\u4ee5\u4e0b\u662f\u5bf9\u8fde\u7eed\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\uff08CTFT\uff09\u4e3b\u8981\u7279\u6027\u7684\u8bc1\u660e\u3002\u5b9a\u4e49\u5085\u91cc\u53f6\u53d8\u6362\u5bf9\u4e3a\uff1a<\/p>\n
\\[
\nX(\\omega) = \\int_{-\\infty}^{\\infty} x(t) e^{-j\\omega t} dt, \\quad x(t) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} X(\\omega) e^{j\\omega t} d\\omega
\n\\]<\/p>\n
\u8bb0\u4f5c \\( x(t) \\leftrightarrow X(\\omega) \\)\u3002<\/p>\n
—<\/p>\n
### 1. \u5ef6\u65f6\uff08\u65f6\u79fb\uff09
\n**\u7279\u6027**\uff1a
\n\\[
\nx(t – t_0) \\leftrightarrow e^{-j\\omega t_0} X(\\omega)
\n\\]<\/p>\n
**\u8bc1\u660e**\uff1a
\n\u4ee4 \\( y(t) = x(t – t_0) \\)\u3002
\n\\[
\nY(\\omega) = \\int_{-\\infty}^{\\infty} x(t – t_0) e^{-j\\omega t} dt
\n\\]
\n\u8bbe \\( \\tau = t – t_0 \\)\uff0c\u5219 \\( t = \\tau + t_0 \\)\uff0c\\( dt = d\\tau \\)\uff1a
\n\\[
\nY(\\omega) = \\int_{-\\infty}^{\\infty} x(\\tau) e^{-j\\omega (\\tau + t_0)} d\\tau = e^{-j\\omega t_0} \\int_{-\\infty}^{\\infty} x(\\tau) e^{-j\\omega \\tau} d\\tau = e^{-j\\omega t_0} X(\\omega)
\n\\]<\/p>\n
—<\/p>\n
### 2. \u9891\u79fb
\n**\u7279\u6027**\uff1a
\n\\[
\ne^{j\\omega_0 t} x(t) \\leftrightarrow X(\\omega – \\omega_0)
\n\\]<\/p>\n
**\u8bc1\u660e**\uff1a
\n\\[
\n\\mathcal{F}[e^{j\\omega_0 t} x(t)] = \\int_{-\\infty}^{\\infty} x(t) e^{j\\omega_0 t} e^{-j\\omega t} dt = \\int_{-\\infty}^{\\infty} x(t) e^{-j(\\omega – \\omega_0) t} dt = X(\\omega – \\omega_0)
\n\\]<\/p>\n
—<\/p>\n
### 3. \u65f6\u57df\u5fae\u5206
\n**\u7279\u6027**\uff1a
\n\\[
\n\\frac{d x(t)}{d t} \\leftrightarrow j\\omega X(\\omega)
\n\\]
\n\uff08\u5047\u8bbe \\( x(t) \\to 0 \\) \u5f53 \\( t \\to \\pm\\infty \\)\uff09<\/p>\n
**\u8bc1\u660e**\uff1a
\n\\[
\n\\mathcal{F}\\left[\\frac{dx}{dt}\\right] = \\int_{-\\infty}^{\\infty} \\frac{dx}{dt} e^{-j\\omega t} dt
\n\\]
\n\u5206\u90e8\u79ef\u5206\uff1a\u4ee4 \\( u = e^{-j\\omega t} \\), \\( dv = \\frac{dx}{dt} dt \\)\uff0c\u5219 \\( du = -j\\omega e^{-j\\omega t} dt \\)\uff0c\\( v = x(t) \\)\uff1a
\n\\[
\n= \\left. x(t) e^{-j\\omega t} \\right|_{-\\infty}^{\\infty} – \\int_{-\\infty}^{\\infty} x(t) (-j\\omega) e^{-j\\omega t} dt
\n\\]
\n\u8fb9\u754c\u9879\u4e3a 0\uff08\\( x \\) \u8870\u51cf\uff09\uff0c\u5f97\uff1a
\n\\[
\n= j\\omega \\int_{-\\infty}^{\\infty} x(t) e^{-j\\omega t} dt = j\\omega X(\\omega)
\n\\]
\n\u63a8\u5e7f\uff1a
\n\\[
\n\\frac{d^n x(t)}{dt^n} \\leftrightarrow (j\\omega)^n X(\\omega)
\n\\]<\/p>\n
—<\/p>\n
### 4. \u9891\u57df\u5fae\u5206
\n**\u7279\u6027**\uff1a
\n\\[
\n(-jt)^n x(t) \\leftrightarrow \\frac{d^n X(\\omega)}{d\\omega^n}
\n\\]
\n\u7279\u522b\u5730\uff1a
\n\\[
\nt x(t) \\leftrightarrow j \\frac{dX(\\omega)}{d\\omega}
\n\\]<\/p>\n
**\u8bc1\u660e**\uff08\u5bf9 \\( n=1 \\)\uff09\uff1a
\n\u7531\u5b9a\u4e49\uff1a
\n\\[
\nX(\\omega) = \\int x(t) e^{-j\\omega t} dt
\n\\]
\n\u4e24\u8fb9\u5bf9 \\( \\omega \\) \u6c42\u5bfc\uff1a
\n\\[
\n\\frac{dX(\\omega)}{d\\omega} = \\int x(t) (-jt) e^{-j\\omega t} dt = \\mathcal{F}[-j t x(t)]
\n\\]
\n\u6240\u4ee5 \\( \\mathcal{F}[t x(t)] = j \\frac{dX(\\omega)}{d\\omega} \\)\u3002\u5bf9 \\( n \\) \u6b21\u6c42\u5bfc\u5f97\uff1a
\n\\[
\n\\mathcal{F}[(-jt)^n x(t)] = \\frac{d^n X(\\omega)}{d\\omega^n}
\n\\]<\/p>\n
—<\/p>\n
### 5. \u65f6\u57df\u5377\u79ef
\n**\u7279\u6027**\uff1a
\n\\[
\n(x * y)(t) \\leftrightarrow X(\\omega) Y(\\omega)
\n\\]
\n\u5176\u4e2d \\( (x*y)(t) = \\int_{-\\infty}^{\\infty} x(\\tau) y(t-\\tau) d\\tau \\)\u3002<\/p>\n
**\u8bc1\u660e**\uff1a
\n\\[
\n\\mathcal{F}[x*y] = \\int_{-\\infty}^{\\infty} \\left[ \\int_{-\\infty}^{\\infty} x(\\tau) y(t-\\tau) d\\tau \\right] e^{-j\\omega t} dt
\n\\]
\n\u4ea4\u6362\u79ef\u5206\u987a\u5e8f\uff1a
\n\\[
\n= \\int_{-\\infty}^{\\infty} x(\\tau) \\left[ \\int_{-\\infty}^{\\infty} y(t-\\tau) e^{-j\\omega t} dt \\right] d\\tau
\n\\]
\n\u5bf9\u5185\u90e8\u79ef\u5206\u53d8\u91cf\u66ff\u6362 \\( u = t – \\tau \\)\uff1a
\n\\[
\n\\int y(u) e^{-j\\omega (u+\\tau)} du = e^{-j\\omega \\tau} Y(\\omega)
\n\\]
\n\u4ee3\u5165\uff1a
\n\\[
\n= \\int x(\\tau) e^{-j\\omega \\tau} Y(\\omega) d\\tau = X(\\omega) Y(\\omega)
\n\\]<\/p>\n
—<\/p>\n
### 6. \u9891\u57df\u5377\u79ef
\n**\u7279\u6027**\uff1a
\n\\[
\nx(t) y(t) \\leftrightarrow \\frac{1}{2\\pi} (X * Y)(\\omega)
\n\\]
\n\u5176\u4e2d \\( (X*Y)(\\omega) = \\int_{-\\infty}^{\\infty} X(\\nu) Y(\\omega-\\nu) d\\nu \\)\u3002<\/p>\n
**\u8bc1\u660e**\uff1a
\n\\[
\n\\mathcal{F}[x(t) y(t)] = \\int_{-\\infty}^{\\infty} x(t) y(t) e^{-j\\omega t} dt
\n\\]
\n\u7528\u53cd\u53d8\u6362\u8868\u793a \\( y(t) \\)\uff1a
\n\\[
\ny(t) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} Y(\\nu) e^{j\\nu t} d\\nu
\n\\]
\n\u4ee3\u5165\uff1a
\n\\[
\n= \\int x(t) \\left[ \\frac{1}{2\\pi} \\int Y(\\nu) e^{j\\nu t} d\\nu \\right] e^{-j\\omega t} dt
\n\\]
\n\u4ea4\u6362\u79ef\u5206\uff1a
\n\\[
\n= \\frac{1}{2\\pi} \\int Y(\\nu) \\left[ \\int x(t) e^{-j(\\omega – \\nu) t} dt \\right] d\\nu
\n\\]
\n\\[
\n= \\frac{1}{2\\pi} \\int Y(\\nu) X(\\omega – \\nu) d\\nu = \\frac{1}{2\\pi} (X * Y)(\\omega)
\n\\]<\/p>\n
—<\/p>\n
### 7. \u5e15\u8428\u74e6\u5c14\u5b9a\u7406
\n**\u7279\u6027**\uff08\u80fd\u91cf\u5b88\u6052\uff09\uff1a
\n\\[
\n\\int_{-\\infty}^{\\infty} |x(t)|^2 dt = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} |X(\\omega)|^2 d\\omega
\n\\]<\/p>\n
**\u8bc1\u660e**\uff1a
\n\\[
\n\\int_{-\\infty}^{\\infty} |x(t)|^2 dt = \\int x(t) x^*(t) dt
\n\\]
\n\u7528\u53cd\u53d8\u6362\u8868\u793a \\( x^*(t) \\)\uff08\u6ce8\u610f\u53d6\u5171\u8f6d\uff09\uff1a
\n\\[
\nx^*(t) = \\frac{1}{2\\pi} \\int X^*(\\omega) e^{-j\\omega t} d\\omega
\n\\]
\n\u4ee3\u5165\uff1a
\n\\[
\n= \\int x(t) \\left[ \\frac{1}{2\\pi} \\int X^*(\\omega) e^{-j\\omega t} d\\omega \\right] dt
\n\\]
\n\u4ea4\u6362\u79ef\u5206\u987a\u5e8f\uff1a
\n\\[
\n= \\frac{1}{2\\pi} \\int X^*(\\omega) \\left[ \\int x(t) e^{-j\\omega t} dt \\right] d\\omega
\n\\]
\n\\[
\n= \\frac{1}{2\\pi} \\int X^*(\\omega) X(\\omega) d\\omega = \\frac{1}{2\\pi} \\int |X(\\omega)|^2 d\\omega
\n\\]<\/p>\n
—<\/p>\n
\u4ee5\u4e0a\u5373\u5e38\u89c1\u8fde\u7eed\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\u7279\u6027\u7684\u4e25\u683c\u8bc1\u660e\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u4ee5\u4e0b\u662f\u5bf9\u8fde\u7eed\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\uff08CTFT\uff09\u4e3b\u8981\u7279\u6027\u7684\u8bc1\u660e\u3002\u5b9a\u4e49\u5085\u91cc\u53f6\u53d8\u6362\u5bf9\u4e3a\uff1a \\[ X(\\omega) = \\i […]<\/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-546","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/546","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=546"}],"version-history":[{"count":1,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/546\/revisions"}],"predecessor-version":[{"id":547,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/546\/revisions\/547"}],"wp:attachment":[{"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/media?parent=546"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/categories?post=546"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/tags?post=546"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}