{"id":5244,"date":"2013-04-08T03:09:19","date_gmt":"2013-04-07T19:09:19","guid":{"rendered":"http:\/\/www.andrewsun.net\/panta_rhei\/?p=5244"},"modified":"2020-02-04T23:45:06","modified_gmt":"2020-02-04T15:45:06","slug":"fourier%e5%8f%98%e6%8d%a2%ef%bc%88%e4%b8%80%ef%bc%89","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5244","title":{"rendered":"Fourier\u53d8\u6362\uff08\u4e00\uff09"},"content":{"rendered":"
\n

[mathjax]<\/p>\n

\u4e3a\u4e86\u5e26\u5e08\u5f1f\u505a\u5b9e\u9a8c\uff0c\u7b80\u8981\u603b\u7ed3\u4e00\u4e0b\u5728MATLAB\u91cc\u8fdb\u884cFourier\u53d8\u6362\u7684\u4e00\u4e9b\u4ee3\u7801\uff0c\u4ee5\u4f9b\u5927\u5bb6\u53c2\u8003\u3002<\/p>\n

Discrete Fourier transform (DFT)<\/a>: The sequence of N<\/i> complex numbers x<\/i>0<\/sub>, …, x<\/i>N<\/i>-1<\/sub> is transformed into an N<\/i>-periodic sequence of complex numbers according to the DFT formula<\/p>\n

\"X_k=\sum_{n=0}^{N-1}x_n.<\/p>\n

\u4e00\u822c\u628a\u53d8\u6362\u524d\u7684\u4fe1\u53f7xn<\/sub><\/i>\u5f53\u6210\u65f6\u57df\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u7684Xk<\/sub><\/i>\u5f53\u6210\u9891\u57df\u4fe1\u53f7\u3002\u53ef\u89c1\uff0c\u65f6\u57df\u662fN<\/i>\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u9891\u57df\u8fd8\u662fN<\/i>\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\u3002\u4ec0\u4e48\u5468\u671f\u554a\u9891\u7387\u554a\u8fd9\u4e9b\u4fe1\u606f\u662f\u6682\u65f6\u6ca1\u6709\u7684\u3002\u4f8b\u5982xn<\/sub><\/i>\u4e3a\u4ee5\u4e0b\u8fd9\u4e2a\u5e8f\u5217\uff08Fig. 1\uff09\uff0c\u6a2a\u5750\u6807\u5c31\u662f\u5e8f\u53f7n<\/i>\u3002\u8fd9\u6bb5\u5e8f\u5217\uff0c\u53ef\u4ee5\u662f\u4e00\u6bb510\u79d2\u949f\u7684\u7684\u4fe1\u53f7\uff0c\u4e5f\u53ef\u4ee5\u662f\u4e00\u6bb5100\u79d2\u949f\u7684\u4fe1\u53f7\u3002\u5f53\u7136\uff0c\u5b9e\u9645\u60c5\u51b5\u4e0b\u4e00\u6bb5\u4fe1\u53f7\u662f\u591a\u5c11\u79d2\u662f\u4e8b\u5148\u77e5\u9053\u7684\uff0c\u51fa\u6765\u7684\u4fe1\u53f7\u4e00\u5171\u6709\u591a\u5c11\u4e2a\u5355\u5143\u503c\uff08N<\/i>\uff09\u5219\u53d6\u51b3\u4e8e\u6211\u4eec\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\u3002\u5982\u679c\u4e00\u6bb510\u79d2\u949f\u7684\u4fe1\u53f7\uff0c\u6211\u4eec\u6bcf\u79d2\u53d6\u4e00\u6b21\u6837\uff0c\u8fd9\u4e2a\u5e8f\u5217\u5c31\u670910\u4e2a\u5355\u5143\uff08N<\/i> = 10\uff09\u3002\u53cd\u4e4b\uff0cFig. 1\u7684\u8fd9\u4e2a\u5e8f\u5217N<\/i> = 93\uff0c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u96940.1\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u7684\u603b\u65f6\u957f\u5c31\u662f9.2\u79d2\uff1b\u800c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u969410\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u603b\u65f6\u957f\u5c31\u662f920\u79d2\u4e86\u3002\u6bcf\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\uff0c\u53ef\u4ee5\u7528\u53d6\u6837\u9891\u7387<\/em>f<\/i>s<\/sub>\u6765\u8868\u793a\u3002f<\/i>s<\/sub> = 10 Hz\u5c31\u662f\u6bcf\u96940.1 s\u53d6\u4e00\u6b21\u6837\u3002\u6240\u4ee5\uff0c\u4e3a\u4e86\u7ed9\u4e00\u5bf9\u79bb\u6563Fourier\u53d8\u6362xn<\/sub><\/i>\u548cXk<\/sub><\/i>\u8d4b\u4ee5\u5468\u671f\u6216\u9891\u7387\u7684\u4fe1\u606f\uff0c\u5fc5\u987b\u7ed9\u65f6\u57df\u4fe1\u53f7\u89c4\u5b9a\u4e00\u4e2af<\/i>s<\/sub>\u3002\u4e00\u65e6f<\/i>s<\/sub>\u89c4\u5b9a\u4e86\uff0c\u5468\u671f\u548c\u9891\u7387\u4e5f\u5c31\u89c4\u5b9a\u4e86\u3002\u8fd8\u62ffFig. 1\u7684\u5e8f\u5217\u4e3a\u4f8b\uff0c\u8bbe\u8be5\u5e8f\u5217\u7684\u53d6\u6837\u9891\u7387f<\/i>s<\/sub> = 10 Hz\uff0c\u5219\u8be5\u5e8f\u5217\u6bcf\u4e2a\u5355\u5143\u9694\u4e860.1 s\u3002\u6211\u4eec\u53ef\u4ee5\u6570\u4e00\u4e0b\u8fd9\u4e2a\u5e8f\u5217\u4e00\u4e2a\u5468\u671f\u6709\u591a\u5c11\u4e2a\u5355\u5143\u2014\u201420\u4e2a\uff0c\u90a3\u4e48\u8be5\u5e8f\u5217\u7684\u5468\u671fT<\/i> = 2 s\uff0c\u9891\u7387f<\/i> = 0.5 Hz\u3002<\/p>\n

\"Figure<\/a>

Figure 1<\/p><\/div>\n

\u4e8b\u5b9e\u4e0aFig. 1\u7684\u5e8f\u5217\u662f\u4e00\u4e2a\u6709\u4e00\u5b9a\u65f6\u5ef6\u7684\u6b63\u5f26\u5e8f\u5217\uff08\u5373\u4e0d\u662f\u4ece\u96f6\u5f00\u59cb\uff09\uff0c\u6211\u4eec\u5df2\u7ecf\u6839\u636e\u89c4\u5b9a\u7684f<\/i>s<\/sub>\u77e5\u9053\u4e86\u5b83\u7684\u5468\u671f\u548c\u9891\u7387\uff0c\u4e0d\u4eff\u5c31\u628a\u5b83\u6539\u4f5c\u5728\u65f6\u95f4\u8f74\u4e0a\uff08Fig. 2\uff09\u3002<\/p>\n

\"Figure<\/a>

Figure 2<\/p><\/div>\n

\u53ef\u4ee5\u770b\u5230\uff0c\u8be5\u6b63\u5f26\u4fe1\u53f7\u7684\u96f6\u4f4d\u5728\u7b2c17\u548c\u7b2c18\u4e2a\u5355\u5143\u4e4b\u95f4\uff0c\u5373\u57281.6 s\u52301.7 s\u5904\uff0c\u4ee51.6 s\u8ba1\u7b97\uff0c\u8be5\u4fe1\u53f7\u53ef\u5199\u6210\"x\left(t\right)=A\sin(\left[2\pi\uff0c\u5219\u76f8\u4f4d\u89d2\"\delta=-1.6\pi\"\uff0c\u6362\u62100\u52302\u03c0<\/i>\u4e4b\u95f4\u5c31\u662f\"\delta=0.4\pi=1.2566\"\u3002\u82e5\u53d6\u65f6\u79fb\u662f1.7 s\u5f97\"\delta\uff0c\u6240\u4ee5\u03b4<\/i>\u5e94\u8be5\u57280.9425\u52301.2566\u4e4b\u95f4\u3002\u4ece\u56fe\u4e2d\u8fd8\u80fd\u8bfb\u51fa\u632f\u5e45A<\/i> = 2\uff0c\u5219\u4fe1\u53f7\"x\left(t\right)=2\sin\left(\pi\u3002\u7528\u6b63\u5f26\u51fd\u6570\u53bb\u62df\u5408Fig. 2\u7684\u5e8f\u5217\u53d1\u73b0\u5176\u5b9e\u6700\u7cbe\u786e\u7684\u51fd\u6570\u63cf\u8ff0\u5e94\u8be5\u662f\"x\left(t\right)=2\sin\left(\pi\uff0c\u89c1Fig. 3\u3002<\/p>\n

\"Figure<\/a>

Figure 3<\/p><\/div>\n

\u73b0\u5728\u6211\u4eec\u8bd5\u7740\u5728MATLAB\u901a\u8fc7Fourier\u53d8\u6362\u6765\u7cbe\u786e\u5730\u5f97\u5230xn<\/sub><\/i>\u7684\u5e45\u503c\u548c\u76f8\u4f4d\uff0c\u91cd\u6784\u4e00\u4e2axn<\/sub><\/i>\u6240\u5bf9\u5e94\u7684\u8fde\u7eed\u4fe1\u53f7\u3002\u9996\u5148\u6784\u9020\u4e0a\u8ff0\u7684\u65f6\u57df\u4fe1\u53f7\uff1a<\/p>\n


\nt = 0:0.1:9.2; % fs = 10 Hz
\nt = t(:); % convert into column vector
\nf = 0.5; % f = 0.5 Hz
\nx = 2 * sin(2 * pi * fi * t + 1)
\n<\/code><\/p>\n

\u524d\u9762\u8bf4\u8fc7\u5bf9\u4e00\u7ec4N<\/i>\u4e2a\u5355\u5143\u7684\u5e8f\u5217\u8fdb\u884cFourier\u53d8\u6362\u4e4b\u540e\u5f97\u5230\u7684Xk<\/sub><\/i>\u4e5f\u662f\u4e00\u4e2aN<\/i>\u4e2a\u590d\u6570\u5355\u5143\u7684\u5e8f\u5217\u3002\u6211\u4eec\u770b\u770b\u5b83\u957f\u4ec0\u4e48\u6837\uff08\u5206\u5f00\u6a21\u5e8f\u5217\u548c\u8f90\u89d2\u5e8f\u5217\uff09\uff1a<\/p>\n


\nXraw = fft(x);
\nstem(abs(Xraw),'o'); % Figure 4
\nstem(angle(Xraw),'o'); % Figure 5
\n<\/code><\/p>\n

\"Figure<\/a>

Figure 4<\/p><\/div>\n

\"Figure<\/a>

Figure 5<\/p><\/div>\n

\u6211\u4eec\u4eceFig. 4\u53ef\u4ee5\u770b\u5230\u7ed3\u679c\u8ddf\u6211\u4eec\u60f3\u50cf\u7684\u5dee\u5f88\u8fdc\u3002\u4e3b\u8981\u6709\u4ee5\u4e0b\u56db\u4e2a\u95ee\u9898\uff1a<\/p>\n

    \n
  1. \u9891\u57df\u8c31\u5206\u6210\u4e86\u5bf9\u79f0\u7684\u4e24\u534a\u3002\u8981\u53d6\u5de6\u8fb9\u4e00\u534a\u3002\u4f46\u603b\u5355\u5143\u4e2a\u6570\u4e3a\u5947\u6570\uff08N<\/i> = 93\uff09\uff0c\u5230\u5e95\u662f\u53d646\u4e2a\u8fd8\u662f47\u4e2a\uff1f<\/li>\n
  2. \u82e5\u53ea\u770b\u4e00\u534a\uff0c\u9891\u8c31\u51fa\u73b0\u4e86\u5355\u5cf0\u662f\u6ca1\u9519\uff08\u56e0\u4e3a\u6211\u4eec\u7684\u4fe1\u53f7\u53ea\u6709\u4e00\u4e2a\u9891\u7387\uff09\uff0c\u4f46\u5b83\u4e0d\u662f\u53ea\u5728\u4e00\u4e2a\u9891\u7387\u4e0a\uff0c\u800c\u662f\u6709\u4e00\u4e2a\u5f88\u5bbd\u7684\u5206\u5e03\uff0c\u4f3c\u4e4e\u6211\u7684\u8f93\u5165\u4fe1\u53f7\u7684\u9891\u7387\u662f\u201c\u4e0d\u7eaf\u201d\u7684\u2014\u2014\u8fd9\u5e76\u4e0d\u7b26\u5408\u4e8b\u5b9e<\/li>\n
  3. \u5e45\u503c\u4e0d\u5bf9\uff0c\u4fe1\u53f7\u5e45\u503c\u662f2\uff0c\u4f46\u9891\u8c31\u7684\u5cf0\u9ad8\u5feb\u523080\u4e86<\/li>\n
  4. \u6a2a\u8f74\u5982\u4f55\u5bf9\u5e94\u5230\u771f\u5b9e\u7684\u9891\u7387\u8f74\uff1f<\/li>\n<\/ol>\n

    \u6211\u4eec\u4e00\u4e2a\u4e00\u4e2a\u95ee\u9898\u5730\u89e3\u51b3\u3002\u7b2c\u4e00\u4e2a\u95ee\u9898\uff0c\u6309\u7167fft\u7b97\u6cd5\u7684\u89c4\u5f8b\uff0c\u82e5N<\/i>\u4e3a\u5076\u6570\uff0c\u5219\u5de6\u53f3\u5206\u6210\u5404N<\/i>\/2\u7684\u4e24\u534a\u5373\u53ef\uff1b\u82e5N<\/i>\u4e3a\u5947\u6570\uff0c\u5219\u5904\u4e8e\u4e2d\u592e\u7684\u90a3\u4e2a\u5355\u5143\u662f\u5de6\u53f3\u4e24\u534a\u8c31\u5171\u7528\u7684\u3002\u6240\u4ee5\uff0c\u65e0\u8bba\u53d6\u5de6\u534a\u8fd8\u662f\u53f3\u534a\uff0c\u90fd\u53d6(N<\/i>+1)\/2+1\u4e2a\u3002\u7efc\u5408\u5947\u6570\u548c\u5076\u6570\u7684\u60c5\u51b5\uff0c\u5bf9\u4efb\u610f\u6b63\u6574\u6570N<\/i>\u7684\u53d6\u6cd5\u5c31\u662f\u5bf9N<\/i>\/2\u5411\u4e0a\u53d6\u6574\u3002<\/p>\n


    \nN = length(x);
    \nXraw = Xraw(1:ceil(N\/2));
    \nstem(abs(Xraw));
    \n<\/code><\/p>\n

    \u7b2c\u4e8c\u4e2a\u95ee\u9898\uff0c\u4e4b\u6240\u4ee5\u9891\u8c31\u7684\u5cf0\u6709\u4e00\u4e2a\u5f88\u5bbd\u7684\u5206\u5e03\uff0c\u662f\u7531\u4e8e\u8c31\u6cc4\u6f0f\uff08spectral leakage<\/a>\uff09\u73b0\u8c61\uff0c\u7b80\u5355\u5730\u8bf4\u5c31\u662f\u6211\u4eec\u7528\u6765\u505aFourier\u53d8\u6362\u7684\u4fe1\u53f7\u5e8f\u5217\u957f\u5ea6\u5e76\u4e0d\u662f\u6070\u597d\u6574\u6570\u4e2a\u5468\u671f\u3002\u53ea\u6709\u622a\u53d6\u6574\u6570\u4e2a\u5468\u671f\u6765\u505aFourier\u53d8\u6362\uff0c\u624d\u80fd\u907f\u514d\u8c31\u6cc4\u6f0f\u3002\u53ef\u662f\u5bf9\u4e8e\u4e00\u4e2a\u672a\u77e5\u9891\u7387\u7684\u4fe1\u53f7\uff0c\u6211\u4eec\u672c\u6765\u5c31\u662f\u8981\u901a\u8fc7Fourier\u53d8\u6362\u6765\u77e5\u9053\u5b83\u7684\u9891\u7387\uff0c\u53c8\u600e\u4e48\u80fd\u5728\u6b64\u4e4b\u524d\u5c31\u5f97\u77e5\u4fe1\u53f7\u7684\u5468\u671f\u5462\uff1f\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u53ea\u80fd\u91c7\u7528\u53e6\u4e00\u79cd\u65b9\u6cd5\uff0c\u90a3\u5c31\u662f\u628a\u4fe1\u53f7\u7684\u9996\u5c3e\u8fdb\u884c\u67d0\u79cd\u8870\u51cf\u5904\u7406\uff0c\u800c\u8fd9\u4e00\u5904\u7406\u5728\u6570\u5b66\u4e0a\u53c8\u8981\u4fdd\u8bc1\u4e0d\u4e25\u91cd\u5f71\u54cdFourier\u53d8\u6362\u7684\u7ed3\u679c\uff0c\u79f0\u4e3a\u52a0\u7a97\u64cd\u4f5c\uff08windowing<\/a>\uff09\u3002\u8be6\u7ec6\u7684\u539f\u7406\u4e0d\u5728\u6b64\u8be6\u8ff0\u3002\u4f5c\u4e3a\u793a\u8303\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u6211\u4eec\u5bf9\u65f6\u57df\u4fe1\u53f7\u7684\u5df2\u77e5\u4fe1\u606f\u6765\u89e3\u51b3\u3002\u5df2\u77e5\u4fe1\u53f7\u5468\u671fT<\/i> = 2 s\uff0c\u53d6\u6837\u9891\u7387f<\/i>s<\/sub> = 10 Hz\uff0c\u5230\u5e95\u622a\u53d6\u591a\u5c11\u4e2a\u70b9\u7b97\u4e00\u4e2a\u5468\u671f\u5462\uff1f\u524d\u9762\u8ba1\u7b97\u8fc7\u4e86\uff0c20\u4e2a\u70b9\u3002\u5e72\u8106\u6211\u4eec\u5c31\u53ea\u53d6\u4e00\u4e2a\u5468\u671f\u5f97\u4e86\uff0c\u8bd5\u8bd5\u770b\uff1a<\/p>\n


    \nx_cut = x(1:20); % N = 20
    \nXraw = fft(x_cut);
    \nXraw = Xraw(1:10); % N=20, so ceil(N\/2) = 10
    \nstem(abs(Xraw)); % Figure 6
    \n<\/code><\/p>\n

    \"Figure<\/a>

    Figure 6<\/p><\/div>\n

    \u8fd9\u6b21\uff0c\u9891\u5cf0\u503c\u7ec8\u4e8e\u53ea\u51fa\u73b0\u5728\u4e00\u4e2a\u9891\u7387\u4e0a\u4e86\u3002\u800c\u7531\u4e8e\u7528\u6765\u505aFourier\u53d8\u6362\u7684\u4fe1\u53f7N<\/i>\u51cf\u5c0f\u5230\u4e8620\u4e2a\uff0c\u6240\u4ee5\u9891\u57df\u4fe1\u53f7\u4e5f\u5c31\u53ea\u670920\u4e2a\u5355\u5143\uff0c\u53d6\u5de6\u534a\u8fb9\u5c31\u662f\u5341\u4e2a\u5355\u5143\u3002\u4f46\u6211\u4eec\u770b\u5230\uff0c\u5cf0\u503c\u4e5f\u53d8\u4e86\uff0c\u539f\u6765\u8fd180\uff0c\u73b0\u5728\u662f\u572820\u3002\u8fd9\u8bf4\u660e\uff0c\u8fd9\u4e2a\u4e0d\u6b63\u786e\u7684\u5cf0\u503c\uff0c\u662f\u8ddfN<\/i>\u503c\u6709\u5173\u7684\u3002\u8fd9\u5c31\u6d89\u53ca\u5230\u7b2c\u4e09\u4e2a\u95ee\u9898\uff1a\u5cf0\u503c\u600e\u4e48\u6821\u6b63\u56de\u6765\u3002\u5176\u5b9e\uff0c\u6211\u4eec\u505a\u79bb\u6563Fourier\u53d8\u6362\uff0c\u9700\u8981\u8fdb\u884c\u5f52\u4e00\u5316\u3002\u4f46\u662f\u600e\u4e48\u5f52\u4e00\u5316\uff0c\u8981\u770b\u79bb\u6563Fourier\u53d8\u6362\u548c\u53cdFourier\u53d8\u6362\u7684\u5177\u4f53\u89c4\u5b9a\uff0c\u4e0d\u540c\u7684\u4e66\u89c4\u5b9a\u4e0d\u4e00\u6837\u3002MATLAB\u91ccfft\u547d\u4ee4\u662f\u9700\u8981\u7528N<\/i>\u6765\u5f52\u4e00\u5316\u7684\u3002\u4f46\u5982\u679c\u51fa\u6765\u53cc\u9762\u7684\u9891\u8c31\uff0c\u5404\u8fb9\u5c31\u8981\u7528N<\/i>\/2\u6765\u5f52\u4e00\u5316\u3002\u6240\u4ee5\u8fdb\u884c\u4e86\u5f52\u4e00\u5316\u7684\u7684Fourier\u53d8\u6362\u64cd\u4f5c\u5e94\u4e3a\uff1a<\/p>\n


    \nXraw = 2 * fft(x_cut) \/ 20;
    \nXraw = Xraw(1:10);
    \nstem(abs(Xraw));
    \n<\/code><\/p>\n

    \u51fa\u6765\u7684\u7ed3\u679c\uff0c\u5cf0\u503c\u5c31\u4f1a\u51c6\u786e\u843d\u57282\uff08\u56fe\u7565\uff09\u3002\u73b0\u5728\u5269\u4e0b\u6700\u540e\u4e00\u4e2a\u95ee\u9898\uff0c\u90a3\u5c31\u662f\u6a2a\u8f74\u600e\u6837\u6362\u6210\u771f\u6b63\u7684\u9891\u7387\uff1f\u8fd9\u5c31\u8981\u5148\u4e86\u89e3Shannon-Nyquist\u5b9a\u7406<\/a>\u3002\u5728\u6b64\u6211\u53ea\u628a\u7ed3\u8bba\u8bf4\u51fa\u6765\uff1a\u9891\u57df\u4fe1\u53f7\u5355\u5143\u7684\u95f4\u683c\u9891\u7387\u662ff<\/i>s<\/sub>\/N\u3002\u56e0\u6b64\u6574\u4e2a\u5de6\u534a\u8fb9\u8c31\u7684\u9891\u7387\u8303\u56f4\u5c31\u662f0 Hz ~ f<\/i>s<\/sub>\/2 – f<\/i>s<\/sub>\/N<\/i>\u3002\u4e3a\u4e86\u89e3\u51b3\u5947\u6570\u5076\u6570\u95ee\u9898\uff08\u5373\u4ee5\u9632f<\/i>s<\/sub>\/2\u4e0d\u4e00\u5b9a\u662ff<\/i>s<\/sub>\/N\u7684\u6574\u6570\u500d\uff09\uff0c\u6700\u7ec8\u6211\u4eec\u5c31\u5982\u4e0b\u6784\u9020\u76f8\u5e94\u7684\u9891\u7387\u8f74\uff1a<\/p>\n


    \nfreq = 0:fs\/20:(ceil(N\/2)-1)*fs\/N;
    \nfreq = freq(:);
    \nstem(freq,abs(Xraw)); % Figure 7
    \n<\/code><\/p>\n

    \"Figure<\/a>

    Figure 7<\/p><\/div>\n

    \u5176\u5b9e\uff0c\u8981\u53d6\u6574\u6570\u4e2a\u5468\u671f\uff0c\u4e0d\u4e00\u5b9a\u8981\u53ea\u53d6\u4e00\u4e2a\u5468\u671f\u3002\u539f\u65f6\u57df\u4fe1\u53f7\u81f3\u5c11\u67094\u4e2a\u5468\u671f\uff08\u89c1Fig. 1\uff09\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u53d6\u6ee1\u65744\u4e2a\u5468\u671f\u6765\u505aFourier\u53d8\u6362\uff1a<\/p>\n


    \nx_cut = x(1:80); % 4 cycles = 80 points
    \nXraw = 2 * fft(x_cut) \/ 80; % N = 80
    \nXraw = Xraw (1:40);
    \nfreq = 0:fs\/80:fs\/2-1\/80;
    \nstem(freq,abs(Xraw)); % Figure 8
    \n<\/code><\/p>\n

    \"Figure<\/a>

    Figure 8<\/p><\/div>\n

    \u53ef\u89c1\uff0c\u5f53\u6211\u4eec\u591a\u53d6\u51e0\u4e2a\u5468\u671f\u65f6\uff0cFourier\u53d8\u6362\u7684\u7ed3\u679c\u9891\u7387\u201c\u5206\u8fa8\u7387\u201d\u66f4\u9ad8\u3002\u56e0\u6b64\u4e3a\u4e86\u83b7\u5f97\u5c3d\u53ef\u80fd\u5b8c\u6574\u5730\u4fe1\u606f\uff0c\u8981\u5728\u4fdd\u6301\u53d6\u6574\u4e2a\u5468\u671f\u7684\u540c\u65f6\uff0c\u5c3d\u91cf\u53d6\u6700\u591a\u4e2a\u5468\u671f\u6765\u8fdb\u884cFourier\u53d8\u6362\u3002\u6211\u4eec\u628a\u76f8\u5e94\u7684\u8f90\u89d2\u8c31\u4e5f\u4f5c\u51fa\u6765\uff1a<\/p>\n


    \nstem(freq,angle(Xraw); % Figure 9
    \n<\/code><\/p>\n

    \"Figure<\/a>

    Figure 9<\/p><\/div>\n

    \u6211\u4eec\u53d1\u73b0\uff0c\u8ddfFig. 5\u76f8\u6bd4\uff0cFig. 9\u7684\u8f90\u89d2\u8c31\u663e\u5f97\u975e\u5e38\u6742\u4e71\u65e0\u7ae0\u3002\u8fd9\u662f\u7531\u4e8e\u6211\u4eec\u5f88\u597d\u7684\u6d88\u9664\u4e86\u8c31\u6cc4\u6f0f\u3002\u89c4\u5f8b\uff1a\u8c31\u6cc4\u6f0f\u8d8a\u4e25\u91cd\uff0c\u8f90\u89d2\u8c31\u8d8a\u6709\u5e8f\uff1b\u8c31\u6cc4\u6f0f\u907f\u514d\u5f97\u8d8a\u5f7b\u5e95\uff0c\u8f90\u89d2\u8c31\u5c31\u8d8a\u6742\u4e71\u65e0\u7ae0\u3002<\/em>\u867d\u7136\u6742\u4e71\u65e0\u7ae0\uff0c\u4f46\u662f\u6a21\u8c31\u7684\u5e45\u503c\u6240\u5728\u76f8\u5e94\u9891\u7387\u5bf9\u5e94\u7684\u8f90\u89d2\u662f\u5e94\u8be5\u662f\u6b63\u786e\u7684\u3002\u6211\u4eec\u770b\u770b0.5 Hz\u5904\u7684\u8f90\u89d2\uff1a<\/p>\n


    \nwrapTo2Pi(angle(Xraw(5))) % wrap result within 0 to 2Pi
    \n<\/code><\/p>\n

    \u7ed3\u679c\u4e0d\u5bf9\uff0c\u8fd9\u662f\u7531\u4e8e\u5728MATLAB\u7684fft\u5f97\u5230\u7684\u8f90\u89d2\u5dee\u4e86\u03c0<\/i>\/2\uff0c\u8981\u52a0\u4e0a\uff1a<\/p>\n


    \nwrapTo2Pi(angle(Xraw(5))+0.5*pi)
    \n<\/code><\/p>\n

    \u8fd9\u6b21\u5c31\u5bf9\u4e86\u3002<\/p>\n

    \u95ee\u9898\u662f\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u4e0d\u53ef\u80fd\u505a\u4e00\u4e2aFourier\u53d8\u6362\uff0c\u5c31\u4f5c\u4e00\u6b21\u56fe\uff0c\u7528\u8089\u773c\u770b\u51fa\u5cf0\u503c\u843d\u5728\u54ea\u4e2a\u9891\u7387\u4e0a\u3002\u5fc5\u987b\u4e8b\u5148\u5c31\u77e5\u9053\u4fe1\u53f7\u9891\u7387f<\/i>\u5728\u9891\u57df\u4fe1\u53f7\u4e2d\u7684\u5bf9\u5e94\u5e8f\u53f7\uff0c\u76f4\u63a5\u5c31\u7ed9\u51fa\u6a21\u7684\u5cf0\u503c\u548c\u76f8\u5e94\u7684\u8f90\u89d2\u3002\u6240\u4ee5\uff0c\u73b0\u5728\u7684\u95ee\u9898\u662f\uff0c\u5982\u4f55\u4e0d\u901a\u8fc7\u4f5c\u56fe\u5c31\u77e5\u9053\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u9891\u7387\u4e3af<\/i>\uff0c\u91c7\u6837\u9891\u7387\u4e3af<\/i>s<\/sub>\uff0cN<\/i>\u4e2a\u5355\u5143\u7684\u4fe1\u53f7xn<\/sub><\/i>\uff0c\u8fdb\u884c\u4e86\u4e0a\u8ff0\u7684Fourier\u53d8\u6362\u540e\u5f97\u5230\u7684Xk<\/sub><\/i>\uff0c\u7b2c\u51e0\u4e2a\u5355\u5143\u5462\u5bf9\u5e94\u7740f<\/i>\uff1f\u8fd9\u5f88\u7b80\u5355\uff0cFourier\u53d8\u6362\u540e\u7684\u9891\u7387\u95f4\u9694\u662ff<\/i>s<\/sub>\/N<\/i>\uff0c\u56e0\u6b64\uff0c\u7b2cNf<\/i>\/f<\/i>s<\/sub>+1\u4e2a\u5355\u5143\u5c31\u662f\u6a21\u7684\u5cf0\u503c\u6240\u5728\u9891\u7387\u3002\u672c\u4f8b\u5b50\u4e2d\u65e2\u53ef\u9a8c\u8bc1\u3002<\/p>\n

    \u4f46\u662f\uff0c\u5047\u5982\u4fe1\u53f7\u7684\u9891\u7387f<\/i>\u5728Fourier\u53d8\u6362\u4e4b\u540e\u7684\u9891\u7387\u8f74\u4e0a\u7684\u4f4d\u7f6e\uff0c\u662f\u6070\u597d\u5728\u9891\u57df\u4fe1\u53f7\u4e24\u4e2a\u5355\u5143\u4e4b\u95f4\u5462\uff1f\u4f8b\u5982\uff0c\u6211\u7684\u4fe1\u53f7\u9891\u7387\u662ff<\/i>=0.51 Hz\uff0cf<\/i>s<\/sub>\u548cN<\/i>\u503c\u4e0d\u53d8\uff0c\u6309\u7167\u4e0a\u8ff0\u7b97\u6cd5\uff0c\u8981\u53d6\u7b2c5.08\u4e2a\u5355\u5143\uff0c\u5f53\u7136\u4e0d\u5b58\u5728\u8fd9\u4e2a\u975e\u6574\u6570\u5355\u5143\u3002\u8fd9\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u8981\u5bf9Nf<\/i>\/f<\/i>s<\/sub>+1\u8fd9\u4e2a\u516c\u5f0f\u52a0\u4e00\u4e2a\u56db\u820d\u4e94\u5165\u7684\u53d6\u6574\u3002\u6ce8\u610f\uff0c\u7531\u4e8e\u4fe1\u53f7\u9891\u7387\u4e0d\u540c\u4e86\uff0c\u201c\u6574\u6570\u4e2a\u5468\u671f\u201d\u7684\u622a\u53d6\u53ef\u80fd\u4f1a\u7565\u6709\u4e0d\u540c\u3002\u4ee5\u4e0b\u662f\u5177\u4f53\u7684\u8fd0\u7b97\u8fc7\u7a0b\uff1a<\/p>\n


    \nf = 0.51
    \nx = 2 * sin(2 * pi * f * t + 1);
    \nN = length(x);
    \nN_per_cycle = fs\/f; % average points per cycle
    \nN_cycles = floor(N \/ N_per_cycle); % maximum number of complete cycles
    \nN_cut = round(N_cycles * N_per_cycle); % number of points for FFT
    \nx_cut = x(1:N_cut);
    \nXraw = 2 * fft(x_cut) \/ N_cut;
    \nXraw = Xraw(1:ceil(N_cut\/2));
    \nfreq = 0:fs\/N_cut:fs\/2-fs\/N_cut;
    \nabs(Xraw(round(N_cut*f\/fs)+1))
    \nwrapTo2Pi(angle(Xraw(round(N_cut*f\/fs)+1))+0.5*pi)
    \nfreq(round(N_cut*f\/fs)+1)
    \n<\/code><\/p>\n

    \u7ed3\u679c\u79bb\u771f\u5b9e\u503c\u5c31\u7a0d\u6709\u504f\u5dee\u3002\u8fd9\u65f6\uff0c\u5982\u679c\u6211\u4eec\u8bd5\u8bd5\u628af<\/i>s<\/sub>\u63d0\u9ad8\u52301000 Hz\u3002<\/p>\n


    \nclear all;
    \nfs = 1000;
    \nt = 0:1\/fs:9.2;
    \nt=t(:);
    \nf = 0.51
    \nx = 2 * sin(2 * pi * f * t + 1);
    \nN = length(x);
    \nN_per_cycle = fs\/f; % average points per cycle
    \nN_cycles = floor(N \/ N_per_cycle); % maximum number of complete cycles
    \nN_cut = round(N_cycles * N_per_cycle); % number of points for FFT
    \nx_cut = x(1:N_cut);
    \nXraw = 2 * fft(x_cut) \/ N_cut;
    \nXraw = Xraw(1:ceil(N_cut\/2));
    \nfreq = 0:fs\/N_cut:fs\/2-fs\/N_cut;
    \nA = abs(Xraw(round(N_cut*f\/fs)+1))
    \ndelta = wrapTo2Pi(angle(Xraw(round(N_cut*f\/fs)+1))+0.5*pi)
    \nfrq = freq(round(N_cut*f\/fs)+1)
    \n<\/code><\/p>\n

    \u7ed3\u679c\u5c31\u6bd4\u4e4b\u524d\u66f4\u63a5\u8fd1\u771f\u5b9e\u503c\u4e86\u3002\u8fd9\u8bf4\u660e\uff0c\u65f6\u57df\u4fe1\u53f7\u7684\u91c7\u6837\u9891\u7387\u8d8a\u9ad8\uff0c\u8d8a\u6709\u5229\u4e8e\u7cbe\u786e\u5730\u91cd\u6784\u3002\u73b0\u5728\u53ef\u4ee5\u770b\u4e00\u4e0b\u91cd\u6784\u7684\u6548\u679c\uff1a<\/p>\n


    \nx_rec = A * sin(2*pi*frq*t+delta);
    \nplot(t,x,'bo',t,x_rec,'r-') % Figure 10
    \n<\/code><\/p>\n

    \"Figure<\/a>

    Figure 10<\/p><\/div>\n

    \u4e0b\u4e00\u7bc7\u6587\u7ae0\u4f1a\u4ecb\u7ecd\u5f53\u539f\u65f6\u57df\u4fe1\u53f7\u7684\u9891\u7387\u201c\u4e0d\u7eaf\u201d\u7684\u65f6\u5019\u5982\u4f55\u901a\u8fc7Fourier\u53d8\u6362\u8fdb\u884c\u91cd\u6784\u3002<\/p>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

    [mathjax] \u4e3a\u4e86\u5e26\u5e08\u5f1f\u505a\u5b9e\u9a8c\uff0c\u7b80\u8981\u603b\u7ed3\u4e00\u4e0b\u5728MATLAB\u91cc\u8fdb\u884cFourier\u53d8\u6362\u7684\u4e00\u4e9b\u4ee3\u7801\uff0c\u4ee5\u4f9b\u5927\u5bb6\u53c2\u8003\u3002 Discrete Fourier transform (DFT): The sequence of N complex numbers x0, …, xN-1 is transformed into an N-periodic sequence of complex numbers according to the DFT formula . \u4e00\u822c\u628a\u53d8\u6362\u524d\u7684\u4fe1\u53f7xn\u5f53\u6210\u65f6\u57df\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u7684Xk\u5f53\u6210\u9891\u57df\u4fe1\u53f7\u3002\u53ef\u89c1\uff0c\u65f6\u57df\u662fN\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u9891\u57df\u8fd8\u662fN\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\u3002\u4ec0\u4e48\u5468\u671f\u554a\u9891\u7387\u554a\u8fd9\u4e9b\u4fe1\u606f\u662f\u6682\u65f6\u6ca1\u6709\u7684\u3002\u4f8b\u5982xn\u4e3a\u4ee5\u4e0b\u8fd9\u4e2a\u5e8f\u5217\uff08Fig. 1\uff09\uff0c\u6a2a\u5750\u6807\u5c31\u662f\u5e8f\u53f7n\u3002\u8fd9\u6bb5\u5e8f\u5217\uff0c\u53ef\u4ee5\u662f\u4e00\u6bb510\u79d2\u949f\u7684\u7684\u4fe1\u53f7\uff0c\u4e5f\u53ef\u4ee5\u662f\u4e00\u6bb5100\u79d2\u949f\u7684\u4fe1\u53f7\u3002\u5f53\u7136\uff0c\u5b9e\u9645\u60c5\u51b5\u4e0b\u4e00\u6bb5\u4fe1\u53f7\u662f\u591a\u5c11\u79d2\u662f\u4e8b\u5148\u77e5\u9053\u7684\uff0c\u51fa\u6765\u7684\u4fe1\u53f7\u4e00\u5171\u6709\u591a\u5c11\u4e2a\u5355\u5143\u503c\uff08N\uff09\u5219\u53d6\u51b3\u4e8e\u6211\u4eec\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\u3002\u5982\u679c\u4e00\u6bb510\u79d2\u949f\u7684\u4fe1\u53f7\uff0c\u6211\u4eec\u6bcf\u79d2\u53d6\u4e00\u6b21\u6837\uff0c\u8fd9\u4e2a\u5e8f\u5217\u5c31\u670910\u4e2a\u5355\u5143\uff08N = 10\uff09\u3002\u53cd\u4e4b\uff0cFig. 1\u7684\u8fd9\u4e2a\u5e8f\u5217N = 93\uff0c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u96940.1\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u7684\u603b\u65f6\u957f\u5c31\u662f9.2\u79d2\uff1b\u800c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u969410\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u603b\u65f6\u957f\u5c31\u662f920\u79d2\u4e86\u3002\u6bcf\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\uff0c\u53ef\u4ee5\u7528\u53d6\u6837\u9891\u7387fs\u6765\u8868\u793a\u3002fs = 10 Hz\u5c31\u662f\u6bcf\u96940.1 s\u53d6\u4e00\u6b21\u6837\u3002\u6240\u4ee5\uff0c\u4e3a\u4e86\u7ed9\u4e00\u5bf9\u79bb\u6563Fourier\u53d8\u6362xn\u548cXk\u8d4b\u4ee5\u5468\u671f\u6216\u9891\u7387\u7684\u4fe1\u606f\uff0c\u5fc5\u987b\u7ed9\u65f6\u57df\u4fe1\u53f7\u89c4\u5b9a\u4e00\u4e2afs\u3002\u4e00\u65e6fs\u89c4\u5b9a\u4e86\uff0c\u5468\u671f\u548c\u9891\u7387\u4e5f\u5c31\u89c4\u5b9a\u4e86\u3002\u8fd8\u62ffFig. 1\u7684\u5e8f\u5217\u4e3a\u4f8b\uff0c\u8bbe\u8be5\u5e8f\u5217\u7684\u53d6\u6837\u9891\u7387fs = 10 Hz\uff0c\u5219\u8be5\u5e8f\u5217\u6bcf\u4e2a\u5355\u5143\u9694\u4e860.1 s\u3002\u6211\u4eec\u53ef\u4ee5\u6570\u4e00\u4e0b\u8fd9\u4e2a\u5e8f\u5217\u4e00\u4e2a\u5468\u671f\u6709\u591a\u5c11\u4e2a\u5355\u5143\u2014\u201420\u4e2a\uff0c\u90a3\u4e48\u8be5\u5e8f\u5217\u7684\u5468\u671fT = 2 s\uff0c\u9891\u7387f = 0.5 Hz\u3002 \u4e8b\u5b9e\u4e0aFig. 1\u7684\u5e8f\u5217\u662f\u4e00\u4e2a\u6709\u4e00\u5b9a\u65f6\u5ef6\u7684\u6b63\u5f26\u5e8f\u5217\uff08\u5373\u4e0d\u662f\u4ece\u96f6\u5f00\u59cb\uff09\uff0c\u6211\u4eec\u5df2\u7ecf\u6839\u636e\u89c4\u5b9a\u7684fs\u77e5\u9053\u4e86\u5b83\u7684\u5468\u671f\u548c\u9891\u7387\uff0c\u4e0d\u4eff\u5c31\u628a\u5b83\u6539\u4f5c\u5728\u65f6\u95f4\u8f74\u4e0a\uff08Fig. 2\uff09\u3002 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[264,303],"tags":[],"class_list":["post-5244","post","type-post","status-publish","format-standard","hentry","category-264","category-rheo"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p1aPvF-1mA","jetpack-related-posts":[{"id":5297,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5297","url_meta":{"origin":5244,"position":0},"title":"Fourier\u53d8\u6362\uff08\u4e8c\uff09","author":"Andrew","date":"2013\u5e744\u670823\u65e5","format":false,"excerpt":"[mathjax] \u4e0a\u4e00\u7bc7\u4ecb\u7ecd\u4e86\u7528Fourier\u53d8\u6362\u6765\u91cd\u6784\u4e00\u4e2a\u672a\u77e5\u6b63\u5f26\u4fe1\u53f7\u7684\u65b9\u6cd5\u3002\u672c\u7bc7\u4ecb\u7ecd\u5bf9\u4e8e\u672a\u77e5\u975e\u6b63\u5f26\u4fe1\u53f7\u7684\u60c5\u51b5\u3002 \u5927\u5e45\u632f\u8361\u526a\u5207\u7684\u53c2\u6570 \u5927\u5e45\u632f\u8361\u526a\u5207\u5b9e\u9a8c\u4ea7\u751f\u7684\u6ce2\u5f62\u662f\u975e\u6b63\u5f26\u4fe1\u53f7\uff0c\u975e\u6b63\u5f26\u662f\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7279\u70b9\u3002\u6240\u4ee5\u5bf9\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7a0b\u5ea6\u7684\u8868\u5f81\uff0c\u5c31\u662f\u5bf9\u6ce2\u5f62\u6b63\u5f26\u5f62\u72b6\u504f\u79bb\u7a0b\u5ea6\u7684\u8868\u5f81\u3002\u5173\u4e8e\u5927\u5e45\u632f\u8361\u526a\u5207\u548c\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u539f\u7406\u4e0e\u5b9e\u8df5\uff0c\u53ef\u89c1Hyhn\u7b49\u7684\u7efc\u8ff0\uff0c\u6216\u8005\u53c2\u9605\u6211\u7684\u535a\u58eb\u5b66\u4f4d\u8bba\u6587\u7b2c1.2.1\u8282\u3001\u7b2c1.3\u8282\u548c\u7b2c2\u7ae0\u3002\u672c\u6587\u53ea\u89e3\u51b3MATLAB\u7f16\u7a0b\u5b9e\u73b0\u95ee\u9898\u3002 \u5927\u5e45\u632f\u8361\u526a\u5207\uff08LAOS\uff09\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u4ece\u6d41\u53d8\u4eea\u540c\u65f6\u83b7\u53d6\u4e24\u4e2a\u4fe1\u53f7\u3002\u4e00\u4e2a\u662f\u5e94\u53d8\u7684\u4fe1\u53f7\uff0c\u4e00\u4e2a\u662f\u5e94\u529b\u7684\u4fe1\u53f7\uff08\u4e8b\u5b9e\u4e0a\uff0c\u4e00\u4e2a\u662f\u9a6c\u8fbe\u4f4d\u79fb\u4fe1\u53f7\uff0c\u53e6\u4e00\u4e2a\u662f\u529b\u77e9\u4fe1\u53f7\uff0c\u5b83\u4eec\u90fd\u8981\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u624d\u662f\u5e94\u53d8\u548c\u5e94\u529b\u3002\u6b64\u6587\u5148\u7565\u53bb\u8fd9\u4e00\u4e8b\u5b9e\u4e86\uff09\u3002\u5728\u5e94\u53d8\u63a7\u5236\u7684\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u65bd\u52a0\u4e00\u4e2a\u6b63\u5f26\u7684\u5e94\u53d8\uff0c\u6d4b\u91cf\u5e94\u529b\u7684\u54cd\u5e94\uff0c\u5373 $$\\begin{cases}\\gamma\\left(t\\right)=\\sin\\omega t\\\\ \\sigma\\left(t\\right)=\\sum_{n=0:\\text{odd}}^\\infty\\sigma_n\\sin\\left(\\omega t+\\delta_n\\right)\\end{cases}$$\u7531\u6b64\u5f0f\u53ef\u5b9a\u4e49LAOS\u5b9e\u9a8c\u7ed3\u679c\u7684\u8868\u5f81\u53c2\u6570\uff0c\u4ee5\u53ca\uff0c\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u504f\u79bb\u6b63\u5f26\u51fd\u6570\u7684\u7a0b\u5ea6\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u4f1a\u4ee5\u5bf9\u4f5c\u56fe\uff08\u53c2\u6570\u65b9\u7a0b\u4f5c\u56fe\uff09\uff0c\u5f97\u5230Lissajous\u66f2\u7ebf\u3002\u7136\u540e\u7528\u66f2\u7ebf\u7684\u5f62\u72b6\u7279\u5f81\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u66f2\u7ebf\u504f\u79bb\u692d\u5706\u7684\u7a0b\u5ea6\uff09\u3002\u63cf\u8ff0\u66f2\u7ebf\u5f62\u72b6\u7279\u5f81\uff0c\u5f80\u5f80\u5c31\u662f\u5404\u79cd\u5404\u6837\u7684\u5207\u7ebf\u659c\u7387\uff08\u53ef\u81ea\u884c\u4e0a\u7f51\u641c\u7d22\u590d\u4e60\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc\uff09\u3002\u5728LAOS\u4e2d\u5e38\u7528\uff1a$$G_\\text{M}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\gamma=0}$$$$G_\\text{L}=\\frac{\\sigma{\\gamma_0}}{\\gamma_0}$$$$G_\\text{k}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\sigma=0}$$ MATLAB\u7f16\u7a0b\u6b65\u9aa4 \u5728\u8be6\u7ec6\u4ecb\u7ecdMATLAB\u4ee3\u7801\u4e4b\u524d\uff0c\u5148\u603b\u7ed3\u4e00\u4e0b\u6211\u4eec\u8981\u5b8c\u6210\u4ec0\u4e48\u4efb\u52a1\uff0c\u8ddf\u4ece\u54ea\u4e9b\u6b65\u9aa4\u3002\u6211\u4eec\u7684\u4efb\u52a1\u5c31\u662f\u8981\u4ece\u5e94\u529b\u548c\u5e94\u53d8\u7684\u4fe1\u53f7\u8ba1\u7b97\u51fa\u4e0a\u8ff0\u7684\u8fd9\u4e48\u591a\u53c2\u6570\u3002\u5b9e\u9a8c\u65f6\uff0c\u6211\u4eec\u4f1a\u89c4\u5b9a\u5e94\u53d8\u7684\u5e45\u5ea6\u548c\u9891\u7387\uff0c\u4f46\u662f\uff0c\u7531\u4e8e\u4eea\u5668\u8bef\u5dee\uff0c\u5b9e\u9645\u65bd\u52a0\u5728\u6837\u54c1\u4e0a\u7684\u5e94\u53d8\u5e45\u5ea6\u548c\u9891\u7387\uff0c\u672a\u5fc5\u5b8c\u5168\u7b49\u4e8e\u7684\u547d\u4ee4\u503c\u3002\u5c3d\u7ba1\u8fd9\u4e00\u8bef\u5dee\u975e\u5e38\u5c0f\uff0c\u4f46\u4ece\u505a\u6cd5\u4e0a\u6211\u4eec\u8fd8\u662f\u8981\u4ee5\u5b9e\u9645\u5e94\u53d8\u4fe1\u53f7\u4e3a\u51c6\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u9996\u5148\u8981\u5bf9\u5b9e\u9645\u5e94\u53d8\u4fe1\u53f7\u8fdb\u884c\u5224\u65ad\uff0c\u83b7\u5f97\u5b9e\u9645\u7684\u5e94\u53d8\u5e45\u5ea6\u548c\u9891\u7387\u3002\u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u672a\u77e5\u6b63\u5f26\u4fe1\u53f7\u7684\u632f\u5e45\u3001\u9891\u7387\u548c\u76f8\u4f4d\u7684\u65b9\u6cd5\u5df2\u7ecf\u5728\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4e2d\u4ecb\u7ecd\u4e86\u3002 \u7136\u540e\uff0c\u5c31\u8981\u5904\u7406\u5e94\u529b\u7684\u4fe1\u53f7\u3002\u4e3a\u4e86\u83b7\u5f97\u548c\u540c\u6837\u9700\u8981\u5bf9\u8fdb\u884cfft\u5904\u7406\uff0c\u65b9\u6cd5\u8ddf\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4ecb\u7ecd\u7684\u7c7b\u4f3c\uff0c\u533a\u522b\u53ea\u5728\u4e8e\u6211\u4eec\u9664\u4e86\u8981\u83b7\u5f97\u57fa\u9891\u7684\u5e45\u503c\u548c\u76f8\u4f4d\u4e4b\u5916\uff0c\u8fd8\u8981\u83b7\u5f97\u9ad8\u6b21\u8c10\u6ce2\u7684\u5e45\u503c\u548c\u76f8\u4f4d\u3002\u7406\u8bba\u4e0a\u6765\u8bf4\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u6211\u4eec\u6839\u636e\u9700\u8981\u53d6\u6709\u9650\u7684n\u4e2a\u9ad8\u6b21\u8c10\u6ce2\uff0c\u5c31\u53ef\u4ee5\u6bd4\u8f83\u51c6\u786e\u5730\u91cd\u6784\u51fa\u7684\u6ce2\u5f62\u4e86\u3002\u63d0\u9192\u4e00\u4e0b\uff0c\u5b9a\u4e49\u5f0f\u91cc\u5e94\u53d8\u662f\u4e00\u4e2a\u96f6\u76f8\u4f4d\u7684\u6807\u51c6\u6b63\u5f26\u51fd\u6570\uff0c\u4f46\u5b9e\u9645\u91cd\u6784\u7684\u5e94\u53d8\u6ce2\u5f62\u53ef\u80fd\u4f1a\u6709\u975e\u96f6\u7684\u76f8\u4f4d\uff0c\u8fd9\u65f6\uff0c\u5e94\u529b\u7684\u5404\u4e2a\u8c10\u6ce2\u7684\u76f8\u4f4d\u8981\u90fd\u51cf\u53bb\u5e94\u53d8\u7684\u76f8\u4f4d\uff0c\u624d\u662f\u6211\u4eec\u8981\u6c42\u7684\u3002 Lissajous\u66f2\u7ebf\u7684\u5404\u79cd\u5bfc\u6570\uff0c\u8981\u540c\u65f6\u5728\u548c\u90fd\u91cd\u6784\u4e86\u4e4b\u540e\uff0c\u518d\u8fdb\u884c\u6c42\u5bfc\u8ba1\u7b97\u5f97\u5230\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u8981\u5b8c\u6210\u4ee5\u4e0b\u4e09\u4e2a\u6b65\u9aa4\uff1a \u91cd\u6784\u5e94\u53d8\uff0c\u83b7\u5f97\u3001\u548c\u76f8\u4f4d \u91cd\u6784\uff0c\u83b7\u5f97\u548c\uff0c\u8ba1\u7b97 \u7528\u91cd\u6784\u7684\u548c\u4f5cLissajous\u66f2\u7ebf\uff0c\u4ee5\u53ca\u8ba1\u7b97\u3001\u548c \u91cd\u6784\u548c MATLAB\u5e2e\u52a9\u6587\u4ef6\u91ccfft\u547d\u4ee4\u7684\u6761\u76ee\u6709\u4e00\u4e2a\u8303\u4f8b\uff0c\u5e94\u8be5\u80fd\u89e3\u51b3\u5927\u90e8\u5206\u95ee\u9898\u4e86\u3002\u5176\u4e2d\u5728coherent sampling\uff08\u89c1\u4e0a\u7bc7\u6587\u7ae0\uff09\u5904\uff0c\u8303\u4f8b\u7528next power of 2\uff0c\u5176\u5b9e\u6ca1\u6709\u5fc5\u8981\u3002\u4e0b\u9762\u7684\u4ee3\u7801\uff0c\u6211\u5c31\u4e0d\u591a\u4f5c\u89e3\u91ca\u4e86\u3002 clear all clc % the strain signal fs=1000; % sampling frequency t=0:1\/fs:30.2; % the time series of the strain signal t=t(:); % convert to colume vector w=5; %\u2026","rel":"","context":"In "\u5de5\u4f5c\u7b14\u8bb0"","block_context":{"text":"\u5de5\u4f5c\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/personal\/%e5%b7%a5%e4%bd%9c%e7%ac%94%e8%ae%b0"},"img":{"alt_text":"Figure 1 Parts of the strain and the stress signals","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=525%2C300 1.5x"},"classes":[]},{"id":2993,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/2993","url_meta":{"origin":5244,"position":1},"title":"\u7b2c\u4e09\u5929\u4ea4\u6d41@KIT","author":"Andrew","date":"2011\u5e745\u670819\u65e5","format":false,"excerpt":"LAOS\u4e0epercolation 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Deepak\u73b0\u5728\u4e3b\u8981\u60f3\u8003\u5bdfpercolation\u7684\u95ee\u9898\u3002Deepak\u53d1\u73b0\uff0c\u5411\u7eaf\u805a\u5408\u7269\u4e2d\u6dfb\u52a0\u78b3\u7eb3\u7c73\u7ba1\uff0c\u4e00\u5f00\u59cbI3\/1\u662f\u968f\u6dfb\u52a0\u6d53\u5ea6\u51cf\u5c0f\u7684\uff0c\u8d85\u8fc7\u4e00\u5b9a\u6d53\u5ea6\u4e4b\u540eI3\/1\u624d\u968f\u78b3\u7eb3\u7c73\u7ba1\u7684\u6d53\u5ea6\u589e\u52a0\u800c\u589e\u52a0\u3002Deepak\u5f88\u56f0\u60d1\u4e3a\u4ec0\u4e48\u4f1a\u6709\u4e00\u6bb5\u968f\u7740\u78b3\u7eb3\u7c73\u7ba1\u6d53\u5ea6\u589e\u52a0\u800c\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u51cf\u5c0f\u7684\u533a\u95f4\uff0c\u800c\u5728\u6b64\u533a\u95f4\u91ccG\u2019\u662f\u4e0d\u65ad\u589e\u5927\u7684\u3002 \u6211\u4e5f\u66fe\u7ecf\u5c1d\u8bd5\u901a\u8fc7LAOS\u7814\u7a76percolation\u95ee\u9898\uff0c\u53ea\u662f\u6211\u7684\u4f53\u7cfb\u662fPAA\/Fe(III)\u51dd\u80f6\uff0c\u53ea\u80fd\u901a\u8fc7\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u65b9\u6cd5\uff08Winter\u6cd5\uff09\u627epercolation threshold\u3002\u5728LAOS\u6761\u4ef6\u4e0bcritical gel\u4f1a\u4e0d\u53ef\u9006\u7834\u574f\uff0c\u770b\u4e0d\u5230\u4ec0\u4e48\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u3002 \u56e0\u6b64\u6211\u7684\u611f\u89c9\u662f\uff0c\u4ec5\u4ec5\u5f62\u6210\u7f51\u7edc\u7ed3\u6784\u5e76\u4e0d\u610f\u5473\u7740\u5728LAOS\u4e0b\u663e\u793a\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u3002\u540e\u8005\u9700\u8981\u6240\u5f62\u6210\u7684\u7f51\u7edc\u5177\u6709\u53ef\u9006\u6062\u590d\u7684\u6027\u8d28\u3002\u800c\u4e14\u6062\u590d\u7684\u677e\u5f1b\u65f6\u95f4\u8981\u8ddfLAOS\u7684\u9891\u7387\u76f8\u8026\u5408\u3002\u4e0d\u53d1\u751f\u7f51\u7edc\u7834\u574f\uff0cLAOS\u7684\u6ce2\u5f62\u5c31\u4e0d\u4f1a\u626d\u66f2\uff1b\u4e0d\u6301\u7eed\u6062\u590d\u548c\u7834\u574f\uff0c\u5c31\u5f97\u4e0d\u5230\u5927\u91cf\u5468\u671f\u7684\u975e\u6b63\u5f26LAOS\u7ed3\u679c\uff08FFT\u4e4b\u540e\u5c31\u4e0d\u51fa\u9ad8\u6b21\u8c10\u6ce2\uff09\u3002 \u53e6\u5916\uff0c\u6211\u89c9\u5f97\u5c31\u7b97\u5728LAOS\u4e0b\u6837\u54c1\u663e\u793a\u51fa\u67d0\u79cd\u4e34\u754c\u8f6c\u53d8\uff0c\u8f6c\u53d8\u70b9\u7684\u4f4d\u7f6e\u4e5f\u662f\u4e0d\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4f1a\u4f9d\u8d56\u5e94\u53d8\u5e45\u5ea6\u3002\u8fd9\u4e2a\u4e5f\u8bb8\u80fd\u7b97\u4f5cpercolation\uff0c\u4f46\u4e5f\u8ddfWinter\u6cd5\u7684\u4e34\u754c\u70b9\u6ca1\u6709\u53ef\u6bd4\u6027\u3002\u603b\u7684\u6765\u8bf4\u6211\u4e5f\u89c9\u5f97LAOS\u548cSAOS\u7684\u7ed3\u679c\u4e4b\u95f4\u662f\u6ca1\u6709\u53ef\u6bd4\u6027\u7684\u3002 \u8ddfDeepak\u8fd8\u8ba8\u8bba\u4e86\u5f88\u591a\u7410\u788e\u7684\u95ee\u9898\uff0c\u4e00\u4e2a\u4e0a\u5348\u5c31\u8fc7\u53bb\u4e86\u3002 \u9080\u8bf7\u62a5\u544a \u4e2d\u5348\u5403\u996d\u7684\u65f6\u5019\u53c8\u9047\u5230\u4e86Madhi\u3002\u6211\u53d1\u73b0\u5f53\u5176\u4e5f\u4eba\u7528\u5fb7\u8bed\u804a\u5929\u65f6\u5019\u4ed6\u603b\u662f\u5f88\u5b89\u9759\uff0c\u52a0\u4e0a\u4ed6\u7684\u540d\u5b57\uff0c\u6211\u5c31\u95ee\u4ed6\u662f\u6765\u81ea\u54ea\u91cc\u7684\uff0c\u539f\u6765\u662f\u4f0a\u6717\u3002\u4ed6\u5728\u4f0a\u6717\u8bfb\u535a\u58eb\u5b66\u4f4d\uff0c\u6765\u5fb7\u56fd\u8fd9\u91cc\u4ea4\u6362\u516d\u4e2a\u6708\u3002\u4ed6\u4e0d\u7ecf\u5e38\u8bf4\u8bdd\u4f46\u6574\u5929\u7b11\uff0c\u5f88\u6e29\u6587\u5c14\u96c5\u3002 \u4e2d\u53481\u70b9\u9080\u8bf7\u4e86\u4e00\u4e2a\u6559\u6388\u8fc7\u6765\u505a\u62a5\u544a\uff0c\u51e0\u4e4e\u5168\u4f53\u7ec4\u5458\u90fd\u8981\u53c2\u52a0\uff0cProf. Whilhelm\u4e3b\u6301\u3002\u62a5\u544a\u8005\u662fThomas Hanemann\uff0c\u6765\u81eaInstitute of Applied Materials\uff0c\u62a5\u544a\u7684\u9898\u76ee\u662fPolymer-based Materials in Microsystem Technologies\u3002\u62a5\u544a\u7684\u4e3b\u8981\u5185\u5bb9\u653e\u5728\u5fae\u7c73\u7ea7\u7684\u5149\u5b66\u5143\u4ef6\u4e0a\u3002\u5bf9\u8fd9\u79cd\u6750\u6599\u7684\u8981\u6c42\u9996\u5148\u662f\u9ad8\u900f\u5149\u7387\u548c\u9ad8\u6298\u5c04\u7387\u3002\u9ad8\u900f\u5149\u7387\u7684\u805a\u5408\u7269\u6709\u5f88\u591a\uff08\u4f8b\u5982PMMA\u7b49\uff09\uff0c\u4f46\u4e3a\u4e86\u63d0\u9ad8\u6298\u5c04\u7387\uff0c\u5c31\u8981\u5f80\u805a\u5408\u7269\u91cc\u6dfb\u52a0\u5176\u4ed6\u6210\u4efd\uff0c\u5f80\u5f80\u5c31\u8981\u964d\u4f4e\u900f\u5149\u7387\u3002\u63d0\u9ad8\u6298\u5c04\u7387\u53ef\u4ee5\u901a\u8fc7\u6dfb\u52a0\u65e0\u673a\u586b\u6599\u6216\u8005\u6709\u673a\u63ba\u6742\u3002\u65e0\u673a\u586b\u6599\u4ea7\u751f\u7684\u95ee\u9898\u5c31\u662f\u6563\u5c04\u3001\u5206\u6563\u6027\u3001\u6d53\u5ea6\u548c\u8868\u9762\u6027\u8d28\u7b49\u7b49\u90fd\u4f1a\u5f71\u54cd\u6750\u6599\u7684\u900f\u5149\u7387\u3002\u6709\u673a\u63ba\u6742\u867d\u7136\u6ca1\u6709\u6563\u5c04\u7684\u95ee\u9898\uff0c\u4f46\u6709\u5f88\u591a\u5206\u5b50\u672c\u8eab\u53c8\u6709\u989c\u8272\uff0c\u800c\u4e14\u63ba\u4e86\u6709\u673a\u5c0f\u5206\u5b50\u5bf9\u805a\u5408\u7269\u6709\u589e\u5851\u4f5c\u7528\uff0c\u964d\u4f4e\u4e86\u6750\u6599\u7684\u529b\u5b66\u7a33\u5b9a\u6027\uff08\u4f8b\u5982\u8010\u9ad8\u6e29\u6027\u8d28\uff09\u3002\u4f46\u8fd9\u4e9b\u90fd\u7ec8\u5f52\u53ea\u662f\u8ba8\u8bba\uff0cProf. Hanemann\u8fd8\u662f\u6210\u529f\u5730\u505a\u51fa\u4e86\u4e00\u4e9b\u6750\u6599\u3002\u53e6\u5916\uff0cProf. Hanemann\u8fd8\u5728\u7814\u7a76\u9676\u74f7\u7684\u5fae\u91cf\u6ce8\u5851\u6280\u672f\uff08\u901a\u8fc7sol-gel\u65b9\u6cd5\uff09\u3002\u8fd9\u4e2a\u6280\u672f\u53ef\u4ee5\u7528\u4e8e\u505a\u975e\u5e38\u7cbe\u7ec6\u7684\u5143\u4ef6\u3002\u4f46\u662f\u5230\u4e86\u5fae\u91cf\u6210\u5f62\u7684\u5c3a\u5ea6\uff0c\u5404\u4e2a\u73af\u8282\u90fd\u6709\u96be\u5ea6\u3002\u4f8b\u5982\u8d77\u59cb\u7684feedstock\u9700\u8981\u4f18\u5316\uff0c\u6d41\u9053\u9700\u8981\u4f18\u5316\u7b49\u7b49\u3002 \u8bb2\u5ea7\u5927\u6982\u82b1\u4e86\u4e00\u4e2a\u5c0f\u65f6\uff0c\u5b8c\u4e86\u4e4b\u540e\u6709Nico\u5e26Hanemann\u6559\u6388\u548c\u4ed6\u7684\u5b66\u751f\u53c2\u89c2\u5b9e\u9a8c\u5ba4\uff0c\u7136\u540e\u53c8\u5230\u4e00\u4e2a\u623f\u95f4\u91cc\u559d\u4e0b\u5348\u8336\u7ee7\u7eed\u95f2\u804a\uff0c\u5404\u81ea\u4ecb\u7ecd\u81ea\u5df1\u7684\u5de5\u4f5c\u5185\u5bb9\uff0c\u5bfb\u627e\u5408\u4f5c\u3002Hanemann\u6709\u4e9b\u5b66\u751f\u662f\u8981\u7814\u7a76feedstock\u7684\u6d41\u4f53\u529b\u5b66\u6027\u8d28\u7684\uff0c\u6070\u597dWhilhelm\u7684\u5b66\u751fRoland\u5c31\u662f\u4e13\u95e8\u6363\u9f13\u6d41\u52a8\u4e0d\u7a33\u5b9a\u6027\u7684\uff0c\u4e8e\u662f\u8fd8\u786e\u5b9e\u6709\u5408\u4f5c\u7684\u7a7a\u95f4\u3002 Fourier\u53d8\u6362 \u56de\u5230\u529e\u516c\u5ba4\u5df2\u7ecf\u662f\u4e0b\u5348\u4e09\u70b9\u591a\u4e86\u3002\u6211\u53c8\u8bf7\u6559Kathrin\u5173\u4e8eLAOS\u65f6\u95f4\u626b\u63cf\u7684\u60f3\u6cd5\u3002\u7531\u4e8e\u6211\u51c6\u5907\u5728\u4e00\u4e2a\u957f\u65f6\u95f4\u4fe1\u53f7\u4e2d\u9010\u6bb5\u622a\u53d6\u90e8\u5206\u4fe1\u53f7\u8fdb\u884c\u5206\u6790\u3002\u56e0\u6b64\u622a\u53d6\u7684\u65f6\u95f4\u7a97\u53e3\u7684\u5927\u5c0f\u662f\u53d7\u9650\u7684\u3002\u5982\u679c\u5728\u6709\u9650\u7684\u65f6\u95f4\u7a97\u53e3\u0394t\u4e2d\u6709N\u4e2a\u6570\u636e\uff08oversampling\u4ee5\u540e\uff09\uff0c\u90a3\u4e48Fourier\u53d8\u6362\u4e4b\u540e\u7684\u9891\u7387\u5206\u8fa8\u7387\u5c31\u662f\u0394f = fs \/ N\uff0cfs\u662fsampling frequency\u3002\u8fdb\u4e00\u6b65\u5206\u6790\u53d1\u73b0\uff0c\u0394f = f0 \/ Ncycle\uff0cf0\u662f\u65f6\u57df\u4fe1\u53f7\u7684\u9891\u7387\uff0cNcycle\u662f\u65f6\u57df\u4fe1\u53f7\u7684\u5468\u671f\u6570\uff0c\u5fc5\u987b\u662f\u975e\u96f6\u6b63\u6574\u6570\u3002\u5728\u7ed9\u5b9af0\u7684\u6761\u4ef6\u4e0b\uff0c\u662f\u4e0d\u80fd\u65e2\u8ba9Ncycle\u5c3d\u53ef\u80fd\u5c0f\u53c8\u8ba9\u0394f\u5c3d\u53ef\u80fd\u5c0f\u7684\u3002\u5982\u679c\u60c5\u613fNcycle\u5c3d\u53ef\u80fd\u5c0f\uff08Ncycle = 1\uff09\u90a3\u4e48\u0394f = f0\u3002\u4e5f\u5c31\u662f\u8bf4Fourier\u53d8\u6362\u4e4b\u540e\u7684\u66f2\u7ebf\u5c31\u603b\u662f\u4ecef = 0\u4e00\u4e0b\u8df3\u5230f = f0\u3002\u5e94\u8be5\u662f\u6700\u5dee\u7684\u60c5\u51b5\u3002\u4ee5\u4e0b\u662ff0 = 1Hz\uff0cNcycle = 1\u7684\u60c5\u51b5\u3002 \u7531\u4e8e\u6211\u662f\u8981\u53d6\u7279\u5b9a\u9891\u7387\u7684\u5e45\u503c\uff08f0\u30013f0\u30015f0\u2026\u2026\uff09\uff0c\u6211\u62c5\u5fc3\u5f53\u0394f\u5f88\u5927\u7684\u65f6\u5019\uff0c\u5728\u6240\u9700\u7684\u9891\u7387\u4f4d\u7f6e\u4f1a\u6ca1\u6709\u6570\u636e\u70b9\u3002\u4f46\u6839\u636e\u4e0a\u9762\u7684\u5206\u6790\u53ea\u8981Ncycle\u662f\u975e\u96f6\u6b63\u6574\u6570\u7684\u8bdd\uff0c\u8fd9\u4e2a\u60c5\u51b5\u662f\u4e0d\u4f1a\u51fa\u73b0\u7684\u3002\u6211\u53c8\u62c5\u5fc3\u5bf9\u4e8e\u540c\u6837\u7684\u65f6\u57df\u4fe1\u53f7\uff0cFourier\u53d8\u6362\u7684\u0394f\u4e0d\u540c\uff0c\u5f97\u5230\u7684\u5e45\u503c\u4f1a\u4e0d\u540c\uff0c\u81f3\u5c11\u5f53\u0394f\u7279\u522b\u4f4e\u7684\u65f6\u5019\u3002Kathrin\u5012\u662f\u7814\u7a76\u8fc7\u8fd9\u4e2a\u95ee\u9898\uff0c\u5979\u7ed9\u6211\u770b\u4e86\u4e00\u4e2a\u7ed3\u679c\uff0c\u5dee\u522b\u5f88\u5c0f\u3002\u56e0\u6b64\u4e5f\u5c31\u4e0d\u7528\u62c5\u5fc3\u8fd9\u4e2a\u95ee\u9898\u3002 \u9891\u7387\u5206\u8fa8\u7387\u7684\u95ee\u9898\u5206\u6790\u5b8c\u4e4b\u540e\uff0c\u4e0b\u4e00\u4e2a\u95ee\u9898\u5c31\u662f\u5b9e\u9645\u4fe1\u53f7\u7684\u57fa\u9891\u5f80\u5f80\u5e76\u4e0d\u7cbe\u786e\u5730\u7b49\u4e8e\u5b9e\u9a8c\u8f93\u5165\u7684\u9891\u7387\u3002\u4f8b\u5982\u6211\u9700\u8981\u9a6c\u8fbe\u8fdb\u884c\u9891\u7387\u4e3a1Hz\u7684\u632f\u8361\uff0c\u4f46\u5b9e\u9645\u9a6c\u8fbe\u4fe1\u53f7\u53d6\u51fa\u6765\u57fa\u9891\u5f88\u53ef\u80fd\u662f0.99Hz\u3002\u8fd9\u4e00\u70b9\u70b9\u5dee\u522b\u57283f0\u30015f0\u7b49\u500d\u9891\u5904\u4f1a\u6210\u500d\u589e\u5927\u3002\u5982\u679c\u6211\u5728\u505aFourier\u53d8\u6362\u7684\u65f6\u5019\u4ee5\u5b9e\u9a8c\u8f93\u5165\u7684\uff08\u81ea\u4ee5\u4e3a\u662f\u7684\uff09\u9891\u7387\u4e3a\u57fa\u9891\u6765\u53d6\u8c10\u6ce2\u5e45\u503c\uff0c\u5c31\u6709\u53ef\u80fd\u4f1a\u51fa\u9519\u3002\u56e0\u6b64\u6709\u5fc5\u8981\u5148\u786e\u5b9a\u4fe1\u53f7\u7684\u771f\u5b9e\u57fa\u9891\uff0c\u518d\u8fdb\u884cFourier\u53d8\u6362\u3002\u4e8e\u662f\u6211\u8bf7\u6559\u786e\u5b9a\u4fe1\u53f7\u7684\u771f\u5b9e\u57fa\u9891\u6709\u6ca1\u6709\u6bd4\u8f83\u201c\u4e13\u4e1a\u201d\u4e00\u70b9\u7684\u65b9\u6cd5\u3002\u8ba8\u8bba\u7684\u7ed3\u679c\u662f\u8c8c\u4f3c\u6ca1\u6709\uff0c\u65e0\u975e\u5c31\u662f\u9009\u62e9\u6070\u5f53\u7684\u7a97\u53e3\u8fdb\u884cfind max\u3002\u56e0\u6b64\uff0c\u5148\u662f\u4e3a\u4e86\u786e\u5b9a\u57fa\u9891\u5c31\u8981\u5360\u7528\u4e00\u5b9a\u7684Ncycle\u3002\u786e\u5b9a\u57fa\u9891\u6240\u7528\u7684Ncycle\u4e5f\u662f\u8d8a\u5c0f\u8d8a\u597d\u7684\u3002\u6240\u4ee5\u5982\u679c\u4e0d\u786e\u5b9a\u4e0a\u8ff0\u7684\u0394f\u592a\u5927\u5bf9\u5e45\u503c\u6ca1\u6709\u5f71\u54cd\uff0c\u57fa\u9891\u7684\u786e\u5b9a\u8fd9\u4e00\u6b65\u5c31\u5f88\u96be\u4fdd\u8bc1\u4e86\u3002\u2026","rel":"","context":"In "AV"","block_context":{"text":"AV","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/personal\/av"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2011\/05\/untitled-300x225.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":9177,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9177","url_meta":{"origin":5244,"position":2},"title":"\u5728\u7535\u5b50\u8ba1\u7b97\u673a\u4e4b\u524d\u7684Fourier\u5206\u6790","author":"Andrew","date":"2025\u5e743\u670830\u65e5","format":"aside","excerpt":"[latexpage] \u5982\u679c\u6211\u4eec\u8981\u5bf9\u4e00\u4e2a\u5468\u671f\u4fe1\u53f7\u8fdb\u884cFourier\u7ea7\u6570\u5206\u6790\uff0c\u90a3\u4e48\u6211\u4eec\u662f\u5047\u5b9a\u8be5\u4fe1\u53f7\u53ef\u4ee5\u8868\u793a\u6210\\[f\\left(x\\right)=\\sum_{n=-\\infty}^\\infty c_n e^{\\mathrm{i}2\\pi n x\/P}\\]\u5176\u4e2d\\[c_n=\\frac{1}{P}\\int_0^P f\\left(x\\right)e^{-\\mathrm{i} 2\\pi n x\/P}\\mathrm{d}x\\] \u5728\u4eca\u5929\uff0c\u6211\u4eec\u7684\u5b9e\u9a8c\u4fe1\u53f7\u57fa\u672c\u90fd\u5df2\u7ecf\u6570\u5b57\u5316\uff0c\u5728\u7535\u5b50\u8ba1\u7b97\u673a\u4e2d\u4e00\u79d2\u949f\u4e0d\u5230\u5c31\u505a\u5b8c\u4e86\u4e00\u4e2a\u5feb\u901fFourier\u53d8\u6362\u8ba1\u7b97\u3002\u4f46\u662f\u5728\u4e0a\u4e16\u7eaa50~60\u5e74\u4ee3\uff0c\u7535\u5b50\u8ba1\u7b97\u673a\u8fd8\u6ca1\u53d1\u660e\u4e4b\u524d\uff0c\u5b9e\u9a8c\u7ed3\u679c\u672c\u8eab\u5c31\u53ef\u80fd\u662f\u901a\u8fc7X-Y\u8bb0\u5f55\u4eea\u6216\u53cc\u7b14\u8bb0\u5f55\u4eea\u753b\u5728\u5750\u6807\u7eb8\u4e0a\u7684\u3002\u5f53\u65f6\u7684\u4eba\u662f\u600e\u6837\u5bf9\u8fd9\u4e48\u4e00\u6bb5\u58a8\u8ff9\u8fdb\u884cFourier\u7ea7\u6570\u5206\u6790\u7684\u5462\uff1f\u6211\u4eec\u53ef\u4ee5\u770b\u5230\uff0cFourier\u7ea7\u6570\u7684\u5404\u4e2a\u7cfb\u6570$c_n$\u662f\u4e00\u4e2a\u79ef\u5206\u3002\u5728\u90a3\u4e2a\u5e74\u4ee3\uff0c\u79ef\u5206\u53ef\u4ee5\u4f7f\u7528\u6b27\u62c9\u6cd5\u6570\u503c\u6c42\u89e3\u3002\u5728\u6b64\u4e4b\u524d\uff0c\u9700\u8981\u5c06\u88ab\u79ef\u51fd\u6570\u8fdb\u884c\u4eba\u5de5\u91c7\u6837\uff08sampling\uff09\u79bb\u6563\u5316\u3002\u56e0\u6b64\u8fd9\u662f\u4e00\u4e2a\u5f88\u7e41\u7410\u7684\u4efb\u52a1\u3002 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Philippoff\u5f88\u65e9\u5c31\u5f00\u53d1\u5bbd\u9891\u8303\u56f4\u7684\u52a8\u6001\u6d4b\u8bd5\u4eea\u5668\u3002\u4ed6\u57281966\u5e74\u53d1\u8868\u7684\u6587\u7ae0\u4e2d\uff0c\u7528\u4e00\u4e2a\u7279\u522b\u6784\u5efa\u7684\u53ef\u8c03\u8282\u6a21\u62df\u8ba1\u7b97\u5668\uff0c\u53bb\u540c\u65f6\u751f\u6210\u4e00\u4e2a\u57fa\u9891\u548c3\u500d\u9891\u7684\u6b63\u5f26\u4fe1\u53f7\uff0c\u7136\u540e\u4f5c\u52a0\u548c\u3002\u4ed6\u901a\u8fc7\u8c03\u8282\u8fd9\u4e2a\u6a21\u62df\u8ba1\u7b97\u5668\uff0c\u51d1\u51fa\u4e0e\u5b9e\u9a8c\u7ed3\u679c\u5f62\u72b6\u76f8\u4f3c\u7684Lissajous\u66f2\u7ebf\uff0c\u4ece\u800c\u5f97\u77e5\u8be5\u66f2\u7ebf\u76843\u500d\u9891\u8c10\u6ce2\u5e45\u503c\u548c\u76f8\u4f4d\u3002 1960\u5e74\u4ee3\uff0c\u6070\u597d\u662f\u6a21\u62df\u6280\u672f\u7684\u98a0\u5cf0\u65f6\u671f\u3002\u6a21\u62df\u8ba1\u7b97\u7535\u8def\u80fd\u80fd\u505a\u5f88\u591a\u4e8b\u60c5\u3002\u7531\u6d41\u53d8\u4eea\u5f97\u51fa\u7684\u6ce2\u5f62\u4fe1\u53f7\uff0c\u53ef\u901a\u8fc7\u76f8\u5173\u5668\u4e0e\u7ed9\u5b9a\u7684\u6ce2\u5f62\u6bd4\u8f83\uff0c\u8f93\u51fa\u7684\u7ed3\u679c\u662f\u4e0e\u7ed9\u5b9a\u6ce2\u5f62\u7684\u5185\u79ef\u3002\u8fd9\u6837\u5c31\u53ef\u4ee5\u901a\u8fc7\u8bbe\u5b9a\u53c2\u8003\u6ce2\u5f62\u4fe1\u53f7\uff0c\u4ece\u539f\u59cb\u4fe1\u53f7\u4e2d\u63d0\u53d6\u76f8\u5e94\u7684\u5206\u91cf\uff0c\u4ece\u800c\u5b9e\u73b0\u8c10\u6ce2\u5206\u6790\u3002\u4f8b\u5982\uff0cQueen's Collage\u673a\u68b0\u5de5\u7a0b\u7cfb\u7684Harris\u548cBogie\u3002 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In A computationally efficient method for accurate spectral testing without requiring coherent sampling, Test Conference, 2004. Proceedings. ITC 2004. International, 26-28 Oct. 2004; 2004; pp 1398-1407. Minshun, W.; Degang, C.; Guican, C. 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