{"id":82,"date":"2026-09-07T16:46:46","date_gmt":"2026-09-07T08:46:46","guid":{"rendered":"https:\/\/www.oldhamburger.cn\/?p=82"},"modified":"2026-09-07T16:46:46","modified_gmt":"2026-09-07T08:46:46","slug":"fourier-transform-guide","status":"publish","type":"post","link":"https:\/\/www.oldhamburger.cn\/index.php\/2026\/09\/07\/fourier-transform-guide\/","title":{"rendered":"\u5085\u91cc\u53f6\u53d8\u6362\u8be6\u89e3\uff1a\u4ece\u76f4\u89c9\u3001\u516c\u5f0f\u5230 FFT\u3001\u91c7\u6837\u4e0e\u5de5\u7a0b\u5b9e\u8df5"},"content":{"rendered":"\n

\u5f53\u6211\u4eec\u542c\u5230\u4e00\u6bb5\u97f3\u4e50\u3001\u6d4b\u91cf\u4e00\u7ec4\u632f\u52a8\u6570\u636e\uff0c\u6216\u89c2\u5bdf\u4e00\u5e45\u56fe\u50cf\u65f6\uff0c\u773c\u524d\u5f97\u5230\u7684\u901a\u5e38\u662f\u201c\u4fe1\u53f7\u968f\u65f6\u95f4\u6216\u7a7a\u95f4\u600e\u6837\u53d8\u5316\u201d\u3002\u5085\u91cc\u53f6\u53d8\u6362\u5219\u63d0\u51fa\u4e86\u53e6\u4e00\u4e2a\u95ee\u9898\uff1a\u8fd9\u4e2a\u590d\u6742\u4fe1\u53f7\u7531\u54ea\u4e9b\u9891\u7387\u6210\u5206\u7ec4\u6210\uff0c\u6bcf\u79cd\u6210\u5206\u6709\u591a\u5f3a\uff0c\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u4f4d\u5173\u7cfb\u662f\u4ec0\u4e48\uff1f<\/strong><\/p>\n\n\n\n

\u5b83\u50cf\u4e00\u526f\u6570\u5b66\u68f1\u955c\uff1a\u68f1\u955c\u628a\u767d\u5149\u5206\u6210\u4e0d\u540c\u989c\u8272\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u628a\u590d\u6742\u4fe1\u53f7\u5206\u89e3\u6210\u4e0d\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\u3002\u8fd9\u4e2a\u60f3\u6cd5\u8fde\u63a5\u4e86\u58f0\u97f3\u3001\u56fe\u50cf\u3001\u901a\u4fe1\u3001\u96f7\u8fbe\u3001\u533b\u5b66\u6210\u50cf\u3001\u673a\u68b0\u6545\u969c\u8bca\u65ad\u3001\u63a7\u5236\u7cfb\u7edf\u3001\u504f\u5fae\u5206\u65b9\u7a0b\u548c\u6570\u636e\u538b\u7f29\u3002\u672c\u6587\u4e0d\u8981\u6c42\u8bfb\u8005\u9884\u5148\u638c\u63e1\u9ad8\u7b49\u6570\u5b66\u8bc1\u660e\uff1b\u6211\u4eec\u4f1a\u5148\u5efa\u7acb\u76f4\u89c9\uff0c\u518d\u9010\u5c42\u8865\u4e0a\u516c\u5f0f\u4e0e\u5de5\u7a0b\u7ec6\u8282\u3002<\/p>\n\n\n

\n
\"\u4e00\u4e2a\u75315<\/div>
\u540c\u4e00\u6bb5\u4fe1\u53f7\u7684\u4e24\u79cd\u89c6\u89d2\uff1a\u5de6\u8fb9\u662f\u65f6\u57df\u6ce2\u5f62\uff0c\u53f3\u8fb9\u662f\u9891\u8c31\u3002\u539f\u672c\u7ea0\u7f20\u5728\u4e00\u8d77\u7684 5 Hz \u4e0e 20 Hz \u6210\u5206\uff0c\u5728\u9891\u57df\u4e2d\u53d8\u6210\u4e24\u4e2a\u6e05\u695a\u7684\u5cf0\u3002<\/figcaption><\/figure>\n<\/div>\n\n\n

\u4e00\u3001\u5148\u522b\u6025\u7740\u7b97\uff1a\u5085\u91cc\u53f6\u53d8\u6362\u5230\u5e95\u5728\u505a\u4ec0\u4e48\uff1f<\/h2>\n\n\n\n

\u5047\u8bbe\u94a2\u7434\u3001\u5409\u4ed6\u548c\u4eba\u58f0\u540c\u65f6\u6f14\u594f\u3002\u9ea6\u514b\u98ce\u53ea\u8bb0\u5f55\u5230\u4e00\u6761\u968f\u65f6\u95f4\u4e0a\u4e0b\u8d77\u4f0f\u7684\u7535\u538b\u66f2\u7ebf\uff0c\u5404\u79cd\u58f0\u97f3\u5df2\u7ecf\u53e0\u5728\u4e00\u8d77\u3002\u4ec5\u770b\u6ce2\u5f62\uff0c\u5f88\u96be\u8bf4\u51fa\u5176\u4e2d\u6709\u54ea\u4e9b\u97f3\u9ad8\uff1b\u5982\u679c\u628a\u5b83\u8f6c\u6362\u5230\u9891\u7387\u57df\uff0c\u57fa\u9891\u3001\u6cdb\u97f3\u548c\u566a\u58f0\u5374\u4f1a\u5728\u4e0d\u540c\u9891\u7387\u4f4d\u7f6e\u663e\u73b0\u51fa\u6765\u3002<\/p>\n\n\n\n

\u56e0\u6b64\u53ef\u4ee5\u628a\u4e24\u79cd\u89c6\u89d2\u6982\u62ec\u4e3a\uff1a<\/p>\n\n\n\n

  • \u65f6\u57df\uff08time domain\uff09<\/strong>\uff1a\u6a2a\u8f74\u662f\u65f6\u95f4\uff0c\u56de\u7b54\u201c\u4fe1\u53f7\u5728\u6bcf\u4e00\u65f6\u523b\u662f\u591a\u5c11\u201d\u3002<\/li>
  • \u9891\u57df\uff08frequency domain\uff09<\/strong>\uff1a\u6a2a\u8f74\u662f\u9891\u7387\uff0c\u56de\u7b54\u201c\u4fe1\u53f7\u542b\u6709\u54ea\u4e9b\u5feb\u6162\u4e0d\u540c\u7684\u632f\u8361\u6210\u5206\u201d\u3002<\/li><\/ul>\n\n\n\n

    \u8fd9\u4e0d\u662f\u628a\u4fe1\u53f7\u6362\u6210\u4e86\u53e6\u4e00\u4e2a\u4fe1\u53f7\uff0c\u800c\u662f\u7528\u53e6\u4e00\u5957\u5750\u6807\u63cf\u8ff0\u540c\u4e00\u4e2a\u5bf9\u8c61\u3002\u5e73\u9762\u4e0a\u7684\u5411\u91cf\u53ef\u4ee5\u7528\u6a2a\u3001\u7eb5\u5750\u6807\u8868\u793a\uff0c\u4e5f\u53ef\u4ee5\u6539\u7528\u65cb\u8f6c\u540e\u7684\u5750\u6807\u8f74\uff1b\u5085\u91cc\u53f6\u53d8\u6362\u6240\u505a\u7684\u4e8b\u60c5\u975e\u5e38\u76f8\u4f3c\uff0c\u53ea\u4e0d\u8fc7\u201c\u5750\u6807\u8f74\u201d\u53d8\u6210\u4e86\u4e0d\u540c\u9891\u7387\u7684\u590d\u6307\u6570\uff08\u6216\u6b63\u5f26\u3001\u4f59\u5f26\uff09\u3002\u53ea\u8981\u4fdd\u7559\u5b8c\u6574\u7684\u5e45\u5ea6\u548c\u76f8\u4f4d\uff0c\u9006\u5085\u91cc\u53f6\u53d8\u6362\u5c31\u80fd\u628a\u9891\u8c31\u65e0\u635f\u5730\u8fd8\u539f\u4e3a\u539f\u4fe1\u53f7\u3002<\/p>\n\n\n\n

    \u4e00\u53e5\u8bdd\u76f4\u89c9\uff1a<\/strong>\u5085\u91cc\u53f6\u53d8\u6362\u662f\u5728\u8be2\u95ee\u4fe1\u53f7\uff1a\u201c\u4f60\u548c\u9891\u7387\u4e3a f \u7684\u6807\u51c6\u632f\u8361\u6709\u591a\u50cf\uff1f\u201d\u628a\u6240\u6709 f \u90fd\u95ee\u4e00\u904d\uff0c\u5c31\u5f97\u5230\u5b8c\u6574\u9891\u8c31\u3002<\/p><\/blockquote>\n\n\n\n

    \u4e8c\u3001\u5085\u91cc\u53f6\u4e16\u754c\u7684\u57fa\u672c\u79ef\u6728\uff1a\u6b63\u5f26\u6ce2<\/h2>\n\n\n\n

    \u6700\u7b80\u5355\u7684\u5468\u671f\u632f\u8361\u53ef\u5199\u4e3a\uff1a<\/p>\n\n\n\n

    x(t) = A cos(2\u03c0ft + \u03c6)<\/strong><\/p><\/div><\/div>\n\n\n\n

    • A\uff08\u5e45\u5ea6\uff09<\/strong>\u51b3\u5b9a\u632f\u8361\u6709\u591a\u5f3a\uff0c\u4f8b\u5982\u58f0\u97f3\u6709\u591a\u54cd\u3002<\/li>
    • f\uff08\u9891\u7387\uff09<\/strong>\u8868\u793a\u6bcf\u79d2\u632f\u8361\u591a\u5c11\u6b21\uff0c\u5355\u4f4d\u662f Hz\u3002<\/li>
    • \u03c6\uff08\u521d\u76f8\u4f4d\uff09<\/strong>\u51b3\u5b9a\u6ce2\u5f62\u5728\u65f6\u95f4\u8f74\u4e0a\u4ece\u54ea\u91cc\u5f00\u59cb\u3002<\/li>
    • 2\u03c0f\uff08\u89d2\u9891\u7387 \u03c9\uff09<\/strong>\u7684\u5355\u4f4d\u662f rad\/s\uff0c\u56e0\u6b64\u4e5f\u5e38\u5199\u4e3a x(t)=A cos(\u03c9t+\u03c6)<\/strong>\u3002<\/li><\/ul>\n\n\n\n

      \u4e3a\u4ec0\u4e48\u504f\u504f\u7528\u6b63\u5f26\u6ce2\uff1f\u56e0\u4e3a\u5b83\u65e2\u8db3\u591f\u7b80\u5355\uff0c\u53c8\u5177\u6709\u6781\u597d\u7684\u7a33\u5b9a\u6027\uff1a\u6b63\u5f26\u6ce2\u7ecf\u8fc7\u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf\u540e\uff0c\u9891\u7387\u4e0d\u4f1a\u6539\u53d8\uff0c\u53ea\u4f1a\u6539\u53d8\u5e45\u5ea6\u548c\u76f8\u4f4d\uff1b\u5bf9\u5b83\u6c42\u5bfc\u3001\u79ef\u5206\uff0c\u4ecd\u7136\u662f\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\uff1b\u4e0d\u540c\u6574\u6570\u9891\u7387\u7684\u6b63\u5f26\u6ce2\u5728\u5b8c\u6574\u5468\u671f\u4e0a\u8fd8\u5177\u6709\u6b63\u4ea4\u6027\u3002\u8fd9\u4e9b\u6027\u8d28\u8ba9\u590d\u6742\u7684\u5fae\u5206\u65b9\u7a0b\u548c\u5377\u79ef\u8fd0\u7b97\u5728\u9891\u57df\u4e2d\u5927\u5e45\u7b80\u5316\u3002<\/p>\n\n\n\n

      \u6b27\u62c9\u516c\u5f0f\uff1a\u628a\u6b63\u5f26\u548c\u4f59\u5f26\u7edf\u4e00\u8d77\u6765<\/h3>\n\n\n\n

      \u590d\u6570\u5f62\u5f0f\u4e0d\u662f\u6545\u5f04\u7384\u865a\uff0c\u800c\u662f\u540c\u65f6\u643a\u5e26\u5e45\u5ea6\u4e0e\u76f8\u4f4d\u7684\u7d27\u51d1\u8bed\u8a00\u3002\u6b27\u62c9\u516c\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n

      ej\u03b8<\/sup> = cos \u03b8 + j sin \u03b8\u3000\u3000\uff08j\u00b2 = \u22121\uff09<\/strong><\/p><\/div><\/div>\n\n\n\n

      \u628a \u03b8 \u6362\u6210 2\u03c0ft\uff0cej2\u03c0ft<\/sup> \u5c31\u50cf\u590d\u5e73\u9762\u4e0a\u4ee5\u9891\u7387 f \u5300\u901f\u65cb\u8f6c\u7684\u6307\u9488\u3002\u5085\u91cc\u53f6\u53d8\u6362\u53ef\u4ee5\u7406\u89e3\u4e3a\uff1a\u8ba9\u4fe1\u53f7\u5206\u522b\u4e0e\u5404\u79cd\u8f6c\u901f\u7684\u6307\u9488\u76f8\u4e58\u5e76\u7d2f\u79ef\uff1b\u5f53\u8f6c\u901f\u5339\u914d\u65f6\uff0c\u5404\u9879\u4f1a\u540c\u5411\u53e0\u52a0\uff0c\u5f97\u5230\u8f83\u5927\u7684\u7ed3\u679c\uff0c\u4e0d\u5339\u914d\u65f6\u5219\u4f1a\u4e92\u76f8\u62b5\u6d88\u3002<\/p>\n\n\n\n

      \u4e09\u3001\u4ece\u201c\u76f8\u5173\u6027\u201d\u7406\u89e3\u9891\u7387\u68c0\u6d4b<\/h2>\n\n\n\n

      \u8981\u68c0\u6d4b\u4fe1\u53f7 x(t) \u4e2d\u662f\u5426\u542b\u6709\u9891\u7387 f\uff0c\u53ef\u4ee5\u628a\u5b83\u4e0e cos(2\u03c0ft) \u548c sin(2\u03c0ft) \u5206\u522b\u76f8\u4e58\uff0c\u518d\u5728\u4e00\u6bb5\u65f6\u95f4\u4e0a\u79ef\u5206\u3002\u82e5\u9891\u7387\u4e00\u81f4\uff0c\u540c\u76f8\u90e8\u5206\u4f1a\u6301\u7eed\u7d2f\u52a0\uff1b\u82e5\u9891\u7387\u4e0d\u540c\uff0c\u6b63\u8d1f\u533a\u57df\u5f80\u5f80\u5f7c\u6b64\u62b5\u6d88\u3002\u8fd9\u4e0e\u5411\u91cf\u70b9\u79ef\u5224\u65ad\u4e24\u4e2a\u65b9\u5411\u7684\u76f8\u4f3c\u7a0b\u5ea6\u5b8c\u5168\u540c\u6784\u3002<\/p>\n\n\n\n

      \u4f7f\u7528\u590d\u6307\u6570\u540e\uff0c\u4e24\u4e2a\u68c0\u6d4b\u4e00\u6b21\u5b8c\u6210\uff1a<\/p>\n\n\n\n

      X(f) = \u222b\u2212\u221e<\/sub>\u221e<\/sup> x(t)e\u2212j2\u03c0ft<\/sup> dt<\/strong><\/p><\/div><\/div>\n\n\n\n

      \u8fd9\u5c31\u662f\u8fde\u7eed\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\u3002\u7ed3\u679c X(f) \u901a\u5e38\u662f\u590d\u6570\uff1a\u5176\u6a21 |X(f)| \u8868\u793a\u8be5\u9891\u7387\u6210\u5206\u7684\u5f3a\u5f31\uff0c\u5176\u8f90\u89d2 arg X(f) \u8868\u793a\u76f8\u4f4d\u3002\u6307\u6570\u524d\u7684\u8d1f\u53f7\u610f\u5473\u7740\u7528\u53cd\u5411\u65cb\u8f6c\u53bb\u201c\u62b5\u6d88\u201d\u5f85\u6d4b\u6210\u5206\u7684\u65cb\u8f6c\uff1b\u62b5\u6d88\u6210\u529f\u65f6\uff0c\u8be5\u6210\u5206\u53d8\u4e3a\u8fd1\u4f3c\u9759\u6b62\u7684\u5411\u91cf\uff0c\u79ef\u5206\u540e\u7559\u4e0b\u8f83\u5927\u7ed3\u679c\u3002<\/p>\n\n\n\n

      \u56db\u3001\u5085\u91cc\u53f6\u7ea7\u6570\uff1a\u5148\u62c6\u89e3\u5468\u671f\u4fe1\u53f7<\/h2>\n\n\n\n

      \u5085\u91cc\u53f6\u7ea7\u6570\u5904\u7406\u5468\u671f\u4fe1\u53f7\u3002\u82e5 x(t) \u7684\u5468\u671f\u4e3a T0<\/sub>\uff0c\u57fa\u9891 f0<\/sub>=1\/T0<\/sub>\uff0c\u5b83\u53ef\u5728\u9002\u5f53\u6761\u4ef6\u4e0b\u8868\u793a\u6210\u57fa\u9891\u53ca\u5176\u6574\u6570\u500d\u8c10\u6ce2\u7684\u53e0\u52a0\uff1a<\/p>\n\n\n\n

      x(t) = \u03a3k=\u2212\u221e<\/sub>\u221e<\/sup> Ck<\/sub>ej2\u03c0kf\u2080t<\/sup><\/strong><\/p>

      Ck<\/sub> = (1\/T\u2080) \u222bt\u2080<\/sub>t\u2080+T\u2080<\/sup> x(t)e\u2212j2\u03c0kf\u2080t<\/sup> dt<\/strong><\/p><\/div><\/div>\n\n\n\n

      \u5468\u671f\u4fe1\u53f7\u7684\u9891\u8c31\u53ea\u51fa\u73b0\u5728 kf0<\/sub> \u8fd9\u4e9b\u79bb\u6563\u9891\u7387\u4e0a\uff0c\u6240\u4ee5\u662f\u201c\u7ebf\u8c31\u201d\u3002\u4f8b\u5982\u7406\u60f3\u65b9\u6ce2\u7531\u57fa\u9891\u548c\u65e0\u7a77\u591a\u4e2a\u5947\u6b21\u8c10\u6ce2\u7ec4\u6210\uff0c\u8c10\u6ce2\u5e45\u5ea6\u968f\u6b21\u6570\u5927\u81f4\u6309 1\/k \u4e0b\u964d\u3002\u4fdd\u7559\u7684\u8c10\u6ce2\u8d8a\u591a\uff0c\u91cd\u5efa\u6ce2\u5f62\u8d8a\u63a5\u8fd1\u65b9\u6ce2\uff0c\u4f46\u8df3\u53d8\u9644\u8fd1\u4ecd\u4f1a\u51fa\u73b0\u56fa\u5b9a\u6bd4\u4f8b\u7684\u8fc7\u51b2\uff0c\u8fd9\u53eb\u5409\u5e03\u65af\u73b0\u8c61<\/strong>\u3002\u5b83\u63d0\u9192\u6211\u4eec\uff1a\u4e0d\u8fde\u7eed\u8fb9\u6cbf\u9700\u8981\u975e\u5e38\u5bbd\u7684\u9891\u7387\u8303\u56f4\u6765\u63cf\u8ff0\u3002<\/p>\n\n\n\n

      \u82e5\u8ba9\u5468\u671f T0<\/sub> \u8d8a\u6765\u8d8a\u957f\uff0c\u8c31\u7ebf\u95f4\u9694 f0<\/sub> \u8d8a\u6765\u8d8a\u5c0f\uff0c\u79bb\u6563\u7684\u5085\u91cc\u53f6\u7ea7\u6570\u9891\u8c31\u9010\u6e10\u53d8\u6210\u8fde\u7eed\u9891\u8c31\uff0c\u4fbf\u81ea\u7136\u8fc7\u6e21\u5230\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/p>\n\n\n\n

      \u4e94\u3001\u8fde\u7eed\u5085\u91cc\u53f6\u53d8\u6362\u4e0e\u9006\u53d8\u6362<\/h2>\n\n\n\n

      \u672c\u6587\u91c7\u7528\u201c\u666e\u901a\u9891\u7387 f\uff08Hz\uff09\u201d\u7684\u7ea6\u5b9a\u3002\u53d8\u6362\u5bf9\u4e3a\uff1a<\/p>\n\n\n\n

      \u6b63\u53d8\u6362\uff1aX(f) = \u222b\u2212\u221e<\/sub>\u221e<\/sup> x(t)e\u2212j2\u03c0ft<\/sup> dt<\/strong><\/p>

      \u9006\u53d8\u6362\uff1ax(t) = \u222b\u2212\u221e<\/sub>\u221e<\/sup> X(f)ej2\u03c0ft<\/sup> df<\/strong><\/p><\/div><\/div>\n\n\n\n

      \u6709\u4e9b\u6559\u6750\u7528\u89d2\u9891\u7387 \u03c9=2\u03c0f\uff0c\u6b64\u65f6\u6307\u6570\u5199\u6210 e\u2212j\u03c9t<\/sup>\uff0c\u9006\u53d8\u6362\u524d\u901a\u5e38\u591a\u4e00\u4e2a 1\/(2\u03c0)\u3002\u4e24\u79cd\u5b9a\u4e49\u90fd\u6b63\u786e\uff0c\u5173\u952e\u662f\u4ece\u5934\u5230\u5c3e\u4fdd\u6301\u540c\u4e00\u7ea6\u5b9a\u3002\u5f88\u591a\u201c\u516c\u5f0f\u5dee\u4e00\u4e2a 2\u03c0\u201d\u7684\u4e89\u8bba\uff0c\u672c\u8d28\u53ea\u662f\u5750\u6807\u548c\u5f52\u4e00\u5316\u4e60\u60ef\u4e0d\u540c\u3002<\/p>\n\n\n\n

      \u5e45\u5ea6\u8c31\u3001\u76f8\u4f4d\u8c31\u4e0e\u529f\u7387\u8c31<\/h3>\n\n\n\n
      • \u5e45\u5ea6\u8c31 |X(f)|<\/strong>\uff1a\u6bcf\u4e2a\u9891\u7387\u6210\u5206\u6709\u591a\u5f3a\u3002<\/li>
      • \u76f8\u4f4d\u8c31 \u2220X(f)<\/strong>\uff1a\u5404\u9891\u7387\u6210\u5206\u76f8\u5bf9\u65f6\u95f4\u539f\u70b9\u7684\u4f4d\u7f6e\u3002<\/li>
      • \u529f\u7387\u8c31 |X(f)|\u00b2<\/strong>\uff1a\u5e38\u7528\u4e8e\u89c2\u5bdf\u80fd\u91cf\u6216\u529f\u7387\u5728\u9891\u7387\u4e0a\u7684\u5206\u5e03\u3002<\/li>
      • \u529f\u7387\u8c31\u5bc6\u5ea6 PSD<\/strong>\uff1a\u5355\u4f4d\u9891\u5bbd\u5185\u7684\u5e73\u5747\u529f\u7387\uff0c\u9002\u5408\u968f\u673a\u3001\u957f\u65f6\u95f4\u4fe1\u53f7\uff1b\u5b83\u4e0d\u662f\u201c\u628a\u4e00\u6b21 FFT \u53d6\u5e73\u65b9\u201d\u8fd9\u4e48\u7b80\u5355\uff0c\u5b9e\u9645\u5e38\u901a\u8fc7\u5206\u6bb5\u3001\u52a0\u7a97\u548c\u5e73\u5747\uff08\u4f8b\u5982 Welch \u65b9\u6cd5\uff09\u6765\u4f30\u8ba1\u3002<\/li><\/ul>\n\n\n\n

        \u76f8\u4f4d\u7edd\u4e0d\u662f\u53ef\u6709\u53ef\u65e0\u3002\u4e24\u4e2a\u4fe1\u53f7\u53ef\u4ee5\u62e5\u6709\u5b8c\u5168\u76f8\u540c\u7684\u5e45\u5ea6\u8c31\uff0c\u5374\u56e0\u76f8\u4f4d\u4e0d\u540c\u800c\u5448\u73b0\u5b8c\u5168\u4e0d\u540c\u7684\u65f6\u57df\u6ce2\u5f62\u548c\u8fb9\u7f18\u4f4d\u7f6e\u3002\u56fe\u50cf\u5904\u7406\u4e2d\uff0c\u5e45\u5ea6\u66f4\u50cf\u201c\u542b\u6709\u54ea\u4e9b\u5c3a\u5ea6\u201d\uff0c\u76f8\u4f4d\u5f80\u5f80\u66f4\u5f3a\u5730\u51b3\u5b9a\u201c\u7269\u4f53\u5728\u54ea\u91cc\u3001\u8f6e\u5ed3\u662f\u4ec0\u4e48\u201d\u3002<\/p>\n\n\n\n

        \u8d1f\u9891\u7387\u662f\u4ec0\u4e48\uff1f<\/h3>\n\n\n\n

        \u8d1f\u9891\u7387\u4e0d\u662f\u7269\u7406\u9891\u7387\u4f4e\u4e8e\u96f6\uff0c\u800c\u662f\u590d\u5e73\u9762\u4e0a\u76f8\u53cd\u65b9\u5411\u7684\u65cb\u8f6c\u3002\u5bf9\u4e8e\u5b9e\u503c\u4fe1\u53f7\uff0c\u6709\u5171\u8f6d\u5bf9\u79f0\u5173\u7cfb X(\u2212f)=X*(f)\uff0c\u56e0\u6b64\u6b63\u8d1f\u9891\u7387\u643a\u5e26\u955c\u50cf\u4fe1\u606f\u3002\u4e00\u4e2a\u5b9e\u4f59\u5f26\u53ef\u5199\u6210\u4e24\u4e2a\u53cd\u5411\u65cb\u8f6c\u7684\u590d\u6307\u6570\u4e4b\u548c\uff1a<\/p>\n\n\n\n

        cos(2\u03c0f\u2080t) = \u00bdej2\u03c0f\u2080t<\/sup> + \u00bde\u2212j2\u03c0f\u2080t<\/sup><\/strong><\/p><\/div><\/div>\n\n\n\n

        \u6240\u4ee5\u5b83\u5728 +f0<\/sub> \u4e0e \u2212f0<\/sub> \u5904\u5404\u6709\u4e00\u4e2a\u5206\u91cf\u3002\u5de5\u7a0b\u4e2d\u753b\u5b9e\u4fe1\u53f7\u7684\u5355\u8fb9\u8c31\u65f6\uff0c\u5e38\u628a\u6b63\u9891\u7387\u975e\u76f4\u6d41\u5206\u91cf\u4e58\u4ee5 2\uff0c\u4ee5\u8865\u56de\u88ab\u7701\u7565\u7684\u8d1f\u9891\u7387\u80fd\u91cf\u3002<\/p>\n\n\n\n

        \u516d\u3001\u6700\u91cd\u8981\u7684\u53d8\u6362\u6027\u8d28<\/h2>\n\n\n\n
        \u6027\u8d28<\/th>\u65f6\u57df<\/th>\u9891\u57df<\/th>\u76f4\u89c2\u610f\u4e49<\/th><\/tr><\/thead>
        \u7ebf\u6027<\/td>ax(t)+by(t)<\/td>aX(f)+bY(f)<\/td>\u53ef\u5206\u522b\u5206\u6790\u540e\u518d\u53e0\u52a0<\/td><\/tr>
        \u65f6\u79fb<\/td>x(t\u2212t\u2080)<\/td>X(f)e\u2212j2\u03c0ft\u2080<\/sup><\/td>\u5ef6\u8fdf\u6539\u53d8\u76f8\u4f4d\uff0c\u4e0d\u6539\u53d8\u5e45\u5ea6\u8c31<\/td><\/tr>
        \u9891\u79fb\/\u8c03\u5236<\/td>x(t)ej2\u03c0f\u2080t<\/sup><\/td>X(f\u2212f\u2080)<\/td>\u4e58\u8f7d\u6ce2\u4f1a\u642c\u79fb\u9891\u8c31<\/td><\/tr>
        \u5c3a\u5ea6\u53d8\u6362<\/td>x(at)<\/td>(1\/|a|)X(f\/a)<\/td>\u65f6\u95f4\u538b\u7f29\uff0c\u9891\u8c31\u5c55\u5bbd<\/td><\/tr>
        \u5fae\u5206<\/td>dx(t)\/dt<\/td>j2\u03c0fX(f)<\/td>\u9ad8\u9891\u88ab\u66f4\u5f3a\u5730\u653e\u5927<\/td><\/tr>
        \u5377\u79ef<\/td>x(t)\u2217h(t)<\/td>X(f)H(f)<\/td>\u590d\u6742\u5377\u79ef\u53d8\u6210\u9010\u9891\u7387\u76f8\u4e58<\/td><\/tr>
        \u76f8\u4e58<\/td>x(t)h(t)<\/td>X(f)\u2217H(f)<\/td>\u622a\u65ad\u3001\u52a0\u7a97\u4f1a\u8ba9\u9891\u8c31\u53d1\u751f\u5377\u79ef<\/td><\/tr>
        \u80fd\u91cf\u5b88\u6052<\/td>\u222b|x(t)|\u00b2dt<\/td>\u222b|X(f)|\u00b2df<\/td>\u80fd\u91cf\u53ea\u662f\u6362\u4e86\u4e00\u79cd\u5750\u6807\u8868\u8fbe<\/td><\/tr><\/tbody><\/table>
        \u7b26\u53f7 \u2217 \u8868\u793a\u5377\u79ef\uff1b\u5177\u4f53\u5e38\u6570\u4ecd\u53d6\u51b3\u4e8e\u6240\u91c7\u7528\u7684\u5085\u91cc\u53f6\u53d8\u6362\u5b9a\u4e49\u3002<\/figcaption><\/figure>\n\n\n\n

        \u5176\u4e2d\u5377\u79ef\u5b9a\u7406<\/strong>\u5c24\u5176\u91cd\u8981\u3002\u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf\u6ee1\u8db3 y(t)=x(t)\u2217h(t)\uff0c\u53d8\u5230\u9891\u57df\u540e\u6210\u4e3a Y(f)=X(f)H(f)\u3002\u7cfb\u7edf\u5bf9\u6bcf\u4e2a\u9891\u7387\u6240\u505a\u7684\u4e8b\uff0c\u90fd\u6d53\u7f29\u5728\u9891\u7387\u54cd\u5e94 H(f) \u91cc\uff1a\u4f4e\u901a\u6ee4\u6ce2\u5668\u4fdd\u7559\u4f4e\u9891\u3001\u8870\u51cf\u9ad8\u9891\uff1b\u9ad8\u901a\u6ee4\u6ce2\u5668\u76f8\u53cd\uff1b\u5e26\u901a\u6ee4\u6ce2\u5668\u53ea\u5141\u8bb8\u4e00\u4e2a\u9891\u6bb5\u901a\u8fc7\u3002\u957f\u5e8f\u5217\u5377\u79ef\u4e5f\u53ef\u7528 FFT \u2192 \u9010\u70b9\u76f8\u4e58 \u2192 IFFT \u9ad8\u6548\u5b8c\u6210\u3002<\/p>\n\n\n\n

        \u4e03\u3001\u4ece\u8fde\u7eed\u4e16\u754c\u5230\u8ba1\u7b97\u673a\uff1a\u91c7\u6837\u4e0e\u5948\u594e\u65af\u7279\u6761\u4ef6<\/h2>\n\n\n\n

        \u8ba1\u7b97\u673a\u4e0d\u80fd\u4fdd\u5b58\u8fde\u7eed\u65f6\u95f4\u7684\u6bcf\u4e00\u70b9\uff0c\u53ea\u80fd\u4ee5\u91c7\u6837\u9891\u7387 fs<\/sub> \u6bcf\u79d2\u8bb0\u5f55 fs<\/sub> \u4e2a\u6837\u672c\u3002\u91c7\u6837\u95f4\u9694\u4e3a \u0394t=1\/fs<\/sub>\u3002\u8981\u5728\u7406\u60f3\u6761\u4ef6\u4e0b\u4ece\u6837\u672c\u552f\u4e00\u91cd\u5efa\u4e00\u4e2a\u5e26\u9650\u4fe1\u53f7\uff0c\u91c7\u6837\u9891\u7387\u5fc5\u987b\u4e25\u683c\u9ad8\u4e8e\u4fe1\u53f7\u6700\u9ad8\u9891\u7387 fmax<\/sub> \u7684\u4e24\u500d\uff1a<\/p>\n\n\n\n

        fs<\/sub> > 2fmax<\/sub>\u3000\u3000\uff0c\u5948\u594e\u65af\u7279\u9891\u7387 fN<\/sub>=fs<\/sub>\/2<\/strong><\/p><\/div><\/div>\n\n\n\n

        \u82e5\u4fe1\u53f7\u542b\u6709\u8d85\u8fc7 fs<\/sub>\/2 \u7684\u9891\u7387\uff0c\u5b83\u4f1a\u6298\u53e0\u5230\u8f83\u4f4e\u9891\u7387\uff0c\u770b\u8d77\u6765\u50cf\u53e6\u4e00\u4e2a\u5b8c\u5168\u4e0d\u540c\u7684\u4fe1\u53f7\uff0c\u8fd9\u5c31\u662f\u6df7\u53e0\uff08aliasing\uff09<\/strong>\u3002\u91c7\u6837\u4e4b\u540e\u901a\u5e38\u65e0\u6cd5\u4ec5\u9760\u6570\u5b57\u7b97\u6cd5\u5224\u65ad\u67d0\u4e2a\u4f4e\u9891\u7a76\u7adf\u662f\u771f\u5b9e\u4f4e\u9891\u8fd8\u662f\u6298\u53e0\u4e0b\u6765\u7684\u9ad8\u9891\uff0c\u6240\u4ee5\u91c7\u6837\u524d\u8981\u7528\u6a21\u62df\u6297\u6df7\u53e0\u4f4e\u901a\u6ee4\u6ce2\u5668\u9650\u5236\u5e26\u5bbd\uff0c\u5e76\u7ed9\u6ee4\u6ce2\u5668\u7559\u51fa\u5b9e\u9645\u8fc7\u6e21\u5e26\u3002<\/p>\n\n\n

        \n
        \"12<\/div>
        12 Hz \u4fe1\u53f7\u7528 16 Hz \u91c7\u6837\u65f6\uff0c\u91c7\u6837\u70b9\u4e5f\u5b8c\u5168\u7b26\u5408 4 Hz \u6b63\u5f26\u6ce2\uff0c\u4e8e\u662f 12 Hz \u88ab\u8bef\u8ba4\u6210 4 Hz\u3002\u8fd9\u91cc 16 Hz \u7684\u5948\u594e\u65af\u7279\u9891\u7387\u53ea\u6709 8 Hz\u3002<\/figcaption><\/figure>\n<\/div>\n\n\n

        \u516b\u3001\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff08DFT\uff09<\/h2>\n\n\n\n

        \u62ff\u5230 N \u4e2a\u79bb\u6563\u6837\u672c x[0],x[1],\u2026,x[N\u22121] \u540e\uff0c\u8ba1\u7b97\u673a\u4f7f\u7528 DFT\uff1a<\/p>\n\n\n\n

        X[k] = \u03a3n=0<\/sub>N\u22121<\/sup> x[n]e\u2212j2\u03c0kn\/N<\/sup>\u3000\uff0ck=0,\u2026,N\u22121<\/strong><\/p>

        x[n] = (1\/N)\u03a3k=0<\/sub>N\u22121<\/sup> X[k]ej2\u03c0kn\/N<\/sup><\/strong><\/p><\/div><\/div>\n\n\n\n

        \u8fd9\u91cc\u6bcf\u4e2a k \u5bf9\u5e94\u4e00\u4e2a\u9891\u7387\u683c\u70b9\uff08frequency bin\uff09\uff1a<\/p>\n\n\n\n

        fk<\/sub> = kfs<\/sub>\/N\u3000\u3000\uff0c\u9891\u7387\u95f4\u9694 \u0394f = fs<\/sub>\/N = 1\/T<\/strong><\/p><\/div><\/div>\n\n\n\n

        T=N\/fs<\/sub> \u662f\u8bb0\u5f55\u603b\u65f6\u957f\u3002\u7531 \u0394f=1\/T \u53ef\u89c1\uff0c\u9891\u7387\u5206\u8fa8\u7387\u4e3b\u8981\u7531\u89c2\u5bdf\u65f6\u957f\u51b3\u5b9a<\/strong>\uff1a\u60f3\u533a\u5206 100 Hz \u4e0e 100.5 Hz\uff0c\u8bb0\u5f55\u65f6\u95f4\u81f3\u5c11\u8981\u8fbe\u5230\u7ea6 2 \u79d2\uff0c\u5b9e\u9645\u8fd8\u8981\u8003\u8651\u7a97\u51fd\u6570\u4e0e\u4fe1\u566a\u6bd4\u3002\u4ec5\u63d0\u9ad8\u91c7\u6837\u7387\u800c\u4e0d\u5ef6\u957f\u8bb0\u5f55\u65f6\u957f\uff0c\u901a\u5e38\u4e0d\u4f1a\u8ba9\u76f8\u90bb\u9891\u7387\u66f4\u5bb9\u6613\u5206\u5f00\u3002<\/p>\n\n\n\n

        DFT \u8fd8\u6709\u4e00\u4e2a\u5bb9\u6613\u5ffd\u89c6\u7684\u5047\u8bbe\uff1a\u8f93\u5165\u7684 N \u4e2a\u6837\u672c\u88ab\u89c6\u4e3a\u67d0\u4e2a\u65e0\u9650\u5468\u671f\u5e8f\u5217\u7684\u4e00\u6bb5\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u7b97\u6cd5\u9ed8\u8ba4\u6700\u540e\u4e00\u4e2a\u6837\u672c\u4e4b\u540e\u4f1a\u65e0\u7f1d\u63a5\u56de\u7b2c\u4e00\u4e2a\u6837\u672c\u3002\u5982\u679c\u9996\u5c3e\u4e0d\u8fde\u7eed\uff0c\u5468\u671f\u62fc\u63a5\u5904\u5c31\u4eba\u4e3a\u5236\u9020\u4e86\u5c16\u9510\u8fb9\u6cbf\uff0c\u4ece\u800c\u4ea7\u751f\u989d\u5916\u7684\u5bbd\u5e26\u9891\u7387\u6210\u5206\u3002<\/p>\n\n\n\n

        \u4e5d\u3001FFT \u662f\u4ec0\u4e48\uff1f\u5b83\u4e0d\u662f\u53e6\u4e00\u79cd\u53d8\u6362<\/h2>\n\n\n\n

        FFT\uff08Fast Fourier Transform\uff0c\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff09\u662f\u8ba1\u7b97 DFT \u7684\u9ad8\u6548\u7b97\u6cd5\u65cf\uff0c\u800c\u4e0d\u662f\u65b0\u7684\u6570\u5b66\u53d8\u6362\u3002<\/strong>\u76f4\u63a5\u6309\u5b9a\u4e49\u8ba1\u7b97 N \u4e2a\u9891\u7387\u3001\u6bcf\u4e2a\u9891\u7387\u7d2f\u52a0 N \u9879\uff0c\u9700\u8981\u7ea6 O(N\u00b2) \u6b21\u8fd0\u7b97\uff1bFFT \u5229\u7528\u590d\u6307\u6570\u7684\u5468\u671f\u6027\u548c\u5bf9\u79f0\u6027\uff0c\u628a\u95ee\u9898\u9012\u5f52\u62c6\u5206\uff0c\u5178\u578b\u590d\u6742\u5ea6\u964d\u4e3a O(N log N)\u3002<\/p>\n\n\n\n

        \u4f8b\u5982 N=1,048,576 \u65f6\uff0cN\u00b2 \u7ea6\u4e3a 1.1 \u4e07\u4ebf\uff0c\u800c Nlog2<\/sub>N \u7ea6\u4e3a 2100 \u4e07\uff0c\u6570\u91cf\u7ea7\u5dee\u8ddd\u5de8\u5927\u3002Cooley\u2013Tukey \u7b97\u6cd5\u53ef\u628a\u957f\u5ea6 N \u5206\u89e3\u6210\u8f83\u5c0f\u56e0\u5b50\u7684\u7ec4\u5408\uff1b2 \u7684\u5e42\u957f\u5ea6\u5f80\u5f80\u5f88\u5feb\uff0c\u4f46\u73b0\u4ee3 FFT \u5e93\u4e5f\u80fd\u6709\u6548\u5904\u7406\u8bb8\u591a\u5408\u6570\u957f\u5ea6\u3002\u9047\u5230\u8f83\u5927\u7684\u8d28\u6570\u957f\u5ea6\u65f6\uff0c\u5e93\u53ef\u80fd\u6539\u7528 Bluestein \u6216 Rader \u7b49\u7b97\u6cd5\u3002<\/p>\n\n\n\n

        \u5173\u7cfb\u603b\u7ed3\uff1a<\/strong>\u5085\u91cc\u53f6\u53d8\u6362\u662f\u601d\u60f3\u4e0e\u8fde\u7eed\u6570\u5b66\uff1bDFT \u662f\u6709\u9650\u79bb\u6563\u6570\u636e\u7684\u53d8\u6362\u5b9a\u4e49\uff1bFFT \u662f\u5feb\u901f\u8ba1\u7b97 DFT \u7684\u7b97\u6cd5\u3002<\/p><\/blockquote>\n\n\n\n

        \u5341\u3001\u771f\u5b9e FFT \u4e3a\u4ec0\u4e48\u4f1a\u201c\u6f0f\u201d\u5f97\u5230\u5904\u90fd\u662f\uff1f<\/h2>\n\n\n\n

        \u5982\u679c\u4fe1\u53f7\u9891\u7387\u521a\u597d\u843d\u5728\u67d0\u4e2a DFT \u683c\u70b9\uff0c\u800c\u4e14\u622a\u53d6\u533a\u95f4\u5305\u542b\u6574\u6570\u4e2a\u5468\u671f\uff0c\u80fd\u91cf\u4f1a\u96c6\u4e2d\u5728\u5bf9\u5e94\u683c\u70b9\u3002\u82e5\u4e0d\u6ee1\u8db3\u8fd9\u4e2a\u6761\u4ef6\uff0c\u9996\u5c3e\u62fc\u63a5\u4ea7\u751f\u4e0d\u8fde\u7eed\uff0c\u80fd\u91cf\u5c31\u6269\u6563\u5230\u5468\u56f4\u9891\u7387\uff0c\u8fd9\u53eb\u9891\u8c31\u6cc4\u6f0f\uff08spectral leakage\uff09<\/strong>\u3002\u5b83\u4e0d\u662f\u8f6f\u4ef6\u7b97\u9519\uff0c\u800c\u662f\u6709\u9650\u89c2\u5bdf\u7a97\u53e3\u7684\u5fc5\u7136\u7ed3\u679c\u3002<\/p>\n\n\n\n

        \u5de5\u7a0b\u4e0a\u4f1a\u5728 FFT \u524d\u5c06\u6570\u636e\u4e58\u4ee5\u7a97\u51fd\u6570\uff0c\u4f7f\u4e24\u7aef\u5e73\u6ed1\u63a5\u8fd1\u96f6\u3002\u5e38\u89c1\u9009\u62e9\u5305\u62ec\uff1a<\/p>\n\n\n\n

        • \u77e9\u5f62\u7a97<\/strong>\uff1a\u7b49\u4e8e\u4e0d\u52a0\u7a97\u3002\u4e3b\u74e3\u7a84\uff0c\u4f46\u65c1\u74e3\u9ad8\uff1b\u9002\u5408\u4e25\u683c\u540c\u6b65\u3001\u6574\u6570\u5468\u671f\u7684\u6570\u636e\u3002<\/li>
        • Hann \u7a97<\/strong>\uff1a\u901a\u7528\u4e14\u7a33\u5065\uff0c\u5e38\u7528\u4e8e\u9891\u7387\u5206\u6790\u3002<\/li>
        • Hamming \u7a97<\/strong>\uff1a\u9996\u4e2a\u65c1\u74e3\u8f83\u4f4e\uff0c\u8bed\u97f3\u7b49\u573a\u666f\u5e38\u89c1\u3002<\/li>
        • Blackman \/ Blackman\u2013Harris \u7a97<\/strong>\uff1a\u65c1\u74e3\u66f4\u4f4e\uff0c\u9002\u5408\u5728\u5f3a\u5206\u91cf\u65c1\u5bfb\u627e\u5f31\u5206\u91cf\uff0c\u4f46\u4e3b\u74e3\u66f4\u5bbd\u3002<\/li>
        • Flat-top \u7a97<\/strong>\uff1a\u5e45\u503c\u6d4b\u91cf\u51c6\u786e\u5ea6\u597d\uff0c\u4f46\u9891\u7387\u5206\u8fa8\u80fd\u529b\u8f83\u5f31\u3002<\/li><\/ul>\n\n\n
          \n
          \"\u77e9\u5f62\u7a97\u4e0eHann\u7a97\u9891\u8c31\u6cc4\u6f0f\u5bf9\u6bd4\uff0c\u4ee5\u53caHann\u7a97\u5e73\u6ed1\u4fe1\u53f7\u7aef\u70b9\u7684\u65f6\u57df\u793a\u610f\"<\/div>
          \u7a97\u51fd\u6570\u662f\u5728\u201c\u5206\u8fa8\u76f8\u8fd1\u9891\u7387\u201d\u548c\u201c\u538b\u4f4e\u8fdc\u5904\u6cc4\u6f0f\u201d\u4e4b\u95f4\u4f5c\u6743\u8861\uff0c\u6ca1\u6709\u5bf9\u6240\u6709\u4efb\u52a1\u90fd\u6700\u4f18\u7684\u7a97\u3002<\/figcaption><\/figure>\n<\/div>\n\n\n

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          \u96f6\u586b\u5145\u80fd\u63d0\u9ad8\u5206\u8fa8\u7387\u5417\uff1f<\/h3>\n\n\n\n

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          \u5341\u4e00\u3001\u600e\u6837\u6b63\u786e\u8bfb\u4e00\u5f20 FFT \u9891\u8c31\uff1f<\/h2>\n\n\n\n
          1. \u5148\u786e\u8ba4\u91c7\u6837\u7387\u4e0e\u5355\u4f4d\u3002<\/strong>\u6a2a\u8f74\u4e0a\u9650\u901a\u5e38\u662f fs<\/sub>\/2\uff1b\u65f6\u95f4\u5355\u4f4d\u82e5\u4e0d\u662f\u79d2\uff0c\u9891\u7387\u5355\u4f4d\u4e5f\u4f1a\u968f\u4e4b\u6539\u53d8\u3002<\/li>
          2. \u68c0\u67e5\u76f4\u6d41\u5206\u91cf\u3002<\/strong>X[0] \u5bf9\u5e94\u4fe1\u53f7\u5e73\u5747\u503c\u3002\u82e5\u53ea\u5173\u5fc3\u4ea4\u6d41\u6210\u5206\uff0c\u53ef\u5148\u51cf\u53bb\u5747\u503c\uff0c\u4f46\u5e94\u660e\u786e\u8bb0\u5f55\u8fd9\u4e00\u9884\u5904\u7406\u3002<\/li>
          3. \u786e\u8ba4\u5355\u8fb9\u8c31\u8fd8\u662f\u53cc\u8fb9\u8c31\u3002<\/strong>\u5b9e\u4fe1\u53f7\u7684\u53cc\u8fb9\u8c31\u5bf9\u79f0\uff1b\u5355\u8fb9\u5e45\u5ea6\u8c31\u901a\u5e38\u9700\u628a\u975e\u76f4\u6d41\u3001\u975e\u5948\u594e\u65af\u7279\u9879\u4e58 2\u3002<\/li>
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          5. \u786e\u8ba4\u5e45\u503c\u7c7b\u578b\u3002<\/strong>\u5cf0\u503c\u3001\u5cf0\u5cf0\u503c\u3001RMS\u3001\u529f\u7387\u3001\u529f\u7387\u8c31\u5bc6\u5ea6\u4e0d\u662f\u540c\u4e00\u4e2a\u91cf\uff1b\u5bf9\u6b63\u5f26\u6ce2\uff0cRMS=\u5cf0\u503c\/\u221a2\u3002<\/li>
          6. \u68c0\u67e5\u7a97\u51fd\u6570\u3002<\/strong>\u7a97\u4f1a\u6539\u53d8\u4e3b\u74e3\u5bbd\u5ea6\u3001\u65c1\u74e3\u4e0e\u5e45\u503c\uff0c\u9700\u8981\u548c\u6d4b\u91cf\u76ee\u7684\u5339\u914d\u3002<\/li>
          7. \u68c0\u67e5\u8bb0\u5f55\u957f\u5ea6\u3002<\/strong>\u0394f=fs<\/sub>\/N=1\/T\u3002\u9891\u7387\u5206\u8fa8\u7387\u4e0d\u8db3\u65f6\uff0c\u4e0d\u8981\u8fc7\u5ea6\u89e3\u91ca\u4e00\u4e2a\u5bbd\u5cf0\u3002<\/li>
          8. \u7559\u610f\u6df7\u53e0\u3002<\/strong>\u9760\u8fd1\u5948\u594e\u65af\u7279\u9891\u7387\u7684\u5f02\u5e38\u5cf0\u5c24\u5176\u503c\u5f97\u68c0\u67e5\u6a21\u62df\u6ee4\u6ce2\u548c\u66f4\u9ad8\u91c7\u6837\u7387\u3002<\/li>
          9. \u7ed3\u5408\u65f6\u57df\u3002<\/strong>\u524a\u9876\u3001\u8109\u51b2\u3001\u6f02\u79fb\u3001\u4e22\u6837\u7b49\u95ee\u9898\u5728\u65f6\u57df\u53ef\u80fd\u66f4\u76f4\u89c2\uff1b\u6c38\u8fdc\u4e0d\u8981\u53ea\u770b\u4e00\u5f20\u9891\u8c31\u4e0b\u7ed3\u8bba\u3002<\/li><\/ol>\n\n\n\n

            \u5341\u4e8c\u3001\u4e00\u4e2a\u53ef\u4ee5\u8fd0\u884c\u7684 Python \/ NumPy \u793a\u4f8b<\/h2>\n\n\n\n

            \u4e0b\u9762\u6784\u9020\u4e00\u4e2a\u542b 50 Hz\u3001120 Hz \u548c\u5c11\u91cf\u566a\u58f0\u7684\u4fe1\u53f7\uff0c\u53bb\u5747\u503c\u3001\u52a0 Hann \u7a97\uff0c\u7136\u540e\u8ba1\u7b97\u5355\u8fb9\u5e45\u5ea6\u8c31\u3002\u4ee3\u7801\u4e2d\u7684\u201c\u9664\u4ee5\u7a97\u51fd\u6570\u4e4b\u548c\u201d\u662f\u5728\u6821\u6b63\u7a97\u7684\u76f8\u5e72\u589e\u76ca\uff0c\u4f7f\u9891\u7387\u6070\u597d\u843d\u5728\u683c\u70b9\u4e0a\u7684\u6b63\u5f26\u5cf0\u503c\u66f4\u63a5\u8fd1\u539f\u59cb\u5cf0\u503c\u5e45\u5ea6\u3002<\/p>\n\n\n\n

            import numpy as np\nimport matplotlib.pyplot as plt\n\n# 1. \u91c7\u6837\u53c2\u6570\nfs = 1000.0                 # \u91c7\u6837\u7387\uff1a1000 Hz\nduration = 2.0              # \u8bb0\u5f55\u65f6\u957f\uff1a2 s\nt = np.arange(0, duration, 1 \/ fs)\n\n# 2. \u6784\u9020\u4fe1\u53f7\uff1a50 Hz \u5e45\u503c1.0\uff0c120 Hz \u5e45\u503c0.4\uff0c\u518d\u52a0\u566a\u58f0\nrng = np.random.default_rng(42)\nx = (1.0 * np.sin(2 * np.pi * 50 * t)\n     + 0.4 * np.sin(2 * np.pi * 120 * t + 0.7)\n     + 0.15 * rng.normal(size=t.size))\n\n# 3. \u9884\u5904\u7406\uff1a\u53bb\u76f4\u6d41\uff0c\u52a0 Hann \u7a97\nx = x - np.mean(x)\nwindow = np.hanning(len(x))\nxw = x * window\n\n# 4. \u5b9e\u4fe1\u53f7\u53ea\u9700 rfft\uff1brfftfreq \u751f\u6210\u5bf9\u5e94\u9891\u7387\u8f74\nX = np.fft.rfft(xw)\nfreq = np.fft.rfftfreq(len(x), d=1 \/ fs)\n\n# 5. \u5355\u8fb9\u5cf0\u503c\u5e45\u5ea6\u8c31\uff1b\u76f4\u6d41\u548c\u5948\u594e\u65af\u7279\u70b9\u4e0d\u5e94\u4e582\namplitude = 2.0 * np.abs(X) \/ np.sum(window)\namplitude[0] \/= 2.0\nif len(x) % 2 == 0:\n    amplitude[-1] \/= 2.0\n\n# 6. \u627e\u51fa\u8f83\u660e\u663e\u7684\u5cf0\uff08\u8fd9\u91cc\u53ea\u4f5c\u7b80\u6613\u6f14\u793a\uff09\nindices = np.argsort(amplitude)[-8:]\nfor i in indices[np.argsort(freq[indices])]:\n    print(f\"{freq[i]:7.2f} Hz : {amplitude[i]:.3f}\")\n\n# 7. \u7ed8\u56fe\nplt.figure(figsize=(10, 4))\nplt.plot(freq, amplitude)\nplt.xlim(0, 250)\nplt.xlabel(\"Frequency (Hz)\")\nplt.ylabel(\"Peak amplitude\")\nplt.title(\"Single-sided amplitude spectrum\")\nplt.grid(alpha=0.3)\nplt.tight_layout()\nplt.show()<\/code><\/pre>\n\n\n\n
            \u56e0\u4e3a\u603b\u65f6\u957f\u4e3a 2 \u79d2\uff0c\u6240\u4ee5 \u0394f=1\/T=0.5 Hz\u3002\u8f93\u51fa\u5e94\u5728 50 Hz \u4e0e 120 Hz \u9644\u8fd1\u51fa\u73b0\u4e3b\u8981\u5cf0\u3002\u566a\u58f0\u4f7f\u5176\u4ed6\u9891\u70b9\u4e0d\u4e3a\u96f6\uff1b\u82e5\u6bcf\u6b21\u66f4\u6362\u968f\u673a\u79cd\u5b50\uff0c\u566a\u58f0\u5e95\u4e5f\u4f1a\u6539\u53d8\uff0c\u800c\u4e24\u4e2a\u786e\u5b9a\u6027\u6b63\u5f26\u5cf0\u4ecd\u7a33\u5b9a\u5b58\u5728\u3002<\/p>\n\n\n\n
            \u4e8c\u7ef4 FFT\uff1a\u56fe\u50cf\u9891\u57df\u7684\u6700\u5c0f\u793a\u4f8b<\/h3>\n\n\n\n
            import numpy as np\nimport matplotlib.pyplot as plt\nfrom PIL import Image\n\nimg = np.asarray(Image.open(\"input.png\").convert(\"L\"), dtype=float)\nF = np.fft.fft2(img)          # \u4e8c\u7ef4 DFT\nFc = np.fft.fftshift(F)       # \u628a\u96f6\u9891\u79fb\u5230\u56fe\u50cf\u4e2d\u592e\nspectrum = np.log1p(np.abs(Fc))\n\nplt.imshow(spectrum, cmap=\"gray\")\nplt.title(\"Log magnitude spectrum\")\nplt.axis(\"off\")\nplt.show()<\/code><\/pre>\n\n\n\n
            \u4e8c\u7ef4\u9891\u8c31\u4e2d\u5fc3\u9644\u8fd1\u4ee3\u8868\u7f13\u6162\u53d8\u5316\u7684\u4eae\u5ea6\u4e0e\u5927\u5c3a\u5ea6\u7ed3\u6784\uff0c\u79bb\u4e2d\u5fc3\u8d8a\u8fdc\u4ee3\u8868\u8d8a\u9ad8\u7684\u7a7a\u95f4\u9891\u7387\uff0c\u901a\u5e38\u5bf9\u5e94\u7ec6\u8282\u3001\u7eb9\u7406\u548c\u8fb9\u7f18\u3002\u56fe\u50cf\u67d0\u65b9\u5411\u4e0a\u7684\u6761\u7eb9\uff0c\u4f1a\u5728\u9891\u57df\u4e2d\u8868\u73b0\u4e3a\u4e0e\u5176\u65b9\u5411\u5782\u76f4\u7684\u6210\u5bf9\u4eae\u70b9\u3002\u663e\u793a\u65f6\u4f7f\u7528 log(1+|F|)\uff0c\u662f\u56e0\u4e3a\u76f4\u6d41\u548c\u4f4e\u9891\u5e38\u6bd4\u9ad8\u9891\u5f3a\u5f88\u591a\uff0c\u76f4\u63a5\u7ebf\u6027\u663e\u793a\u4f1a\u8ba9\u5f31\u6210\u5206\u51e0\u4e4e\u5168\u9ed1\u3002<\/p>\n\n\n\n
            \u5341\u4e09\u3001\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362\uff1a\u5f53\u9891\u7387\u968f\u65f6\u95f4\u53d8\u5316<\/h2>\n\n\n\n
            \u666e\u901a\u5085\u91cc\u53f6\u53d8\u6362\u544a\u8bc9\u6211\u4eec\u6574\u6bb5\u4fe1\u53f7\u201c\u603b\u4f53\u542b\u6709\u54ea\u4e9b\u9891\u7387\u201d\uff0c\u5374\u4e0d\u544a\u8bc9\u6211\u4eec\u8fd9\u4e9b\u9891\u7387\u5728\u4ec0\u4e48\u65f6\u5019\u51fa\u73b0\u3002\u5bf9\u7a33\u5b9a\u7684\u7535\u673a\u8f6c\u901f\u4fe1\u53f7\u8fd9\u53ef\u80fd\u591f\u7528\uff0c\u4f46\u8bed\u97f3\u3001\u97f3\u4e50\u3001\u9e1f\u9e23\u3001\u53d8\u901f\u673a\u68b0\u548c\u77ac\u6001\u6545\u969c\u90fd\u662f\u975e\u5e73\u7a33\u4fe1\u53f7\u3002\u89e3\u51b3\u65b9\u6cd5\u4e4b\u4e00\u662f\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362\uff08STFT\uff09<\/strong>\uff1a\u7528\u4e00\u4e2a\u77ed\u7a97\u6cbf\u65f6\u95f4\u8f74\u6ed1\u52a8\uff0c\u5bf9\u6bcf\u4e2a\u5c40\u90e8\u7247\u6bb5\u505a FFT\u3002<\/p>\n\n\n\n
            STFT{x}(\u03c4,f) = \u222b x(t)w(t\u2212\u03c4)e\u2212j2\u03c0ft<\/sup>dt<\/strong><\/p><\/div><\/div>\n\n\n\n
            \u628a\u6bcf\u4e2a\u65f6\u95f4\u7a97\u7684\u9891\u8c31\u6309\u65f6\u95f4\u6392\u5217\uff0c\u5c31\u5f97\u5230\u58f0\u8c31\u56fe\uff08spectrogram\uff09<\/strong>\uff1a\u6a2a\u8f74\u65f6\u95f4\u3001\u7eb5\u8f74\u9891\u7387\u3001\u989c\u8272\u8868\u793a\u5f3a\u5ea6\u3002\u7a97\u8d8a\u77ed\uff0c\u8d8a\u80fd\u5b9a\u4f4d\u77ac\u95f4\u53d8\u5316\uff0c\u4f46\u9891\u7387\u5206\u8fa8\u7387\u8d8a\u5dee\uff1b\u7a97\u8d8a\u957f\uff0c\u9891\u7387\u66f4\u7cbe\u7ec6\uff0c\u4f46\u4e8b\u4ef6\u53d1\u751f\u65f6\u523b\u66f4\u6a21\u7cca\u3002\u8fd9\u662f\u65f6\u9891\u4e0d\u786e\u5b9a\u6027\u7684\u4e00\u79cd\u8868\u73b0\uff0c\u4e0d\u5b58\u5728\u540c\u65f6\u65e0\u9650\u7cbe\u786e\u7684\u65f6\u95f4\u4e0e\u9891\u7387\u5206\u8fa8\u7387\u3002<\/p>\n\n\n\n
            \u5341\u56db\u3001\u5085\u91cc\u53f6\u53d8\u6362\u5728\u73b0\u5b9e\u4e2d\u600e\u6837\u53d1\u6325\u4f5c\u7528\uff1f<\/h2>\n\n\n\n
            1. \u97f3\u9891\u4e0e\u8bed\u97f3<\/h3>\n\n\n\n
            \u5747\u8861\u5668\u5728\u4e0d\u540c\u9891\u5e26\u4e0a\u8c03\u6574\u589e\u76ca\uff1b\u964d\u566a\u7b97\u6cd5\u4f30\u8ba1\u566a\u58f0\u9891\u8c31\u5e76\u6291\u5236\u5b83\uff1b\u97f3\u9ad8\u68c0\u6d4b\u5bfb\u627e\u57fa\u9891\u4e0e\u8c10\u6ce2\uff1bMP3\/AAC \u7b49\u611f\u77e5\u7f16\u7801\u867d\u7136\u5e38\u4f7f\u7528\u4e0e\u5085\u91cc\u53f6\u601d\u60f3\u5bc6\u5207\u76f8\u5173\u7684\u53d8\u6362\uff08\u4f8b\u5982 MDCT\uff09\uff0c\u6838\u5fc3\u4ecd\u662f\u628a\u4fe1\u53f7\u8f6c\u6362\u5230\u66f4\u6613\u6309\u542c\u89c9\u91cd\u8981\u6027\u5206\u914d\u6bd4\u7279\u7684\u57df\u3002<\/p>\n\n\n\n
            2. \u901a\u4fe1\u7cfb\u7edf<\/h3>\n\n\n\n
            \u8c03\u5236\u628a\u57fa\u5e26\u9891\u8c31\u642c\u79fb\u5230\u8f7d\u6ce2\u9644\u8fd1\uff1b\u6ee4\u6ce2\u9009\u62e9\u9700\u8981\u7684\u4fe1\u9053\uff1bOFDM \u628a\u9ad8\u901f\u6570\u636e\u5206\u914d\u5230\u8bb8\u591a\u6b63\u4ea4\u5b50\u8f7d\u6ce2\u4e0a\uff0c\u53d1\u9001\u7aef\u4f7f\u7528 IFFT \u5408\u6210\u65f6\u57df\u6ce2\u5f62\uff0c\u63a5\u6536\u7aef\u7528 FFT \u5206\u79bb\u5b50\u8f7d\u6ce2\u3002Wi\u2011Fi\u30014G\/5G \u548c\u6570\u5b57\u7535\u89c6\u4e2d\u90fd\u80fd\u770b\u5230\u8fd9\u79cd\u601d\u60f3\u3002<\/p>\n\n\n\n
            3. \u56fe\u50cf\u5904\u7406<\/h3>\n\n\n\n
            \u4e8c\u7ef4\u5085\u91cc\u53f6\u53d8\u6362\u628a\u56fe\u50cf\u5206\u89e3\u4e3a\u7a7a\u95f4\u9891\u7387\uff1a\u4f4e\u9891\u5bf9\u5e94\u7f13\u6162\u53d8\u5316\u7684\u533a\u57df\uff0c\u9ad8\u9891\u5bf9\u5e94\u8fb9\u7f18\u548c\u7ec6\u7eb9\u7406\u3002\u53bb\u9664\u5468\u671f\u6027\u6761\u7eb9\u3001\u9891\u57df\u6ee4\u6ce2\u3001\u6a21\u7cca\u6838\u5206\u6790\u3001\u56fe\u50cf\u914d\u51c6\u548c\u5149\u5b66\u884d\u5c04\u90fd\u79bb\u4e0d\u5f00\u5b83\u3002\u9700\u8981\u6ce8\u610f\uff0c\u7b80\u5355\u7c97\u66b4\u5730\u5207\u6389\u9ad8\u9891\u4f1a\u4ea7\u751f\u632f\u94c3\uff0c\u56e0\u4e3a\u9891\u57df\u786c\u622a\u65ad\u76f8\u5f53\u4e8e\u5728\u7a7a\u95f4\u57df\u4e0e\u957f\u5c3e sinc \u6838\u5377\u79ef\u3002<\/p>\n\n\n\n
            4. \u673a\u68b0\u632f\u52a8\u4e0e\u6545\u969c\u8bca\u65ad<\/h3>\n\n\n\n
            \u65cb\u8f6c\u673a\u68b0\u6545\u969c\u5e38\u5728\u8f6c\u9891\u3001\u500d\u9891\u3001\u9f7f\u8f6e\u556e\u5408\u9891\u7387\u6216\u8f74\u627f\u7279\u5f81\u9891\u7387\u9644\u8fd1\u7559\u4e0b\u5cf0\u503c\u53ca\u8fb9\u5e26\u3002\u4f8b\u5982\u4e0d\u5e73\u8861\u5e38\u589e\u5f3a 1\u00d7 \u8f6c\u9891\uff0c\u5bf9\u4e2d\u4e0d\u826f\u53ef\u80fd\u589e\u5f3a 2\u00d7 \u5206\u91cf\uff0c\u5468\u671f\u51b2\u51fb\u4f1a\u6fc0\u53d1\u5bbd\u5e26\u5171\u632f\u3002\u4f46\u771f\u5b9e\u8bca\u65ad\u4e0d\u80fd\u53ea\u51ed\u201c\u4e00\u6839\u5cf0\u201d\u5224\u6545\u969c\uff0c\u8fd8\u8981\u7ed3\u5408\u5de5\u51b5\u3001\u4f20\u611f\u5668\u65b9\u5411\u3001\u8f6c\u901f\u540c\u6b65\u3001\u5305\u7edc\u89e3\u8c03\u3001\u8d8b\u52bf\u6570\u636e\u4e0e\u8bbe\u5907\u7ed3\u6784\u3002<\/p>\n\n\n\n
            5. \u533b\u5b66\u4e0e\u79d1\u5b66\u8ba1\u7b97<\/h3>\n\n\n\n
            MRI \u91c7\u96c6\u7684\u6570\u636e\u4f4d\u4e8e\u6240\u8c13 k \u7a7a\u95f4\uff0c\u7ecf\u8fc7\u5085\u91cc\u53f6\u91cd\u5efa\u5f97\u5230\u56fe\u50cf\uff1bX \u5c04\u7ebf CT \u7684\u91cd\u5efa\u4e5f\u4e0e\u5085\u91cc\u53f6\u5207\u7247\u601d\u60f3\u5bc6\u5207\u76f8\u5173\u3002\u5728\u504f\u5fae\u5206\u65b9\u7a0b\u4e2d\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u53ef\u628a\u7a7a\u95f4\u5fae\u5206\u53d8\u6210\u9891\u57df\u4e58\u6cd5\uff0c\u4f8b\u5982\u4e8c\u9636\u5bfc\u6570\u5bf9\u5e94 \u2212(2\u03c0f)\u00b2\uff0c\u4ece\u800c\u628a\u67d0\u4e9b\u5fae\u5206\u65b9\u7a0b\u8f6c\u5316\u4e3a\u66f4\u5bb9\u6613\u6c42\u89e3\u7684\u4ee3\u6570\u65b9\u7a0b\u3002<\/p>\n\n\n\n
            \u5341\u4e94\u3001\u51e0\u4e2a\u7ecf\u5178\u53d8\u6362\u5bf9\u5e26\u6765\u7684\u76f4\u89c9<\/h2>\n\n\n\n
            \u65f6\u57df\u5bf9\u8c61<\/th>\u9891\u57df\u5f62\u6001<\/th>\u542f\u793a<\/th><\/tr><\/thead>
            \u65e0\u9650\u957f\u5355\u9891\u6b63\u5f26<\/td>\u00b1f\u2080 \u5904\u7684\u51b2\u6fc0<\/td>\u65f6\u95f4\u4e0a\u65e0\u9650\u957f\uff0c\u9891\u7387\u53ef\u65e0\u9650\u7cbe\u786e<\/td><\/tr>
            \u5f88\u77ed\u7684\u8109\u51b2<\/td>\u5f88\u5bbd\u7684\u9891\u8c31<\/td>\u53d8\u5316\u8d8a\u5feb\uff0c\u9700\u8981\u7684\u9ad8\u9891\u8d8a\u591a<\/td><\/tr>
            \u77e9\u5f62\u8109\u51b2<\/td>sinc \u5f62\u9891\u8c31<\/td>\u65f6\u95f4\u786c\u622a\u65ad\u4ea7\u751f\u9891\u57df\u65c1\u74e3<\/td><\/tr>
            \u9ad8\u65af\u51fd\u6570<\/td>\u4ecd\u662f\u9ad8\u65af\u51fd\u6570<\/td>\u65f6\u9891\u96c6\u4e2d\u6027\u8d28\u975e\u5e38\u597d<\/td><\/tr>
            \u51b2\u6fc0\u5e8f\u5217<\/td>\u53e6\u4e00\u4e2a\u51b2\u6fc0\u5e8f\u5217<\/td>\u63ed\u793a\u91c7\u6837\u9020\u6210\u7684\u9891\u8c31\u5468\u671f\u590d\u5236<\/td><\/tr>
            \u5e38\u6570<\/td>\u96f6\u9891\u5904\u7684\u51b2\u6fc0<\/td>\u4e0d\u968f\u65f6\u95f4\u53d8\u5316\u5c31\u662f\u7eaf\u76f4\u6d41<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
            \u8fd9\u91cc\u5b58\u5728\u4e00\u6761\u8d2f\u7a7f\u5168\u6587\u7684\u89c4\u5f8b\uff1a\u4e00\u4e2a\u57df\u91cc\u8d8a\u96c6\u4e2d\uff0c\u53e6\u4e00\u4e2a\u57df\u91cc\u901a\u5e38\u8d8a\u5206\u6563\u3002<\/strong>\u6781\u77ed\u8109\u51b2\u9700\u8981\u5bbd\u9891\u5e26\uff1b\u6781\u7a84\u9891\u5e26\u4fe1\u53f7\u5fc5\u987b\u6301\u7eed\u8f83\u957f\u65f6\u95f4\u3002\u901a\u4fe1\u8109\u51b2\u6574\u5f62\u3001\u96f7\u8fbe\u8ddd\u79bb\u5206\u8fa8\u7387\u3001\u77ed\u65f6\u9891\u8c31\u5206\u6790\u548c\u6ee4\u6ce2\u5668\u8bbe\u8ba1\uff0c\u90fd\u5728\u4e0e\u8fd9\u79cd\u65f6\u9891\u6743\u8861\u6253\u4ea4\u9053\u3002<\/p>\n\n\n\n
            \u5341\u516d\u3001\u5e38\u89c1\u8bef\u533a\u4e0e\u6392\u67e5\u6e05\u5355<\/h2>\n\n\n\n
            • \u8bef\u533a\uff1aFFT \u70b9\u6570\u8d8a\u591a\uff0c\u4efb\u4f55\u60c5\u51b5\u4e0b\u5206\u8fa8\u7387\u90fd\u8d8a\u9ad8\u3002<\/strong>
              \u82e5\u53ea\u662f\u96f6\u586b\u5145\uff0c\u56fe\u66f4\u5e73\u6ed1\u4f46\u771f\u5b9e\u5206\u8fa8\u80fd\u529b\u4e0d\u53d8\uff1b\u589e\u52a0\u5b9e\u9645\u91c7\u6837\u65f6\u957f\u624d\u589e\u52a0\u4fe1\u606f\u3002<\/li>
            • \u8bef\u533a\uff1a\u5948\u594e\u65af\u7279\u9891\u7387\u5c31\u662f\u53ef\u653e\u5fc3\u6d4b\u91cf\u7684\u6700\u9ad8\u9891\u7387\u3002<\/strong>
              \u5b9e\u9645\u6297\u6df7\u53e0\u6ee4\u6ce2\u5668\u9700\u8981\u8fc7\u6e21\u5e26\uff0c\u76ee\u6807\u4fe1\u53f7\u5e26\u5bbd\u901a\u5e38\u5e94\u660e\u663e\u4f4e\u4e8e fs<\/sub>\/2\u3002<\/li>
            • \u8bef\u533a\uff1a\u9891\u8c31\u5cf0\u9ad8\u5ea6\u5c31\u662f\u539f\u59cb\u6b63\u5f26\u5e45\u503c\u3002<\/strong>
              \u5b83\u8fd8\u53d6\u51b3\u4e8e N\u3001\u5355\u53cc\u8fb9\u8c31\u3001FFT \u5f52\u4e00\u5316\u3001\u7a97\u51fd\u6570\u3001\u5cf0\u503c\/RMS\u5b9a\u4e49\u4ee5\u53ca\u9891\u7387\u662f\u5426\u843d\u5728\u683c\u70b9\u4e0a\u3002<\/li>
            • \u8bef\u533a\uff1a\u52a0\u7a97\u603b\u4f1a\u8ba9\u9891\u8c31\u66f4\u51c6\u786e\u3002<\/strong>
              \u7a97\u662f\u5728\u65c1\u74e3\u3001\u4e3b\u74e3\u5bbd\u5ea6\u548c\u5e45\u503c\u8bef\u5dee\u4e4b\u95f4\u4ea4\u6362\uff0c\u4e0d\u540c\u76ee\u7684\u5e94\u9009\u4e0d\u540c\u7a97\u3002<\/li>
            • \u8bef\u533a\uff1a\u5220\u9664\u9891\u8c31\u4e2d\u7684\u9ad8\u9891\u5c31\u4e00\u5b9a\u80fd\u5e72\u51c0\u964d\u566a\u3002<\/strong>
              \u4fe1\u53f7\u672c\u8eab\u7684\u8fb9\u7f18\u548c\u77ac\u6001\u4e5f\u9700\u8981\u9ad8\u9891\uff0c\u786c\u5207\u9891\u8c31\u8fd8\u53ef\u80fd\u5f15\u5165\u65f6\u57df\u632f\u94c3\u3002<\/li>
            • \u8bef\u533a\uff1a\u5e45\u5ea6\u8c31\u76f8\u540c\uff0c\u4fe1\u53f7\u5c31\u76f8\u540c\u3002<\/strong>
              \u4e22\u5f03\u76f8\u4f4d\u540e\u65e0\u6cd5\u4e00\u822c\u6027\u91cd\u5efa\u539f\u4fe1\u53f7\u3002<\/li>
            • \u8bef\u533a\uff1a\u4e00\u4e2a\u5cf0\u5c31\u5bf9\u5e94\u4e00\u4e2a\u7269\u7406\u539f\u56e0\u3002<\/strong>
              \u8c10\u6ce2\u3001\u8c03\u5236\u8fb9\u5e26\u3001\u7ed3\u6784\u5171\u632f\u3001\u7a97\u6cc4\u6f0f\u3001\u6df7\u53e0\u3001\u7535\u6e90\u5e72\u6270\u548c\u975e\u7ebf\u6027\u90fd\u53ef\u80fd\u4ea7\u751f\u5cf0\uff0c\u5fc5\u987b\u7ed3\u5408\u7cfb\u7edf\u77e5\u8bc6\u9a8c\u8bc1\u3002<\/li>
            • \u8bef\u533a\uff1aFFT \u9002\u5408\u6240\u6709\u975e\u5e73\u7a33\u4fe1\u53f7\u3002<\/strong>
              \u9891\u7387\u968f\u65f6\u95f4\u660e\u663e\u53d8\u5316\u65f6\uff0c\u5e94\u8003\u8651 STFT\u3001\u5c0f\u6ce2\u53d8\u6362\u3001\u540c\u6b65\u5206\u6790\u6216\u5176\u4ed6\u65f6\u9891\u65b9\u6cd5\u3002<\/li><\/ul>\n\n\n\n
              \u5341\u4e03\u3001\u8fdb\u4e00\u6b65\u7406\u89e3\uff1a\u5085\u91cc\u53f6\u53d8\u6362\u5176\u5b9e\u662f\u4e00\u79cd\u201c\u6362\u57fa\u201d<\/h2>\n\n\n\n
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                    \n\n\n\n
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