{"id":1170566,"date":"2025-01-15T16:24:08","date_gmt":"2025-01-15T08:24:08","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/1170566.html"},"modified":"2025-01-15T16:24:12","modified_gmt":"2025-01-15T08:24:12","slug":"python%e5%a6%82%e4%bd%95%e5%81%9a%e5%82%85%e9%87%8c%e5%8f%b6%e5%8f%98%e6%8d%a2","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/1170566.html","title":{"rendered":"python\u5982\u4f55\u505a\u5085\u91cc\u53f6\u53d8\u6362"},"content":{"rendered":"

\"python\u5982\u4f55\u505a\u5085\u91cc\u53f6\u53d8\u6362\"<\/p>\n

\u5728Python\u4e2d\uff0c\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u4e3b\u8981\u4f9d\u8d56\u4e8eNumPy\u5e93\u4e2d\u7684fft<\/code>\uff08\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff09\u6a21\u5757\u3002\u4f7f\u7528NumPy\u5e93\u7684numpy.fft<\/code>\u6a21\u5757\u3001SciPy\u5e93\u7684scipy.fftpack<\/code>\u6a21\u5757\u3001Matplotlib\u5e93\u7684\u53ef\u89c6\u5316\u529f\u80fd<\/strong>\uff0c\u90fd\u53ef\u4ee5\u8f7b\u677e\u5730\u5b9e\u73b0\u5085\u91cc\u53f6\u53d8\u6362\u3002\u4e0b\u9762\u8be6\u7ec6\u4ecb\u7ecd\u5176\u4e2d\u4e00\u4e2a\u70b9\uff0c\u5373NumPy\u5e93\u7684numpy.fft<\/code>\u6a21\u5757\u3002<\/p>\n<\/p>\n

NumPy\u5e93\u7684numpy.fft<\/code>\u6a21\u5757<\/strong>\uff1a<\/p>\n<\/p>\n

NumPy\u662fPython\u4e2d\u6700\u5e38\u7528\u7684\u79d1\u5b66\u8ba1\u7b97\u5e93\u4e4b\u4e00\uff0c\u63d0\u4f9b\u4e86\u4e00\u4e2a\u540d\u4e3anumpy.fft<\/code>\u7684\u6a21\u5757\uff0c\u7528\u4e8e\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u3002\u4f60\u53ef\u4ee5\u4f7f\u7528numpy.fft.fft<\/code>\u51fd\u6570\u5bf9\u4e00\u7ef4\u6570\u7ec4\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u4f7f\u7528numpy.fft.ifft<\/code>\u51fd\u6570\u8fdb\u884c\u9006\u5085\u91cc\u53f6\u53d8\u6362\u3002\u4ee5\u4e0b\u662f\u8be6\u7ec6\u7684\u6b65\u9aa4\u548c\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n<\/p>\n

\u6b65\u9aa4<\/strong>\uff1a<\/p>\n<\/p>\n

    \n
  1. \u5bfc\u5165\u5fc5\u8981\u7684\u5e93\u3002<\/li>\n
  2. \u751f\u6210\u6216\u52a0\u8f7d\u4e00\u4e2a\u4fe1\u53f7\u3002<\/li>\n
  3. \u4f7f\u7528numpy.fft.fft<\/code>\u51fd\u6570\u5bf9\u4fe1\u53f7\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/li>\n
  4. \u4f7f\u7528numpy.fft.ifft<\/code>\u51fd\u6570\u5bf9\u5085\u91cc\u53f6\u53d8\u6362\u7684\u7ed3\u679c\u8fdb\u884c\u9006\u53d8\u6362\uff0c\u4ee5\u9a8c\u8bc1\u7ed3\u679c\u3002<\/li>\n
  5. \u53ef\u89c6\u5316\u4fe1\u53f7\u548c\u9891\u8c31\u3002<\/li>\n<\/ol>\n

    \u793a\u4f8b\u4ee3\u7801<\/strong>\uff1a<\/p>\n<\/p>\n

    import numpy as np<\/p>\n
    import matplotlib.pyplot as plt<\/p>\n
    1. \u751f\u6210\u4e00\u4e2a\u4fe1\u53f7<\/strong><\/h2>\n
    sampling_rate = 1000  # \u91c7\u6837\u7387<\/p>\n
    T = 1.0 \/ sampling_rate  # \u91c7\u6837\u95f4\u9694<\/p>\n
    x = np.linspace(0.0, 1.0, sampling_rate)<\/p>\n
    y = np.sin(50.0 * 2.0 * np.pi * x) + 0.5 * np.sin(80.0 * 2.0 * np.pi * x)<\/p>\n
    2. \u4f7f\u7528 numpy.fft.fft \u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362<\/strong><\/h2>\n
    yf = np.fft.fft(y)<\/p>\n
    xf = np.fft.fftfreq(sampling_rate, T)<\/p>\n
    3. \u53ef\u89c6\u5316\u4fe1\u53f7\u548c\u9891\u8c31<\/strong><\/h2>\n
    plt.figure(figsize=(12, 6))<\/p>\n
    plt.subplot(2, 1, 1)<\/p>\n
    plt.plot(x, y)<\/p>\n
    plt.title('Original Signal')<\/p>\n
    plt.xlabel('Time (s)')<\/p>\n
    plt.ylabel('Amplitude')<\/p>\n
    plt.subplot(2, 1, 2)<\/p>\n
    plt.plot(xf, np.abs(yf))<\/p>\n
    plt.title('Fourier Transform')<\/p>\n
    plt.xlabel('Frequency (Hz)')<\/p>\n
    plt.ylabel('Amplitude')<\/p>\n
    plt.tight_layout()<\/p>\n
    plt.show()<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u8be6\u7ec6\u63cf\u8ff0<\/strong>\uff1a<\/p>\n<\/p>\n
    \u5728\u4e0a\u9762\u7684\u4ee3\u7801\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u5bfc\u5165\u4e86NumPy\u548cMatplotlib\u5e93\u3002\u7136\u540e\u751f\u6210\u4e86\u4e00\u4e2a\u7531\u4e24\u4e2a\u4e0d\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\u7ec4\u6210\u7684\u4fe1\u53f7\u3002\u63a5\u7740\u4f7f\u7528numpy.fft.fft<\/code>\u51fd\u6570\u5bf9\u4fe1\u53f7\u8fdb\u884c\u4e86\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u5e76\u4f7f\u7528numpy.fft.fftfreq<\/code>\u51fd\u6570\u751f\u6210\u76f8\u5e94\u7684\u9891\u7387\u8f74\u3002\u6700\u540e\uff0c\u6211\u4eec\u4f7f\u7528Matplotlib\u5e93\u5c06\u539f\u59cb\u4fe1\u53f7\u548c\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u9891\u8c31\u8fdb\u884c\u4e86\u53ef\u89c6\u5316\u3002<\/p>\n<\/p>\n
    \u4e00\u3001NUMPY\u5e93\u7684numpy.fft<\/code>\u6a21\u5757<\/h3>\n<\/p>\n
    1.1 \u751f\u6210\u548c\u52a0\u8f7d\u4fe1\u53f7<\/h4>\n<\/p>\n
    \u5728\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u4e4b\u524d\uff0c\u9996\u5148\u9700\u8981\u751f\u6210\u6216\u52a0\u8f7d\u4e00\u4e2a\u4fe1\u53f7\u3002\u4fe1\u53f7\u53ef\u4ee5\u662f\u4efb\u4f55\u5f62\u5f0f\u7684\u65f6\u95f4\u5e8f\u5217\u6570\u636e\uff0c\u4f8b\u5982\u97f3\u9891\u4fe1\u53f7\u3001\u56fe\u50cf\u4fe1\u53f7\u7b49\u3002\u5728\u4e0a\u9762\u7684\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u751f\u6210\u4e86\u4e00\u4e2a\u7531\u4e24\u4e2a\u4e0d\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\u7ec4\u6210\u7684\u4fe1\u53f7\u3002<\/p>\n<\/p>\n
    sampling_rate = 1000  # \u91c7\u6837\u7387<\/p>\n
    T = 1.0 \/ sampling_rate  # \u91c7\u6837\u95f4\u9694<\/p>\n
    x = np.linspace(0.0, 1.0, sampling_rate)<\/p>\n
    y = np.sin(50.0 * 2.0 * np.pi * x) + 0.5 * np.sin(80.0 * 2.0 * np.pi * x)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0csampling_rate<\/code>\u8868\u793a\u91c7\u6837\u7387\uff0cT<\/code>\u8868\u793a\u91c7\u6837\u95f4\u9694\uff0cx<\/code>\u8868\u793a\u65f6\u95f4\u8f74\uff0cy<\/code>\u8868\u793a\u4fe1\u53f7\u3002\u6211\u4eec\u4f7f\u7528np.linspace<\/code>\u51fd\u6570\u751f\u6210\u65f6\u95f4\u8f74\uff0c\u5e76\u4f7f\u7528np.sin<\/code>\u51fd\u6570\u751f\u6210\u6b63\u5f26\u6ce2\u4fe1\u53f7\u3002<\/p>\n<\/p>\n
    1.2 \u4f7f\u7528numpy.fft.fft<\/code>\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362<\/h4>\n<\/p>\n
    \u751f\u6210\u4fe1\u53f7\u540e\uff0c\u4f7f\u7528numpy.fft.fft<\/code>\u51fd\u6570\u5bf9\u4fe1\u53f7\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/p>\n<\/p>\n
    yf = np.fft.fft(y)<\/p>\n
    xf = np.fft.fftfreq(sampling_rate, T)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0cyf<\/code>\u8868\u793a\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u7ed3\u679c\uff0cxf<\/code>\u8868\u793a\u9891\u7387\u8f74\u3002numpy.fft.fft<\/code>\u51fd\u6570\u5bf9\u4fe1\u53f7\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\uff0cnumpy.fft.fftfreq<\/code>\u51fd\u6570\u751f\u6210\u9891\u7387\u8f74\u3002<\/p>\n<\/p>\n
    1.3 \u53ef\u89c6\u5316\u4fe1\u53f7\u548c\u9891\u8c31<\/h4>\n<\/p>\n
    \u4e3a\u4e86\u66f4\u597d\u5730\u7406\u89e3\u5085\u91cc\u53f6\u53d8\u6362\u7684\u7ed3\u679c\uff0c\u53ef\u4ee5\u4f7f\u7528Matplotlib\u5e93\u5c06\u539f\u59cb\u4fe1\u53f7\u548c\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u9891\u8c31\u8fdb\u884c\u53ef\u89c6\u5316\u3002<\/p>\n<\/p>\n
    plt.figure(figsize=(12, 6))<\/p>\n
    plt.subplot(2, 1, 1)<\/p>\n
    plt.plot(x, y)<\/p>\n
    plt.title('Original Signal')<\/p>\n
    plt.xlabel('Time (s)')<\/p>\n
    plt.ylabel('Amplitude')<\/p>\n
    plt.subplot(2, 1, 2)<\/p>\n
    plt.plot(xf, np.abs(yf))<\/p>\n
    plt.title('Fourier Transform')<\/p>\n
    plt.xlabel('Frequency (Hz)')<\/p>\n
    plt.ylabel('Amplitude')<\/p>\n
    plt.tight_layout()<\/p>\n
    plt.show()<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521b\u5efa\u4e86\u4e00\u4e2a\u5305\u542b\u4e24\u4e2a\u5b50\u56fe\u7684\u56fe\u5f62\u3002\u7b2c\u4e00\u4e2a\u5b50\u56fe\u663e\u793a\u539f\u59cb\u4fe1\u53f7\uff0c\u7b2c\u4e8c\u4e2a\u5b50\u56fe\u663e\u793a\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u9891\u8c31\u3002\u4f7f\u7528plt.plot<\/code>\u51fd\u6570\u7ed8\u5236\u56fe\u5f62\uff0c\u4f7f\u7528plt.title<\/code>\u3001plt.xlabel<\/code>\u548cplt.ylabel<\/code>\u51fd\u6570\u8bbe\u7f6e\u56fe\u5f62\u7684\u6807\u9898\u548c\u6807\u7b7e\u3002<\/p>\n<\/p>\n
    \u4e8c\u3001SCIPY\u5e93\u7684scipy.fftpack<\/code>\u6a21\u5757<\/h3>\n<\/p>\n
    SciPy\u5e93\u662f\u4e00\u4e2a\u57fa\u4e8eNumPy\u7684\u79d1\u5b66\u8ba1\u7b97\u5e93\uff0c\u63d0\u4f9b\u4e86\u66f4\u591a\u9ad8\u7ea7\u7684\u6570\u5b66\u3001\u79d1\u5b66\u548c\u5de5\u7a0b\u529f\u80fd\u3002SciPy\u5e93\u4e2d\u7684scipy.fftpack<\/code>\u6a21\u5757\u4e5f\u63d0\u4f9b\u4e86\u5085\u91cc\u53f6\u53d8\u6362\u7684\u529f\u80fd\u3002<\/p>\n<\/p>\n
    2.1 \u5bfc\u5165\u5fc5\u8981\u7684\u5e93<\/h4>\n<\/p>\n
    \u9996\u5148\uff0c\u5bfc\u5165SciPy\u5e93\u548cMatplotlib\u5e93\u3002<\/p>\n<\/p>\n
    from scipy.fftpack import fft, ifft<\/p>\n
    import numpy as np<\/p>\n
    import matplotlib.pyplot as plt<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    2.2 \u751f\u6210\u548c\u52a0\u8f7d\u4fe1\u53f7<\/h4>\n<\/p>\n
    \u4e0eNumPy\u5e93\u7c7b\u4f3c\uff0c\u9996\u5148\u9700\u8981\u751f\u6210\u6216\u52a0\u8f7d\u4e00\u4e2a\u4fe1\u53f7\u3002<\/p>\n<\/p>\n
    sampling_rate = 1000  # \u91c7\u6837\u7387<\/p>\n
    T = 1.0 \/ sampling_rate  # \u91c7\u6837\u95f4\u9694<\/p>\n
    x = np.linspace(0.0, 1.0, sampling_rate)<\/p>\n
    y = np.sin(50.0 * 2.0 * np.pi * x) + 0.5 * np.sin(80.0 * 2.0 * np.pi * x)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    2.3 \u4f7f\u7528scipy.fftpack.fft<\/code>\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362<\/h4>\n<\/p>\n
    \u751f\u6210\u4fe1\u53f7\u540e\uff0c\u4f7f\u7528scipy.fftpack.fft<\/code>\u51fd\u6570\u5bf9\u4fe1\u53f7\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/p>\n<\/p>\n
    yf = fft(y)<\/p>\n
    xf = np.linspace(0.0, 1.0\/(2.0*T), sampling_rate\/\/2)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0cyf<\/code>\u8868\u793a\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u7ed3\u679c\uff0cxf<\/code>\u8868\u793a\u9891\u7387\u8f74\u3002scipy.fftpack.fft<\/code>\u51fd\u6570\u5bf9\u4fe1\u53f7\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\uff0cnp.linspace<\/code>\u51fd\u6570\u751f\u6210\u9891\u7387\u8f74\u3002<\/p>\n<\/p>\n
    2.4 \u53ef\u89c6\u5316\u4fe1\u53f7\u548c\u9891\u8c31<\/h4>\n<\/p>\n
    \u4e3a\u4e86\u66f4\u597d\u5730\u7406\u89e3\u5085\u91cc\u53f6\u53d8\u6362\u7684\u7ed3\u679c\uff0c\u53ef\u4ee5\u4f7f\u7528Matplotlib\u5e93\u5c06\u539f\u59cb\u4fe1\u53f7\u548c\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u9891\u8c31\u8fdb\u884c\u53ef\u89c6\u5316\u3002<\/p>\n<\/p>\n
    plt.figure(figsize=(12, 6))<\/p>\n
    plt.subplot(2, 1, 1)<\/p>\n
    plt.plot(x, y)<\/p>\n
    plt.title('Original Signal')<\/p>\n
    plt.xlabel('Time (s)')<\/p>\n
    plt.ylabel('Amplitude')<\/p>\n
    plt.subplot(2, 1, 2)<\/p>\n
    plt.plot(xf, 2.0\/sampling_rate * np.abs(yf[:sampling_rate\/\/2]))<\/p>\n
    plt.title('Fourier Transform')<\/p>\n
    plt.xlabel('Frequency (Hz)')<\/p>\n
    plt.ylabel('Amplitude')<\/p>\n
    plt.tight_layout()<\/p>\n
    plt.show()<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521b\u5efa\u4e86\u4e00\u4e2a\u5305\u542b\u4e24\u4e2a\u5b50\u56fe\u7684\u56fe\u5f62\u3002\u7b2c\u4e00\u4e2a\u5b50\u56fe\u663e\u793a\u539f\u59cb\u4fe1\u53f7\uff0c\u7b2c\u4e8c\u4e2a\u5b50\u56fe\u663e\u793a\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u9891\u8c31\u3002\u4f7f\u7528plt.plot<\/code>\u51fd\u6570\u7ed8\u5236\u56fe\u5f62\uff0c\u4f7f\u7528plt.title<\/code>\u3001plt.xlabel<\/code>\u548cplt.ylabel<\/code>\u51fd\u6570\u8bbe\u7f6e\u56fe\u5f62\u7684\u6807\u9898\u548c\u6807\u7b7e\u3002<\/p>\n<\/p>\n
    \u4e09\u3001\u5085\u91cc\u53f6\u53d8\u6362\u7684\u57fa\u672c\u7406\u8bba<\/h3>\n<\/p>\n
    \u5085\u91cc\u53f6\u53d8\u6362\u662f\u5c06\u65f6\u95f4\u57df\u4fe1\u53f7\u8f6c\u6362\u4e3a\u9891\u7387\u57df\u4fe1\u53f7\u7684\u6570\u5b66\u5de5\u5177\u3002\u5b83\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u590d\u6742\u7684\u4fe1\u53f7\u5206\u89e3\u4e3a\u4e0d\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\u548c\u4f59\u5f26\u6ce2\u7684\u53e0\u52a0\u3002\u5085\u91cc\u53f6\u53d8\u6362\u5728\u4fe1\u53f7\u5904\u7406\u3001\u56fe\u50cf\u5904\u7406\u3001\u97f3\u9891\u5904\u7406\u7b49\u9886\u57df\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n<\/p>\n
    3.1 \u8fde\u7eed\u5085\u91cc\u53f6\u53d8\u6362<\/h4>\n<\/p>\n
    \u5bf9\u4e8e\u4e00\u4e2a\u8fde\u7eed\u7684\u65f6\u95f4\u4fe1\u53f7x(t)<\/code>\uff0c\u5176\u5085\u91cc\u53f6\u53d8\u6362\u5b9a\u4e49\u4e3a\uff1a<\/p>\n<\/p>\n
    X(f) = \u222b x(t) * e^(-j2\u03c0ft) dt<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5176\u4e2d\uff0cX(f)<\/code>\u8868\u793a\u4fe1\u53f7\u5728\u9891\u7387f<\/code>\u4e0a\u7684\u5206\u91cf\uff0cj<\/code>\u662f\u865a\u6570\u5355\u4f4d\u3002<\/p>\n<\/p>\n
    3.2 \u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362<\/h4>\n<\/p>\n
    \u5bf9\u4e8e\u4e00\u4e2a\u79bb\u6563\u7684\u65f6\u95f4\u4fe1\u53f7x[n]<\/code>\uff0c\u5176\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff08DFT\uff09\u5b9a\u4e49\u4e3a\uff1a<\/p>\n<\/p>\n
    X[k] = \u2211 x[n] * e^(-j2\u03c0kn\/N)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5176\u4e2d\uff0cX[k]<\/code>\u8868\u793a\u4fe1\u53f7\u5728\u9891\u7387k<\/code>\u4e0a\u7684\u5206\u91cf\uff0cN<\/code>\u662f\u4fe1\u53f7\u7684\u957f\u5ea6\u3002<\/p>\n<\/p>\n
    3.3 \u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362<\/h4>\n<\/p>\n
    \u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff08FFT\uff09\u662f\u4e00\u79cd\u9ad8\u6548\u8ba1\u7b97\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\u7684\u65b9\u6cd5\u3002\u5b83\u5229\u7528\u4fe1\u53f7\u7684\u5bf9\u79f0\u6027\u548c\u5468\u671f\u6027\uff0c\u5c06\u8ba1\u7b97\u590d\u6742\u5ea6\u4eceO(N^2)<\/code>\u964d\u4f4e\u5230O(N log N)<\/code>\u3002<\/p>\n<\/p>\n
    \u56db\u3001\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5e94\u7528<\/h3>\n<\/p>\n
    \u5085\u91cc\u53f6\u53d8\u6362\u5728\u8bb8\u591a\u9886\u57df\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a<\/p>\n<\/p>\n
    4.1 \u4fe1\u53f7\u5904\u7406<\/h4>\n<\/p>\n
    \u5728\u4fe1\u53f7\u5904\u7406\u9886\u57df\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u7528\u4e8e\u5206\u6790\u548c\u5904\u7406\u65f6\u95f4\u57df\u4fe1\u53f7\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u5085\u91cc\u53f6\u53d8\u6362\u6765\u5206\u6790\u97f3\u9891\u4fe1\u53f7\u7684\u9891\u8c31\uff0c\u53bb\u9664\u566a\u58f0\uff0c\u6216\u8005\u8fdb\u884c\u6ee4\u6ce2\u3002<\/p>\n<\/p>\n
    4.2 \u56fe\u50cf\u5904\u7406<\/h4>\n<\/p>\n
    \u5728\u56fe\u50cf\u5904\u7406\u9886\u57df\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u7528\u4e8e\u5206\u6790\u548c\u5904\u7406\u56fe\u50cf\u7684\u9891\u7387\u5206\u91cf\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u5085\u91cc\u53f6\u53d8\u6362\u6765\u53bb\u9664\u56fe\u50cf\u4e2d\u7684\u5468\u671f\u6027\u566a\u58f0\uff0c\u589e\u5f3a\u56fe\u50cf\u7684\u7ec6\u8282\uff0c\u6216\u8005\u8fdb\u884c\u56fe\u50cf\u538b\u7f29\u3002<\/p>\n<\/p>\n
    4.3 \u97f3\u9891\u5904\u7406<\/h4>\n<\/p>\n
    \u5728\u97f3\u9891\u5904\u7406\u9886\u57df\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u7528\u4e8e\u5206\u6790\u548c\u5904\u7406\u97f3\u9891\u4fe1\u53f7\u7684\u9891\u8c31\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u5085\u91cc\u53f6\u53d8\u6362\u6765\u5206\u6790\u97f3\u9891\u4fe1\u53f7\u7684\u9891\u7387\u6210\u5206\uff0c\u53bb\u9664\u566a\u58f0\uff0c\u6216\u8005\u8fdb\u884c\u97f3\u9891\u6548\u679c\u5904\u7406\u3002<\/p>\n<\/p>\n
    \u4e94\u3001\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5b9e\u73b0\u7ec6\u8282<\/h3>\n<\/p>\n
    \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5b9e\u73b0\u7ec6\u8282\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u4ee5\u4e0b\u662f\u4e00\u4e9b\u5e38\u89c1\u7684\u5b9e\u73b0\u7ec6\u8282\u548c\u6ce8\u610f\u4e8b\u9879\u3002<\/p>\n<\/p>\n
    5.1 \u4fe1\u53f7\u9884\u5904\u7406<\/h4>\n<\/p>\n
    \u5728\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u4e4b\u524d\uff0c\u901a\u5e38\u9700\u8981\u5bf9\u4fe1\u53f7\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u7a97\u51fd\u6570\uff08\u5982\u6c49\u5b81\u7a97\u3001\u6d77\u660e\u7a97\u7b49\uff09\u6765\u51cf\u5c11\u9891\u8c31\u6cc4\u6f0f\uff0c\u4f7f\u7528\u53bb\u5747\u503c\u6765\u51cf\u5c11\u76f4\u6d41\u5206\u91cf\u7684\u5f71\u54cd\u3002<\/p>\n<\/p>\n
    from scipy.signal import hanning<\/p>\n
    window = hanning(len(y))<\/p>\n
    y_windowed = y * window<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u6c49\u5b81\u7a97\u5bf9\u4fe1\u53f7\u8fdb\u884c\u4e86\u52a0\u7a97\u5904\u7406\u3002<\/p>\n<\/p>\n
    5.2 \u9891\u8c31\u5f52\u4e00\u5316<\/h4>\n<\/p>\n
    \u5728\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u540e\uff0c\u901a\u5e38\u9700\u8981\u5bf9\u9891\u8c31\u8fdb\u884c\u5f52\u4e00\u5316\u5904\u7406\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u4fe1\u53f7\u957f\u5ea6\u5bf9\u9891\u8c31\u8fdb\u884c\u5f52\u4e00\u5316\uff0c\u4ee5\u4fbf\u4e8e\u6bd4\u8f83\u4e0d\u540c\u957f\u5ea6\u7684\u4fe1\u53f7\u3002<\/p>\n<\/p>\n
    yf_normalized = yf \/ len(y)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4fe1\u53f7\u957f\u5ea6\u5bf9\u9891\u8c31\u8fdb\u884c\u4e86\u5f52\u4e00\u5316\u5904\u7406\u3002<\/p>\n<\/p>\n
    5.3 \u9006\u5085\u91cc\u53f6\u53d8\u6362<\/h4>\n<\/p>\n
    \u5085\u91cc\u53f6\u53d8\u6362\u7684\u9006\u53d8\u6362\u53ef\u4ee5\u5c06\u9891\u7387\u57df\u4fe1\u53f7\u8f6c\u6362\u56de\u65f6\u95f4\u57df\u4fe1\u53f7\u3002\u5728NumPy\u5e93\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528numpy.fft.ifft<\/code>\u51fd\u6570\u8fdb\u884c\u9006\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/p>\n<\/p>\n
    y_reconstructed = np.fft.ifft(yf)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0cy_reconstructed<\/code>\u8868\u793a\u9006\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u91cd\u5efa\u4fe1\u53f7\u3002<\/p>\n<\/p>\n
    \u516d\u3001\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5c40\u9650\u6027<\/h3>\n<\/p>\n
    \u5c3d\u7ba1\u5085\u91cc\u53f6\u53d8\u6362\u5728\u8bb8\u591a\u9886\u57df\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u4f46\u5b83\u4e5f\u6709\u4e00\u4e9b\u5c40\u9650\u6027\u3002<\/p>\n<\/p>\n
    6.1 \u9891\u8c31\u6cc4\u6f0f<\/h4>\n<\/p>\n
    \u9891\u8c31\u6cc4\u6f0f\u662f\u5085\u91cc\u53f6\u53d8\u6362\u7684\u4e00\u4e2a\u5e38\u89c1\u95ee\u9898\u3002\u5f53\u4fe1\u53f7\u7684\u9891\u7387\u6210\u5206\u4e0d\u662f\u6574\u6570\u500d\u7684\u57fa\u672c\u9891\u7387\u65f6\uff0c\u9891\u8c31\u4f1a\u51fa\u73b0\u6cc4\u6f0f\u73b0\u8c61\u3002\u53ef\u4ee5\u4f7f\u7528\u7a97\u51fd\u6570\u6765\u51cf\u5c11\u9891\u8c31\u6cc4\u6f0f\u3002<\/p>\n<\/p>\n
    6.2 \u65f6\u95f4\u5206\u8fa8\u7387<\/h4>\n<\/p>\n
    \u5085\u91cc\u53f6\u53d8\u6362\u5c06\u65f6\u95f4\u57df\u4fe1\u53f7\u8f6c\u6362\u4e3a\u9891\u7387\u57df\u4fe1\u53f7\uff0c\u4f46\u4e27\u5931\u4e86\u65f6\u95f4\u4fe1\u606f\u3002\u5bf9\u4e8e\u975e\u5e73\u7a33\u4fe1\u53f7\uff0c\u53ef\u4ee5\u4f7f\u7528\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362\uff08STFT\uff09\u6216\u5c0f\u6ce2\u53d8\u6362\u6765\u540c\u65f6\u5206\u6790\u4fe1\u53f7\u7684\u65f6\u95f4\u548c\u9891\u7387\u6210\u5206\u3002<\/p>\n<\/p>\n
    6.3 \u8ba1\u7b97\u590d\u6742\u5ea6<\/h4>\n<\/p>\n
    \u5c3d\u7ba1\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u5927\u5927\u964d\u4f4e\u4e86\u8ba1\u7b97\u590d\u6742\u5ea6\uff0c\u4f46\u5bf9\u4e8e\u5927\u89c4\u6a21\u6570\u636e\uff0c\u8ba1\u7b97\u4ecd\u7136\u53ef\u80fd\u975e\u5e38\u8017\u65f6\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u4f7f\u7528\u9ad8\u6548\u7684FFT\u5e93\uff08\u5982FFTW\uff09\u6216\u5e76\u884c\u8ba1\u7b97\u6765\u52a0\u901f\u8ba1\u7b97\u3002<\/p>\n<\/p>\n
    \u4e03\u3001Python\u4e2d\u7684\u5085\u91cc\u53f6\u53d8\u6362\u5e93<\/h3>\n<\/p>\n
    \u9664\u4e86NumPy\u548cSciPy\u5e93\uff0cPython\u4e2d\u8fd8\u6709\u5176\u4ed6\u4e00\u4e9b\u5085\u91cc\u53f6\u53d8\u6362\u5e93\uff0c\u53ef\u4ee5\u6839\u636e\u5177\u4f53\u9700\u6c42\u9009\u62e9\u5408\u9002\u7684\u5e93\u3002<\/p>\n<\/p>\n
    7.1 PyFFTW<\/h4>\n<\/p>\n
    PyFFTW\u662f\u4e00\u4e2a\u9ad8\u6548\u7684FFT\u5e93\uff0c\u57fa\u4e8eFFTW\uff08\u4e00\u4e2aC\u8bed\u8a00\u5199\u7684\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u5e93\uff09\u3002\u5b83\u652f\u6301\u591a\u7ebf\u7a0b\u548c\u5e76\u884c\u8ba1\u7b97\uff0c\u53ef\u4ee5\u5927\u5927\u52a0\u901f\u5085\u91cc\u53f6\u53d8\u6362\u7684\u8ba1\u7b97\u3002<\/p>\n<\/p>\n
    import numpy as np<\/p>\n
    import pyfftw<\/p>\n
    y = np.sin(50.0 * 2.0 * np.pi * x) + 0.5 * np.sin(80.0 * 2.0 * np.pi * x)<\/p>\n
    yf = pyfftw.interfaces.numpy_fft.fft(y)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528PyFFTW\u5e93\u5bf9\u4fe1\u53f7\u8fdb\u884c\u4e86\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/p>\n<\/p>\n
    7.2 CuPy<\/h4>\n<\/p>\n
    CuPy\u662f\u4e00\u4e2a\u57fa\u4e8eNumPy\u7684GPU\u52a0\u901f\u5e93\uff0c\u652f\u6301CUDA\u5e76\u884c\u8ba1\u7b97\u3002\u5b83\u53ef\u4ee5\u5927\u5927\u52a0\u901f\u5927\u89c4\u6a21\u6570\u636e\u7684\u5085\u91cc\u53f6\u53d8\u6362\u8ba1\u7b97\u3002<\/p>\n<\/p>\n
    import cupy as cp<\/p>\n
    y = cp.sin(50.0 * 2.0 * np.pi * x) + 0.5 * cp.sin(80.0 * 2.0 * np.pi * x)<\/p>\n
    yf = cp.fft.fft(y)<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u5728\u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528CuPy\u5e93\u5bf9\u4fe1\u53f7\u8fdb\u884c\u4e86\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/p>\n<\/p>\n
    \u516b\u3001\u5085\u91cc\u53f6\u53d8\u6362\u7684\u9ad8\u7ea7\u5e94\u7528<\/h3>\n<\/p>\n
    \u5085\u91cc\u53f6\u53d8\u6362\u5728\u8bb8\u591a\u9ad8\u7ea7\u5e94\u7528\u4e2d\u90fd\u6709\u91cd\u8981\u4f5c\u7528\uff0c\u4f8b\u5982\u538b\u7f29\u611f\u77e5\u3001\u6a21\u5f0f\u8bc6\u522b\u3001\u673a\u5668\u5b66\u4e60<\/a>\u7b49\u3002<\/p>\n<\/p>\n
    8.1 \u538b\u7f29\u611f\u77e5<\/h4>\n<\/p>\n
    \u538b\u7f29\u611f\u77e5\u662f\u4e00\u79cd\u4fe1\u53f7\u5904\u7406\u6280\u672f\uff0c\u53ef\u4ee5\u5728\u8fdc\u4f4e\u4e8e\u5948\u594e\u65af\u7279\u91c7\u6837\u7387\u7684\u60c5\u51b5\u4e0b\u91cd\u5efa\u4fe1\u53f7\u3002\u5085\u91cc\u53f6\u53d8\u6362\u5728\u538b\u7f29\u611f\u77e5\u4e2d\u8d77\u7740\u91cd\u8981\u4f5c\u7528\uff0c\u53ef\u4ee5\u7528\u4e8e\u4fe1\u53f7\u7684\u7a00\u758f\u8868\u793a\u548c\u91cd\u5efa\u3002<\/p>\n<\/p>\n
    8.2 \u6a21\u5f0f\u8bc6\u522b<\/h4>\n<\/p>\n
    \u5728\u6a21\u5f0f\u8bc6\u522b\u9886\u57df\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u53ef\u4ee5\u7528\u4e8e\u7279\u5f81\u63d0\u53d6\u548c\u6a21\u5f0f\u5339\u914d\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u5085\u91cc\u53f6\u53d8\u6362\u6765\u5206\u6790\u56fe\u50cf\u7684\u9891\u7387\u7279\u5f81\uff0c\u5b9e\u73b0\u56fe\u50cf\u7684\u5339\u914d\u548c\u5206\u7c7b\u3002<\/p>\n<\/p>\n
    8.3 \u673a\u5668\u5b66\u4e60<\/h4>\n<\/p>\n
    \u5728\u673a\u5668\u5b66\u4e60\u9886\u57df\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u53ef\u4ee5\u7528\u4e8e\u7279\u5f81\u63d0\u53d6\u548c\u6570\u636e\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u5085\u91cc\u53f6\u53d8\u6362\u6765\u63d0\u53d6\u97f3\u9891\u4fe1\u53f7\u7684\u9891\u7387\u7279\u5f81\uff0c\u4f5c\u4e3a\u673a\u5668\u5b66\u4e60\u6a21\u578b\u7684\u8f93\u5165\u7279\u5f81\u3002<\/p>\n<\/p>\n
    \u4e5d\u3001\u5085\u91cc\u53f6\u53d8\u6362\u7684\u672a\u6765\u53d1\u5c55<\/h3>\n<\/p>\n
    \u968f\u7740\u8ba1\u7b97\u673a\u6280\u672f\u7684\u53d1\u5c55\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u5728\u8bb8\u591a\u9886\u57df\u7684\u5e94\u7528\u4e0d\u65ad\u6269\u5c55\u548c\u6df1\u5316\u3002\u672a\u6765\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u6709\u671b\u5728\u5927\u6570\u636e\u3001\u4eba\u5de5\u667a\u80fd<\/a>\u3001\u91cf\u5b50\u8ba1\u7b97\u7b49\u9886\u57df\u53d1\u6325\u66f4\u91cd\u8981\u7684\u4f5c\u7528\u3002<\/p>\n<\/p>\n
    9.1 \u5927\u6570\u636e<\/h4>\n<\/p>\n
    \u5728\u5927\u6570\u636e\u9886\u57df\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u53ef\u4ee5\u7528\u4e8e\u6570\u636e\u7684\u538b\u7f29\u3001\u5206\u6790\u548c\u5904\u7406\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u5085\u91cc\u53f6\u53d8\u6362\u6765\u5206\u6790\u5927\u89c4\u6a21\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u7684\u9891\u7387\u7279\u5f81\uff0c\u5b9e\u73b0\u6570\u636e\u7684\u9ad8\u6548\u5b58\u50a8\u548c\u4f20\u8f93\u3002<\/p>\n<\/p>\n
    9.2 \u4eba\u5de5\u667a\u80fd<\/h4>\n<\/p>\n
    \u5728\u4eba\u5de5\u667a\u80fd\u9886\u57df\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u53ef\u4ee5\u7528\u4e8e\u7279\u5f81\u63d0\u53d6\u548c\u6570\u636e\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u5085\u91cc\u53f6\u53d8\u6362\u6765\u63d0\u53d6\u56fe\u50cf\u3001\u97f3\u9891\u3001\u6587\u672c\u7b49\u6570\u636e\u7684\u9891\u7387\u7279\u5f81\uff0c\u63d0\u9ad8\u673a\u5668\u5b66\u4e60\u6a21\u578b\u7684\u6027\u80fd\u548c\u7cbe\u5ea6\u3002<\/p>\n<\/p>\n
    9.3 \u91cf\u5b50\u8ba1\u7b97<\/h4>\n<\/p>\n
    \u5728\u91cf\u5b50\u8ba1\u7b97\u9886\u57df\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u662f\u4e00\u79cd\u91cd\u8981\u7684\u91cf\u5b50\u7b97\u6cd5\uff0c\u53ef\u4ee5\u7528\u4e8e\u91cf\u5b50\u6001\u7684\u53d8\u6362\u548c\u6d4b\u91cf\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u91cf\u5b50\u5085\u91cc\u53f6\u53d8\u6362\u6765\u5b9e\u73b0\u91cf\u5b50\u6001\u7684\u5206\u89e3\u548c\u91cd\u6784\uff0c\u63d0\u9ad8\u91cf\u5b50\u8ba1\u7b97\u7684\u6548\u7387\u548c\u51c6\u786e\u6027\u3002<\/p>\n<\/p>\n
    \u7efc\u4e0a\u6240\u8ff0\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u5728Python\u4e2d\u7684\u5b9e\u73b0\u4e3b\u8981<\/p>\n<\/p>\n
    \u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n
     \u5085\u91cc\u53f6\u53d8\u6362\u5728Python\u4e2d\u6709\u4ec0\u4e48\u5e94\u7528\u573a\u666f\uff1f<\/strong>
    \u5085\u91cc\u53f6\u53d8\u6362\u4e3b\u8981\u7528\u4e8e\u4fe1\u53f7\u5904\u7406\u3001\u56fe\u50cf\u5206\u6790\u548c\u6570\u636e\u538b\u7f29\u7b49\u9886\u57df\u3002\u5728\u4fe1\u53f7\u5904\u7406\u4e2d\uff0c\u5b83\u53ef\u4ee5\u5e2e\u52a9\u5206\u6790\u9891\u7387\u6210\u5206\uff0c\u63d0\u53d6\u4fe1\u53f7\u7279\u5f81\u3002\u5728\u56fe\u50cf\u5904\u7406\u4e2d\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u53ef\u4ee5\u7528\u4e8e\u53bb\u566a\u548c\u56fe\u50cf\u590d\u539f\u3002\u5728\u6570\u636e\u538b\u7f29\u9886\u57df\uff0c\u5e38\u89c1\u7684JPEG\u538b\u7f29\u5c31\u5229\u7528\u4e86\u5085\u91cc\u53f6\u53d8\u6362\u6765\u964d\u4f4e\u56fe\u50cf\u7684\u5b58\u50a8\u9700\u6c42\u3002<\/p>\n
    Python\u4e2d\u6709\u54ea\u4e9b\u5e93\u53ef\u4ee5\u7528\u4e8e\u5085\u91cc\u53f6\u53d8\u6362\uff1f<\/strong>
    \u5728Python\u4e2d\uff0cNumPy\u548cSciPy\u662f\u6700\u5e38\u7528\u7684\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5e93\u3002NumPy\u63d0\u4f9b\u4e86numpy.fft<\/code>\u6a21\u5757\uff0c\u5176\u4e2d\u5305\u542b\u4e86\u591a\u79cd\u5085\u91cc\u53f6\u53d8\u6362\u51fd\u6570\u3002\u800cSciPy\u7684scipy.fftpack<\/code>\u6a21\u5757\u5219\u63d0\u4f9b\u4e86\u66f4\u591a\u9ad8\u7ea7\u529f\u80fd\uff0c\u5305\u62ec\u5bf9\u4e0d\u540c\u7ef4\u5ea6\u7684\u5085\u91cc\u53f6\u53d8\u6362\u652f\u6301\u3002\u6b64\u5916\uff0cMatplotlib\u53ef\u4ee5\u7528\u6765\u53ef\u89c6\u5316\u5085\u91cc\u53f6\u53d8\u6362\u7684\u7ed3\u679c\uff0c\u5e2e\u52a9\u7528\u6237\u66f4\u597d\u5730\u7406\u89e3\u6570\u636e\u3002<\/p>\n
    \u5982\u4f55\u5904\u7406\u5085\u91cc\u53f6\u53d8\u6362\u540e\u7684\u6570\u636e\u4ee5\u83b7\u5f97\u6709\u6548\u7684\u9891\u57df\u4fe1\u606f\uff1f<\/strong>
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