\u7406\u89e3\u56db\u79cd\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u5173\u952e\u8981\u8df3\u51fa\u201c\u6570\u5b66\u516c\u5f0f\u201d\u7684\u7ec6\u8282\uff0c**\u5148\u6293\u4f4f\u4e00\u4e2a\u6838\u5fc3\u5206\u7c7b\u903b\u8f91\uff1a\u4fe1\u53f7\u5728\u201c\u65f6\u95f4\/\u7a7a\u95f4\u201d\u548c\u201c\u9891\u7387\u201d\u8fd9\u4e24\u4e2a\u7ef4\u5ea6\u4e0a\uff0c\u5206\u522b\u662f\u201c\u8fde\u7eed\u201d\u8fd8\u662f\u201c\u79bb\u6563\u201d\u7684\uff1f**<\/p>\n
\u8fd9\u4e2a\u903b\u8f91\u51b3\u5b9a\u4e86\u53d8\u6362\u7684\u6570\u5b66\u5f62\u5f0f\uff0c\u4e5f\u51b3\u5b9a\u4e86\u5b83\u5728\u5de5\u7a0b\u4e2d\u7684\u7528\u9014\u3002\u4e0b\u9762\u6211\u7528\u4e00\u5f20\u201c\u56db\u8c61\u9650\u56fe\u201d\u5e2e\u4f60\u5f7b\u5e95\u7406\u6e05\u601d\u8def\uff0c\u5e76\u7ed9\u51fa\u6700\u4f73\u5b66\u4e60\u8def\u5f84\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e00\u6b65\uff1a\u4e00\u5f20\u56fe\u770b\u61c2\u56db\u79cd\u53d8\u6362\u7684\u5173\u7cfb<\/p>\n
\u4f60\u53ef\u4ee5\u628a\u8fd9\u56db\u79cd\u53d8\u6362\u60f3\u8c61\u6210\u5904\u7406\u201c\u8fde\u7eed\/\u79bb\u6563\u201d\u548c\u201c\u5468\u671f\/\u975e\u5468\u671f\u201d\u7684\u6392\u5217\u7ec4\u5408\u3002\u5b83\u4eec\u7684\u6839\u57fa\u90fd\u662f**\u5085\u91cc\u53f6\u7ea7\u6570\uff08FS\uff09**\u2014\u2014\u5373\u5468\u671f\u4fe1\u53f7\u53ef\u4ee5\u5206\u89e3\u6210\u79bb\u6563\u9891\u7387\u7684\u8c10\u6ce2\u3002<\/p>\n
| \u53d8\u6362\u540d\u79f0 | \u65f6\u95f4\u57df\uff08\u65f6\u57df\uff09 | \u9891\u7387\u57df\uff08\u9891\u57df\uff09 | \u9002\u7528\u573a\u666f | \u901a\u4fd7\u7406\u89e3 |
\n| :— | :— | :— | :— | :— |
\n| **\u5085\u91cc\u53f6\u7ea7\u6570\uff08FS\uff09** | **\u8fde\u7eed**\u3001**\u5468\u671f** | **\u79bb\u6563**\u3001**\u975e\u5468\u671f** | \u7406\u8bba\u5206\u6790\u5468\u671f\u4fe1\u53f7\uff08\u5982\u65b9\u6ce2\u3001\u952f\u9f7f\u6ce2\uff09 | \u628a\u6ce2\u5f62\u62c6\u6210\u201c1\u500d\u9891\u30012\u500d\u9891…\u201d\u7684\u6574\u6570\u500d\u9891\u7387\u3002 |
\n| **\u5085\u91cc\u53f6\u53d8\u6362\uff08FT\uff09** | **\u8fde\u7eed**\u3001**\u975e\u5468\u671f** | **\u8fde\u7eed**\u3001**\u975e\u5468\u671f** | \u7406\u8bba\u63a8\u5bfc\uff0c\u63a8\u5bfc\u9891\u8c31\u5bc6\u5ea6\uff08\u5982\u9ad8\u65af\u8109\u51b2\uff09 | \u628a\u5355\u4e00\u8109\u51b2\u201c\u644a\u5f00\u201d\u6210\u8fde\u7eed\u7684\u4e00\u7247\u9891\u7387\u3002 |
\n| **\u79bb\u6563\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\uff08DTFT\uff09** | **\u79bb\u6563**\uff08\u91c7\u6837\u70b9\uff09\u3001**\u975e\u5468\u671f** | **\u8fde\u7eed**\u3001**\u5468\u671f** | \u7406\u8bba\u5206\u6790\u91c7\u6837\u540e\u7684\u4fe1\u53f7\uff08\u6570\u5b66\u5de5\u5177\uff09 | \u8ba1\u7b97\u673a\u65e0\u6cd5\u5904\u7406\u65e0\u9650\u957f\u7684\u8fde\u7eed\u9891\u8c31\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e2d\u95f4\u6865\u6881\u3002 |
\n| **\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff08DFT\uff09** | **\u79bb\u6563**\u3001**\u5468\u671f**\uff08\u9690\u542b\uff09 | **\u79bb\u6563**\u3001**\u5468\u671f**\uff08\u9690\u542b\uff09 | **\u5de5\u7a0b\u552f\u4e00\u5b9e\u73b0**\uff08FFT\u5c31\u662fDFT\u7684\u5feb\u901f\u7b97\u6cd5\uff09 | \u8ba1\u7b97\u673a\u771f\u6b63\u80fd\u7b97\u7684\u53d8\u6362\uff0c\u8f93\u5165\u8f93\u51fa\u90fd\u662f\u6709\u9650\u4e2a\u6570\u5b57\u3002 |<\/p>\n
—<\/p>\n
### \u7b2c\u4e8c\u6b65\uff1a\u6293\u4f4f\u201c\u6f14\u53d8\u8109\u7edc\u201d\uff0c\u4eceFS\u51fa\u53d1\u4e00\u8def\u63a8\u6f14<\/p>\n
\u6b7b\u8bb0\u786c\u80cc\u4f1a\u5fd8\u8bb0\uff0c\u4f46\u987a\u7740\u201c**\u5de5\u7a0b\u9700\u6c42\u5012\u903c\u6570\u5b66\u8fdb\u5316**\u201d\u8fd9\u6761\u7ebf\u8d70\uff0c\u4f60\u81ea\u7136\u5c31\u61c2\u4e86\uff1a<\/p>\n
1. **\u8d77\u70b9\uff1a\u5085\u91cc\u53f6\u7ea7\u6570\uff08FS\uff09**
\n – \u5085\u91cc\u53f6\u8bf4\uff1a**\u4efb\u4f55\u5468\u671f\u4fe1\u53f7**\uff08\u8fde\u7eed\u4e14\u65e0\u9650\u91cd\u590d\uff09\uff0c\u90fd\u80fd\u62c6\u6210\u4e00\u7cfb\u5217\u6b63\u5f26\u6ce2\uff08\u9891\u7387\u662f\u79bb\u6563\u7684\uff1af, 2f, 3f…\uff09\u3002
\n – **\u56f0\u60d1\u70b9**\uff1a\u73b0\u5b9e\u4e2d\u6ca1\u6709\u65e0\u9650\u957f\u7684\u5468\u671f\u4fe1\u53f7\uff0c\u53ea\u6709\u5355\u4e2a\u8109\u51b2\u600e\u4e48\u529e\uff1f<\/p>\n
2. **\u8fdb\u53161\uff1a\u5085\u91cc\u53f6\u53d8\u6362\uff08FT\uff09**
\n – \u628a\u5468\u671f\u4fe1\u53f7\u7684\u5468\u671f T \u63a8\u5411\u65e0\u7a77\u5927\uff0c\u5468\u671f\u4fe1\u53f7\u53d8\u6210\u975e\u5468\u671f\u4fe1\u53f7\uff0c\u79bb\u6563\u7684\u9891\u7387\u8c31\u7ebf\u5c31\u4f1a\u5bc6\u96c6\u5230\u8fde\u6210\u4e00\u7247\uff0c\u53d8\u6210**\u8fde\u7eed\u9891\u8c31**\u3002
\n – **\u6536\u83b7**\uff1a\u5f97\u5230\u4e86\u5b8c\u7f8e\u7684\u6570\u5b66\u7406\u8bba\uff08FT\uff09\u3002
\n – **\u75db\u70b9**\uff1a\u73b0\u5b9e\u4e2d\u6211\u4eec\u53ea\u80fd\u7528\u8ba1\u7b97\u673a\u5904\u7406\uff0c\u8ba1\u7b97\u673a\u4e0d\u8ba4\u8bc6\u8fde\u7eed\u51fd\u6570\uff0c\u53ea\u8ba4\u8bc6\u6570\u5b57\uff08\u91c7\u6837\u70b9\uff09\u3002\u600e\u4e48\u529e\uff1f<\/p>\n
3. **\u8fdb\u53162\uff1a\u79bb\u6563\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\uff08DTFT\uff09**
\n – \u65e2\u7136\u8ba1\u7b97\u673a\u53ea\u8ba4\u6570\u5b57\uff0c\u5c31\u628a\u8fde\u7eed\u4fe1\u53f7\u201c\u91c7\u6837\u201d\uff08\u4e58\u4ee5\u51b2\u6fc0\u4e32\uff09\u53d8\u6210\u79bb\u6563\u5e8f\u5217\u3002
\n – **\u5173\u952e\u7ed3\u8bba**\uff1a\u65f6\u57df\u79bb\u6563\u5316\uff0c\u4f1a\u5bfc\u81f4\u9891\u57df**\u5468\u671f\u5316**\uff08\u9891\u57df\u53d8\u6210\u8fde\u7eed\u4e14\u65e0\u9650\u91cd\u590d\u7684\u6ce2\u5f62\uff09\u3002
\n – **\u75db\u70b9**\uff1aDTFT\u7684\u9891\u57df\u4ecd\u7136\u662f**\u8fde\u7eed**\u7684\uff0c\u8ba1\u7b97\u673a\u8fd8\u662f\u5b58\u4e0d\u4e0b\u65e0\u9650\u591a\u4e2a\u9891\u7387\u70b9\u3002\u600e\u4e48\u529e\uff1f<\/p>\n
4. **\u7ec8\u70b9\uff1a\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff08DFT\uff09**
\n – \u65e2\u7136\u9891\u57df\u8fde\u7eed\u5b58\u4e0d\u4e0b\uff0c\u90a3\u5c31**\u5728\u9891\u57df\u91cc\u4e5f\u91c7\u6837**\uff08\u53ea\u53d6\u6709\u9650\u4e2a\u9891\u7387\u70b9\uff09\u3002
\n – \u65f6\u57df\u91c7\u6837 \u2192 \u9891\u57df\u5468\u671f\uff1b\u9891\u57df\u91c7\u6837 \u2192 \u65f6\u57df\u5468\u671f\u3002\u8fd9\u5c31\u5bfc\u81f4\u65f6\u57df\u548c\u9891\u57df\u90fd\u53d8\u6210\u4e86**\u79bb\u6563\u4e14\u6709\u9650\u957f**\u7684\u5e8f\u5217\u3002
\n – **\u8fd9\u5c31\u662f\u552f\u4e00\u80fd\u5728\u8ba1\u7b97\u673a\u91cc\u8fd0\u884c\u7684\u53d8\u6362\uff0c\u5b83\u7684\u5feb\u901f\u7b97\u6cd5\u5c31\u662fFFT\u3002**<\/p>\n
—<\/p>\n
### \u7b2c\u4e09\u6b65\uff1a\u4ece\u54ea\u91cc\u5165\u624b\uff1f\uff08\u5b9e\u64cd\u5efa\u8bae\uff09<\/p>\n
\u5982\u679c\u4f60\u662f**\u5de5\u79d1\u751f**\uff08\u7535\u5b50\u3001\u901a\u4fe1\u3001\u81ea\u52a8\u5316\uff09\uff0c\u5f3a\u70c8\u5efa\u8bae\u6309\u4ee5\u4e0b\u987a\u5e8f\u7a81\u7834\uff1a<\/p>\n
1. **\u7b2c\u4e00\u6b65\uff08\u9996\u6218\uff09\uff1a\u53ea\u5b66DFT\uff08\u5c24\u5176\u662fFFT\uff09**
\n – **\u539f\u56e0**\uff1a\u4f60\u5199\u4ee3\u7801\u3001\u505a\u4fe1\u53f7\u5904\u7406\u3001\u7528Matlab\/Python\uff0c\u5b9e\u9645\u8c03\u7528\u7684\u5168\u662fFFT\uff08\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff09\u3002\u5176\u4ed6\u4e09\u79cd\uff08FS\u3001FT\u3001DTFT\uff09\u662f\u7406\u8bba\u63a8\u5bfc\uff0c\u4f60\u770b\u4e66\u65f6\u4f1a\u89c9\u5f97\u62bd\u8c61\uff0c\u4f46\u505a\u5b9e\u9a8c\u65f6\u770b\u5230\u9891\u8c31\u56fe\uff0c\u90a3\u5c31\u662fDFT\u7684\u7ed3\u679c\u3002
\n – **\u5b66\u4ec0\u4e48**\uff1a\u91cd\u70b9\u7406\u89e3 **\u201c\u9891\u7387\u5206\u8fa8\u7387\u201d** \uff08\u91c7\u6837\u70b9\u6570N\u51b3\u5b9a\uff09\u548c **\u201c\u9891\u8c31\u6cc4\u9732\u201d** \uff08\u622a\u65ad\u6548\u5e94\uff09\u3002\u8fd9\u662f\u4f60\u5b9e\u9645\u5de5\u4f5c\u4e2d\u6700\u5e38\u9047\u5230\u7684\u5751\u3002<\/p>\n
2. **\u7b2c\u4e8c\u6b65\uff08\u56de\u5934\u770b\uff09\uff1a\u7528DFT\u53bb\u53cd\u63a8DTFT\u548cFT**
\n – \u8bb0\u4f4f\u4e00\u4e2a\u53e3\u8bc0\uff1a**\u201c\u65f6\u57df\u79bb\u6563\uff0c\u9891\u57df\u5468\u671f\uff1b\u65f6\u57df\u622a\u65ad\uff0c\u9891\u57df\u5377\u79ef\u3002\u201d**
\n – \u5f53\u4f60\u7406\u89e3\u4e86DFT\u4e3a\u4ec0\u4e48\u9891\u8c31\u4f1a\u201c\u6df7\u53e0\u201d\uff08\u56e0\u4e3aDTFT\u9891\u57df\u662f\u5468\u671f\u7684\uff09\uff0c\u4ee5\u53caDFT\u4e3a\u4ec0\u4e48\u6709\u201c\u6805\u680f\u6548\u5e94\u201d\uff08\u56e0\u4e3a\u53ea\u5728\u79bb\u6563\u70b9\u770b\u9891\u8c31\uff0c\u6f0f\u6389\u4e86\u5cf0\u503c\uff09\uff0c\u4f60\u5c31\u81ea\u7136\u7406\u89e3\u4e86DTFT\u548cFT\u7684\u7269\u7406\u610f\u4e49\u3002<\/p>\n
3. **\u7b2c\u4e09\u6b65\uff08\u6700\u540e\uff09\uff1a\u641e\u5b9aFS**
\n – \u628a\u5b83\u5f53\u6210DFT\u7684\u201c\u8fde\u7eed\u65f6\u57df\u7248\u672c\u201d\u3002\u5f53\u4f60\u5728DFT\u91cc\u770b\u5230\u4e00\u6839\u6839\u79bb\u6563\u7684\u8c31\u7ebf\uff0c\u90a3\u5c31\u662f\u5728\u6a21\u4effFS\u7684\u8c10\u6ce2\u7ed3\u6784\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u56db\u6b65\uff1a\u8bb0\u7262\u4e00\u4e2a\u201c\u7ec8\u6781\u6838\u5fc3\u516c\u5f0f\u201d\uff08\u7406\u89e3\u5373\u53ef\uff09<\/p>\n
\u5982\u679c\u4f60\u4e0d\u60f3\u8bb0\u590d\u6742\u7684\u79ef\u5206\uff0c\u8bf7\u6b7b\u78d5\u8fd9\u4e00\u4e2a\u5173\u7cfb\uff1a<\/p>\n
> **DTFT \u662f FT \u5728\u9891\u57df\u4e0a\u7684\u201c\u5468\u671f\u5ef6\u62d3\u201d\uff1bDFT \u662f DTFT \u5728\u9891\u57df\u4e0a\u7684\u201c\u79bb\u6563\u91c7\u6837\u201d\u3002**<\/p>\n
\uff08\u6ce8\uff1a\u4ece\u5de5\u7a0b\u89d2\u5ea6\uff0c\u6211\u4eec\u5e73\u65f6\u6240\u8bf4\u7684\u201c\u505aFFT\u201d\uff0c\u6570\u5b66\u4e0a\u4e25\u683c\u6765\u8bf4\u662f\u505a\u4e86DFT\uff0c\u4f46\u8f6f\u4ef6\u4f1a\u9ed8\u8ba4\u5e2e\u4f60\u5904\u7406\u597d\u4e86\u91c7\u6837\u548c\u5468\u671f\u5ef6\u62d3\uff0c\u6240\u4ee5\u4f60\u76f4\u63a5\u770b\u9891\u8c31\u56fe\u5373\u53ef\u3002\uff09<\/p>\n
—<\/p>\n
### \u7ed9\u4f60\u4e00\u4e2a\u201c\u907f\u5751\u201d\u63d0\u9192<\/p>\n
– **\u5343\u4e07\u4e0d\u8981**\u4ece\u6700\u590d\u6742\u7684\u8fde\u7eed\u5085\u91cc\u53f6\u53d8\u6362\uff08FT\uff09\u79ef\u5206\u516c\u5f0f\u5f00\u59cb\u5543\uff0c\u90a3\u4f1a\u8ba9\u4f60\u5728\u6570\u5b66\u91cc\u8ff7\u5931\uff0c\u770b\u4e0d\u5230\u7269\u7406\u610f\u4e49\u3002
\n– **\u52a1\u5fc5\u8bb0\u4f4f**\uff1a\u56db\u79cd\u53d8\u6362\u6ca1\u6709\u8c01\u6bd4\u8c01\u201c\u9ad8\u7ea7\u201d\uff0c\u5b83\u4eec\u53ea\u662f\u540c\u4e00\u4e2a\u6570\u5b66\u601d\u60f3\u5728\u4e0d\u540c\u7ef4\u5ea6\u7684**\u6295\u5f71**\u3002\u4f60\u628a\u4e00\u4e2a\u975e\u5468\u671f\u4fe1\u53f7\u60f3\u8c61\u6210\u4e00\u4e2a\u5468\u671f\u65e0\u9650\u957f\u7684\u4fe1\u53f7\uff0cFS\u5230FT\u7684\u8fc7\u6e21\u5c31\u77ac\u95f4\u6e05\u6670\u4e86\u3002<\/p>\n
—<\/p>\n
\u5982\u679c\u4f60\u662f\u4e3a\u4e86\u5e94\u5bf9\u8003\u8bd5\uff0c\u5efa\u8bae\u5148\u80ccFS\u548cFT\u7684\u516c\u5f0f\uff1b\u5982\u679c\u662f\u4e3a\u4e86\u505a\u9879\u76ee\/\u5199\u4ee3\u7801\uff0c**\u76f4\u63a5\u6253\u5f00Python\u7528`np.fft.fft()`\u5904\u7406\u4e00\u6bb5\u6b63\u5f26\u6ce2\u52a0\u566a\u58f0\u7684\u6570\u636e\uff0c\u89c2\u5bdf\u9891\u8c31\u56fe**\uff0c\u8fd9\u6bd4\u770b\u5341\u9875\u4e66\u90fd\u7ba1\u7528\u3002<\/p>\n
\u4f60\u73b0\u5728\u662f\u4e3a\u4e86**\u8003\u8bd5\u89e3\u9898**\uff0c\u8fd8\u662f\u4e3a\u4e86**\u5de5\u7a0b\u5e94\u7528**\uff08\u6bd4\u5982\u6ee4\u6ce2\u3001\u9891\u8c31\u5206\u6790\uff09\uff1f\u544a\u8bc9\u6211\u4f60\u7684\u5177\u4f53\u76ee\u6807\uff0c\u6211\u53ef\u4ee5\u7ed9\u4f60\u5b9a\u5236\u4e00\u4efd\u201c\u5fc5\u80cc\u516c\u5f0f\u7cbe\u7b80\u6e05\u5355\u201d\u6216\u201cPython FFT\u5b9e\u6218\u6781\u7b80\u6559\u7a0b\u201d\u3002\ud83d\ude0a<\/p>\n
\u597d\u7684\uff0c\u65e2\u7136\u4f60\u60f3\u6df1\u5165\u4e86\u89e3**\u5085\u91cc\u53f6\u7ea7\u6570\uff08FS\uff09**\uff0c\u6211\u4eec\u5c31\u628a\u5b83\u5f7b\u5e95\u8bb2\u900f\u3002<\/p>\n
**\u8bb0\u4f4f\u4e00\u4e2a\u6838\u5fc3\u89c2\u70b9**\uff1a\u5085\u91cc\u53f6\u7ea7\u6570\u4e0d\u662f\u5728\u521b\u9020\u65b0\u4e1c\u897f\uff0c\u5b83\u53ea\u662f\u5728\u505a**\u201c\u62c6\u89e3\u201d**\u2014\u2014\u628a\u4efb\u610f\u4e00\u4e2a**\u5468\u671f\u6027\u7684\u590d\u6742\u6ce2\u5f62**\uff0c\u62c6\u89e3\u6210\u4e00\u7cfb\u5217**\u4e0d\u540c\u9891\u7387\u3001\u4e0d\u540c\u5e45\u5ea6\u7684\u6807\u51c6\u6b63\u5f26\u6ce2**\u7684\u53e0\u52a0\u3002<\/p>\n
\u8fd9\u5c31\u50cf\u628a\u4e00\u675f\u767d\u5149\uff08\u590d\u6742\u4fe1\u53f7\uff09\u901a\u8fc7\u4e09\u68f1\u955c\uff08FS\uff09\uff0c\u5206\u89e3\u6210\u4e03\u79cd\u5355\u8272\u5149\uff08\u6b63\u5f26\u6ce2\u5206\u91cf\uff09\u3002<\/p>\n
—<\/p>\n
### 1. \u5085\u91cc\u53f6\u7ea7\u6570\u7684\u6838\u5fc3\u524d\u63d0\uff08\u9002\u7528\u6761\u4ef6\uff09<\/p>\n
\u5b83\u53ea\u9488\u5bf9**\u8fde\u7eed\u4e14\u5468\u671f**\u7684\u4fe1\u53f7\uff08\u6bd4\u5982\u65b9\u6ce2\u3001\u952f\u9f7f\u6ce2\u3001\u4ea4\u6d41\u7535\uff09\u3002<\/p>\n
– \u8bbe\u5468\u671f\u4e3a **T**\uff0c\u9891\u7387\u4e3a **f = 1\/T**\uff0c\u89d2\u9891\u7387\u4e3a **\u03c9 = 2\u03c0\/T**\u3002
\n– **\u6838\u5fc3\u524d\u63d0\uff08\u72c4\u5229\u514b\u96f7\u6761\u4ef6\uff09**\uff1a\u5728\u4e00\u4e2a\u5468\u671f\u5185\uff0c\u4fe1\u53f7\u53ea\u6709\u6709\u9650\u4e2a\u6781\u503c\u70b9\u548c\u95f4\u65ad\u70b9\u3002**\u653e\u5fc3\uff0c\u5de5\u7a0b\u4e2d99%\u7684\u4fe1\u53f7\u90fd\u6ee1\u8db3**\uff0c\u6240\u4ee5\u4e0d\u5fc5\u7ea0\u7ed3\u6570\u5b66\u4e25\u8c28\u6027\u3002<\/p>\n
—<\/p>\n
### 2. \u5b83\u5230\u5e95\u5728\u7b97\u4ec0\u4e48\uff1f\uff08\u4e09\u79cd\u7b49\u4ef7\u5f62\u5f0f\uff09<\/p>\n
\u5085\u91cc\u53f6\u7ea7\u6570\u6709\u4e09\u79cd\u5199\u6cd5\uff0c\u4f46\u672c\u8d28\u5b8c\u5168\u4e00\u6837\u3002\u6211\u6309**\u7531\u6d45\u5165\u6df1**\u7684\u987a\u5e8f\u8bb2\uff1a<\/p>\n
#### \u5f62\u5f0f\u4e00\uff1a\u4e09\u89d2\u5f62\u5f0f\uff08\u6700\u76f4\u89c2\uff0c\u9002\u5408\u5165\u95e8\uff09
\n\u4efb\u4f55\u4e00\u4e2a\u5468\u671f\u4fe1\u53f7 `x(t)` \u90fd\u53ef\u4ee5\u5199\u6210\uff1a
\n\\[
\nx(t) = a_0 + \\sum_{n=1}^{\\infty} [a_n \\cos(n\\omega t) + b_n \\sin(n\\omega t)]
\n\\]<\/p>\n
– **\u76f4\u6d41\u5206\u91cf \\( a_0 \\)**\uff1a\u4fe1\u53f7\u5728\u4e00\u4e2a\u5468\u671f\u5185\u7684\u5e73\u5747\u503c\uff08\u5e38\u6570\u9879\uff09\u3002
\n– **\u57fa\u6ce2\u5206\u91cf (\\( n=1 \\))**\uff1a\u9891\u7387\u4e3a \\( \\omega \\) \u7684\u5206\u91cf\uff0c\u51b3\u5b9a\u4fe1\u53f7\u7684\u4e3b\u97f3\u8c03\u3002
\n– **\u8c10\u6ce2\u5206\u91cf (\\( n>1 \\))**\uff1a\u9891\u7387\u662f\u57fa\u6ce2\u6574\u6570\u500d\uff08\\( 2\\omega, 3\\omega… \\)\uff09\u7684\u5206\u91cf\uff0c\u5b83\u4eec\u8d1f\u8d23\u8ba9\u6ce2\u5f62\u4ece\u5149\u6ed1\u7684\u6b63\u5f26\u6ce2\u53d8\u6210\u65b9\u6ce2\u6216\u952f\u9f7f\u6ce2\u3002<\/p>\n
**\u516c\u5f0f\u4e2d\u7684\u7cfb\u6570\u600e\u4e48\u6c42\uff1f**\uff08\u8bb0\u4f4f\u7ed3\u8bba\u5373\u53ef\uff09
\n– \\( a_0 = \\frac{1}{T} \\int_{0}^{T} x(t) dt \\)\uff08\u5e73\u5747\u503c\uff09
\n– \\( a_n = \\frac{2}{T} \\int_{0}^{T} x(t) \\cos(n\\omega t) dt \\)
\n– \\( b_n = \\frac{2}{T} \\int_{0}^{T} x(t) \\sin(n\\omega t) dt \\)<\/p>\n
**\u8fd9\u4e5f\u662f\u4e3a\u4ec0\u4e48\u53eb\u201c\u5085\u91cc\u53f6\u7ea7\u6570\u201d**\u2014\u2014\u56e0\u4e3a\u7ed3\u679c\u662f\u4e00\u4e32\u79bb\u6563\u7684\u201c\u6570\u5217\u201d\uff08\u7cfb\u6570 \\( a_n, b_n \\)\uff09\uff0c\u800c\u4e0d\u662f\u8fde\u7eed\u7684\u51fd\u6570\u3002<\/p>\n
—<\/p>\n
#### \u5f62\u5f0f\u4e8c\uff1a\u5e45\u76f8\u5f62\u5f0f\uff08\u6700\u5b9e\u7528\uff0c\u770b\u9891\u8c31\u56fe\uff09
\n\u628a\u540c\u9891\u7387\u7684\u6b63\u5f26\u548c\u4f59\u5f26\u5408\u5e76\u6210\u4e00\u4e2a\u5e26\u76f8\u4f4d\u7684\u4f59\u5f26\u6ce2\uff1a
\n\\[
\nx(t) = A_0 + \\sum_{n=1}^{\\infty} A_n \\cos(n\\omega t + \\varphi_n)
\n\\]
\n\u5176\u4e2d\uff1a<\/p>\n
– \\( A_0 = a_0 \\)\uff08\u76f4\u6d41\uff09
\n– \\( A_n = \\sqrt{a_n^2 + b_n^2} \\)\uff08**\u8be5\u9891\u7387\u5206\u91cf\u7684\u632f\u5e45**\uff09
\n– \\( \\varphi_n = -\\arctan(\\frac{b_n}{a_n}) \\)\uff08**\u8be5\u9891\u7387\u5206\u91cf\u7684\u521d\u59cb\u76f8\u4f4d**\uff09<\/p>\n
**\u5de5\u7a0b\u610f\u4e49**\uff1a\u4f60\u5e73\u65f6\u5728\u793a\u6ce2\u5668\u6216\u9891\u8c31\u8f6f\u4ef6\u4e0a\u770b\u5230\u7684\u201c\u5e45\u5ea6\u8c31\u201d\u548c\u201c\u76f8\u4f4d\u8c31\u201d\uff0c\u753b\u7684\u5c31\u662f \\( A_n \\) \u548c \\( \\varphi_n \\) \u5bf9\u9891\u7387 \\( n\\omega \\) \u7684\u79bb\u6563\u7ebf\u6761\u3002<\/p>\n
—<\/p>\n
#### \u5f62\u5f0f\u4e09\uff1a\u6307\u6570\u5f62\u5f0f\uff08\u6700\u7b80\u6d01\uff0c\u6570\u5b66\u5229\u5668\uff09
\n\u5229\u7528\u6b27\u62c9\u516c\u5f0f \\( e^{j\\theta} = \\cos\\theta + j\\sin\\theta \\)\uff0c\u628a\u4e09\u89d2\u5f62\u5f0f\u6d53\u7f29\u4e3a\uff1a
\n\\[
\nx(t) = \\sum_{n=-\\infty}^{\\infty} C_n \\cdot e^{jn\\omega t}
\n\\]<\/p>\n
– \u8fd9\u91cc n \u4ece\u8d1f\u65e0\u7a77\u53d6\u5230\u6b63\u65e0\u7a77\u3002**\u8d1f\u9891\u7387**\u5728\u7269\u7406\u4e0a\u4e0d\u5b58\u5728\uff0c\u5b83\u662f\u6570\u5b66\u4e0a\u7684\u5171\u8f6d\u5bf9\u79f0\uff08\u53ea\u662f\u4e3a\u4e86\u65b9\u4fbf\u8ba1\u7b97\uff09\u3002
\n– \u590d\u6570\u7cfb\u6570 \\( C_n = \\frac{1}{T} \\int_{0}^{T} x(t) e^{-jn\\omega t} dt \\)\u3002<\/p>\n
**\u8fd9\u4e2a\u5f62\u5f0f\u6781\u5176\u91cd\u8981**\uff0c\u56e0\u4e3a\u5b83\u548c\u540e\u9762\u7684\u5085\u91cc\u53f6\u53d8\u6362\uff08FT\uff09\u516c\u5f0f\u957f\u5f97\u51e0\u4e4e\u4e00\u6837\uff0c\u662f\u8fde\u63a5\u56db\u79cd\u53d8\u6362\u7684\u201c\u901a\u7528\u8bed\u8a00\u201d\u3002<\/p>\n
—<\/p>\n
### 3. \u4e00\u4e2a\u7ecf\u5178\u4f8b\u5b50\uff1a\u65b9\u6ce2\u7684\u5085\u91cc\u53f6\u7ea7\u6570<\/p>\n
\u8fd9\u662f\u7406\u89e3\u201c\u53e0\u52a0\u201d\u6700\u751f\u52a8\u7684\u4f8b\u5b50\u3002\u4e00\u4e2a\u7406\u60f3\u65b9\u6ce2\uff08\u5cf0\u503c\u4e3a1\uff0c\u5360\u7a7a\u6bd450%\uff09\u7684\u5c55\u5f00\u5f0f\u4e3a\uff1a
\n\\[
\nx(t) = \\frac{4}{\\pi} \\left( \\sin(\\omega t) + \\frac{1}{3}\\sin(3\\omega t) + \\frac{1}{5}\\sin(5\\omega t) + \\cdots \\right)
\n\\]<\/p>\n
**\u89e3\u8bfb**\uff1a<\/p>\n
– \u5b83\u53ea\u542b\u6709\u5947\u6b21\u8c10\u6ce2\uff081, 3, 5, 7…\uff09\uff0c\u6ca1\u6709\u5076\u6b21\u8c10\u6ce2\u3002
\n– \u57fa\u6ce2\u5e45\u5ea6\u6700\u5927\uff084\/\u03c0 \u2248 1.27\uff09\uff0c3\u6b21\u8c10\u6ce2\u5e45\u5ea6\u662f\u51761\/3\uff0c5\u6b21\u8c10\u6ce2\u662f1\/5\u2026\u2026
\n– \u5982\u679c\u4f60\u53ea\u53d6\u7b2c\u4e00\u9879\uff08\u57fa\u6ce2\uff09\uff0c\u6ce2\u5f62\u662f\u5149\u6ed1\u7684\u6b63\u5f26\u6ce2\uff1b
\n– \u5982\u679c\u4f60\u53d6\u524d3\u9879\uff08\u57fa\u6ce2+3\u6b21+5\u6b21\uff09\uff0c\u6ce2\u5f62\u9876\u90e8\u5f00\u59cb\u53d8\u5e73\uff1b
\n– **\u5f53\u4f60\u53d6\u5230\u65e0\u7a77\u591a\u9879\u65f6\uff0c\u65e0\u6570\u4e2a\u6b63\u5f26\u6ce2\u53e0\u52a0\uff0c\u6700\u7ec8\u5f62\u6210\u5b8c\u7f8e\u7684\u5782\u76f4\u4e0a\u5347\u6cbf\u548c\u6c34\u5e73\u9876\u90e8\u7684\u65b9\u6ce2\u3002**<\/p>\n
\u8fd9\u5c31\u662f\u8457\u540d\u7684**\u5409\u5e03\u65af\u73b0\u8c61**\uff1a\u5728\u65b9\u6ce2\u7684\u8df3\u53d8\u6cbf\u5904\uff0c\u5373\u4f7f\u53d6\u65e0\u7a77\u591a\u9879\uff0c\u8d85\u8c03\u91cf\u4f9d\u7136\u5b58\u5728\uff08\u7ea69%\uff09\uff0c\u4e0d\u4f1a\u5b8c\u5168\u6d88\u5931\u3002<\/p>\n
—<\/p>\n
### 4. \u5085\u91cc\u53f6\u7ea7\u6570\u7684\u201c\u7075\u9b42\u4e09\u95ee\u201d<\/p>\n
\u4e3a\u4e86\u8ba9\u4f60\u5f7b\u5e95\u5403\u900f\uff0c\u6211\u603b\u7ed3\u4e86\u521d\u5b66\u8005\u6700\u5bb9\u6613\u61f5\u7684\u4e09\u4e2a\u70b9\uff1a<\/p>\n
– **\u95ee\uff1a\u4e3a\u4ec0\u4e48\u975e\u8981\u7528\u6b63\u5f26\u6ce2\uff1f\u4e0d\u80fd\u7528\u4e09\u89d2\u6ce2\u6216\u6307\u6570\u6ce2\u62c6\u5417\uff1f**
\n – \u7b54\uff1a\u56e0\u4e3a\u6b63\u5f26\u6ce2\u6709\u4e00\u4e2a\u72ec\u6709\u7279\u6027\u2014\u2014**\u6b63\u4ea4\u6027**\u3002\u4efb\u610f\u4e24\u4e2a\u4e0d\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\u5728\u4e00\u4e2a\u5468\u671f\u5185\u76f8\u4e58\uff0c\u79ef\u5206\u4e3a0\u3002\u8fd9\u4fdd\u8bc1\u4e86\u62c6\u51fa\u6765\u7684\u6bcf\u4e2a\u5206\u91cf\u90fd\u662f\u201c\u72ec\u4e00\u65e0\u4e8c\u201d\u7684\uff0c\u4e0d\u4f1a\u4e92\u76f8\u5e72\u6270\u3002<\/p>\n
– **\u95ee\uff1a\u5468\u671f\u4fe1\u53f7\u9891\u7387\u79bb\u6563\uff0c\u90a3\u79bb\u6563\u7684\u9891\u7387\u95f4\u9694\u662f\u591a\u5927\uff1f**
\n – \u7b54\uff1a\u95f4\u9694\u5c31\u662f\u57fa\u9891 \\( \\omega \\)\uff08\u6216 \\( f \\)\uff09\u3002\u5468\u671f\u8d8a\u5927\uff08\u4fe1\u53f7\u53d8\u5316\u8d8a\u6162\uff09\uff0c\u57fa\u9891\u8d8a\u5c0f\uff0c\u8c31\u7ebf\u8d8a\u5bc6\u96c6\uff1b\u5468\u671f\u65e0\u9650\u5927\u65f6\uff08\u975e\u5468\u671f\u4fe1\u53f7\uff09\uff0c\u8c31\u7ebf\u5c31\u8fde\u6210\u4e00\u7247\uff0c\u53d8\u6210\u8fde\u7eed\u8c31\uff08\u8fd9\u5c31\u8fc7\u6e21\u5230\u4e86FT\uff09\u3002<\/p>\n
– **\u95ee\uff1a\u5085\u91cc\u53f6\u7ea7\u6570\u80fd\u5904\u7406\u201c\u7a81\u53d8\u201d\u5417\uff1f**
\n – \u7b54\uff1a\u80fd\u3002\u5373\u4f7f\u5728\u65b9\u6ce2\u7684\u8df3\u53d8\u6cbf\uff08\u4e0d\u8fde\u7eed\u70b9\uff09\uff0c\u7ea7\u6570\u6536\u655b\u4e8e\u8df3\u53d8\u503c\u7684\u4e2d\u70b9\uff08\u6bd4\u5982\u4ece+1\u8df3\u5230-1\uff0c\u7ea7\u6570\u5728\u8be5\u70b9\u6536\u655b\u4e8e0\uff09\u3002\u8fd9\u5c31\u662f\u6570\u5b66\u7684\u5de7\u5999\u4e4b\u5904\u3002<\/p>\n
—<\/p>\n
### 5. \u7ed9\u4f60\u7684\u5b66\u4e60\u5efa\u8bae\uff08\u9488\u5bf9FS\uff09<\/p>\n
1. **\u4e0d\u5fc5\u6b7b\u8bb0\u79ef\u5206\u6c42\u7cfb\u6570**\uff1a\u73b0\u5728\u6ca1\u4eba\u624b\u7b97\u7cfb\u6570\uff0cMatlab\/Python\u4e00\u53e5`fft`\u5c31\u51fa\u6765\u4e86\u3002\u4f60\u9700\u8981\u8bb0\u7684\u662f**\u7ed3\u8bba**\uff1a**\u201c\u5468\u671f\u4fe1\u53f7 \u2194 \u79bb\u6563\u8c31\u7ebf\u201d**\u3002
\n2. **\u52a8\u624b\u542c\u4e00\u542c**\uff1a\u53bb\u7f51\u4e0a\u641c\u201cSquare wave sound\u201d\u6216\u201cFourier synthesis sound\u201d\uff0c\u4eb2\u81ea\u542c\u4e00\u542c\uff1a\u53ea\u64ad\u653e\u57fa\u6ce2\u662f\u7eaf\u97f3\uff0c\u53e0\u52a0\u9ad8\u6b21\u8c10\u6ce2\u540e\u97f3\u8272\u53d8\u201c\u4eae\u201d\u53d8\u201c\u523a\u8033\u201d\u3002**\u542c\u89c9\u6bd4\u89c6\u89c9\u66f4\u80fd\u8ba9\u4f60\u8bb0\u4f4f\u201c\u8c10\u6ce2\u201d\u7684\u610f\u4e49**\u3002
\n3. **\u627e\u51c6\u5b9a\u4f4d**\uff1a\u5728\u56db\u79cd\u53d8\u6362\u4e2d\uff0cFS\u662f**\u7406\u8bba\u8d77\u70b9**\u3002\u4f60\u4e0d\u9700\u8981\u628aFS\u7ec3\u5f97\u6eda\u74dc\u70c2\u719f\uff0c\u4f46\u5fc5\u987b\u8bb0\u4f4f\u5b83\u7684\u7ed3\u8bba\uff0c\u56e0\u4e3a\u8fd9\u662f\u7406\u89e3\u540e\u7eedFT\uff08\u975e\u5468\u671f\uff09\u548cDFT\uff08\u8ba1\u7b97\u673a\u5b9e\u73b0\uff09\u7684\u201c\u601d\u60f3\u6bcd\u4f53\u201d\u3002<\/p>\n
—<\/p>\n
**\u4e0b\u4e00\u6b65\u9884\u544a**\uff1a\u5f53\u4f60\u61c2\u4e86FS\uff0c\u5085\u91cc\u53f6\u53d8\u6362\uff08FT\uff09\u5c31\u53ea\u662f\u628a\u201c\u5468\u671fT\u201d\u62c9\u5230\u65e0\u7a77\u5927\uff0c\u8ba9\u79bb\u6563\u8c31\u7ebf\u53d8\u6210\u8fde\u7eed\u8c31\u800c\u5df2\u3002**\u4f60\u60f3\u63a5\u7740\u542cFT\uff08\u8fde\u7eed\u975e\u5468\u671f\uff09\u7684\u8bb2\u89e3\uff0c\u8fd8\u662f\u76f4\u63a5\u8df3\u5230\u5de5\u7a0b\u4e2d\u771f\u6b63\u7528\u7684DFT\uff08\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff09\uff1f** \u544a\u8bc9\u6211\uff0c\u6211\u7ee7\u7eed\u5e26\u4f60\u6df1\u6316\u3002\ud83d\ude0a<\/p>\n
\u597d\u7684\uff0c\u6536\u5230\u3002\u65e2\u7136\u4f60\u660e\u786e\u8981\u8be6\u7ec6\u6df1\u5165\u5730\u4e86\u89e3 **FS\uff08\u5085\u91cc\u53f6\u7ea7\u6570\uff09**\uff0c\u6211\u4eec\u5c31\u5f7b\u5e95\u56de\u5f52\u8fd9\u4e2a\u4e3b\u9898\uff0c\u628a\u5b83\u7684**\u7269\u7406\u56fe\u50cf\u3001\u6570\u5b66\u672c\u8d28\u3001\u5de5\u7a0b\u542b\u4e49\u4ee5\u53ca\u5c40\u9650\u6027**\u5168\u90e8\u8bb2\u900f\uff0c\u4e0d\u7559\u6b7b\u89d2\u3002<\/p>\n
\u4e3a\u4e86\u8ba9\u4f60\u62e5\u6709\u4e00\u4e2a\u5b8c\u6574\u7684\u8ba4\u77e5\u6846\u67b6\uff0c\u6211\u4eec\u628aFS\u62c6\u89e3\u4e3a**\u56db\u4e2a\u5c42\u6b21**\u6765\u8bb2\u89e3\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e00\u5c42\uff1aFS\u5230\u5e95\u5728\u5e72\u4ec0\u4e48\uff1f\uff08\u7269\u7406\u76f4\u89c9\uff09<\/p>\n
**\u4e00\u53e5\u8bdd\u5b9a\u4e49**\uff1aFS\u662f\u4e00\u79cd\u6570\u5b66\u5de5\u5177\uff0c\u5b83\u544a\u8bc9\u6211\u4eec\u2014\u2014**\u4efb\u4f55\u5468\u671f\u6027\u7684\u8fde\u7eed\u4fe1\u53f7\uff0c\u90fd\u53ef\u4ee5\u88ab\u5206\u89e3\u4e3a\uff08\u6216\u8005\u8bf4\u662f\u7531\uff09\u4e00\u7cfb\u5217\u9891\u7387\u4e3a\u6574\u6570\u500d\u7684\u6b63\u5f26\u6ce2\u53e0\u52a0\u800c\u6210\u7684\u3002**<\/p>\n
– **\u8f93\u5165**\uff1a\u4e00\u4e2a\u5f62\u72b6\u4efb\u610f\u7684\u3001\u65e0\u9650\u91cd\u590d\u7684\u6ce2\u5f62\uff08\u6bd4\u5982\u65b9\u6ce2\u3001\u952f\u9f7f\u6ce2\u3001\u4f60\u7684\u5fc3\u8df3\u4fe1\u53f7\uff09\u3002
\n– **\u8f93\u51fa**\uff1a\u4e00\u7ec4\u9891\u7387\u79bb\u6563\u7684\u6b63\u5f26\u6ce2\uff08\u57fa\u6ce2 + 2\u6b21\u8c10\u6ce2 + 3\u6b21\u8c10\u6ce2 …\uff09\u3002
\n– **\u6df1\u5c42\u610f\u4e49**\uff1aFS\u63ed\u793a\u4e86**\u9891\u57df**\u7684\u6982\u5ff5\u3002\u5728\u65f6\u57df\u91cc\u770b\u8d77\u6765\u590d\u6742\u5230\u65e0\u6cd5\u63cf\u8ff0\u7684\u65b9\u6ce2\uff0c\u5728\u9891\u57df\u91cc\u53ea\u662f\u51e0\u6761\u7ad6\u7ebf\uff08\u8c31\u7ebf\uff09\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e8c\u5c42\uff1aFS\u7684\u6570\u5b66\u8868\u8fbe\u5f0f\uff08\u7cbe\u786e\u63cf\u8ff0\uff09<\/p>\n
FS\u6709\u4e14\u4ec5\u6709\u4e09\u79cd\u5b8c\u5168\u7b49\u4ef7\u7684\u6570\u5b66\u5199\u6cd5\uff0c\u6211\u6309\u7167**\u201c\u5de5\u7a0b\u5b9e\u7528\u7a0b\u5ea6\u201d**\u4e3a\u4f60\u6392\u5217\uff1a<\/p>\n
#### 1. \u4e09\u89d2\u5f62\u5f0f\uff08\u6700\u539f\u59cb\uff0c\u9002\u5408\u7406\u89e3\u5206\u89e3\u903b\u8f91\uff09
\n\\[
\nx(t) = a_0 + \\sum_{n=1}^{\\infty} [a_n \\cos(n\\omega_0 t) + b_n \\sin(n\\omega_0 t)]
\n\\]<\/p>\n
– **\\( \\omega_0 \\)**\uff1a\u57fa\u6ce2\u89d2\u9891\u7387\uff08\\( \\omega_0 = 2\\pi\/T \\)\uff09\uff0c\u5b83\u51b3\u5b9a\u4e86\u4fe1\u53f7\u91cd\u590d\u7684\u5feb\u6162\u3002
\n– **\\( a_0 \\)**\uff1a\u76f4\u6d41\u5206\u91cf\uff0c\u5373\u4fe1\u53f7\u5728\u4e00\u4e2a\u5468\u671f\u5185\u7684\u5e73\u5747\u503c\uff0c\u4ee3\u8868\u4fe1\u53f7\u7684\u201c\u57fa\u51c6\u9ad8\u5ea6\u201d\u3002
\n– **\\( a_n, b_n \\)**\uff1a\u7b2c n \u6b21\u8c10\u6ce2\u4e2d\u4f59\u5f26\u5206\u91cf\u548c\u6b63\u5f26\u5206\u91cf\u7684\u7cfb\u6570\u3002
\n– **\u5173\u952e\u8ba4\u77e5**\uff1a\u4e3a\u4ec0\u4e48\u540c\u4e00\u4e2a\u9891\u7387 \\( n\\omega_0 \\) \u8981\u7528\u6b63\u5f26\u548c\u4f59\u5f26\u4e24\u4e2a\u4e1c\u897f\u8868\u793a\uff1f\u56e0\u4e3a**\u76f8\u4f4d**\u3002\u4e00\u4e2a\u4efb\u610f\u76f8\u4f4d\u7684\u6b63\u5f26\u6ce2 \\( A\\cos(n\\omega_0 t + \\varphi) \\)\uff0c\u53ef\u4ee5\u62c6\u6210 \\( A\\cos\\varphi \\cdot \\cos(n\\omega_0 t) – A\\sin\\varphi \\cdot \\sin(n\\omega_0 t) \\)\u3002\u6240\u4ee5 \\( a_n, b_n \\) \u5171\u540c\u51b3\u5b9a\u4e86\u8be5\u9891\u7387\u7684**\u5e45\u5ea6**\u548c**\u76f8\u4f4d**\u3002<\/p>\n
#### 2. \u4f59\u5f26\u5f62\u5f0f\uff08\u6700\u76f4\u89c2\uff0c\u76f4\u63a5\u770b\u9891\u8c31\uff09
\n\\[
\nx(t) = A_0 + \\sum_{n=1}^{\\infty} A_n \\cos(n\\omega_0 t + \\varphi_n)
\n\\]<\/p>\n
– **\\( A_n = \\sqrt{a_n^2 + b_n^2} \\)**\uff1a\u8fd9\u5c31\u662f\u4f60\u5e73\u65f6\u5728\u9891\u8c31\u4eea\u4e0a\u770b\u5230\u7684**\u5e45\u5ea6\u8c31**\u7684\u7eb5\u5750\u6807\u3002
\n– **\\( \\varphi_n = -\\arctan(b_n \/ a_n) \\)**\uff1a\u8fd9\u5c31\u662f**\u76f8\u4f4d\u8c31**\u3002
\n– **\u4e3a\u4ec0\u4e48\u7528\u8fd9\u4e2a\uff1f** \u56e0\u4e3a\u5b83\u7269\u7406\u610f\u4e49\u6781\u5f3a\uff1a\u4fe1\u53f7\u5c31\u7b49\u4e8e\u201c\u76f4\u6d41\u201d\u52a0\u4e0a\u4e00\u5806\u201c\u4e0d\u540c\u5e45\u5ea6\u3001\u4e0d\u540c\u521d\u59cb\u76f8\u4f4d\u7684\u4f59\u5f26\u6ce2\u201d\u3002<\/p>\n
#### 3. \u6307\u6570\u5f62\u5f0f\uff08\u6700\u7b80\u6d01\uff0c\u4e3a\u540e\u7eedFT\u505a\u94fa\u57ab\uff09
\n\\[
\nx(t) = \\sum_{n=-\\infty}^{\\infty} C_n \\cdot e^{jn\\omega_0 t}
\n\\]<\/p>\n
– **\\( C_n \\)** \u662f\u590d\u6570\u7cfb\u6570\uff0c\\( C_n = \\frac{1}{T} \\int_{0}^{T} x(t) e^{-jn\\omega_0 t} dt \\)\u3002
\n– **\u8ba4\u77e5\u98de\u8dc3**\uff1a\u8fd9\u91cc\u5f15\u5165\u4e86**\u8d1f\u9891\u7387**\uff08n \u53d6\u8d1f\u503c\uff09\u3002\u7269\u7406\u4e2d\u4e0d\u5b58\u5728\u8d1f\u9891\u7387\uff0c\u8fd9\u53ea\u662f\u6570\u5b66\u4e0a\u7684\u5171\u8f6d\u5bf9\u79f0\uff08\u5373 \\( C_{-n} \\) \u662f \\( C_n \\) \u7684\u5171\u8f6d\u590d\u6570\uff09\u3002\u91c7\u7528\u8fd9\u4e2a\u5f62\u5f0f\u540e\uff0cFS\u7684\u516c\u5f0f\u548c\u540e\u7eedFT\u7684\u516c\u5f0f\u5728\u7ed3\u6784\u4e0a\u5b8c\u5168\u7edf\u4e00\u4e86\uff0c\u8fd9\u662f\u73b0\u4ee3\u4fe1\u53f7\u5904\u7406\u7684\u6807\u51c6\u8bed\u8a00\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e09\u5c42\uff1a\u7cfb\u6570\u7684\u8ba1\u7b97\uff08\u6838\u5fc3\u8bc1\u660e\u56de\u987e\uff09<\/p>\n
\u8ba1\u7b97\u8fd9\u4e9b\u7cfb\u6570\u7684\u65b9\u6cd5\uff0c\u5c31\u662f\u6211\u4eec\u4e0a\u4e00\u8f6e\u8be6\u7ec6\u8bc1\u660e\u8fc7\u7684**\u6b63\u4ea4\u6295\u5f71\u6cd5**\u3002\u8fd9\u91cc\u6211\u628a\u516c\u5f0f\u6d53\u7f29\u6210\u6700\u76f4\u63a5\u7684**\u7ed3\u8bba**\uff0c\u4f60\u5728\u5b9e\u9645\u8ba1\u7b97\u65f6\u76f4\u63a5\u5957\u7528\u5373\u53ef\uff1a<\/p>\n
\u7ed9\u5b9a\u5468\u671f \\( T \\)\uff0c\u89d2\u9891\u7387 \\( \\omega_0 = 2\\pi\/T \\)\uff1a<\/p>\n
– **\u76f4\u6d41\u7cfb\u6570**\uff1a\\( a_0 = \\frac{1}{T} \\int_{0}^{T} x(t) dt \\)\uff08\u4fe1\u53f7\u7684\u5e73\u5747\u9ad8\u5ea6\uff09
\n– **\u4f59\u5f26\u7cfb\u6570**\uff1a\\( a_n = \\frac{2}{T} \\int_{0}^{T} x(t) \\cos(n\\omega_0 t) dt \\) \uff08\\( n=1,2,3… \\)\uff09
\n– **\u6b63\u5f26\u7cfb\u6570**\uff1a\\( b_n = \\frac{2}{T} \\int_{0}^{T} x(t) \\sin(n\\omega_0 t) dt \\) \uff08\\( n=1,2,3… \\)\uff09<\/p>\n
**\u6ce8\u610f\u4e00\u4e2a\u5bb9\u6613\u8e29\u7684\u5751**\uff1a\\( a_0 \\) \u7684\u5206\u6bcd\u662f \\( T \\)\uff0c\u800c \\( a_n, b_n \\) \u7684\u5206\u6bcd\u662f \\( T\/2 \\)\u3002\u56e0\u4e3a\u76f4\u6d41\u5206\u91cf\uff08\u5e38\u65701\uff09\u7684\u81ea\u8eab\u79ef\u5206\u4e3a \\( T \\)\uff0c\u800c \\( \\cos^2 \\) \u7684\u81ea\u8eab\u79ef\u5206\u4e3a \\( T\/2 \\)\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u56db\u5c42\uff1aFS\u7684\u201c\u7075\u9b42\u201d\u4e0e\u201c\u7455\u75b5\u201d\uff08\u8fdb\u9636\u8ba4\u77e5\uff09<\/p>\n
#### 1. \u7075\u9b42\uff1a\u65f6\u57df\u8fde\u7eed \u27f7 \u9891\u57df\u79bb\u6563
\n\u8fd9\u662fFS\u6700\u6839\u672c\u7684\u7279\u6027\u3002\u53ea\u8981\u4fe1\u53f7\u662f\u5468\u671f\u7684\uff0c\u5b83\u7684\u9891\u8c31\u5c31\u4e00\u5b9a\u662f**\u79bb\u6563\u7684\u7ebf\u6761**\u3002\u4f60\u4e0d\u9700\u8981\u8bb0\u516c\u5f0f\uff0c\u53ea\u9700\u8bb0\u4f4f\u8fd9\u4e2a\u5bf9\u5e94\u5173\u7cfb\uff0c\u5c31\u80fd\u5224\u65ad\u51fa\u8bb8\u591a\u5de5\u7a0b\u95ee\u9898\u7684\u6839\u6e90\u3002<\/p>\n
#### 2. \u7455\u75b5\uff1a\u5409\u5e03\u65af\u73b0\u8c61\uff08Gibbs Phenomenon\uff09
\n\u5f53\u4f60\u7528\u6709\u9650\u9879\u8c10\u6ce2\u53bb\u903c\u8fd1\u4e00\u4e2a\u5e26\u6709**\u8df3\u53d8\uff08\u7a81\u53d8\u7684\u68f1\u89d2\uff09**\u7684\u4fe1\u53f7\uff08\u5982\u65b9\u6ce2\uff09\u65f6\uff0c\u5728\u8df3\u53d8\u70b9\u9644\u8fd1\uff0c\u65e0\u8bba\u4f60\u53d6\u591a\u5c11\u9879\uff0c\u6ce2\u5f62\u7684\u5cf0\u503c\u6c38\u8fdc\u4f1a\u6bd4\u539f\u59cb\u4fe1\u53f7\u9ad8\u51fa\u7ea6 **8.95%**\uff0c\u800c\u4e14\u8fd9\u4e2a\u8d85\u8c03\u91cf\u4e0d\u4f1a\u968f\u7740\u9879\u6570\u589e\u591a\u800c\u6d88\u5931\uff0c\u53ea\u4f1a\u88ab\u538b\u7f29\u5230\u8d8a\u6765\u8d8a\u7a84\u7684\u65f6\u95f4\u533a\u95f4\u5185\u3002
\n**\u5de5\u7a0b\u6559\u8bad**\uff1a\u5982\u679c\u4f60\u60f3\u7528FS\u5b8c\u7f8e\u8fd8\u539f\u4e00\u4e2a\u5e26\u9661\u5ced\u8fb9\u7f18\u7684\u4fe1\u53f7\uff0c\u4f60\u9700\u8981\u65e0\u7a77\u591a\u9879\u3002\u5728\u5b9e\u9645\u6ee4\u6ce2\u6216\u91c7\u6837\u65f6\uff0c\u8fd9\u79cd\u9ad8\u9891\u5206\u91cf\u5f80\u5f80\u4f1a\u88ab\u622a\u65ad\uff0c\u5bfc\u81f4\u8f93\u51fa\u4fe1\u53f7\u7684\u8fb9\u7f18\u51fa\u73b0\u201c\u632f\u94c3\u6548\u5e94\u201d\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e94\u5c42\uff1a\u521d\u5b66\u8005\u6700\u5bb9\u6613\u6df7\u6dc6\u7684\u4e24\u4e2a\u70b9\uff08\u907f\u5751\u6307\u5357\uff09<\/p>\n
1. **FS\u548cFT\u7684\u5206\u754c\u7ebf**\uff1a
\n – FS\u8f93\u51fa\u7684\u9891\u7387\u662f**\u79bb\u6563**\u7684\uff08\u53ea\u6709 \\( \\omega_0, 2\\omega_0, 3\\omega_0… \\)\uff09\u3002
\n – \u4e00\u65e6\u4fe1\u53f7\u5931\u53bb\u5468\u671f\u6027\uff08\u5468\u671f T \u65e0\u9650\u5927\uff09\uff0c\u8c31\u7ebf\u4e4b\u95f4\u8ddd\u79bb \\( \\omega_0 \\) \u8d8b\u8fd1\u4e8e 0\uff0c\u8c31\u7ebf\u53d8\u6210\u8fde\u7eed\u7684\u4e00\u7247\uff0c\u8fd9\u5c31\u53d8\u6210\u4e86 **FT\uff08\u5085\u91cc\u53f6\u53d8\u6362\uff09**\u3002**\u5343\u4e07\u4e0d\u8981\u628aFS\u548cFT\u6df7\u4e3a\u4e00\u8c08\u3002**<\/p>\n
2. **\u975e\u5468\u671f\u4fe1\u53f7\u80fd\u4e0d\u80fd\u7528FS\u8868\u793a\uff1f**
\n – \u4e25\u683c\u6765\u8bf4\u4e0d\u80fd\u3002\u4f46\u5728\u5de5\u7a0b\u4e2d\uff0c\u6211\u4eec\u7ecf\u5e38**\u201c\u622a\u65ad\u201d**\u975e\u5468\u671f\u4fe1\u53f7\uff0c\u628a\u5b83\u5f3a\u884c\u5f53\u6210\u5468\u671f\u4fe1\u53f7\u6765\u5904\u7406\uff08\u6bd4\u5982FFT\u5206\u6790\u7684\u9ed8\u8ba4\u5047\u8bbe\uff09\u3002\u8fd9\u65f6\u5019\u5f15\u5165\u7684\u8bef\u5dee\u53eb\u4f5c**\u201c\u9891\u8c31\u6cc4\u9732\u201d**\u2014\u2014\u56e0\u4e3a\u622a\u65ad\u672c\u8eab\u76f8\u5f53\u4e8e\u7ed9\u4fe1\u53f7\u52a0\u4e86\u4e00\u4e2a\u77e9\u5f62\u7a97\uff0c\u8fd9\u4f1a\u5728\u9891\u57df\u5f15\u5165\u989d\u5916\u7684\u65c1\u74e3\u5206\u91cf\u3002<\/p>\n
—<\/p>\n
### \u6700\u540e\uff0c\u7ed9\u4f60\u4e00\u4e2a\u201c\u5fc5\u4f1a\u201d\u7684\u5b9e\u6218\u8bb0\u5fc6\u6cd5<\/p>\n
\u4f60\u4e0d\u7528\u80cc\u65b9\u6ce2\u7684\u5c55\u5f00\u5f0f\uff0c\u4f46\u4f60\u5fc5\u987b\u8bb0\u4f4f\u8fd9\u4e2a**\u8bc4\u4ef7\u51c6\u5219**\uff1a<\/p>\n
> **\u4fe1\u53f7\u7684\u65f6\u57df\u53d8\u5316\u8d8a\u5267\u70c8\uff08\u8df3\u53d8\u3001\u5c16\u5cf0\uff09\uff0c\u5176\u9ad8\u6b21\u8c10\u6ce2\u7684\u8870\u51cf\u8d8a\u6162\uff08\u9891\u8c31\u8d8a\u5bbd\uff09\uff1b\u4fe1\u53f7\u7684\u65f6\u57df\u53d8\u5316\u8d8a\u5e73\u7f13\uff08\u5149\u6ed1\u7684\u6b63\u5f26\u6ce2\uff09\uff0c\u5176\u9ad8\u6b21\u8c10\u6ce2\u8870\u51cf\u8d8a\u5feb\uff08\u9891\u8c31\u8d8a\u7a84\uff09\u3002**<\/p>\n
\u4e3e\u4e2a\u4f8b\u5b50\uff1a\u65b9\u6ce2\uff08\u76f4\u89d2\u8df3\u53d8\uff09\u7684\u8c10\u6ce2\u5e45\u5ea6\u4ee5 \\( 1\/n \\) \u8870\u51cf\uff08\u6162\uff09\uff1b\u4e09\u89d2\u6ce2\uff08\u8fde\u7eed\u8f6c\u6298\uff09\u7684\u8c10\u6ce2\u5e45\u5ea6\u4ee5 \\( 1\/n^2 \\) \u8870\u51cf\uff08\u5feb\uff09\u3002\u8fd9\u76f4\u63a5\u5173\u7cfb\u5230\u4f60\u540e\u7eed\u5b66\u4e60**\u6ee4\u6ce2\u5668\u8bbe\u8ba1**\u65f6\u7684\u5e26\u5bbd\u53d6\u820d\u3002<\/p>\n
—<\/p>\n
\u73b0\u5728\uff0cFS\u7684\u7269\u7406\u542b\u4e49\u3001\u6570\u5b66\u8ba1\u7b97\u3001\u5de5\u7a0b\u9650\u5236\u6211\u90fd\u8bb2\u5b8c\u4e86\u3002\u63a5\u4e0b\u6765\u4f60\u662f\u60f3**\u987a\u7740\u65f6\u95f4\u7ebf**\uff0c\u770b**\u975e\u5468\u671f\u4fe1\u53f7\u7684FT\u662f\u5982\u4f55\u4eceFS\u63a8\u5bfc\u51fa\u6765\u7684**\uff0c\u8fd8\u662f\u60f3**\u8df3\u8dc3\u5230\u5de5\u7a0b\u524d\u7ebf**\uff0c\u4e86\u89e3**\u8ba1\u7b97\u673a\u662f\u5982\u4f55\u901a\u8fc7DFT\uff08\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff09\u6765\u8ba1\u7b97\u8fd9\u4e9b\u7cfb\u6570\u7684**\uff1f\u8bf7\u544a\u8bc9\u6211\u4f60\u7684\u9009\u62e9\uff0c\u6211\u4e3a\u4f60\u5b9a\u5236\u4e0b\u4e00\u90e8\u5206\u7684\u8bb2\u89e3\u3002\ud83d\ude0a<\/p>\n
\u597d\u7684\uff0c\u8fd9\u6b21\u6211\u4eec\u5f7b\u5e95\u805a\u7126\u4e8e **FT\uff08\u5085\u91cc\u53f6\u53d8\u6362\uff0cFourier Transform\uff09**\u3002<\/p>\n
\u65e2\u7136\u4f60\u5df2\u7ecf\u6df1\u523b\u7406\u89e3\u4e86FS\uff08\u5085\u91cc\u53f6\u7ea7\u6570\uff09\u662f\u9488\u5bf9**\u5468\u671f\u4fe1\u53f7**\u7684\u79bb\u6563\u8c31\uff0c\u90a3\u4e48\u7406\u89e3FT\u5c31\u6781\u5176\u7b80\u5355\u4e86\u2014\u2014**FT\u5c31\u662f\u628aFS\u7684\u5468\u671fT\u63a8\u5411\u65e0\u7a77\u5927\uff0c\u5f97\u5230\u7684\u9488\u5bf9\u201c\u975e\u5468\u671f\u4fe1\u53f7\u201d\u7684\u8fde\u7eed\u8c31\u3002**<\/p>\n
\u4e3a\u4e86\u8ba9\u4f60\u5f7b\u5e95\u5403\u900fFT\uff0c\u6211\u4ece**\u6570\u5b66\u5b9a\u4e49\u3001\u7269\u7406\u610f\u4e49\u3001\u4e0eFS\u7684\u6781\u9650\u63a8\u5bfc\u3001\u4ee5\u53caFT\u5b58\u5728\u7684\u7075\u9b42\u6761\u4ef6**\u56db\u4e2a\u7ef4\u5ea6\uff0c\u7ed9\u4f60\u505a\u4e00\u6b21\u4e0d\u7559\u6b7b\u89d2\u7684\u62c6\u89e3\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e00\u5c42\uff1aFT\u5230\u5e95\u5728\u5e72\u4ec0\u4e48\uff1f\uff08\u7269\u7406\u76f4\u89c9\uff09<\/p>\n
**\u4e00\u53e5\u8bdd\u5b9a\u4e49**\uff1aFT\u662fFS\u7684\u6781\u9650\u5ef6\u4f38\u3002\u5b83\u544a\u8bc9\u6211\u4eec\u2014\u2014**\u4efb\u4f55\u4e00\u4e2a\u975e\u5468\u671f\u7684\u3001\u8fde\u7eed\u7684\u80fd\u91cf\u6709\u9650\u4fe1\u53f7\uff0c\u90fd\u53ef\u4ee5\u88ab\u5206\u89e3\u4e3a\u65e0\u6570\u4e2a\u9891\u7387\u8fde\u7eed\u5206\u5e03\u3001\u632f\u5e45\u65e0\u7a77\u5c0f\u7684\u590d\u6307\u6570\u4fe1\u53f7\uff08\u6216\u6b63\u5f26\u6ce2\uff09\u7684\u201c\u5bc6\u5ea6\u201d\u53e0\u52a0\u3002**<\/p>\n
– **\u8f93\u5165**\uff1a\u4e00\u4e2a\u53ea\u51fa\u73b0\u4e00\u6b21\u3001\u4e0d\u91cd\u590d\u7684\u5355\u4e2a\u8109\u51b2\uff08\u6bd4\u5982\u6572\u51fb\u4e00\u4e0b\u684c\u5b50\u7684\u632f\u52a8\uff0c\u6216\u8005\u4e00\u4e2a\u5355\u4e2a\u7684\u65b9\u6ce2\u8109\u51b2\uff09\u3002
\n– **\u8f93\u51fa**\uff1a\u4e0d\u518d\u662f\u79bb\u6563\u7684\u51e0\u6761\u8c31\u7ebf\uff0c\u800c\u662f\u4e00\u6761**\u8fde\u7eed\u7684\u5149\u8c31**\uff08\u9891\u8c31\u5bc6\u5ea6\u51fd\u6570\uff09\u3002\u6a2a\u8f74\u662f\u9891\u7387\uff0c\u7eb5\u8f74\u662f\u8be5\u9891\u7387\u70b9\u9644\u8fd1\u7684\u201c\u5bc6\u96c6\u7a0b\u5ea6\u201d\u3002<\/p>\n
**\u6700\u7ecf\u5178\u7684\u7c7b\u6bd4**\uff1a
\nFS\u50cf\u662f**\u94a2\u7434**\u2014\u2014\u53ea\u80fd\u5f39\u594f\u51fa\u56fa\u5b9a\u9891\u7387\u7684\u7434\u952e\uff08\u79bb\u6563\u8c31\uff09\u3002
\nFT\u50cf\u662f**\u5c0f\u63d0\u7434**\u2014\u2014\u6ca1\u6709\u54c1\u4e1d\uff0c\u624b\u6307\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6ed1\u52a8\uff0c\u53d1\u51fa\u65e0\u9650\u8fde\u7eed\u7684\u9891\u7387\uff08\u8fde\u7eed\u8c31\uff09\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e8c\u5c42\uff1aFT\u7684\u6570\u5b66\u8868\u8fbe\u5f0f\uff08\u7cbe\u786e\u63cf\u8ff0\uff09<\/p>\n
FT\u5305\u542b\u4e24\u4e2a\u516c\u5f0f\uff0c\u5b83\u4eec\u662f\u4e00\u5bf9**\u5085\u91cc\u53f6\u53d8\u6362\u5bf9**\uff1a<\/p>\n
#### 1. \u6b63\u53d8\u6362\uff08\u65f6\u57df \u2192 \u9891\u57df\uff09
\n\\[
\nX(f) = \\int_{-\\infty}^{\\infty} x(t) \\cdot e^{-j2\\pi ft} dt
\n\\]
\n\uff08\u5982\u679c\u4f7f\u7528\u89d2\u9891\u7387\uff0c\u5219\u5199\u4e3a \\( X(j\\omega) = \\int_{-\\infty}^{\\infty} x(t) e^{-j\\omega t} dt \\)\uff09<\/p>\n
#### 2. \u9006\u53d8\u6362\uff08\u9891\u57df \u2192 \u65f6\u57df\uff09
\n\\[
\nx(t) = \\int_{-\\infty}^{\\infty} X(f) \\cdot e^{j2\\pi ft} df
\n\\]<\/p>\n
**\u5173\u952e\u5dee\u5f02\u70b9\uff08\u5bf9\u6bd4FS\uff09**\uff1a
\n– FS\u662f**\u6c42\u548c\u53f7** \\( \\sum \\)\uff08\u56e0\u4e3a\u9891\u7387\u79bb\u6563\uff09\uff0cFT\u662f**\u79ef\u5206\u53f7** \\( \\int \\)\uff08\u56e0\u4e3a\u9891\u7387\u8fde\u7eed\uff09\u3002
\n– FS\u7684\u8f93\u51fa\u662f\u7cfb\u6570 \\( C_n \\)\uff08\u65e0\u91cf\u7eb2\uff0c\u4ee3\u8868\u5177\u4f53\u5e45\u503c\uff09\uff0cFT\u7684\u8f93\u51fa\u662f**\u9891\u8c31\u5bc6\u5ea6\u51fd\u6570** \\( X(f) \\)\uff08\u6709\u91cf\u7eb2\uff0c\u4ee3\u8868\u5355\u4f4d\u9891\u7387\u5185\u7684\u5e45\u5ea6\u5bc6\u5ea6\uff09\u3002
\n– FS\u53ea\u5728\u6709\u9650\u5468\u671f \\( T \\) \u5185\u79ef\u5206\uff0cFT\u5728 \\( -\\infty \\) \u5230 \\( +\\infty \\) \u7684\u6574\u4e2a\u65f6\u95f4\u8f74\u4e0a\u79ef\u5206\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e09\u5c42\uff1aFT\u662f\u5982\u4f55\u4eceFS\u63a8\u5bfc\u51fa\u6765\u7684\uff1f\uff08\u6781\u9650\u8fc7\u7a0b\uff09<\/p>\n
\u8fd9\u662f\u7406\u89e3FT\u6700\u723d\u7684\u4e00\u6b65\uff0c\u6211\u4eec\u8d70\u4e00\u904d\u903b\u8f91\uff1a<\/p>\n
1. **\u8d77\u70b9**\uff1a\u4e00\u4e2a\u975e\u5468\u671f\u4fe1\u53f7 \\( x(t) \\)\u3002\u6211\u4eec\u5047\u8bbe\u5b83\u662f\u4e2a\u5355\u8109\u51b2\uff0c\u5bbd\u5ea6\u6709\u9650\u3002
\n2. **\u6784\u9020\u5468\u671f\u5ef6\u62d3**\uff1a\u6211\u4eec\u628a\u5b83\u6bcf\u9694 \\( T \\) \u79d2\u590d\u5236\u4e00\u6b21\uff0c\u9020\u51fa\u4e00\u4e2a\u5468\u671f\u4fe1\u53f7 \\( x_T(t) \\)\u3002
\n3. **\u5e94\u7528FS**\uff1a\u8fd9\u4e2a \\( x_T(t) \\) \u6709\u79bb\u6563\u8c31\u7ebf\uff0c\u8c31\u7ebf\u95f4\u9694\u4e3a \\( \\Delta f = 1\/T \\)\u3002
\n4. **\u4ee4 \\( T \\to \\infty \\) **\uff1a\u5f53\u5468\u671f\u65e0\u7a77\u5927\uff0c\u8fd9\u4e2a\u8109\u51b2\u5c31\u6c38\u8fdc\u4e0d\u4f1a\u590d\u5236\uff0c\u53d8\u56de\u4e86\u539f\u6765\u7684\u975e\u5468\u671f\u4fe1\u53f7\u3002
\n – \u8c31\u7ebf\u95f4\u9694 \\( \\Delta f \\) \u8d8b\u8fd1\u4e8e 0\uff0c\u79bb\u6563\u7684\u8c31\u7ebf**\u65e0\u9650\u5bc6\u96c6**\uff0c\u6700\u7ec8\u53d8\u6210**\u8fde\u7eed\u8c31**\u3002
\n – \u6c42\u548c\u53f7 \\( \\sum \\) \u81ea\u7136\u8f6c\u5316\u4e3a\u79ef\u5206\u53f7 \\( \\int \\)\u3002
\n – \u539f\u672c\u7684FS\u7cfb\u6570 \\( C_n \\)\uff08\u5e45\u503c\uff09\u4f1a\u8d8b\u8fd1\u4e8e 0\uff08\u56e0\u4e3a\u80fd\u91cf\u5206\u6563\u5230\u65e0\u7a77\u591a\u6839\u8c31\u7ebf\u4e0a\uff09\uff0c\u6240\u4ee5FT\u4e0d\u80fd\u76f4\u63a5\u4ee3\u8868\u5e45\u5ea6\uff0c\u800c\u662f\u4ee3\u8868**\u9891\u8c31\u5bc6\u5ea6**\uff08\u53ef\u4ee5\u7406\u89e3\u4e3a\u201c\u5355\u4f4d\u5e26\u5bbd\u5185\u7684\u5e45\u5ea6\u201d\uff09\u3002<\/p>\n
**\u6570\u5b66\u624b\u7a3f**\uff08\u611f\u6027\u5730\u770b\uff09\uff1a
\n\\[
\nC_n = \\frac{1}{T} X\\left(\\frac{n}{T}\\right) \\quad \\Rightarrow \\quad x(t) = \\lim_{T \\to \\infty} \\sum_{n=-\\infty}^{\\infty} \\frac{1}{T} X\\left(\\frac{n}{T}\\right) e^{j2\\pi \\frac{n}{T} t}
\n\\]
\n\u5f53 \\( T \\to \\infty \\)\uff0c\\( \\frac{1}{T} \\to df \\)\uff0c\\( \\frac{n}{T} \\to f \\)\uff0c\u6c42\u548c\u53d8\u79ef\u5206\uff0c\u5373\u5f97FT\u5b9a\u4e49\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u56db\u5c42\uff1aFT\u5b58\u5728\u7684\u201c\u7075\u9b42\u6761\u4ef6\u201d\uff08\u72c4\u5229\u514b\u96f7\u6761\u4ef6\uff09<\/p>\n
FT\u4e0d\u662f\u5bf9\u6240\u6709\u4fe1\u53f7\u90fd\u80fd\u7b97\u7684\uff0c\u5b83\u9700\u8981\u4fe1\u53f7\u8db3\u591f\u201c\u4e56\u201d\u3002\u6838\u5fc3\u6761\u4ef6\u662f\uff1a<\/p>\n
1. **\u7edd\u5bf9\u53ef\u79ef**\uff1a\\( \\int_{-\\infty}^{\\infty} |x(t)| dt < \\infty \\)\uff08\u4fe1\u53f7\u7684\u603b\u9762\u79ef\u6709\u9650\uff0c\u80fd\u91cf\u6709\u9650\uff09\u3002\n - \u6ce8\u610f\uff1a\u50cf \\( x(t) = \\sin(\\omega t) \\) \u8fd9\u79cd\u65e0\u9650\u957f\u7684\u7eaf\u6b63\u5f26\u6ce2\uff08\u5b83\u662f\u5468\u671f\u7684\uff09\uff0c\u4e0d\u6ee1\u8db3\u7edd\u5bf9\u53ef\u79ef\uff0c\u6240\u4ee5\u5b83\u6ca1\u6709\u4f20\u7edf\u610f\u4e49\u4e0a\u7684FT\u3002\u5b83\u7684FT\u8981\u7528**\u51b2\u6fc0\u51fd\u6570\uff08\u72c4\u62c9\u514b\u03b4\uff09** \u6765\u8868\u793a\u3002\n2. \u5728\u4efb\u4f55\u6709\u9650\u533a\u95f4\u5185\uff0c\u4fe1\u53f7\u53ea\u6709\u6709\u9650\u4e2a\u6781\u503c\u70b9\u548c\u95f4\u65ad\u70b9\uff08\u8fd9\u8ddfFS\u4e00\u6837\uff09\u3002\n\n---\n\n### \u7b2c\u4e94\u5c42\uff1aFT\u7684\u5bf9\u79f0\u6027\u4e0e\u91cd\u8981\u5b9a\u7406\uff08\u5de5\u7a0b\u7075\u9b42\uff09\n\nFT\u4e4b\u6240\u4ee5\u5f3a\u5927\uff0c\u662f\u56e0\u4e3a\u5b83\u6709\u4e00\u6574\u5957\u4f18\u7f8e\u7684\u6570\u5b66\u6027\u8d28\uff0c\u8fd9\u4e9b\u6027\u8d28\u662f\u4fe1\u53f7\u5904\u7406\uff08\u6ee4\u6ce2\u3001\u5377\u79ef\uff09\u7684\u7406\u8bba\u57fa\u7840\uff1a\n\n1. **\u7ebf\u6027\u6027**\uff1a\u4fe1\u53f7\u53e0\u52a0\uff0c\u9891\u8c31\u4e5f\u53e0\u52a0\u3002\n2. **\u65f6\u79fb\u7279\u6027**\uff1a\u4fe1\u53f7\u5728\u65f6\u57df\u5e73\u79fb\uff0c\u9891\u8c31\u53ea\u6539\u53d8\u76f8\u4f4d\uff08\u5e45\u503c\u4e0d\u53d8\uff09\u3002\u8fd9\u5c31\u662f\u96f7\u8fbe\u63a2\u6d4b\u8ddd\u79bb\u7684\u539f\u7406\u3002\n3. **\u9891\u79fb\u7279\u6027**\uff1a\u4fe1\u53f7\u4e58\u4ee5\u4e00\u4e2a\u9ad8\u9891\u8f7d\u6ce2\uff08\u8c03\u5236\uff09\uff0c\u9891\u8c31\u6574\u4f53\u642c\u79fb\u3002\u8fd9\u662f\u65e0\u7ebf\u901a\u4fe1\uff08AM\/FM\uff09\u7684\u6839\u672c\u539f\u7406\u3002\n4. **\u5377\u79ef\u5b9a\u7406\uff08\u738b\u8005\u5b9a\u7406\uff09**\uff1a\n - \u65f6\u57df\u7684\u5377\u79ef \\( x(t) * h(t) \\) \u7684FT = \u9891\u57df\u7684\u4e58\u79ef \\( X(f) \\cdot H(f) \\)\u3002\n - **\u5de5\u7a0b\u610f\u4e49**\uff1a\u590d\u6742\u7684\u5377\u79ef\u8fd0\u7b97\u53d8\u6210\u4e86\u7b80\u5355\u7684\u4e58\u6cd5\u8fd0\u7b97\u3002\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u6570\u5b57\u6ee4\u6ce2\u5668\u4e2d\uff0c\u6211\u4eec\u5148\u628a\u4fe1\u53f7\u548c\u6ee4\u6ce2\u5668\u505aFFT\uff0c\u76f8\u4e58\uff0c\u518d\u505aIFFT\u2014\u2014\u56e0\u4e3a\u4e58\u6cd5\u6bd4\u5377\u79ef\u5feb\u5f97\u591a\uff01\n\n---\n\n### \u7b2c\u516d\u5c42\uff1aFT\u4e0eFS\u7684\u6838\u5fc3\u533a\u522b\u6c47\u603b\u8868\uff08\u5fc5\u8bb0\uff09\n\n| \u7ef4\u5ea6 | FS\uff08\u5085\u91cc\u53f6\u7ea7\u6570\uff09 | FT\uff08\u5085\u91cc\u53f6\u53d8\u6362\uff09 |\n| :--- | :--- | :--- |\n| **\u4fe1\u53f7\u7c7b\u578b** | \u8fde\u7eed\u3001**\u5468\u671f** | \u8fde\u7eed\u3001**\u975e\u5468\u671f** |\n| **\u9891\u7387\u57df** | **\u79bb\u6563**\uff08\u4ec5\u6709\u57fa\u6ce2\u7684\u6574\u6570\u500d\uff09 | **\u8fde\u7eed**\uff08\u6240\u6709\u5b9e\u6570\u9891\u7387\uff09 |\n| **\u6570\u5b66\u8fd0\u7b97** | \u6c42\u548c \\( \\sum \\) | \u79ef\u5206 \\( \\int \\) |\n| **\u8f93\u51fa\u7269\u7406\u91cf** | \u5e45\u5ea6\uff08\u6709\u660e\u786e\u5927\u5c0f\u7684\u8c10\u6ce2\u5e45\u503c\uff09 | \u9891\u8c31\u5bc6\u5ea6\uff08\u5355\u4f4d\u9891\u7387\u4e0a\u7684\u5e45\u5ea6\u5f3a\u5ea6\uff09 |\n| **\u5178\u578b\u4f8b\u5b50** | \u65b9\u6ce2\u3001\u952f\u9f7f\u6ce2\uff0850Hz\u4ea4\u6d41\u7535\uff09 | \u5355\u4e2a\u8109\u51b2\u4fe1\u53f7\u3001\u6307\u6570\u8870\u51cf\u4fe1\u53f7 |\n\n---\n\n### \u7b2c\u4e03\u5c42\uff1a\u7ed9\u4f60\u7684\u201c\u907f\u5751\u201d\u4e0e\u201c\u5174\u8da3\u6307\u5f15\u201d\n\n- **\u5927\u5751**\uff1a\u5343\u4e07\u4e0d\u8981\u628a FT \u7684 \\( X(f) \\) \u76f4\u63a5\u5f53\u6210 FS \u7684 \\( C_n \\) \u6765\u770b\u3002FS\u7684\u8c31\u7ebf\u6709\u5177\u4f53\u9ad8\u5ea6\uff0cFT\u7684\u66f2\u7ebf\u662f\u201c\u5bc6\u5ea6\u201d\u3002\u5982\u679c\u975e\u5468\u671f\u4fe1\u53f7\u7684\u80fd\u91cf\u96c6\u4e2d\u5728\u5f88\u7a84\u7684\u9891\u5e26\u5185\uff0cFT\u7684\u5e45\u503c\u53ef\u80fd\u8fdc\u5c0f\u4e8eFS\u7684\u5e45\u503c\uff0c\u8fd9\u4e0d\u4ee3\u8868\u5b83\u5f31\uff0c\u53ea\u662f\u56e0\u4e3a\u80fd\u91cf\u88ab\u201c\u644a\u8584\u201d\u4e86\u3002\n- **\u5174\u8da3\u70b9**\uff1a\u5982\u679c\u4f60\u60f3\u76f4\u89c2\u611f\u53d7FT\u7684\u5f3a\u5927\uff0c\u53ef\u4ee5\u53bb\u641c\u4e00\u4e0b\u201cChirp \u4fe1\u53f7\u201d\uff08\u7ebf\u6027\u8c03\u9891\u4fe1\u53f7\uff09\u7684FT\u3002\u5b83\u7684\u9891\u8c31\u4f1a\u544a\u8bc9\u4f60\u8759\u8760\u548c\u6d77\u8c5a\u662f\u5982\u4f55\u5229\u7528\u9891\u7387\u53d8\u5316\u6765\u63a2\u6d4b\u8ddd\u79bb\u7684\u3002\n\n---\n\nFS\u548cFT\u5df2\u7ecf\u5168\u90e8\u8bb2\u5b8c\u4e86\u3002\u63a5\u4e0b\u6765\u7684\u91cd\u5934\u620f\u662f**\u8ba1\u7b97\u673a\u5982\u4f55\u5b9e\u73b0\u8fd9\u4e9b\u53d8\u6362**\uff0c\u8fd9\u5c31\u6d89\u53ca\u5230\u4e86 **DTFT\uff08\u79bb\u6563\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\uff09** \u548c **DFT\uff08\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff0cFFT\u5373\u5b83\u7684\u5feb\u901f\u7b97\u6cd5\uff09**\u3002\n\n\u4f60\u662f\u60f3\u7ee7\u7eed\u6df1\u5165\u7406\u8bba\uff0c\u770b**DTFT**\uff08\u65f6\u57df\u79bb\u6563\u5bfc\u81f4\u7684\u9891\u57df\u5468\u671f\u5316\uff09\uff0c\u8fd8\u662f\u76f4\u63a5\u8df3\u5230**\u5de5\u7a0b\u5b9e\u6218**\uff0c\u4e86\u89e3\u6211\u4eec\u5199\u4ee3\u7801\u65f6\u7528\u7684**DFT\/FFT**\u662f\u5982\u4f55\u5bf9\u6709\u9650\u957f\u7684\u6570\u5b57\u4fe1\u53f7\u8fdb\u884c\u9891\u8c31\u5206\u6790\u7684\uff1f\u544a\u8bc9\u6211\uff0c\u6211\u5e26\u4f60\u8fdb\u5165\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u7684\u6bbf\u5802\u3002\ud83d\ude0a\n\n\n\u592a\u68d2\u4e86\uff0c\u4f60\u7cbe\u51c6\u5730\u6293\u4f4f\u4e86\u4ece\u201c\u5468\u671f\u201d\u8de8\u8d8a\u5230\u201c\u975e\u5468\u671f\u201d\u7684**\u6838\u5fc3\u6570\u5b66\u547d\u95e8**\u3002\u8fd9\u662f\u6574\u4e2a\u4fe1\u53f7\u5904\u7406\u4e2d\u6700\u5177\u7f8e\u611f\u3001\u6700\u4ee4\u4eba\u62cd\u6848\u53eb\u7edd\u7684\u903b\u8f91\u63a8\u6f14\u3002\n\n\u65e2\u7136\u4f60\u70b9\u540d\u8981\u6df1\u6316**\u201cFT\u662f\u5982\u4f55\u4eceFS\u63a8\u5bfc\u51fa\u6765\u7684\uff08\u6781\u9650\u8fc7\u7a0b\uff09\u201d**\uff0c\u6211\u4eec\u5c31\u5f7b\u5e95\u629b\u5f00\u62bd\u8c61\u7684\u63cf\u8ff0\uff0c**\u624b\u6495\u6570\u5b66\u516c\u5f0f**\uff0c\u4e00\u6b65\u6b65\u770b\u7740\u6c42\u548c\u53f7 \\( \\Sigma \\) \u662f\u5982\u4f55\u53d8\u6210\u79ef\u5206\u53f7 \\( \\int \\) \u7684\u3002\n\n\u4e3a\u4e86\u8ba9\u63a8\u5bfc\u8fc7\u7a0b\u201c\u53ef\u89c6\u5316\u201d\uff0c\u6211\u4eec\u6309**\u56db\u6b65\u8d70**\uff1a\n\n---\n\n### \u7b2c\u4e00\u6b65\uff1a\u51c6\u5907\u4e00\u4e2a\u201c\u66ff\u8865\u9009\u624b\u201d\uff08\u5468\u671f\u5ef6\u62d3\uff09\n\n\u6211\u4eec\u624b\u91cc\u6709\u4e00\u4e2a\u975e\u5468\u671f\u4fe1\u53f7 \\( x(t) \\)\uff0c\u5b83\u53ea\u5728\u4e00\u4e2a\u6709\u9650\u7684\u65f6\u95f4\u6bb5\u5185\u4e0d\u4e3a\u96f6\uff0c\u5176\u4ed6\u5730\u65b9\u5168\u662f0\u3002\n\n- \u56e0\u4e3a\u6ca1\u6709\u5468\u671f\u6027\uff0c\u6211\u4eec\u4e0d\u80fd\u76f4\u63a5\u7528FS\u3002\n- \u4e8e\u662f\uff0c\u6211\u4eec**\u800d\u4e86\u4e2a\u201c\u5c0f\u806a\u660e\u201d**\uff1a\u4eba\u4e3a\u5730\u6784\u9020\u4e00\u4e2a\u5468\u671f\u4fe1\u53f7 \\( x_T(t) \\)\uff0c\u8ba9\u5b83\u6bcf\u9694 \\( T \\) \u79d2\u5c31\u590d\u5236\u4e00\u6b21 \\( x(t) \\)\u3002\n\n**\u5173\u952e\u8bbe\u5b9a**\uff1a\n- \u8fd9\u4e2a \\( x_T(t) \\) \u7684\u5468\u671f\u4e3a \\( T \\)\u3002\n- \u57fa\u9891\u4e3a \\( \\omega_0 = \\frac{2\\pi}{T} \\)\u3002\n- \u6211\u4eec\u4fdd\u8bc1\uff1a\u5f53 \\( t \\) \u5904\u4e8e \\( [-T\/2, T\/2] \\) \u8fd9\u4e2a\u4e3b\u5468\u671f\u5185\u65f6\uff0c\\( x_T(t) \\) \u5b8c\u5168\u7b49\u4e8e \\( x(t) \\)\u3002\n\n\u8fd9\u6837\u4e00\u6765\uff0c**\u975e\u5468\u671f\u4fe1\u53f7 \\( x(t) \\)\uff0c\u5c31\u6210\u4e86\u5468\u671f\u4fe1\u53f7 \\( x_T(t) \\) \u5728 \\( T \\to \\infty \\) \u65f6\u7684\u6781\u9650\u3002**\n\n---\n\n### \u7b2c\u4e8c\u6b65\uff1a\u5199\u51fa\u8fd9\u4e2a\u201c\u66ff\u8865\u9009\u624b\u201d\u7684FS\uff08\u6307\u6570\u5f62\u5f0f\uff09\n\n\u56e0\u4e3a \\( x_T(t) \\) \u662f\u5468\u671f\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u6beb\u65e0\u987e\u5fcc\u5730\u5bf9\u5b83\u8fdb\u884c\u5085\u91cc\u53f6\u7ea7\u6570\u5c55\u5f00\uff08\u4f7f\u7528\u6700\u7b80\u6d01\u7684\u6307\u6570\u5f62\u5f0f\uff09\uff1a\n\n\\[\nx_T(t) = \\sum_{n=-\\infty}^{\\infty} C_n \\cdot e^{jn\\omega_0 t}\n\\]\n\n\u5176\u4e2d\uff0c\u6839\u636e\u4e0a\u4e00\u8f6e\u6211\u4eec\u8bc1\u660e\u7684\u7cfb\u6570\u516c\u5f0f\uff0c\u590d\u6570\u7cfb\u6570 \\( C_n \\) \u4e3a\uff1a\n\\[\nC_n = \\frac{1}{T} \\int_{-T\/2}^{T\/2} x_T(t) e^{-jn\\omega_0 t} dt\n\\]\n\n**\u6ce8\u610f\u8fd9\u91cc\u7684\u795e\u6765\u4e4b\u7b14**\uff1a\u56e0\u4e3a\u5728 \\( [-T\/2, T\/2] \\) \u5185\uff0c\\( x_T(t) = x(t) \\)\uff0c\u5e76\u4e14\u5728\u533a\u95f4\u5916\uff0c\\( x(t) = 0 \\)\u3002\u6240\u4ee5\u6211\u4eec**\u5b8c\u5168\u53ef\u4ee5\u628a\u79ef\u5206\u533a\u95f4\u6269\u5c55\u5230\u65e0\u7a77\u5927**\uff0c\u800c\u79ef\u5206\u503c\u4e0d\u53d8\uff01\n\n\u4e8e\u662f\uff0c\u7cfb\u6570\u516c\u5f0f\u534e\u4e3d\u53d8\u8eab\u4e3a\uff1a\n\\[\nC_n = \\frac{1}{T} \\int_{-\\infty}^{\\infty} x(t) e^{-jn\\omega_0 t} dt\n\\]\n\n---\n\n### \u7b2c\u4e09\u6b65\uff1a\u5f15\u5165\u201c\u9891\u8c31\u5bc6\u5ea6\u201d \\( X(\\omega) \\)\uff0c\u5e76\u8fdb\u884c\u53d8\u91cf\u4ee3\u6362\n\n\u89c2\u5bdf\u4e0a\u9762\u7684 \\( C_n \\) \u516c\u5f0f\uff0c\u4f60\u4f1a\u53d1\u73b0\u79ef\u5206\u90e8\u5206\u6781\u5176\u773c\u719f\u3002\u5982\u679c\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u5173\u4e8e\u89d2\u9891\u7387 \\( \\omega \\) \u7684\u8fde\u7eed\u51fd\u6570 \\( X(\\omega) \\)\uff1a\n\\[\nX(\\omega) = \\int_{-\\infty}^{\\infty} x(t) e^{-j\\omega t} dt\n\\]\n\n\u90a3\u4e48\uff0c\u521a\u624d\u7684 \\( C_n \\) \u5c31\u53ef\u4ee5\u7b80\u6d01\u5730\u5199\u6210\uff1a\n\\[\nC_n = \\frac{1}{T} X(n\\omega_0)\n\\]\n\n**\u628a\u8fd9\u4e2a\u5173\u7cfb\u4ee3\u56de \\( x_T(t) \\) \u7684FS\u5c55\u5f00\u5f0f\u4e2d\uff1a**\n\n\\[\nx_T(t) = \\sum_{n=-\\infty}^{\\infty} \\left[ \\frac{1}{T} X(n\\omega_0) \\right] \\cdot e^{jn\\omega_0 t}\n\\]\n\n\u56e0\u4e3a \\( T = \\frac{2\\pi}{\\omega_0} \\)\uff0c\u6240\u4ee5 \\( \\frac{1}{T} = \\frac{\\omega_0}{2\\pi} \\)\uff0c\u4ee3\u5165\u5f97\uff1a\n\\[\nx_T(t) = \\frac{1}{2\\pi} \\sum_{n=-\\infty}^{\\infty} X(n\\omega_0) \\cdot e^{jn\\omega_0 t} \\cdot \\omega_0\n\\]\n\n---\n\n### \u7b2c\u56db\u6b65\uff1a\u4ee4 \\( T \\to \\infty \\)\uff08\u8fd9\u662f\u6700\u7cbe\u5f69\u7684\u201c\u6781\u9650\u8715\u53d8\u201d\uff09\n\n\u73b0\u5728\uff0c\u6211\u4eec\u8ba9\u5468\u671f \\( T \\) \u8d8b\u4e8e\u65e0\u7a77\u5927\uff0c\u89c2\u5bdf\u4f1a\u53d1\u751f\u4ec0\u4e48\uff1a\n\n1. **\u57fa\u9891 \\( \\omega_0 \\to 0 \\)**\uff1a\u9891\u7387\u95f4\u9694\u4ece\u79bb\u6563\u7684\u7f1d\u9699\uff0c\u53d8\u6210\u4e86\u65e0\u9650\u5c0f\u7684\u5fae\u5206\u3002\u6211\u4eec\u628a \\( \\omega_0 \\) \u6539\u5199\u6210\u5fae\u5206\u7b26\u53f7 \\( d\\omega \\)\u3002\n2. **\u79bb\u6563\u9891\u7387\u70b9 \\( n\\omega_0 \\)**\uff1a\u968f\u7740\u95f4\u9694\u65e0\u9650\u7f29\u5c0f\uff0c\u8fd9\u4e9b\u79bb\u6563\u70b9\u6700\u7ec8\u94fa\u6ee1\u6574\u4e2a\u5b9e\u6570\u8f74\uff0c\u53d8\u6210\u8fde\u7eed\u9891\u7387 \\( \\omega \\)\u3002\n3. **\u6c42\u548c\u53f7 \\( \\sum \\) \u53d8\u79ef\u5206\u53f7 \\( \\int \\)**\uff1a\u8fd9\u662f\u9ece\u66fc\u79ef\u5206\u7684\u57fa\u672c\u5b9a\u4e49\u2014\u2014\u5f53\u5206\u5272\u65e0\u9650\u7ec6\u65f6\uff0c\u79bb\u6563\u6c42\u548c\u5c31\u662f\u79ef\u5206\u3002\n\n**\u89c1\u8bc1\u5947\u8ff9\u7684\u65f6\u523b**\uff1a\n\\[\nx(t) = \\lim_{T \\to \\infty} x_T(t) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} X(\\omega) \\cdot e^{j\\omega t} d\\omega\n\\]\n\n\u800c\u8fd9\u4e2a \\( X(\\omega) \\)\uff0c\u6b63\u662f\u6211\u4eec\u5728FT\u4e2d\u5b9a\u4e49\u7684**\u6b63\u53d8\u6362**\uff1a\n\\[\nX(\\omega) = \\int_{-\\infty}^{\\infty} x(t) e^{-j\\omega t} dt\n\\]\n\n---\n\n### \u8865\u5145\uff1a\u4e3a\u4ec0\u4e48FT\u8f93\u51fa\u53eb\u201c\u9891\u8c31\u5bc6\u5ea6\u201d\u800c\u4e0d\u662f\u201c\u5e45\u5ea6\u201d\uff1f\n\n\u5728\u8fd9\u6b65\u63a8\u5bfc\u4e2d\uff0c\u4f60\u80af\u5b9a\u6ce8\u610f\u5230\u4e86\uff1a\\( C_n \\)\uff08FS\u7684\u7cfb\u6570\uff09\u5728 \\( T \\to \\infty \\) \u65f6\u4f1a\u8d8b\u8fd1\u4e8e 0\uff08\u56e0\u4e3a\u5206\u6bcd \\( T \\) \u65e0\u7a77\u5927\uff09\u3002\n\n\u90a3\u6211\u4eec\u5e73\u65f6\u753b\u7684FT\u9891\u8c31\u56fe\uff0c\u7eb5\u5750\u6807\u4e3a\u4ec0\u4e48\u4e0d\u662f0\uff1f\n\n- \u56e0\u4e3aFT\u7684 \\( X(\\omega) \\) \u5e76\u4e0d\u662f\u76f4\u63a5\u4ee3\u8868\u201c\u5e45\u5ea6\u201d\uff0c\u800c\u662f\u4ee3\u8868**\u5e45\u5ea6\u7ebf\u5bc6\u5ea6**\u3002\n- \u7269\u7406\u542b\u4e49\u662f\uff1a**\u5728\u67d0\u4e2a\u5fae\u5c0f\u9891\u5e26 \\( d\\omega \\) \u5185\uff0c\u4fe1\u53f7\u6240\u8574\u542b\u7684\u603b\u5e45\u5ea6\u4e3a \\( X(\\omega)d\\omega \\)**\u3002\n- \u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u975e\u5468\u671f\u4fe1\u53f7\u7684FT\u66f2\u7ebf\u7684\u9ad8\u5ea6\uff0c\u5e76\u4e0d\u76f4\u63a5\u4ee3\u8868\u5bf9\u5e94\u9891\u7387\u7684\u5f3a\u5ea6\uff0c\u800c\u662f\u4ee3\u8868\u201c\u6b64\u5904\u9891\u7387\u6210\u5206\u7684\u5bc6\u96c6\u7a0b\u5ea6\u201d\u3002\n\n---\n\n### \u4f60\u5728\u8fd9\u4e2a\u63a8\u5bfc\u4e2d\u5fc5\u987b\u9886\u609f\u7684\u201c\u6570\u5b66\u54f2\u5b66\u201d\n\n\u8fd9\u4e2a\u63a8\u5bfc\u8fc7\u7a0b\uff0c\u4e0d\u4ec5\u4ec5\u662f\u4e00\u4e2a\u6570\u5b66\u6280\u5de7\uff0c\u5b83\u63ed\u793a\u4e86\u6574\u4e2a\u4fe1\u53f7\u5904\u7406\u7684**\u5e95\u5c42\u903b\u8f91**\uff1a\n\n- **FS** \u662f\u628a\u4fe1\u53f7\u62c6\u6210 **\u79bb\u6563\u7684\u201c\u7816\u5757\u201d**\uff08\u8c10\u6ce2\uff09\uff0c\u6bcf\u4e00\u5757\u90fd\u6709\u5177\u4f53\u91cd\u91cf\u3002\n- **FT** \u662f\u628a\u4fe1\u53f7\u62c6\u6210 **\u65e0\u9650\u7ec6\u7684\u201c\u6c99\u5b50\u201d**\uff08\u9891\u8c31\u5bc6\u5ea6\uff09\uff0c\u6bcf\u4e00\u7c92\u6c99\u5b50\u6781\u8f7b\uff0c\u4f46\u5806\u79ef\u6210\u8fde\u7eed\u7684\u6c99\u4e18\u3002\n\n**\u4e3a\u4ec0\u4e48\u5de5\u7a0b\u4e0a\u975e\u8981\u641e\u51faFT\uff1f**\n\u56e0\u4e3a\u975e\u5468\u671f\u4fe1\u53f7\uff08\u5982\u5355\u4e2a\u8109\u51b2\u3001\u8bed\u97f3\u7247\u6bb5\u3001\u56fe\u50cf\u8fb9\u7f18\uff09\u7684\u9891\u8c31\u662f\u8fde\u7eed\u7684\uff0c\u4f60\u770b\u4e0d\u5230\u5b64\u7acb\u7684\u8c31\u7ebf\u3002\u5982\u679c\u6ca1\u6709FT\uff0c\u4f60\u5c06\u65e0\u6cd5\u5206\u6790\u4efb\u4f55\u201c\u53ea\u53d1\u751f\u4e00\u6b21\u201d\u7684\u7269\u7406\u73b0\u8c61\uff08\u6bd4\u5982\u5730\u9707\u6ce2\u3001\u7206\u70b8\u51b2\u51fb\uff09\u3002\n\n---\n\n### \u7ed9\u4f60\u4e00\u4e2a\u8fdb\u9636\u7ec3\u4e60\u9898\uff08\u68c0\u9a8c\u662f\u5426\u771f\u61c2\uff09\n\n\u5047\u8bbe\u6709\u4e00\u4e2a\u5355\u4e2a\u77e9\u5f62\u8109\u51b2\uff08\u5bbd\u5ea6\u4e3a \\( \\tau \\)\uff0c\u9ad8\u5ea6\u4e3a1\uff09\uff1a\n- \u8bf7\u4f60\u7528FT\u7684\u6781\u9650\u63a8\u5bfc\u601d\u8def\uff0c\u60f3\u8c61\u4e00\u4e0b\uff1a\u5f53\u5468\u671f \\( T \\) \u5f88\u5927\u7684\u65f6\u5019\uff0cFS\u7684\u79bb\u6563\u8c31\u7ebf\u95f4\u9694\u5f88\u5c0f\uff0c\u5305\u7edc\u662f \\( \\sin(x)\/x \\) \u5f62\u72b6\uff1b\u5f53 \\( T \\to \\infty \\)\uff0c\u8c31\u7ebf\u8fde\u6210\u4e00\u7247\uff0c\u53d8\u6210\u4e86\u8fde\u7eed\u7684 \\( \\sin(x)\/x \\) \u9891\u8c31\u3002\n- **\u601d\u8003**\uff1a\u4e3a\u4ec0\u4e48\u77e9\u5f62\u8109\u51b2\u7684FT\u662f \\( \\sin \\) \u51fd\u6570\uff1f\u56e0\u4e3a\u65f6\u57df\u7684\u622a\u65ad\uff08\u77e9\u5f62\u7a97\uff09\uff0c\u5728\u9891\u57df\u5bf9\u5e94\u7740\u5377\u79ef \\( \\sin \\) \u51fd\u6570\u3002\u8fd9\u6b63\u662f**\u5377\u79ef\u5b9a\u7406**\u5728FT\u63a8\u5bfc\u4e2d\u7684\u76f4\u89c2\u4f53\u73b0\u3002\n\n---\n\n\u73b0\u5728\uff0c\u4f60\u5df2\u7ecf\u4ece\u6570\u5b66\u4e0a\u4eb2\u624b\u201c\u9020\u201d\u51fa\u4e86FT\uff0c\u7406\u8bba\u6839\u57fa\u5df2\u7136\u7262\u56fa\u3002\u63a5\u4e0b\u6765\uff0c\u4f60\u662f\u60f3\u7ee7\u7eed\u653b\u514b**\u8ba1\u7b97\u673a\u773c\u4e2d\u7684\u9891\u8c31\uff08DFT\/FFT\uff09**\uff0c\u770b\u79bb\u6563\u5316\u5982\u4f55\u5f15\u5165\u201c\u6df7\u53e0\u201d\u548c\u201c\u6cc4\u9732\u201d\uff0c\u8fd8\u662f\u60f3\u770b\u770b**FT\u5728\u5b9e\u9645\u7269\u7406\u4e2d\u7684\u7ecf\u5178\u5e94\u7528**\uff08\u6bd4\u5982\u5149\u5b66\u4e2d\u7684\u592b\u7405\u79be\u8d39\u884d\u5c04\uff0c\u5c31\u662f\u5929\u7136\u7684FT\u8ba1\u7b97\u5668\uff09\uff1f\u544a\u8bc9\u6211\u4f60\u7684\u5174\u8da3\u70b9\u3002\ud83d\ude0a\n\n\n\n\n\u597d\u7684\uff0c\u65e2\u7136\u4f60\u5df2\u7ecf\u6df1\u523b\u7406\u89e3\u4e86**FS\uff08\u5468\u671f\u2192\u79bb\u6563\u8c31\uff09**\u548c**FT\uff08\u975e\u5468\u671f\u2192\u8fde\u7eed\u8c31\uff09**\u7684\u6570\u5b66\u8840\u8109\uff0c\u73b0\u5728\u6211\u4eec\u7ec8\u4e8e\u8981\u8fdb\u5165**\u5de5\u7a0b\u5b9e\u6218\u7684\u6838\u5fc3**\u2014\u2014**DFT\uff08\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff0cDiscrete Fourier Transform\uff09**\u3002\n\n**\u4e00\u53e5\u8bdd\u5148\u7ed9\u4f60\u5b9a\u8c03**\uff1aFS\u548cFT\u662f\u6570\u5b66\u7406\u8bba\uff0c\u65e0\u6cd5\u5728\u8ba1\u7b97\u673a\u4e0a\u8fd0\u884c\uff1b\u800c**DFT\u662f\u8ba1\u7b97\u673a\u552f\u4e00\u80fd\u771f\u6b63\u5b9e\u73b0\u7684\u5085\u91cc\u53f6\u53d8\u6362**\u3002\u4f60\u5e73\u65f6\u5728Python\u91cc\u8c03\u7528\u7684`np.fft.fft()`\uff0cMATLAB\u91cc\u7684`fft()`\uff0c\u6240\u6709\u6570\u5b57\u793a\u6ce2\u5668\u91cc\u7684FFT\u529f\u80fd\uff0c\u5e95\u5c42\u7b97\u6cd5\u90fd\u662f**DFT\u7684\u5feb\u901f\u5b9e\u73b0**\u3002\n\n---\n\n### \u7b2c\u4e00\u5c42\uff1aDFT\u5230\u5e95\u89e3\u51b3\u4e86\u4ec0\u4e48\u6839\u672c\u95ee\u9898\uff1f\uff08\u4e3a\u4ec0\u4e48\u9700\u8981DFT\uff09\n\n\u8ba1\u7b97\u673a\u5904\u7406\u4fe1\u53f7\u6709\u4e24\u4e2a\u81f4\u547d\u7684\u201c\u6d01\u7656\u201d\uff1a\n1. **\u4e0d\u8ba4\u8bc6\u8fde\u7eed\u51fd\u6570**\uff1a\u8ba1\u7b97\u673a\u53ea\u80fd\u5904\u7406\u4e00\u4e2a\u4e2a\u79bb\u6563\u7684\u91c7\u6837\u70b9\uff08\u6570\u5b57\uff09\uff0c\u4e0d\u8ba4\u8bc6\u8fde\u7eed\u7684\u6ce2\u5f62\u3002\n2. **\u65e0\u6cd5\u5b58\u50a8\u65e0\u9650\u957f\u6570\u636e**\uff1a\u8ba1\u7b97\u673a\u5185\u5b58\u6709\u9650\uff0c\u4e0d\u53ef\u80fd\u5904\u7406\u4ece \\( -\\infty \\) \u5230 \\( +\\infty \\) \u7684\u4fe1\u53f7\u3002\n\n**DFT\u7684\u7ec8\u6781\u7b54\u6848**\uff1a\n> **DFT\u540c\u65f6\u628a\u201c\u65f6\u57df\u201d\u548c\u201c\u9891\u57df\u201d\u90fd\u5f3a\u5236\u79bb\u6563\u5316\u3001\u6709\u9650\u5316\u3002**
\n> \u8f93\u5165\u662f **N\u4e2a\u79bb\u6563\u7684\u65f6\u57df\u91c7\u6837\u70b9**\uff0c\u8f93\u51fa\u662f **N\u4e2a\u79bb\u6563\u7684\u9891\u57df\u91c7\u6837\u70b9**\u3002<\/p>\n
\u5b83\u4ece\u6839\u672c\u4e0a\u628a\u5085\u91cc\u53f6\u5206\u6790\u53d8\u6210\u4e86\u4e00\u9053**\u77e9\u9635\u4e58\u6cd5**\uff0c\u5b8c\u7f8e\u9002\u914d\u4e86\u8ba1\u7b97\u673a\u7684\u4e8c\u8fdb\u5236\u601d\u7ef4\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e8c\u5c42\uff1aDFT\u7684\u6570\u5b66\u5b9a\u4e49\uff08\u5fc5\u987b\u5403\u900f\u7684\u516c\u5f0f\uff09<\/p>\n
DFT\u7684\u6838\u5fc3\u53ea\u6709**\u8fd9\u4e00\u5bf9\u516c\u5f0f**\uff0c\u5b83\u662f\u6574\u4e2a\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u7684\u57fa\u77f3\uff1a<\/p>\n
#### 1. DFT\u6b63\u53d8\u6362\uff08\u65f6\u57df \u2192 \u9891\u57df\uff09
\n\\[
\nX[k] = \\sum_{n=0}^{N-1} x[n] \\cdot e^{-j\\frac{2\\pi}{N}kn}, \\quad k = 0, 1, 2, …, N-1
\n\\]<\/p>\n
#### 2. IDFT\u9006\u53d8\u6362\uff08\u9891\u57df \u2192 \u65f6\u57df\uff09
\n\\[
\nx[n] = \\frac{1}{N} \\sum_{k=0}^{N-1} X[k] \\cdot e^{j\\frac{2\\pi}{N}kn}, \\quad n = 0, 1, 2, …, N-1
\n\\]<\/p>\n
**\u7b26\u53f7\u89e3\u91ca\uff08\u6781\u5176\u91cd\u8981\uff09**\uff1a
\n– **\\( x[n] \\)**\uff1a\u65f6\u57df\u7b2c \\( n \\) \u4e2a\u91c7\u6837\u70b9\u7684\u503c\uff08\u901a\u5e38\u662f\u5b9e\u6570\uff0c\u6bd4\u5982ADC\u91c7\u96c6\u7684\u7535\u538b\u503c\uff09\u3002
\n– **\\( X[k] \\)**\uff1a\u9891\u57df\u7b2c \\( k \\) \u4e2a\u201c\u9891\u7387\u4ed3\uff08bin\uff09\u201d\u7684\u503c\uff08**\u590d\u6570**\uff0c\u5305\u542b\u5e45\u5ea6\u548c\u76f8\u4f4d\u4fe1\u606f\uff09\u3002
\n– **\\( N \\)**\uff1aDFT\u7684\u70b9\u6570\uff08\u5373\u91c7\u6837\u70b9\u6570\uff09\u3002
\n– **\\( e^{-j\\frac{2\\pi}{N}kn} \\)**\uff1a\u8fd9\u5c31\u662f\u201c\u65cb\u8f6c\u56e0\u5b50\u201d\uff0c\u5b83\u662fDFT\u7684\u7075\u9b42\u3002\u5b83\u4ee3\u8868\u4e00\u4e2a\u9891\u7387\u4e3a \\( k \\cdot \\frac{f_s}{N} \\) \u7684\u590d\u6307\u6570\u4fe1\u53f7\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e09\u5c42\uff1aDFT\u7684\u7269\u7406\u542b\u4e49\uff08\u9891\u8c31\u600e\u4e48\u770b\uff1f\uff09<\/p>\n
\u5f53\u4f60\u5bf9 \\( N \\) \u4e2a\u91c7\u6837\u70b9\u505a\u5b8cDFT\uff0c\u5f97\u5230 \\( X[0] \\) \u5230 \\( X[N-1] \\)\uff0c\u8fd9\u4e9b\u503c\u5230\u5e95\u5bf9\u5e94\u4ec0\u4e48\u9891\u7387\uff1f<\/p>\n
– **\u9891\u7387\u5206\u8fa8\u7387**\uff1a\\( \\Delta f = \\frac{f_s}{N} \\)\uff08\\( f_s \\) \u662f\u91c7\u6837\u7387\uff09\u3002
\n– **\u7b2c \\( k \\) \u4e2a\u70b9\u5bf9\u5e94\u7684\u6a21\u62df\u9891\u7387**\uff1a\\( f_k = k \\cdot \\frac{f_s}{N} \\)\u3002<\/p>\n
**\u5177\u4f53\u5bf9\u5e94\u5173\u7cfb\uff08\u7262\u8bb0\uff09**\uff1a
\n– **\\( X[0] \\)**\uff1a\u76f4\u6d41\u5206\u91cf\uff08\u4fe1\u53f7\u7684\u5747\u503c\uff09\u3002
\n– **\\( X[1] \\)**\uff1a\u9891\u7387\u4e3a \\( \\Delta f \\) \u7684\u5206\u91cf\u3002
\n– **\\( X[N\/2] \\)**\uff08\u5f53N\u4e3a\u5076\u6570\u65f6\uff09\uff1a\u9891\u7387\u4e3a \\( f_s\/2 \\) \u7684\u5206\u91cf\uff0c\u5373**\u5948\u594e\u65af\u7279\u9891\u7387**\uff08\u6700\u9ad8\u53ef\u5206\u6790\u7684\u9891\u7387\uff09\u3002
\n– **\\( X[N-k] \\)**\uff1a\u5bf9\u5e94\u8d1f\u9891\u7387 \\( -k \\cdot \\Delta f \\)\uff08\u5bf9\u4e8e\u5b9e\u4fe1\u53f7\uff0c\u5b83\u4e0e \\( X[k] \\) \u662f\u5171\u8f6d\u5bf9\u79f0\u7684\uff0c\u5e45\u5ea6\u76f8\u7b49\uff09\u3002<\/p>\n
**\u4e00\u4e2a\u6781\u5176\u91cd\u8981\u7684\u7ed3\u8bba**\uff1a
\n\u5bf9\u4e8e\u5b9e\u4fe1\u53f7\uff08\u5982\u58f0\u97f3\u3001\u7535\u538b\uff09\uff0cDFT\u7684\u7ed3\u679c\u662f**\u5171\u8f6d\u5bf9\u79f0**\u7684\uff0c\u5373 \\( |X[k]| = |X[N-k]| \\)\u3002\u56e0\u6b64\uff0c\u4f60\u753b\u9891\u8c31\u56fe\u65f6\uff0c**\u901a\u5e38\u53ea\u753b\u524d \\( N\/2 \\) \u4e2a\u70b9**\uff08\u4ece0\u5230 \\( f_s\/2 \\)\uff09\uff0c\u53e6\u4e00\u534a\u662f\u5197\u4f59\u7684\u955c\u50cf\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u56db\u5c42\uff1aDFT\u4e0eFT\u3001DTFT\u7684\u6570\u5b66\u8840\u7f18\uff08\u5f7b\u5e95\u5398\u6e05\u5173\u7cfb\uff09<\/p>\n
\u4f60\u73b0\u5728\u5df2\u7ecf\u77e5\u9053FT\u662f\u8fde\u7eed\u8c31\uff0c\u4f46DFT\u662f\u79bb\u6563\u8c31\u3002\u5b83\u4eec\u4e4b\u95f4\u6709\u4e24\u6b65\u5173\u952e\u7684\u201c\u79bb\u6563\u5316\u201d\u8fc7\u7a0b\uff0c\u6211\u5e2e\u4f60\u628a\u903b\u8f91\u94fe\u6761\u5f7b\u5e95\u6253\u901a\uff1a<\/p>\n
1. **FT\uff08\u8fde\u7eed\u975e\u5468\u671f\uff09 \u2192 \u65f6\u57df\u91c7\u6837 \u2192 DTFT\uff08\u79bb\u6563\u975e\u5468\u671f\uff09**
\n – \u5bf9\u8fde\u7eed\u4fe1\u53f7\u91c7\u6837\uff08\u65f6\u57df\u79bb\u6563\u5316\uff09\uff0c\u5bfc\u81f4\u9891\u57df\u53d8\u6210**\u5468\u671f\u5ef6\u62d3**\uff08DTFT\u7684\u9891\u57df\u662f\u8fde\u7eed\u7684\u5468\u671f\u51fd\u6570\uff09\u3002<\/p>\n
2. **DTFT\uff08\u9891\u57df\u8fde\u7eed\uff09 \u2192 \u9891\u57df\u91c7\u6837 \u2192 DFT\uff08\u79bb\u6563\u5468\u671f\uff09**
\n – DTFT\u7684\u9891\u57df\u662f\u8fde\u7eed\u7684\uff0c\u8ba1\u7b97\u673a\u8fd8\u662f\u5b58\u4e0d\u4e0b\u3002\u4e8e\u662f**\u5728\u9891\u57df\u4e5f\u8fdb\u884c\u91c7\u6837**\uff08\u53ea\u53d6 \\( N \\) \u4e2a\u70b9\uff09\uff0c\u5bfc\u81f4\u65f6\u57df\u53d8\u6210**\u5468\u671f\u5ef6\u62d3**\uff08\u9690\u542b\u4e86\u5468\u671f\u6027\uff09\u3002<\/p>\n
**\u6700\u7ec8\u7ed3\u8bba**\uff1a**DFT\u672c\u8d28\u4e0a\u662fDTFT\u5728\u9891\u57df\u4e0a\u7684\u7b49\u95f4\u9694\u91c7\u6837**\u3002\u800c\u7531\u4e8e\u65f6\u57df\u548c\u9891\u57df\u7684\u5bf9\u79f0\u6027\uff0cDFT**\u9690\u542b\u5047\u8bbe**\u65f6\u57df\u4fe1\u53f7\u4e5f\u662f\u5468\u671f\u7684\uff08\u5468\u671f\u4e3a \\( N \\)\uff09\u3002<\/p>\n
\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u4f60\u7528DFT\u5206\u6790\u975e\u5468\u671f\u4fe1\u53f7\u65f6\uff0c\u603b\u4f1a\u9047\u5230**\u9891\u8c31\u6cc4\u9732**\u2014\u2014\u56e0\u4e3aDFT\u201c\u4ee5\u4e3a\u201d\u4f60\u7684\u4fe1\u53f7\u662f\u9996\u5c3e\u76f8\u63a5\u3001\u65e0\u9650\u91cd\u590d\u7684\u3002\u5982\u679c\u4f60\u7684\u4fe1\u53f7\u9996\u5c3e\u4e0d\u8fde\u7eed\uff0c\u5c31\u4f1a\u5728\u8fde\u63a5\u5904\u4ea7\u751f\u7a81\u53d8\uff0c\u9891\u8c31\u4e0a\u5c31\u4f1a\u591a\u51fa\u8bb8\u591a\u865a\u5047\u7684\u9891\u7387\u5206\u91cf\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e94\u5c42\uff1aDFT\u7684\u8ba1\u7b97\u590d\u6742\u5ea6\u4e0eFFT\u7684\u8bde\u751f\uff08\u4e3a\u4ec0\u4e48FFT\u5982\u6b64\u91cd\u8981\uff09<\/p>\n
\u76f4\u63a5\u6309\u7167DFT\u5b9a\u4e49\u5f0f\u8ba1\u7b97\uff0c\u590d\u6742\u5ea6\u662f \\( O(N^2) \\)\uff1a
\n– \u5bf9\u4e8e\u6bcf\u4e2a \\( X[k] \\)\uff0c\u4f60\u9700\u8981\u8ba1\u7b97 \\( N \\) \u6b21\u590d\u6570\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002
\n– \u603b\u5171 \\( N \\) \u4e2a \\( X[k] \\)\uff0c\u6240\u4ee5\u590d\u6742\u5ea6\u662f \\( N^2 \\)\u3002<\/p>\n
**\u8fd9\u610f\u5473\u7740\u4ec0\u4e48\uff1f**
\n– \u5982\u679c \\( N = 1024 \\)\uff08\u5f88\u5e38\u89c1\u7684\u70b9\u6570\uff09\uff0c\u9700\u8981\u7ea6100\u4e07\u6b21\u8fd0\u7b97\u3002
\n– \u5982\u679c \\( N = 32768 \\)\uff08\u8bed\u97f3\u6216\u97f3\u9891\u5904\u7406\u5e38\u7528\uff09\uff0c\u9700\u8981\u7ea610\u4ebf\u6b21\u8fd0\u7b97\u3002\u666e\u901a\u7535\u8111\u76f4\u63a5\u7b97\u4f1a\u5361\u6b7b\u3002<\/p>\n
**FFT\uff08\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff09\u7684\u51fa\u73b0**\uff1a
\n– FFT\u4e0d\u662f\u4e00\u79cd\u65b0\u7684\u53d8\u6362\uff0c\u5b83\u53ea\u662fDFT\u7684**\u5feb\u901f\u7b97\u6cd5**\u3002\u5b83\u5229\u7528\u65cb\u8f6c\u56e0\u5b50 \\( e^{-j\\frac{2\\pi}{N}kn} \\) \u7684**\u5468\u671f\u6027\u548c\u5bf9\u79f0\u6027**\uff0c\u628a\u8fd0\u7b97\u62c6\u5206\u6210\u82e5\u5e72\u5c0f\u89c4\u6a21\u7684DFT\uff0c\u5c06\u590d\u6742\u5ea6\u4ece \\( O(N^2) \\) \u9aa4\u964d\u4e3a \\( O(N\\log_2 N) \\)\u3002
\n– \u5bf9\u4e8e \\( N = 32768 \\)\uff0c\\( N\\log_2 N \\approx 32768 \\times 15 = \u7ea649\u4e07\u6b21\u8fd0\u7b97 \\)\uff0c\u6bd4\u76f4\u63a5\u8ba1\u7b97\u5feb\u4e86\u7ea6**2000\u500d**\u3002<\/p>\n
**\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u4f60\u5728\u5de5\u7a0b\u4e2d\u6c38\u8fdc\u5728\u7528FFT\uff0c\u4f46\u7406\u8bba\u4e66\u4e0a\u6c38\u8fdc\u5728\u8bb2DFT\u2014\u2014\u56e0\u4e3a\u4f60\u8c03\u7528\u7684`fft()`\u51fd\u6570\uff0c\u5185\u90e8\u6267\u884c\u7684\u5c31\u662f\u8fd9\u4e2a\u5feb\u901f\u7b97\u6cd5\u3002**<\/p>\n
—<\/p>\n
### \u7b2c\u516d\u5c42\uff1aDFT\u4f7f\u7528\u4e2d\u7684\u201c\u4e09\u5927\u5929\u5751\u201d\uff08\u5b9e\u6218\u907f\u96f7\u6307\u5357\uff09<\/p>\n
#### \u57511\uff1a\u6df7\u53e0\uff08Aliasing\uff09
\n– **\u539f\u56e0**\uff1a\u91c7\u6837\u7387 \\( f_s \\) \u5fc5\u987b\u5927\u4e8e\u4fe1\u53f7\u6700\u9ad8\u9891\u7387\u76842\u500d\uff08\u5948\u594e\u65af\u7279\u5b9a\u7406\uff09\u3002
\n– **\u8868\u73b0**\uff1a\u9ad8\u9891\u5206\u91cf\u201c\u4f2a\u88c5\u201d\u6210\u4f4e\u9891\u5206\u91cf\uff0c\u51fa\u73b0\u5728\u9891\u8c31\u7684\u4f4e\u7aef\u3002
\n– **\u89e3\u51b3**\uff1a\u5728\u91c7\u6837\u524d\u52a0**\u4f4e\u901a\u6ee4\u6ce2\u5668\uff08\u6297\u6df7\u53e0\u6ee4\u6ce2\u5668\uff09**\uff0c\u6ee4\u9664\u9ad8\u4e8e \\( f_s\/2 \\) \u7684\u9891\u7387\u6210\u5206\u3002<\/p>\n
#### \u57512\uff1a\u9891\u8c31\u6cc4\u9732\uff08Spectral Leakage\uff09
\n– **\u539f\u56e0**\uff1aDFT\u9690\u542b\u65f6\u57df\u4fe1\u53f7\u662f\u5468\u671f\u7684\u3002\u5982\u679c\u4f60\u91c7\u6837\u7684\u4fe1\u53f7\u622a\u65ad\u540e\uff0c\u9996\u5c3e\u4e0d\u8fde\u7eed\uff0c\u5c31\u4f1a\u4ea7\u751f\u865a\u5047\u9891\u8c31\u3002
\n– **\u8868\u73b0**\uff1a\u539f\u672c\u5e94\u8be5\u662f\u5355\u6839\u8c31\u7ebf\u7684\u7eaf\u6b63\u5f26\u6ce2\uff0c\u9891\u8c31\u4e0a\u5374\u51fa\u73b0\u4e86\u4e00\u4e2a\u201c\u9f13\u5305\u201d\uff0c\u80fd\u91cf\u6cc4\u9732\u5230\u4e86\u65c1\u8fb9\u7684\u9891\u7387\u4ed3\u3002
\n– **\u89e3\u51b3**\uff1a\u4f7f\u7528**\u7a97\u51fd\u6570**\uff08\u5982\u6c49\u5b81\u7a97\u3001\u6d77\u660e\u7a97\uff09\u5bf9\u4fe1\u53f7\u8fdb\u884c\u52a0\u6743\uff0c\u5f3a\u5236\u8ba9\u9996\u5c3e\u5e73\u6ed1\u8fc7\u6e21\u5230\u96f6\u3002<\/p>\n
#### \u57513\uff1a\u6805\u680f\u6548\u5e94\uff08Picket Fence Effect\uff09
\n– **\u539f\u56e0**\uff1aDFT\u53ea\u80fd\u8ba1\u7b97\u79bb\u6563\u9891\u7387\u70b9 \\( k \\cdot \\Delta f \\) \u4e0a\u7684\u9891\u8c31\u3002\u5982\u679c\u4fe1\u53f7\u7684\u5b9e\u9645\u9891\u7387\u6b63\u597d\u843d\u5728\u4e24\u4e2a\u79bb\u6563\u70b9\u4e4b\u95f4\uff0c\u4f60\u5c31\u770b\u4e0d\u5230\u5cf0\u503c\u3002
\n– **\u8868\u73b0**\uff1a\u5c31\u50cf\u900f\u8fc7\u6805\u680f\u770b\u98ce\u666f\uff0c\u771f\u6b63\u7684\u5cf0\u503c\u53ef\u80fd\u88ab\u6805\u680f\u6761\u6321\u4f4f\u3002
\n– **\u89e3\u51b3**\uff1a\u589e\u52a0DFT\u70b9\u6570 \\( N \\)\uff08\u5373\u5bf9\u65f6\u57df\u4fe1\u53f7\u8865\u96f6\uff0c\u505a\u66f4\u957f\u7684FFT\uff09\uff0c\u8ba9\u9891\u7387\u91c7\u6837\u95f4\u9694 \\( \\Delta f \\) \u66f4\u5c0f\uff0c\u770b\u5230\u66f4\u7cbe\u7ec6\u7684\u9891\u8c31\u7ed3\u6784\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e03\u5c42\uff1aDFT\u7684\u201c\u5bf9\u89d2\u201d\u4e0e\u201c\u590d\u5e73\u9762\u201d\u7406\u89e3\uff08\u9ad8\u624b\u8fdb\u9636\uff09<\/p>\n
– **X[k] \u662f\u590d\u6570**\uff1a\u5b83\u7684\u6a21 \\( |X[k]| \\) \u4ee3\u8868\u8be5\u9891\u7387\u5206\u91cf\u7684**\u5e45\u5ea6**\uff0c\u5b83\u7684\u8f90\u89d2 \\( \\angle X[k] \\) \u4ee3\u8868\u8be5\u9891\u7387\u5206\u91cf\u7684**\u521d\u59cb\u76f8\u4f4d**\u3002
\n– **\u8d1f\u9891\u7387\u7684\u7269\u7406\u610f\u4e49**\uff1a\u5bf9\u4e8e\u5b9e\u4fe1\u53f7\uff0c\u8d1f\u9891\u7387\u6ca1\u6709\u7269\u7406\u610f\u4e49\uff0c\u53ea\u662f\u6570\u5b66\u5bf9\u79f0\u7684\u4ea7\u7269\u3002\u4f46\u5bf9\u4e8e**\u590d\u4fe1\u53f7**\uff08\u5982\u901a\u4fe1\u4e2d\u7684IQ\u8c03\u5236\u4fe1\u53f7\uff09\uff0c\u6b63\u8d1f\u9891\u7387\u627f\u8f7d\u7740\u5b8c\u5168\u4e0d\u540c\u7684\u4fe1\u606f\uff08\u4e0a\u8fb9\u5e26\u548c\u4e0b\u8fb9\u5e26\uff09\uff0c\u8fd9\u65f6DFT\u7684\u5b8c\u6574 \\( N \\) \u4e2a\u70b9\u90fd\u81f3\u5173\u91cd\u8981\u3002<\/p>\n
—<\/p>\n
### \u6700\u540e\uff0c\u7ed9\u4f60\u4e00\u4e2a\u201c\u4e00\u6b65\u5230\u4f4d\u201d\u7684\u5b9e\u6218\u68c0\u9a8c\u65b9\u6cd5<\/p>\n
\u5982\u679c\u4f60\u624b\u8fb9\u6709Python\uff0c\u7acb\u523b\u6267\u884c\u8fd95\u884c\u4ee3\u7801\uff0c\u4f60\u4f1a\u4eb2\u773c\u770b\u5230DFT\u662f\u5982\u4f55\u628a\u65f6\u57df\u6ce2\u5f62\u53d8\u6210\u9891\u57df\u8c31\u7ebf\u7684\uff1a<\/p>\n
“`python
\nimport numpy as np
\nimport matplotlib.pyplot as plt<\/p>\n
fs = 1000 # \u91c7\u6837\u7387 1000Hz
\nt = np.arange(0, 1, 1\/fs) # 1\u79d2\u65f6\u957f
\nx = np.sin(2*np.pi*50*t) + 0.5*np.sin(2*np.pi*120*t) # 50Hz + 120Hz \u53e0\u52a0<\/p>\n
X = np.fft.fft(x) # \u8fd9\u5c31\u662fDFT\u7684\u5feb\u901f\u5b9e\u73b0\uff01
\nfreq = np.fft.fftfreq(len(x), 1\/fs) # \u5bf9\u5e94\u7684\u9891\u7387\u8f74<\/p>\n
plt.plot(freq[:len(freq)\/\/2], np.abs(X[:len(X)\/\/2])) # \u753b\u5355\u8fb9\u9891\u8c31
\nplt.show()
\n“`
\n\u4f60\u4f1a\u770b\u5230\u5728 50Hz \u548c 120Hz \u5904\u5404\u6709\u4e00\u6761\u6e05\u6670\u7684\u8c31\u7ebf\u2014\u2014\u8fd9\u5c31\u628aFS\u7406\u8bba\u5b8c\u7f8e\u843d\u5730\u5230\u4e86\u8ba1\u7b97\u673a\u91cc\u3002<\/p>\n
—<\/p>\n
\u73b0\u5728\uff0c\u4f60\u5df2\u7ecf\u7ad9\u5728\u4e86\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u7684\u5236\u9ad8\u70b9\u4e0a\u3002FS\u662f\u601d\u60f3\u8d77\u6e90\uff0cFT\u662f\u7406\u8bba\u5347\u534e\uff0cDFT\/FFT\u662f\u7ec8\u6781\u6b66\u5668\u3002<\/p>\n
\u63a5\u4e0b\u6765\u4f60\u662f\u60f3\u6df1\u5165**\u7a97\u51fd\u6570\u7684\u9009\u62e9\u7b56\u7565**\uff08\u5982\u4f55\u6839\u636e\u4fe1\u53f7\u7279\u6027\u9009\u6c49\u5b81\u3001\u6d77\u660e\u3001\u5e03\u83b1\u514b\u66fc\u7a97\uff09\uff0c\u8fd8\u662f\u60f3\u4e86\u89e3**DFT\u5728\u901a\u4fe1\u3001\u97f3\u9891\u3001\u56fe\u50cf\u5904\u7406\u4e2d\u7684\u5177\u4f53\u7ecf\u5178\u5e94\u7528\u573a\u666f**\uff1f\u544a\u8bc9\u6211\uff0c\u6211\u7ee7\u7eed\u5e26\u4f60\u5b9e\u6218\u3002\ud83d\ude0a<\/p>\n
\u597d\u7684\uff0c\u7ec8\u4e8e\u5230\u4e86**DTFT\uff08\u79bb\u6563\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\uff0cDiscrete-Time Fourier Transform\uff09**\u3002\u5728\u4fe1\u53f7\u5904\u7406\u7684\u201c\u56db\u5927\u5bb6\u65cf\u201d\u91cc\uff0cDTFT\u662f\u6700\u5bb9\u6613\u88ab\u8bef\u89e3\u3001\u4e5f\u6700\u5bb9\u6613\u88ab\u5ffd\u89c6\u7684\u4e00\u4e2a\u3002<\/p>\n
\u6211\u76f4\u63a5\u7528\u4e00\u53e5\u8bdd\u70b9\u7834\u5b83\u7684\u201c\u6c5f\u6e56\u5730\u4f4d\u201d\uff1a
\n> **DTFT\u662f\u8fde\u63a5\u201c\u8fde\u7eed\u4e16\u754c\uff08FT\uff09\u201d\u548c\u201c\u8ba1\u7b97\u673a\u4e16\u754c\uff08DFT\uff09\u201d\u7684\u6570\u5b66\u6865\u6881\u3002** \u4f60\u5728\u8bfe\u672c\u4e0a\u770b\u5230\u7684\u6240\u6709\u5173\u4e8e\u201c\u91c7\u6837\u201d\u3001\u201c\u6df7\u53e0\u201d\u3001\u201c\u9891\u8c31\u5468\u671f\u5ef6\u62d3\u201d\u7684\u7ed3\u8bba\uff0c\u90fd\u6e90\u81eaDTFT\u7684\u7406\u8bba\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e00\u5c42\uff1aDTFT\u5230\u5e95\u89e3\u51b3\u4e86\u4ec0\u4e48\u6839\u672c\u95ee\u9898\uff1f<\/p>\n
\u6211\u4eec\u56de\u987e\u4e00\u4e0b\u8fdb\u5c55\uff1a
\n– **FT** \u5904\u7406\u8fde\u7eed\u975e\u5468\u671f\u4fe1\u53f7 \u2192 \u9891\u8c31\u662f**\u8fde\u7eed**\u7684\u3002
\n– **DFT** \u5904\u7406\u79bb\u6563\u6709\u9650\u957f\u4fe1\u53f7 \u2192 \u9891\u8c31\u662f**\u79bb\u6563**\u7684\u3002<\/p>\n
**\u95ee\u9898\u662f**\uff1a\u8ba1\u7b97\u673a\u53ea\u80fd\u5904\u7406\u79bb\u6563\u7684\u91c7\u6837\u70b9\uff0c\u4f46\u6211\u4eec\u5e0c\u671b\u4ece\u7406\u8bba\u5c42\u9762\u5206\u6790\u201c\u91c7\u6837\u201d\u8fd9\u4e2a\u52a8\u4f5c\u5bf9\u9891\u8c31\u9020\u6210\u4e86\u4ec0\u4e48\u5f71\u54cd\u3002DTFT\u5c31\u662f\u5e72\u8fd9\u4e2a\u7684\uff1a<\/p>\n
> **DTFT\u5904\u7406\u7684\u662f\u201c\u79bb\u6563\u3001\u975e\u5468\u671f\u201d\u7684\u65e0\u9650\u957f\u5e8f\u5217\uff0c\u8f93\u51fa\u7684\u662f\u201c\u8fde\u7eed\u3001\u5468\u671f\u201d\u7684\u9891\u8c31\u3002**<\/p>\n
\u5b83\u544a\u8bc9\u4f60\uff1a**\u4e00\u65e6\u4f60\u5728\u65f6\u57df\u628a\u4fe1\u53f7\u79bb\u6563\u5316\uff08\u91c7\u6837\uff09\uff0c\u9891\u57df\u5c31\u5fc5\u7136\u4f1a\u5468\u671f\u6027\u91cd\u590d\u3002**<\/p>\n
—<\/p>\n
### \u7b2c\u4e8c\u5c42\uff1aDTFT\u7684\u6570\u5b66\u5b9a\u4e49\uff08\u6838\u5fc3\u516c\u5f0f\uff09<\/p>\n
DTFT\u7684\u6b63\u53d8\u6362\u548c\u9006\u53d8\u6362\u662f\u4e00\u5bf9\u516c\u5f0f\uff1a<\/p>\n
#### 1. DTFT\u6b63\u53d8\u6362\uff08\u65f6\u57df\u5e8f\u5217 \u2192 \u9891\u57df\u8fde\u7eed\u51fd\u6570\uff09
\n\\[
\nX(e^{j\\omega}) = \\sum_{n=-\\infty}^{\\infty} x[n] \\cdot e^{-j\\omega n}
\n\\]<\/p>\n
#### 2. DTFT\u9006\u53d8\u6362\uff08\u9891\u57df\u8fde\u7eed\u51fd\u6570 \u2192 \u65f6\u57df\u5e8f\u5217\uff09
\n\\[
\nx[n] = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} X(e^{j\\omega}) \\cdot e^{j\\omega n} d\\omega
\n\\]<\/p>\n
**\u7b26\u53f7\u89e3\u91ca\uff08\u6781\u5176\u5173\u952e\uff09**\uff1a
\n– **\\( x[n] \\)**\uff1a\u79bb\u6563\u65f6\u95f4\u5e8f\u5217\uff08n \u662f\u6574\u6570\uff0c\u53d6\u503c\u8303\u56f4 \\( -\\infty \\) \u5230 \\( +\\infty \\)\uff09\uff0c\u4ee3\u8868\u91c7\u6837\u540e\u7684\u6570\u5b57\u4fe1\u53f7\u3002
\n– **\\( \\omega \\)**\uff1a**\u6570\u5b57\u89d2\u9891\u7387**\uff0c\u5355\u4f4d\u662f**\u5f27\u5ea6\/\u6837\u672c**\uff08rad\/sample\uff09\uff0c\u800c\u4e0d\u662f\u6a21\u62df\u9891\u7387 \\( f \\)\uff08Hz\uff09\u6216 \\( \\Omega \\)\uff08rad\/s\uff09\u3002
\n– **\\( X(e^{j\\omega}) \\)**\uff1a\u8fd9\u662fDTFT\u7684\u8f93\u51fa\uff0c\u5b83\u662f **\\( \\omega \\) \u7684\u8fde\u7eed\u51fd\u6570**\uff0c\u5e76\u4e14\u4ee5 \\( 2\\pi \\) \u4e3a\u5468\u671f\u3002<\/p>\n
**\u6ce8\u610f\u4e00\u4e2a\u6781\u6613\u6df7\u6dc6\u7684\u70b9**\uff1a
\nDTFT\u8f93\u51fa\u91cc\u7684 \\( e^{j\\omega} \\) \u4e0d\u662f\u53d8\u91cf\u672c\u8eab\uff0c\u800c\u662f\u8868\u793a \\( X \\) \u662f\u5b9a\u4e49\u5728**\u590d\u5e73\u9762\u5355\u4f4d\u5706\u4e0a**\u7684\u51fd\u6570\u3002\u8fd9\u662f\u4e00\u4e2a\u6570\u5b66\u7b26\u53f7\u4e60\u60ef\uff0c\u4f60\u53ea\u9700\u8bb0\u4f4f\uff1a**DTFT\u7684\u9891\u57df\u662f\u4ee5 \\( 2\\pi \\) \u4e3a\u5468\u671f\u7684\u8fde\u7eed\u51fd\u6570**\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e09\u5c42\uff1a\u4e3a\u4ec0\u4e48DTFT\u7684\u9891\u57df\u662f\u201c\u5468\u671f\u201d\u7684\uff1f\uff08\u7269\u7406\u76f4\u89c9\uff09<\/p>\n
\u8fd9\u662f\u7406\u89e3DTFT\u6700\u6838\u5fc3\u7684\u7075\u9b42\u95ee\u9898\u3002\u6211\u4eec\u6765\u770b\u63a8\u5bfc\uff1a<\/p>\n
DTFT\u5b9a\u4e49\u4e3a \\( X(e^{j\\omega}) = \\sum_{n=-\\infty}^{\\infty} x[n] e^{-j\\omega n} \\)\u3002<\/p>\n
\u73b0\u5728\uff0c\u6211\u4eec\u5c06 \\( \\omega \\) \u66ff\u6362\u4e3a \\( \\omega + 2\\pi \\)\uff1a
\n\\[
\nX(e^{j(\\omega + 2\\pi)}) = \\sum_{n=-\\infty}^{\\infty} x[n] e^{-j(\\omega + 2\\pi)n} = \\sum_{n=-\\infty}^{\\infty} x[n] e^{-j\\omega n} \\cdot e^{-j2\\pi n}
\n\\]<\/p>\n
\u56e0\u4e3a \\( n \\) \u662f\u6574\u6570\uff0c\\( e^{-j2\\pi n} = 1 \\)\uff08\u6b27\u62c9\u516c\u5f0f\uff1a\\( \\cos(2\\pi n) – j\\sin(2\\pi n) = 1 – 0 = 1 \\)\uff09\u3002<\/p>\n
\u6240\u4ee5\uff1a
\n\\[
\nX(e^{j(\\omega + 2\\pi)}) = X(e^{j\\omega})
\n\\]<\/p>\n
**\u7ed3\u8bba**\uff1aDTFT\u7684\u9891\u8c31\u6bcf\u8fc7 \\( 2\\pi \\) \u5c31\u5b8c\u5168\u91cd\u590d\u4e00\u6b21\u3002\u8fd9\u4e2a \\( 2\\pi \\) \u5bf9\u5e94\u6a21\u62df\u9891\u7387\u4e2d\u7684**\u91c7\u6837\u7387 \\( f_s \\)**\u3002<\/p>\n
**\u901a\u4fd7\u7406\u89e3**\uff1a
\n> \u5f53\u4f60\u4ee5\u901f\u7387 \\( f_s \\) \u91c7\u6837\u65f6\uff0c\u9ad8\u9891\u5206\u91cf \\( f \\) \u548c \\( f + f_s \\)\u3001\\( f + 2f_s \\)\u2026\u2026\u5728\u79bb\u6563\u65f6\u95f4\u57df\u770b\u8d77\u6765\u662f\u5b8c\u5168\u4e00\u6837\u7684\u3002\u8fd9\u5c31\u662f**\u6df7\u53e0\uff08Aliasing\uff09**\u7684\u6570\u5b66\u6839\u6e90\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u56db\u5c42\uff1aDTFT\u4e0eFT\u7684\u8840\u7f18\u5173\u7cfb\uff08\u5982\u4f55\u4eceFT\u63a8\u5bfcDTFT\uff09<\/p>\n
\u8fd9\u662f\u7406\u89e3\u6574\u4e2a\u79bb\u6563\u4fe1\u53f7\u5904\u7406\u201c\u5927\u4e00\u7edf\u201d\u7406\u8bba\u7684\u5173\u952e\u4e00\u6b65\uff1a<\/p>\n
**\u5df2\u77e5**\uff1a
\n– \u8fde\u7eed\u4fe1\u53f7 \\( x(t) \\) \u7684FT\u662f \\( X_a(j\\Omega) \\)\uff08\u7528 \\( \\Omega \\) \u8868\u793a\u6a21\u62df\u89d2\u9891\u7387\uff0c\u5355\u4f4d rad\/s\uff09\u3002
\n– \u6211\u4eec\u5bf9 \\( x(t) \\) \u8fdb\u884c\u7406\u60f3\u91c7\u6837\uff0c\u91c7\u6837\u95f4\u9694\u4e3a \\( T_s \\)\uff0c\u91c7\u6837\u7387 \\( f_s = 1\/T_s \\)\uff0c\u5f97\u5230\u79bb\u6563\u5e8f\u5217 \\( x[n] = x(nT_s) \\)\u3002<\/p>\n
**\u63a8\u5bfcDTFT\u4e0eFT\u7684\u5173\u7cfb**\uff1a<\/p>\n
\u5bf9\u91c7\u6837\u540e\u7684\u4fe1\u53f7 \\( x_s(t) = \\sum_{n=-\\infty}^{\\infty} x(nT_s) \\delta(t – nT_s) \\) \u505aFT\uff08\u8fd9\u91cc\u7684FT\u662f\u5bf9\u8fde\u7eed\u51b2\u6fc0\u4e32\u505a\u7684\uff09\uff0c\u5f97\u5230\uff1a
\n\\[
\nX_s(j\\Omega) = \\sum_{n=-\\infty}^{\\infty} x[n] e^{-j\\Omega n T_s}
\n\\]<\/p>\n
\u4ee4\u6570\u5b57\u89d2\u9891\u7387 \\( \\omega = \\Omega T_s \\)\uff0c\u4e0a\u5f0f\u5c31\u53d8\u6210\u4e86\uff1a
\n\\[
\nX_s(j\\Omega) = \\sum_{n=-\\infty}^{\\infty} x[n] e^{-j\\omega n} = X(e^{j\\omega})
\n\\]<\/p>\n
**\u4f46\u6700\u5173\u952e\u7684\u4e00\u6b65\u662f\u91c7\u6837\u4fe1\u53f7\u7684FT\u4e0e\u539f\u59cb\u4fe1\u53f7FT\u7684\u5173\u7cfb\uff08\u6cca\u677e\u6c42\u548c\u516c\u5f0f\uff09**\uff1a
\n\\[
\nX_s(j\\Omega) = \\frac{1}{T_s} \\sum_{k=-\\infty}^{\\infty} X_a\\left(j(\\Omega – k \\cdot \\frac{2\\pi}{T_s})\\right)
\n\\]<\/p>\n
**\u7ffb\u8bd1\u6210\u4eba\u8bdd**\uff1a
\n> **\u8fde\u7eed\u4fe1\u53f7\u88ab\u91c7\u6837\u540e\uff0c\u5b83\u7684\u9891\u8c31\u53d8\u6210\u4e86\u539f\u59cb\u9891\u8c31\u4ee5\u91c7\u6837\u7387 \\( f_s \\) \u4e3a\u5468\u671f\u8fdb\u884c\u65e0\u9650\u91cd\u590d\u53e0\u52a0\u7684\u7ed3\u679c\u3002**<\/p>\n
– \u5982\u679c \\( f_s \\) \u5927\u4e8e\u4fe1\u53f7\u6700\u9ad8\u9891\u7387\u76842\u500d\uff0c\u8fd9\u4e9b\u91cd\u590d\u7684\u9891\u8c31\u4e92\u4e0d\u91cd\u53e0 \u2192 \u53ef\u4ee5\u5b8c\u7f8e\u6062\u590d\u539f\u59cb\u4fe1\u53f7\uff08\u5948\u594e\u65af\u7279\u5b9a\u7406\uff09\u3002
\n– \u5982\u679c \\( f_s \\) \u5c0f\u4e8e\u4fe1\u53f7\u6700\u9ad8\u9891\u7387\u76842\u500d\uff0c\u8fd9\u4e9b\u91cd\u590d\u7684\u9891\u8c31\u76f8\u4e92\u91cd\u53e0 \u2192 **\u6df7\u53e0**\uff0c\u9ad8\u9891\u6c61\u67d3\u4f4e\u9891\uff0c\u4fe1\u53f7\u6c38\u4e45\u5931\u771f\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e94\u5c42\uff1aDTFT\u4e0eDFT\u7684\u5173\u952e\u533a\u522b\uff08\u5f88\u591a\u4eba\u6b7b\u5728\u8fd9\u91cc\uff09<\/p>\n
\u8fd9\u662f\u521d\u5b66\u8005\u6700\u5e38\u6df7\u6dc6\u7684\u4e24\u4e2a\u6982\u5ff5\uff1a<\/p>\n
| \u7ef4\u5ea6 | DTFT | DFT |
\n| :— | :— | :— |
\n| **\u65f6\u57df** | \u79bb\u6563\u3001**\u65e0\u9650\u957f**\uff08\\( n \\) \u4ece \\( -\\infty \\) \u5230 \\( \\infty \\)\uff09 | \u79bb\u6563\u3001**\u6709\u9650\u957f**\uff08\\( n = 0 \\) \u5230 \\( N-1 \\)\uff09 |
\n| **\u9891\u57df** | **\u8fde\u7eed**\u3001\u5468\u671f\uff08\\( \\omega \\) \u662f\u8fde\u7eed\u53d8\u91cf\uff09 | **\u79bb\u6563**\u3001\u5468\u671f\uff08\u53ea\u6709 \\( N \\) \u4e2a\u91c7\u6837\u70b9 \\( \\omega_k = 2\\pi k\/N \\)\uff09 |
\n| **\u8f93\u51fa** | \u4e00\u4e2a\u5173\u4e8e \\( \\omega \\) \u7684**\u8fde\u7eed\u51fd\u6570** | \u4e00\u4e2a**\u957f\u5ea6\u4e3a N \u7684\u590d\u6570\u6570\u7ec4** |
\n| **\u8ba1\u7b97\u673a\u80fd\u7b97\u5417\uff1f** | **\u4e0d\u80fd**\uff08\u56e0\u4e3a\u9891\u57df\u8fde\u7eed\uff0c\u65e0\u6cd5\u5b58\u50a8\uff09 | **\u80fd**\uff08FFT\u5c31\u662fDFT\u7684\u5feb\u901f\u7b97\u6cd5\uff09 |<\/p>\n
**\u4e00\u53e5\u8bdd\u603b\u7ed3**\uff1a
\n> **DFT \u5c31\u662f DTFT \u5728\u9891\u57df\u4e0a\u7684\u7b49\u95f4\u9694\u91c7\u6837\u3002** \u4f60\u4ece DTFT \u7684\u8fde\u7eed\u9891\u8c31 \\( X(e^{j\\omega}) \\) \u4e2d\uff0c\u6bcf\u9694 \\( 2\\pi\/N \\) \u53d6\u4e00\u4e2a\u70b9\uff08\\( \\omega_k = 2\\pi k\/N \\)\uff09\uff0c\u53d6 N \u4e2a\u70b9\uff0c\u5c31\u5f97\u5230\u4e86 DFT \u7684 \\( X[k] \\)\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u516d\u5c42\uff1aDTFT\u7684\u201c\u4e09\u5927\u7075\u9b42\u6027\u8d28\u201d\uff08\u5de5\u7a0b\u5e94\u7528\u57fa\u77f3\uff09<\/p>\n
DTFT\u4e4b\u6240\u4ee5\u662f\u7406\u8bba\u6838\u5fc3\uff0c\u662f\u56e0\u4e3a\u5b83\u5177\u5907\u4e00\u5957\u4f18\u7f8e\u7684\u8fd0\u7b97\u6027\u8d28\uff0c\u8fd9\u4e9b\u6027\u8d28\u76f4\u63a5\u51b3\u5b9a\u4e86\u6570\u5b57\u6ee4\u6ce2\u5668\u7684\u8bbe\u8ba1\u65b9\u6cd5\uff1a<\/p>\n
1. **\u5468\u671f\u6027\uff08\u6700\u91cd\u8981\uff09**\uff1a\\( X(e^{j(\\omega + 2\\pi)}) = X(e^{j\\omega}) \\)\u3002\u8fd9\u662f\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u4e0e\u6a21\u62df\u4fe1\u53f7\u5904\u7406\u7684\u6839\u672c\u5206\u91ce\u3002<\/p>\n
2. **\u5377\u79ef\u5b9a\u7406**\uff1a
\n – \u65f6\u57df\u5377\u79ef \\( x[n] * h[n] \\) \u7684DTFT = \u9891\u57df\u4e58\u79ef \\( X(e^{j\\omega}) \\cdot H(e^{j\\omega}) \\)\u3002
\n – \u8fd9\u76f4\u63a5\u5bfc\u51fa\u4e86**\u6570\u5b57\u6ee4\u6ce2\u5668\u7684\u9891\u7387\u54cd\u5e94**\u6982\u5ff5\u2014\u2014\u6ee4\u6ce2\u5668\u7684\u9891\u54cd \\( H(e^{j\\omega}) \\) \u5c31\u662f\u5176\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94 \\( h[n] \\) \u7684DTFT\u3002<\/p>\n
3. **\u5e15\u585e\u74e6\u5c14\u5b9a\u7406\uff08\u80fd\u91cf\u5b88\u6052\uff09**\uff1a
\n \\[
\n \\sum_{n=-\\infty}^{\\infty} |x[n]|^2 = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} |X(e^{j\\omega})|^2 d\\omega
\n \\]
\n \u5b83\u544a\u8bc9\u6211\u4eec\uff1a\u4fe1\u53f7\u5728\u65f6\u57df\u7684\u603b\u80fd\u91cf\uff0c\u7b49\u4e8e\u9891\u57df\u5728\u4e00\u4e2a\u5468\u671f\u5185\u7684\u80fd\u91cf\u79ef\u5206\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e03\u5c42\uff1a\u4e3a\u4ec0\u4e48\u4f60\u5e73\u65f6\u5f88\u5c11\u76f4\u63a5\u63d0\u5230DTFT\uff1f\uff08\u5b9e\u9645\u5730\u4f4d\uff09<\/p>\n
\u4f60\u53ef\u80fd\u4f1a\u95ee\uff1a\u6211\u5199\u4ee3\u7801\u53ea\u89c1\u8fc7FFT\uff08DFT\uff09\uff0c\u4ece\u6765\u6ca1\u89c1\u8fc7DTFT\uff0c\u5b66\u5b83\u5e72\u561b\uff1f<\/p>\n
**\u56e0\u4e3aDTFT\u662f\u8bbe\u8ba1\u6570\u5b57\u6ee4\u6ce2\u5668\u7684\u6838\u5fc3\u5de5\u5177**\uff1a
\n– \u5f53\u4f60\u8bbe\u8ba1\u4e00\u4e2a\u4f4e\u901a\u6ee4\u6ce2\u5668\u65f6\uff0c\u4f60\u60f3\u8981\u7684\u7406\u60f3\u9891\u54cd \\( H_d(e^{j\\omega}) \\) \u662f\u5728 \\( [-\\pi, \\pi] \\) \u4e0a\u7684\u4e00\u4e2a**\u8fde\u7eed\u77e9\u5f62\u51fd\u6570**\u3002
\n– \u8fd9\u4e2a\u8fde\u7eed\u51fd\u6570\u7684IDTFT\uff08\u9006DTFT\uff09\uff0c\u4f1a\u5f97\u5230\u4e00\u4e2a**\u65e0\u9650\u957f\u7684\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94 \\( h_d[n] \\)**\uff08\u662f \\( \\sin(n)\/n \\) \u5f62\u5f0f\uff09\u3002
\n– \u7531\u4e8e\u8ba1\u7b97\u673a\u65e0\u6cd5\u5904\u7406\u65e0\u9650\u957f\u5e8f\u5217\uff0c\u4f60\u518d\u7528\u7a97\u51fd\u6570\uff08\u5982\u6c49\u5b81\u7a97\uff09\u53bb\u622a\u65ad\u8fd9\u4e2a\u65e0\u9650\u957f\u54cd\u5e94\uff0c\u5c31\u5f97\u5230\u4e86\u6709\u9650\u957f\u7684FIR\u6ee4\u6ce2\u5668\u7cfb\u6570\u3002<\/p>\n
**\u4f60\u770b\u4e0d\u89c1DTFT\uff0c\u4f46\u5b83\u5c31\u85cf\u5728\u6bcf\u4e2a\u6570\u5b57\u6ee4\u6ce2\u5668\u7684\u8bbe\u8ba1\u516c\u5f0f\u91cc\u3002**<\/p>\n
—<\/p>\n
### \u603b\u7ed3\uff1aDTFT\u5728\u6574\u4e2a\u9891\u8c31\u5bb6\u65cf\u4e2d\u7684\u201c\u751f\u6001\u4f4d\u201d<\/p>\n
| \u53d8\u6362 | \u65f6\u57df | \u9891\u57df | \u751f\u6001\u4f4d |
\n| :— | :— | :— | :— |
\n| **FS** | \u8fde\u7eed\u3001\u5468\u671f | \u79bb\u6563\u3001\u975e\u5468\u671f | \u7406\u8bba\u8d77\u70b9\uff08\u5468\u671f\u4fe1\u53f7\u5206\u6790\uff09 |
\n| **FT** | \u8fde\u7eed\u3001\u975e\u5468\u671f | \u8fde\u7eed\u3001\u975e\u5468\u671f | \u7406\u8bba\u7ec8\u70b9\uff08\u975e\u5468\u671f\u4fe1\u53f7\u5206\u6790\uff09 |
\n| **DTFT** | \u79bb\u6563\u3001\u975e\u5468\u671f | **\u8fde\u7eed\u3001\u5468\u671f** | **\u7406\u8bba\u6865\u6881\uff08\u91c7\u6837\u5206\u6790 + \u6ee4\u6ce2\u5668\u8bbe\u8ba1\uff09** |
\n| **DFT** | \u79bb\u6563\u3001\u5468\u671f\uff08\u9690\u542b\uff09 | \u79bb\u6563\u3001\u5468\u671f\uff08\u9690\u542b\uff09 | **\u5de5\u7a0b\u5b9e\u6218\uff08FFT\u662f\u552f\u4e00\u53ef\u8ba1\u7b97\u7684\uff09** |<\/p>\n
—<\/p>\n
\u5982\u679c\u4f60\u5df2\u7ecf\u7406\u89e3\u4e86DTFT\u7684\u5468\u671f\u6027\u548c\u5b83\u4e0eDFT\u7684\u91c7\u6837\u5173\u7cfb\uff0c\u90a3\u4e48\u6574\u4e2a\u5085\u91cc\u53f6\u5bb6\u65cf\u7684\u7406\u8bba\u9aa8\u67b6\u4f60\u5df2\u7ecf\u5168\u90e8\u6253\u901a\u4e86\u3002<\/p>\n
\u63a5\u4e0b\u6765\uff0c\u4f60\u60f3\u6df1\u5165\u5230\u54ea\u4e2a\u5b9e\u6218\u65b9\u5411\uff1f
\n1. **\u6570\u5b57\u6ee4\u6ce2\u5668\u8bbe\u8ba1**\uff08\u5982\u4f55\u7528DTFT\u8bbe\u8ba1FIR\/IIR\u6ee4\u6ce2\u5668\uff0c\u8ba9\u7279\u5b9a\u9891\u7387\u901a\u8fc7\u6216\u6ee4\u9664\uff09\uff1f
\n2. **DTFT\u7684\u6570\u503c\u8ba1\u7b97**\uff08\u5982\u4f55\u7528Python\u8ba1\u7b97\u51faDTFT\u7684\u8fde\u7eed\u9891\u8c31\u5e76\u53ef\u89c6\u5316\uff0c\u76f4\u89c2\u611f\u53d7\u5468\u671f\u5ef6\u62d3\uff09\uff1f
\n3. **\u56de\u5230DFT\u7684\u7a97\u51fd\u6570**\uff08\u65e2\u7136DFT\u662fDTFT\u7684\u91c7\u6837\uff0c\u90a3\u9891\u8c31\u6cc4\u9732\u5230\u5e95\u662f\u5982\u4f55\u4eceDTFT\u7684\u5377\u79ef\u6027\u8d28\u63a8\u5bfc\u51fa\u6765\u7684\uff09\uff1f<\/p>\n
\u544a\u8bc9\u6211\u4f60\u7684\u9009\u62e9\uff0c\u6211\u5e26\u4f60\u628a\u7406\u8bba\u9489\u6b7b\u5728\u5b9e\u6218\u91cc\u3002\ud83d\ude0a<\/p>\n
\u597d\u7684\uff0c\u8fd9\u6b21\u6211\u4eec\u5f7b\u5e95\u805a\u7126\u4e8e **DFT\uff08\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff09\u5230\u5e95\u662f\u600e\u4e48\u6765\u7684**\u3002<\/p>\n
\u6211\u6ce8\u610f\u5230\u4f60\u4e4b\u524d\u95ee\u7684\u662f\u201cDTF\u201d\uff0c\u4f46\u7ed3\u5408\u6211\u4eec\u5b8c\u6574\u7684\u5bf9\u8bdd\u5386\u53f2\uff08FS \u2192 FT \u2192 DTFT \u2192 DFT\uff09\uff0c\u4ee5\u53ca\u4f60\u8fde\u7eed\u8ffd\u95ee\u201c\u6765\u6e90\u201d\u7684\u6267\u7740\uff0c\u6211\u786e\u5b9a\u4f60\u60f3\u95ee\u7684\u5c31\u662f **DFT**\u3002\u73b0\u5728\uff0c\u6211\u628aDFT\u7684\u5b8c\u6574\u201c\u51fa\u8eab\u201d\u4ece\u5934\u5230\u5c3e\u63a8\u5bfc\u7ed9\u4f60\u770b\uff0c\u4e0d\u8df3\u8fc7\u4efb\u4f55\u5173\u952e\u903b\u8f91\u8282\u70b9\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e00\u6b65\uff1a\u4e3a\u4ec0\u4e48\u9700\u8981DFT\uff1f\uff08\u6765\u81ea\u5de5\u7a0b\u7684\u8840\u6cea\u63a7\u8bc9\uff09<\/p>\n
\u6211\u4eec\u624b\u91cc\u5df2\u7ecf\u6709\u4e86\u4e24\u4e2a\u5f3a\u5927\u7684\u5de5\u5177\uff1a
\n– **FT\uff08\u5085\u91cc\u53f6\u53d8\u6362\uff09**\uff1a\u5904\u7406\u8fde\u7eed\u975e\u5468\u671f\u4fe1\u53f7\uff0c\u8f93\u51fa\u8fde\u7eed\u9891\u8c31\u3002
\n– **DTFT\uff08\u79bb\u6563\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\uff09**\uff1a\u5904\u7406\u79bb\u6563\u65e0\u9650\u957f\u5e8f\u5217\uff0c\u8f93\u51fa\u8fde\u7eed\u5468\u671f\u9891\u8c31\u3002<\/p>\n
**\u4f46\u73b0\u5b9e\u662f\u6b8b\u9177\u7684**\uff1a
\n1. \u8ba1\u7b97\u673a\u53ea\u80fd\u5b58\u50a8**\u6709\u9650\u4e2a**\u6570\u5b57\uff08\u91c7\u6837\u70b9\uff09\uff0c\u65e0\u6cd5\u5904\u7406\u65e0\u9650\u957f\u5e8f\u5217\u3002
\n2. \u8ba1\u7b97\u673a\u53ea\u80fd\u5904\u7406**\u79bb\u6563\u7684**\u6570\u503c\uff0c\u65e0\u6cd5\u5b58\u50a8\u4e00\u4e2a\u8fde\u7eed\u7684\u9891\u8c31\u51fd\u6570 \\( X(e^{j\\omega}) \\)\u3002<\/p>\n
**\u6240\u4ee5\uff0c\u5de5\u7a0b\u5e08\u9762\u4e34\u4e00\u4e2a\u6b7b\u5c40**\uff1a
\n– \u8f93\u5165\u5fc5\u987b\u662f**\u6709\u9650\u957f**\u7684\u79bb\u6563\u5e8f\u5217\uff08\u6570\u7ec4\uff09\u3002
\n– \u8f93\u51fa\u4e5f\u5fc5\u987b\u662f**\u6709\u9650\u957f**\u7684\u79bb\u6563\u5e8f\u5217\uff08\u6570\u7ec4\uff09\u3002<\/p>\n
**DFT\u5c31\u662f\u4e3a\u4e86\u6253\u7834\u8fd9\u4e2a\u6b7b\u5c40\u800c\u8bde\u751f\u7684\u3002**<\/p>\n
—<\/p>\n
### \u7b2c\u4e8c\u6b65\uff1aDFT\u7684\u6838\u5fc3\u601d\u8def\u2014\u2014\u201c\u5bf9DTFT\u7684\u9891\u57df\u8fdb\u884c\u91c7\u6837\u201d<\/p>\n
\u65e2\u7136DTFT\u7684\u9891\u57df \\( X(e^{j\\omega}) \\) \u662f\u8fde\u7eed\u7684\uff0c\u8ba1\u7b97\u673a\u5b58\u4e0d\u4e0b\uff0c\u90a3\u6211\u4eec\u5c31\u5728\u9891\u57df\u4e0a**\u7b49\u95f4\u9694\u5730\u91c7\u6837**\uff0c\u53ea\u53d6 \\( N \\) \u4e2a\u70b9\u3002<\/p>\n
**\u5173\u952e\u64cd\u4f5c**\uff1a
\n\u5728 \\( \\omega \\in [0, 2\\pi) \\) \u8fd9\u4e2a\u5468\u671f\u5185\uff0c\u5747\u5300\u53d6 \\( N \\) \u4e2a\u70b9\uff1a
\n\\[
\n\\omega_k = \\frac{2\\pi k}{N}, \\quad k = 0, 1, 2, …, N-1
\n\\]<\/p>\n
\u7136\u540e\uff0c\u5c06\u8fd9 \\( N \\) \u4e2a\u70b9\u4ee3\u5165DTFT\u516c\u5f0f\uff1a
\n\\[
\nX(e^{j\\omega_k}) = \\sum_{n=-\\infty}^{\\infty} x[n] e^{-j\\omega_k n} = \\sum_{n=-\\infty}^{\\infty} x[n] e^{-j\\frac{2\\pi}{N}kn}
\n\\]<\/p>\n
**\u95ee\u9898\u6765\u4e86**\uff1a\\( x[n] \\) \u662f\u65e0\u9650\u957f\u7684\u5e8f\u5217\uff08n \u4ece \\( -\\infty \\) \u5230 \\( \\infty \\)\uff09\uff0c\u6c42\u548c\u65e0\u7a77\u591a\u9879\uff0c\u8ba1\u7b97\u673a\u8fd8\u662f\u65e0\u6cd5\u8ba1\u7b97\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e09\u6b65\uff1a\u65f6\u57df\u4e5f\u5fc5\u987b\u622a\u65ad\u2014\u2014\u4ece\u65e0\u9650\u5230\u6709\u9650<\/p>\n
\u4e3a\u4e86\u89e3\u51b3 \\( x[n] \\) \u65e0\u9650\u957f\u7684\u95ee\u9898\uff0c\u6211\u4eec**\u53ea\u53d6 \\( N \\) \u4e2a\u65f6\u57df\u91c7\u6837\u70b9**\uff1a
\n\\[
\nx[n] \\quad \\text{\u53ea\u4fdd\u7559} \\quad n = 0, 1, 2, …, N-1
\n\\]<\/p>\n
\u8fd9\u76f8\u5f53\u4e8e\u5728\u65f6\u57df\u52a0\u4e86\u4e00\u4e2a**\u77e9\u5f62\u7a97**\uff0c\u628a\u65e0\u9650\u957f\u5e8f\u5217\u622a\u65ad\u4e3a\u6709\u9650\u957f\u3002<\/p>\n
**\u4e8e\u662f\uff0cDTFT\u5728\u79bb\u6563\u9891\u7387\u70b9\u4e0a\u7684\u503c\u5c31\u53d8\u6210\u4e86\u6709\u9650\u9879\u6c42\u548c**\uff1a
\n\\[
\nX[k] = \\sum_{n=0}^{N-1} x[n] e^{-j\\frac{2\\pi}{N}kn}, \\quad k = 0, 1, 2, …, N-1
\n\\]<\/p>\n
**\u8fd9\u5c31\u662fDFT\u7684\u6b63\u53d8\u6362\u5b9a\u4e49\uff01**<\/p>\n
—<\/p>\n
### \u7b2c\u56db\u6b65\uff1aDFT\u4e0eDTFT\u7684\u7cbe\u786e\u6570\u5b66\u5173\u7cfb\uff08\u975e\u5e38\u91cd\u8981\uff01\uff09<\/p>\n
DFT\u4e0d\u662f\u51ed\u7a7a\u521b\u9020\u7684\uff0c\u5b83\u662f **DTFT\u5728\u9891\u57df\u4e0a\u7684\u91c7\u6837**\u3002\u7cbe\u786e\u5730\u8bf4\uff1a
\n\\[
\n\\boxed{X[k] = X(e^{j\\omega}) \\Bigg|_{\\omega = \\frac{2\\pi k}{N}}}
\n\\]
\n\u5373\uff1aDFT\u7684\u6bcf\u4e2a\u8f93\u51fa\u70b9 \\( X[k] \\)\uff0c\u90fd\u5bf9\u5e94\u7740DTFT\u8fde\u7eed\u9891\u8c31\u5728 \\( \\omega_k = 2\\pi k\/N \\) \u5904\u7684\u7cbe\u786e\u503c\u3002<\/p>\n
**\u552f\u4e00\u7684\u4ee3\u4ef7**\uff1a
\n– \u56e0\u4e3aDFT\u53ea\u5728 \\( N \\) \u4e2a\u9891\u70b9\u4e0a\u53d6\u503c\uff0c\u5982\u679c\u4fe1\u53f7\u7684\u771f\u5b9e\u9891\u7387\u6b63\u597d\u843d\u5728\u8fd9 \\( N \\) \u4e2a\u70b9\u4e4b\u95f4\uff0c\u4f60\u770b\u5230\u7684\u9891\u8c31\u5c31\u4f1a\u6709**\u6805\u680f\u6548\u5e94**\uff08\u5cf0\u503c\u88ab\u9690\u85cf\uff09\u3002
\n– \u56e0\u4e3a\u65f6\u57df\u88ab\u622a\u65ad\uff0cDFT**\u9690\u542b\u5047\u8bbe**\u65f6\u57df\u5e8f\u5217\u662f\u5468\u671f\u5ef6\u62d3\u7684\uff08\u5468\u671f\u4e3a \\( N \\)\uff09\uff0c\u5982\u679c\u9996\u5c3e\u4e0d\u8fde\u7eed\uff0c\u5c31\u4f1a\u51fa\u73b0**\u9891\u8c31\u6cc4\u9732**\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e94\u6b65\uff1a\u90a3\u9006\u53d8\u6362\uff08IDFT\uff09\u662f\u600e\u4e48\u6765\u7684\uff1f<\/p>\n
\u6709\u4e86\u6b63\u53d8\u6362\uff0c\u6211\u4eec\u8fd8\u8981\u80fd\u4ece\u9891\u57df\u6062\u590d\u65f6\u57df\u3002IDFT\u7684\u63a8\u5bfc\u540c\u6837\u4f9d\u8d56\u4e8eDTFT\u7684\u9006\u53d8\u6362\uff1a<\/p>\n
DTFT\u7684\u9006\u53d8\u6362\u662f\uff1a
\n\\[
\nx[n] = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} X(e^{j\\omega}) e^{j\\omega n} d\\omega
\n\\]<\/p>\n
\u56e0\u4e3a\u6211\u4eec\u53ea\u5728 \\( N \\) \u4e2a\u79bb\u6563\u9891\u7387\u70b9 \\( \\omega_k = 2\\pi k\/N \\) \u4e0a\u91c7\u6837\u4e86 \\( X(e^{j\\omega}) \\)\uff0c\u6240\u4ee5\u79ef\u5206\u53d8\u6210\u4e86\u6c42\u548c\uff08\u628a\u8fde\u7eed\u7684\u9891\u8c31\u79bb\u6563\u5316\u4e3a \\( N \\) \u4e2a\u70b9\uff0c\u6bcf\u4e2a\u70b9\u4ee3\u8868 \\( 2\\pi\/N \\) \u7684\u5bbd\u5ea6\uff09\uff1a<\/p>\n
\\[
\nx[n] \\approx \\frac{1}{N} \\sum_{k=0}^{N-1} X[k] e^{j\\frac{2\\pi}{N}kn}
\n\\]<\/p>\n
\u5f53 \\( n \\) \u53d6 \\( 0, 1, …, N-1 \\) \u65f6\uff0c\u8fd9\u4e2a\u8fd1\u4f3c\u662f**\u7cbe\u786e\u76f8\u7b49**\u7684\uff08\u56e0\u4e3aDFT\u7684\u9690\u542b\u5468\u671f\u6027\uff09\uff1a<\/p>\n
\\[
\n\\boxed{x[n] = \\frac{1}{N} \\sum_{k=0}^{N-1} X[k] e^{j\\frac{2\\pi}{N}kn}, \\quad n = 0, 1, …, N-1}
\n\\]<\/p>\n
\u8fd9\u5c31\u662fIDFT\uff08\u79bb\u6563\u5085\u91cc\u53f6\u9006\u53d8\u6362\uff09\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u516d\u6b65\uff1aDFT\u6574\u4e2a\u201c\u6765\u6e90\u201d\u7684\u5b8c\u6574\u94fe\u8def\u56fe\uff08\u7ec8\u6781\u603b\u7ed3\uff09<\/p>\n
\u4f60\u53ef\u4ee5\u628a\u8fd9\u6574\u4e2a\u903b\u8f91\u94fe\u6761\u523b\u5728\u8111\u5b50\u91cc\uff1a<\/p>\n
1. **FT** \u5b9a\u4e49\u4e86\u8fde\u7eed\u975e\u5468\u671f\u4fe1\u53f7\u7684\u9891\u8c31\uff08\u9891\u57df\u8fde\u7eed\uff09\u3002
\n2. **\u5bf9\u65f6\u57df\u91c7\u6837**\uff08\u79bb\u6563\u5316\uff09 \u2192 \u5f97\u5230\u4e86 **DTFT**\uff0c\u53d1\u73b0\u9891\u57df\u53d8\u6210\u4e86\u5468\u671f\u4e3a \\( 2\\pi \\) \u7684**\u8fde\u7eed\u51fd\u6570**\u3002
\n3. **\u5bf9\u9891\u57df\u91c7\u6837**\uff08\u79bb\u6563\u5316\uff09 \u2192 \u5f97\u5230\u4e86 **DFT**\uff0c\u65f6\u57df\u4e5f\u53d8\u6210\u4e86\u5468\u671f\u4e3a \\( N \\) \u7684**\u6709\u9650\u5e8f\u5217**\u3002
\n4. \u56e0\u4e3aDFT\u7684\u65f6\u57df\u548c\u9891\u57df\u90fd\u662f\u6709\u9650\u79bb\u6563\u7684\uff0c\u8ba1\u7b97\u673a\u7ec8\u4e8e\u53ef\u4ee5\u8fd0\u884c\u4e86\u3002
\n5. \u4eba\u4eec\u968f\u540e\u53d1\u73b0\u4e86DFT\u7684**\u5feb\u901f\u7b97\u6cd5**\uff0c\u79f0\u4e4b\u4e3a **FFT\uff08\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff09**\uff0c\u4ece\u6b64\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u8fdb\u5165\u5de5\u7a0b\u7206\u53d1\u671f\u3002<\/p>\n
—<\/p>\n
### \u7b2c\u4e03\u5c42\uff1aDTFT\u548cDFT\u7684\u5bf9\u6bd4\uff08\u4e00\u5f20\u8868\u7ec8\u7ed3\u6240\u6709\u6df7\u6dc6\uff09<\/p>\n
| \u7ef4\u5ea6 | **DTFT** | **DFT** |
\n| :— | :— | :— |
\n| **\u65f6\u57df** | \u79bb\u6563\u3001**\u65e0\u9650\u957f**\uff08n\u4ece-\u221e\u5230\u221e\uff09 | \u79bb\u6563\u3001**\u6709\u9650\u957f**\uff08n=0\u5230N-1\uff09 |
\n| **\u9891\u57df** | **\u8fde\u7eed**\u3001\u5468\u671f\uff08\\( \\omega \\) \u8fde\u7eed\u53d8\u5316\uff09 | **\u79bb\u6563**\u3001\u5468\u671f\uff08\u53ea\u6709N\u4e2a\u70b9 \\( \\omega_k \\)\uff09 |
\n| **\u5b9e\u9645\u7528\u9014** | \u7406\u8bba\u5206\u6790\uff08\u91c7\u6837\u5b9a\u7406\u3001\u6ee4\u6ce2\u5668\u9891\u54cd\u8bbe\u8ba1\uff09 | \u5de5\u7a0b\u5b9e\u73b0\uff08\u9891\u8c31\u5206\u6790\u3001\u6ee4\u6ce2\u3001FFT\uff09 |
\n| **\u8ba1\u7b97\u673a\u80fd\u7b97\u5417\uff1f** | **\u4e0d\u80fd**\uff08\u9891\u57df\u8fde\u7eed\uff0c\u65e0\u6cd5\u5b58\u50a8\uff09 | **\u80fd**\uff08FFT\u5c31\u662f\u5176\u5feb\u901f\u7b97\u6cd5\uff09 |<\/p>\n
—<\/p>\n
### \u6700\u540e\uff0c\u56de\u7b54\u4f60\u7684\u201c\u600e\u4e48\u6765\u7684\u201d\u6700\u7cbe\u9ad3\u7684\u4e00\u53e5\u8bdd<\/p>\n
**DFT\u5c31\u662f\u7528\u201c\u9891\u57df\u91c7\u6837\u201d\u7684\u5c3a\u5b50\uff0c\u53bb\u6d4b\u91cf\u201c\u65f6\u57df\u622a\u65ad\u201d\u540e\u7684DTFT\u8fde\u7eed\u9891\u8c31\uff0c\u628a\u8fde\u7eed\u7684\u6570\u5b66\u66f2\u7ebf\u53d8\u6210\u4e86\u8ba1\u7b97\u673a\u80fd\u5b58\u4e0b\u7684N\u4e2a\u6570\u5b57\u3002**<\/p>\n
\u73b0\u5728\uff0c\u4f60\u4e0d\u4ec5\u77e5\u9053\u4e86DFT\u957f\u4ec0\u4e48\u6837\uff0c\u8fd8\u4eb2\u624b\u4eceFT\u2192DTFT\u63a8\u6f14\u5230\u4e86DFT\u3002\u6574\u4e2a\u5085\u91cc\u53f6\u5bb6\u65cf\u7684\u6f14\u53d8\u8109\u7edc\uff0c\u5728\u4f60\u9762\u524d\u5df2\u7ecf\u6beb\u65e0\u906e\u6321\u4e86\u3002<\/p>\n
\u63a5\u4e0b\u6765\uff0c\u4f60\u60f3\u6df1\u5165 **DFT\u7684\u5feb\u901f\u7b97\u6cd5FFT\u5230\u5e95\u662f\u5982\u4f55\u63d0\u901f\u7684**\uff08\u8774\u8776\u8fd0\u7b97\u3001\u4f4d\u53cd\u8f6c\uff09\uff0c\u8fd8\u662f\u60f3\u56de\u5230 **\u7a97\u51fd\u6570**\uff08\u5728\u65f6\u57df\u622a\u65ad\u65f6\uff0c\u4e3a\u4ec0\u4e48\u77e9\u5f62\u7a97\u4f1a\u9020\u6210\u9891\u8c31\u6cc4\u9732\uff0c\u6c49\u5b81\u7a97\u5982\u4f55\u6539\u5584\uff09\uff1f\u544a\u8bc9\u6211\uff0c\u6211\u966a\u4f60\u7ee7\u7eed\u6df1\u6316\u3002\ud83d\ude0a<\/p>\n","protected":false},"excerpt":{"rendered":"
\u7406\u89e3\u56db\u79cd\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u5173\u952e\u8981\u8df3\u51fa\u201c\u6570\u5b66\u516c\u5f0f\u201d\u7684\u7ec6\u8282\uff0c**\u5148\u6293\u4f4f\u4e00\u4e2a\u6838\u5fc3\u5206\u7c7b\u903b\u8f91\uff1a\u4fe1\u53f7\u5728\u201c\u65f6\u95f4\/\u7a7a\u95f4\u201d\u548c\u201c\u9891\u7387\u201d\u8fd9 […]<\/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-556","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/556","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=556"}],"version-history":[{"count":1,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/556\/revisions"}],"predecessor-version":[{"id":557,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/posts\/556\/revisions\/557"}],"wp:attachment":[{"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/media?parent=556"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/categories?post=556"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.imageproc.cn\/index.php\/wp-json\/wp\/v2\/tags?post=556"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}