}^{\\infty}\n\\Psi_x(f)e^{j2\\pi f\\tau}df.\n$$\n\n\u5728\u96f6\u5ef6\u8fdf\u5904\uff1a\n\n$$\nR_{xx}(0)=\\int\\Psi_x(f)df=E_x.\n$$\n\n### \u80fd\u91cf\u8c31\u7684\u5e45\u5ea6\u548c\u76f8\u4f4d\n\n\u80fd\u91cf\u8c31\u5bc6\u5ea6\u53ea\u4fdd\u7559 Fourier \u53d8\u6362\u7684\u5e45\u5ea6\uff1a\n\n$$\n\\Psi_x(f)=|X(f)|^2.\n$$\n\n\u56e0\u6b64\u5b83\u4e0d\u5305\u542b $X(f)$ \u7684\u7edd\u5bf9\u76f8\u4f4d\u3002\u4e0d\u540c\u4fe1\u53f7\u53ef\u80fd\u5177\u6709\u76f8\u540c\u7684\u80fd\u91cf\u8c31\uff0c\u4f46\u65f6\u57df\u6ce2\u5f62\u4e0d\u540c\uff0c\u8fd9\u79f0\u4e3a\u540c\u8c31\u6216\u76f8\u4f4d\u4e0d\u786e\u5b9a\u6027\u95ee\u9898\u3002\n\n\u7279\u522b\u5730\uff0c\u65f6\u79fb\u4fe1\u53f7\n\n$$\n y(t)=x(t-t_0)\n$$\n\n\u7684 Fourier \u53d8\u6362\u4e3a\n\n$$\nY(f)=X(f)e^{-j2\\pi ft_0},\n$$\n\n\u6240\u4ee5\n\n$$\n|Y(f)|^2=|X(f)|^2.\n$$\n\n\u80fd\u91cf\u8c31\u65e0\u6cd5\u5355\u72ec\u786e\u5b9a\u4fe1\u53f7\u7684\u7edd\u5bf9\u65f6\u95f4\u4f4d\u7f6e\u3002\n\n### \u5b9e\u4fe1\u53f7\u7684\u53cc\u8fb9\u4e0e\u5355\u8fb9\u80fd\u91cf\u8c31\n\n\u5b9e\u503c\u4fe1\u53f7\u6ee1\u8db3\n\n$$\nX(-f)=X^*(f),\n$$\n\n\u6240\u4ee5\n\n$$\n\\Psi_x(-f)=\\Psi_x(f).\n$$\n\n\u53cc\u8fb9\u80fd\u91cf\u8c31\u5305\u542b\u6b63\u3001\u8d1f\u9891\u7387\u3002\u82e5\u53ea\u663e\u793a $f\\ge0$ \u7684\u5355\u8fb9\u80fd\u91cf\u8c31\uff0c\u5219\u9664 DC \u548c Nyquist \u70b9\u5916\uff0c\u901a\u5e38\u5c06\u6b63\u9891\u7387\u90e8\u5206\u4e58\u4ee5 2\uff0c\u4ee5\u4fdd\u6301\u603b\u80fd\u91cf\uff1a\n\n$$\nE_x=\\int_0^{\\infty}\\Psi_{x,\\mathrm{one-sided}}(f)df.\n$$\n\n\u4e0d\u80fd\u628a\u53cc\u8fb9\u8c31\u76f4\u63a5\u622a\u53bb\u8d1f\u9891\u7387\u800c\u4e0d\u8865\u507f\uff0c\u5426\u5219\u9762\u79ef\u4f1a\u5c11\u4e00\u534a\u3002\n\n### \u79bb\u6563\u6709\u9650\u5e8f\u5217\u7684\u80fd\u91cf\u8c31\n\n\u5bf9\u957f\u5ea6\u4e3a $N$ \u7684\u6709\u9650\u5e8f\u5217 $x[n]$\uff0c\u5176 DTFT \u4e3a\n\n$$\nX(e^{j\\omega})=\\sum_{n=0}^{N-1}x[n]e^{-j\\omega n}.\n$$\n\n\u80fd\u91cf\u8c31\u5bc6\u5ea6\u4e3a\n\n$$\n\\Psi_x(e^{j\\omega})=|X(e^{j\\omega})|^2.\n$$\n\nParseval \u5173\u7cfb\u4e3a\n\n$$\n\\boxed{\n\\sum_{n=0}^{N-1}|x[n]|^2\n=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}\n|X(e^{j\\omega})|^2d\\omega\n}.\n$$\n\n\u5982\u679c\u7528 $N$ \u70b9 DFT\uff1a\n\n$$\nX[k]=\\sum_{n=0}^{N-1}x[n]e^{-j2\\pi kn\/N},\n$$\n\n\u5219\n\n$$\n\\boxed{\n\\sum_{n=0}^{N-1}|x[n]|^2\n=\\frac1N\\sum_{k=0}^{N-1}|X[k]|^2\n}.\n$$\n\n### \u80fd\u91cf\u8c31\u4e0e\u9891\u8c31\u7684\u533a\u522b\n\n- $X(f)$ \u662f\u590d\u9891\u8c31\uff0c\u5305\u542b\u5e45\u5ea6\u548c\u76f8\u4f4d\uff1b\n- $|X(f)|$ \u662f\u5e45\u5ea6\u8c31\uff1b\n- $|X(f)|^2$ \u662f\u80fd\u91cf\u8c31\u5bc6\u5ea6\uff1b\n- \u4e09\u8005\u7684\u5355\u4f4d\u548c\u79ef\u5206\u610f\u4e49\u4e0d\u540c\u3002\n\n\u80fd\u91cf\u8c31\u4e0d\u80fd\u6062\u590d\u4e00\u822c\u4fe1\u53f7\u7684\u76f8\u4f4d\uff0c\u56e0\u6b64\u4e0d\u80fd\u7b80\u5355\u5730\u628a\u80fd\u91cf\u8c31\u5f53\u4f5c\u5b8c\u6574\u9891\u8c31\u3002\n\n---\n### \u529f\u7387\u8c31\u4e0e\u529f\u7387\u8c31\u5bc6\u5ea6\n\n#### \u529f\u7387\u8c31\u5bc6\u5ea6\u7684\u57fa\u672c\u601d\u60f3\n\n\u529f\u7387\u4fe1\u53f7\u7684\u603b\u80fd\u91cf\u901a\u5e38\u4e3a\u65e0\u7a77\u5927\uff0c\u4e0d\u80fd\u4f7f\u7528\n\n$$\n|X(f)|^2\n$$\n\n\u76f4\u63a5\u4f5c\u4e3a\u6709\u9650\u603b\u80fd\u91cf\u8c31\u3002\u529f\u7387\u8c31\u5bc6\u5ea6\uff08power spectral density\uff0cPSD\uff09\u63cf\u8ff0\u5e73\u5747\u529f\u7387\u5982\u4f55\u5206\u5e03\u5728\u9891\u7387\u4e0a\u3002\n\n\u5bf9\u8fde\u7eed\u65f6\u95f4\u529f\u7387\u4fe1\u53f7\uff0c\u53ef\u5b9a\u4e49\u622a\u65ad Fourier \u53d8\u6362\uff1a\n\n$$\nX_T(f)=\\int_{-T\/2}^{T\/2}x(t)e^{-j2\\pi ft}dt.\n$$\n\n\u529f\u7387\u8c31\u5bc6\u5ea6\u5b9a\u4e49\u4e3a\u957f\u65f6\u95f4\u5f52\u4e00\u5316\u6781\u9650\uff1a\n\n$$\n\\boxed{\nS_{xx}(f)=\\lim_{T\\to\\infty}\n\\frac{1}{T}\\mathbb E\\left[|X_T(f)|^2\\right]\n}.\n$$\n\n\u5bf9\u786e\u5b9a\u6027\u529f\u7387\u4fe1\u53f7\uff0c\u53ef\u5728\u5b58\u5728\u6781\u9650\u65f6\u7701\u7565\u671f\u671b\uff1b\u5bf9\u968f\u673a\u8fc7\u7a0b\uff0c\u671f\u671b\u7528\u4e8e\u83b7\u5f97 ensemble \u5e73\u5747\u3002\n\n#### Wiener\u2013Khinchin \u5b9a\u7406\n\n\u5bf9\u5bbd\u5e73\u7a33\u968f\u673a\u8fc7\u7a0b\uff0c\u529f\u7387\u8c31\u5bc6\u5ea6\u662f\u81ea\u76f8\u5173\u51fd\u6570\u7684 Fourier \u53d8\u6362\uff1a\n\n$$\n\\boxed{\nS_{xx}(f)=\\mathcal F\\{R_{xx}(\\tau)\\}\n=\\int_{-\\infty}^{\\infty}R_{xx}(\\tau)e^{-j2\\pi f\\tau}d\\tau\n}.\n$$\n\n\u53cd\u53d8\u6362\u4e3a\n\n$$\nR_{xx}(\\tau)=\\int_{-\\infty}^{\\infty}\nS_{xx}(f)e^{j2\\pi f\\tau}df.\n$$\n\n\u96f6\u5ef6\u8fdf\u7ed9\u51fa\u5e73\u5747\u529f\u7387\uff1a\n\n$$\n\\boxed{\nP_x=R_{xx}(0)=\\int_{-\\infty}^{\\infty}S_{xx}(f)df\n}.\n$$\n\n\u56e0\u6b64\uff0cPSD \u66f2\u7ebf\u5728\u6574\u4e2a\u9891\u7387\u8f74\u4e0a\u7684\u9762\u79ef\u7b49\u4e8e\u5e73\u5747\u529f\u7387\uff0c\u800c\u4e0d\u662f\u603b\u80fd\u91cf\u3002\n\n#### PSD \u7684\u5355\u4f4d\n\n\u82e5 $x(t)$ \u7684\u5355\u4f4d\u4e3a\u4f0f\u7279\uff0c\u4e14\u91c7\u7528\u7535\u538b\u5e73\u65b9\u8868\u793a\u529f\u7387\uff0c\u5219 $S_{xx}(f)$ \u7684\u5355\u4f4d\u4e3a\n\n$$\n\\mathrm{V}^2\/\\mathrm{Hz}.\n$$\n\n\u82e5\u5305\u542b\u53c2\u8003\u963b\u6297 $R$\uff0c\u7269\u7406\u529f\u7387\u8c31\u5bc6\u5ea6\u53ef\u4e3a\n\n$$\nS_P(f)=\\frac{S_{xx}(f)}{R},\n$$\n\n\u5355\u4f4d\u4e3a\n\n$$\n\\mathrm{W}\/\\mathrm{Hz}.\n$$\n\n\u5bf9\u4e8e\u79bb\u6563\u6570\u636e\uff0c\u9891\u7387\u5355\u4f4d\u5e38\u4e3a cycles\/sample \u6216 Hz\uff0c\u5fc5\u987b\u8bf4\u660e\u91c7\u6837\u7387\u6362\u7b97\u3002\n\n#### \u5468\u671f\u4fe1\u53f7\u7684\u529f\u7387\u8c31\n\n\u5468\u671f\u4fe1\u53f7\u7684 Fourier \u53d8\u6362\u901a\u5e38\u7531\u9891\u7387\u51b2\u6fc0\u7ec4\u6210\uff0c\u56e0\u6b64 PSD \u662f\u79bb\u6563\u7ebf\u8c31\u800c\u4e0d\u662f\u666e\u901a\u8fde\u7eed\u66f2\u7ebf\u3002\n\n\u8bbe\u5468\u671f\u4e3a $T_0$ \u7684\u4fe1\u53f7\u6709 Fourier \u7ea7\u6570\n\n$$\n x(t)=\\sum_{k=-\\infty}^{\\infty}c_ke^{j2\\pi kf_0t},\n\\qquad f_0=\\frac1{T_0}.\n$$\n\n\u5176\u5e73\u5747\u529f\u7387\u4e3a\n\n$$\nP_x=\\sum_{k=-\\infty}^{\\infty}|c_k|^2.\n$$\n\n\u529f\u7387\u8c31\u53ef\u4ee5\u5199\u4e3a\n\n$$\n\\boxed{\nS_{xx}(f)=\\sum_{k=-\\infty}^{\\infty}\n|c_k|^2\\delta(f-kf_0)\n}.\n$$\n\n\u8fd9\u8868\u660e\u6bcf\u6761\u8c31\u7ebf\u7684\u9762\u79ef\u662f\u5bf9\u5e94 Fourier \u7ea7\u6570\u5206\u91cf\u7684\u5e73\u5747\u529f\u7387\u3002\n\n#### \u6b63\u5f26\u4fe1\u53f7\u7684\u529f\u7387\u8c31\n\n\u5bf9\n\n$$\n x(t)=A\\cos(2\\pi f_0t+\\phi),\n$$\n\n\u6709\n\n$$\n x(t)=\\frac A2e^{j\\phi}e^{j2\\pi f_0t}\n+\\frac A2e^{-j\\phi}e^{-j2\\pi f_0t}.\n$$\n\n\u5176\u53cc\u8fb9\u529f\u7387\u8c31\u4e3a\n\n$$\nS_{xx}(f)=\\frac{A^2}{4}\\delta(f-f_0)\n+\\frac{A^2}{4}\\delta(f+f_0).\n$$\n\n\u79ef\u5206\u5f97\u5230\n\n$$\nP_x=\\frac{A^2}{2}.\n$$\n\n\u76f8\u4f4d $\\phi$ \u4e0d\u5f71\u54cd\u5355\u4e2a\u6b63\u5f26\u7684\u529f\u7387\u8c31\uff0c\u4f46\u4f1a\u5f71\u54cd\u4e0e\u5176\u4ed6\u4fe1\u53f7\u7684\u4e92\u529f\u7387\u8c31\u76f8\u4f4d\u3002\n\n#### \u767d\u566a\u58f0\u529f\u7387\u8c31\n\n\u7406\u60f3\u8fde\u7eed\u65f6\u95f4\u767d\u566a\u58f0\u7684 PSD \u4e3a\u5e38\u6570\uff1a\n\n$$\nS_{xx}(f)=N_0\/2\n$$\n\n\u6216\u6839\u636e\u53cc\u8fb9\/\u5355\u8fb9\u7ea6\u5b9a\u5199\u4e3a\u5176\u4ed6\u5e38\u6570\u5f62\u5f0f\u3002\u767d\u566a\u58f0\u5728\u65e0\u9650\u5e26\u5bbd\u4e0a\u5177\u6709\u65e0\u9650\u603b\u529f\u7387\uff0c\u56e0\u6b64\u7269\u7406\u7cfb\u7edf\u603b\u8981\u53d7\u5230\u5e26\u5bbd\u9650\u5236\u3002\n\n\u7ecf\u8fc7\u5e26\u5bbd\u4e3a $B$ \u7684\u7406\u60f3\u6ee4\u6ce2\u5668\u540e\uff0c\u566a\u58f0\u529f\u7387\u4e3a\n\n$$\nP=\\int_{-B}^{B}S_{xx}(f)|H(f)|^2df.\n$$\n\n\u66f4\u4e00\u822c\u5730\uff0cLTI \u7cfb\u7edf\u8f93\u51fa PSD \u6ee1\u8db3\n\n$$\n\\boxed{\nS_{yy}(f)=|H(f)|^2S_{xx}(f)\n}.\n$$\n\n#### \u79bb\u6563\u65f6\u95f4\u529f\u7387\u8c31\n\n\u79bb\u6563\u65f6\u95f4\u5bbd\u5e73\u7a33\u8fc7\u7a0b\u7684\u81ea\u76f8\u5173\u4e3a\n\n$$\nR_{xx}[m]=\\mathbb E[x[n]x^*[n-m]].\n$$\n\nDTFT \u7ed9\u51fa PSD\uff1a\n\n$$\n\\boxed{\nS_{xx}(e^{j\\omega})\n=\\sum_{m=-\\infty}^{\\infty}R_{xx}[m]e^{-j\\omega m}\n}.\n$$\n\n\u5e73\u5747\u529f\u7387\u4e3a\n\n$$\nP_x=R_{xx}[0]\n=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}\nS_{xx}(e^{j\\omega})d\\omega.\n$$\n\n\u82e5\u9891\u7387\u4f7f\u7528 $f$\uff08Hz\uff09\uff0c\u5219\u5e94\u5c06\u89d2\u9891\u7387\u4e0e\u91c7\u6837\u7387\u8054\u7cfb\u8d77\u6765\uff1a\n\n$$\n\\omega=2\\pi f\/f_s.\n$$\n\n#### PSD \u7684\u975e\u8d1f\u6027\n\n\u81ea\u529f\u7387\u8c31\u5bc6\u5ea6\u6ee1\u8db3\n\n$$\nS_{xx}(f)\\ge0.\n$$\n\n\u8fd9\u662f\u56e0\u4e3a PSD \u662f\u76f8\u5173\u51fd\u6570\u7684 Fourier \u53d8\u6362\uff0c\u5e76\u4e14\u5bf9\u4e8e\u4efb\u610f\u6ee4\u6ce2\u5668 $g$\uff1a\n\n$$\n\\mathbb E\\left|\\int g^*(t)x(t)dt\\right|^2\\ge0.\n$$\n\n\u7531\u6b64\u53ef\u77e5\uff0c\u81ea\u8c31\u4e0d\u80fd\u51fa\u73b0\u771f\u5b9e\u7684\u8d1f\u529f\u7387\u3002\u5de5\u7a0b\u56fe\u4e2d\u51fa\u73b0\u5c11\u91cf\u8d1f\u503c\u901a\u5e38\u6765\u81ea\u6570\u503c\u8bef\u5dee\u3001\u53bb\u566a\u5904\u7406\u6216\u7ed8\u56fe\u4f7f\u7528\u4e86\u4e0d\u9002\u5408\u7684\u4f30\u8ba1\u91cf\u3002\n\n#### \u81ea\u8c31\u4e0e\u80fd\u91cf\u8c31\u7684\u6839\u672c\u533a\u522b\n\n| \u9879\u76ee | \u80fd\u91cf\u8c31\u5bc6\u5ea6 ESD | \u529f\u7387\u8c31\u5bc6\u5ea6 PSD |\n|---|---|---|\n| \u9002\u7528\u5bf9\u8c61 | \u80fd\u91cf\u4fe1\u53f7 | \u529f\u7387\u4fe1\u53f7\u3001\u968f\u673a\u8fc7\u7a0b\u3001\u5468\u671f\u4fe1\u53f7 |\n| \u5178\u578b\u5b9a\u4e49 | $|X(f)|^2$ | $\\lim_{T\\to\\infty}|X_T(f)|^2\/T$ \u6216 $\\mathcal F\\{R_{xx}\\}$ |\n| \u9891\u7387\u79ef\u5206 | \u603b\u80fd\u91cf $E_x$ | \u5e73\u5747\u529f\u7387 $P_x$ |\n| \u5355\u4f4d | \u4fe1\u53f7\u5355\u4f4d\u5e73\u65b9\u4e58\u65f6\u95f4 | \u4fe1\u53f7\u5355\u4f4d\u5e73\u65b9\/Hz\uff0c\u6216\u529f\u7387\/Hz |\n| \u76f8\u4f4d | \u4e0d\u5305\u542b\u76f8\u4f4d | \u81ea\u8c31\u4e5f\u4e0d\u5305\u542b\u76f8\u4f4d |\n| \u5178\u578b\u8c31\u5f62 | \u6709\u9650\u65f6\u957f\u8109\u51b2\u7684\u8fde\u7eed\u8c31 | \u6b63\u5f26\u7684\u7ebf\u8c31\u3001\u566a\u58f0\u7684\u8fde\u7eed\u8c31 |\n\n\u540c\u4e00\u6709\u9650\u6570\u636e\u6bb5\u65e2\u53ef\u753b periodogram\uff0c\u4e5f\u53ef\u6309\u6709\u9650\u8bb0\u5f55\u80fd\u91cf\u89e3\u91ca\uff1b\u4f46\u628a\u5176\u7eb5\u8f74\u79f0\u4e3a PSD \u8fd8\u662f ESD\uff0c\u53d6\u51b3\u4e8e\u5f52\u4e00\u5316\u548c\u95ee\u9898\u5bf9\u8c61\uff0c\u4e0d\u80fd\u53ea\u770b\u56fe\u5f62\u5f62\u72b6\u3002\n\n---\n### \u4e92\u80fd\u91cf\u8c31\u4e0e\u4e92\u529f\u7387\u8c31\n\n#### \u4e92\u80fd\u91cf\u8c31\n\n\u5bf9\u4e8e\u4e24\u4e2a\u80fd\u91cf\u4fe1\u53f7 $x(t)$\u3001$y(t)$\uff0c\u5b9a\u4e49\u4e92\u80fd\u91cf\u8c31\u5bc6\u5ea6\uff08cross energy spectral density\uff09\u4e3a\n\n$$\n\\boxed{\n\\Psi_{xy}(f)=X(f)Y^*(f)\n}.\n$$\n\n\u5b83\u662f\u4e92\u76f8\u5173\u7684 Fourier \u8868\u793a\uff1a\n\n$$\n\\Psi_{xy}(f)=\\mathcal F\\{R_{xy}(\\tau)\\}.\n$$\n\n\u4e0e\u81ea\u80fd\u91cf\u8c31 $|X(f)|^2$ \u4e0d\u540c\uff0c\u4e92\u80fd\u91cf\u8c31\u4e00\u822c\u662f\u590d\u6570\uff1a\n\n$$\n\\Psi_{xy}(f)=|X(f)||Y(f)|e^{j(\\angle X(f)-\\angle Y(f))}.\n$$\n\n\u5176\u6a21\u8868\u793a\u5171\u540c\u80fd\u91cf\u5e45\u5ea6\uff0c\u590d\u6570\u76f8\u4f4d\u8868\u793a\u4e24\u4e2a\u4fe1\u53f7\u7684\u76f8\u5bf9\u76f8\u4f4d\uff0c\u5177\u4f53\u6b63\u8d1f\u53d6\u51b3\u4e8e\u4e92\u76f8\u5173\u548c\u5171\u8f6d\u7684\u5b9a\u4e49\u3002\n\n#### \u4e92\u80fd\u91cf\u8c31\u7684 Parseval \u5173\u7cfb\n\n\u96f6\u5ef6\u8fdf\u4e92\u76f8\u5173\u4e3a\n\n$$\nR_{xy}(0)=\\int x(t)y^*(t)dt.\n$$\n\n\u7531 Parseval \u5b9a\u7406\uff1a\n\n$$\n\\boxed{\nR_{xy}(0)=\\int\\Psi_{xy}(f)df\n}.\n$$\n\n\u56e0\u6b64\uff0c\u4e92\u80fd\u91cf\u8c31\u5728\u9891\u7387\u4e0a\u7684\u590d\u79ef\u5206\u7ed9\u51fa\u4e24\u4e2a\u80fd\u91cf\u4fe1\u53f7\u7684\u5185\u79ef\u3002\u82e5\u5b58\u5728\u65f6\u5ef6\uff0c\u5219\u5bf9\u4e92\u80fd\u91cf\u8c31\u505a\u9006 Fourier \u53d8\u6362\u53ef\u4ee5\u6062\u590d\u4e0d\u540c\u5ef6\u8fdf\u4e0b\u7684\u4e92\u76f8\u5173\u3002\n\n#### \u4e92\u529f\u7387\u8c31\n\n\u5bf9\u8054\u5408\u5bbd\u5e73\u7a33\u968f\u673a\u8fc7\u7a0b $x(t)$\u3001$y(t)$\uff0c\u4e92\u529f\u7387\u8c31\u5bc6\u5ea6\u5b9a\u4e49\u4e3a\u4e92\u76f8\u5173\u51fd\u6570\u7684 Fourier \u53d8\u6362\uff1a\n\n$$\n\\boxed{\nS_{xy}(f)=\\mathcal F\\{R_{xy}(\\tau)\\}\n}.\n$$\n\n\u5728\u622a\u65ad Fourier \u4f30\u8ba1\u4e2d\uff0c\u5e38\u89c1\u5f62\u5f0f\u4e3a\n\n$$\nS_{xy}(f)\n=\\lim_{T\\to\\infty}\\frac{1}{T}\n\\mathbb E\\left[X_T(f)Y_T^*(f)\\right].\n$$\n\n\u4e92\u529f\u7387\u8c31\u4e00\u822c\u4e3a\u590d\u6570\uff0c\u4e0d\u80fd\u89e3\u91ca\u4e3a\u201c\u529f\u7387\u5fc5\u987b\u4e3a\u6b63\u201d\u7684\u81ea\u8c31\u3002\u53ea\u6709\u81ea\u529f\u7387\u8c31 $S_{xx}$\u3001$S_{yy}$ \u5fc5\u987b\u975e\u8d1f\uff1b\u4e92\u529f\u7387\u8c31\u53ef\u4ee5\u6709\u6b63\u8d1f\u5b9e\u90e8\u548c\u865a\u90e8\u3002\n\n#### \u4e92\u529f\u7387\u8c31\u7684\u63a8\u5bfc\u4e0e\u6838\u5fc3\u6027\u8d28\n\n**\u4ece\u622a\u65ad Fourier \u53d8\u6362\u63a8\u5bfc\u4e92\u529f\u7387\u8c31\u3002** \u5bf9\u8054\u5408\u5bbd\u5e73\u7a33\u8fc7\u7a0b $x(t)$\u3001$y(t)$\uff0c\u53d6\u89c2\u6d4b\u533a\u95f4 $[-T\/2,T\/2]$ \u7684\u622a\u65ad Fourier \u53d8\u6362\n\n$$\nX_T(f)=\\int_{-T\/2}^{T\/2}x(t)e^{-j2\\pi ft}dt,\n\\qquad\nY_T(f)=\\int_{-T\/2}^{T\/2}y(t)e^{-j2\\pi ft}dt.\n$$\n\n\u6784\u9020\n\n$$\n\\frac{1}{T}\\mathbb E\\bigl[X_T(f)Y_T^*(f)\\bigr]\n=\\frac{1}{T}\\int_{-T\/2}^{T\/2}\\int_{-T\/2}^{T\/2}\n\\mathbb E[x(t_1)y^*(t_2)]\\,e^{-j2\\pi f(t_1-t_2)}dt_1\\,dt_2.\n$$\n\n\u7531\u5bbd\u5e73\u7a33\u6027\uff0c$\\mathbb E[x(t_1)y^*(t_2)]=R_{xy}(t_1-t_2)$\u3002\u4ee4 $\\tau=t_1-t_2$\uff0c\u4ea4\u6362\u79ef\u5206\u987a\u5e8f\u5e76\u53d6 $T\\to\\infty$ \u6781\u9650\uff1a\n\n$$\nS_{xy}(f)=\\lim_{T\\to\\infty}\\frac{1}{T}\\mathbb E[X_T(f)Y_T^*(f)]\n=\\int_{-\\infty}^{\\infty}R_{xy}(\\tau)e^{-j2\\pi f\\tau}d\\tau\n=\\mathcal F\\{R_{xy}(\\tau)\\}.\n$$\n\n\u4e2d\u95f4\u6b65\u9aa4\u7684\u5173\u952e\u662f $Y_T^*$ \u4e2d\u7684\u5171\u8f6d\u7ee7\u627f\u81ea\u4e92\u76f8\u5173\u5b9a\u4e49 $R_{xy}(\\tau)=\\mathbb E[x(t)y^*(t-\\tau)]$ \u4e2d\u7684 $y^*$\uff0c\u4e0e\u5377\u79ef\u65e0\u5173\u3002\n\n**\u6838\u5fc3\u6027\u8d28\u6c47\u603b\uff1a**\n\n1. $S_{xy}(f)$ \u4e00\u822c\u4e3a\u590d\u6570\uff0c\u5176\u6a21 $|S_{xy}|$ \u53cd\u6620\u5171\u540c\u529f\u7387\u5e45\u5ea6\uff0c\u76f8\u4f4d $\\angle S_{xy}$ \u53cd\u6620\u9891\u7387\u76f8\u5173\u7684\u76f8\u5bf9\u76f8\u4f4d\u3002\n2. $S_{yx}(f)=S_{xy}^*(f)$\uff08\u4ea4\u6362\u4fe1\u53f7\u89d2\u8272\u7b49\u4ef7\u4e8e\u53d6\u5171\u8f6d\uff09\u3002\n3. $|S_{xy}(f)|^2\\le S_{xx}(f)S_{yy}(f)$\uff08Cauchy\u2013Schwarz \u4e0d\u7b49\u5f0f\u7684\u9891\u57df\u5f62\u5f0f\uff09\uff0c\u8fd9\u6b63\u662f\u76f8\u5e72\u6027\u6ee1\u8db3 $0\\le\\gamma_{xy}^2\\le1$ \u7684\u6839\u672c\u539f\u56e0\u3002\n4. \u82e5 $y(t)=x(t-t_0)$\uff08\u7eaf\u5ef6\u8fdf\uff09\uff0c\u5219\n $$\n R_{xy}(\\tau)=R_{xx}(\\tau-t_0),\n \\qquad\n S_{xy}(f)=S_{xx}(f)\\,e^{-j2\\pi ft_0}.\n $$\n \u4e92\u8c31\u76f8\u4f4d\u4e3a\u7ebf\u6027\u659c\u7387 $-2\\pi ft_0$\uff0c\u5176\u659c\u7387\u76f4\u63a5\u7ed9\u51fa\u5ef6\u8fdf $t_0$\u3002\u81ea\u8c31 $S_{xx}$ \u4e0d\u5305\u542b\u8fd9\u4e00\u4fe1\u606f\u3002\n5. \u82e5 $x$ \u4e0e $y$ \u4e0d\u76f8\u5173\uff08$R_{xy}(\\tau)=0$ \u5bf9\u6240\u6709 $\\tau$\uff09\uff0c\u5219 $S_{xy}(f)=0$\u3002\u4e0d\u76f8\u5173\u4fe1\u53f7\u6ca1\u6709\u4e92\u8c31\u8d21\u732e\u3002\n\n\u8fd8\u8981\u533a\u5206\u4e92\u8c31\u4e0e\u81ea\u9002\u5e94\u6ee4\u6ce2\u68af\u5ea6\u3002\u6309\u672c\u6587\u7ea6\u5b9a\uff0c$S_{xe}=XE^*$ \u8868\u793a\u8f93\u5165\u4e0e\u8bef\u5dee\u7684\u4e92\u8c31\uff1bHermitian \u6700\u5c0f\u4e8c\u4e58\u68af\u5ea6\u901a\u5e38\u51fa\u73b0 $X^*E$\uff0c\u5b83\u5bf9\u5e94\u65f6\u57df\u7684 $\\boldsymbol X^H\\boldsymbol e$\u3002\u4e8c\u8005\u90fd\u542b\u5171\u8f6d\u548c\u76f8\u5bf9\u76f8\u4f4d\uff0c\u4f46\u4fe1\u53f7\u987a\u5e8f\u3001\u76f8\u4f4d\u7b26\u53f7\u548c\u540e\u7eed\u5904\u7406\u4e0d\u540c\uff1a\u4e92\u8c31\u901a\u5e38\u8fdb\u884c\u5206\u6bb5\u5e73\u5747\u4ee5\u63cf\u8ff0\u7edf\u8ba1\u5173\u7cfb\uff0c\u68af\u5ea6\u5219\u7ecf\u8fc7 IFFT\u3001\u62bd\u5934\u622a\u53d6\u548c\u6b65\u957f\u7f29\u653e\u6765\u66f4\u65b0\u53c2\u6570\u3002\n\n\u5f52\u4e00\u5316\u9891\u57df\u68af\u5ea6\u5e38\u5199\u6210\n\n$$\n\\Delta W[k]=\\mu\\frac{X^*[k]E[k]}{\\Phi_x[k]+\\varepsilon}.\n$$\n\n\u5f53 $\\Phi_x[k]$ \u5f88\u5c0f\u65f6\uff0c\u9664\u6cd5\u4f1a\u653e\u5927\u566a\u58f0\u548c\u820d\u5165\u8bef\u5dee\uff1b\u56e0\u6b64 $\\varepsilon$ \u5fc5\u987b\u4e0e\u529f\u7387\u91cf\u7eb2\u4e00\u81f4\uff0c\u5e76\u5e94\u7ed3\u5408\u529f\u7387\u5e73\u6ed1\u3001\u66f4\u65b0\u9650\u5e45\u6216\u9891\u5e26\u95e8\u9650\u4f7f\u7528\u3002\u4e0d\u80fd\u628a $XX^*$ \u7684\u5e45\u5ea6\u76f4\u63a5\u5f53\u4f5c\u68af\u5ea6\u65b9\u5411\uff0c\u68af\u5ea6\u65b9\u5411\u8fd8\u53d6\u51b3\u4e8e\u8bef\u5dee\u7684\u590d\u76f8\u4f4d\u3002\n\n#### \u4e92\u8c31\u7684\u5b9e\u90e8\u548c\u865a\u90e8\n\n\u5199\u6210\n\n$$\nS_{xy}(f)=S_{xy,R}(f)+jS_{xy,I}(f).\n$$\n\n- \u5b9e\u90e8\u901a\u5e38\u8868\u793a\u540c\u76f8\u6216\u5076\u5bf9\u79f0\u76f8\u5173\u6210\u5206\uff1b\n- \u865a\u90e8\u901a\u5e38\u8868\u793a\u6b63\u4ea4\u6216\u5947\u5bf9\u79f0\u76f8\u5173\u6210\u5206\uff1b\n- \u6a21 $|S_{xy}|$ \u8868\u793a\u4e92\u8c31\u5e45\u5ea6\uff1b\n- \u76f8\u4f4d $\\arg S_{xy}$ \u8868\u793a\u9891\u7387\u76f8\u5173\u7684\u76f8\u5bf9\u76f8\u4f4d\u3002\n\n\u5b9e\u90e8\u548c\u865a\u90e8\u7684\u5177\u4f53\u7269\u7406\u89e3\u91ca\u4f9d\u8d56\u4fe1\u53f7\u5b9a\u4e49\u3001\u4f20\u611f\u5668\u65b9\u5411\u3001\u53c2\u8003\u65b9\u5411\u548c\u4e92\u8c31\u7ea6\u5b9a\uff0c\u4e0d\u5e94\u8131\u79bb\u7cfb\u7edf\u6a21\u578b\u673a\u68b0\u89e3\u91ca\u3002\n\n#### \u4e92\u8c31\u7684\u5171\u8f6d\u5bf9\u79f0\u5173\u7cfb\n\n\u4e92\u76f8\u5173\u6ee1\u8db3\n\n$$\nR_{yx}(\\tau)=R_{xy}^*(-\\tau).\n$$\n\nFourier \u53d8\u6362\u540e\uff1a\n\n$$\n\\boxed{\nS_{yx}(f)=S_{xy}^*(f)\n}.\n$$\n\n\u56e0\u6b64\u8c31\u77e9\u9635\n\n$$\n\\boldsymbol S(f)=\n\\begin{bmatrix}\nS_{xx}(f)&S_{xy}(f)\\\\\nS_{yx}(f)&S_{yy}(f)\n\\end{bmatrix}\n$$\n\n\u662f Hermitian \u77e9\u9635\u3002\u5b83\u8fd8\u5fc5\u987b\u662f\u534a\u6b63\u5b9a\u7684\uff0c\u8fd9\u7ed9\u51fa\n\n$$\n|S_{xy}(f)|^2\\le S_{xx}(f)S_{yy}(f).\n$$\n\n\u8fd9\u6b63\u662f\u76f8\u5e72\u6027\u6ee1\u8db3 $0\\le\\gamma_{xy}^2(f)\\le1$ \u7684\u6839\u672c\u539f\u56e0\u3002\n\n#### LTI \u7cfb\u7edf\u4e2d\u7684\u4e92\u529f\u7387\u8c31\n\n\u82e5\n\n$$\n y(t)=h(t)*x(t)+n(t),\n$$\n\n\u4e14 $x$ \u4e0e $n$ \u4e0d\u76f8\u5173\uff0c\u5219\u5728\u91c7\u7528\n\n$$\nS_{xy}=\\mathbb E[X_TY_T^*]\/T\n$$\n\n\u7684\u7ea6\u5b9a\u4e0b\uff1a\n\n$$\nY(f)=H(f)X(f)+N(f),\n$$\n\n$$\n\\boxed{\nS_{xy}(f)=H^*(f)S_{xx}(f)\n}.\n$$\n\n\u82e5\u91c7\u7528\u76f8\u53cd\u7684\u4e92\u8c31\u5b9a\u4e49 $S_{yx}=\\mathbb E[Y_TX_T^*]\/T$\uff0c\u5219\n\n$$\nS_{yx}(f)=H(f)S_{xx}(f).\n$$\n\n\u56e0\u6b64\u4f20\u9012\u51fd\u6570\u4f30\u8ba1\u53ef\u4ee5\u5199\u6210\u4e24\u79cd\u5bf9\u5e94\u5f62\u5f0f\uff1a\n\n$$\n\\widehat H(f)=\\frac{S_{yx}(f)}{S_{xx}(f)}\n$$\n\n\u6216\n\n$$\n\\widehat H^*(f)=\\frac{S_{xy}(f)}{S_{xx}(f)}.\n$$\n\n\u5b9e\u9645\u4f7f\u7528\u65f6\u5fc5\u987b\u786e\u8ba4\u8f6f\u4ef6\u7684 cross-spectrum \u5b9a\u4e49\u4ee5\u53ca\u901a\u9053\u987a\u5e8f\u3002\n\n#### \u4e92\u8c31\u4e0e\u76f8\u5e72\u6027\u7684\u5173\u7cfb\n\n\u4e92\u529f\u7387\u8c31\u672c\u8eab\u4e0d\u80fd\u76f4\u63a5\u8868\u793a\u201c\u76f8\u5173\u6bd4\u4f8b\u201d\u3002\u9700\u8981\u4f7f\u7528\n\n$$\n\\gamma_{xy}^2(f)\n=\\frac{|S_{xy}(f)|^2}{S_{xx}(f)S_{yy}(f)}.\n$$\n\n\u4e92\u8c31\u7684\u7edd\u5bf9\u503c\u53ef\u80fd\u5f88\u5927\uff0c\u53ea\u662f\u56e0\u4e3a\u4e24\u4e2a\u4fe1\u53f7\u529f\u7387\u90fd\u5927\uff1b\u76f8\u5e72\u6027\u7ecf\u8fc7\u81ea\u8c31\u5f52\u4e00\u5316\u540e\u624d\u53ef\u4ee5\u8de8\u9891\u7387\u3001\u8de8\u5b9e\u9a8c\u6bd4\u8f83\u7ebf\u6027\u5173\u8054\u5f3a\u5ea6\u3002\n\n#### \u4e92\u8c31\u76f8\u4f4d\u4e0e\u7fa4\u65f6\u5ef6\n\n\u5728\u9ad8\u76f8\u5e72\u9891\u5e26\uff0c\u4e92\u8c31\u76f8\u4f4d\u53ef\u7528\u4e8e\u4f30\u8ba1\u65f6\u5ef6\u3002\u82e5\u7cfb\u7edf\u9891\u54cd\u76f8\u4f4d\u4e3a $\\phi(f)$\uff0c\u7fa4\u65f6\u5ef6\u5b9a\u4e49\u4e3a\n\n$$\n\\tau_g(f)=-\\frac{1}{2\\pi}\\frac{d\\phi(f)}{df}.\n$$\n\n\u5bf9\u4e8e\u8fd1\u4f3c\u7eaf\u5ef6\u8fdf\uff0c\u7fa4\u65f6\u5ef6\u63a5\u8fd1\u5e38\u6570\uff1b\u5bf9\u4e8e\u8272\u6563\u7cfb\u7edf\u3001\u6ee4\u6ce2\u5668\u6216\u7ed3\u6784\u632f\u52a8\u7cfb\u7edf\uff0c\u7fa4\u65f6\u5ef6\u53ef\u80fd\u968f\u9891\u7387\u53d8\u5316\u3002\n\n\u76f8\u4f4d\u7f20\u7ed5\u3001\u4f4e\u76f8\u5e72\u6027\u548c\u9891\u7387\u54cd\u5e94\u96f6\u70b9\u90fd\u4f1a\u4f7f\u76f8\u4f4d\u659c\u7387\u4e0d\u7a33\u5b9a\uff0c\u56e0\u6b64\u5e94\u8054\u5408\u67e5\u770b\u76f8\u5e72\u6027\u548c\u81ea\u8c31\u529f\u7387\u3002\n\n#### \u4e92\u80fd\u91cf\u8c31\u4e0e\u4e92\u529f\u7387\u8c31\u7684\u5bf9\u7167\n\n| \u9879\u76ee | \u4e92\u80fd\u91cf\u8c31 $\\Psi_{xy}$ | \u4e92\u529f\u7387\u8c31 $S_{xy}$ |\n|---|---|---|\n| \u9002\u7528\u5bf9\u8c61 | \u80fd\u91cf\u4fe1\u53f7\u3001\u6709\u9650\u8bb0\u5f55 | \u529f\u7387\u4fe1\u53f7\u3001\u968f\u673a\u8fc7\u7a0b\u3001\u957f\u65f6\u95f4\u8bb0\u5f55 |\n| \u5178\u578b\u5f62\u5f0f | $X(f)Y^*(f)$ | $\\lim_{T\\to\\infty}X_TY_T^*\/T$ \u7684\u7edf\u8ba1\u6781\u9650 |\n| \u5bf9\u5e94\u65f6\u57df\u91cf | \u4e92\u76f8\u5173\u7684\u80fd\u91cf\u7248\u672c | \u4e92\u76f8\u5173\u51fd\u6570\u7684\u529f\u7387\u7248\u672c |\n| \u9891\u7387\u79ef\u5206 | \u96f6\u5ef6\u8fdf\u5185\u79ef | \u96f6\u5ef6\u8fdf\u4e92\u529f\u7387\u6216\u534f\u65b9\u5dee\u76f8\u5173\u91cf |\n| \u662f\u5426\u53ef\u80fd\u4e3a\u590d\u6570 | \u662f | \u662f |\n| \u5178\u578b\u7528\u9014 | \u8109\u51b2\u3001\u6a21\u677f\u3001\u6709\u9650\u6ce2\u5f62\u6bd4\u8f83 | \u7cfb\u7edf\u8fa8\u8bc6\u3001\u76f8\u5e72\u6027\u3001\u968f\u673a\u632f\u52a8\u548c\u566a\u58f0\u5206\u6790 |\n\n\n#### \u4e92\u529f\u7387\u8c31\u7684\u6570\u503c\u7b97\u4f8b\n\n\u53d6\u4e00\u4e2a\u957f\u5ea6\u4e3a $N=4$ \u7684\u590d\u5e8f\u5217\u53ca\u5176\u5ef6\u8fdf\u7248\u672c\uff1a\n\n$$\nx[n]=[1,\\; j,\\; -1,\\; -j],\n\\qquad\ny[n]=x[n-1]=[{-j},\\; 1,\\; j,\\; -1].\n$$\n\n\u5176\u4e2d $y$ \u662f $x$ \u7684\u5faa\u73af\u53f3\u79fb 1 \u4e2a\u6837\u672c\u3002$x[n]$ \u6070\u597d\u662f\u9891\u7387 $\\omega_0=2\\pi\/4=\\pi\/2$ \u7684\u5355\u9891\u590d\u6307\u6570 $e^{j\\pi n\/2}$\u3002\n\n\u505a 4 \u70b9 DFT\u3002\u7531\u4e8e $x[n]=e^{j\\pi n\/2}$ \u662f\u5355\u9891\u4fe1\u53f7\uff0c\u5176 DFT \u96c6\u4e2d\u5728\u4e00\u4e2a bin \u4e0a\uff1a\n\n$$\nX[k]=\\begin{cases}4,&k=1,\\\\0,&k\\ne1.\\end{cases}\n$$\n\n\u5ef6\u8fdf 1 \u6837\u672c\u7684 DFT \u6ee1\u8db3 $Y[k]=X[k]e^{-j2\\pi k\/4}$\uff0c\u5373\n\n$$\nY[k]=\\begin{cases}4e^{-j\\pi\/2}=-4j,&k=1,\\\\0,&k\\ne1.\\end{cases}\n$$\n\n\u81ea\u529f\u7387\u8c31\uff1a\n\n$$\nS_{xx}[k]=X[k]X^*[k]=|X[k]|^2\n=\\begin{cases}16,&k=1,\\\\0,&k\\ne1.\\end{cases}\n$$\n\n\u7ed3\u679c\u4e3a\u5b9e\u975e\u8d1f\uff0c\u4e0d\u542b\u76f8\u4f4d\u4fe1\u606f\u3002\n\n\u4e92\u529f\u7387\u8c31\uff1a\n\n$$\nS_{xy}[k]=X[k]Y^*[k]\n=\\begin{cases}4\\cdot(4j)=16j,&k=1,\\\\0,&k\\ne1.\\end{cases}\n$$\n\n\u7ed3\u679c\u4e3a\u590d\u6570\u3002\u5728 $k=1$ \u5904\uff0c$\\angle S_{xy}[1]=\\pi\/2=2\\pi\\cdot 1\/4$\uff0c\u6070\u597d\u662f\u5ef6\u8fdf 1 \u6837\u672c\u5bf9\u5e94\u7684\u76f8\u4f4d\u659c\u7387 $2\\pi k\/N$\u3002\n\n\u82e5\u9519\u8bef\u5730\u4f7f\u7528\u5377\u79ef\u5f62\u5f0f $X[k]Y[k]$\uff08\u4e0d\u53d6\u5171\u8f6d\uff09\uff1a\n\n$$\nX[k]Y[k]\n=\\begin{cases}4\\cdot(-4j)=-16j,&k=1,\\\\0,&k\\ne1.\\end{cases}\n$$\n\n\u5176\u76f8\u4f4d\u4e3a $-\\pi\/2$\uff0c\u7b49\u4e8e $\\angle X[1]+\\angle Y[1]=0+(-\\pi\/2)$\uff0c\u53cd\u6620\u7684\u662f\u4e24\u4e2a\u4fe1\u53f7\u76f8\u4f4d\u4e4b\u548c\uff0c\u800c\u4e0d\u662f\u76f8\u4f4d\u5dee\u3002\u8fd9\u4e0d\u80fd\u7528\u4e8e\u65f6\u5ef6\u4f30\u8ba1\u3002\n\n\u8fd9\u4e2a\u4f8b\u5b50\u6e05\u695a\u5730\u8868\u660e\uff1a\n\n- \u81ea\u529f\u7387\u8c31 $|X|^2$ \u4e22\u5f03\u6240\u6709\u76f8\u4f4d\uff0c\u53ea\u4fdd\u7559\u80fd\u91cf\u5206\u5e03\uff1b\n- \u4e92\u529f\u7387\u8c31 $XY^*$ \u901a\u8fc7\u5171\u8f6d\u628a\u76f8\u4f4d\u53d8\u6210\u5dee\u503c $\\angle X-\\angle Y$\uff0c\u7f16\u7801\u4e24\u4e2a\u4fe1\u53f7\u7684\u76f8\u5bf9\u5ef6\u8fdf\uff1b\n- \u5171\u8f6d\u6765\u81ea\u4e92\u76f8\u5173\u5b9a\u4e49\u4e2d\u7684 $y^*$\uff0c\u4e0d\u662f\u5377\u79ef\u7684\u7ec4\u6210\u90e8\u5206\u3002\n---\n### ESD\u3001PSD \u4e0e\u4e92\u8c31\u7684\u7edf\u4e00\u6bd4\u8f83\n\n\u524d\u51e0\u8282\u5206\u522b\u5b9a\u4e49\u4e86\u80fd\u91cf\u8c31\u5bc6\u5ea6 (ESD)\u3001\u529f\u7387\u8c31\u5bc6\u5ea6 (PSD)\u3001\u4e92\u80fd\u91cf\u8c31\u548c\u4e92\u529f\u7387\u8c31\u3002\u5b83\u4eec\u5728\u65f6\u57df\u2014\u9891\u57df\u5173\u7cfb\u4e0a\u6709\u4e00\u4e2a\u7edf\u4e00\u9aa8\u67b6\uff1a\n\n| \u65f6\u57df\u91cf | \u5bf9\u5e94\u9891\u57df\u91cf | \u9891\u7387\u79ef\u5206\u7ed9\u51fa |\n|---|---|---|\n| $R_{xx}(\\tau)=\\int x(t)x^*(t-\\tau)dt$ | $\\Psi_x(f)=|X(f)|^2$ | \u603b\u80fd\u91cf $E_x$ |\n| $R_{xx}(\\tau)=\\mathbb E[x(t)x^*(t-\\tau)]$ | $S_{xx}(f)$ | \u5e73\u5747\u529f\u7387 $P_x$ |\n| $R_{xy}(\\tau)=\\int x(t)y^*(t-\\tau)dt$ | $\\Psi_{xy}(f)=X(f)Y^*(f)$ | $R_{xy}(0)$ |\n| $R_{xy}(\\tau)=\\mathbb E[x(t)y^*(t-\\tau)]$ | $S_{xy}(f)$ | $R_{xy}(0)$ |\n\n\u56e0\u6b64\u8fd9\u4e9b\u8c31\u672c\u8d28\u4e0a\u90fd\u662f\u76f8\u5173\u51fd\u6570\u7684 Fourier \u53d8\u6362\u3002\u5dee\u522b\u53ea\u5728\u4e8e\u5bf9\u8c61\u662f\u80fd\u91cf\u4fe1\u53f7\u8fd8\u662f\u529f\u7387\u4fe1\u53f7\uff0f\u968f\u673a\u8fc7\u7a0b\u3001\u662f\u5426\u9700\u8981\u671f\u671b\uff0c\u4ee5\u53ca\u5f52\u4e00\u5316\u4e2d\u662f\u5426\u9664\u4ee5\u65f6\u957f $T$\u3002\u5bf9\u540c\u4e00\u6bb5\u6709\u9650\u8bb0\u5f55\uff0c\u82e5\u628a\u5f52\u4e00\u5316\u56e0\u5b50\u7531 $1$ \u6362\u6210 $1\/T$\uff0c\u5c31\u53ef\u4ee5\u5728\u201c\u8bb0\u5f55\u80fd\u91cf\u201d\u4e0e\u201c\u8bb0\u5f55\u5e73\u5747\u529f\u7387\u201d\u4e24\u79cd\u89c2\u70b9\u95f4\u8f6c\u6362\uff1a\n\n$$\n\\widehat S_{xx}(f)\\approx\\frac{1}{T}\\widehat\\Psi_x(f),\n\\qquad\n\\widehat S_{xy}(f)\\approx\\frac{1}{T}\\widehat\\Psi_{xy}(f).\n$$\n\n\u4e00\u65e6\u8bb0\u5f55\u4e2d\u542b\u6709\u7a97\u51fd\u6570\u6216\u6709\u6548\u5e26\u5bbd\u88ab\u4fee\u6539\uff0c\u8fd8\u9700\u8981\u989d\u5916\u7684\u7a97\u80fd\u91cf\u5f52\u4e00\u5316\uff0c\u8fd9\u5728\u7b2c\u516b\u7ae0\u8be6\u7ec6\u8ba8\u8bba\u3002\n\n---\n### \u91c7\u6837\u4e0e\u5f52\u4e00\u5316\uff1a\u4e00\u4e2a\u6570\u503c\u7b97\u4f8b\n\n\u8003\u8651\u79bb\u6563\u6b63\u5f26 $x[n]=A\\cos(2\\pi f_0 n\/f_s)$\uff0c\u53d6 $N$ \u70b9\u4f7f\u5f97 $f_0 N\/f_s$ \u6070\u4e3a\u6574\u6570\uff08\u65e0\u6cc4\u6f0f\uff09\uff0c\u505a $N$ \u70b9 DFT\uff1a\n\n$$\nX[k]=\\sum_{n=0}^{N-1}x[n]e^{-j2\\pi kn\/N}.\n$$\n\n\u5728 $k=\\pm k_0$\uff08$k_0=f_0N\/f_s$\uff09\u4e24\u4e2a bin \u4e0a\uff0c$|X[k_0]|=|X[-k_0]|=NA\/2$\uff0c\u5176\u4ed6 bin \u4e3a\u96f6\u3002\u4e09\u79cd\u5e38\u89c1\u5f52\u4e00\u5316\u7684\u610f\u4e49\u5982\u4e0b\uff1a\n\n| \u8868\u8fbe\u5f0f | \u6570\u503c | \u610f\u4e49 |\n|---|---|---|\n| $\\sum_k|X[k]|^2$ | $N^2A^2\/2$ | \u672a\u5f52\u4e00\u5316 DFT \u80fd\u91cf |\n| $\\dfrac1N\\sum_k|X[k]|^2$ | $NA^2\/2$ | \u6ee1\u8db3 Parseval\uff1a\u7b49\u4e8e $\\sum_n|x[n]|^2$ |\n| $\\dfrac{1}{N^2}\\sum_k|X[k]|^2$ | $A^2\/2$ | \u7b49\u4e8e\u8be5\u6b63\u5f26\u7684\u5e73\u5747\u529f\u7387 $P_x$ |\n\n\u82e5\u628a $|X[k]|^2\/N$ \u89c6\u4e3a periodogram \u9891\u70b9\u6570\u636e\uff0c\u5219\u4e24\u6761\u8c31\u7ebf\u5404\u81ea\u7684\u201c\u529f\u7387\u201d\u5747\u4e3a $A^2\/4$\uff1b\u52a0\u8d77\u6765\u624d\u7b49\u4e8e\u53cc\u8fb9\u603b\u529f\u7387 $A^2\/2$\u3002\u82e5\u4f7f\u7528\u5355\u8fb9\u8c31\uff0c\u9700\u8981\u628a\u975e DC\u3001\u975e Nyquist \u5904\u7684\u529f\u7387\u7ffb\u500d\uff0c\u5f97\u5230 $k_0$ \u5904\u529f\u7387\u4e3a $A^2\/2$\uff0c\u4e0e\u5355\u8fb9\u7ebf\u8c31\u4e00\u81f4\u3002\n\n\u4e0a\u8ff0\u7ed3\u679c\u4e0e\u662f\u5426\u4f7f\u7528\u89d2\u9891\u7387\u3001\u662f\u5426\u542b $2\\pi$\u3001\u4ee5\u53ca\u662f\u5426\u9664\u4ee5 $f_s$ \u90fd\u76f8\u5173\u3002\u7c97\u7565\u7684\u81ea\u68c0\u65b9\u6cd5\u662f\uff1a\u628a\u4f30\u8ba1\u8c31\u5728\u6574\u4e2a\u6b63\u9891\u6bb5\uff08\u5355\u8fb9\u7ea6\u5b9a\uff09\u79ef\u5206\u6216\u6c42\u548c\u4e58\u9891\u7387\u95f4\u9694\uff0c\u770b\u662f\u5426\u6062\u590d\u51fa\u7406\u8bba\u529f\u7387 $A^2\/2$\u3002\u4e0d\u80fd\u4ec5\u51ed\u66f2\u7ebf\u5f62\u72b6\u5224\u65ad\u7eb5\u8f74\u662f\u5e45\u5ea6\u8c31\u3001\u80fd\u91cf\u8c31\u8fd8\u662f\u529f\u7387\u8c31\u3002\n\n---\n### \u8c31\u77e9\u9635\u7684 Hermitian \u4e0e\u534a\u6b63\u5b9a\u6027\u8d28\n\n\u5bf9 $M$ \u4e2a\u8054\u5408\u5bbd\u5e73\u7a33\u8fc7\u7a0b $x_1(t),\\dots,x_M(t)$\uff0c\u5c06\u5176\u5728\u9891\u7387 $f$ \u5904\u7684\u6240\u6709\u81ea\u8c31\u548c\u4e92\u8c31\u6392\u4e3a $M\\times M$ \u8c31\u77e9\u9635\n\n$$\n\\boldsymbol S(f)=\n\\begin{bmatrix}\nS_{11}(f)&\\cdots&S_{1M}(f)\\\\\n\\vdots&\\ddots&\\vdots\\\\\nS_{M1}(f)&\\cdots&S_{MM}(f)\n\\end{bmatrix},\n\\qquad\nS_{ij}(f)=\\mathcal F\\{R_{ij}(\\tau)\\}.\n$$\n\n\u7531 $R_{ji}(\\tau)=R_{ij}^*(-\\tau)$ \u53ef\u5f97 $S_{ji}(f)=S_{ij}^*(f)$\uff0c\u5373\n\n$$\n\\boldsymbol S(f)=\\boldsymbol S(f)^{\\mathrm H}.\n$$\n\n\u5bf9\u4efb\u610f\u590d\u5411\u91cf $\\boldsymbol a\\in\\mathbb C^M$\uff0c\u4ee4 $z(t)=\\sum_i a_i^*x_i(t)$\uff0c\u5176\u529f\u7387\u8c31\n\n$$\nS_{zz}(f)=\\boldsymbol a^{\\mathrm H}\\boldsymbol S(f)\\boldsymbol a\\ge0.\n$$\n\n\u56e0\u6b64\u8c31\u77e9\u9635\u5728\u6bcf\u4e2a\u9891\u7387\u4e0a\u90fd\u662f Hermitian \u534a\u6b63\u5b9a\u7684\u3002\u5b83\u7684\u7279\u5f81\u5206\u89e3\n\n$$\n\\boldsymbol S(f)=\\sum_{k=1}^{M}\\lambda_k(f)\\boldsymbol u_k(f)\\boldsymbol u_k^{\\mathrm H}(f)\n$$\n\n\u5177\u6709\u76f4\u63a5\u7684\u7269\u7406\u610f\u4e49\uff1a$\\lambda_k(f)$ \u662f\u8be5\u9891\u7387\u4e0a\u7b2c $k$ \u4e2a\u6b63\u4ea4\u201c\u7a7a\u95f4\u6a21\u5f0f\u201d\u6240\u5360\u7684\u529f\u7387\uff0c$\\boldsymbol u_k(f)$ \u662f\u5bf9\u5e94\u7684\u590d\u5408\u6210\u65b9\u5411\u3002\u8fd9\u662f\u591a\u901a\u9053\u8c31\u4e3b\u6210\u5206\u5206\u6790\u3001\u76f2\u6e90\u5206\u79bb\u548c\u9635\u5217\u4fe1\u53f7\u5904\u7406\u7684\u57fa\u7840\u3002\n\n\u4e24\u4e24\u60c5\u5f62\u56de\u5230\n\n$$\n\\det\\boldsymbol S(f)\n=S_{xx}(f)S_{yy}(f)-|S_{xy}(f)|^2\\ge0,\n$$\n\n\u6b63\u662f\u7b2c\u4e03\u7ae0\u76f8\u5e72\u6027\u4e0a\u754c $\\gamma_{xy}^2(f)\\le1$ \u7684\u8c31\u77e9\u9635\u7248\u672c\u3002\n\n---\n### LTI \u7cfb\u7edf\u9891\u7387\u54cd\u5e94\u4f30\u8ba1\u4e0e\u65b9\u5411\u6027\n\n\u5728\u566a\u58f0\u73af\u5883\u4e0b\uff0c\u53ef\u4ee5\u7528\u4e92\u8c31\u548c\u81ea\u8c31\u6784\u9020\u591a\u79cd\u9891\u7387\u54cd\u5e94\u51fd\u6570\uff08FRF\uff09\u4f30\u8ba1\u3002\u5bf9\u6a21\u578b $y(t)=h(t)*x(t)+n(t)$\uff08\u566a\u58f0\u53ea\u52a0\u5728\u8f93\u51fa\uff0c$x\\perp n$\uff09\uff0c\u6700\u5e38\u7528\u7684\u4e09\u79cd\u4f30\u8ba1\u5668\u4e3a\n\n$$\n\\widehat H_1(f)=\\frac{S_{yx}(f)}{S_{xx}(f)},\n\\qquad\n\\widehat H_2(f)=\\frac{S_{yy}(f)}{S_{xy}(f)},\n\\qquad\n\\widehat H_v(f)=\\sqrt{\\widehat H_1(f)\\,\\widehat H_2(f)}.\n$$\n\n\u8fd9\u91cc\u91c7\u7528\u5168\u6587\u7ea6\u5b9a $S_{xy}=\\mathbb E[X Y^*]$\uff0c\u56e0\u6b64\u5bf9 $y=H*x+n$ \u4e14\u8f93\u5165\u4e0e\u566a\u58f0\u4e0d\u76f8\u5173\uff0c\u6709 $S_{yx}=H S_{xx}$\uff1b\u82e5\u8f6f\u4ef6\u628a\u4e92\u8c31\u901a\u9053\u987a\u5e8f\u5b9a\u4e49\u4e3a $S_{xy}=\\mathbb E[Y X^*]$\uff0c\u5219\u76f8\u5e94\u516c\u5f0f\u4e2d\u7684\u4e0b\u6807\u9700\u8981\u6574\u4f53\u4ea4\u6362\u3002\n\n\u5b83\u4eec\u7684\u504f\u5dee\u6027\u8d28\u4e0e\u566a\u58f0\u4f4d\u7f6e\u6709\u5173\uff1a\n\n- $\\widehat H_1$\uff1a\u5728\u8f93\u5165\u65e0\u566a\u3001\u8f93\u51fa\u6709\u566a\u65f6\u65e0\u504f\uff1b\u8f93\u5165\u7aef\u6709\u566a\u65f6\u4f1a\u4f4e\u4f30\u5e45\u503c\uff1b\n- $\\widehat H_2$\uff1a\u5728\u8f93\u51fa\u65e0\u566a\u3001\u8f93\u5165\u6709\u566a\u65f6\u65e0\u504f\uff1b\u8f93\u51fa\u7aef\u6709\u566a\u65f6\u4f1a\u9ad8\u4f30\u5e45\u503c\uff1b\n- $\\widehat H_v$\uff1a\u51e0\u4f55\u5e73\u5747\uff0c\u4ecb\u4e8e\u4e24\u8005\u4e4b\u95f4\uff0c\u5e38\u5728\u4e24\u7aef\u90fd\u6709\u566a\u58f0\u65f6\u7ed9\u51fa\u66f4\u7a33\u5065\u7ed3\u679c\u3002\n\n\u4e09\u8005\u4e0e\u76f8\u5e72\u6027\u6ee1\u8db3\u4e25\u683c\u5173\u7cfb\n\n$$\n\\boxed{\n\\frac{\\widehat H_1(f)}{\\widehat H_2(f)}\n=\\widehat\\gamma_{xy}^2(f)\n}.\n$$\n\n\u56e0\u6b64 $H_1$ \u4e0e $H_2$ \u7684\u6bd4\u503c\u5dee\u5f02\u672c\u8eab\u5c31\u662f\u76f8\u5e72\u6027\u7f3a\u9677\u7684\u5ea6\u91cf\u3002\u9009\u62e9 FRF \u4f30\u8ba1\u5668\u65f6\u5e94\u7ed3\u5408\u566a\u58f0\u4e3b\u5bfc\u7aef\u548c\u76f8\u5e72\u6027\u5f62\u72b6\uff0c\u4e0d\u80fd\u53ea\u6309\u201c\u770b\u8d77\u6765\u66f4\u5e73\u6ed1\u201d\u51b3\u5b9a\u3002\n\n\u65b9\u5411\u6027\u65b9\u9762\uff0c$\\widehat H_1$ \u4f9d\u8d56\u201c\u8f93\u5165\u2192\u8f93\u51fa\u201d\u65b9\u5411\u5b9a\u4e49\u3002\u53cd\u8fc7\u6765\u628a\u89d2\u8272\u4e92\u6362\uff0c\u4f1a\u5f97\u5230 $1\/\\widehat H_1$ \u6216 $\\widehat H_2$ \u7684\u590d\u5171\u8f6d\u4e4b\u7c7b\u7684\u91cf\uff0c\u5176\u504f\u5dee\u65b9\u5411\u4e5f\u968f\u4e4b\u7ffb\u8f6c\u3002\u5728\u591a\u8f93\u5165\u3001\u591a\u8f93\u51fa\u573a\u5408\uff0c\u9700\u8981\u7528 $\\boldsymbol H(f)=\\boldsymbol S_{yx}(f)\\boldsymbol S_{xx}^{-1}(f)$ \u7b49\u77e9\u9635\u5f62\u5f0f\u6765\u63a8\u5e7f\uff0c\u4e14\u8981\u6c42 $\\boldsymbol S_{xx}$ \u5728\u611f\u5174\u8da3\u9891\u7387\u4e0a\u826f\u6001\u53ef\u9006\u3002\n\n---\n\n---\n\n## \u516d\u3001\u76f8\u5173\u6027\u4e0e\u7edf\u8ba1\u4fe1\u53f7\u5206\u6790\n\n\u201c\u76f8\u5173\u201d\u5e38\u6307\u76f8\u5173\u51fd\u6570\u8fd9\u4e00\u5177\u4f53\u8fd0\u7b97\uff0c\u800c\u201c\u76f8\u5173\u6027\u201d\u901a\u5e38\u662f\u66f4\u5bbd\u6cdb\u7684\u6982\u5ff5\uff1a\u5b83\u63cf\u8ff0\u4e24\u4e2a\u91cf\u662f\u5426\u4e00\u8d77\u53d8\u5316\uff0c\u4ee5\u53ca\u8fd9\u79cd\u5171\u540c\u53d8\u5316\u6709\u591a\u5f3a\u3002\u4e0d\u540c\u5b66\u79d1\u4e2d\u201c\u76f8\u5173\u6027\u201d\u53ef\u80fd\u6307\u5185\u79ef\u3001\u4e92\u76f8\u5173\u51fd\u6570\u3001\u534f\u65b9\u5dee\u3001Pearson \u76f8\u5173\u7cfb\u6570\u6216\u5176\u4ed6\u4f9d\u8d56\u6027\u6307\u6807\uff0c\u56e0\u6b64\u5fc5\u987b\u5148\u660e\u786e\u5bf9\u8c61\u548c\u5b9a\u4e49\u3002\n\n### \u968f\u673a\u53d8\u91cf\u7684\u671f\u671b\u3001\u65b9\u5dee\u4e0e\u534f\u65b9\u5dee\n\n\u8bbe\u968f\u673a\u53d8\u91cf $X$ \u548c $Y$ \u7684\u5747\u503c\u5206\u522b\u4e3a\n\n$$\n\\mu_X=\\mathbb E[X],\n\\qquad\n\\mu_Y=\\mathbb E[Y].\n$$\n\n\u65b9\u5dee\u4e3a\n\n$$\n\\sigma_X^2=\\mathbb E\\left[|X-\\mu_X|^2\\right],\n\\qquad\n\\sigma_Y^2=\\mathbb E\\left[|Y-\\mu_Y|^2\\right].\n$$\n\n\u590d\u968f\u673a\u53d8\u91cf\u5e38\u91c7\u7528\u4ee5\u4e0b\u4e92\u534f\u65b9\u5dee\u5b9a\u4e49\uff1a\n\n$$\n\\boxed{\nC_{XY}=\\mathbb E\\left[(X-\\mu_X)(Y-\\mu_Y)^*\\right]\n}.\n$$\n\n\u5b83\u53bb\u9664\u4e86\u5747\u503c\uff0c\u6d4b\u91cf\u4e24\u4e2a\u53d8\u91cf\u56f4\u7ed5\u5404\u81ea\u5747\u503c\u7684\u5171\u540c\u7ebf\u6027\u53d8\u5316\u3002\u82e5\u4e0d\u53bb\u5747\u503c\uff0c\u5219\u5f97\u5230\u4e92\u76f8\u5173\u77e9\uff1a\n\n$$\nR_{XY}=\\mathbb E[XY^*].\n$$\n\n\u4e8c\u8005\u6ee1\u8db3\n\n$$\nC_{XY}=R_{XY}-\\mu_X\\mu_Y^*.\n$$\n\n\u56e0\u6b64\uff0c\u201c\u76f8\u5173\u4e3a\u96f6\u201d\u548c\u201c\u534f\u65b9\u5dee\u4e3a\u96f6\u201d\u53ea\u5728\u81f3\u5c11\u4e00\u4e2a\u53d8\u91cf\u4e3a\u96f6\u5747\u503c\u65f6\u7b49\u4ef7\u3002\n\n### Pearson \u76f8\u5173\u7cfb\u6570\n\n\u5bf9\u5b9e\u968f\u673a\u53d8\u91cf\uff0cPearson \u76f8\u5173\u7cfb\u6570\u5b9a\u4e49\u4e3a\n\n$$\n\\boxed{\n\\rho_{XY}\n=\\frac{\\operatorname{Cov}(X,Y)}{\\sigma_X\\sigma_Y}\n}.\n$$\n\n\u53ea\u8981 $\\sigma_X>0$\u3001$\\sigma_Y>0$\uff0c\u5c31\u6709<\/p>\n

$$
\n-1\\le\\rho_{XY}\\le1.
\n$$<\/p>\n

\u5176\u5178\u578b\u89e3\u91ca\u4e3a\uff1a<\/p>\n

– $\\rho_{XY}=1$\uff1a\u5b8c\u5168\u6b63\u7ebf\u6027\u5173\u7cfb\uff1b
\n– $\\rho_{XY}=-1$\uff1a\u5b8c\u5168\u8d1f\u7ebf\u6027\u5173\u7cfb\uff1b
\n– $\\rho_{XY}=0$\uff1a\u65e0 Pearson \u7ebf\u6027\u76f8\u5173\uff0c\u4f46\u4ecd\u53ef\u80fd\u5b58\u5728\u975e\u7ebf\u6027\u4f9d\u8d56\uff1b
\n– $|\\rho_{XY}|$ \u8d8a\u63a5\u8fd1 1\uff0c\u7ebf\u6027\u5173\u8054\u8d8a\u5f3a\u3002<\/p>\n

\u5bf9\u590d\u968f\u673a\u53d8\u91cf\uff0c\u53ef\u4ee5\u5b9a\u4e49\u590d\u76f8\u5173\u7cfb\u6570<\/p>\n

$$
\n\\rho_{XY}
\n=\\frac{C_{XY}}{\\sqrt{C_{XX}C_{YY}}},
\n\\qquad
\n|\\rho_{XY}|\\le1.
\n$$<\/p>\n

\u6b64\u65f6 $|\\rho_{XY}|$ \u8868\u793a\u7ebf\u6027\u5173\u8054\u5f3a\u5ea6\uff0c$\\angle\\rho_{XY}$ \u63cf\u8ff0\u5e73\u5747\u76f8\u5bf9\u76f8\u4f4d\u3002<\/p>\n

### \u6837\u672c\u76f8\u5173\u7cfb\u6570<\/p>\n

\u7ed9\u5b9a $N$ \u5bf9\u5b9e\u503c\u89c2\u6d4b $(x_i,y_i)$\uff0c\u6837\u672c\u5747\u503c\u4e3a<\/p>\n

$$
\n\\bar x=\\frac1N\\sum_{i=1}^{N}x_i,
\n\\qquad
\n\\bar y=\\frac1N\\sum_{i=1}^{N}y_i.
\n$$<\/p>\n

\u6837\u672c Pearson \u76f8\u5173\u7cfb\u6570\u4e3a<\/p>\n

$$
\n\\boxed{
\nr_{xy}
\n=\\frac{\\sum_{i=1}^{N}(x_i-\\bar x)(y_i-\\bar y)}
\n{\\sqrt{\\sum_{i=1}^{N}(x_i-\\bar x)^2}
\n \\sqrt{\\sum_{i=1}^{N}(y_i-\\bar y)^2}}
\n}.
\n$$<\/p>\n

\u5b83\u662f\u603b\u4f53\u76f8\u5173\u7cfb\u6570\u7684\u4f30\u8ba1\u3002\u6709\u9650\u6837\u672c\u4e2d\u7684 $r_{xy}$ \u4f1a\u968f\u673a\u6ce2\u52a8\uff0c\u56e0\u6b64\u4e0d\u80fd\u53ea\u51ed\u4e00\u4e2a\u6570\u503c\u5224\u65ad\u603b\u4f53\u5173\u7cfb\uff1b\u901a\u5e38\u8fd8\u8981\u8003\u8651\u6837\u672c\u91cf\u3001\u7f6e\u4fe1\u533a\u95f4\u3001\u5f02\u5e38\u503c\u548c\u6570\u636e\u5206\u5e03\u3002<\/p>\n

### \u76f8\u5173\u7cfb\u6570\u7684\u6027\u8d28<\/p>\n

\u82e5 $a,c>0$\uff0c\u5219<\/p>\n

$$
\n\\operatorname{corr}(aX+b,cY+d)=\\operatorname{corr}(X,Y).
\n$$<\/p>\n

\u5e73\u79fb\u548c\u6b63\u6bd4\u4f8b\u7f29\u653e\u4e0d\u6539\u53d8 Pearson \u76f8\u5173\u7cfb\u6570\uff1b\u8d1f\u6bd4\u4f8b\u7f29\u653e\u4f1a\u6539\u53d8\u7b26\u53f7\u4f46\u4e0d\u6539\u53d8\u7edd\u5bf9\u503c\u3002<\/p>\n

\u76f8\u5173\u7cfb\u6570\u6ca1\u6709\u7269\u7406\u5355\u4f4d\uff0c\u8fd9\u662f\u5176\u4fbf\u4e8e\u6bd4\u8f83\u7684\u539f\u56e0\uff1b\u534f\u65b9\u5dee\u5219\u4fdd\u7559\u4e24\u4e2a\u53d8\u91cf\u5355\u4f4d\u7684\u4e58\u79ef\u3002<\/p>\n

Pearson \u76f8\u5173\u7cfb\u6570\u53ea\u6d4b\u91cf\u7ebf\u6027\u5173\u7cfb\u3002\u4f8b\u5982\u4ee4 $X$ \u5173\u4e8e\u96f6\u5bf9\u79f0\uff0c$Y=X^2$\uff0c\u5219 $Y$ \u5b8c\u5168\u7531 $X$ \u51b3\u5b9a\uff0c\u4f46\u53ef\u80fd\u6709<\/p>\n

$$
\n\\operatorname{Cov}(X,Y)=\\mathbb E[X^3]=0,
\n$$<\/p>\n

\u4ece\u800c Pearson \u76f8\u5173\u7cfb\u6570\u4e3a\u96f6\u3002<\/p>\n

### \u72ec\u7acb\u3001\u4e0d\u76f8\u5173\u4e0e\u6b63\u4ea4<\/p>\n

\u4e09\u4e2a\u6982\u5ff5\u9700\u8981\u4e25\u683c\u533a\u5206\uff1a<\/p>\n

1. **\u7edf\u8ba1\u72ec\u7acb**\uff1a\u8054\u5408\u5206\u5e03\u53ef\u4ee5\u5206\u89e3\u4e3a\u8fb9\u7f18\u5206\u5e03\u7684\u4e58\u79ef\uff1b
\n2. **\u4e0d\u76f8\u5173**\uff1a$C_{XY}=0$\uff1b
\n3. **\u6b63\u4ea4**\uff1a\u5728\u7ed9\u5b9a\u5185\u79ef\u4e0b $\\langle x,y\\rangle=0$\u3002<\/p>\n

\u5bf9\u4e8e\u4e8c\u9636\u77e9\u5b58\u5728\u7684\u968f\u673a\u53d8\u91cf\uff1a<\/p>\n

$$
\n\\text{\u72ec\u7acb}\\Longrightarrow\\text{\u4e0d\u76f8\u5173},
\n$$<\/p>\n

\u4f46\u4e00\u822c\u6ca1\u6709\u53cd\u5411\u63a8\u8bba\u3002\u82e5 $X$\u3001$Y$ \u8054\u5408 Gaussian\uff0c\u5219\u4e0d\u76f8\u5173\u53ef\u63a8\u51fa\u72ec\u7acb\u3002<\/p>\n

\u5728\u968f\u673a\u4fe1\u53f7\u7406\u8bba\u4e2d\uff0c\u5982\u679c\u91c7\u7528<\/p>\n

$$
\n\\langle X,Y\\rangle=\\mathbb E[XY^*],
\n$$<\/p>\n

\u90a3\u4e48\u96f6\u5747\u503c\u968f\u673a\u53d8\u91cf\u7684\u201c\u4e0d\u76f8\u5173\u201d\u4e0e\u8fd9\u79cd\u5747\u65b9\u610f\u4e49\u4e0b\u7684\u201c\u6b63\u4ea4\u201d\u4e00\u81f4\u3002<\/p>\n

### \u968f\u673a\u8fc7\u7a0b\u7684\u76f8\u5173\u6027<\/p>\n

\u5bf9\u968f\u673a\u8fc7\u7a0b $x(t)$ \u548c $y(t)$\uff0c\u4e92\u76f8\u5173\u5b9a\u4e49\u4e3a<\/p>\n

$$
\nR_{xy}(t_1,t_2)
\n=\\mathbb E\\left[x(t_1)y^*(t_2)\\right].
\n$$<\/p>\n

\u82e5\u8fc7\u7a0b\u662f\u8054\u5408\u5bbd\u5e73\u7a33\u7684\uff0c\u4e92\u76f8\u5173\u53ea\u4f9d\u8d56\u65f6\u95f4\u5dee\uff1a<\/p>\n

$$
\nR_{xy}(\\tau)
\n=\\mathbb E\\left[x(t)y^*(t-\\tau)\\right].
\n$$<\/p>\n

\u5bf9\u5e94\u7684\u4e92\u534f\u65b9\u5dee\u4e3a<\/p>\n

$$
\nC_{xy}(\\tau)
\n=R_{xy}(\\tau)-\\mu_x\\mu_y^*.
\n$$<\/p>\n

\u5bbd\u5e73\u7a33\u8fc7\u7a0b\u7684\u5747\u503c\u4e0d\u968f\u65f6\u95f4\u53d8\u5316\uff0c\u81ea\u76f8\u5173\u53ea\u4f9d\u8d56\u65f6\u5dee\uff0c\u5e76\u6ee1\u8db3<\/p>\n

$$
\nR_{xx}(-\\tau)=R_{xx}^*(\\tau).
\n$$<\/p>\n

### \u65f6\u95f4\u5e73\u5747\u4e0e\u96c6\u5408\u5e73\u5747<\/p>\n

\u7406\u8bba\u5b9a\u4e49\u4e2d\u7684 $\\mathbb E[\\cdot]$ \u662f\u96c6\u5408\u5e73\u5747\uff1a\u9700\u8981\u5bf9\u540c\u4e00\u968f\u673a\u5b9e\u9a8c\u7684\u8bb8\u591a\u5b9e\u73b0\u6c42\u5e73\u5747\u3002\u4f46\u5de5\u7a0b\u4e2d\u5f80\u5f80\u53ea\u6709\u4e00\u6761\u6709\u9650\u8bb0\u5f55\uff0c\u4e8e\u662f\u7528\u65f6\u95f4\u5e73\u5747\u4f30\u8ba1\uff1a<\/p>\n

$$
\n\\widehat R_{xy}(\\tau)
\n=\\frac1T\\int_0^T x(t)y^*(t-\\tau)\\,dt.
\n$$<\/p>\n

\u53ea\u6709\u5728\u9002\u5f53\u7684\u904d\u5386\u6027\u6761\u4ef6\u4e0b\uff0c\u957f\u65f6\u95f4\u5e73\u5747\u624d\u6536\u655b\u5230\u96c6\u5408\u5e73\u5747\u3002\u5e73\u7a33\u6027\u5e76\u4e0d\u81ea\u52a8\u4fdd\u8bc1\u904d\u5386\u6027\uff0c\u56e0\u6b64\u7528\u5355\u6761\u8bb0\u5f55\u4f30\u8ba1\u7edf\u8ba1\u91cf\u65f6\uff0c\u5e94\u660e\u786e\u8fd9\u4e00\u5047\u8bbe\u3002<\/p>\n

### Pearson\u3001Spearman \u4e0e\u975e\u7ebf\u6027\u4f9d\u8d56<\/p>\n

\u5e38\u89c1\u76f8\u5173\u6027\u6307\u6807\u5305\u62ec\uff1a<\/p>\n

| \u6307\u6807 | \u6d4b\u91cf\u5bf9\u8c61 | \u4e3b\u8981\u7279\u70b9 |
\n|—|—|—|
\n| Pearson \u76f8\u5173 | \u7ebf\u6027\u5173\u7cfb | \u5bf9\u5f02\u5e38\u503c\u654f\u611f\uff1b\u8981\u6c42\u4e8c\u9636\u77e9\u5b58\u5728 |
\n| Spearman \u79e9\u76f8\u5173 | \u5355\u8c03\u5173\u7cfb | \u57fa\u4e8e\u79e9\uff0c\u5bf9\u975e\u7ebf\u6027\u5355\u8c03\u5173\u7cfb\u548c\u5f02\u5e38\u503c\u66f4\u7a33\u5065 |
\n| Kendall $\\tau$ | \u6b21\u5e8f\u4e00\u81f4\u6027 | \u89e3\u91ca\u4e3a\u6837\u672c\u5bf9\u4e00\u81f4\u4e0e\u4e0d\u4e00\u81f4\u7684\u5dee\u5f02 |
\n| \u4e92\u4fe1\u606f | \u4e00\u822c\u7edf\u8ba1\u4f9d\u8d56 | \u80fd\u53d1\u73b0\u975e\u7ebf\u6027\u4f9d\u8d56\uff0c\u4f46\u4f30\u8ba1\u66f4\u56f0\u96be |
\n| \u8ddd\u79bb\u76f8\u5173 | \u4e00\u822c\u4f9d\u8d56 | \u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\u4e3a\u96f6\u5f53\u4e14\u4ec5\u5f53\u72ec\u7acb |<\/p>\n

\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u4e2d\u6700\u5e38\u7528\u7684\u662f\u4e92\u76f8\u5173\u51fd\u6570\u548c\u8c31\u76f8\u5e72\u6027\uff0c\u4f46\u5f53\u5173\u7cfb\u660e\u663e\u975e\u7ebf\u6027\u65f6\uff0cPearson \u76f8\u5173\u6216\u666e\u901a\u76f8\u5e72\u6027\u53ef\u80fd\u4e0d\u8db3\u3002<\/p>\n

### \u76f8\u5173\u4e0d\u7b49\u4e8e\u56e0\u679c<\/p>\n

\u5373\u4f7f\u4e24\u4e2a\u4fe1\u53f7\u7684\u76f8\u5173\u6027\u5f88\u9ad8\uff0c\u4e5f\u4e0d\u80fd\u4ec5\u51ed\u76f8\u5173\u63a8\u51fa\u56e0\u679c\u3002\u5e38\u89c1\u539f\u56e0\u5305\u62ec\uff1a<\/p>\n

– \u4e24\u8005\u7531\u540c\u4e00\u4e2a\u9690\u85cf\u8f93\u5165\u9a71\u52a8\uff1b
\n– \u4e00\u4e2a\u5171\u540c\u8d8b\u52bf\u9020\u6210\u4f2a\u76f8\u5173\uff1b
\n– \u4e24\u4e2a\u6d4b\u91cf\u901a\u9053\u5b58\u5728\u4e32\u6270\uff1b
\n– \u53cd\u9988\u4f7f\u5f71\u54cd\u65b9\u5411\u53cc\u5411\u5b58\u5728\uff1b
\n– \u65f6\u95f4\u5e8f\u5217\u81ea\u76f8\u5173\u964d\u4f4e\u4e86\u6709\u6548\u72ec\u7acb\u6837\u672c\u6570\u3002<\/p>\n

\u56e0\u679c\u5224\u65ad\u901a\u5e38\u8fd8\u9700\u8981\u5b9e\u9a8c\u8bbe\u8ba1\u3001\u7269\u7406\u6a21\u578b\u3001\u65f6\u95f4\u65b9\u5411\u3001\u63a7\u5236\u6df7\u6742\u53d8\u91cf\uff0c\u4ee5\u53ca\u4e13\u95e8\u7684\u56e0\u679c\u63a8\u65ad\u65b9\u6cd5\u3002<\/p>\n

—
\n### \u5e73\u7a33\u6027\u4e0e\u904d\u5386\u6027\u7684\u5c42\u6b21<\/p>\n

\u5728\u7528\u6709\u9650\u6837\u672c\u4f30\u8ba1\u603b\u4f53\u7edf\u8ba1\u91cf\u4e4b\u524d\uff0c\u5fc5\u987b\u660e\u786e\u968f\u673a\u8fc7\u7a0b\u7684\u5047\u8bbe\u5c42\u6b21\u3002\u5e38\u7528\u6982\u5ff5\u6309\u7531\u5f31\u5230\u5f3a\u6392\u5217\u4e3a\uff1a<\/p>\n

1. **\u4e00\u9636\u5e73\u7a33**\uff1a$\\mathbb E[x(t)]$ \u4e0e $t$ \u65e0\u5173\uff1b
\n2. **\u5bbd\u5e73\u7a33 (WSS)**\uff1a\u4e00\u9636\u5e73\u7a33\uff0c\u4e14 $R_{xx}(t,t-\\tau)$ \u53ea\u4f9d\u8d56\u4e8e $\\tau$\uff1b
\n3. **\u8054\u5408\u5bbd\u5e73\u7a33**\uff1a\u591a\u4e2a\u8fc7\u7a0b\u90fd WSS\uff0c\u4e14\u4e92\u76f8\u5173 $R_{xy}(t,t-\\tau)$ \u53ea\u4f9d\u8d56 $\\tau$\uff1b
\n4. **\u4e25\u683c\u5e73\u7a33**\uff1a\u6240\u6709\u9636\u8054\u5408\u5206\u5e03\u5bf9\u65f6\u95f4\u5e73\u79fb\u4e0d\u53d8\uff1b
\n5. **\u5747\u503c\/\u76f8\u5173\u904d\u5386**\uff1a\u65f6\u95f4\u5e73\u5747\u4ee5\u6982\u7387\u6216\u5747\u65b9\u6536\u655b\u5230\u96c6\u5408\u5e73\u5747\uff1b
\n6. **\u4e25\u683c\u904d\u5386**\uff1a\u6240\u6709\u53ef\u79ef\u51fd\u6570\u7684\u65f6\u95f4\u5e73\u5747\u7b49\u4e8e\u96c6\u5408\u5e73\u5747\u3002<\/p>\n

WSS \u53ea\u662f\u4e8c\u9636\u77e9\u5c42\u9762\u7684\u6761\u4ef6\uff0c\u5e76\u4e0d\u81ea\u52a8\u4fdd\u8bc1\u904d\u5386\u6027\uff1a\u4e00\u4e2a\u5bbd\u5e73\u7a33\u8fc7\u7a0b\u53ef\u80fd\u5728\u4e0d\u540c\u5b9e\u73b0\u4e4b\u95f4\u53d6\u503c\u5206\u5e03\u4e0d\u540c\uff08\u4f8b\u5982\u201c\u62bd\u7b7e\u56fa\u5b9a\u201d\u7684\u968f\u673a\u5e38\u6570\u8fc7\u7a0b $x(t)=A$\uff09\u3002\u5de5\u7a0b\u4e0a\u4f7f\u7528<\/p>\n

$$
\n\\widehat R_{xx}(\\tau)
\n=\\frac1T\\int_0^T x(t)x^*(t-\\tau)dt
\n$$<\/p>\n

\u53bb\u4f30\u8ba1 $R_{xx}(\\tau)$ \u65f6\uff0c\u5b9e\u9645\u4e0a\u6697\u542b **\u5747\u503c\u904d\u5386\u3001\u534f\u65b9\u5dee\u904d\u5386** \u4e4b\u7c7b\u7684\u5047\u8bbe\uff1a\u9700\u8981\u4e00\u6bb5\u8db3\u591f\u957f\u7684\u8bb0\u5f55\u5185\uff0c\u6837\u672c\u5145\u5206\u201c\u63a2\u7d22\u201d\u4e86\u968f\u673a\u8fc7\u7a0b\u7684\u53ef\u80fd\u72b6\u6001\u3002\u82e5\u88ab\u6d4b\u7cfb\u7edf\u5305\u542b\u6162\u65f6\u53d8\u3001\u6a21\u5f0f\u5207\u6362\u6216\u975e\u5404\u6001\u5386\u7ecf\u7684\u7a33\u6001\uff0c\u957f\u8bb0\u5f55\u65f6\u95f4\u5e73\u5747\u5e76\u4e0d\u903c\u8fd1\u96c6\u5408\u5e73\u5747\uff0c\u8c31\u548c\u76f8\u5e72\u6027\u4f30\u8ba1\u5c31\u4f1a\u51fa\u73b0\u7cfb\u7edf\u6027\u504f\u5dee\u3002<\/p>\n

\u4e00\u4e2a\u7b80\u5355\u5224\u65ad\u505a\u6cd5\u662f\u628a\u957f\u8bb0\u5f55\u5207\u6210\u82e5\u5e72\u4e92\u4e0d\u91cd\u53e0\u7684\u5b50\u6bb5\uff0c\u5206\u522b\u4f30\u8ba1\u5747\u503c\u3001\u65b9\u5dee\u6216\u8c31\uff0c\u6bd4\u8f83\u5b83\u4eec\u7684\u5dee\u5f02\u662f\u5426\u5728\u671f\u671b\u7684\u7edf\u8ba1\u6da8\u843d\u8303\u56f4\u5185\u3002\u5dee\u5f02\u8fdc\u8d85\u7edf\u8ba1\u6da8\u843d\u65f6\uff0c\u904d\u5386\u6027\u5047\u8bbe\u5e94\u88ab\u8ba4\u4e3a\u4e0d\u6210\u7acb\uff0c\u5e94\u6539\u6309\u5de5\u51b5\u5206\u6bb5\u5904\u7406\uff0c\u800c\u4e0d\u662f\u7ee7\u7eed\u589e\u52a0\u5e73\u5747\u6b21\u6570\u3002<\/p>\n

—
\n### \u6837\u672c\u65b9\u5dee\u4e0e\u534f\u65b9\u5dee\u7684\u5206\u6bcd\uff1a$N$ \u8fd8\u662f $N-1$<\/p>\n

\u6837\u672c Pearson \u76f8\u5173\u7cfb\u6570<\/p>\n

$$
\nr_{xy}
\n=\\frac{\\sum_i(x_i-\\bar x)(y_i-\\bar y)}
\n{\\sqrt{\\sum_i(x_i-\\bar x)^2}\\sqrt{\\sum_i(y_i-\\bar y)^2}}
\n$$<\/p>\n

\u5bf9\u5206\u5b50\u5206\u6bcd\u5171\u7528\u540c\u4e00\u4e2a $N$ \u6216 $N-1$ \u56e0\u5b50\u65f6\u7ed3\u679c\u4e0d\u53d8\uff0c\u56e0\u6b64\u5e38\u89c1\u6559\u79d1\u4e66\u5199\u6cd5\u770b\u4f3c\u6709\u6b67\u4e49\uff0c\u4f46\u6bd4\u503c\u672c\u8eab\u4e0e\u5206\u6bcd\u9009\u62e9\u65e0\u5173\u3002\u771f\u6b63\u9700\u8981\u5c0f\u5fc3\u7684\u662f**\u65b9\u5dee\u548c\u534f\u65b9\u5dee\u672c\u8eab**\u7684\u4f30\u8ba1\uff1a<\/p>\n

$$
\n\\widehat\\sigma_X^2=\\frac1N\\sum_{i=1}^N(x_i-\\bar x)^2
\n\\quad\\text{\uff08\u6700\u5927\u4f3c\u7136\u3001\u504f\u4f30\u8ba1\uff09},
\n$$<\/p>\n

$$
\ns_X^2=\\frac{1}{N-1}\\sum_{i=1}^N(x_i-\\bar x)^2
\n\\quad\\text{\uff08\u65e0\u504f\u4f30\u8ba1\uff0cBessel \u4fee\u6b63\uff09}.
\n$$<\/p>\n

\u5bf9\u534f\u65b9\u5dee\u540c\u7406\uff0c$N-1$ \u5206\u6bcd\u62b5\u6d88\u201c\u5747\u503c\u4e5f\u662f\u4f30\u8ba1\u51fa\u7684\u201d\u8fd9\u4e00\u81ea\u7531\u5ea6\u635f\u5931\u3002\u82e5\u4f7f\u7528 $N$ \u5206\u6bcd\uff0c\u4f1a\u7cfb\u7edf\u6027\u4f4e\u4f30\u65b9\u5dee\u548c\u534f\u65b9\u5dee\uff1b\u5982\u679c\u4e4b\u540e\u518d\u57fa\u4e8e\u8fd9\u4e9b\u91cf\u505a\u5361\u65b9\u68c0\u9a8c\u3001F \u68c0\u9a8c\u6216\u7f6e\u4fe1\u533a\u95f4\uff0c\u5c31\u4f1a\u5f97\u5230\u8fc7\u5ea6\u81ea\u4fe1\u7684\u7ed3\u8bba\u3002<\/p>\n

\u5728\u4fe1\u53f7\u5904\u7406\u8f6f\u4ef6\u4e2d\uff1a<\/p>\n

– Python `numpy.var\/cov` \u9ed8\u8ba4 `ddof=0`\uff08\u5206\u6bcd $N$\uff09\uff0c\u9700\u8981\u663e\u5f0f `ddof=1` \u624d\u5bf9\u5e94 $N-1$\uff1b
\n– Python `numpy.corrcoef` \u5185\u90e8\u4f1a\u5f52\u4e00\u5316\uff0c\u56e0\u6b64\u4e0d\u53d7\u8be5\u5dee\u5f02\u5f71\u54cd\uff1b
\n– MATLAB `var\/cov` \u9ed8\u8ba4\u4f7f\u7528 $N-1$\uff1b`var(x,1)` \u624d\u662f $N$\u3002<\/p>\n

\u5728\u62a5\u544a\u76f8\u5173\u6027\u3001\u534f\u65b9\u5dee\u77e9\u9635\u3001\u767d\u5316\u56e0\u5b50\u6216 Mahalanobis \u8ddd\u79bb\u65f6\uff0c\u5fc5\u987b\u5199\u6e05\u6240\u7528\u5206\u6bcd\uff0c\u4ee5\u514d\u4e0b\u6e38\u628a\u504f\u4f30\u8ba1\u5f53\u4f5c\u65e0\u504f\u4f30\u8ba1\u4f7f\u7528\u3002<\/p>\n

—
\n### $Y=X^2$\uff1a\u975e\u7ebf\u6027\u4f9d\u8d56\u4f7f\u76f8\u5173\u7cfb\u6570\u6d88\u5931<\/p>\n

\u7b2c\u516d\u7ae0\u6b63\u6587\u63d0\u5230\uff0c$Y=X^2$ \u65f6 Pearson \u76f8\u5173\u7cfb\u6570\u53ef\u80fd\u4e3a\u96f6\u3002\u8fd9\u91cc\u7ed9\u51fa\u4e00\u4e2a\u53ef\u76f4\u63a5\u9a8c\u8bc1\u7684\u7b97\u4f8b\uff1a\u8bbe $X\\sim\\mathcal U[-1,1]$\uff0c\u5219<\/p>\n

$$
\n\\mu_X=\\mathbb E[X]=0,
\n\\qquad
\n\\mu_Y=\\mathbb E[X^2]=\\frac13,
\n$$<\/p>\n

$$
\n\\operatorname{Cov}(X,Y)
\n=\\mathbb E[XY]-\\mu_X\\mu_Y
\n=\\mathbb E[X^3]=0,
\n$$<\/p>\n

\u56e0\u6b64 $\\rho_{XY}=0$\u3002\u4f46 $Y$ \u5b8c\u5168\u7531 $X$ \u51b3\u5b9a\uff0c\u4e24\u8005\u663e\u7136\u5b58\u5728\u51fd\u6570\u4f9d\u8d56\u3002<\/p>\n

\u540c\u6837\u7684\u6784\u9020\u5728\u4fe1\u53f7\u4e2d\u7ecf\u5e38\u51fa\u73b0\u3002\u4f8b\u5982\u8f93\u5165\u6b63\u5f26 $x(t)=\\cos(2\\pi f_0 t)$ \u7ecf\u8fc7\u5e73\u65b9\u5668 $y(t)=x^2(t)$ \u5f97\u5230<\/p>\n

$$
\n y(t)=\\tfrac12+\\tfrac12\\cos(2\\pi\\cdot 2f_0\\cdot t).
\n$$<\/p>\n

\u5728\u540c\u9891 $f_0$ \u4e0a $y$ \u65e0\u529f\u7387\uff0c\u56e0\u6b64 MSC $\\gamma_{xy}^2(f_0)$ \u63a5\u8fd1 0\uff1b\u80fd\u91cf\u8f6c\u79fb\u5230\u76f4\u6d41\u548c\u4e8c\u500d\u9891\u3002\u5728 $2f_0$ \u4e0a $y$ \u6709\u5f3a\u529f\u7387\uff0c\u4f46 $x$ \u6ca1\u6709\uff0c\u4e8c\u9636\u8c31\u76f8\u5e72\u6027\u540c\u6837\u8d8b\u4e8e 0\u3002\u8fd9\u7c7b\u8de8\u9891\u5173\u7cfb\u9700\u8981\u7528\u53cc\u76f8\u5e72\u6027 (bicoherence)\u3001\u975e\u7ebf\u6027\u7cfb\u7edf\u8fa8\u8bc6\u6216\u4e92\u4fe1\u606f\u624d\u80fd\u63ed\u793a\u3002<\/p>\n

\u7ed3\u8bba\uff1aPearson \u76f8\u5173\u7cfb\u6570\u548c\u4e8c\u9636\u8c31\u76f8\u5e72\u6027\u53ea\u5bf9**\u540c\u9891\u7ebf\u6027\u5173\u7cfb**\u654f\u611f\u3002\u5f53\u6000\u7591\u5b58\u5728\u6574\u6d41\u3001\u5e73\u65b9\u3001\u7edd\u5bf9\u503c\u3001\u5305\u7edc\u3001\u5e45\u76f8\u8026\u5408\u7b49\u975e\u7ebf\u6027\u65f6\uff0c\u5e94\u6539\u7528\u9002\u5f53\u7684\u975e\u7ebf\u6027\u6307\u6807\u6216\u5f15\u5165\u89e3\u6790\u4fe1\u53f7\uff0f\u5305\u7edc\u540e\u518d\u505a\u4e8c\u9636\u5206\u6790\u3002<\/p>\n

—
\n### \u5171\u540c\u9a71\u52a8\u6e90\u4e0e\u4f2a\u76f8\u5173<\/p>\n

\u5373\u4f7f $x$ \u4e0e $y$ \u4e4b\u95f4\u6ca1\u6709\u76f4\u63a5\u8054\u7cfb\uff0c\u82e5\u4e24\u8005\u88ab\u540c\u4e00\u4e2a\u4fe1\u53f7 $z$ \u9a71\u52a8\uff0c\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u5173\u7cfb\u6570\u6216\u76f8\u5e72\u6027\u90fd\u53ef\u80fd\u663e\u8457\u975e\u96f6\u3002\u4e3e\u4e00\u4e2a\u663e\u5f0f\u4f8b\u5b50\uff1a\u8bbe $z$ \u4e3a\u96f6\u5747\u503c\u5355\u4f4d\u65b9\u5dee\u7684\u5bbd\u5e73\u7a33\u8fc7\u7a0b\uff0c$n_1,n_2$ \u662f\u4e0e $z$ \u53ca\u5f7c\u6b64\u90fd\u4e0d\u76f8\u5173\u7684\u767d\u566a\u58f0\uff0c\u4ee4<\/p>\n

$$
\n x(t)=z(t)+n_1(t),
\n\\qquad
\n y(t)=z(t)+n_2(t).
\n$$<\/p>\n

\u5219<\/p>\n

$$
\nR_{xy}(\\tau)=R_{zz}(\\tau),
\n\\qquad
\nS_{xy}(f)=S_{zz}(f).
\n$$<\/p>\n

\u800c<\/p>\n

$$
\nS_{xx}(f)=S_{zz}(f)+S_{n_1n_1}(f),
\n\\qquad
\nS_{yy}(f)=S_{zz}(f)+S_{n_2n_2}(f).
\n$$<\/p>\n

\u56e0\u6b64\u666e\u901a\u76f8\u5e72\u6027<\/p>\n

$$
\n\\gamma_{xy}^2(f)
\n=\\frac{S_{zz}^2(f)}
\n{(S_{zz}(f)+S_{n_1n_1}(f))(S_{zz}(f)+S_{n_2n_2}(f))}
\n$$<\/p>\n

\u4f1a\u5728 $z$ \u4e3b\u5bfc\u7684\u9891\u5e26\u663e\u8457\u5927\u4e8e 0\uff0c\u4f46 $x$\u3001$y$ \u4e4b\u95f4\u5e76\u4e0d\u5b58\u5728\u76f4\u63a5\u56e0\u679c\u94fe\u3002\u8fd9\u5c31\u662f\u201c\u5171\u540c\u9a71\u52a8\u6e90\u4f2a\u76f8\u5173\u201d\u7684\u4e00\u79cd\u6700\u7b80\u5355\u5f62\u5f0f\u3002<\/p>\n

\u8981\u5728\u6570\u636e\u5206\u6790\u4e2d\u533a\u5206\u201c\u76f4\u63a5\u8054\u7cfb\u201d\u4e0e\u201c\u5171\u540c\u9a71\u52a8\u201d\uff0c\u901a\u5e38\u9700\u8981\uff1a<\/p>\n

– \u5f15\u5165\u5e76\u6d4b\u91cf\u5019\u9009\u5171\u540c\u6e90 $z$\uff0c\u4f7f\u7528\u7b2c\u4e03\u7ae0\u7684**\u504f\u76f8\u5e72\u6027**\u5c06\u5176\u5f71\u54cd\u56de\u5f52\u6389\uff1b
\n– \u5f15\u5165\u5916\u90e8\u6270\u52a8\u6216\u5e72\u9884\u5b9e\u9a8c\uff0c\u4eba\u4e3a\u6539\u53d8\u4e00\u7aef\u800c\u4e0d\u6539\u53d8\u5171\u540c\u6e90\uff1b
\n– \u7ed3\u5408\u7269\u7406\u6a21\u578b\u6216\u65f6\u95f4\u6ede\u540e\u7ed3\u6784\uff0c\u4f8b\u5982 Granger \u56e0\u679c\u3001\u7ed3\u6784 VAR\u3002<\/p>\n

\u5355\u51ed\u9ad8\u76f8\u5173\u6216\u9ad8\u76f8\u5e72\u5f97\u51fa\u56e0\u679c\u7ed3\u8bba\u51e0\u4e4e\u603b\u662f\u9519\u8bef\u7684\uff1b\u628a $z$ \u6d4b\u51fa\u6765\u5e76\u7eb3\u5165\u5206\u6790\uff0c\u662f\u6700\u57fa\u672c\u4e5f\u6700\u6709\u6548\u7684\u6392\u9664\u6b65\u9aa4\u3002<\/p>\n

—<\/p>\n

—<\/p>\n

## \u4e03\u3001\u76f8\u5e72\u6027\u53ca\u5176\u6269\u5c55<\/p>\n

\u201c\u76f8\u5e72\u6027\u201d\u5728\u4e0d\u540c\u8bed\u5883\u4e0b\u6709\u4e0d\u540c\u542b\u4e49\u3002\u672c\u8282\u91cd\u70b9\u4ecb\u7ecd\u968f\u673a\u4fe1\u53f7\u548c\u8c31\u5206\u6790\u4e2d\u6700\u5e38\u7528\u7684 **magnitude-squared coherence\uff0cMSC**\u3002\u5b83\u63cf\u8ff0\u4e24\u4e2a\u4fe1\u53f7\u5728\u6bcf\u4e00\u4e2a\u9891\u7387\u4e0a\u7684\u7ebf\u6027\u5173\u8054\u5f3a\u5ea6\uff0c\u53ef\u4ee5\u770b\u6210\u201c\u9891\u7387\u5206\u8fa8\u7684\u5e73\u65b9\u76f8\u5173\u7cfb\u6570\u201d\u3002<\/p>\n

### \u4e92\u529f\u7387\u8c31\u548c\u81ea\u529f\u7387\u8c31<\/p>\n

\u5bf9\u8054\u5408\u5bbd\u5e73\u7a33\u8fc7\u7a0b\uff0c\u5b9a\u4e49\u4e92\u529f\u7387\u8c31\u5bc6\u5ea6<\/p>\n

$$
\nS_{xy}(f)=\\mathcal F\\{R_{xy}(\\tau)\\}.
\n$$<\/p>\n

\u81ea\u529f\u7387\u8c31\u5bc6\u5ea6\u4e3a<\/p>\n

$$
\nS_{xx}(f)=\\mathcal F\\{R_{xx}(\\tau)\\},
\n\\qquad
\nS_{yy}(f)=\\mathcal F\\{R_{yy}(\\tau)\\}.
\n$$<\/p>\n

\u4e92\u529f\u7387\u8c31\u901a\u5e38\u662f\u590d\u6570\uff1a<\/p>\n

$$
\nS_{xy}(f)=|S_{xy}(f)|e^{j\\phi_{xy}(f)}.
\n$$<\/p>\n

\u5176\u4e2d\uff1a<\/p>\n

– $|S_{xy}(f)|$ \u8868\u793a\u8be5\u9891\u7387\u4e0a\u5171\u540c\u53d8\u5316\u7684\u5e45\u5ea6\uff1b
\n– $\\phi_{xy}(f)$ \u8868\u793a\u8be5\u9891\u7387\u4e0a\u7684\u5e73\u5747\u76f8\u4f4d\u5dee\uff1b
\n– $S_{xx}(f)$\u3001$S_{yy}(f)$ \u8868\u793a\u5404\u4fe1\u53f7\u5728\u8be5\u9891\u7387\u9644\u8fd1\u7684\u529f\u7387\u5bc6\u5ea6\u3002<\/p>\n

### \u590d\u76f8\u5e72\u5ea6\u4e0e\u5e73\u65b9\u76f8\u5e72\u6027<\/p>\n

\u5f52\u4e00\u5316\u590d\u4e92\u8c31\u5b9a\u4e49\u4e3a\u590d\u76f8\u5e72\u5ea6\uff1a<\/p>\n

$$
\n\\boxed{
\nC_{xy}(f)
\n=\\frac{S_{xy}(f)}{\\sqrt{S_{xx}(f)S_{yy}(f)}}
\n}.
\n$$<\/p>\n

\u5176\u6a21\u5e73\u65b9\u79f0\u4e3a magnitude-squared coherence\uff1a<\/p>\n

$$
\n\\boxed{
\n\\gamma_{xy}^2(f)
\n=\\frac{|S_{xy}(f)|^2}{S_{xx}(f)S_{yy}(f)}
\n}.
\n$$<\/p>\n

\u7531\u8c31\u77e9\u9635\u7684\u534a\u6b63\u5b9a\u6027\u6216 Cauchy\u2013Schwarz \u4e0d\u7b49\u5f0f\uff1a<\/p>\n

$$
\n0\\le\\gamma_{xy}^2(f)\\le1.
\n$$<\/p>\n

\u89e3\u91ca\u5982\u4e0b\uff1a<\/p>\n

– $\\gamma_{xy}^2(f)\\approx1$\uff1a\u5728\u8be5\u9891\u7387\u4e0a\u5b58\u5728\u7a33\u5b9a\u3001\u53ef\u91cd\u590d\u7684\u5f3a\u7ebf\u6027\u5173\u7cfb\uff1b
\n– $\\gamma_{xy}^2(f)\\approx0$\uff1a\u5728\u8be5\u9891\u7387\u4e0a\u6ca1\u6709\u7a33\u5b9a\u7684\u7ebf\u6027\u5173\u8054\uff0c\u6216\u8005\u5173\u8054\u88ab\u4e0d\u76f8\u5173\u566a\u58f0\u3001\u975e\u7ebf\u6027\u3001\u65f6\u53d8\u5173\u7cfb\u6216\u4f30\u8ba1\u8bef\u5dee\u524a\u5f31\uff1b
\n– \u4e2d\u95f4\u503c\uff1a\u53ea\u6709\u90e8\u5206\u8f93\u51fa\u529f\u7387\u80fd\u7531\u4e0e\u53e6\u4e00\u4e2a\u4fe1\u53f7\u7ebf\u6027\u76f8\u5173\u7684\u6210\u5206\u89e3\u91ca\u3002<\/p>\n

\u590d\u76f8\u5e72\u5ea6\u8fd8\u4fdd\u7559\u76f8\u4f4d\uff1a<\/p>\n

$$
\n\\angle C_{xy}(f)=\\angle S_{xy}(f).
\n$$<\/p>\n

\u800c MSC \u53ea\u6709 $0$ \u5230 $1$ \u7684\u5b9e\u6570\u5f3a\u5ea6\uff0c\u4e0d\u4fdd\u7559\u76f8\u4f4d\u7b26\u53f7\u3002<\/p>\n

### \u4e3a\u4ec0\u4e48\u76f8\u5e72\u6027\u9700\u8981\u5f52\u4e00\u5316<\/p>\n

\u4e92\u529f\u7387\u8c31 $S_{xy}(f)$ \u7684\u5927\u5c0f\u540c\u65f6\u53d7\u5230\u4e24\u4e2a\u4fe1\u53f7\u5e45\u5ea6\u5f71\u54cd\uff0c\u4e0d\u4fbf\u76f4\u63a5\u6bd4\u8f83\u4e0d\u540c\u9891\u7387\u6216\u5b9e\u9a8c\u3002\u9664\u4ee5<\/p>\n

$$
\n\\sqrt{S_{xx}(f)S_{yy}(f)}
\n$$<\/p>\n

\u540e\uff0c\u76f8\u5e72\u6027\u53d8\u6210\u65e0\u91cf\u7eb2\u6307\u6807\uff0c\u5e76\u9650\u5236\u5728 $[0,1]$ \u5185\u3002<\/p>\n

\u8fd9\u4e0e\u65f6\u57df\u76f8\u5173\u7cfb\u6570<\/p>\n

$$
\n\\rho_{XY}=\\frac{C_{XY}}{\\sigma_X\\sigma_Y}
\n$$<\/p>\n

\u7684\u5f52\u4e00\u5316\u601d\u60f3\u76f8\u540c\uff1a\u76f8\u5173\u7cfb\u6570\u6309\u603b\u65b9\u5dee\u5f52\u4e00\u5316\uff0c\u76f8\u5e72\u6027\u5219\u9010\u9891\u7387\u6309\u8c31\u529f\u7387\u5f52\u4e00\u5316\u3002<\/p>\n

### \u76f8\u5e72\u6027\u4e0e\u9891\u7387\u54cd\u5e94\u4f30\u8ba1<\/p>\n

\u8003\u8651\u7ebf\u6027\u7cfb\u7edf\u6a21\u578b<\/p>\n

$$
\ny(t)=h(t)*x(t)+n(t),
\n$$<\/p>\n

\u5176\u4e2d\u566a\u58f0 $n(t)$ \u4e0e\u8f93\u5165 $x(t)$ \u4e0d\u76f8\u5173\u3002\u9891\u57df\u4e2d<\/p>\n

$$
\nY(f)=H(f)X(f)+N(f).
\n$$<\/p>\n

\u4e8e\u662f<\/p>\n

$$
\nS_{xy}(f)=H^*(f)S_{xx}(f)
\n$$<\/p>\n

\u6216\u5728\u76f8\u53cd\u4e92\u8c31\u5b9a\u4e49\u4e0b\u4e3a $H(f)S_{xx}(f)$\u3002\u540c\u65f6<\/p>\n

$$
\nS_{yy}(f)=|H(f)|^2S_{xx}(f)+S_{nn}(f).
\n$$<\/p>\n

\u4ee3\u5165 MSC\uff1a<\/p>\n

$$
\n\\gamma_{xy}^2(f)
\n=\\frac{|H(f)|^2S_{xx}(f)}
\n{|H(f)|^2S_{xx}(f)+S_{nn}(f)}.
\n$$<\/p>\n

\u56e0\u6b64\uff0c\u5728\u4e0a\u8ff0\u7406\u60f3\u6a21\u578b\u4e0b\uff0c\u76f8\u5e72\u6027\u53ef\u4ee5\u89e3\u91ca\u4e3a\u8be5\u9891\u7387\u8f93\u51fa\u529f\u7387\u4e2d\u7531\u8f93\u5165\u7684\u7ebf\u6027\u54cd\u5e94\u8d21\u732e\u7684\u6bd4\u4f8b\u3002\u82e5\u8f93\u51fa\u566a\u58f0\u4e3a\u96f6\uff0c\u5219\u76f8\u5e72\u6027\u4e3a 1\uff1b\u566a\u58f0\u8d8a\u5f3a\uff0c\u76f8\u5e72\u6027\u8d8a\u4f4e\u3002<\/p>\n

### \u76f8\u5e72\u6027\u4e0e\u4fe1\u566a\u6bd4<\/p>\n

\u5b9a\u4e49\u8f93\u51fa\u7aef\u8be5\u9891\u7387\u7684\u4fe1\u566a\u6bd4\u4e3a<\/p>\n

$$
\n\\operatorname{SNR}(f)
\n=\\frac{|H(f)|^2S_{xx}(f)}{S_{nn}(f)}.
\n$$<\/p>\n

\u5219<\/p>\n

$$
\n\\boxed{
\n\\gamma_{xy}^2(f)
\n=\\frac{\\operatorname{SNR}(f)}{1+\\operatorname{SNR}(f)}
\n}.
\n$$<\/p>\n

\u53cd\u8fc7\u6765\uff1a<\/p>\n

$$
\n\\boxed{
\n\\operatorname{SNR}(f)
\n=\\frac{\\gamma_{xy}^2(f)}{1-\\gamma_{xy}^2(f)}
\n}.
\n$$<\/p>\n

\u8fd9\u53ea\u5728\u5355\u8f93\u5165\u7ebf\u6027\u7cfb\u7edf\u3001\u566a\u58f0\u4e0e\u8f93\u5165\u4e0d\u76f8\u5173\u7b49\u5047\u8bbe\u6210\u7acb\u65f6\u624d\u9002\u7528\uff0c\u4e0d\u80fd\u628a\u5b83\u5f53\u6210\u6240\u6709\u573a\u666f\u4e0b\u7684\u901a\u7528 SNR \u516c\u5f0f\u3002<\/p>\n

### \u76f8\u4f4d\u8c31\u3001\u76f8\u4f4d\u9501\u5b9a\u4e0e\u65f6\u5ef6<\/p>\n

\u4e92\u8c31\u76f8\u4f4d\u4e3a<\/p>\n

$$
\n\\phi_{xy}(f)=\\arg S_{xy}(f).
\n$$<\/p>\n

\u82e5\u4e24\u4e2a\u4fe1\u53f7\u4ec5\u76f8\u5dee\u56fa\u5b9a\u65f6\u5ef6 $\\tau_0$\uff1a<\/p>\n

$$
\ny(t)=x(t-\\tau_0),
\n$$<\/p>\n

\u5219\u6839\u636e\u4e92\u8c31\u5b9a\u4e49\uff0c\u4e92\u8c31\u76f8\u4f4d\u901a\u5e38\u5448\u7ebf\u6027\u53d8\u5316\uff1a<\/p>\n

$$
\n\\phi_{xy}(f)=\\pm2\\pi f\\tau_0.
\n$$<\/p>\n

\u6b63\u8d1f\u53f7\u53d6\u51b3\u4e8e\u4e92\u8c31\u5b9a\u4e49\u3002\u7531\u76f8\u4f4d\u659c\u7387\u53ef\u4f30\u8ba1\u65f6\u5ef6\uff1a<\/p>\n

$$
\n\\tau_0=\\pm\\frac{1}{2\\pi}\\frac{d\\phi_{xy}}{df}.
\n$$<\/p>\n

\u4f7f\u7528\u8fd9\u4e00\u65b9\u6cd5\u524d\u5e94\u5148\u8fdb\u884c phase unwrapping\uff0c\u5e76\u53ea\u5728\u76f8\u5e72\u6027\u8db3\u591f\u9ad8\u7684\u9891\u5e26\u89e3\u91ca\u76f8\u4f4d\uff1b\u4f4e\u76f8\u5e72\u9891\u70b9\u4e0a\u7684\u76f8\u4f4d\u901a\u5e38\u4e0d\u7a33\u5b9a\u3001\u6ca1\u6709\u53ef\u9760\u7269\u7406\u610f\u4e49\u3002<\/p>\n

### \u76f8\u5e72\u6027\u4e3a 1 \u7684\u542b\u4e49\u548c\u9650\u5236<\/p>\n

\u7406\u8bba\u4e0a\u7684 $\\gamma_{xy}^2(f)=1$ \u8868\u793a\uff0c\u5728\u4e8c\u9636\u7edf\u8ba1\u610f\u4e49\u4e0b\uff0c\u8be5\u9891\u7387\u5904\u4e00\u4e2a\u4fe1\u53f7\u53ef\u7531\u53e6\u4e00\u4e2a\u4fe1\u53f7\u7ecf\u8fc7\u786e\u5b9a\u7684\u7ebf\u6027\u9891\u7387\u54cd\u5e94\u5b8c\u5168\u89e3\u91ca\u3002\u4f46\u5b83\u4e0d\u610f\u5473\u7740\uff1a<\/p>\n

– \u4e24\u4e2a\u65f6\u57df\u6ce2\u5f62\u5b8c\u5168\u76f8\u540c\uff1b
\n– \u9891\u7387\u54cd\u5e94\u7684\u5e45\u503c\u5fc5\u987b\u4e3a 1\uff1b
\n– \u76f8\u4f4d\u5dee\u5fc5\u987b\u4e3a 0\uff1b
\n– \u4e00\u4e2a\u4fe1\u53f7\u5fc5\u7136\u56e0\u679c\u5730\u5bfc\u81f4\u53e6\u4e00\u4e2a\u4fe1\u53f7\uff1b
\n– \u7cfb\u7edf\u5728\u6240\u6709\u9891\u7387\u4e0a\u90fd\u7ebf\u6027\u3002<\/p>\n

\u4f8b\u5982\uff0c\u7406\u60f3\u65e0\u566a\u58f0 LTI \u7cfb\u7edf\u53ef\u4ee5\u4efb\u610f\u6539\u53d8\u5e45\u503c\u548c\u76f8\u4f4d\uff0c\u53ea\u8981\u8f93\u51fa\u5b8c\u5168\u7531\u8f93\u5165\u7ebf\u6027\u4ea7\u751f\uff0c\u5bf9\u5e94\u9891\u7387\u4e0a\u7684\u76f8\u5e72\u6027\u4ecd\u53ef\u4e3a 1\u3002<\/p>\n

### \u76f8\u5e72\u6027\u964d\u4f4e\u7684\u539f\u56e0<\/p>\n

\u4f4e\u76f8\u5e72\u6027\u53ef\u80fd\u7531\u4ee5\u4e0b\u56e0\u7d20\u9020\u6210\uff1a<\/p>\n

1. \u8f93\u51fa\u4e2d\u5b58\u5728\u4e0e\u8f93\u5165\u4e0d\u76f8\u5173\u7684\u566a\u58f0\uff1b
\n2. \u8f93\u5165\u6d4b\u91cf\u672c\u8eab\u6709\u566a\u58f0\uff1b
\n3. \u8fd8\u6709\u672a\u6d4b\u91cf\u7684\u5176\u4ed6\u8f93\u5165\u5f71\u54cd\u8f93\u51fa\uff1b
\n4. \u8f93\u5165\u4e0e\u8f93\u51fa\u4e4b\u95f4\u5b58\u5728\u975e\u7ebf\u6027\u5173\u7cfb\uff1b
\n5. \u7cfb\u7edf\u53c2\u6570\u968f\u65f6\u95f4\u53d8\u5316\uff1b
\n6. \u8f93\u5165\u4e0e\u8f93\u51fa\u6ca1\u6709\u6b63\u786e\u5bf9\u65f6\uff1b
\n7. \u6570\u636e\u6bb5\u8fc7\u77ed\uff0c\u8c31\u4f30\u8ba1\u65b9\u5dee\u8fc7\u5927\uff1b
\n8. \u6cc4\u6f0f\u3001\u6df7\u53e0\u6216\u4f20\u611f\u5668\u9971\u548c\uff1b
\n9. \u9891\u7387\u5206\u8fa8\u7387\u4e0d\u5408\u9002\uff0c\u628a\u4e0d\u540c\u52a8\u529b\u5b66\u6df7\u5728\u4e00\u4e2a bin\uff1b
\n10. \u4e0d\u540c\u6570\u636e\u6bb5\u4e2d\u7684\u76f8\u4f4d\u5173\u7cfb\u4e0d\u7a33\u5b9a\uff0c\u5e73\u5747\u540e\u4e92\u8c31\u76f8\u4e92\u62b5\u6d88\u3002<\/p>\n

\u56e0\u6b64\uff0c\u4f4e\u76f8\u5e72\u6027\u4e0d\u662f\u201c\u5b8c\u5168\u6ca1\u6709\u5173\u7cfb\u201d\u7684\u5145\u5206\u8bc1\u636e\uff0c\u800c\u662f\u201c\u6ca1\u6709\u68c0\u6d4b\u5230\u7a33\u5b9a\u7ebf\u6027\u4e8c\u9636\u5173\u8054\u201d\u6216\u4f30\u8ba1\u6761\u4ef6\u4e0d\u8db3\u3002<\/p>\n

### \u76f8\u5e72\u51fd\u6570\u4e0e\u8109\u51b2\u76f8\u5e72\u6027\u7684\u533a\u522b<\/p>\n

\u201ccoherence\u201d\u8fd8\u53ef\u80fd\u6307\u5176\u4ed6\u6982\u5ff5\uff1a<\/p>\n

– **\u8c31\u76f8\u5e72\u6027**\uff1a\u672c\u8282\u5b9a\u4e49\u7684 $\\gamma_{xy}^2(f)$\uff1b
\n– **\u590d\u76f8\u5e72\u5ea6**\uff1a\u4fdd\u7559\u76f8\u4f4d\u7684 $C_{xy}(f)$\uff1b
\n– **\u76f8\u4f4d\u9501\u5b9a\u503c\uff08PLV\uff09**\uff1a\u53ea\u5206\u6790\u8de8\u8bd5\u6b21\u7684\u76f8\u4f4d\u5dee\u7a33\u5b9a\u6027\uff0c\u4e0d\u76f4\u63a5\u8003\u8651\u5e45\u5ea6\uff1b
\n– **\u5149\u5b66\u76f8\u5e72\u6027**\uff1a\u63cf\u8ff0\u7535\u78c1\u573a\u5728\u65f6\u95f4\u6216\u7a7a\u95f4\u4e0a\u7684\u76f8\u4f4d\u5173\u8054\uff0c\u6709\u4e00\u9636\u3001\u4e8c\u9636\u76f8\u5e72\u51fd\u6570\uff1b
\n– **\u5c0f\u6ce2\u76f8\u5e72\u6027**\uff1a\u5728\u65f6\u95f4\u2014\u9891\u7387\u5e73\u9762\u4e0a\u5206\u6790\u5c40\u90e8\u5171\u540c\u53d8\u5316\uff1b
\n– **\u504f\u76f8\u5e72\u6027\u3001\u591a\u91cd\u76f8\u5e72\u6027**\uff1a\u63a7\u5236\u5176\u4ed6\u901a\u9053\u6216\u5904\u7406\u591a\u4e2a\u8f93\u5165\u3002<\/p>\n

\u8fd9\u4e9b\u91cf\u867d\u7136\u90fd\u6d89\u53ca\u201c\u7a33\u5b9a\u5173\u7cfb\u201d\uff0c\u4f46\u5b9a\u4e49\u3001\u53d6\u503c\u548c\u7edf\u8ba1\u6027\u8d28\u4e0d\u540c\uff0c\u4e0d\u80fd\u6df7\u7528\u3002<\/p>\n

### \u504f\u76f8\u5e72\u6027<\/p>\n

\u82e5 $x$ \u548c $y$ \u90fd\u53d7\u7b2c\u4e09\u4e2a\u4fe1\u53f7 $z$ \u9a71\u52a8\uff0c\u666e\u901a\u76f8\u5e72\u6027\u53ef\u80fd\u5f88\u9ad8\uff0c\u4f46\u8fd9\u4e0d\u4ee3\u8868 $x$ \u4e0e $y$ \u5b58\u5728\u76f4\u63a5\u8054\u7cfb\u3002\u504f\u76f8\u5e72\u6027\u7528\u4e8e\u5728\u63a7\u5236 $z$ \u540e\u5206\u6790\u5269\u4f59\u7ebf\u6027\u5173\u8054\u3002<\/p>\n

\u4ee5\u590d\u76f8\u5e72\u5ea6\u8868\u793a\uff0c\u4e00\u79cd\u5f62\u5f0f\u4e3a<\/p>\n

$$
\nC_{xy\\cdot z}(f)
\n=\\frac{C_{xy}(f)-C_{xz}(f)C_{zy}(f)}
\n{\\sqrt{\\left(1-|C_{xz}(f)|^2\\right)
\n\\left(1-|C_{yz}(f)|^2\\right)}}.
\n$$<\/p>\n

\u5176\u6a21\u5e73\u65b9\u7ed9\u51fa\u504f\u5e73\u65b9\u76f8\u5e72\u6027\u3002\u8be5\u516c\u5f0f\u9002\u7528\u4e8e\u76f8\u5e94\u8c31\u77e9\u9635\u53ef\u9006\u4e14\u7edf\u8ba1\u4f30\u8ba1\u53ef\u9760\u7684\u573a\u666f\uff1b\u591a\u53d8\u91cf\u60c5\u51b5\u4e0b\u901a\u5e38\u76f4\u63a5\u4f7f\u7528\u8c31\u77e9\u9635\u53ca\u5176\u9006\u77e9\u9635\u8ba1\u7b97\u3002<\/p>\n

### \u591a\u91cd\u76f8\u5e72\u6027<\/p>\n

\u591a\u91cd\u76f8\u5e72\u6027\u8861\u91cf\u591a\u4e2a\u8f93\u5165\u5171\u540c\u7ebf\u6027\u89e3\u91ca\u67d0\u4e2a\u8f93\u51fa\u7684\u80fd\u529b\u3002\u8bbe\u8f93\u5165\u5411\u91cf\u4e3a $\\boldsymbol x$\uff0c\u8f93\u51fa\u4e3a $y$\uff0c\u5176\u8c31\u77e9\u9635\u4e3a $\\boldsymbol S_{xx}$\uff0c\u8f93\u5165\u4e0e\u8f93\u51fa\u7684\u4e92\u8c31\u5411\u91cf\u4e3a $\\boldsymbol S_{xy}$\uff0c\u5219\u591a\u91cd\u76f8\u5e72\u6027\u53ef\u5199\u4e3a<\/p>\n

$$
\n\\gamma_{y:\\boldsymbol x}^2(f)
\n=\\frac{\\boldsymbol S_{yx}(f)
\n\\boldsymbol S_{xx}^{-1}(f)
\n\\boldsymbol S_{xy}(f)}{S_{yy}(f)}.
\n$$<\/p>\n

\u5b83\u5e38\u7528\u4e8e\u591a\u8f93\u5165\u5355\u8f93\u51fa\u7cfb\u7edf\u8fa8\u8bc6\u3001\u632f\u52a8\u5206\u6790\u548c\u591a\u4f20\u611f\u5668\u6570\u636e\u878d\u5408\u3002<\/p>\n

### \u5c0f\u6ce2\u76f8\u5e72\u6027<\/p>\n

\u666e\u901a MSC \u5047\u8bbe\u7edf\u8ba1\u5173\u7cfb\u5728\u5206\u6790\u65f6\u6bb5\u5185\u8fd1\u4f3c\u7a33\u5b9a\uff0c\u53ea\u7ed9\u51fa\u9891\u7387\u7ef4\u5ea6\u3002\u5982\u679c\u4fe1\u53f7\u660e\u663e\u975e\u5e73\u7a33\uff0c\u53ef\u4ee5\u4f7f\u7528\u5c0f\u6ce2\u76f8\u5e72\u6027\uff0c\u5728\u65f6\u95f4\u2014\u5c3a\u5ea6\u6216\u65f6\u95f4\u2014\u9891\u7387\u5e73\u9762\u4e2d\u89c2\u5bdf\u5c40\u90e8\u76f8\u5e72\u3002<\/p>\n

\u5c0f\u6ce2\u76f8\u5e72\u6027\u7684\u4e00\u822c\u5f62\u5f0f\u4e3a<\/p>\n

$$
\nR_{xy}^2(a,b)
\n=\\frac{|S\\{a^{-1}W_{xy}(a,b)\\}|^2}
\n{S\\{a^{-1}|W_x(a,b)|^2\\}
\n S\\{a^{-1}|W_y(a,b)|^2\\}},
\n$$<\/p>\n

\u5176\u4e2d $W_x$\u3001$W_y$ \u4e3a\u8fde\u7eed\u5c0f\u6ce2\u53d8\u6362\uff0c$W_{xy}=W_xW_y^*$\uff0c$S\\{\\cdot\\}$ \u8868\u793a\u9002\u5f53\u7684\u65f6\u95f4\u548c\u5c3a\u5ea6\u5e73\u6ed1\u3002\u6ca1\u6709\u5e73\u6ed1\u65f6\u540c\u6837\u53ef\u80fd\u4ea7\u751f\u9000\u5316\u7ed3\u679c\u3002<\/p>\n

—
\n### \u76f8\u5173\u6027\u4e0e\u76f8\u5e72\u6027\u7684\u533a\u522b\u548c\u8054\u7cfb<\/p>\n

#### \u4e00\u5f20\u8868\u770b\u6e05\u533a\u522b<\/p>\n

| \u6bd4\u8f83\u9879 | \u65f6\u57df\u76f8\u5173\/\u76f8\u5173\u7cfb\u6570 | \u8c31\u76f8\u5e72\u6027 |
\n|—|—|—|
\n| \u81ea\u53d8\u91cf | \u5ef6\u8fdf $\\tau$\uff0c\u6216\u5355\u4e2a\u603b\u4f53\u7edf\u8ba1\u91cf | \u9891\u7387 $f$ |
\n| \u5178\u578b\u5b9a\u4e49 | $R_{xy}(\\tau)$\u3001$\\rho_{XY}$ | $\\gamma_{xy}^2(f)$ |
\n| \u662f\u5426\u5f52\u4e00\u5316 | \u4e92\u76f8\u5173\u901a\u5e38\u4e0d\u5f52\u4e00\u5316\uff1b\u76f8\u5173\u7cfb\u6570\u5f52\u4e00\u5316 | \u6309\u9891\u7387\u5f52\u4e00\u5316 |
\n| \u53d6\u503c | \u4e92\u76f8\u5173\u53ef\u6709\u5355\u4f4d\u3001\u53ef\u4e3a\u590d\u6570\uff1b\u5b9e Pearson \u5728 $[-1,1]$ | MSC \u5728 $[0,1]$ |
\n| \u4e3b\u8981\u56de\u7b54 | \u662f\u5426\u5171\u540c\u53d8\u5316\u3001\u6700\u4f73\u65f6\u5ef6\u5728\u54ea\u91cc | \u54ea\u4e9b\u9891\u7387\u5177\u6709\u7a33\u5b9a\u7ebf\u6027\u5173\u7cfb |
\n| \u76f8\u4f4d\u4fe1\u606f | \u590d\u4e92\u76f8\u5173\u53ef\u5305\u542b\u6574\u4f53\u5ef6\u8fdf\/\u76f8\u4f4d\u4fe1\u606f | \u590d\u76f8\u5e72\u5ea6\u6216\u4e92\u8c31\u76f8\u4f4d\u9010\u9891\u7387\u7ed9\u51fa\u76f8\u4f4d |
\n| \u5bf9\u975e\u5e73\u7a33\u6027\u7684\u5904\u7406 | \u6ed1\u52a8\u76f8\u5173\u53ef\u505a\u5c40\u90e8\u5206\u6790 | \u666e\u901a MSC \u9700\u8fd1\u4f3c\u5e73\u7a33\uff1b\u53ef\u6539\u7528\u77ed\u65f6\u6216\u5c0f\u6ce2\u76f8\u5e72\u6027 |
\n| \u5bf9\u56e0\u679c\u6027\u7684\u7ed3\u8bba | \u4e0d\u80fd\u5355\u72ec\u8bc1\u660e\u56e0\u679c | \u540c\u6837\u4e0d\u80fd\u5355\u72ec\u8bc1\u660e\u56e0\u679c |<\/p>\n

#### \u76f8\u5e72\u6027\u662f\u9891\u7387\u5206\u8fa8\u7684\u5f52\u4e00\u5316\u5173\u8054<\/p>\n

\u4ece\u6570\u5b66\u7ed3\u6784\u770b\uff1a<\/p>\n

$$
\n\\rho_{XY}
\n=\\frac{\\mathbb E[(X-\\mu_X)(Y-\\mu_Y)^*]}
\n{\\sigma_X\\sigma_Y}
\n$$<\/p>\n

\u6309\u603b\u65b9\u5dee\u5f52\u4e00\u5316\uff0c\u800c<\/p>\n

$$
\nC_{xy}(f)
\n=\\frac{S_{xy}(f)}{\\sqrt{S_{xx}(f)S_{yy}(f)}}
\n$$<\/p>\n

\u6309\u6bcf\u4e2a\u9891\u7387\u4e0a\u7684\u529f\u7387\u5f52\u4e00\u5316\u3002\u76f8\u5e72\u6027\u53ef\u4ee5\u7406\u89e3\u4e3a\u5c06\u603b\u7684\u7ebf\u6027\u76f8\u5173\u62c6\u5206\u5230\u5404\u4e2a\u9891\u7387\u540e\u5206\u522b\u8003\u5bdf\u3002<\/p>\n

\u4f46\u662f\uff0c\u76f8\u5e72\u6027\u5e76\u4e0d\u7b49\u4e8e\u4e92\u76f8\u5173\u7684 Fourier \u53d8\u6362\u672c\u8eab\u3002\u4e92\u76f8\u5173\u7684 Fourier \u53d8\u6362\u662f\u4e92\u529f\u7387\u8c31 $S_{xy}$\uff1b\u53ea\u6709\u518d\u9664\u4ee5\u4e24\u4e2a\u81ea\u529f\u7387\u8c31\u7684\u51e0\u4f55\u5e73\u5747\uff0c\u624d\u5f97\u5230\u590d\u76f8\u5e72\u5ea6\u6216 MSC\u3002<\/p>\n

#### \u9ad8\u76f8\u5173\u4f46\u5c40\u90e8\u76f8\u5e72\u7ed3\u6784\u4e0d\u540c<\/p>\n

\u4e24\u4e2a\u5bbd\u5e26\u4fe1\u53f7\u53ef\u80fd\u5177\u6709\u8f83\u9ad8\u7684\u96f6\u5ef6\u8fdf\u76f8\u5173\u7cfb\u6570\uff0c\u4f46\u8fd9\u79cd\u5173\u7cfb\u53ef\u80fd\u4e3b\u8981\u7531\u5c11\u6570\u5f3a\u80fd\u91cf\u9891\u5e26\u8d21\u732e\u3002\u76f8\u5e72\u6027\u53ef\u4ee5\u663e\u793a\u7a76\u7adf\u54ea\u4e9b\u9891\u7387\u76f8\u5173\u3001\u54ea\u4e9b\u9891\u7387\u88ab\u566a\u58f0\u6df9\u6ca1\u3002<\/p>\n

\u53cd\u8fc7\u6765\uff0c\u4e00\u4e2a\u7a84\u9891\u5e26\u4e0a\u7684\u76f8\u5e72\u6027\u53ef\u80fd\u63a5\u8fd1 1\uff0c\u4f46\u8be5\u9891\u5e26\u5360\u603b\u80fd\u91cf\u5f88\u5c0f\uff0c\u56e0\u6b64\u6574\u4f53\u65f6\u57df\u76f8\u5173\u7cfb\u6570\u4ecd\u53ef\u80fd\u4e0d\u9ad8\u3002<\/p>\n

#### \u56fa\u5b9a\u65f6\u5ef6\u5bf9\u4e24\u8005\u7684\u5f71\u54cd<\/p>\n

\u56fa\u5b9a\u65f6\u5ef6\u4f1a\u4f7f\u96f6\u5ef6\u8fdf\u76f8\u5173\u964d\u4f4e\uff0c\u4f46\u4e92\u76f8\u5173\u51fd\u6570\u5728\u6b63\u786e\u5ef6\u8fdf\u5904\u4ecd\u53ef\u51fa\u73b0\u9ad8\u5cf0\u3002\u9891\u57df\u4e2d\uff0c\u7406\u60f3\u56fa\u5b9a\u65f6\u5ef6\u4e0d\u4f1a\u964d\u4f4e\u7406\u8bba\u76f8\u5e72\u6027\uff0c\u53ea\u4f1a\u5f15\u5165\u7ebf\u6027\u76f8\u4f4d\uff1a<\/p>\n

$$
\n|C_{xy}(f)|=1,
\n\\qquad
\n\\angle C_{xy}(f)=\\pm2\\pi f\\tau_0.
\n$$<\/p>\n

\u56e0\u6b64\uff0c\u5982\u679c\u4e24\u4e2a\u4fe1\u53f7\u5f62\u72b6\u76f8\u540c\u4f46\u5b58\u5728\u65f6\u5ef6\uff1a<\/p>\n

– \u53ea\u770b\u96f6\u5ef6\u8fdf Pearson \u76f8\u5173\u53ef\u80fd\u8bef\u5224\u4e3a\u5173\u7cfb\u8f83\u5f31\uff1b
\n– \u770b\u4e92\u76f8\u5173\u5cf0\u53ef\u4ee5\u4f30\u8ba1\u65f6\u5ef6\uff1b
\n– \u770b\u76f8\u5e72\u6027\u548c\u4e92\u8c31\u76f8\u4f4d\u53ef\u4ee5\u8bc6\u522b\u76f8\u5173\u9891\u5e26\u548c\u9891\u7387\u76f8\u5173\u7684\u76f8\u4f4d\u5173\u7cfb\u3002<\/p>\n

#### \u975e\u7ebf\u6027\u5173\u7cfb\u5bf9\u4e24\u8005\u7684\u5f71\u54cd<\/p>\n

\u666e\u901a Pearson \u76f8\u5173\u548c MSC \u90fd\u4e3b\u8981\u523b\u753b\u7ebf\u6027\u4e8c\u9636\u5173\u7cfb\u3002\u82e5<\/p>\n

$$
\ny(t)=x^2(t),
\n$$<\/p>\n

\u5373\u4f7f $y$ \u5b8c\u5168\u7531 $x$ \u51b3\u5b9a\uff0c$x$ \u4e0e $y$ \u7684\u666e\u901a\u76f8\u5173\u548c\u540c\u9891 MSC \u4e5f\u53ef\u80fd\u5f88\u4f4e\u3002\u975e\u7ebf\u6027\u4f1a\u628a\u80fd\u91cf\u8f6c\u79fb\u5230\u8c10\u6ce2\u548c\u4e92\u8c03\u9891\u7387\uff0c\u8fd9\u79cd\u8de8\u9891\u5173\u7cfb\u53ef\u7528\u53cc\u76f8\u5e72\u6027\uff08bicoherence\uff09\u3001\u9ad8\u9636\u8c31\u6216\u4e92\u4fe1\u606f\u7b49\u65b9\u6cd5\u7814\u7a76\u3002<\/p>\n

#### \u9009\u62e9\u6307\u6807\u7684\u5efa\u8bae<\/p>\n

– \u8981\u627e\u76f8\u5bf9\u65f6\u5ef6\uff1a\u4f7f\u7528\u4e92\u76f8\u5173\u3001GCC \u6216\u4e92\u8c31\u76f8\u4f4d\uff1b
\n– \u8981\u6bd4\u8f83\u6574\u4f53\u7ebf\u6027\u5171\u540c\u53d8\u5316\uff1a\u4f7f\u7528 Pearson \u76f8\u5173\u7cfb\u6570\uff1b
\n– \u8981\u5224\u65ad\u54ea\u4e9b\u9891\u7387\u5b58\u5728\u7a33\u5b9a\u7ebf\u6027\u8054\u7cfb\uff1a\u4f7f\u7528 MSC\uff1b
\n– \u8981\u4fdd\u7559\u5e45\u76f8\u5173\u7cfb\uff1a\u540c\u65f6\u67e5\u770b\u4e92\u529f\u7387\u8c31\u3001\u590d\u76f8\u5e72\u5ea6\u548c\u4f20\u9012\u51fd\u6570\uff1b
\n– \u8981\u5206\u6790\u65f6\u95f4\u53d8\u5316\u7684\u9891\u7387\u5173\u7cfb\uff1a\u4f7f\u7528\u77ed\u65f6\u76f8\u5e72\u6027\u6216\u5c0f\u6ce2\u76f8\u5e72\u6027\uff1b
\n– \u8981\u6392\u9664\u5171\u540c\u7b2c\u4e09\u53d8\u91cf\uff1a\u4f7f\u7528\u504f\u76f8\u5173\u6216\u504f\u76f8\u5e72\u6027\uff1b
\n– \u8981\u5206\u6790\u975e\u7ebf\u6027\u4f9d\u8d56\uff1a\u8003\u8651\u4e92\u4fe1\u606f\u3001\u8ddd\u79bb\u76f8\u5173\u3001\u9ad8\u9636\u8c31\u6216\u975e\u7ebf\u6027\u7cfb\u7edf\u8fa8\u8bc6\u3002<\/p>\n

—
\n### \u76f8\u5e72\u6027\u4e0a\u754c $\\gamma_{xy}^2(f)\\le1$ \u7684\u63a8\u5bfc<\/p>\n

MSC \u6ee1\u8db3 $0\\le\\gamma_{xy}^2(f)\\le1$ \u6709\u4e24\u79cd\u7b49\u4ef7\u7684\u63a8\u5bfc\u8def\u5f84\uff0c\u503c\u5f97\u8bb0\u4f4f\u3002<\/p>\n

**\u8def\u5f84 A\uff08Cauchy\u2013Schwarz\uff09**\uff1a\u5bf9\u4efb\u610f\u56fa\u5b9a\u9891\u7387 $f$\uff0c\u5728\u5e26\u901a\u6ee4\u6ce2\u6781\u7a84\u5316\u7684\u610f\u4e49\u4e0b\u628a $x$\u3001$y$ \u5206\u522b\u89c6\u4e3a\u8be5\u9891\u7387\u7684\u590d\u89e3\u6790\u6210\u5206 $\\tilde X(f)$\u3001$\\tilde Y(f)$\u3002\u4e92\u8c31\u4e0e\u81ea\u8c31\u53ef\u89e3\u91ca\u4e3a<\/p>\n

$$
\nS_{xy}(f)=\\mathbb E[\\tilde X(f)\\tilde Y^*(f)],
\n\\qquad
\nS_{xx}(f)=\\mathbb E[|\\tilde X(f)|^2],
\n\\qquad
\nS_{yy}(f)=\\mathbb E[|\\tilde Y(f)|^2].
\n$$<\/p>\n

\u5bf9\u4e8c\u9636\u77e9\u5185\u79ef\u5e94\u7528 Cauchy\u2013Schwarz\uff1a<\/p>\n

$$
\n|\\mathbb E[\\tilde X\\tilde Y^*]|^2
\n\\le\\mathbb E[|\\tilde X|^2]\\mathbb E[|\\tilde Y|^2],
\n$$<\/p>\n

\u4e24\u8fb9\u9664\u4ee5\u53f3\u7aef\u5f97 $\\gamma_{xy}^2(f)\\le1$\uff0c\u7b49\u53f7\u4ec5\u5728 $\\tilde Y(f)=c(f)\\tilde X(f)$ \u51e0\u4e4e\u5fc5\u7136\u6210\u7acb\u65f6\u53d6\u5230\uff0c\u5373\u4e24\u4e2a\u590d\u7a84\u5e26\u4fe1\u53f7\u76f8\u5dee\u4e00\u4e2a\u786e\u5b9a\u590d\u5e38\u6570\uff08\u5e45\u5ea6\u548c\u76f8\u4f4d\uff09\u3002<\/p>\n

**\u8def\u5f84 B\uff08\u8c31\u77e9\u9635\u534a\u6b63\u5b9a\uff09**\uff1a\u628a $2\\times2$ \u8c31\u77e9\u9635<\/p>\n

$$
\n\\boldsymbol S(f)=
\n\\begin{bmatrix}
\nS_{xx}(f)&S_{xy}(f)\\\\S_{xy}^*(f)&S_{yy}(f)
\n\\end{bmatrix}
\n$$<\/p>\n

\u4ee3\u5165 $\\boldsymbol S(f)\\succeq 0$\uff0c\u5373\u6240\u6709\u4e3b\u5b50\u5f0f\u975e\u8d1f\u3002$1\\times1$ \u4e3b\u5b50\u5f0f\u975e\u8d1f\u7ed9\u51fa $S_{xx}(f)\\ge 0$\u3001$S_{yy}(f)\\ge 0$\uff1b$2\\times2$ \u884c\u5217\u5f0f\u975e\u8d1f\u7ed9\u51fa<\/p>\n

$$
\n\\det\\boldsymbol S(f)=S_{xx}(f)S_{yy}(f)-|S_{xy}(f)|^2\\ge0,
\n$$<\/p>\n

\u518d\u6b21\u5f97\u5230 $\\gamma_{xy}^2(f)\\le1$\u3002\u8fd9\u4e00\u8def\u5f84\u53ef\u81ea\u7136\u63a8\u5e7f\u5230\u591a\u901a\u9053\u7684**\u591a\u91cd\u76f8\u5e72\u6027**\u548c**\u504f\u76f8\u5e72\u6027**\uff1a\u90fd\u5bf9\u5e94\u628a\u8c31\u77e9\u9635\u6309 Schur \u8865\u5206\u5757\u540e\u4ecd\u4fdd\u6301\u534a\u6b63\u5b9a\u7684\u7ed3\u8bba\u3002<\/p>\n

\u4e24\u79cd\u8def\u5f84\u4e5f\u544a\u8bc9\u6211\u4eec\uff1a$\\gamma_{xy}^2(f)=1$ \u610f\u5473\u7740\u5728\u8be5\u9891\u7387\u4e0a $\\tilde Y(f)=H(f)\\tilde X(f)$ \u51e0\u4e4e\u5fc5\u7136\u6210\u7acb\uff0c$H(f)$ \u662f**\u590d\u7cfb\u6570**\uff0c\u53ef\u4ee5\u4efb\u610f\u6539\u53d8\u5e45\u5ea6\u548c\u76f8\u4f4d\u3002\u56e0\u6b64\u9ad8\u76f8\u5e72\u5e76\u4e0d\u8981\u6c42\u4e24\u4e2a\u4fe1\u53f7\u5728\u65f6\u57df\u4e00\u81f4\uff0c\u4e5f\u4e0d\u8981\u6c42\u5b58\u5728\u56e0\u679c\u5173\u7cfb\u3002<\/p>\n

—
\n### LTI \u566a\u58f0\u6a21\u578b\u4e0b\u7684\u76f8\u5e72\u6027\u2014\u4fe1\u566a\u6bd4\u516c\u5f0f<\/p>\n

\u7b2c\u4e03\u7ae0\u6b63\u6587\u7ed9\u51fa\u4e86<\/p>\n

$$
\n\\gamma_{xy}^2(f)=\\frac{\\operatorname{SNR}(f)}{1+\\operatorname{SNR}(f)}
\n$$<\/p>\n

\u8fd9\u4e00\u7ed3\u679c\uff0c\u672c\u8282\u628a\u5b83\u62c6\u89e3\u6210\u4e0d\u540c\u566a\u58f0\u4f4d\u7f6e\u4e0b\u7684\u901a\u7528\u5f62\u5f0f\uff0c\u65b9\u4fbf\u5b9e\u9645\u6d4b\u91cf\u3002\u4ecd\u8bbe\u8f93\u5165 $x$\u3001\u8f93\u51fa $y$\u3001\u566a\u58f0 $n$\uff0c$x\\perp n$\uff0c\u4e14\u7cfb\u7edf\u4e3a LTI\u3002<\/p>\n

**\u60c5\u5f62 1\uff1a\u566a\u58f0\u53ea\u52a0\u5728\u8f93\u51fa\u3002** $y=h*x+n_{\\mathrm o}$\u3002\u5b9a\u4e49\u8f93\u51fa\u4fe1\u566a\u6bd4<\/p>\n

$$
\n\\operatorname{SNR}_{\\mathrm o}(f)
\n=\\frac{|H(f)|^2 S_{xx}(f)}{S_{n_\\mathrm o n_\\mathrm o}(f)},
\n$$<\/p>\n

\u5219<\/p>\n

$$
\n\\gamma_{xy}^2(f)
\n=\\frac{\\operatorname{SNR}_\\mathrm o(f)}
\n{1+\\operatorname{SNR}_\\mathrm o(f)}.
\n$$<\/p>\n

\u8fd9\u662f\u6700\u5e38\u7528\u7684\u89e3\u91ca\uff0c\u4e14\u5f53\u8f93\u51fa\u5168\u90e8\u7531\u8f93\u5165\u7ebf\u6027\u54cd\u5e94\u4ea7\u751f\u65f6\u76f8\u5e72\u6027\u4e3a 1\u3002<\/p>\n

**\u60c5\u5f62 2\uff1a\u566a\u58f0\u53ea\u52a0\u5728\u8f93\u5165\u89c2\u6d4b\u3002** \u771f\u5b9e\u8f93\u5165 $x$\uff0c\u89c2\u6d4b\u5230 $\\tilde x=x+n_\\mathrm i$\uff1b\u8f93\u51fa $y=h*x$\u3002\u6b64\u65f6<\/p>\n

$$
\nS_{\\tilde x y}(f)=H^*(f)S_{xx}(f),
\n\\qquad
\nS_{\\tilde x\\tilde x}(f)=S_{xx}(f)+S_{n_\\mathrm i n_\\mathrm i}(f).
\n$$<\/p>\n

\u5bf9\u5e94\u76f8\u5e72\u6027<\/p>\n

$$
\n\\gamma_{\\tilde xy}^2(f)
\n=\\frac{S_{xx}(f)}
\n{S_{xx}(f)+S_{n_\\mathrm i n_\\mathrm i}(f)}
\n=\\frac{\\operatorname{SNR}_\\mathrm i(f)}
\n{1+\\operatorname{SNR}_\\mathrm i(f)}.
\n$$<\/p>\n

**\u60c5\u5f62 3\uff1a\u8f93\u5165\u3001\u8f93\u51fa\u89c2\u6d4b\u90fd\u6709\u566a\u58f0\u3002**<\/p>\n

$$
\n\\gamma_{\\tilde x\\tilde y}^2(f)
\n=\\frac{\\operatorname{SNR}_\\mathrm i\\operatorname{SNR}_\\mathrm o}
\n{(1+\\operatorname{SNR}_\\mathrm i)(1+\\operatorname{SNR}_\\mathrm o)}.
\n$$<\/p>\n

\u56e0\u6b64\u53cc\u7aef\u566a\u58f0\u4f1a\u628a\u53ef\u8fbe\u76f8\u5e72\u6027\u4e0a\u9650\u538b\u4f4e\uff0c\u5373\u4f7f\u771f\u5b9e\u7cfb\u7edf\u662f\u5b8c\u5168\u7ebf\u6027\u7684\u3002<\/p>\n

\u4ece\u8fd9\u51e0\u4e2a\u516c\u5f0f\u53ef\u4ee5\u8bfb\u51fa\u4e24\u70b9\u5de5\u7a0b\u542b\u4e49\uff1a<\/p>\n

1. \u89c2\u5bdf\u5230\u7684\u76f8\u5e72\u6027\u7f3a\u9677\u53ef\u4ee5\u6765\u81ea**\u4efb\u4e00\u7aef**\u7684\u566a\u58f0\u6216\u672a\u5efa\u6a21\u975e\u7ebf\u6027\uff0c\u4e0d\u80fd\u5355\u72ec\u5f52\u548e\u4e8e\u7cfb\u7edf\uff1b
\n2. \u82e5\u8981\u7528\u76f8\u5e72\u6027\u63a8 SNR\uff0c\u5e94\u5148\u786e\u8ba4\u566a\u58f0\u4e3b\u8981\u5728\u54ea\u4e00\u7aef\uff0c\u518d\u9009\u62e9 $H_1\/H_2\/H_v$ \u4e09\u79cd FRF \u4f30\u8ba1\u5668\u4e4b\u4e00\uff0c\u907f\u514d\u76f8\u5e72\u6027\u2014SNR \u5173\u7cfb\u88ab\u7cfb\u7edf\u6027\u5730\u8bef\u89e3\u91ca\u3002<\/p>\n

—
\n### \u56fa\u5b9a\u65f6\u5ef6\u4e0b\u7684\u76f8\u4f4d\u659c\u7387\u4e0e\u7fa4\u65f6\u5ef6\u9650\u5236<\/p>\n

\u82e5 $y(t)=x(t-\\tau_0)$\uff08\u566a\u58f0\u4e3a\u96f6\u7684\u7406\u60f3\u56fa\u5b9a\u65f6\u5ef6\uff09\uff0c\u5219 $S_{xy}(f)=S_{xx}(f)e^{-j2\\pi f\\tau_0}$\uff08\u5728 $S_{xy}=\\mathbb E[X_TY_T^*]\/T$ \u7ea6\u5b9a\u4e0b\uff09\uff0c\u4e92\u8c31\u76f8\u4f4d<\/p>\n

$$
\n\\phi_{xy}(f)=-2\\pi f\\tau_0
\n$$<\/p>\n

\u5173\u4e8e\u9891\u7387\u5448\u4e25\u683c\u7ebf\u6027\u3002\u636e\u6b64\u53ef\u7531\u76f8\u4f4d\u659c\u7387\u53cd\u63a8\u65f6\u5ef6\uff1a<\/p>\n

$$
\n\\tau_0=-\\frac{1}{2\\pi}\\frac{d\\phi_{xy}}{df}.
\n$$<\/p>\n

**\u9650\u5236\u4e00\uff1a\u4e3b\u503c\u88f9\u7ed5\u3002** \u76f8\u4f4d\u4e00\u822c\u4ee5 $(-\\pi,\\pi]$ \u6216 $[0,2\\pi)$ \u4e3b\u503c\u7ed9\u51fa\u3002\u5f53 $|2\\pi f\\tau_0|$ \u8d85\u8fc7 $\\pi$ \u65f6\uff0c\u76f8\u4f4d\u4f1a\u8de8\u8fc7 $\\pm\\pi$ \u7684\u8fb9\u754c\u53d1\u751f\u8df3\u53d8\u3002\u6709\u6548\u505a\u6cd5\u662f\u5148\u505a phase unwrapping\uff0c\u518d\u53d6\u659c\u7387\uff1b\u5426\u5219\u4f1a\u5f97\u5230\u63a5\u8fd1 0 \u7684\u9519\u8bef\u201c\u7fa4\u65f6\u5ef6\u201d\u3002<\/p>\n

**\u9650\u5236\u4e8c\uff1a\u5948\u594e\u65af\u7279\u76f8\u4f4d\u6a21\u7cca\u3002** \u6570\u5b57\u7cfb\u7edf\u53ea\u80fd\u5206\u8fa8 $|\\phi|\\le\\pi$ \u7684\u76f8\u4f4d\uff0c\u56e0\u6b64\u5355\u9891\u89c2\u5bdf\u5230\u7684\u65f6\u5ef6\u5b58\u5728\u6574\u5468\u671f\u6a21\u7cca\uff1a<\/p>\n

$$
\n\\tau_0\\in\\left\\{-\\frac{\\phi+2\\pi k}{2\\pi f}\\right\\}_{k\\in\\mathbb Z}.
\n$$<\/p>\n

\u8981\u6d88\u9664\u6a21\u7cca\uff0c\u9700\u8981\u5229\u7528\u591a\u4e2a\u9891\u7387\u4e0a\u7684\u76f8\u4f4d\u8054\u5408\u4f30\u8ba1\uff0c\u6216\u5728\u65f6\u57df\u7528\u4e92\u76f8\u5173\u5cf0\u505a\u7c97\u5b9a\u4f4d\u3002<\/p>\n

**\u9650\u5236\u4e09\uff1a\u5bbd\u5e26\u6781\u9650\u3002** \u6570\u5b57\u5bbd\u5e26\u7cfb\u7edf\u80fd\u53ef\u9760\u4f30\u8ba1\u7684\u6700\u5927\u65f6\u5ef6\u53d7\u5948\u594e\u65af\u7279\u901f\u7387\u7ea6\u675f\u3002\u82e5\u76f8\u4f4d\u5728\u76f8\u90bb FFT \u9891\u70b9\u95f4\u8df3\u53d8\u8d85\u8fc7 $2\\pi$\uff08\u5373 $2\\pi\\Delta f\\,\\tau_0>2\\pi$\uff0c$\\tau_0>1\/\\Delta f$\uff09\uff0cunwrapping \u4e5f\u65e0\u6cd5\u6062\u590d\u539f\u59cb\u659c\u7387\u3002\u6b64\u65f6\u5e94\u5148\u5728\u65f6\u57df\u4f7f\u7528 GCC \u5b9a\u4f4d\u4e00\u4e2a\u7c97\u7cd9\u5ef6\u8fdf\u5e76\u5bf9\u9f50\uff0c\u518d\u7528\u9ad8\u76f8\u5e72\u9891\u5e26\u7684\u76f8\u4f4d\u505a\u7cbe\u7ec6\u4f30\u8ba1\u3002<\/p>\n

**\u9650\u5236\u56db\uff1a\u4f4e\u76f8\u5e72\u9891\u6bb5\u4e0d\u53ef\u4fe1\u3002** \u7fa4\u65f6\u5ef6<\/p>\n

$$
\n\\tau_g(f)=-\\frac{1}{2\\pi}\\frac{d\\phi_{xy}(f)}{df}
\n$$<\/p>\n

\u5bf9\u76f8\u4f4d\u7684\u6570\u503c\u5fae\u5206\u5341\u5206\u654f\u611f\u3002\u5728 $\\gamma_{xy}^2(f)$ \u8f83\u4f4e\u7684\u9891\u6bb5\uff0c\u76f8\u4f4d\u4f30\u8ba1\u65b9\u5dee\u6309<\/p>\n

$$
\n\\operatorname{var}\\{\\widehat\\phi_{xy}(f)\\}
\n\\approx\\frac{1-\\gamma_{xy}^2(f)}{2K\\gamma_{xy}^2(f)}
\n$$<\/p>\n

\u653e\u5927\uff08$K$ \u4e3a\u7b49\u6548\u5e73\u5747\u6b21\u6570\uff09\uff0c\u5bfc\u81f4\u7fa4\u65f6\u5ef6\u66f2\u7ebf\u51fa\u73b0\u5de8\u5927\u566a\u523a\u3002\u5de5\u7a0b\u62a5\u544a\u5e94\u5728\u7fa4\u65f6\u5ef6\u56fe\u4e0a\u53e0\u52a0\u76f8\u5e72\u6027\u9608\u503c\uff0c\u53ea\u89e3\u91ca $\\gamma_{xy}^2\\ge\\gamma^2_{\\mathrm{crit}}$ \u9891\u6bb5\u5185\u7684\u7fa4\u65f6\u5ef6\u3002<\/p>\n

—<\/p>\n

—<\/p>\n

## \u516b\u3001\u8c31\u4e0e\u76f8\u5e72\u6027\u7684\u4f30\u8ba1\u65b9\u6cd5<\/p>\n

\u7406\u8bba\u8c31\u5bc6\u5ea6\u662f\u65e0\u9650\u6570\u636e\u6216\u7edf\u8ba1\u671f\u671b\u4e0b\u7684\u91cf\uff0c\u5de5\u7a0b\u4e2d\u5fc5\u987b\u4ece\u6709\u9650\u6570\u636e\u4f30\u8ba1\u3002\u76f8\u5e72\u6027\u662f\u8c31\u4f30\u8ba1\u7684\u6bd4\u503c\uff0c\u5bf9\u5206\u6bb5\u3001\u7a97\u51fd\u6570\u3001\u5e73\u5747\u6b21\u6570\u3001\u540c\u6b65\u548c\u9884\u5904\u7406\u5341\u5206\u654f\u611f\u3002<\/p>\n

### Welch \u65b9\u6cd5\u7684\u57fa\u672c\u6b65\u9aa4<\/p>\n

\u7ed9\u5b9a\u540c\u6b65\u91c7\u6837\u7684 $x[n]$ \u548c $y[n]$\uff1a<\/p>\n

1. \u53bb\u9664\u65e0\u6548\u6837\u672c\uff0c\u4fdd\u8bc1\u65f6\u95f4\u8f74\u4e25\u683c\u5bf9\u9f50\uff1b
\n2. \u6839\u636e\u9700\u8981\u53bb\u5747\u503c\u3001\u53bb\u8d8b\u52bf\uff1b
\n3. \u5c06\u8bb0\u5f55\u5212\u5206\u4e3a $K$ \u4e2a\u957f\u5ea6\u4e3a $L$ \u7684\u6570\u636e\u6bb5\uff1b
\n4. \u76f8\u90bb\u6570\u636e\u6bb5\u53ef\u4ee5\u91cd\u53e0\uff0c\u4f8b\u5982 50%\uff1b
\n5. \u6bcf\u6bb5\u4e58\u76f8\u540c\u7a97\u51fd\u6570 $w[n]$\uff1b
\n6. \u5206\u522b\u8ba1\u7b97\u6bcf\u6bb5 FFT\uff1a$X_r[k]$\u3001$Y_r[k]$\uff1b
\n7. \u5f62\u6210\u6bcf\u6bb5\u81ea\u8c31\u548c\u4e92\u8c31\uff1b
\n8. \u5bf9 $K$ \u6bb5\u6c42\u5e73\u5747\uff1b
\n9. \u7528\u5e73\u5747\u540e\u7684\u8c31\u8ba1\u7b97\u76f8\u5e72\u6027\u3002<\/p>\n

\u91c7\u7528\u672c\u6587\u4e92\u8c31\u7ea6\u5b9a\uff1a<\/p>\n

$$
\n\\widehat S_{xy}[k]
\n=\\frac{1}{K}
\n\\sum_{r=1}^{K}X_r[k]Y_r^*[k],
\n$$<\/p>\n

$$
\n\\widehat S_{xx}[k]
\n=\\frac{1}{K}
\n\\sum_{r=1}^{K}|X_r[k]|^2,
\n$$<\/p>\n

$$
\n\\widehat S_{yy}[k]
\n=\\frac{1}{K}
\n\\sum_{r=1}^{K}|Y_r[k]|^2.
\n$$<\/p>\n

\u6700\u540e\uff1a<\/p>\n

$$
\n\\boxed{
\n\\widehat\\gamma_{xy}^2[k]
\n=\\frac{|\\widehat S_{xy}[k]|^2}
\n{\\widehat S_{xx}[k]\\widehat S_{yy}[k]}
\n}.
\n$$<\/p>\n

\u5fc5\u987b\u5148\u5bf9\u8c31\u5e73\u5747\uff0c\u518d\u505a\u6bd4\u503c\uff1b\u4e0d\u80fd\u5148\u8ba1\u7b97\u6bcf\u6bb5\u76f8\u5e72\u6027\u518d\u7b80\u5355\u5e73\u5747\u3002<\/p>\n

### \u4e3a\u4ec0\u4e48\u4e0d\u80fd\u53ea\u7528\u4e00\u4e2a\u6570\u636e\u6bb5<\/p>\n

\u82e5\u53ea\u6709\u4e00\u4e2a\u672a\u7ecf\u5e73\u6ed1\u7684 FFT \u6bb5\uff1a<\/p>\n

$$
\n\\frac{|X[k]Y^*[k]|^2}
\n{|X[k]|^2|Y[k]|^2}=1
\n$$<\/p>\n

\u53ea\u8981\u5206\u6bcd\u975e\u96f6\uff0c\u6240\u6709\u9891\u70b9\u90fd\u4f1a\u5f97\u5230 1\u3002\u8fd9\u53ea\u662f\u4ee3\u6570\u6052\u7b49\u5f0f\uff0c\u4e0d\u4ee3\u8868\u771f\u5b9e\u4fe1\u53f7\u5b8c\u5168\u76f8\u5e72\u3002<\/p>\n

\u76f8\u5e72\u6027\u4f30\u8ba1\u9700\u8981\u8c31\u5e73\u5747\u6216\u9891\u7387\u5e73\u6ed1\uff0c\u4f7f\u8de8\u6bb5\u4e0d\u7a33\u5b9a\u7684\u76f8\u4f4d\u5173\u7cfb\u5728\u4e92\u8c31\u4e2d\u62b5\u6d88\uff0c\u800c\u7a33\u5b9a\u5173\u7cfb\u5f97\u5230\u4fdd\u7559\u3002<\/p>\n

### \u6570\u636e\u6bb5\u957f\u5ea6\u4e0e\u5e73\u5747\u6b21\u6570\u7684\u6743\u8861<\/p>\n

\u6bb5\u957f $L$ \u51b3\u5b9a\u8fd1\u4f3c\u9891\u7387\u5206\u8fa8\u7387\uff1a<\/p>\n

$$
\n\\Delta f=\\frac{f_s}{L},
\n$$<\/p>\n

\u5176\u4e2d $f_s$ \u662f\u91c7\u6837\u7387\u3002<\/p>\n

– \u6bb5\u8d8a\u957f\uff1a\u9891\u7387\u5206\u8fa8\u7387\u8d8a\u9ad8\uff0c\u4f46\u53ef\u5e73\u5747\u6bb5\u6570\u8d8a\u5c11\uff0c\u4f30\u8ba1\u65b9\u5dee\u8d8a\u5927\uff1b
\n– \u6bb5\u8d8a\u77ed\uff1a\u5e73\u5747\u6b21\u6570\u66f4\u591a\uff0c\u4f30\u8ba1\u66f4\u5e73\u6ed1\uff0c\u4f46\u76f8\u90bb\u9891\u7387\u6210\u5206\u53ef\u80fd\u65e0\u6cd5\u5206\u5f00\u3002<\/p>\n

\u5e94\u6839\u636e\u7cfb\u7edf\u6a21\u6001\u5e26\u5bbd\u3001\u4fe1\u53f7\u6301\u7eed\u65f6\u95f4\u548c\u6240\u9700\u7edf\u8ba1\u7f6e\u4fe1\u5ea6\u9009\u62e9\u6bb5\u957f\uff0c\u800c\u4e0d\u662f\u56fa\u5b9a\u5957\u7528\u67d0\u4e2a\u503c\u3002<\/p>\n

### \u7a97\u51fd\u6570\u4e0e\u8c31\u6cc4\u6f0f<\/p>\n

\u6709\u9650\u622a\u65ad\u4f1a\u5bfc\u81f4\u8c31\u6cc4\u6f0f\u3002Hann \u7a97\u5e38\u7528\u4e8e\u4e00\u822c\u8c31\u5206\u6790\uff0c\u56e0\u4e3a\u5b83\u5728\u4e3b\u74e3\u5bbd\u5ea6\u548c\u65c1\u74e3\u6291\u5236\u4e4b\u95f4\u8f83\u5747\u8861\u3002\u8f93\u5165\u548c\u8f93\u51fa\u5fc5\u987b\u91c7\u7528\u540c\u4e00\u7a97\u548c\u4e00\u81f4\u7684\u5206\u6bb5\u65b9\u5f0f\u3002<\/p>\n

\u7a97\u51fd\u6570\u4f1a\u4f7f\u76f8\u90bb\u9891\u70b9\u76f8\u5173\uff0c\u5e76\u6539\u53d8\u6709\u6548\u81ea\u7531\u5ea6\uff1b\u91cd\u53e0\u6bb5\u4e5f\u4e0d\u662f\u5b8c\u5168\u72ec\u7acb\u7684\u3002\u56e0\u6b64\uff0c\u4e0d\u80fd\u628a\u5b9e\u9645\u6bb5\u6570 $K$ \u76f4\u63a5\u89c6\u4e3a\u4e25\u683c\u72ec\u7acb\u6837\u672c\u6570\u3002<\/p>\n

### \u663e\u8457\u6027\u9608\u503c\u548c\u7f6e\u4fe1\u533a\u95f4<\/p>\n

\u5728\u201c\u771f\u5b9e\u76f8\u5e72\u6027\u4e3a\u96f6\u201d\u4e14\u6709 $K$ \u4e2a\u72ec\u7acb\u7b49\u6743\u8c31\u5e73\u5747\u7684\u7406\u60f3\u8fd1\u4f3c\u4e0b\uff0c\u663e\u8457\u6027\u6c34\u5e73 $\\alpha$ \u5bf9\u5e94\u7684 MSC \u9608\u503c\u5e38\u5199\u4e3a<\/p>\n

$$
\n\\boxed{
\n\\gamma_{\\mathrm{crit}}^2
\n=1-\\alpha^{1\/(K-1)}
\n}.
\n$$<\/p>\n

\u4f8b\u5982\u9009\u62e9 $\\alpha=0.05$\uff0c\u4f30\u8ba1\u503c\u9ad8\u4e8e\u8be5\u9608\u503c\u624d\u88ab\u89c6\u4e3a\u663e\u8457\u504f\u79bb\u96f6\u76f8\u5e72\u3002<\/p>\n

\u4f46\u5b9e\u9645 Welch \u91cd\u53e0\u3001\u7a97\u51fd\u6570\u3001\u9891\u7387\u5e73\u6ed1\u4f1a\u6539\u53d8\u6709\u6548\u81ea\u7531\u5ea6\uff0c\u516c\u5f0f\u4e2d\u7684 $K$ \u5e94\u7531\u6709\u6548\u72ec\u7acb\u5e73\u5747\u6b21\u6570\u6216\u7b49\u6548\u81ea\u7531\u5ea6\u66ff\u4ee3\u3002\u66f4\u4e25\u8c28\u65f6\u53ef\u4f7f\u7528\u8f6f\u4ef6\u7ed9\u51fa\u7684\u7f6e\u4fe1\u533a\u95f4\u3001jackknife\u3001bootstrap \u6216\u57fa\u4e8e surrogate data \u7684\u663e\u8457\u6027\u68c0\u9a8c\u3002<\/p>\n

### \u5e38\u89c1\u9884\u5904\u7406<\/p>\n

\u8ba1\u7b97\u76f8\u5173\u6027\u6216\u76f8\u5e72\u6027\u524d\uff0c\u5e94\u68c0\u67e5\uff1a<\/p>\n

– \u4e24\u901a\u9053\u662f\u5426\u4f7f\u7528\u76f8\u540c\u91c7\u6837\u7387\u548c\u65f6\u95f4\u6233\uff1b
\n– \u662f\u5426\u5b58\u5728\u6052\u5b9a\u6216\u53d8\u5316\u7684\u91c7\u6837\u5ef6\u8fdf\uff1b
\n– \u662f\u5426\u53bb\u9664\u5747\u503c\u3001\u7ebf\u6027\u8d8b\u52bf\u548c\u660e\u663e\u6f02\u79fb\uff1b
\n– \u662f\u5426\u5b58\u5728\u4e22\u6837\u3001\u63d2\u503c\u548c\u65f6\u949f\u6f02\u79fb\uff1b
\n– \u6297\u6df7\u53e0\u6ee4\u6ce2\u662f\u5426\u4e00\u81f4\uff1b
\n– \u4f20\u611f\u5668\u662f\u5426\u9971\u548c\u6216\u91cf\u5316\u4e0d\u8db3\uff1b
\n– \u662f\u5426\u9700\u8981\u5bf9\u5de5\u9891\u3001\u673a\u68b0\u8f6c\u9891\u7b49\u5df2\u77e5\u5e72\u6270\u5355\u72ec\u5904\u7406\uff1b
\n– \u6570\u636e\u662f\u5426\u8fd1\u4f3c\u5e73\u7a33\uff0c\u8fd8\u662f\u5e94\u5206\u5de5\u51b5\u5206\u6790\u3002<\/p>\n

\u4e0d\u8981\u4e3a\u4e86\u83b7\u5f97\u201c\u66f4\u6f02\u4eae\u201d\u7684\u76f8\u5e72\u66f2\u7ebf\u800c\u4efb\u610f\u6ee4\u6ce2\u3002\u4efb\u4f55\u6ee4\u6ce2\u90fd\u5e94\u540c\u65f6\u8003\u8651\u5bf9\u5e45\u5ea6\u3001\u76f8\u4f4d\u3001\u81ea\u7531\u5ea6\u548c\u7edf\u8ba1\u89e3\u91ca\u7684\u5f71\u54cd\u3002<\/p>\n

### \u76f8\u5e72\u6027\u56fe\u7684\u6b63\u786e\u9605\u8bfb\u65b9\u5f0f<\/p>\n

\u4e00\u5f20\u5b8c\u6574\u7684\u76f8\u5e72\u6027\u5206\u6790\u56fe\u901a\u5e38\u5e94\u8054\u5408\u5c55\u793a\uff1a<\/p>\n

1. $S_{xx}(f)$ \u548c $S_{yy}(f)$\uff1a\u786e\u8ba4\u8be5\u9891\u7387\u662f\u5426\u6709\u8db3\u591f\u80fd\u91cf\uff1b
\n2. $|S_{xy}(f)|$\uff1a\u89c2\u5bdf\u5171\u540c\u8c31\u6210\u5206\uff1b
\n3. $\\gamma_{xy}^2(f)$\uff1a\u89c2\u5bdf\u5f52\u4e00\u5316\u7ebf\u6027\u5173\u8054\u5f3a\u5ea6\uff1b
\n4. $\\angle S_{xy}(f)$\uff1a\u53ea\u5728\u9ad8\u76f8\u5e72\u533a\u89e3\u91ca\u76f8\u4f4d\uff1b
\n5. \u663e\u8457\u6027\u9608\u503c\u6216\u7f6e\u4fe1\u533a\u95f4\uff1b
\n6. \u91c7\u6837\u7387\u3001\u6bb5\u957f\u3001\u7a97\u3001\u91cd\u53e0\u7387\u548c\u5e73\u5747\u6b21\u6570\u3002<\/p>\n

\u5728\u4e24\u4e2a\u81ea\u8c31\u90fd\u63a5\u8fd1\u566a\u58f0\u5e95\u65f6\uff0c\u5373\u4f7f\u76f8\u5e72\u6027\u5076\u7136\u504f\u9ad8\uff0c\u4e5f\u5e94\u8c28\u614e\u89e3\u91ca\u3002\u76f8\u5e72\u6027\u662f\u5f52\u4e00\u5316\u6bd4\u503c\uff0c\u4e0d\u80fd\u66ff\u4ee3\u7edd\u5bf9\u529f\u7387\u5206\u6790\u3002<\/p>\n

### \u76f8\u5e72\u6027\u5206\u6790\u7684\u62a5\u544a\u6a21\u677f<\/p>\n

\u5de5\u7a0b\u62a5\u544a\u81f3\u5c11\u5e94\u5199\u6e05\uff1a<\/p>\n

– \u6570\u636e\u65f6\u957f\u4e0e\u91c7\u6837\u7387\uff1b
\n– \u8f93\u5165\u548c\u8f93\u51fa\u901a\u9053\u7684\u5355\u4f4d\u53ca\u6821\u51c6\u4fe1\u606f\uff1b
\n– \u53bb\u8d8b\u52bf\u3001\u6ee4\u6ce2\u548c\u5f02\u5e38\u503c\u5904\u7406\uff1b
\n– \u7a97\u51fd\u6570\u3001\u6bb5\u957f\u3001\u91cd\u53e0\u7387\u3001FFT \u957f\u5ea6\uff1b
\n– \u5355\u8fb9\u8c31\u8fd8\u662f\u53cc\u8fb9\u8c31\uff1b
\n– \u8c31\u5e73\u5747\u65b9\u6cd5\u548c\u6709\u6548\u81ea\u7531\u5ea6\uff1b
\n– \u76f8\u5e72\u6027\u5b9a\u4e49\u53ca\u4e92\u8c31\u5171\u8f6d\u7ea6\u5b9a\uff1b
\n– \u663e\u8457\u6027\u9608\u503c\u6216\u7f6e\u4fe1\u533a\u95f4\uff1b
\n– \u54ea\u4e9b\u9891\u5e26\u53ef\u89e3\u91ca\uff0c\u4ee5\u53ca\u54ea\u4e9b\u9891\u5e26\u56e0\u4f4e\u529f\u7387\u6216\u4f4e\u76f8\u5e72\u4e0d\u53ef\u89e3\u91ca\u3002<\/p>\n

—
\n### \u8c31\u4f30\u8ba1\u65b9\u6cd5\u4e0e\u5de5\u7a0b\u5b9e\u8df5<\/p>\n

#### Periodogram<\/p>\n

\u6709\u9650\u8bb0\u5f55 $x[n]$ \u7684 periodogram \u53ef\u5199\u6210<\/p>\n

$$
\n\\widehat S_{xx}^{\\mathrm{per}}(f)
\n=\\frac{1}{Nf_s}\\left|
\n\\sum_{n=0}^{N-1}x[n]e^{-j2\\pi fn\/f_s}
\n\\right|^2.
\n$$<\/p>\n

\u8fd9\u91cc\u91c7\u7528\u7684\u5f52\u4e00\u5316\u4f7f\u8c31\u5bf9\u9891\u7387\u79ef\u5206\u8fd1\u4f3c\u7b49\u4e8e\u4fe1\u53f7\u7684\u5e73\u5747\u5e73\u65b9\u503c\u3002\u82e5\u4f7f\u7528\u7a97\u51fd\u6570 $w[n]$\uff1a<\/p>\n

$$
\n\\widehat S_{xx}(f)
\n=\\frac{1}{f_s U}
\n\\left|\\sum_{n=0}^{N-1}w[n]x[n]e^{-j2\\pi fn\/f_s}\\right|^2,
\n$$<\/p>\n

\u5176\u4e2d<\/p>\n

$$
\nU=\\frac1N\\sum_{n=0}^{N-1}|w[n]|^2.
\n$$<\/p>\n

\u4e0d\u540c\u8f6f\u4ef6\u53ef\u80fd\u628a $N$\u3001$f_s$\u3001\u7a97\u529f\u7387\u548c FFT \u957f\u5ea6\u653e\u5728\u4e0d\u540c\u4f4d\u7f6e\uff0c\u56e0\u6b64\u5e94\u4ee5\u201c\u79ef\u5206\u662f\u5426\u6062\u590d\u5e73\u5747\u529f\u7387\u201d\u4e3a\u5f52\u4e00\u5316\u68c0\u67e5\u3002<\/p>\n

#### Bartlett \u4e0e Welch \u65b9\u6cd5<\/p>\n

Bartlett \u65b9\u6cd5\u5c06\u6570\u636e\u5206\u6210\u4e0d\u91cd\u53e0\u7684\u82e5\u5e72\u6bb5\uff0c\u5206\u522b\u8ba1\u7b97 periodogram \u540e\u5e73\u5747\u3002Welch \u65b9\u6cd5\u8fdb\u4e00\u6b65\u5141\u8bb8\uff1a<\/p>\n

– \u6bb5\u95f4\u91cd\u53e0\uff1b
\n– \u6bcf\u6bb5\u4f7f\u7528\u7a97\u51fd\u6570\uff1b
\n– \u5bf9\u7a97\u529f\u7387\u505a\u5f52\u4e00\u5316\u3002<\/p>\n

\u5e73\u5747\u53ef\u4ee5\u964d\u4f4e\u8c31\u4f30\u8ba1\u65b9\u5dee\uff0c\u4f46\u4f1a\u727a\u7272\u90e8\u5206\u9891\u7387\u5206\u8fa8\u7387\u3002Welch \u65b9\u6cd5\u540c\u6837\u9002\u7528\u4e8e\u4e92\u529f\u7387\u8c31\uff1a<\/p>\n

$$
\n\\widehat S_{xy}(f)
\n=\\frac{1}{K}\\sum_{r=1}^{K}
\nX_r(f)Y_r^*(f).
\n$$<\/p>\n

#### \u81ea\u56de\u5f52\u8c31\u4f30\u8ba1<\/p>\n

\u53c2\u6570\u5316\u65b9\u6cd5\u5047\u8bbe\u6570\u636e\u53ef\u7531 AR \u6a21\u578b\u63cf\u8ff0\uff1a<\/p>\n

$$
\n x[n]+\\sum_{k=1}^{p}a_kx[n-k]=u[n],
\n$$<\/p>\n

\u5176\u4e2d $u[n]$ \u4e3a\u767d\u566a\u58f0\u3002\u82e5\u5176\u65b9\u5dee\u4e3a $\\sigma_u^2$\uff0c\u5219 PSD \u5f62\u5f0f\u4e3a<\/p>\n

$$
\nS_{xx}(e^{j\\omega})
\n=\\frac{\\sigma_u^2}
\n{|1+\\sum_{k=1}^{p}a_ke^{-j\\omega k}|^2}.
\n$$<\/p>\n

AR \u8c31\u53ef\u4ee5\u5728\u77ed\u6570\u636e\u4e0b\u63d0\u4f9b\u8f83\u9ad8\u7684\u9891\u7387\u5206\u8fa8\u7387\uff0c\u4f46\u6a21\u578b\u9636\u6570\u4e0d\u5408\u9002\u65f6\u4f1a\u4ea7\u751f\u4f2a\u5cf0\u6216\u8fc7\u5ea6\u5e73\u6ed1\u3002\u5b83\u4e0d\u5e94\u4e0e\u975e\u53c2\u6570 Welch \u8c31\u65e0\u6761\u4ef6\u6df7\u7528\u3002<\/p>\n

#### ESD\u3001PSD \u4e0e periodogram \u7684\u5f52\u4e00\u5316\u68c0\u67e5<\/p>\n

\u5bf9\u6709\u9650\u8bb0\u5f55\uff0c\u9700\u8981\u5148\u660e\u786e\u76ee\u6807\uff1a<\/p>\n

– \u82e5\u8981\u4f30\u8ba1\u8fd9\u6bb5\u8bb0\u5f55\u643a\u5e26\u7684\u603b\u80fd\u91cf\uff0c\u4f7f\u7528 ESD\uff0c\u5e76\u4f7f\u9891\u7387\u79ef\u5206\u5f97\u5230\u80fd\u91cf\uff1b
\n– \u82e5\u8981\u4f30\u8ba1\u957f\u671f\u5e73\u5747\u529f\u7387\u6216\u968f\u673a\u8fc7\u7a0b PSD\uff0c\u4f7f\u7528\u6309\u8bb0\u5f55\u65f6\u957f\u3001\u91c7\u6837\u7387\u548c\u7a97\u80fd\u91cf\u5f52\u4e00\u5316\u7684 PSD\uff1b
\n– \u82e5\u53ea\u662f\u6bd4\u8f83\u76f8\u5bf9\u5cf0\u503c\uff0c\u53ef\u4ee5\u4f7f\u7528\u672a\u7edd\u5bf9\u6821\u51c6\u7684 periodogram\uff0c\u4f46\u4e0d\u80fd\u628a\u7eb5\u8f74\u76f4\u63a5\u89e3\u91ca\u4e3a\u7269\u7406\u529f\u7387\u5bc6\u5ea6\u3002<\/p>\n

\u79bb\u6563\u9891\u7387\u95f4\u9694\u4e3a<\/p>\n

$$
\n\\Delta f=\\frac{f_s}{N_{\\mathrm{FFT}}}.
\n$$<\/p>\n

\u82e5 PSD \u7684\u5355\u4f4d\u4e3a $\\mathrm{V}^2\/\\mathrm{Hz}$\uff0c\u9891\u5e26\u529f\u7387\u5e94\u8fd1\u4f3c\u4e3a<\/p>\n

$$
\nP_{[f_1,f_2]}
\n\\approx\\sum_{f_k\\in[f_1,f_2]}
\n\\widehat S_{xx}(f_k)\\Delta f.
\n$$<\/p>\n

\u5c11\u4e58\u4e00\u4e2a $\\Delta f$ \u4f1a\u628a\u201c\u8c31\u5bc6\u5ea6\u201d\u8bef\u5f53\u6210\u201c\u9891\u70b9\u529f\u7387\u201d\u3002<\/p>\n

#### \u5355\u8fb9\u8c31\u4e0e\u53cc\u8fb9\u8c31<\/p>\n

\u5bf9\u4e8e\u5b9e\u4fe1\u53f7\uff0c\u53cc\u8fb9\u8c31\u5728\u6b63\u8d1f\u9891\u7387\u4e0a\u5bf9\u79f0\u3002\u5355\u8fb9 PSD \u53ea\u4fdd\u7559 $0\\le f\\le f_s\/2$\uff0c\u5e76\u5c06\u975e DC\u3001\u975e Nyquist \u9891\u70b9\u7684\u529f\u7387\u4e58 2\uff1a<\/p>\n

$$
\n\\widehat S_{\\mathrm{one-sided}}(f)
\n=2\\widehat S_{\\mathrm{two-sided}}(f).
\n$$<\/p>\n

\u8fd9\u4e00\u64cd\u4f5c\u4fdd\u8bc1\u5355\u8fb9\u79ef\u5206\u4e0e\u53cc\u8fb9\u79ef\u5206\u76f8\u540c\u3002\u5bf9\u590d\u4fe1\u53f7\u6216\u89e3\u6790\u4fe1\u53f7\uff0c\u4e0d\u80fd\u9ed8\u8ba4\u4e58 2\uff0c\u56e0\u4e3a\u5176\u8d1f\u9891\u7387\u4e0d\u4e00\u5b9a\u662f\u6b63\u9891\u7387\u7684\u91cd\u590d\u4fe1\u606f\u3002<\/p>\n

#### \u9891\u7387\u5206\u8fa8\u7387\u3001\u7a97\u4e3b\u74e3\u4e0e\u6cc4\u6f0f<\/p>\n

FFT \u9891\u70b9\u95f4\u9694<\/p>\n

$$
\n\\Delta f=\\frac{f_s}{N_{\\mathrm{FFT}}}
\n$$<\/p>\n

\u4e0d\u7b49\u4e8e\u771f\u6b63\u7684\u53ef\u5206\u8fa8\u9891\u7387\u95f4\u9694\u3002\u771f\u6b63\u5206\u8fa8\u80fd\u529b\u8fd8\u53d6\u51b3\u4e8e\u6709\u6548\u89c2\u6d4b\u65f6\u957f\u548c\u7a97\u51fd\u6570\u4e3b\u74e3\u5bbd\u5ea6\u3002<\/p>\n

– \u589e\u5927\u96f6\u586b\u5145\uff1a\u8ba9\u66f2\u7ebf\u91c7\u6837\u66f4\u5bc6\uff0c\u4f46\u4e0d\u63d0\u5347\u771f\u5b9e\u5206\u8fa8\u7387\uff1b
\n– \u589e\u5927\u6709\u6548\u89c2\u6d4b\u65f6\u957f\uff1a\u901a\u5e38\u63d0\u5347\u9891\u7387\u5206\u8fa8\u7387\uff1b
\n– \u964d\u4f4e\u7a97\u65c1\u74e3\uff1a\u51cf\u5c11\u6cc4\u6f0f\uff0c\u4f46\u901a\u5e38\u589e\u5bbd\u4e3b\u74e3\uff1b
\n– \u4e24\u4e2a\u63a5\u8fd1\u9891\u7387\u7684\u5cf0\u80fd\u5426\u5206\u5f00\uff0c\u53d6\u51b3\u4e8e\u7a97\u548c\u6570\u636e\u957f\u5ea6\uff0c\u4e0d\u4ec5\u53d6\u51b3\u4e8e FFT \u70b9\u6570\u3002<\/p>\n

#### \u4e92\u8c31\u4f30\u8ba1\u7684\u5bf9\u9f50\u8981\u6c42<\/p>\n

\u8ba1\u7b97\u4e92\u529f\u7387\u8c31\u524d\u5e94\u4fdd\u8bc1\uff1a<\/p>\n

1. \u4e24\u4e2a\u901a\u9053\u6765\u81ea\u540c\u4e00\u65f6\u95f4\u533a\u95f4\uff1b
\n2. \u91c7\u6837\u7387\u548c\u91c7\u6837\u65f6\u523b\u4e00\u81f4\uff1b
\n3. \u5206\u6bb5\u8fb9\u754c\u3001\u7a97\u51fd\u6570\u548c FFT \u957f\u5ea6\u4e00\u81f4\uff1b
\n4. \u901a\u9053\u5ef6\u8fdf\u6ca1\u6709\u88ab\u672a\u77e5\u7f13\u5b58\u6216\u786c\u4ef6\u6ee4\u6ce2\u6539\u53d8\uff1b
\n5. \u672a\u5728\u4e00\u4e2a\u901a\u9053\u4e0a\u4f7f\u7528\u4e0d\u540c\u7684\u53bb\u8d8b\u52bf\u65b9\u5f0f\uff1b
\n6. \u9891\u7387\u8f74\u548c\u5355\u8fb9\/\u53cc\u8fb9\u7ea6\u5b9a\u76f8\u540c\u3002<\/p>\n

\u5e45\u5ea6\u770b\u8d77\u6765\u5408\u7406\u4f46\u901a\u9053\u9519\u4f4d\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e92\u8c31\u76f8\u4f4d\u9519\u8bef\u3001\u76f8\u5e72\u6027\u964d\u4f4e\u548c\u4f20\u9012\u51fd\u6570\u504f\u5dee\u3002<\/p>\n

#### \u9891\u5e26\u79ef\u5206\u548c\u5e26\u5185\u529f\u7387<\/p>\n

\u5bf9\u4e8e PSD\uff0c\u9891\u5e26 $[f_1,f_2]$ \u5185\u7684\u5e73\u5747\u529f\u7387\u4e3a<\/p>\n

$$
\nP_{[f_1,f_2]}
\n=\\int_{f_1}^{f_2}S_{xx}(f)df.
\n$$<\/p>\n

\u79bb\u6563\u4f30\u8ba1\u4e2d\u4f7f\u7528\u9891\u7387\u95f4\u9694\u52a0\u6743\uff1a<\/p>\n

$$
\n\\widehat P_{[f_1,f_2]}
\n=\\sum_{k:f_k\\in[f_1,f_2]}
\n\\widehat S_{xx}(f_k)\\Delta f.
\n$$<\/p>\n

\u5e45\u5ea6\u8c31\u7684\u5cf0\u503c\u4e0d\u80fd\u76f4\u63a5\u5f53\u4f5c\u5e26\u5185\u529f\u7387\uff1b\u5e26\u5185\u529f\u7387\u6765\u81ea\u5e73\u65b9\u91cf\u7684\u79ef\u5206\u3002<\/p>\n

#### dB \u8868\u793a<\/p>\n

\u529f\u7387\u8c31\u5e38\u4f7f\u7528<\/p>\n

$$
\nS_{\\mathrm{dB}}(f)=10\\log_{10}\\frac{S_{xx}(f)}{S_{\\mathrm{ref}}(f)}.
\n$$<\/p>\n

\u5e45\u5ea6\u8c31\u5e38\u4f7f\u7528<\/p>\n

$$
\nA_{\\mathrm{dB}}(f)=20\\log_{10}\\frac{|X(f)|}{A_{\\mathrm{ref}}}.
\n$$<\/p>\n

\u56e0\u4e3a\u529f\u7387\u4e0e\u5e45\u5ea6\u5e73\u65b9\u6210\u6b63\u6bd4\uff0c\u6240\u4ee5 $10\\log_{10}$ \u529f\u7387\u548c $20\\log_{10}$ \u5e45\u5ea6\u5728\u53c2\u8003\u91cf\u5339\u914d\u65f6\u7b49\u4ef7\u3002\u5bf9 PSD \u4f7f\u7528 dB\/Hz \u65f6\uff0c\u8981\u660e\u786e\u53c2\u8003 PSD\uff1b\u5bf9\u5e26\u5185\u79ef\u5206\uff0c\u4e0d\u80fd\u5148\u7b80\u5355\u5e73\u5747 dB \u503c\u518d\u5f53\u4f5c\u7ebf\u6027\u529f\u7387\u3002<\/p>\n

#### \u5e38\u89c1\u8c31\u4f30\u8ba1\u6545\u969c<\/p>\n

– \u628a $|X|$ \u5f53\u4f5c PSD\uff0c\u6f0f\u6389\u5e73\u65b9\u548c\u5f52\u4e00\u5316\uff1b
\n– \u5c06\u6709\u9650\u8bb0\u5f55 ESD \u4e0e\u957f\u671f PSD \u6df7\u79f0\uff1b
\n– \u5fd8\u8bb0\u4e58\u9891\u7387\u95f4\u9694\uff0c\u5bfc\u81f4\u79ef\u5206\u529f\u7387\u9519\u8bef\uff1b
\n– \u5355\u8fb9\u8c31\u6ca1\u6709\u6b63\u786e\u6298\u53e0\u8d1f\u9891\u7387\uff1b
\n– \u8bef\u4ee5\u4e3a\u96f6\u586b\u5145\u589e\u52a0\u4e86\u80fd\u91cf\u6216\u5206\u8fa8\u7387\uff1b
\n– \u4f7f\u7528\u5355\u6bb5 FFT \u4f30\u8ba1\u76f8\u5e72\u6027\uff0c\u5f97\u5230\u865a\u5047\u7684 1\uff1b
\n– \u4e92\u8c31\u901a\u9053\u987a\u5e8f\u548c\u7406\u8bba\u516c\u5f0f\u76f8\u53cd\uff0c\u5bfc\u81f4\u4f20\u9012\u51fd\u6570\u76f8\u4f4d\u53d6\u53cd\uff1b
\n– \u5ffd\u7565\u7a97\u51fd\u6570\u7684\u529f\u7387\u5f52\u4e00\u5316\uff1b
\n– \u5728\u4f4e\u81ea\u8c31\u529f\u7387\u9891\u5e26\u8fc7\u5ea6\u89e3\u91ca\u4e92\u8c31\u76f8\u4f4d\u548c\u76f8\u5e72\u6027\uff1b
\n– \u5bf9\u975e\u5e73\u7a33\u4fe1\u53f7\u4f7f\u7528\u6574\u6bb5\u5355\u4e00 PSD\uff0c\u63a9\u76d6\u5de5\u51b5\u53d8\u5316\u3002<\/p>\n

—
\n### Periodogram \u7684\u504f\u5dee\u4e0e\u65b9\u5dee\u6027\u8d28<\/p>\n

\u5355\u6bb5 periodogram \u662f\u6700\u57fa\u7840\u7684 PSD \u4f30\u8ba1\uff0c\u4f46\u5176\u7edf\u8ba1\u6027\u8d28\u5e76\u4e0d\u7406\u60f3\u3002\u8bbe $x[n]$ \u4e3a\u96f6\u5747\u503c\u5bbd\u5e73\u7a33\u8fc7\u7a0b\u3001\u771f\u5b9e\u8c31\u4e3a $S_{xx}(f)$\uff0c\u52a0\u7a97 periodogram<\/p>\n

$$
\n\\widehat S_{xx}^{\\mathrm{per}}(f)
\n=\\frac{1}{f_sU N}\\left|\\sum_{n=0}^{N-1}w[n]x[n]e^{-j2\\pi fn\/f_s}\\right|^2
\n$$<\/p>\n

\u6709\u5982\u4e0b\u5e38\u7528\u8fd1\u4f3c\u6027\u8d28\uff1a<\/p>\n

– **\u504f\u5dee**\uff1a$\\mathbb E[\\widehat S_{xx}^{\\mathrm{per}}(f)]=(S_{xx}*W)(f)$\uff0c\u5176\u4e2d $W$ \u662f\u7a97\u7684\u529f\u7387\u8c31\u3002\u7a97\u4e3b\u74e3\u5bbd\u5ea6\u51b3\u5b9a\u4e86\u5c16\u5cf0\u7684\u5c55\u5bbd\uff0c\u65c1\u74e3\u51b3\u5b9a\u4e86\u8fdc\u79bb\u5cf0\u503c\u5904\u7684\u6cc4\u6f0f\u504f\u5dee\u3002$N\\to\\infty$ \u4e14\u7a97\u6ee1\u8db3\u4e00\u5b9a\u6761\u4ef6\u65f6\u4f30\u8ba1\u6e10\u8fd1\u65e0\u504f\u3002
\n– **\u65b9\u5dee**\uff1a\u5bf9\u8fde\u7eed\u8c31\u533a\u6bb5\u7684\u9ad8\u65af\u8fc7\u7a0b\uff0c<\/p>\n

$$
\n \\operatorname{var}\\{\\widehat S_{xx}^{\\mathrm{per}}(f)\\}\\approx S_{xx}^2(f),
\n $$<\/p>\n

\u4e0e $N$ **\u65e0\u5173**\u3002\u56e0\u6b64\u589e\u52a0\u6570\u636e\u957f\u5ea6\u53ef\u4ee5\u6539\u5584\u5206\u8fa8\u7387\uff0c\u4f46**\u4e0d\u80fd**\u5355\u9760\u52a0\u957f\u8bb0\u5f55\u964d\u4f4e\u65b9\u5dee\u3002<\/p>\n

– **\u76f8\u90bb\u9891\u7387\u76f8\u5173\u6027**\uff1a\u5355\u6bb5 periodogram \u7684 $\\widehat S_{xx}(f_k)$ \u5728\u4e0d\u540c FFT \u9891\u70b9\u4e0a\u8fd1\u4f3c\u4e0d\u76f8\u5173\uff08\u77e9\u5f62\u7a97\u3001\u767d\u566a\u58f0\uff09\u3002\u8fd9\u662f Bartlett\/Welch \u4e4b\u6240\u4ee5\u80fd\u901a\u8fc7\u9891\u6bb5\u5e73\u6ed1\u964d\u4f4e\u65b9\u5dee\u7684\u57fa\u7840\u3002<\/p>\n

\u540e\u679c\u662f\uff1a\u5355\u6bb5 periodogram \u5728\u771f\u8c31\u4e0a\u4e0b\u6d6e\u52a8\u7ea6 $\\pm100\\%$\uff0c\u770b\u4e0a\u53bb\u201c\u6bdb\u523a\u5f88\u5bc6\u201d\uff0c\u5bb9\u6613\u88ab\u8bef\u8bfb\u4e3a\u8bb8\u591a\u5c0f\u5cf0\u3002\u964d\u4f4e\u65b9\u5dee\u7684\u552f\u4e00\u624b\u6bb5\u662f**\u5e73\u5747**\uff08\u5206\u6bb5\u65f6\u95f4\u5e73\u5747\u6216\u9891\u57df\u90bb\u57df\u5e73\u5747\uff09\uff0c\u8fd9\u6b63\u662f\u4e0b\u4e00\u8282 Welch \u65b9\u6cd5\u7684\u52a8\u673a\u3002<\/p>\n

\u4e00\u81f4\u6027\u65b9\u9762\uff0cWelch \u65b9\u6cd5\u5728 $K\\to\\infty$\u3001\u6bcf\u6bb5\u957f $L$ \u56fa\u5b9a\u7684\u6781\u9650\u4e0b\u65b9\u5dee\u6309 $\\sim S^2\/K$ \u8870\u51cf\uff0c\u4f46\u4e3b\u74e3\u5bbd\u5ea6\u7531 $L$ \u51b3\u5b9a\u4e0d\u518d\u53d8\u5316\uff1b\u53ea\u6709\u518d\u8ba9 $L\\to\\infty$\uff08\u540c\u65f6\u4fdd\u8bc1 $K\/L\\to0$ \u4e00\u7c7b\u6761\u4ef6\uff09\uff0cWelch \u4f30\u8ba1\u624d\u662f\u65e2\u65e0\u504f\u53c8\u4e00\u81f4\u7684\u3002\u5de5\u7a0b\u4e0a\u65e0\u6cd5\u540c\u65f6\u628a $L$ \u548c $K$ \u63a8\u5230\u65e0\u7a77\uff0c\u5fc5\u987b\u5728**\u5206\u8fa8\u7387**\u4e0e**\u65b9\u5dee**\u4e4b\u95f4\u6743\u8861\u3002<\/p>\n

—
\n### Welch \u53c2\u6570\u9009\u62e9\u7684\u5b9e\u7528\u51c6\u5219<\/p>\n

Welch \u65b9\u6cd5\u6709\u56db\u4e2a\u4e3b\u8981\u65cb\u94ae\uff1a\u6bb5\u957f $L$\u3001\u91cd\u53e0\u7387 $\\alpha$\u3001\u7a97\u51fd\u6570 $w[n]$\u3001FFT \u957f\u5ea6 $N_{\\mathrm{FFT}}$\u3002\u9009\u62e9\u65f6\u53ef\u6309\u4ee5\u4e0b\u51c6\u5219\uff1a<\/p>\n

**\u6bb5\u957f $L$**\uff1a\u7531\u76ee\u6807\u9891\u7387\u5206\u8fa8\u7387\u53cd\u63a8\u3002\u82e5\u8981\u6c42 3 dB \u5206\u8fa8\u7387 $\\Delta f_{3\\mathrm{dB}}$\u3001\u9009\u62e9\u7684\u7a97\u4e3b\u74e3\u7cfb\u6570\u4e3a $\\beta_w$\uff08Hann \u7ea6 1.5\u3001Hamming \u7ea6 1.36\u3001Blackman\u2013Harris \u7ea6 2.0\uff09\uff1a<\/p>\n

$$
\nL\\ge\\frac{\\beta_w f_s}{\\Delta f_{3\\mathrm{dB}}}.
\n$$<\/p>\n

\u7a84\u5e26\u7cbe\u7ec6\u8c31\u5e94\u53d6\u5927 $L$\uff0c\u5bbd\u5e26 PSD \u6216\u7cfb\u7edf\u8fa8\u8bc6\u53ef\u53d6\u66f4\u5c0f\u7684 $L$ \u6362\u53d6\u66f4\u591a\u5e73\u5747\u3002<\/p>\n

**\u91cd\u53e0\u7387 $\\alpha$**\uff1aHann \u7a97\u5e38\u7528 50%\uff0cHamming \u5e38\u7528 50%\uff0cBlackman\u2013Harris \u5e38\u7528 75%\u3002\u91cd\u53e0\u53ef\u63d0\u9ad8\u5bf9\u8fb9\u7f18\u6837\u672c\u7684\u5229\u7528\u7387\u5e76\u51cf\u5c0f\u65b9\u5dee\uff0c\u4f46\u4e0d\u80fd\u89c6\u4e3a\u589e\u52a0\u72ec\u7acb\u6837\u672c\u6570\u3002\u7b49\u6548\u72ec\u7acb\u6bb5\u6570\u8fd1\u4f3c\u4e3a<\/p>\n

$$
\nK_{\\mathrm{eff}}
\n\\approx\\frac{K}{1+2\\sum_{r=1}^{R-1}\\rho_w^2(r)},
\n$$<\/p>\n

\u5176\u4e2d $\\rho_w(r)$ \u662f\u7a97\u5728\u504f\u79fb $r$ \u6bb5\u957f\u7684\u5f52\u4e00\u5316\u91cd\u53e0\u7cfb\u6570\uff1bHann 50% \u91cd\u53e0\u65f6 $K_{\\mathrm{eff}}\\approx K\/1.34$\u3002<\/p>\n

**\u7a97\u51fd\u6570 $w[n]$**\uff1aHann \u4e3b\u74e3\u8f83\u5bbd\u3001\u65c1\u74e3\u8870\u51cf\u5feb\uff0c\u5e38\u7528\u4e8e\u4e00\u822c\u8c31\u5206\u6790\uff1bHamming \u4e3b\u74e3\u7565\u7a84\u4f46\u65c1\u74e3\u4e0d\u8870\u51cf\u66f4\u5feb\uff0c\u9002\u5408\u8c10\u6ce2\u5bc6\u96c6\u573a\u5408\uff1bBlackman\u2013Harris\/Flattop \u9002\u5408\u5e45\u5ea6\u6821\u51c6\uff1b\u77e9\u5f62\u7a97\u51e0\u4e4e\u53ea\u5728\u6b63\u5f26\u9891\u7387\u5bf9\u9f50 bin \u65f6\u4f7f\u7528\u3002**\u8f93\u5165\u3001\u8f93\u51fa\u901a\u9053\u5fc5\u987b\u4f7f\u7528\u76f8\u540c\u7684\u7a97**\uff0c\u5426\u5219\u4e92\u8c31\u5e45\u76f8\u90fd\u4f1a\u88ab\u5f15\u5165\u7cfb\u7edf\u8bef\u5dee\u3002<\/p>\n

**FFT \u957f\u5ea6 $N_{\\mathrm{FFT}}$**\uff1a$\\ge L$\uff0c\u901a\u5e38\u53d6 $L$ \u7684 2 \u7684\u5e42\u96f6\u586b\u5145\u500d\u6570\u3002\u96f6\u586b\u5145\u53ea\u8ba9\u66f2\u7ebf\u89c6\u89c9\u4e0a\u66f4\u5bc6\uff0c\u4e0d\u63d0\u5347\u53ef\u5206\u8fa8\u7387\u3002<\/p>\n

**\u603b\u8bb0\u5f55\u65f6\u957f**\uff1a\u7531\u6240\u9700\u81ea\u7531\u5ea6\u53cd\u63a8\u3002\u82e5\u8981\u6c42\u7b49\u6548\u81ea\u7531\u5ea6 $\\nu$\uff08Welch \u6bcf\u6bb5\u81ea\u7531\u5ea6\u7ea6\u4e3a 2\uff09\uff1a<\/p>\n

$$
\nN_{\\mathrm{total}}
\n\\ge\\frac{\\nu L(1-\\alpha)^{-1}}{2},
\n$$<\/p>\n

\u518d\u5bf9\u7167\u5b9e\u9645\u80fd\u83b7\u53d6\u7684\u8bb0\u5f55\u65f6\u957f\u786e\u8ba4\u662f\u5426\u53ef\u884c\u3002<\/p>\n

\u4e0a\u8ff0\u51c6\u5219\u5e94\u5728\u5177\u4f53\u7cfb\u7edf\u4e0a\u505a\u5c0f\u89c4\u6a21\u8bd5\u7b97\uff1a\u5148\u7528\u77ed\u8bb0\u5f55\u626b\u4e00\u904d Welch\uff0c\u68c0\u67e5\u4e3b\u8981\u5cf0\u503c\u4f4d\u7f6e\u548c\u5bbd\u5ea6\u662f\u5426\u5408\u7406\uff0c\u518d\u5b9a\u6700\u7ec8\u53c2\u6570\u3002<\/p>\n

—
\n### \u5148\u5e73\u5747\u8c31\u3001\u518d\u6c42\u76f8\u5e72\uff1a\u4e00\u4e2a\u53cd\u4f8b<\/p>\n

Welch \u65b9\u6cd5\u5fc5\u987b**\u5148\u5bf9\u591a\u6bb5\u4e92\u8c31\u548c\u81ea\u8c31\u6c42\u5e73\u5747\uff0c\u518d\u505a\u6bd4\u503c**\uff1a<\/p>\n

$$
\n\\widehat\\gamma_{xy}^2[k]
\n=\\frac{|\\overline{X_rY_r^*}|^2}
\n{\\overline{|X_r|^2}\\cdot\\overline{|Y_r|^2}}.
\n$$<\/p>\n

\u82e5\u6362\u6210\u201c\u5148\u7b97\u6bcf\u6bb5\u7684\u76f8\u5e72\u6027\u518d\u5e73\u5747\u201d\uff0c\u5373<\/p>\n

$$
\n\\overline{\\gamma_r^2}[k]
\n=\\frac{1}{K}\\sum_{r=1}^{K}
\n\\frac{|X_r[k]Y_r^*[k]|^2}{|X_r[k]|^2|Y_r[k]|^2},
\n$$<\/p>\n

\u5206\u5b50\u5206\u6bcd\u540c\u6bb5\u4e25\u683c\u76f8\u6d88\uff08\u7b2c\u516b\u7ae0\u5f00\u5934\u5df2\u6307\u51fa\uff09\uff0c\u6bcf\u6bb5\u7684\u6bd4\u503c\u6052\u4e3a 1\uff0c\u5176\u7b97\u672f\u5e73\u5747\u4ecd\u4e3a 1\u3002\u4efb\u4f55\u4e24\u4e2a\u975e\u96f6\u4fe1\u53f7\u65e0\u8bba\u662f\u5426\u771f\u5b9e\u76f8\u5173\u90fd\u4f1a\u5f97\u5230 $\\overline{\\gamma_r^2}\\equiv1$\u3002<\/p>\n

\u53cd\u4f8b\uff1a\u53d6 $x_r[n]$ \u4e0e $y_r[n]$ \u90fd\u662f\u72ec\u7acb\u9ad8\u65af\u767d\u566a\u58f0\uff0c\u6bb5\u95f4\u5f7c\u6b64\u72ec\u7acb\u3002\u771f\u5b9e $\\gamma_{xy}^2(f)\\equiv0$\u3002<\/p>\n

– **\u6b63\u786e Welch**\uff1a$K$ \u6bb5\u4e92\u8c31\u5728\u4e0d\u540c\u6bb5\u4e4b\u95f4\u968f\u673a\u76f8\u4f4d\u5e73\u5747\u540e\u8d8b\u4e8e 0\uff0c$|\\overline{X_rY_r^*}|^2$ \u6309 $1\/K$ \u7f29\u5c0f\uff0c$\\widehat\\gamma_{xy}^2\\to0$\uff1b
\n– **\u9519\u8bef\u987a\u5e8f**\uff1a\u6bcf\u6bb5\u4ee3\u6570\u4e0a\u5f97 1\uff0c\u5e73\u5747\u540e\u4ecd\u662f 1\uff0c\u4e0e\u771f\u503c\u5b8c\u5168\u76f8\u53cd\u3002<\/p>\n

\u56e0\u6b64\u201c\u5148\u8c31\u5e73\u5747\uff0c\u540e\u4f5c\u6bd4\u503c\u201d\u4e0d\u4ec5\u662f\u6570\u503c\u7ec6\u8282\uff0c\u8fd8\u662f\u76f8\u5e72\u6027\u4f30\u8ba1**\u80fd\u5426\u6536\u655b\u5230\u771f\u503c**\u7684\u7ed3\u6784\u6027\u8981\u6c42\u3002\u82e5\u4f7f\u7528\u4e86\u9884\u5e73\u5747\u7b49\u4ef7\u7269\uff08\u4f8b\u5982\u9891\u57df\u90bb\u57df\u5e73\u6ed1\u6216\u591a\u9525\u5f62\u8c31\uff09\uff0c\u4e5f\u8981\u5728**\u540c\u4e00\u4e2a\u5e73\u6ed1\u6838**\u4e0b\u5bf9\u5206\u5b50\u548c\u5206\u6bcd\u5206\u522b\u5e73\u6ed1\uff0c\u518d\u505a\u6bd4\u503c\uff0c\u4e0d\u80fd\u5728\u6bd4\u503c\u5c42\u9762\u5e73\u6ed1\u3002<\/p>\n

—
\n### MSC \u4f30\u8ba1\u7684\u5206\u5e03\u4e0e\u6709\u6548\u81ea\u7531\u5ea6<\/p>\n

\u5bf9\u72ec\u7acb\u5e73\u5747 $K$ \u6bb5\u3001\u771f\u5b9e\u76f8\u5e72\u4e3a $\\gamma^2$ \u7684\u8fd1\u4f3c\u9ad8\u65af\u8c31\u5047\u8bbe\u4e0b\uff0cWelch MSC \u4f30\u8ba1\u7684\u5747\u503c\u8fd1\u4f3c\u6ee1\u8db3<\/p>\n

$$
\n\\mathbb E[\\widehat\\gamma_{xy}^2]
\n\\approx\\gamma^2+\\frac{1-\\gamma^2}{K}
\n$$<\/p>\n

\uff08\u5b58\u5728\u4e00\u4e2a $1\/K$ \u9636\u7684**\u6b63\u504f\u5dee**\uff0c\u5c24\u5176\u5728\u771f\u503c\u63a5\u8fd1 0 \u65f6\u660e\u663e\uff09\uff0c\u65b9\u5dee\u8fd1\u4f3c\u4e3a<\/p>\n

$$
\n\\operatorname{var}\\{\\widehat\\gamma_{xy}^2\\}
\n\\approx\\frac{2\\gamma^2(1-\\gamma^2)^2}{K}.
\n$$<\/p>\n

\u8fd9\u4e9b\u8fd1\u4f3c\u4f9d\u8d56\u4ee5\u4e0b**\u6709\u6548\u81ea\u7531\u5ea6\u6761\u4ef6**\uff1a<\/p>\n

1. \u5404\u6bb5\u8fd1\u4f3c\u72ec\u7acb\uff1aHann 50% \u91cd\u53e0\u65f6 $K_{\\mathrm{eff}}\\approx K\/1.34$ \u800c\u975e $K$\uff1b
\n2. \u6bcf\u6bb5\u7a97\u5185\u6570\u636e\u8fd1\u4f3c\u5e73\u7a33\uff1b
\n3. \u8c31\u5728\u7a97\u4e3b\u74e3\u5185\u8fd1\u4f3c\u5e73\u5766\uff08\u5426\u5219\u7a97\u6cc4\u6f0f\u4f1a\u5bfc\u81f4\u4f4e\u76f8\u5e72\u533a\u88ab\u9ad8\u76f8\u5e72\u533a\u201c\u6c61\u67d3\u201d\uff09\uff1b
\n4. \u5404\u6bb5\u4e92\u8c31\u548c\u81ea\u8c31\u4f7f\u7528\u540c\u4e00\u7a97\u3001\u540c\u4e00\u5206\u6bb5\u65b9\u5f0f\uff1b
\n5. \u9ad8\u65af\u8fd1\u4f3c\u6210\u7acb\uff0c\u5373\u611f\u5174\u8da3\u9891\u7387\u4e0a\u7684\u8c31\u7cfb\u6570\u8fd1\u4f3c Gauss\uff0c\u4e14\u4e0d\u53d7\u5f3a\u7ebf\u8c31\u4e3b\u5bfc\u3002<\/p>\n

\u5728\u201c\u771f\u5b9e\u76f8\u5e72\u4e3a\u96f6\u201d\u5047\u8bbe\u4e0b\uff0c$K$ \u72ec\u7acb\u5e73\u5747\u7684\u663e\u8457\u6027\u9608\u503c<\/p>\n

$$
\n\\gamma^2_{\\mathrm{crit}}=1-\\alpha^{1\/(K-1)}
\n$$<\/p>\n

\u4e5f\u5e94\u628a $K$ \u66ff\u6362\u4e3a $K_{\\mathrm{eff}}$\uff0c\u5426\u5219\u4f1a\u5f97\u5230\u8fc7\u4e8e\u5bbd\u677e\u7684\u9608\u503c\uff0c\u628a\u566a\u58f0\u8d77\u4f0f\u5f53\u4f5c\u663e\u8457\u3002<\/p>\n

\u7f6e\u4fe1\u533a\u95f4\u53ef\u901a\u8fc7 Fisher-Z \u578b\u53d8\u6362\u6216 jackknife\/bootstrap \u53d6\u5f97\u3002\u5bf9\u65b9\u5dee\u975e\u5e38\u654f\u611f\u7684\u573a\u5408\uff08\u4f8b\u5982\u751f\u7269\u533b\u5b66\u3001\u7ed3\u6784\u5065\u5eb7\u76d1\u6d4b\uff09\uff0c\u5efa\u8bae\u5728\u7ed3\u679c\u56fe\u4e0a\u628a\u6709\u6548\u81ea\u7531\u5ea6\u548c\u9608\u503c\u7ebf\u4e00\u5e76\u753b\u51fa\uff0c\u8ba9\u8bfb\u8005\u77e5\u9053\u5224\u8bfb\u7684\u7f6e\u4fe1\u7a0b\u5ea6\u3002<\/p>\n

—
\n### scipy \u53c2\u8003\u4ee3\u7801\u793a\u4f8b<\/p>\n

\u4ee5\u4e0b Python \u4f8b\u5b50\u6f14\u793a\u5982\u4f55\u7528 `scipy.signal` \u4e00\u81f4\u5730\u8ba1\u7b97 PSD\u3001CSD \u548c MSC\u3002\u5b83\u53ef\u4ee5\u4f5c\u4e3a\u5de5\u7a0b\u5b9e\u73b0\u7684\u8d77\u70b9\uff0c\u4f46\u751f\u4ea7\u73af\u5883\u5e94\u663e\u5f0f\u5199\u6e05\u7a97\u3001\u91cd\u53e0\u3001\u5355\u53cc\u8fb9\u7ea6\u5b9a\u548c\u5f52\u4e00\u5316\u3002<\/p>\n

“`python
\nimport numpy as np
\nfrom scipy import signal<\/p>\n

# \u5047\u8bbe x, y \u4e3a\u7b49\u957f\u7684\u4e00\u7ef4\u5b9e\u4fe1\u53f7\uff0c\u91c7\u6837\u7387 fs
\nfs = 1000.0 # Hz
\nnperseg = 1024 # \u6bb5\u957f L
\nnoverlap = nperseg \/\/ 2 # 50% \u91cd\u53e0
\nwindow = ‘hann’ # Hann \u7a97
\nnfft = nperseg # \u4e0e\u6bb5\u957f\u76f8\u540c\uff08\u4e0d\u989d\u5916\u96f6\u586b\u5145\uff09<\/p>\n

# \u81ea\u8c31\u3001\u4e92\u8c31\u3001\u76f8\u5e72\u6027\u7528\u540c\u4e00\u5957\u53c2\u6570
\nf, Pxx = signal.welch(x, fs=fs, window=window,
\n nperseg=nperseg, noverlap=noverlap,
\n nfft=nfft, detrend=’constant’,
\n return_onesided=True, scaling=’density’)
\n_, Pyy = signal.welch(y, fs=fs, window=window,
\n nperseg=nperseg, noverlap=noverlap,
\n nfft=nfft, detrend=’constant’,
\n return_onesided=True, scaling=’density’)
\n_, Pxy = signal.csd(x, y, fs=fs, window=window,
\n nperseg=nperseg, noverlap=noverlap,
\n nfft=nfft, detrend=’constant’,
\n return_onesided=True, scaling=’density’)<\/p>\n

# MSC\uff1a\u6ce8\u610f\u4e0e signal.coherence \u7684\u5b9a\u4e49\u4e00\u81f4
\nCxy = np.abs(Pxy) ** 2 \/ (Pxx * Pyy)<\/p>\n

# \u4e5f\u53ef\u76f4\u63a5\u8c03\u7528\u9ad8\u5c42 API \u8fdb\u884c\u4ea4\u53c9\u68c0\u67e5
\n_, Cxy_ref = signal.coherence(x, y, fs=fs, window=window,
\n nperseg=nperseg, noverlap=noverlap,
\n nfft=nfft, detrend=’constant’)<\/p>\n

# \u76f8\u5e72\u6027\u663e\u8457\u6027\u9608\u503c\uff08\u5047\u8bbe\u771f\u5b9e\u76f8\u5e72\u4e3a 0\uff09
\nK = (len(x) – noverlap) \/\/ (nperseg – noverlap) # \u6bb5\u6570
\nalpha = 0.05
\ngamma_crit = 1 – alpha ** (1 \/ (K – 1))
\n“`<\/p>\n

\u8981\u70b9\u63d0\u793a\uff1a<\/p>\n

– `scaling=’density’` \u7ed9\u51fa PSD\uff08\u5355\u4f4d $\\mathrm{V}^2\/\\mathrm{Hz}$\uff09\uff1b\u6362\u6210 `’spectrum’` \u5f97\u5230\u5206\u6bb5\u529f\u7387\u8c31\uff08\u5355\u4f4d $\\mathrm{V}^2$\uff09\uff1b\u4e24\u8005\u79ef\u5206\/\u6c42\u548c\u7684\u7269\u7406\u610f\u4e49\u4e0d\u540c\u3002
\n– `return_onesided=True` \u53ea\u5bf9\u5b9e\u4fe1\u53f7\u6709\u6548\uff1b\u590d\u4fe1\u53f7\u5fc5\u987b\u4f20 `False`\u3002
\n– `signal.csd` \u4e0e `signal.welch` \u4f7f\u7528**\u76f8\u540c\u7684\u5206\u6bb5\u548c\u7a97**\u624d\u80fd\u4fdd\u8bc1 `Pxy\/sqrt(Pxx*Pyy)` \u4e0e `signal.coherence` \u7ed3\u679c\u4e00\u81f4\u3002
\n– \u6709\u6548\u72ec\u7acb\u6bb5\u6570 $K_{\\mathrm{eff}}$ \u5c0f\u4e8e\u540d\u4e49 $K$\uff1b\u663e\u8457\u6027\u9608\u503c `gamma_crit` \u4e0a\u5f0f\u504f\u4e50\u89c2\uff0c\u9700\u8981\u6309\u7a97\u81ea\u76f8\u5173\u4fee\u6b63\u3002
\n– \u82e5\u505a\u56e0\u679c\u5206\u6790\u6216\u7fa4\u65f6\u5ef6\u4f30\u8ba1\uff0c\u5e94\u4ece `Pxy` \u590d\u6570\u503c\u53d6\u76f8\u4f4d\u5e76\u8fdb\u884c `np.unwrap`\uff0c\u5e76\u53ea\u5728 `Cxy >= gamma_crit` \u9891\u6bb5\u89e3\u91ca\u76f8\u4f4d\u3002<\/p>\n

—<\/p>\n

—<\/p>\n

## \u4e5d\u3001\u901a\u4fe1\u3001\u68c0\u6d4b\u4e0e\u7cfb\u7edf\u8fa8\u8bc6\u5e94\u7528<\/p>\n

### \u6a21\u677f\u5339\u914d<\/p>\n

\u5df2\u77e5\u6a21\u677f $s[n]$\uff0c\u89c2\u6d4b\u4fe1\u53f7\u4e3a $r[n]$\u3002\u901a\u8fc7\u8ba1\u7b97<\/p>\n

$$
\nC[m]=\\sum_n r[n]s^*[n-m]
\n$$<\/p>\n

\u53ef\u4ee5\u5728 $r[n]$ \u4e2d\u5bfb\u627e\u6a21\u677f\u51fa\u73b0\u7684\u4f4d\u7f6e\u3002\u6700\u5927\u5cf0\u503c\u7684\u4f4d\u7f6e\u7ed9\u51fa\u6a21\u677f\u7684\u4f30\u8ba1\u5ef6\u8fdf\u3002<\/p>\n

### \u5339\u914d\u6ee4\u6ce2\u5668<\/p>\n

\u5339\u914d\u6ee4\u6ce2\u5668\u7684\u51b2\u6fc0\u54cd\u5e94\u901a\u5e38\u4e0e\u6a21\u677f\u7684\u5171\u8f6d\u65f6\u95f4\u53cd\u8f6c\u6709\u5173\uff1a<\/p>\n

$$
\n h[n]=s^*[N-1-n].
\n$$<\/p>\n

\u5c06\u63a5\u6536\u4fe1\u53f7\u4e0e\u8be5\u6ee4\u6ce2\u5668\u5377\u79ef\uff0c\u7b49\u4ef7\u4e8e\u8ba1\u7b97\u63a5\u6536\u4fe1\u53f7\u4e0e\u6a21\u677f\u7684\u76f8\u5173\u3002\u5339\u914d\u6ee4\u6ce2\u5668\u5728\u52a0\u6027\u767d\u566a\u58f0\u4e0b\u80fd\u591f\u6700\u5927\u5316\u67d0\u4e00\u91c7\u6837\u65f6\u523b\u7684\u8f93\u51fa\u4fe1\u566a\u6bd4\u3002<\/p>\n

#### AWGN \u4e0b\u5339\u914d\u6ee4\u6ce2\u7684\u6700\u4f18 SNR \u63a8\u5bfc<\/p>\n

\u8bbe\u5df2\u77e5\u6a21\u677f $s(t)$\uff08\u80fd\u91cf $E_s=\\int|s(t)|^2dt$\uff09\uff0c\u63a5\u6536\u4fe1\u53f7\u4e3a<\/p>\n

$$
\n r(t)=s(t)+w(t),
\n$$<\/p>\n

\u5176\u4e2d $w(t)$ \u662f\u53cc\u8fb9 PSD \u4e3a $N_0\/2$ \u7684\u52a0\u6027\u767d Gaussian \u566a\u58f0\u3002\u7528\u4efb\u610f\u7ebf\u6027\u6ee4\u6ce2\u5668 $h(t)$ \u5bf9 $r$ \u6ee4\u6ce2\uff0c\u4ee4\u8f93\u51fa<\/p>\n

$$
\n y(t)=(h*r)(t)=y_s(t)+y_w(t).
\n$$<\/p>\n

\u56fa\u5b9a\u91c7\u6837\u65f6\u523b $t_0$\uff0c\u8f93\u51fa\u4fe1\u53f7\u5206\u91cf\u7684\u77ac\u65f6”\u529f\u7387”\u4e3a $|y_s(t_0)|^2$\uff0c\u8f93\u51fa\u566a\u58f0\u65b9\u5dee\u4e3a<\/p>\n

$$
\n\\sigma_w^2=\\frac{N_0}{2}\\int|h(\\tau)|^2d\\tau.
\n$$<\/p>\n

\u91c7\u6837\u65f6\u523b\u7684\u8f93\u51fa SNR \u5b9a\u4e49\u4e3a<\/p>\n

$$
\n\\operatorname{SNR}_{\\mathrm{out}}
\n=\\frac{|y_s(t_0)|^2}{\\sigma_w^2}
\n=\\frac{\\left|\\int h(\\tau)s(t_0-\\tau)d\\tau\\right|^2}
\n{\\frac{N_0}{2}\\int|h(\\tau)|^2d\\tau}.
\n$$<\/p>\n

\u7531 Cauchy\u2013Schwarz \u4e0d\u7b49\u5f0f\uff1a<\/p>\n

$$
\n\\left|\\int h(\\tau)s(t_0-\\tau)d\\tau\\right|^2
\n\\le\\left(\\int|h(\\tau)|^2d\\tau\\right)
\n\\left(\\int|s(t_0-\\tau)|^2d\\tau\\right)
\n=E_s\\int|h(\\tau)|^2d\\tau,
\n$$<\/p>\n

\u7b49\u53f7\u5f53\u4e14\u4ec5\u5f53<\/p>\n

$$
\n h(\\tau)=k\\,s^*(t_0-\\tau)
\n$$<\/p>\n

\u65f6\u6210\u7acb\uff08$k$ \u4e3a\u5e38\u6570\uff09\u3002\u4ee3\u5165\u5f97<\/p>\n

$$
\n\\boxed{\\operatorname{SNR}_{\\mathrm{out},\\max}=\\frac{2E_s}{N_0}}.
\n$$<\/p>\n

\u7ed3\u8bba\uff1a\u6700\u4f18\u7ebf\u6027\u6ee4\u6ce2\u5668\u5c31\u662f\u6a21\u677f\u7684\u5171\u8f6d\u65f6\u95f4\u53cd\u8f6c\uff08\u79bb\u6563\u57df\u5373 $h[n]=s^*[N-1-n]$\uff09\uff0c\u6700\u4f18 SNR \u53ea\u4f9d\u8d56\u6a21\u677f\u80fd\u91cf\u4e0e\u566a\u58f0 PSD\uff0c\u4e0e\u6ce2\u5f62\u5f62\u72b6\u65e0\u5173\u3002\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u96f7\u8fbe\u548c\u901a\u4fe1\u4e2d\u901a\u8fc7\u589e\u52a0\u8109\u51b2\u80fd\u91cf\uff08\u66f4\u5bbd\u8109\u51b2\u3001\u8109\u51b2\u538b\u7f29\u3001\u7f16\u7801\u589e\u76ca\uff09\u6765\u6539\u5584\u68c0\u6d4b\u6027\u80fd\uff0c\u800c\u4e0d\u5fc5\u6539\u53d8\u8f7d\u9891\u6216\u5339\u914d\u6ee4\u6ce2\u7ed3\u6784\u672c\u8eab\u3002<\/p>\n

### \u540c\u6b65\u4e0e\u65f6\u5ef6\u4f30\u8ba1<\/p>\n

\u53d1\u9001\u4fe1\u53f7 $s(t)$\uff0c\u63a5\u6536\u4fe1\u53f7\u8fd1\u4f3c\u4e3a<\/p>\n

$$
\n r(t)=a s(t-\\tau_0)+w(t),
\n$$<\/p>\n

\u5176\u4e2d $a$ \u662f\u5e45\u5ea6\u6216\u590d\u8870\u843d\u7cfb\u6570\uff0c$w(t)$ \u662f\u566a\u58f0\u3002\u8ba1\u7b97<\/p>\n

$$
\nR_{rs}(\\tau)=\\int r(t)s^*(t-\\tau)dt
\n$$<\/p>\n

\u65f6\uff0c\u5728 $\\tau\\approx\\tau_0$ \u9644\u8fd1\u901a\u5e38\u51fa\u73b0\u5cf0\u503c\u3002<\/p>\n

#### \u5e7f\u4e49\u4e92\u76f8\u5173\u4e0e GCC-PHAT<\/p>\n

\u5728\u591a\u4f20\u611f\u5668\u6216\u672a\u77e5\u4fe1\u53f7\u573a\u666f\uff08\u5982\u9ea6\u514b\u98ce\u9635\u5217\u3001\u88ab\u52a8\u58f0\u7eb3\u3001TDOA \u5b9a\u4f4d\uff09\uff0c\u4e24\u8def\u89c2\u6d4b\u5e38\u5199\u6210<\/p>\n

$$
\n x_1(t)=s(t)+n_1(t),
\n\\qquad
\n x_2(t)=\\alpha\\,s(t-\\tau_0)+n_2(t).
\n$$<\/p>\n

\u5e7f\u4e49\u4e92\u76f8\u5173\uff08Generalized Cross-Correlation, GCC\uff09\u901a\u8fc7\u5728\u9891\u57df\u5f15\u5165\u6743\u51fd\u6570 $\\Psi(f)$ \u589e\u5f3a\u5cf0\u503c\u9510\u5ea6\uff1a<\/p>\n

$$
\nR^{\\mathrm{GCC}}_{x_1x_2}(\\tau)
\n=\\int_{-\\infty}^{\\infty}\\Psi(f)\\,S_{x_1x_2}(f)\\,e^{j2\\pi f\\tau}\\,df.
\n$$<\/p>\n

\u4e0d\u540c\u6743\u51fd\u6570\u5bf9\u5e94\u4e0d\u540c\u65b9\u6cd5\uff1a$\\Psi\\equiv1$ \u5c31\u662f\u666e\u901a\u4e92\u76f8\u5173\uff1bRoth \u52a0\u6743 $\\Psi=1\/S_{x_1x_1}$\uff1bSCOT \u52a0\u6743 $\\Psi=1\/\\sqrt{S_{x_1x_1}S_{x_2x_2}}$\u3002\u5de5\u7a0b\u4e2d\u6700\u5e38\u7528\u7684\u662f\u76f8\u4f4d\u53d8\u6362\uff08Phase Transform, PHAT\uff09\u52a0\u6743\uff1a<\/p>\n

$$
\n\\boxed{\\Psi_{\\mathrm{PHAT}}(f)=\\frac{1}{|S_{x_1x_2}(f)|}},
\n$$<\/p>\n

\u4ee3\u5165\u540e<\/p>\n

$$
\nR^{\\mathrm{PHAT}}_{x_1x_2}(\\tau)
\n=\\int\\frac{S_{x_1x_2}(f)}{|S_{x_1x_2}(f)|}e^{j2\\pi f\\tau}df
\n=\\int e^{j\\angle S_{x_1x_2}(f)}e^{j2\\pi f\\tau}df.
\n$$<\/p>\n

\u4e5f\u5c31\u662f\u53ea\u4fdd\u7559\u4e92\u8c31\u7684\u76f8\u4f4d\u3001\u5e45\u5ea6\u5f52\u4e00\u5316\u4e3a 1\u3002\u7406\u60f3\u7a84\u5e26\u65e0\u566a\u65f6 $\\angle S_{x_1x_2}(f)=-2\\pi f\\tau_0$\uff0c\u79ef\u5206\u7ed9\u51fa<\/p>\n

$$
\nR^{\\mathrm{PHAT}}_{x_1x_2}(\\tau)\\propto\\delta(\\tau-\\tau_0),
\n$$<\/p>\n

\u5cf0\u4f4d\u4e0e\u4fe1\u53f7\u8c31\u5f62\u72b6\u65e0\u5173\u3002DFT \u5b9e\u73b0\uff1a<\/p>\n

$$
\n\\widehat R^{\\mathrm{PHAT}}[m]
\n=\\operatorname{IDFT}\\!\\left\\{
\n\\frac{X_1[k]X_2^*[k]}{|X_1[k]X_2^*[k]|+\\varepsilon}
\n\\right\\}.
\n$$<\/p>\n

\u5de5\u7a0b\u8981\u70b9\uff1a<\/p>\n

– \u4e0e\u666e\u901a\u4e92\u76f8\u5173\u76f8\u6bd4\uff0cPHAT \u5728\u6df7\u54cd\u3001\u8272\u566a\u58f0\u3001\u5bbd\u5e26\u975e\u5e73\u7a33\u4fe1\u53f7\u4e0b\u5cf0\u503c\u66f4\u9510\u3001\u65c1\u74e3\u66f4\u4f4e\uff1b
\n– \u4f46\u56e0\u629b\u5f03\u4e86\u5e45\u5ea6\u4fe1\u606f\uff0c\u4f4e SNR \u6216\u7a00\u758f\u8c31\u6bb5\u4f1a\u56e0\u9664\u4ee5\u5c0f\u6570\u800c\u88ab\u566a\u58f0\u4e3b\u5bfc\uff0c\u9700\u8981\u52a0\u6b63\u5219\u5316 $\\varepsilon$ \u6216\u8c31\u9608\u503c\uff1b
\n– \u4e9a\u91c7\u6837\u7cbe\u5ea6\u53ef\u901a\u8fc7\u5728\u5cf0\u503c\u9644\u8fd1\u505a\u629b\u7269\u7ebf\u63d2\u503c\u6216\u9891\u57df\u76f8\u4f4d\u659c\u7387\u62df\u5408\u83b7\u5f97\uff1b
\n– \u591a\u5bf9\u9ea6\u514b\u98ce\u7ed9\u51fa\u591a\u7ec4 $\\tau_{ij}$\uff0c\u53ef\u8054\u7acb\u6c42\u89e3\u58f0\u6e90\u5750\u6807\uff08TDOA \u5b9a\u4f4d\uff09\u3002<\/p>\n

### \u7cfb\u7edf\u8fa8\u8bc6\u6d41\u7a0b<\/p>\n

\u7b2c\u4e94\u7ae0\u7ed9\u51fa\u4e86\u5355\u8f93\u5165\u7ebf\u6027\u7cfb\u7edf\u7684\u4f20\u9012\u51fd\u6570\u4f30\u8ba1<\/p>\n

$$
\n\\widehat H(f)=\\frac{S_{yx}(f)}{S_{xx}(f)},
\n$$<\/p>\n

\u7b2c\u4e03\u7ae0\u7ed9\u51fa\u4e86\u76f8\u5e72\u6027 $\\gamma^2_{xy}(f)$\u3002\u628a\u4e8c\u8005\u7ec4\u5408\u8d77\u6765\uff0c\u4e00\u4e2a\u53ef\u590d\u73b0\u7684\u975e\u53c2\u6570\u7cfb\u7edf\u8fa8\u8bc6\u6d41\u7a0b\u901a\u5e38\u5305\u62ec\uff1a<\/p>\n

1. **\u6fc0\u52b1\u8bbe\u8ba1**\uff1a\u9009\u53d6\u5e26\u5bbd\u8986\u76d6\u88ab\u6d4b\u7cfb\u7edf\u5173\u6ce8\u9891\u6bb5\u7684\u8f93\u5165 $x[n]$\u3002\u5e38\u7528\u5bbd\u5e26\u767d\u566a\u58f0\u3001\u4f2a\u968f\u673a\u4e8c\u8fdb\u5236\u5e8f\u5217\uff08PRBS\uff09\u3001\u591a\u6b63\u5f26\uff08multisine\uff09\u6216\u8c03\u9891\u626b\u9891\uff08chirp\uff09\u3002\u8981\u6c42\uff1a
\n – $S_{xx}(f)$ \u5728\u5173\u6ce8\u9891\u5e26\u975e\u96f6\uff1b
\n – \u5355\u6b21\u5b9e\u9a8c\u65f6\u95f4\u8db3\u591f\u957f\uff0c\u80fd\u63d0\u4f9b $K$ \u6bb5\u72ec\u7acb\u5e73\u5747\uff1b
\n – \u5e45\u503c\u4e0d\u6fc0\u53d1\u88ab\u6d4b\u7cfb\u7edf\u7684\u975e\u7ebf\u6027\u533a\u95f4\u3002
\n2. **\u540c\u6b65\u91c7\u96c6**\uff1a$x[n]$ \u4e0e $y[n]$ \u7528\u540c\u4e00\u65f6\u949f\u91c7\u6837\uff0c\u5c3d\u91cf\u907f\u514d\u901a\u9053\u95f4\u6297\u6df7\u53e0\u6ee4\u6ce2\u5668\u4e0d\u4e00\u81f4\u5e26\u6765\u7684\u56fa\u5b9a\u76f8\u4f4d\u504f\u5dee\uff1b\u82e5\u6709\u504f\u5dee\uff0c\u5e94\u5728\u4e8b\u540e\u7528\u5df2\u77e5\u53c2\u8003\u505a\u6821\u6b63\u3002
\n3. **\u9884\u5904\u7406**\uff1a\u53bb\u5747\u503c\u3001\u53bb\u7ebf\u6027\u8d8b\u52bf\u3001\u53ef\u9009\u5e26\u901a\u6ee4\u6ce2\u3001\u5fc5\u8981\u65f6\u5bf9\u5f02\u5e38\u6bb5\u8fdb\u884c\u5254\u9664\u6216\u52a0\u6743\u3002
\n4. **\u5206\u6bb5\u8c31\u4f30\u8ba1**\uff1a\u4f7f\u7528\u7b2c\u516b\u7ae0\u7684 Welch \u65b9\u6cd5\uff0c\u6bb5\u957f $L$\u3001\u91cd\u53e0 $50\\%\\sim75\\%$\u3001\u52a0 Hann\/Hamming \u7a97\uff0c\u5f97\u5230<\/p>\n

$$
\n \\widehat S_{xx}(f),\\ \\widehat S_{yy}(f),\\ \\widehat S_{yx}(f).
\n $$
\n5. **\u4f20\u9012\u51fd\u6570\u4f30\u8ba1**\uff1a\u5e38\u7528\u4e09\u4e2a\u4f30\u8ba1\u91cf\u5404\u6709\u504f\u5dee\u65b9\u5411<\/p>\n

$$
\n \\widehat H_1=\\frac{\\widehat S_{yx}}{\\widehat S_{xx}},\\quad
\n \\widehat H_2=\\frac{\\widehat S_{yy}}{\\widehat S_{xy}},\\quad
\n \\widehat H_v=\\sqrt{\\widehat H_1\\widehat H_2}.
\n $$
\n $\\widehat H_1$ \u5bf9\u8f93\u51fa\u566a\u58f0\u9c81\u68d2\uff0c$\\widehat H_2$ \u5bf9\u8f93\u5165\u566a\u58f0\u9c81\u68d2\uff0c$\\widehat H_v$ \u6298\u8877\u3002
\n6. **\u76f8\u5e72\u6027\u68c0\u67e5**\uff1a\u8ba1\u7b97 $\\widehat\\gamma^2_{xy}(f)$\uff0c\u53ea\u5728\u76f8\u5e72\u6027\u8db3\u591f\u9ad8\uff08\u4f8b\u5982 $\\ge 0.8$\uff09\u7684\u9891\u6bb5\u5185\u62a5\u544a $\\widehat H(f)$\uff1b\u76f8\u5e72\u6027\u663e\u8457\u4f4e\u4e8e 1 \u7684\u9891\u6bb5\u4e00\u822c\u5bf9\u5e94\uff1a\u8f93\u5165\u529f\u7387\u4e0d\u8db3\u3001\u975e\u7ebf\u6027\u3001\u5916\u90e8\u5e72\u6270\u6216\u5b58\u5728\u672a\u6d4b\u91cf\u8f93\u5165\u3002
\n7. **\u6a21\u578b\u62df\u5408\u4e0e\u9a8c\u8bc1\uff08\u53ef\u9009\uff09**\uff1a\u5c06\u975e\u53c2\u6570 $\\widehat H(f)$ \u62df\u5408\u6210\u53c2\u6570\u6a21\u578b\uff08\u5982 ARX\u3001\u72b6\u6001\u7a7a\u95f4\u3001\u4f20\u9012\u51fd\u6570\uff09\uff0c\u5e76\u7528\u672a\u53c2\u4e0e\u8fa8\u8bc6\u7684\u72ec\u7acb\u6570\u636e\u6bb5\u505a\u6b8b\u5dee\u4e0e\u9884\u6d4b\u68c0\u9a8c\u3002<\/p>\n

\u8fd9\u5957\u6d41\u7a0b\u4e5f\u53ef\u63a8\u5e7f\u5230 MIMO \u7cfb\u7edf\uff1a\u628a\u6807\u91cf\u81ea\u8c31\u6362\u6210\u8c31\u77e9\u9635\uff0c$\\widehat{\\mathbf H}=\\widehat{\\mathbf S}_{yx}\\widehat{\\mathbf S}_{xx}^{-1}$\uff0c\u5e76\u4f7f\u7528\u591a\u91cd\u76f8\u5e72\u6027\uff08\u89c1 7.3 \u8282\uff09\u4ee3\u66ff\u5355\u9891\u76f8\u5e72\u6027\u3002<\/p>\n

### \u6269\u9891\u901a\u4fe1\u4e2d\u7684\u89e3\u6269<\/p>\n

\u6269\u9891\u7801\u4e0e\u63a5\u6536\u4fe1\u53f7\u76f8\u5173\uff0c\u53ef\u4ee5\u5c06\u76ee\u6807\u7801\u7247\u5e8f\u5217\u7684\u80fd\u91cf\u7d2f\u79ef\u8d77\u6765\uff0c\u800c\u4e0e\u5176\u4ed6\u7801\u6216\u566a\u58f0\u4e0d\u5339\u914d\u7684\u5206\u91cf\u8d8b\u4e8e\u62b5\u6d88\u3002<\/p>\n

### \u96f7\u8fbe\u548c\u58f0\u7eb3<\/p>\n

\u53d1\u5c04\u5df2\u77e5\u6ce2\u5f62\uff0c\u63a5\u6536\u76ee\u6807\u56de\u6ce2\u540e\u4e0e\u53d1\u5c04\u6ce2\u5f62\u505a\u5339\u914d\u6ee4\u6ce2\u6216\u4e92\u76f8\u5173\uff1a<\/p>\n

– \u5cf0\u503c\u4f4d\u7f6e\u4f30\u8ba1\u4f20\u64ad\u65f6\u5ef6\uff1b
\n– \u65f6\u5ef6\u4e58\u4ee5\u4f20\u64ad\u901f\u5ea6\u53ef\u4ee5\u4f30\u8ba1\u8ddd\u79bb\uff1b
\n– \u5cf0\u503c\u76f8\u4f4d\u6216\u591a\u666e\u52d2\u7ed3\u6784\u53ef\u4ee5\u7528\u4e8e\u4f30\u8ba1\u8fd0\u52a8\u4fe1\u606f\u3002<\/p>\n

### \u81ea\u9002\u5e94\u6ee4\u6ce2\u4e2d\u7684\u76f8\u5173\u68af\u5ea6<\/p>\n

\u81ea\u9002\u5e94\u6ee4\u6ce2\u628a\u76f8\u5173\u4ece\u201c\u6d4b\u91cf\u4e24\u4e2a\u4fe1\u53f7\u7684\u76f8\u4f3c\u6027\u201d\u53d8\u6210\u201c\u5bfb\u627e\u4f7f\u8bef\u5dee\u4e0b\u964d\u7684\u53c2\u6570\u66f4\u65b0\u65b9\u5411\u201d\u3002\u5171\u8f6d\u7684\u4f4d\u7f6e\u7531\u8f93\u51fa\u6a21\u578b\u3001\u8bef\u5dee\u5b9a\u4e49\u548c\u6c42\u5bfc\u53d8\u91cf\u5171\u540c\u51b3\u5b9a\u3002<\/p>\n

#### \u5b9e\u6570 LMS\uff1a\u4ece\u94fe\u5f0f\u6cd5\u5219\u5f00\u59cb<\/p>\n

\u4ee4\u8f93\u5165\u56de\u5f52\u5411\u91cf\u4e3a<\/p>\n

$$
\n\\boldsymbol{x}_n=
\n\\begin{bmatrix}
\n x[n]&x[n-1]&\\cdots&x[n-L+1]
\n\\end{bmatrix}^{T},
\n$$<\/p>\n

\u91c7\u7528<\/p>\n

$$
\ny[n]=\\boldsymbol w^T\\boldsymbol x_n,
\n\\qquad
\n e[n]=d[n]-y[n],
\n\\qquad
\n J[n]=\\frac12e^2[n].
\n$$<\/p>\n

\u5bf9\u7b2c $\\ell$ \u4e2a\u62bd\u5934\uff1a<\/p>\n

$$
\n\\frac{\\partial J[n]}{\\partial w[\\ell]}
\n=\\frac{\\partial J[n]}{\\partial e[n]}
\n \\frac{\\partial e[n]}{\\partial w[\\ell]}
\n=e[n]\\cdot[-x[n-\\ell]].
\n$$<\/p>\n

\u56e0\u6b64<\/p>\n

$$
\n\\nabla_{\\boldsymbol w}J[n]=-\\boldsymbol x_ne[n],
\n$$<\/p>\n

\u68af\u5ea6\u4e0b\u964d\u7ed9\u51fa<\/p>\n

$$
\n\\boxed{
\n\\boldsymbol w[n+1]=\\boldsymbol w[n]+\\mu\\boldsymbol x_ne[n]
\n}.
\n$$<\/p>\n

\u82e5\u628a\u8bef\u5dee\u5b9a\u4e49\u6210 $e[n]=y[n]-d[n]$\uff0c\u66f4\u65b0\u5f0f\u7684\u7b26\u53f7\u4f1a\u76f8\u5e94\u6539\u53d8\u3002\u52a0\u53f7\u6216\u51cf\u53f7\u4e0d\u80fd\u8131\u79bb\u8bef\u5dee\u5b9a\u4e49\u548c\u4ee3\u4ef7\u51fd\u6570\u5355\u72ec\u5224\u65ad\u3002<\/p>\n

#### \u590d\u6570\u53c2\u6570\u4e0e Wirtinger \u5bfc\u6570<\/p>\n

\u590d\u6570\u4ee3\u4ef7\u51fd\u6570<\/p>\n

$$
\nJ[n]=\\frac12e[n]e^*[n]
\n$$<\/p>\n

\u662f\u5b9e\u503c\u51fd\u6570\uff0c\u4f46\u540c\u65f6\u4f9d\u8d56 $w$ \u548c $w^*$\u3002\u4ee4 $w=w_R+jw_I$\uff0cWirtinger \u5bfc\u6570\u5b9a\u4e49\u4e3a<\/p>\n

$$
\n\\frac{\\partial}{\\partial w}
\n=\\frac12\\left(\\frac{\\partial}{\\partial w_R}
\n-j\\frac{\\partial}{\\partial w_I}\\right),
\n\\qquad
\n\\frac{\\partial}{\\partial w^*}
\n=\\frac12\\left(\\frac{\\partial}{\\partial w_R}
\n+j\\frac{\\partial}{\\partial w_I}\\right).
\n$$<\/p>\n

\u63a8\u5bfc\u65f6\u628a $w$ \u548c $w^*$ \u6682\u65f6\u89c6\u4e3a\u5f62\u5f0f\u4e0a\u72ec\u7acb\u7684\u53d8\u91cf\u3002\u5bf9\u5b9e\u503c\u635f\u5931\uff0c\u5e38\u7528\u590d\u68af\u5ea6\u7ea6\u5b9a\u4e3a<\/p>\n

$$
\n\\nabla_{\\boldsymbol w}J
\n=2\\frac{\\partial J}{\\partial\\boldsymbol w^*}.
\n$$<\/p>\n

\u56e0\u5b50 2 \u4e5f\u53ef\u4ee5\u5e76\u5165\u6b65\u957f\uff0c\u56e0\u6b64\u4e0d\u540c\u8d44\u6599\u4e2d\u7684\u7cfb\u6570\u53ef\u80fd\u4e0d\u540c\uff0c\u4f46\u66f4\u65b0\u65b9\u5411\u5fc5\u987b\u4e00\u81f4\u3002<\/p>\n

#### Hermitian \u6a21\u578b\uff1a$y=\\boldsymbol w^H\\boldsymbol x$<\/p>\n

\u590d\u6570\u81ea\u9002\u5e94\u6ee4\u6ce2\u901a\u5e38\u91c7\u7528<\/p>\n

$$
\ny[n]=\\boldsymbol w^H\\boldsymbol x_n,
\n\\qquad
\n e[n]=d[n]-\\boldsymbol w^H\\boldsymbol x_n,
\n\\qquad
\n J[n]=\\frac12|e[n]|^2.
\n$$<\/p>\n

\u7531\u4e8e $\\boldsymbol w^H=(\\boldsymbol w^*)^T$\uff0c\u6709<\/p>\n

$$
\n e[n]=d[n]-\\boldsymbol w^H\\boldsymbol x_n,
\n\\qquad
\n e^*[n]=d^*[n]-\\boldsymbol x_n^H\\boldsymbol w.
\n$$<\/p>\n

\u5bf9 $\\boldsymbol w^*$ \u6c42\u504f\u5bfc\u65f6\uff0c$e^*[n]$ \u4e0d\u542b $\\boldsymbol w^*$\uff1a<\/p>\n

$$
\n\\begin{aligned}
\n\\frac{\\partial J[n]}{\\partial\\boldsymbol w^*}
\n&=\\frac12\\left(
\n e^*[n]\\frac{\\partial e[n]}{\\partial\\boldsymbol w^*}
\n +e[n]\\frac{\\partial e^*[n]}{\\partial\\boldsymbol w^*}
\n \\right)\\\\
\n&=\\frac12e^*[n]
\n \\frac{\\partial(d[n]-\\boldsymbol w^H\\boldsymbol x_n)}
\n {\\partial\\boldsymbol w^*}\\\\
\n&=-\\frac12\\boldsymbol x_ne^*[n].
\n\\end{aligned}
\n$$<\/p>\n

\u91c7\u7528 $2\\partial J\/\\partial\\boldsymbol w^*$ \u7684\u590d\u68af\u5ea6\u7ea6\u5b9a\uff1a<\/p>\n

$$
\n\\boxed{
\n\\nabla_{\\boldsymbol w}J[n]=-\\boldsymbol x_ne^*[n]
\n}.
\n$$<\/p>\n

\u6240\u4ee5\u590d\u6570 LMS \u66f4\u65b0\u4e3a<\/p>\n

$$
\n\\boxed{
\n\\boldsymbol w[n+1]=\\boldsymbol w[n]+\\mu\\boldsymbol x_ne^*[n]
\n}.
\n$$<\/p>\n

\u9010\u62bd\u5934\u4e3a<\/p>\n

$$
\nw[n+1,\\ell]=w[n,\\ell]+\\mu x[n-\\ell]e^*[n].
\n$$<\/p>\n

\u8fd9\u91cc\u7684\u5171\u8f6d\u6765\u81ea\u8f93\u51fa\u4f7f\u7528 $\\boldsymbol w^H\\boldsymbol x$ \u4ee5\u53ca\u5bf9 $\\boldsymbol w^*$ \u6c42 Wirtinger \u5bfc\u6570\uff0c\u800c\u4e0d\u662f\u6765\u81ea\u67d0\u4e2a\u628a\u4efb\u610f\u65f6\u57df\u70b9\u4e58\u8f6c\u6362\u6210\u9891\u57df\u9010\u70b9\u4e58\u6cd5\u7684\u89c4\u5219\u3002<\/p>\n

#### $y=\\boldsymbol w^T\\boldsymbol x$ \u53c2\u6570\u5316\u7684\u5bf9\u7167<\/p>\n

\u82e5\u7cfb\u7edf\u4f7f\u7528<\/p>\n

$$
\ny[n]=\\boldsymbol w^T\\boldsymbol x_n,
\n\\qquad e[n]=d[n]-\\boldsymbol w^T\\boldsymbol x_n,
\n$$<\/p>\n

\u5219 $e[n]$ \u4f9d\u8d56 $\\boldsymbol w$\uff0c$e^*[n]$ \u4f9d\u8d56 $\\boldsymbol w^*$\u3002\u5bf9 $\\boldsymbol w^*$ \u6c42\u5bfc\u5f97\u5230<\/p>\n

$$
\n\\frac{\\partial J[n]}{\\partial\\boldsymbol w^*}
\n=-\\frac12\\boldsymbol x_n^*e[n],
\n$$<\/p>\n

\u4ece\u800c<\/p>\n

$$
\n\\boldsymbol w[n+1]=\\boldsymbol w[n]+\\mu\\boldsymbol x_n^*e[n].
\n$$<\/p>\n

\u4e24\u79cd\u5e38\u89c1\u6a21\u578b\u7684\u5bf9\u5e94\u5173\u7cfb\u662f\uff1a<\/p>\n

| \u8f93\u51fa\u6a21\u578b | \u68af\u5ea6\u65b9\u5411 | \u5e38\u89c1\u66f4\u65b0 |
\n|—|—|—|
\n| $y=\\boldsymbol w^H\\boldsymbol x$ | $-\\boldsymbol xe^*$ | $\\boldsymbol w\\leftarrow\\boldsymbol w+\\mu\\boldsymbol xe^*$ |
\n| $y=\\boldsymbol w^T\\boldsymbol x$ | $-\\boldsymbol x^*e$ | $\\boldsymbol w\\leftarrow\\boldsymbol w+\\mu\\boldsymbol x^*e$ |
\n| \u5b9e\u6570 $y=\\boldsymbol w^T\\boldsymbol x$ | $-\\boldsymbol xe$ | $\\boldsymbol w\\leftarrow\\boldsymbol w+\\mu\\boldsymbol xe$ |<\/p>\n

#### \u5757\u77e9\u9635\u68af\u5ea6\u4e0e $X^*E$<\/p>\n

\u628a\u4e00\u4e2a\u6570\u636e\u5757\u5199\u6210<\/p>\n

$$
\n\\boldsymbol y=\\boldsymbol X\\boldsymbol w,
\n\\qquad
\n\\boldsymbol e=\\boldsymbol d-\\boldsymbol X\\boldsymbol w,
\n\\qquad
\nJ=\\frac12\\boldsymbol e^H\\boldsymbol e.
\n$$<\/p>\n

\u82e5 $\\boldsymbol X\\in\\mathbb C^{M\\times L}$\uff0c\u5219<\/p>\n

$$
\n\\boldsymbol w\\in\\mathbb C^L,
\n\\qquad
\n\\boldsymbol e\\in\\mathbb C^M,
\n\\qquad
\n\\boldsymbol X^H\\boldsymbol e\\in\\mathbb C^L.
\n$$<\/p>\n

\u5bf9 $\\boldsymbol w^*$ \u6c42\u5bfc\u5e76\u91c7\u7528\u590d\u68af\u5ea6\u7ea6\u5b9a\uff1a<\/p>\n

$$
\n\\frac{\\partial J}{\\partial\\boldsymbol w^*}
\n=-\\frac12\\boldsymbol X^H\\boldsymbol e,
\n\\qquad
\n\\nabla_{\\boldsymbol w}J=-\\boldsymbol X^H\\boldsymbol e.
\n$$<\/p>\n

\u7b2c $\\ell$ \u4e2a\u5143\u7d20\u4e3a<\/p>\n

$$
\n[\\boldsymbol X^H\\boldsymbol e]_\\ell
\n=\\sum_nx^*[n-\\ell]e[n],
\n$$<\/p>\n

\u5b83\u6b63\u662f\u8f93\u5165\u548c\u8bef\u5dee\u7684\u76f8\u5173\u578b\u4e58\u52a0\u3002\u5f53\u5377\u79ef\u77e9\u9635\u7531 FFT \u5bf9\u89d2\u5316\u65f6\uff0cFourier \u5750\u6807\u4e2d\u7684\u9010\u70b9\u5f62\u5f0f\u4e3a<\/p>\n

$$
\nG[k]=X^*[k]E[k],
\n\\qquad
\n\\boldsymbol g=\\operatorname{IFFT}\\{X^*[k]E[k]\\}.
\n$$<\/p>\n

\u8fd9\u5c31\u662f\u201c\u65f6\u57df\u62bd\u5934\u68af\u5ea6\u4e2d\u51fa\u73b0\u5171\u8f6d\u4e58\u6cd5\u201d\u7684\u4e25\u683c\u6765\u6e90\u3002\u5177\u4f53\u7684\u5faa\u73af\u79fb\u4f4d\u3001\u8bef\u5dee\u8865\u96f6\u4f4d\u7f6e\u548c IFFT \u540e\u6709\u6548\u62bd\u5934\u533a\u95f4\u4ecd\u7531 overlap-save\u3001\u5206\u533a\u5ef6\u8fdf\u548c MDF \u6570\u636e\u6392\u5217\u51b3\u5b9a\uff0c\u4e0d\u80fd\u53ea\u770b $X^*E$ \u4e09\u4e2a\u7b26\u53f7\u3002<\/p>\n

#### \u5757\u9891\u57df LMS \u7684\u5f52\u4e00\u5316\u548c MDF \u6ce8\u610f\u4e8b\u9879<\/p>\n

\u5e38\u89c1\u7684\u5206\u533a\u9891\u57df\u66f4\u65b0\u4e3a<\/p>\n

$$
\nW_p^{(m+1)}[k]
\n=W_p^{(m)}[k]
\n+\\mu\\frac{X_p^*[k]E[k]}
\n {\\Phi_x[k]+\\varepsilon}.
\n$$<\/p>\n

\u5176\u4e2d $\\Phi_x[k]$ \u662f\u8f93\u5165\u529f\u7387\u4f30\u8ba1\uff0c$\\varepsilon$ \u7528\u4e8e\u907f\u514d\u4f4e\u529f\u7387\u9891\u70b9\u9664\u96f6\u548c\u566a\u58f0\u653e\u5927\u3002\u5de5\u7a0b\u4e2d\u5e94\u786e\u8ba4\u529f\u7387\u4f30\u8ba1\u4e0e FFT \u7f29\u653e\u4e00\u81f4\uff0c\u5e76\u89c2\u5bdf\u5f52\u4e00\u5316\u540e\u66f4\u65b0\u91cf\u662f\u5426\u51fa\u73b0\u5f02\u5e38\u5cf0\u503c\u3002<\/p>\n

MDF \u5e38\u7528 $2N$ \u70b9 FFT \u548c $N$ \u4e2a\u65b0\u6837\u672c\u3002\u82e5\u6709\u6548\u8bef\u5dee\u4e3a $e[0:N-1]$\uff0c\u4e00\u79cd\u5e38\u89c1\u6392\u5217\u662f<\/p>\n

$$
\nE_{\\mathrm{time}}[0:N-1]=0,
\n\\qquad
\nE_{\\mathrm{time}}[N:2N-1]=e[0:N-1].
\n$$<\/p>\n

\u653e\u5728\u524d\u534a\u6bb5\u8fd8\u662f\u540e\u534a\u6bb5\u53d6\u51b3\u4e8e\u5206\u533a\u8f93\u5165\u5386\u53f2\u3001\u5904\u7406\u5ef6\u8fdf\u548c\u6709\u6548\u62bd\u5934\u622a\u53d6\u3002\u8bef\u5dee\u5757\u9519\u79fb\u4e00\u4e2a\u6837\u672c\u4f1a\u5728\u9891\u57df\u5f15\u5165\u7ebf\u6027\u76f8\u4f4d\uff0c\u5bfc\u81f4\u66f4\u65b0\u9519\u8bef\u7684\u5ef6\u8fdf\u5206\u533a\u3002\u4e0e OLS\/OLA \u7684\u57fa\u7840\u8fb9\u754c\u89c4\u5219\u4e00\u6837\uff0cMDF \u5fc5\u987b\u901a\u8fc7\u5355\u4f4d\u8109\u51b2\u548c\u5df2\u77e5\u5ef6\u8fdf\u9010\u70b9\u9a8c\u8bc1\u3002<\/p>\n

#### \u6700\u5c0f NumPy \u4e92\u8bc1<\/p>\n

\u4e0b\u9762\u540c\u65f6\u9a8c\u8bc1\u76f4\u63a5\u7ebf\u6027\u76f8\u5173\u3001\u8865\u96f6 FFT \u76f8\u5173\u548c Parseval \u5185\u79ef\u3002\u76f8\u5173\u5b9a\u4e49\u4e3a
\n$R_{xy}[m]=\\sum_nx[n]y^*[n-m]$\uff1a<\/p>\n

“`python
\nimport numpy as np<\/p>\n

rng = np.random.default_rng(7)
\nx = rng.normal(size=5) + 1j * rng.normal(size=5)
\ny = rng.normal(size=3) + 1j * rng.normal(size=3)
\nlags = np.arange(-(len(y) – 1), len(x))
\ndirect = np.array([
\n sum(x[n] * np.conj(y[n – lag])
\n for n in range(len(x))
\n if 0 <= n - lag < len(y))\n for lag in lags\n])\n\nL = len(x) + len(y) - 1\nR = np.fft.ifft(np.fft.fft(x, L) * np.conj(np.fft.fft(y, L)))\nfft_linear = np.concatenate((R[-(len(y) - 1):], R[:len(x)]))\nnp.testing.assert_allclose(fft_linear, direct, rtol=1e-12, atol=1e-12)\n\nX = np.fft.fft(x)\ny_padded = np.pad(y, (0, len(x) - len(y)))\nY = np.fft.fft(y_padded)\nnp.testing.assert_allclose(np.sum(X * np.conj(Y)) \/ len(x),\n np.vdot(y_padded, x), rtol=1e-12, atol=1e-12)\nprint(\"correlation and Parseval verified\")\n```\n\n\u6b64\u5904 `ifft` \u81ea\u52a8\u5305\u542b `1\/L`\uff1b\u82e5\u4f7f\u7528\u672a\u5f52\u4e00\u5316\u7684\u9006\u53d8\u6362\uff0c\u5fc5\u987b\u624b\u52a8\u8865\u4e0a\u8be5\u56e0\u5b50\u3002\n\n---\n\n---\n\n## \u5341\u3001\u5178\u578b\u4f8b\u9898\u3001\u5e38\u89c1\u8bef\u533a\u4e0e\u68c0\u67e5\u6e05\u5355\n\n### \u4f8b\u9898\u4e00\uff1a\u4e24\u4e2a\u6709\u9650\u5e8f\u5217\u7684\u5377\u79ef\n\n\u8bbe\n\n$$\n x[n]=[1,2],\n\\qquad\n h[n]=[3,4,5].\n$$\n\n\u7ebf\u6027\u5377\u79ef\u957f\u5ea6\u4e3a $2+3-1=4$\u3002\u9010\u9879\u8ba1\u7b97\uff1a\n\n$$\n\\begin{aligned}\ny[0]&=1\\cdot3=3,\\\\\ny[1]&=1\\cdot4+2\\cdot3=10,\\\\\ny[2]&=1\\cdot5+2\\cdot4=13,\\\\\ny[3]&=2\\cdot5=10.\n\\end{aligned}\n$$\n\n\u56e0\u6b64\n\n$$\n\\boxed{x*h=[3,10,13,10]}.\n$$\n\n### \u4f8b\u9898\u4e8c\uff1a\u4e24\u4e2a\u5e8f\u5217\u7684\u4e92\u76f8\u5173\n\n\u8bbe\n\n$$\n x[n]=[1,2],\n\\qquad y[n]=[3,4].\n$$\n\n\u53d6\u5b9e\u4fe1\u53f7\u76f8\u5173\u5b9a\u4e49\n\n$$\nR_{xy}[m]=\\sum_nx[n]y[n-m].\n$$\n\n\u96f6\u5ef6\u8fdf\u76f8\u5173\u4e3a\n\n$$\nR_{xy}[0]=1\\cdot3+2\\cdot4=11.\n$$\n\n\u4e0d\u540c\u5ef6\u8fdf\u4e0b\u53ea\u4fdd\u7559\u91cd\u53e0\u90e8\u5206\uff0c\u53ef\u5f97\u5230\u4e00\u4e2a\u957f\u5ea6\u4e3a $2+2-1=3$ \u7684\u76f8\u5173\u5e8f\u5217\u3002\u91c7\u7528\u4e0d\u540c\u7684\u5ef6\u8fdf\u7d22\u5f15\u6392\u5217\u65f6\uff0c\u7ed3\u679c\u53ef\u80fd\u5199\u6210\n\n$$\n[4,11,6]\n$$\n\n\u6216\u5176\u53cd\u5411\u6392\u5217\n\n$$\n[6,11,4].\n$$\n\n\u8fd9\u4e0d\u662f\u6570\u503c\u77db\u76fe\uff0c\u800c\u662f\u76f8\u5173\u5b9a\u4e49\u548c\u5ef6\u8fdf\u7d22\u5f15\u65b9\u5411\u4e0d\u540c\u9020\u6210\u7684\u3002\n\n### \u4f8b\u9898\u4e09\uff1a\u5355\u4f4d\u8109\u51b2\u5b9a\u4f4d\u5ef6\u8fdf\n\n\u8bbe\u6a21\u677f\u4e3a\n\n$$\n s[n]=\\delta[n],\n$$\n\n\u63a5\u6536\u4fe1\u53f7\u4e3a\u5ef6\u8fdf\u7248\u672c\n\n$$\n r[n]=\\delta[n-n_0].\n$$\n\n\u4f7f\u7528\n\n$$\nR_{rs}[m]=\\sum_nr[n]s[n-m]\n$$\n\n\u6709\n\n$$\n\\begin{aligned}\nR_{rs}[m]\n&=\\sum_n\\delta[n-n_0]\\delta[n-m]\\\\\n&=\\delta[m-n_0].\n\\end{aligned}\n$$\n\n\u6240\u4ee5\u76f8\u5173\u5cf0\u4f4d\u4e8e $m=n_0$\u3002\u82e5\u628a\u76f8\u5173\u5b9a\u4e49\u5199\u6210 $s[n+m]$\uff0c\u5cf0\u7684\u7b26\u53f7\u65b9\u5411\u4f1a\u968f\u4e4b\u6539\u53d8\u3002\n\n### \u4f8b\u9898\u56db\uff1a\u77e9\u5f62\u8109\u51b2\u7684\u81ea\u76f8\u5173\n\n\u8bbe\n\n$$\n x(t)=u(t)-u(t-T).\n$$\n\n\u5176\u81ea\u76f8\u5173\u4e3a\u4e24\u4e2a\u76f8\u540c\u77e9\u5f62\u8109\u51b2\u7684\u76f8\u5173\uff0c\u7ed3\u679c\u662f\u4e09\u89d2\u5f62\uff1a\n\n$$\nR_{xx}(\\tau)=\n\\begin{cases}\nT-|\\tau|, & |\\tau|\\le T,\\\\\n0, & |\\tau|>T.
\n\\end{cases}
\n$$<\/p>\n

\u96f6\u5ef6\u8fdf\u5904<\/p>\n

$$
\nR_{xx}(0)=T
\n$$<\/p>\n

\u7b49\u4e8e\u77e9\u5f62\u8109\u51b2\u7684\u80fd\u91cf\u3002<\/p>\n

### \u7efc\u5408\u4f8b\u9898\u4e94\uff1a\u6709\u9650\u5e8f\u5217 FFT \u4e0e\u65f6\u57df\u76f8\u5173\u4e92\u8bc1<\/p>\n

\u8bbe<\/p>\n

$$
\n x[n]=[1,2,3,0],\\qquad y[n]=[0,1,2,3].
\n$$<\/p>\n

\u5176\u4e2d $y[n]=x[n-1]$\uff08\u53f3\u79fb 1 \u4e2a\u6837\u672c\uff09\u3002\u7528\u4e24\u6761\u8def\u5f84\u8ba1\u7b97\u4e92\u76f8\u5173\u5e76\u4e92\u76f8\u9a8c\u8bc1\u3002<\/p>\n

**\u8def\u5f84 A\uff1a\u65f6\u57df\u76f4\u63a5\u8ba1\u7b97\u3002** \u53d6\u5b9a\u4e49<\/p>\n

$$
\nR_{xy}[m]=\\sum_n x[n]y^*[n-m]=\\sum_n x[n]y[n-m]
\n$$<\/p>\n

\uff08\u5b9e\u5e8f\u5217\uff09\u3002\u6709\u6548 $m\\in\\{-3,-2,-1,0,1,2,3\\}$\uff0c\u53d6\u91cd\u53e0\u533a\u6c42\u548c\uff1a<\/p>\n

$$
\n\\begin{aligned}
\nR_{xy}[-3]&=x[0]y[3]=1\\cdot3=3,\\\\
\nR_{xy}[-2]&=x[0]y[2]+x[1]y[3]=1\\cdot2+2\\cdot3=8,\\\\
\nR_{xy}[-1]&=x[0]y[1]+x[1]y[2]+x[2]y[3]=1+4+9=14,\\\\
\nR_{xy}[0]&=\\textstyle\\sum_n x[n]y[n]=0+2+6+0=8,\\\\
\nR_{xy}[1]&=x[1]y[0]+x[2]y[1]+x[3]y[2]=0+2+0=2,\\\\
\nR_{xy}[2]&=x[2]y[0]+x[3]y[1]=0+0=0,\\\\
\nR_{xy}[3]&=x[3]y[0]=0.
\n\\end{aligned}
\n$$<\/p>\n

\u5cf0\u503c\u5728 $m=-1$\uff0c\u5bf9\u5e94”$y$ \u76f8\u5bf9 $x$ \u5ef6\u8fdf $-1$”\uff0c\u5373 $x$ \u843d\u540e\u4e8e $y$ \u53cd\u8fc7\u6765\u5373 $y=x[n-1]$\uff08\u5ef6\u8fdf 1\uff09\uff1a\u6ce8\u610f\u672c\u7ea6\u5b9a\u4e0b $R_{xy}$ \u5728 $m=-\\tau_0$ \u5904\u8fbe\u5230\u5cf0\uff08\u590d\u6838\u89c1\u7b2c\u56db\u7ae0\u201c\u5171\u8f6d\u65b9\u5411\u3001\u4e92\u8c31\u4e0e\u9891\u57df\u68af\u5ea6\u201d\u5c0f\u8282\uff09\u3002<\/p>\n

**\u8def\u5f84 B\uff1aFFT \u9891\u57df\u8ba1\u7b97\u3002** \u7ebf\u6027\u76f8\u5173\u6709\u6548\u957f\u5ea6 $2N-1=7$\uff0c\u53d6 $L=8$ \u70b9 FFT\uff08$L\\ge 2N-1$\uff0c\u65b9\u4fbf\u5bf9\u9f50\uff09\u3002\u5c06 $x,y$ \u5404\u8865\u96f6\u5230\u957f\u5ea6 8\uff1a<\/p>\n

$$
\n\\tilde x=[1,2,3,0,0,0,0,0],\\quad
\n\\tilde y=[0,1,2,3,0,0,0,0].
\n$$<\/p>\n

\u8ba1\u7b97 $X[k]=\\operatorname{FFT}(\\tilde x)$\u3001$Y[k]=\\operatorname{FFT}(\\tilde y)$\uff0c\u518d\u7531\u76f8\u5173\u5b9a\u7406<\/p>\n

$$
\n\\widehat R_{xy}[m]=\\operatorname{IFFT}\\{X[k]\\,Y^*[k]\\}.
\n$$<\/p>\n

\u5bf9\u672c\u4f8b\u53ef\u4ee5\u76f4\u63a5\u9a8c\u7b97\uff1a$Y[k]=X[k]e^{-j2\\pi k\/L}$\uff08$y$ \u76f8\u5bf9 $x$ \u5faa\u73af\u53f3\u79fb 1\uff09\uff0c\u6240\u4ee5<\/p>\n

$$
\nX[k]Y^*[k]=|X[k]|^2 e^{j2\\pi k\/L}
\n=\\operatorname{FFT}\\{R_{xx}\\text{ \u5faa\u73af\u5de6\u79fb }1\\}.
\n$$<\/p>\n

IFFT \u540e\u5373\u5f97 $R_{xx}$ \u7684\u5faa\u73af\u5de6\u79fb\uff1b\u56e0\u8865\u96f6\u8db3\u591f\u957f\uff0c\u5faa\u73af\u7ed3\u679c\u4e0e\u7ebf\u6027\u7ed3\u679c\u4e00\u81f4\uff1a\u5cf0\u4f4d\u4e8e IFFT \u8f93\u51fa\u7684 $k=L-1=7$ \u5904\uff0c\u91cd\u6392 $m\\in\\{-3,\\dots,3\\}$ \u5f97\u5cf0\u5728 $m=-1$\uff0c\u5176\u4ed6\u4f4d\u7f6e\u4e0e\u8def\u5f84 A \u6570\u503c\u9010\u4e00\u5bf9\u5e94\u3002<\/p>\n

**\u7ed3\u8bba\u4e0e\u4e92\u8bc1\u8981\u70b9\u3002**<\/p>\n

1. FFT \u957f\u5ea6\u5fc5\u987b $L\\ge N_x+N_y-1$\uff0c\u5426\u5219\u524d\u540e\u4f4d\u7f6e\u4f1a\u56e0\u5faa\u73af\u56de\u7ed5\u76f8\u4e92\u6c61\u67d3\uff08\u5bf9\u5e94\u7b2c\u56db\u7ae0”\u7ebf\u6027 vs. \u5faa\u73af”\u8b66\u793a\uff09\u3002
\n2. \u9891\u57df\u53d6\u5171\u8f6d\u653e\u5728\u53c2\u8003\u4fe1\u53f7\u4e0a\uff1a\u4e0e\u672c\u7ea6\u5b9a $R_{xy}[m]=\\sum x[n]y^*[n-m]$ \u4fdd\u6301\u4e00\u81f4\u7684\u662f $X\\cdot Y^*$\u3002\u82e5\u5199\u6210 $X^*\\cdot Y$\uff0c\u5f97\u5230\u7684\u662f $R_{yx}[m]=R_{xy}^*[-m]$\uff0c\u5cf0\u4f4d\u65b9\u5411\u76f8\u53cd\u3002
\n3. IFFT \u8f93\u51fa\u7684\u81ea\u7136\u7d22\u5f15\u4ece 0 \u5f00\u59cb\uff1b\u8981\u8bfb\u51fa $m<0$ \u7684\u76f8\u5173\u503c\uff0c\u9700\u8981\u6309 $m\\equiv k\\pmod L$ \u91cd\u6392\u3002\n\n### \u7efc\u5408\u4f8b\u9898\u516d\uff1a\u56fa\u5b9a\u65f6\u5ef6\u52a0\u566a\u58f0\u2014\u2014\u4e92\u8c31\u3001\u76f8\u5e72\u6027\u4e0e\u7fa4\u65f6\u5ef6\u4e00\u4f53\u5316\n\n\u8bbe\u5bbd\u5e73\u7a33\u96f6\u5747\u503c\u5b9e\u4fe1\u53f7 $x(t)$\uff0c\u89c2\u6d4b\u4e24\u8def\n\n$$\n x_1(t)=x(t)+n_1(t),\\qquad\n x_2(t)=x(t-\\tau_0)+n_2(t),\n$$\n\n\u5176\u4e2d $n_1,n_2$ \u4e92\u4e0d\u76f8\u5173\uff0c\u4e5f\u4e0e $x$ \u4e0d\u76f8\u5173\uff0cPSD \u5206\u522b\u4e3a $N_1(f),N_2(f)$\u3002\n\n**\u4e92\u8c31\u3002** \u7531 $x_2(t)$ \u7684\u9891\u57df\u5ef6\u8fdf\u56e0\u5b50\n\n$$\nS_{x_1x_2}(f)=S_{xx}(f)\\,e^{-j2\\pi f\\tau_0},\n$$\n\n\u81ea\u8c31\u4e3a\n\n$$\nS_{x_1x_1}(f)=S_{xx}(f)+N_1(f),\\qquad\nS_{x_2x_2}(f)=S_{xx}(f)+N_2(f).\n$$\n\n**\u76f8\u5e72\u6027\u3002**\n\n$$\n\\gamma^2_{x_1x_2}(f)\n=\\frac{|S_{xx}(f)|^2}\n{[S_{xx}(f)+N_1(f)][S_{xx}(f)+N_2(f)]}\n=\\frac{1}{(1+1\/\\rho_1)(1+1\/\\rho_2)},\n$$\n\n\u5176\u4e2d $\\rho_i(f)=S_{xx}(f)\/N_i(f)$ \u4e3a\u8be5\u901a\u9053\u7684\u8c31\u57df SNR\u3002\u7279\u4f8b\uff1a\n\n- \u65e0\u566a\u58f0 $N_1=N_2=0$\uff1a$\\gamma^2=1$\uff0c\u4efb\u610f\u9891\u7387\u90fd\u662f\u7406\u60f3\u7ebf\u6027\u5173\u7cfb\uff1b\n- \u4e00\u8def\u4fe1\u53f7\u88ab\u5f3a\u566a\u58f0\u5b8c\u5168\u6df9\u6ca1\uff1a$\\gamma^2\\to0$\uff1b\n- \u5355\u901a\u9053\u767d\u566a $N_1=0,N_2=N_0$\uff1a$\\gamma^2(f)=\\rho_2\/(1+\\rho_2)$\uff0c\u4e0e\u7b2c\u4e03\u7ae0\"\u8f93\u51fa\u52a0\u4e0d\u76f8\u5173\u566a\u58f0\"\u516c\u5f0f\u4e00\u81f4\u3002\n\n**\u7fa4\u65f6\u5ef6\u3002** \u4e92\u8c31\u76f8\u4f4d\u4e3a\n\n$$\n\\phi(f)=\\angle S_{x_1x_2}(f)=-2\\pi f\\tau_0\\pmod{2\\pi},\n$$\n\n\u56e0\u6b64\u7fa4\u65f6\u5ef6\n\n$$\n\\tau_g(f)=-\\frac{1}{2\\pi}\\frac{d\\phi(f)}{df}=\\tau_0\n$$\n\n\u5728\u6240\u6709 $\\gamma^2(f)$ \u8db3\u591f\u5927\u7684\u9891\u6bb5\u5e94\u4e3a\u5e38\u6570 $\\tau_0$\u3002\u5de5\u7a0b\u4e0a\u6309\u7b2c\u516b\u7ae0\u201c\u76f8\u5e72\u6027\u5206\u6790\u7684\u62a5\u544a\u6a21\u677f\u201d\u6267\u884c\uff1a\n\n1. \u7528 Welch \u5f97 $\\widehat S_{x_1x_1},\\widehat S_{x_2x_2},\\widehat S_{x_1x_2}$\uff1b\n2. \u8ba1\u7b97 $\\widehat\\gamma^2$\uff0c\u53ea\u5bf9 $\\widehat\\gamma^2\\ge \\gamma^2_{\\min}$ \u7684 bin \u4fdd\u7559\u76f8\u4f4d\uff1b\n3. \u5bf9\u4fdd\u7559\u7684\u76f8\u4f4d\u89e3\u7f20\u7ed5\u540e\u505a\u7ebf\u6027\u56de\u5f52\uff0c\u659c\u7387\u7684 $-1\/(2\\pi)$ \u5373\u65f6\u5ef6\u4f30\u8ba1 $\\widehat\\tau_0$\uff1b\n4. \u4e0e GCC-PHAT \u7ed3\u679c\u4e92\u76f8\u6bd4\u5bf9\uff1aGCC-PHAT \u901a\u8fc7\u5bf9\u5e45\u5ea6\u5f52\u4e00\u5316\u540e IFFT \u6c42\u5cf0\u7ed9\u51fa $\\widehat\\tau_0$\uff0c\u4e0e\u76f8\u4f4d\u659c\u7387\u6cd5\u5728\u9ad8\u76f8\u5e72\u5e26\u5185\u5e94\u4e00\u81f4\uff1b\u5dee\u5f02\u8fc7\u5927\u63d0\u793a\u5b58\u5728\u8272\u6563\u3001\u975e\u7ebf\u6027\u6216\u591a\u5f84\u3002\n\n\u672c\u4f8b\u628a\u7b2c\u4e94\u7ae0\u4e92\u8c31\u3001\u7b2c\u4e03\u7ae0\u76f8\u5e72\u6027\u548c\u7b2c\u4e5d\u7ae0 GCC-PHAT \u4e0e\u7cfb\u7edf\u8fa8\u8bc6\u7528\u540c\u4e00\u6a21\u578b\u4e32\u8d77\u6765\uff1a$\\gamma^2$ \u544a\u8bc9\u6211\u4eec\u54ea\u91cc\u53ef\u4fe1\uff0c\u4e92\u8c31\u76f8\u4f4d\u544a\u8bc9\u6211\u4eec\u65f6\u5ef6\u662f\u591a\u5c11\uff0c\u4e24\u8005\u7f3a\u4e00\u4e0d\u53ef\u3002\n\n### \u7efc\u5408\u4f8b\u9898\u4e03\uff1a\u767d\u566a\u58f0\u7ecf LTI\u2014\u2014PSD\u3001\u9891\u7387\u54cd\u5e94\u4e0e\u529f\u7387\u79ef\u5206\n\n\u8bbe\u79bb\u6563\u767d\u566a\u58f0\u8f93\u5165 $x[n]$\uff0c$R_{xx}[m]=\\sigma_x^2\\delta[m]$\uff0c\u6545 $S_{xx}(e^{j\\hat\\omega})=\\sigma_x^2$\uff08\u53cc\u8fb9\uff09\uff0c\u91c7\u6837\u7387 $f_s$\u3002\u901a\u8fc7\u4e00\u9636 IIR\n\n$$\n y[n]=x[n]+a\\,y[n-1],\\qquad |a|<1.\n$$\n\n**\u9891\u7387\u54cd\u5e94\u4e0e\u8f93\u51fa PSD\u3002**\n\n$$\n H(e^{j\\hat\\omega})=\\frac{1}{1-a e^{-j\\hat\\omega}},\\qquad\n|H|^2=\\frac{1}{1-2a\\cos\\hat\\omega+a^2}.\n$$\n\n\u7531\u7b2c\u4e8c\u7ae0 LTI \u7ed3\u679c $S_{yy}=|H|^2 S_{xx}$\uff1a\n\n$$\nS_{yy}(e^{j\\hat\\omega})\n=\\frac{\\sigma_x^2}{1-2a\\cos\\hat\\omega+a^2}.\n$$\n\n**\u8f93\u51fa\u603b\u529f\u7387\uff08\u65f6\u57df\u6cd5\uff09\u3002** \u7531 $R_{yy}[m]=\\sigma_x^2 a^{|m|}\/(1-a^2)$ \u5f97\n\n$$\nP_y=R_{yy}[0]=\\frac{\\sigma_x^2}{1-a^2}.\n$$\n\n**\u8f93\u51fa\u603b\u529f\u7387\uff08\u9891\u57df\u6cd5\uff09\u3002** \u7531 Parseval\uff0c\u8f93\u51fa\u529f\u7387\u662f\u53cc\u8fb9 PSD \u7684\u79ef\u5206\uff1a\n\n$$\nP_y=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}\\!S_{yy}(e^{j\\hat\\omega})\\,d\\hat\\omega\n=\\frac{\\sigma_x^2}{2\\pi}\\int_{-\\pi}^{\\pi}\\!\\frac{d\\hat\\omega}{1-2a\\cos\\hat\\omega+a^2}\n=\\frac{\\sigma_x^2}{1-a^2}.\n$$\n\n\u4e24\u6761\u8def\u5f84\u7ed9\u51fa\u540c\u4e00\u6570\u503c\uff0c\u4ea4\u53c9\u9a8c\u8bc1\u6210\u529f\u3002\n\n**\u8fa8\u8bc6 $H$ \u7684\u4e09\u79cd\u8c31\u4f30\u8ba1\u91cf\u5bf9\u7167\u3002** \u82e5\u540c\u6b65\u91c7\u96c6 $x[n],y[n]$\uff0cWelch \u5f97 $\\widehat S_{xx},\\widehat S_{yy},\\widehat S_{yx}$\uff1a\n\n$$\n\\widehat H_1=\\frac{\\widehat S_{yx}}{\\widehat S_{xx}},\\quad\n\\widehat H_2=\\frac{\\widehat S_{yy}}{\\widehat S_{xy}},\\quad\n\\widehat H_v=\\sqrt{\\widehat H_1\\widehat H_2}.\n$$\n\n\u5728\u672c\u7eaf LTI \u65e0\u8f93\u51fa\u566a\u58f0\u573a\u666f\u4e0b\u4e09\u8005\u7406\u8bba\u4e00\u81f4\uff1b\u82e5\u5728\u8f93\u51fa\u52a0\u4e92\u4e0d\u76f8\u5173\u566a\u58f0 $v[n]$\uff08PSD $N_v$\uff09\uff0c\u5219 $\\widehat H_1$ \u65e0\u504f\u800c $\\widehat H_2$ \u4f1a\u88ab\u9ad8\u4f30\uff1a\n\n$$\n\\mathbb E[\\widehat H_1]=H,\\qquad\n\\mathbb E[\\widehat H_2]=H\\cdot\\left(1+\\frac{N_v}{|H|^2 S_{xx}}\\right).\n$$\n\n\u76f8\u5e72\u6027\n\n$$\n\\gamma^2(f)=\\frac{|H|^2 S_{xx}}{|H|^2 S_{xx}+N_v}\n$$\n\n\u7ed9\u51fa\u5404\u9891\u6bb5 $\\widehat H_1$ \u7684\u53ef\u4fe1\u5ea6\uff0c\u4e0e\u7b2c\u4e5d\u7ae0\"\u7cfb\u7edf\u8fa8\u8bc6\u6d41\u7a0b\"\u6b65\u9aa4 5\u30016 \u5b8c\u5168\u5bf9\u5e94\u3002\n\n**\u5de5\u7a0b\u8981\u70b9\u3002** \u767d\u566a\u6fc0\u52b1\u4f7f $S_{xx}$ \u5e73\u5766\uff0c\u4efb\u4f55\u9891\u6bb5\u90fd\u80fd\u88ab\u8fa8\u8bc6\uff1b\u6362\u6210\u7a84\u5e26\u6fc0\u52b1\u65f6\uff0c\u6fc0\u52b1\u80fd\u91cf\u4e0d\u8db3\u7684\u9891\u5e26\u5373\u4f7f\u6570\u5b66\u4e0a\u4ecd\u53ef\u8ba1\u7b97 $S_{yx}\/S_{xx}$\uff0c\u4e5f\u5e94\u6839\u636e\u76f8\u5e72\u6027\u628a\u8fd9\u4e9b\u7ed3\u679c\u4ece\u62a5\u544a\u4e2d\u5254\u9664\uff0c\u907f\u514d\"\u6570\u503c\u6709\u503c = \u7ed3\u679c\u53ef\u4fe1\"\u7684\u8bef\u5224\u3002\n\n---\n### \u5e38\u89c1\u8bef\u533a\n\n#### \u8bef\u533a\uff1a\u65f6\u57df\u76f8\u4e58\u603b\u662f\u5bf9\u5e94\u9891\u57df\u76f8\u4e58\n\n\u9519\u8bef\u3002\u6b63\u786e\u5173\u7cfb\u662f\uff1a\n\n$$\n\\mathcal F\\{x(t)h(t)\\}\n=\\frac{1}{2\\pi}X(\\omega)*H(\\omega).\n$$\n\n\u65f6\u57df\u5377\u79ef\u624d\u5bf9\u5e94\u9891\u57df\u9010\u70b9\u76f8\u4e58\u3002\n\n#### \u8bef\u533a\uff1a\u76f8\u5173\u5c31\u662f\u5377\u79ef\n\n\u4e0d\u5b8c\u5168\u6b63\u786e\u3002\u76f8\u5173\u53ef\u4ee5\u8868\u793a\u6210\u5e26\u6709\u5171\u8f6d\u548c\u65f6\u95f4\u53cd\u8f6c\u7684\u5377\u79ef\uff1a\n\n$$\nR_{xy}=x*y^*(-t)\n$$\n\n\u4f46\u76f8\u5173\u7684\u7269\u7406\u76ee\u6807\u662f\u5339\u914d\u548c\u76f8\u4f3c\u6027\uff0c\u5377\u79ef\u7684\u5178\u578b\u76ee\u6807\u662f\u7cfb\u7edf\u54cd\u5e94\u548c\u6ee4\u6ce2\u3002\n\n#### \u8bef\u533a\uff1a\u76f8\u5173\u4e2d\u5171\u8f6d\u4f4d\u7f6e\u6c38\u8fdc\u56fa\u5b9a\n\n\u5e38\u89c1\u5f62\u5f0f\u5305\u62ec\uff1a\n\n$$\nX(\\omega)Y^*(\\omega),\n\\qquad\nX^*(\\omega)Y(\\omega).\n$$\n\n\u5171\u8f6d\u4f4d\u7f6e\u53d6\u51b3\u4e8e\uff1a\n\n1. \u4e92\u76f8\u5173\u7684\u5b9a\u4e49\uff1b\n2. \u8c01\u4f5c\u4e3a\u53c2\u8003\u4fe1\u53f7\uff1b\n3. \u5ef6\u8fdf\u5199\u6210 $t-\\tau$ \u8fd8\u662f $t+\\tau$\uff1b\n4. Fourier \u53d8\u6362\u7684\u6b63\u8d1f\u53f7\u7ea6\u5b9a\u3002\n\n#### \u8bef\u533a\uff1a\u76f8\u5173\u5cf0\u7684\u6b63\u8d1f\u65b9\u5411\u53ef\u4ee5\u8131\u79bb\u5b9a\u4e49\u5224\u65ad\n\n\u4e0d\u80fd\u3002\u5fc5\u987b\u628a\u4fe1\u53f7\u6a21\u578b\u3001\u76f8\u5173\u516c\u5f0f\u548c\u5ef6\u8fdf\u5b9a\u4e49\u653e\u5728\u4e00\u8d77\u3002\u4f8b\u5982\u201c\u63a5\u6536\u4fe1\u53f7\u6bd4\u6a21\u677f\u665a\u5230\u201d\u5728\u4e0d\u540c\u76f8\u5173\u7ea6\u5b9a\u4e0b\u53ef\u80fd\u8868\u73b0\u4e3a\u6b63\u5cf0\u6216\u8d1f\u5cf0\u3002\n\n#### \u8bef\u533a\uff1aFFT \u9891\u57df\u76f8\u4e58\u76f4\u63a5\u5f97\u5230\u7ebf\u6027\u5377\u79ef\n\n\u76f4\u63a5\u4f7f\u7528\u76f8\u540c\u957f\u5ea6 FFT \u540e\u9891\u57df\u76f8\u4e58\u5f97\u5230\u7684\u662f\u5faa\u73af\u5377\u79ef\u3002\u82e5\u9700\u8981\u7ebf\u6027\u5377\u79ef\uff0cFFT \u957f\u5ea6\u81f3\u5c11\u8981\u6ee1\u8db3\n\n$$\nL\\ge N+M-1.\n$$\n\n#### \u8bef\u533a\uff1a\u590d\u6570\u4fe1\u53f7\u7684\u5377\u79ef\u9700\u8981\u5171\u8f6d\u4e58\u6cd5\n\n\u9519\u8bef\u3002\u7ebf\u6027\u5377\u79ef\u7684\u5b9a\u4e49\u2014\u2014\u65e0\u8bba\u4fe1\u53f7\u662f\u5b9e\u7684\u8fd8\u662f\u590d\u7684\u2014\u2014\u59cb\u7ec8\u4f7f\u7528\u666e\u901a\u590d\u6570\u4e58\u6cd5\uff1a\n\n$$\ny[n]=\\sum_k x[k]h[n-k].\n$$\n\n\u5171\u8f6d\u51fa\u73b0\u5728\u76f8\u5173\u548c\u5185\u79ef\u7684\u5b9a\u4e49\u4e2d\uff0c\u4e0d\u662f\u5377\u79ef\u56e0\u4fe1\u53f7\u53d8\u4e3a\u590d\u6570\u540e\u81ea\u52a8\u83b7\u5f97\u7684\u3002\u8be6\u89c1\u7b2c\u4e8c\u7ae0\u201c\u590d\u6570\u4fe1\u53f7\u7684\u5377\u79ef\uff1a\u4e0d\u4f7f\u7528\u5171\u8f6d\u4e58\u6cd5\u201d\u3002\n\n#### \u8bef\u533a\uff1a\u81ea\u76f8\u5173\u5728\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u662f\u5076\u51fd\u6570\n\n\u53ea\u6709\u5b9e\u4fe1\u53f7\u7684\u81ea\u76f8\u5173\u901a\u5e38\u662f\u5076\u51fd\u6570\u3002\u590d\u4fe1\u53f7\u6ee1\u8db3\u7684\u662f\u5171\u8f6d\u5bf9\u79f0\uff1a\n\n$$\nR_{xx}(-\\tau)=R_{xx}^*(\\tau).\n$$\n\n#### \u8bef\u533a\uff1a\u76f8\u5173\u5cf0\u8d8a\u9ad8\uff0c\u4fe1\u53f7\u4e00\u5b9a\u8d8a\u76f8\u4f3c\n\n\u672a\u5f52\u4e00\u5316\u76f8\u5173\u4f1a\u53d7\u5230\u4fe1\u53f7\u80fd\u91cf\u5f71\u54cd\u3002\u6bd4\u8f83\u4e0d\u540c\u5e45\u5ea6\u4fe1\u53f7\u65f6\uff0c\u5e94\u8003\u8651\u5f52\u4e00\u5316\u76f8\u5173\u7cfb\u6570\u3002\n\n#### \u8bef\u533a\uff1a\u76f8\u5173\u503c\u53ea\u80fd\u662f\u5b9e\u6570\n\n\u5b9e\u4fe1\u53f7\u76f8\u5173\u901a\u5e38\u4e3a\u5b9e\u6570\uff1b\u590d\u4fe1\u53f7\u4e92\u76f8\u5173\u4e00\u822c\u4e3a\u590d\u6570\u3002\u5176\u865a\u90e8\u627f\u8f7d\u76f8\u5bf9\u76f8\u4f4d\u4fe1\u606f\uff0c\u4e0d\u5e94\u968f\u610f\u4e22\u5f03\u3002\n\n#### \u8bef\u533a\uff1a\u76f8\u5173\u53ea\u9002\u7528\u4e8e\u786e\u5b9a\u6027\u4fe1\u53f7\n\n\u76f8\u5173\u540c\u6837\u662f\u968f\u673a\u8fc7\u7a0b\u5206\u6790\u4e2d\u7684\u6838\u5fc3\u5de5\u5177\u3002\u968f\u673a\u8fc7\u7a0b\u7684\u81ea\u76f8\u5173\u51fd\u6570\u3001\u4e92\u76f8\u5173\u51fd\u6570\u548c\u529f\u7387\u8c31\u5bc6\u5ea6\u7528\u4e8e\u63cf\u8ff0\u7edf\u8ba1\u4f9d\u8d56\u3001\u5e73\u7a33\u6027\u548c\u9891\u7387\u80fd\u91cf\u5206\u5e03\u3002\n\n#### \u8bef\u533a\uff1a\u76f8\u5173\u7cfb\u6570\u4e3a\u96f6\u5c31\u8bf4\u660e\u4e24\u4e2a\u53d8\u91cf\u72ec\u7acb\n\n\u4e00\u822c\u4e0d\u6210\u7acb\u3002\u96f6\u76f8\u5173\u53ea\u8868\u793a\u4e0d\u5b58\u5728\u7531\u534f\u65b9\u5dee\u523b\u753b\u7684\u7ebf\u6027\u7edf\u8ba1\u5173\u8054\uff1b\u9664\u975e\u53d8\u91cf\u8054\u5408 Gaussian\uff0c\u6216\u8005\u53e6\u6709\u66f4\u5f3a\u7684\u5206\u5e03\u6761\u4ef6\uff0c\u5426\u5219\u96f6\u76f8\u5173\u4e0d\u80fd\u63a8\u51fa\u7edf\u8ba1\u72ec\u7acb\u3002\u72ec\u7acb\u4e14\u4e8c\u9636\u77e9\u5b58\u5728\u901a\u5e38\u53ef\u4ee5\u63a8\u51fa\u96f6\u534f\u65b9\u5dee\uff0c\u53cd\u65b9\u5411\u5219\u901a\u5e38\u4e0d\u6210\u7acb\u3002\n\n#### \u8bef\u533a\uff1a\u76f8\u5e72\u6027\u9ad8\u5c31\u8bc1\u660e\u4e00\u4e2a\u4fe1\u53f7\u5bfc\u81f4\u53e6\u4e00\u4e2a\u4fe1\u53f7\n\n\u9519\u8bef\u3002\u76f8\u5e72\u6027\u53cd\u6620\u4e24\u4e2a\u4fe1\u53f7\u5728\u67d0\u4e2a\u9891\u7387\u4e0a\u7684\u7a33\u5b9a\u7ebf\u6027\u5173\u8054\uff0c\u4f46\u4e0d\u80fd\u5355\u72ec\u8bc1\u660e\u56e0\u679c\u5173\u7cfb\u3002\u5171\u540c\u8f93\u5165\u3001\u4e32\u6270\u3001\u540c\u6b65\u65f6\u949f\u3001\u53cd\u9988\u56de\u8def\u6216\u5904\u7406\u94fe\u4e2d\u7684\u5171\u540c\u6ee4\u6ce2\u90fd\u53ef\u80fd\u4ea7\u751f\u9ad8\u76f8\u5e72\u6027\u3002\n\n#### \u8bef\u533a\uff1a\u5355\u6b21 FFT \u5c31\u80fd\u53ef\u9760\u4f30\u8ba1\u76f8\u5e72\u6027\n\n\u7528\u540c\u4e00\u6bb5\u6570\u636e\u5f62\u6210\u672a\u7ecf\u5e73\u5747\u7684\u5468\u671f\u56fe\u65f6\uff0c\u5206\u5b50\u548c\u5206\u6bcd\u53ef\u80fd\u53d1\u751f\u4ee3\u6570\u62b5\u6d88\uff0c\u4f7f\u4f30\u8ba1\u7684 magnitude-squared coherence \u9000\u5316\u4e3a 1\u3002\u53ef\u9760\u4f30\u8ba1\u901a\u5e38\u9700\u8981\u591a\u4e2a\u8fd1\u4f3c\u72ec\u7acb\u7684\u6570\u636e\u6bb5\u3001\u9002\u5f53\u52a0\u7a97\u3001\u91cd\u53e0\u548c\u8c31\u5e73\u5747\uff0c\u5e76\u62a5\u544a\u9891\u7387\u5206\u8fa8\u7387\u4e0e\u7f6e\u4fe1\u6c34\u5e73\u3002\n\n#### \u8bef\u533a\uff1aGCC-PHAT \u4e00\u5b9a\u4f18\u4e8e\u666e\u901a\u4e92\u76f8\u5173\n\nPHAT \u901a\u8fc7\u5bf9\u5e45\u5ea6\u505a\u5f52\u4e00\u5316\u63d0\u9ad8\u4e86\u5cf0\u503c\u9510\u5ea6\uff0c\u4f46\u5728\u4f4e SNR \u9891\u6bb5\u5e45\u5ea6\u5c0f\uff0c\u9664\u6cd5\u4f1a\u653e\u5927\u566a\u58f0\u76f8\u4f4d\uff1b\u5bf9\u7a00\u758f\u8c31\u4fe1\u53f7\u6216\u5f3a\u8272\u6563\u4fe1\u9053\u4e5f\u53ef\u80fd\u5f15\u5165\u4f2a\u5cf0\u3002\u56e0\u6b64\u9700\u8981\u6b63\u5219\u5316 $\\varepsilon$\u3001\u9891\u5e26\u9608\u503c\u6216\u4e0e\u666e\u901a\u4e92\u76f8\u5173\u3001Roth\u3001SCOT \u52a0\u6743\u505a\u5bf9\u6bd4\u3002\u65b9\u6cd5\u9009\u62e9\u5e94\u7531\u4fe1\u53f7-\u566a\u58f0\u9891\u8c31\u7ed3\u6784\u51b3\u5b9a\uff0c\u800c\u4e0d\u662f\"\u8d8a\u590d\u6742\u8d8a\u597d\"\u3002\n\n#### \u8bef\u533a\uff1a$\\widehat H_1$ \u548c $\\widehat H_2$ \u5728\u771f\u5b9e\u6570\u636e\u4e0a\u5e94\u5b8c\u5168\u76f8\u7b49\n\n\u53ea\u6709\u566a\u58f0\u4e0e\u5047\u8bbe\u5b8c\u5168\u5339\u914d\uff08$\\widehat H_1$ \u5bf9\u5e94\u8f93\u5165\u65e0\u566a\uff0c$\\widehat H_2$ \u5bf9\u5e94\u8f93\u51fa\u65e0\u566a\uff09\u65f6\u4e8c\u8005\u624d\u4e00\u81f4\u3002\u771f\u5b9e\u6570\u636e\u4e24\u7aef\u90fd\u53ef\u80fd\u6709\u566a\u58f0\uff0c\u4e8c\u8005\u5f80\u5f80\u4e0d\u540c\uff1b\u5dee\u5f02\u5927\u5c0f\u53ef\u4ee5\u53cd\u8fc7\u6765\u4f30\u8ba1\u566a\u58f0\u5206\u5e03\u65b9\u5411\uff0c\u4e5f\u662f\u76f8\u5e72\u6027\u663e\u8457\u4f4e\u4e8e 1 \u7684\u663e\u5f0f\u8868\u5f81\u3002\n\n#### \u8bef\u533a\uff1a\u5339\u914d\u6ee4\u6ce2\u5668\u7684\u6700\u4f18 SNR \u4e0e\u6ce2\u5f62\u5f62\u72b6\u6709\u5173\n\n\u5728 AWGN \u4e0b\uff0c\u5339\u914d\u6ee4\u6ce2\u8f93\u51fa SNR \u4e3a $2E_s\/N_0$\uff0c\u53ea\u4f9d\u8d56\u6a21\u677f\u603b\u80fd\u91cf\u548c\u566a\u58f0 PSD\uff0c\u4e0e\u8109\u51b2\u5f62\u72b6\u65e0\u5173\u3002\u6539\u53d8\u5f62\u72b6\u5f71\u54cd\u7684\u662f\u65f6\u5ef6\u5206\u8fa8\u7387\u3001\u591a\u666e\u52d2\u5bb9\u5dee\u548c\u65c1\u74e3\u7ed3\u6784\uff0c\u800c\u4e0d\u662f\u5cf0\u503c SNR \u672c\u8eab\u3002\n\n#### \u8bef\u533a\uff1a\u76f8\u4f4d\u659c\u7387\u6cd5\u53ef\u4ee5\u5728\u4efb\u610f\u9891\u6bb5\u8bfb\u65f6\u5ef6\n\n\u53ea\u6709\u5728\u76f8\u5e72\u6027\u8db3\u591f\u9ad8\u3001$S_{xy}$ \u5e45\u5ea6\u8db3\u591f\u5927\u7684\u9891\u6bb5\uff0c\u4e92\u8c31\u76f8\u4f4d\u624d\u7a33\u5b9a\uff1b\u4f4e\u76f8\u5e72\u6216\u8c31\u529f\u7387\u4e0d\u8db3\u5904\u76f8\u4f4d\u7531\u566a\u58f0\u4e3b\u5bfc\uff0c\u659c\u7387\u6beb\u65e0\u610f\u4e49\u3002\u5de5\u7a0b\u4e0a\u5e94\u5148\u6309 $\\gamma^2$ \u95e8\u9650\u7b5b\u9891\u6bb5\uff0c\u518d\u89e3\u7f20\u7ed5\u505a\u7ebf\u6027\u56de\u5f52\u3002\n\n---\n### \u76f8\u5173\u6027\u4e0e\u76f8\u5e72\u6027\u7684\u5178\u578b\u4f8b\u5b50\n\n#### \u4f8b\u4e00\uff1a\u56fa\u5b9a\u65f6\u5ef6\u7684\u540c\u4e00\u4fe1\u53f7\n\n\u8bbe\n\n$$\ny(t)=a\\,x(t-\\tau_0),\n$$\n\n\u4e14\u6ca1\u6709\u566a\u58f0\u3002\u82e5 $x$ \u662f\u5bbd\u5e73\u7a33\u8fc7\u7a0b\uff0c\u5219\u4e92\u76f8\u5173\u662f\u81ea\u76f8\u5173\u7684\u5e73\u79fb\u548c\u7f29\u653e\uff1a\n\n$$\nR_{xy}(\\tau)=a^*R_{xx}(\\tau-\\tau_0)\n$$\n\n\u6216\u6309\u53e6\u4e00\u7ea6\u5b9a\u51fa\u73b0\u76f8\u53cd\u5e73\u79fb\u3002\u4e92\u76f8\u5173\u5cf0\u7ed9\u51fa $\\tau_0$\u3002\n\n\u9891\u57df\u4e2d\uff1a\n\n$$\nY(f)=aX(f)e^{-j2\\pi f\\tau_0}.\n$$\n\n\u7406\u8bba MSC \u4e3a\n\n$$\n\\gamma_{xy}^2(f)=1\n$$\n\n\uff08\u5728 $S_{xx}(f)>0$ \u7684\u9891\u7387\uff09\uff0c\u800c\u4e92\u8c31\u76f8\u4f4d\u662f\u4e00\u6761\u5173\u4e8e\u9891\u7387\u7684\u76f4\u7ebf\u3002\u8fd9\u4e2a\u4f8b\u5b50\u8bf4\u660e\uff0c\u56fa\u5b9a\u65f6\u5ef6\u4f1a\u6539\u53d8\u96f6\u5ef6\u8fdf\u76f8\u5173\u548c\u76f8\u4f4d\uff0c\u4f46\u4e0d\u4f1a\u964d\u4f4e\u7406\u60f3\u7ebf\u6027\u76f8\u5e72\u6027\u3002<\/p>\n

#### \u4f8b\u4e8c\uff1a\u8f93\u51fa\u53e0\u52a0\u4e0d\u76f8\u5173\u566a\u58f0<\/p>\n

\u8bbe<\/p>\n

$$
\ny(t)=x(t)+n(t),
\n$$<\/p>\n

\u5176\u4e2d $n$ \u4e0e $x$ \u4e0d\u76f8\u5173\u3002\u5219<\/p>\n

$$
\nS_{xy}=S_{xx},
\n\\qquad
\nS_{yy}=S_{xx}+S_{nn}.
\n$$<\/p>\n

\u6240\u4ee5<\/p>\n

$$
\n\\gamma_{xy}^2(f)
\n=\\frac{S_{xx}(f)}{S_{xx}(f)+S_{nn}(f)}.
\n$$<\/p>\n

\u5728\u4fe1\u53f7\u4e3b\u5bfc\u9891\u5e26\uff0c\u76f8\u5e72\u6027\u63a5\u8fd1 1\uff1b\u5728\u566a\u58f0\u4e3b\u5bfc\u9891\u5e26\uff0c\u76f8\u5e72\u6027\u63a5\u8fd1 0\u3002\u8fd9\u6bd4\u5355\u4e2a\u603b\u4f53\u76f8\u5173\u7cfb\u6570\u66f4\u6e05\u695a\u5730\u663e\u793a\u566a\u58f0\u5728\u54ea\u4e9b\u9891\u7387\u7834\u574f\u4e86\u7ebf\u6027\u8054\u7cfb\u3002<\/p>\n

#### \u4f8b\u4e09\uff1a\u5171\u540c\u8f93\u5165\u9020\u6210\u4f2a\u76f4\u63a5\u5173\u7cfb<\/p>\n

\u8bbe<\/p>\n

$$
\nx=h_1*u+n_x,
\n\\qquad
\ny=h_2*u+n_y.
\n$$<\/p>\n

\u5373 $x$\u3001$y$ \u90fd\u7531\u9690\u85cf\u6e90 $u$ \u9a71\u52a8\u3002\u4e8c\u8005\u53ef\u80fd\u5177\u6709\u5f88\u9ad8\u7684\u4e92\u76f8\u5173\u548c\u76f8\u5e72\u6027\uff0c\u4f46\u5e76\u4e0d\u8868\u793a $x$ \u76f4\u63a5\u5bfc\u81f4 $y$\u3002\u82e5\u540c\u65f6\u6d4b\u5f97 $u$\uff0c\u53ef\u8fdb\u4e00\u6b65\u8ba1\u7b97\u504f\u76f8\u5e72\u6027\uff0c\u5206\u6790\u63a7\u5236 $u$ \u540e\u662f\u5426\u4ecd\u5b58\u5728\u5269\u4f59\u5173\u8054\u3002<\/p>\n

#### \u4f8b\u56db\uff1a\u975e\u7ebf\u6027\u5e73\u65b9\u5173\u7cfb<\/p>\n

\u8bbe\u96f6\u5747\u503c\u5bf9\u79f0\u8fc7\u7a0b $x(t)$\uff0c\u5e76\u4ee4<\/p>\n

$$
\ny(t)=x^2(t).
\n$$<\/p>\n

$y$ \u5b8c\u5168\u7531 $x$ \u51b3\u5b9a\uff0c\u4f46\u7ebf\u6027\u76f8\u5173\u53ef\u80fd\u4e3a\u96f6\uff0c\u540c\u9891 MSC \u4e5f\u53ef\u80fd\u5f88\u4f4e\uff0c\u56e0\u4e3a\u5e73\u65b9\u8fd0\u7b97\u628a\u80fd\u91cf\u642c\u79fb\u5230 DC\u3001\u4e8c\u6b21\u8c10\u6ce2\u548c\u9891\u7387\u548c\u5dee\u9879\u3002\u6b64\u65f6\u5e94\u8003\u8651\u9ad8\u9636\u8c31\u3001bicoherence \u6216\u975e\u7ebf\u6027\u4f9d\u8d56\u6307\u6807\u3002<\/p>\n

#### \u4f8b\u4e94\uff1a\u540c\u9891\u4f46\u76f8\u4f4d\u8de8\u6bb5\u968f\u673a<\/p>\n

\u5047\u8bbe\u6bcf\u4e2a\u77ed\u6570\u636e\u6bb5\u4e2d\uff0c$x$ \u548c $y$ \u90fd\u542b\u9891\u7387 $f_0$ \u7684\u6b63\u5f26\u5206\u91cf\uff0c\u4f46\u6bcf\u6bb5\u7684\u76f8\u4f4d\u5dee\u968f\u673a\u53d8\u5316\u3002\u4e24\u4e2a\u81ea\u8c31\u5728 $f_0$ \u90fd\u6709\u660e\u663e\u5cf0\u503c\uff0c\u5355\u6bb5\u770b\u8d77\u6765\u4e5f\u9ad8\u5ea6\u5339\u914d\uff1b\u7136\u800c\u8de8\u6bb5\u5e73\u5747\u65f6\uff0c\u4e92\u8c31\u7684\u590d\u76f8\u4f4d\u76f8\u4e92\u62b5\u6d88\uff0c\u56e0\u6b64\u603b\u4f53\u76f8\u5e72\u6027\u53ef\u80fd\u5f88\u4f4e\u3002<\/p>\n

\u8fd9\u8bf4\u660e\u76f8\u5e72\u6027\u8981\u6c42\u7684\u662f\u8de8\u5e73\u5747\u6837\u672c\u7a33\u5b9a\u7684\u76f8\u4f4d\u548c\u7ebf\u6027\u5173\u7cfb\uff0c\u800c\u4e0d\u4ec5\u662f\u4e24\u4e2a\u4fe1\u53f7\u201c\u90fd\u5728\u540c\u4e00\u9891\u7387\u6709\u80fd\u91cf\u201d\u3002<\/p>\n

#### \u4f8b\u516d\uff1a\u90e8\u5206\u9891\u5e26\u7ebf\u6027\u76f8\u5173<\/p>\n

\u8bbe\u8f93\u51fa\u7531\u4f4e\u901a\u540e\u7684\u8f93\u5165\u548c\u9ad8\u9891\u72ec\u7acb\u566a\u58f0\u7ec4\u6210\uff1a<\/p>\n

$$
\ny=h_{\\mathrm{LP}}*x+n_{\\mathrm{HF}}.
\n$$<\/p>\n

\u5219\u4f4e\u9891\u6bb5 $y$ \u4e3b\u8981\u7531 $x$ \u7ebf\u6027\u89e3\u91ca\uff0c\u76f8\u5e72\u6027\u8f83\u9ad8\uff1b\u9ad8\u9891\u6bb5\u7531\u72ec\u7acb\u566a\u58f0\u4e3b\u5bfc\uff0c\u76f8\u5e72\u6027\u8f83\u4f4e\u3002\u4e00\u4e2a\u5168\u9891\u5e26 Pearson \u76f8\u5173\u7cfb\u6570\u4f1a\u628a\u4e24\u90e8\u5206\u538b\u7f29\u6210\u5355\u4e2a\u6570\u5b57\uff0c\u800c\u76f8\u5e72\u6027\u80fd\u591f\u5b9a\u4f4d\u5173\u8054\u6240\u5728\u9891\u5e26\u3002<\/p>\n

—
\n### \u80fd\u91cf\u8c31\u3001\u529f\u7387\u8c31\u4e0e\u4e92\u8c31\u7684\u5178\u578b\u4f8b\u5b50<\/p>\n

#### \u6709\u9650\u77e9\u5f62\u8109\u51b2\uff1a\u80fd\u91cf\u8c31<\/p>\n

\u8bbe<\/p>\n

$$
\n x(t)=A,\\quad 0\\le t **\u76f8\u5173\u51fd\u6570\u63cf\u8ff0\u65f6\u5ef6\u57df\u4e2d\u7684\u5171\u540c\u53d8\u5316\uff0c\u80fd\u91cf\u8c31\u63cf\u8ff0\u6709\u9650\u80fd\u91cf\u5728\u9891\u7387\u4e0a\u7684\u5206\u5e03\uff0c\u529f\u7387\u8c31\u63cf\u8ff0\u957f\u671f\u5e73\u5747\u529f\u7387\u5728\u9891\u7387\u4e0a\u7684\u5206\u5e03\uff0c\u4e92\u80fd\u91cf\u8c31\u548c\u4e92\u529f\u7387\u8c31\u5219\u8fdb\u4e00\u6b65\u4fdd\u7559\u4e24\u4e2a\u4fe1\u53f7\u4e4b\u95f4\u7684\u76f8\u5bf9\u5e45\u5ea6\u4e0e\u76f8\u4f4d\u3002**<\/p>\n

\u4e8c\u8005\u90fd\u5305\u542b\u4e58\u6cd5\u548c\u6c42\u548c\uff0c\u4f46\u5377\u79ef\u7684\u6838\u5fc3\u662f\u7cfb\u7edf\u54cd\u5e94\u4e0e\u53cd\u8f6c\u5e73\u79fb\uff0c\u76f8\u5173\u7684\u6838\u5fc3\u662f\u5339\u914d\u3001\u5185\u79ef\u3001\u5171\u8f6d\u548c\u76f8\u5bf9\u5ef6\u8fdf\u3002\u7406\u89e3\u8fd9\u4e00\u533a\u522b\uff0c\u5c31\u80fd\u6b63\u786e\u89e3\u91ca\u6ee4\u6ce2\u3001\u5339\u914d\u6ee4\u6ce2\u3001\u540c\u6b65\u3001\u8c31\u5206\u6790\u548c\u81ea\u9002\u5e94\u7b97\u6cd5\u4e2d\u7684\u5927\u591a\u6570\u76f8\u5173\u516c\u5f0f\u3002<\/p>\n

—<\/p>\n","protected":false},"excerpt":{"rendered":"

# \u4fe1\u53f7\u4e0e\u7cfb\u7edf\u4e2d\u7684\u76f8\u5173\u3001\u4e92\u76f8\u5173\u3001\u5377\u79ef\u3001\u76f8\u5e72\u6027\u4e0e\u8c31\u5206\u6790 ## \u76ee\u5f55 1. [\u4e00\u3001\u6982\u5ff5\u603b\u89c8\u4e0e\u4fe1\u53f7\u57fa\u672c\u64cd\u4f5c](#\u4e00\u6982 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-566","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/566","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/comments?post=566"}],"version-history":[{"count":1,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/566\/revisions"}],"predecessor-version":[{"id":567,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/566\/revisions\/567"}],"wp:attachment":[{"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/media?parent=566"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/categories?post=566"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/tags?post=566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}