\u70ba\u4e86\u6aa2\u67e5\u7a0b\u5f0f\u662f\u5426\u6b63\u5e38\u904b\u4f5c\uff0c\u6211\u5011\u69cb\u5efa\u4e00\u500b\u7531\u5169\u500b\u6b63\u5f26\u51fd\u6578\u76f8\u52a0\u800c\u6210\u7684\u6a23\u672c\u9663\u5217 sin(10*2*pi*x)+0.5*sin(5*2*pi*x) \u4e26\u8f38\u5165\u7a0b\u5f0f\u3002\u7a0b\u5f0f\u7e6a\u88fd\u51fa\u4ee5\u4e0b\u7d50\u679c\uff1a
\n
![\"\u5085\u7acb\u8449\u8b8a\u63db\u8207\u8a0a\u865f\u983b\u8b5c\"](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US725.webp\")
\u5716 1 \u8a0a\u865f\u6642\u9593\u51fd\u6578\u5716<\/span><\/i><\/em><\/p><\/div><\/p>\n <\/p>\n \u5716 2 \u8a0a\u865f\u983b\u8b5c\u5716<\/p><\/div>\n <\/p>\n \u6709\u5169\u500b \u8ae7\u6ce2<\/a> \u5728\u983b\u8b5c\u5716\u4e0a \u2013 5 Hz \u632f\u5e45\u70ba 0.5 V\uff0c10 Hz \u632f\u5e45\u70ba 1 V\uff0c\u4e00\u5207\u8207\u539f\u59cb\u8a0a\u865f\u516c\u5f0f\u4e2d\u7684\u60c5\u6cc1\u4e00\u81f4\u3002\u6c92\u554f\u984c\uff0c\u7a0b\u5f0f\u904b\u4f5c\u6b63\u78ba\u3002<\/p>\n \u9019\u610f\u5473\u8457\uff0c\u5982\u679c\u6211\u5011\u5c07\u7531\u5169\u500b\u6b63\u5f26\u6ce2\u6df7\u5408\u800c\u6210\u7684\u771f\u5be6\u8a0a\u865f\u8f38\u5165 ADC\uff0c\u6211\u5011\u5c07\u7372\u5f97\u4e00\u500b\u7531\u5169\u500b\u8ae7\u6ce2\u7d44\u6210\u7684\u985e\u4f3c\u983b\u8b5c\u3002<\/p>\n \u56e0\u6b64\uff0c\u6211\u5011\u7684 \u771f\u5be6 <\/span><\/b><\/strong>\u6e2c\u91cf\u8a0a\u865f \u6301\u7e8c\u6642\u9593\u70ba 5 \u79d2<\/span><\/b><\/strong>\uff0c\u7d93 ADC \u6578\u4f4d\u5316\uff0c\u5373\u8868\u793a\u70ba \u96e2\u6563 <\/span><\/b><\/strong>\u6a23\u672c\uff0c\u5177\u6709 \u96e2\u6563\u975e\u9031\u671f\u6027 <\/span><\/b><\/strong>\u983b\u8b5c\u3002 \u73fe\u5728\u8b93\u6211\u5011\u5617\u8a66\u6e2c\u91cf\u540c\u4e00\u8a0a\u865f\uff0c\u6301\u7e8c\u6642\u9593\u70ba 0.5 \u79d2\u3002<\/p>\n \u5716 3 \u6e2c\u91cf\u9031\u671f\u70ba 0.5 \u79d2\u7684\u51fd\u6578 sin(10*2*pi*x)+0.5*sin(5*2*pi*x) \u5716<\/p><\/div>\n <\/p>\n \u5716 4 \u8a72\u51fd\u6578\u7684\u983b\u8b5c<\/p><\/div>\n <\/p>\n \u9019\u88e1\u6709\u4e9b\u554f\u984c\uff0110 Hz \u7684\u8ae7\u6ce2\u7e6a\u88fd\u6b63\u5e38\uff0c\u4f46 5 Hz \u7684\u8ae7\u6ce2\u4f4d\u7f6e\u51fa\u73fe\u4e86\u4e00\u4e9b\u4e0d\u660e\u78ba\u7684\u8ae7\u6ce2\u3002<\/p>\n \u7db2\u8def\u4e0a\u8aaa\u9700\u8981\u5728\u6a23\u672c\u672b\u5c3e\u88dc\u96f6\uff0c\u983b\u8b5c\u5c31\u6703\u6b63\u5e38\u7e6a\u88fd\u3002<\/p>\n \u5716 5 \u6211\u5011\u5df2\u5728\u6a23\u672c\u672b\u5c3e\u88dc\u96f6\u81f3 5 \u79d2<\/p><\/div>\n <\/p>\n \u5716 6. \u7372\u5f97\u7684\u983b\u8b5c\u3002<\/p><\/div>\n <\/p>\n \u9019\u5b8c\u5168\u4e0d\u662f\u90a3\u9ebc\u56de\u4e8b\u3002\u6211\u5fc5\u9808\u6df1\u5165\u7814\u7a76\u7406\u8ad6\u3002\u8b93\u6211\u5011\u524d\u5f80 wikipedia<\/b><\/strong><\/a><\/span>\u00a0– \u77e5\u8b58\u7684\u4f86\u6e90\u3002<\/p>\n \u5f9e\u6578\u5b78\u89d2\u5ea6\u4f86\u770b\uff0c\u6211\u5011\u6301\u7e8c\u6642\u9593\u70ba T \u79d2\u7684\u8a0a\u865f\u662f\u5b9a\u7fa9\u5728\u5340\u9593 {0, T} \u4e0a\u7684\u67d0\u500b\u51fd\u6578 f(x)\uff08\u6b64\u8655 X \u70ba\u6642\u9593\uff09\u3002\u6b64\u985e\u51fd\u6578\u7e3d\u53ef\u4ee5\u8868\u793a\u70ba\u4ee5\u4e0b\u5f62\u5f0f\u7684\u8ae7\u6ce2\u51fd\u6578\uff08\u6b63\u5f26\u6216\u9918\u5f26\uff09\u4e4b\u548c\uff1a<\/p>\n \u00a0\uff081\uff09\uff0c\u5176\u4e2d\uff1a<\/p>\n <\/p><\/div>\n k \u70ba\u4e09\u89d2\u51fd\u6578\u7684\u5e8f\u6578\uff08\u8ae7\u6ce2\u5206\u91cf\u7684\u5e8f\u6578\uff0c\u8ae7\u6ce2\u7684\u5e8f\u6578\uff09 (2),<\/p><\/div>\n <\/p>\n <\/p>\n <\/p>\n \u5176\u4e2d <\/p>\n \u6216<\/p>\n (3)<\/p><\/div>\n <\/p>\n <\/p>\n <\/p>\n \u4fc2\u6578\uff081\uff09\u8207\uff083\uff09\u4e4b\u9593\u7684\u95dc\u4fc2\u7531\u4ee5\u4e0b\u516c\u5f0f\u8868\u793a\uff1a<\/p>\n <\/p>\n <\/p>\n <\/p>\n <\/p>\n \u6ce8\u610f\uff0c\u5085\u7acb\u8449\u7d1a\u6578\u7684\u9019\u4e09\u7a2e\u8868\u793a\u6cd5\u5b8c\u5168\u7b49\u6548\u3002\u6709\u6642\u5728\u8655\u7406\u5085\u7acb\u8449\u7d1a\u6578\u6642\uff0c\u4f7f\u7528\u865b\u6578\u53c3\u6578\u7684\u6307\u6578\u800c\u975e\u5f26\u6ce2\u8207\u9918\u5f26\u6ce2\uff08\u5373\u4f7f\u7528\u8907\u6578\u5f62\u5f0f\u7684\u5085\u7acb\u8449\u8b8a\u63db\uff09\u6703\u66f4\u65b9\u4fbf\u3002\u4f46\u5c0d\u6211\u5011\u800c\u8a00\uff0c\u4f7f\u7528\u516c\u5f0f (1) \u66f4\u70ba\u65b9\u4fbf\uff0c\u5176\u4e2d\u5085\u7acb\u8449\u7d1a\u6578\u8868\u793a\u70ba\u5177\u6709\u5c0d\u61c9\u632f\u5e45\u8207\u76f8\u4f4d\u7684\u9918\u5f26\u6ce2\u4e4b\u548c\u3002\u56b4\u683c\u4f86\u8aaa\uff0c\u771f\u5be6\u8a0a\u865f\u7684\u5085\u7acb\u8449\u8b8a\u63db\u78ba\u5be6\u6703\u7522\u751f\u8907\u6578\u4fc2\u6578\uff08\u5f62\u5f0f (3)\uff09\uff1a\u6bcf\u500b\u4fc2\u6578\u540c\u6642\u651c\u5e36\u5176\u8ae7\u6ce2\u7684\u632f\u5e45\u8207\u76f8\u4f4d\u3002\u5c0d\u65bc\u771f\u5be6\u8a0a\u865f\uff0c\u9019\u4e9b\u8907\u6578\u4fc2\u6578\u5177\u6709\u5171\u8edb\uff08\u5384\u7c73\u7279\uff09\u5c0d\u7a31\u6027 \u2013 \u8ca0\u983b\u7387\u534a\u90e8\u55ae\u7d14\u662f\u6b63\u983b\u7387\u534a\u90e8\u7684\u93e1\u50cf\uff0c\u4e14\u4e0d\u651c\u5e36\u984d\u5916\u8cc7\u8a0a\u3002\u56e0\u6b64\uff0c\u5f9e\u8907\u6578\u4fc2\u6578\u4e2d\uff0c\u6211\u5011\u7e3d\u80fd\u8f49\u63db\u81f3\u516c\u5f0f (1) \u7684\u5be6\u6578\u975e\u8ca0\u632f\u5e45 Ak \u8207\u76f8\u4f4d \u03b8k \u2013 \u800c\u5206\u6790\u7a0b\u5f0f\u7e6a\u88fd\u7684\u6b63\u662f\u6b64\u632f\u5e45\u983b\u8b5c\u3002<\/p>\n <\/p>\n \u7e3d\u7d50\uff1a<\/span><\/b><\/u><\/strong> \u8acb\u6ce8\u610f\u66f4\u591a\u8981\u9ede\uff0c\u7406\u89e3\u9019\u4e9b\u8981\u9ede\u5c0d\u65bc\u6b63\u78ba\u5c07\u5085\u7acb\u8449\u8b8a\u63db\u61c9\u7528\u65bc\u8a0a\u865f\u5206\u6790\u662f\u5fc5\u9700\u7684\u3002\u5982\u679c\u6211\u5011\u5728\u6574\u500b X \u8ef8\u4e0a\u8003\u616e\u5085\u7acb\u8449\u7d1a\u6578\uff08\u6b63\u5f26\u6ce2\u4e4b\u548c\uff09\uff0c\u6211\u5011\u6703\u767c\u73fe\uff0c\u5728\u5340\u9593 {0, T} \u4e4b\u5916\uff0c\u5085\u7acb\u8449\u7d1a\u6578\u51fd\u6578\u5c07\u9031\u671f\u6027\u5730\u91cd\u8907\u6211\u5011\u7684\u51fd\u6578\u3002<\/p>\n \u4f8b\u5982\uff0c\u5728\u5716 7 \u7684\u5716\u8868\u4e2d\uff0c\u539f\u59cb\u51fd\u6578\u5b9a\u7fa9\u5728\u5340\u9593 {-T\\2, +T\\2} \u4e0a\uff0c\u800c\u5085\u7acb\u8449\u7d1a\u6578\u8868\u793a\u5b9a\u7fa9\u5728\u6574\u500b x \u8ef8\u4e0a\u7684\u9031\u671f\u51fd\u6578\u3002<\/p>\n This is because the sinusoids themselves are periodic functions, so their sum will also be a periodic function.<\/p>\n Figure 7 Representation of a non-periodic source function by a Fourier series<\/p><\/div>\n Thus:<\/p>\n Our original function is a continuous, non-periodic function defined on some segment of length T. Next.<\/p>\n The periods of harmonic components are multiples of the interval {0, T}, on which the initial function f(x) is defined. In other words, the periods of harmonics are multiples of the duration of the signal measurement. For example, the period of the first harmonic in a Fourier series is equal to the interval T in which the function f(x) is defined. The period of the second harmonic in a Fourier series is equal to the interval T\/2. And so on (see Figure 8).<\/p>\n Fig. 8 Periods (frequencies) of harmonic components of the Fourier series (here T=2\u03c0)<\/p><\/div>\n Accordingly, the frequencies of harmonic components are multiples of 1\/T. That is, frequencies of harmonic components Fk are Fk= k\\T, where k runs values from 0 to \u221e, for example, k=0 F0=0; k=1 F1=1\\T; k=2 F2=2\\T;k=3 F3=3\\T;…. Fk= k\\T (at zero frequency, a constant component).<\/p>\n Let our initial function, is a signal recorded during T=1 sec. Then the period of the first harmonic will be equal to the duration of our signal T1=T=1 sec and the frequency of the harmonic is equal to 1 Hz. The period of the second harmonic will be equal to the duration of our signal divided by 2 (T2=T\/2=0.5 sec.) and the frequency is equal to 2 Hz. For the third harmonic, T3=T\/3 sec and the frequency is 3 Hz. And so on.<\/p>\n The step between harmonics in this case is 1 Hz.<\/p>\n Thus, a signal with a duration of 1 sec can be decomposed into harmonic components (to obtain a spectrum) with a frequency resolution of 1 Hz. There is a way to artificially increase the signal duration by adding zeros to the array of samples. But it does not increase the real frequency resolution.<\/p>\n With the development of digital technology the ways of measurement data (signals) storage have changed. Whereas previously a signal could be recorded on a tape recorder and stored on a tape in analog form, now signals are digitized and stored in files in computer memory as a set of numbers (counts).<\/p>\n The usual scheme of signal measurement and digitization looks as follows.<\/p>\n Measuring transducer —- Signal normalizer —- ADC —– Computer <\/span><\/i><\/em><\/p>\n The signal from the measuring transducer goes to the ADC for a period of time T. The signal readings (sampling) received during time T are transmitted to the computer and saved in the memory.<\/p>\n Fig.10 Digitized signal – N samples received for time T<\/p><\/div>\n What are the requirements to signal digitization parameters? A device which converts the input analog signal into a discrete code (digital signal) is called an analog-to-digital converter (ADC) (\u00a9 Wiki).<\/p>\n One of the basic parameters of ADC is the maximum sampling rate – the frequency of sampling of a signal which is continuous in time. Sample rate is measured in hertz. ((\u00a9 Wiki))<\/p>\n \u6839\u64da\u79d1\u6377\u723e\u5c3c\u79d1\u592b\u5b9a\u7406\uff0c\u82e5\u9023\u7e8c\u8a0a\u865f\u7684\u983b\u8b5c\u53d7\u9650\u65bc\u983b\u7387 Fmax\uff0c\u5247\u53ef\u5f9e\u4ee5\u6642\u9593\u9593\u9694 \u0394t \u2264 1\/(2*Fmax)\uff08\u5373\u63a1\u6a23\u983b\u7387 Fd \u2265 2*Fmax\uff09\u53d6\u5f97\u7684\u96e2\u6563\u6a23\u672c\u4e2d\u5b8c\u6574\u4e14\u552f\u4e00\u5730\u91cd\u5efa\u8a72\u8a0a\u865f\uff0c\u5176\u4e2d Fd \u2013 \u63a1\u6a23\u983b\u7387\uff1bFmax \u2013 \u8a0a\u865f\u983b\u8b5c\u7684\u6700\u5927\u983b\u7387\u3002\u63db\u8a00\u4e4b\uff0c\u8a0a\u865f\u6578\u4f4d\u5316\u7684\u983b\u7387\uff08ADC \u7684\u63a1\u6a23\u983b\u7387\uff09\u5fc5\u9808\u81f3\u5c11\u70ba\u6211\u5011\u6b32\u6e2c\u91cf\u8a0a\u865f\u6700\u5927\u983b\u7387\u7684\u5169\u500d\u3002<\/p>\n And what will happen if we take samples with lower frequency than required by Kotelnikov’s theorem?<\/p>\n \u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\u5b58\u5728\u300c\u6df7\u758a<\/a>\u300d\u6548\u61c9\uff08\u4ea6\u7a31\u983b\u9583\u6548\u61c9\u3001\u83ab\u723e\u6548\u61c9\uff09\uff0c\u5176\u4e2d\u9ad8\u983b\u8a0a\u865f\u5728\u6578\u4f4d\u5316\u5f8c\u8f49\u8b8a\u70ba\u5be6\u969b\u4e0a\u4e26\u4e0d\u5b58\u5728\u4f4e\u983b\u8a0a\u865f\u3002\u5728\u5716 11 \u4e2d\uff0c\u7d05\u8272\u9ad8\u983b\u5f26\u6ce2\u70ba\u771f\u5be6\u8a0a\u865f\u3002\u85cd\u8272\u4f4e\u983b\u5f26\u6ce2\u70ba\u865b\u69cb\u8a0a\u865f\uff0c\u5176\u7522\u751f\u539f\u56e0\u662f\u63a1\u6a23\u6642\u9593\u5167\u5df2\u8d85\u904e\u9ad8\u983b\u8a0a\u865f\u7684\u4e00\u534a\u9031\u671f\u3002<\/p>\n Fig. 11. Appearance of a spurious low frequency signal at an insufficiently high sampling rate<\/p><\/div>\n <\/p>\n \u70ba\u907f\u514d\u6df7\u758a\u6548\u61c9\uff0c\u6703\u5728 ADC \u4e4b\u524d\u653e\u7f6e\u7279\u6b8a\u7684\u6297\u6df7\u758a\u6ffe\u6ce2\u5668\uff08\u4f4e\u901a\u6ffe\u6ce2\u5668<\/a>\uff09\u3002\u5b83\u5141\u8a31\u4f4e\u65bc ADC \u63a1\u6a23\u983b\u7387\u4e00\u534a\u7684\u983b\u7387\u901a\u904e\uff0c\u4e26\u5207\u65b7\u66f4\u9ad8\u7684\u983b\u7387\u3002<\/p>\n \u70ba\u4e86\u6839\u64da\u96e2\u6563\u6a23\u672c\u8a08\u7b97\u8a0a\u865f\u983b\u8b5c\uff0c\u9700\u4f7f\u7528\u96e2\u6563 \u5085\u7acb\u8449\u8b8a\u63db\uff08DFT\uff09<\/a> \u3002\u518d\u6b21\u6ce8\u610f\uff0c\u96e2\u6563\u8a0a\u865f\u7684\u983b\u8b5c\u300c\u6839\u64da\u5b9a\u7fa9\u300d\u53d7\u9650\u65bc\u5c0f\u65bc\u63a1\u6a23\u983b\u7387 Fd \u4e00\u534a\u7684\u983b\u7387 Fmax\u3002\u56e0\u6b64\uff0c\u96e2\u6563\u8a0a\u865f\u7684\u983b\u8b5c\u53ef\u8868\u793a\u70ba a finite <\/u>number of harmonics, in contrast to the infinite sum for the Fourier series of a continuous signal, whose spectrum can be unlimited. According to Kotelnikov’s theorem, the maximum frequency of a harmonic must be such that it accounts for at least two samples, so the number of harmonics is equal to half the number of samples of a discrete signal. That is, if there are N samples in the sample, the number of harmonics in the spectrum will be N\/2.<\/p>\n Consider now the discrete Fourier transform (DFT).<\/p>\n Comparing it with the Fourier series<\/p>\n <\/p>\n As we can see, they coincide, except for the fact that time in the FFT is discrete and the number of harmonics is limited to N\/2, which is half the number of samples.<\/p>\n DFT formulas are written in dimensionless integer variables k, s, where k is the number of signal samples, s is the number of spectral components. As it was already said above, when decomposing a non-periodic function (our signal) into Fourier series, the resulting Fourier series actually corresponds to a periodic function with period T (Fig.12).<\/p>\n <\/p>\n Fig.12. Periodic function f(x) with period T0, with period T>T0<\/p><\/div>\n <\/p>\n \u5982\u5716 12 \u6240\u793a\uff0c\u51fd\u6578 f(x) \u70ba\u9031\u671f\u70ba T0 \u7684\u9031\u671f\u51fd\u6578\u3002\u7136\u800c\uff0c\u7531\u65bc\u6e2c\u91cf\u6a23\u672c\u9577\u5ea6 T \u4e0d\u7b49\u65bc\u51fd\u6578\u9031\u671f T0\uff0c\u4f5c\u70ba\u5085\u7acb\u8449\u7d1a\u6578\u7372\u5f97\u7684\u51fd\u6578\u5728 T \u9ede\u8655\u5b58\u5728\u4e0d\u9023\u7e8c\u6027\u3002\u7d50\u679c\uff0c\u8a72\u51fd\u6578\u7684\u983b\u8b5c\u5c07\u5305\u542b\u5927\u91cf\u9ad8\u983b\u8ae7\u6ce2\u3002\u6b64\u73fe\u8c61\u7a31\u70ba \u983b\u8b5c\u6d29\u6f0f<\/a>\uff0c\u5728\u5be6\u52d9\u4e0a\u53ef\u900f\u904e \u52a0\u7a97<\/a> \u8b8a\u63db\u524d\u7684\u8a0a\u865f\u4f86\u6e1b\u5c11\u3002\u82e5\u6e2c\u91cf\u6a23\u672c\u6642\u9577 T \u8207\u51fd\u6578\u9031\u671f T0 \u91cd\u5408\uff0c\u5247\u5085\u7acb\u8449\u8b8a\u63db\u5f8c\u7372\u5f97\u7684\u983b\u8b5c\u5c07\u50c5\u5305\u542b\u7b2c\u4e00\u8ae7\u6ce2\uff08\u9031\u671f\u7b49\u65bc\u6a23\u672c\u6642\u9577\u7684\u5f26\u6ce2\uff09\uff0c\u56e0\u70ba\u51fd\u6578 f(x) \u70ba\u5f26\u6ce2\u3002<\/p>\n In other words, the DFT program “does not know” that our signal is a “slice of a sine wave”, but tries to represent as a series a periodic function which has a discontinuity due to the discontinuity of separate pieces of the sine wave.<\/p>\n As a result, harmonics appear in the spectrum, which should, in total, represent the shape of the function, including this discontinuity.<\/p>\n Thus, to get a “correct” spectrum of a signal which is a sum of several sinusoids with different periods, it is necessary that an integer number of periods of <\/u>each sinusoid should be present on the measuring period of the signal. In practice, this condition can be met with a sufficiently long duration of signal measurement.<\/p>\n <\/p>\n Fig.13 Example of the kinematic error signal function and spectrum of a gearbox<\/p><\/div>\n <\/p>\n At shorter duration the picture will look “worse”:<\/p>\n <\/p>\n \u5716 14 \u8f49\u5b50\u632f\u52d5\u51fd\u6578\u8207\u983b\u8b5c\u7bc4\u4f8b<\/p><\/div>\n <\/p>\n <\/p>\n <\/p>\n In practice, it can be difficult to understand where the “real components” and where the “artifacts” caused by inconsistency of component periods and signal sampling durations or “jumps and breaks” in the waveform. Of course, the words “real components” and “artifacts” are put in quotes for a reason. The presence of many harmonics on the spectrum graph does not mean that our signal actually consists of them. It is like thinking that number 7 “consists” of numbers 3 and 4. The number 7 can be thought of as the sum of 3 and 4 – that is correct.<\/p>\n So also our signal… or rather not even “our signal”, but a periodic function composed by repeating our signal (sample) can be represented as a sum of harmonics (sine waves) with certain amplitudes and phases. But in many cases important for practice (see figures above) it is indeed possible to relate the harmonics obtained in the spectrum also to real processes having cyclic character and contributing significantly to the form of the signal.<\/p>\n 1. \u7d93 ADC \u6578\u4f4d\u5316\u3001\u6642\u9577\u70ba T \u79d2\u7684\u5be6\u969b\u6e2c\u91cf\u8a0a\u865f\uff0c\u5373\u7531\u4e00\u7d44\u96e2\u6563\u6a23\u672c\uff08N \u500b\uff09\u8868\u793a\uff0c\u5176\u983b\u8b5c\u70ba\u96e2\u6563\u975e\u9031\u671f\u6027\uff0c\u7531\u4e00\u7d44\u8ae7\u6ce2\u5206\u91cf\uff08N\/2 \u500b\uff09\u8868\u793a\u3002<\/p>\n 2. \u8a0a\u865f\u7531\u4e00\u7d44\u5be6\u6578\u503c\u8868\u793a\u3002\u5176 DFT \u983b\u8b5c\u70ba\u4e00\u7d44\u5177\u6709\u5171\u8edb\u5c0d\u7a31\u6027\u7684\u8907\u6578\u4fc2\u6578\uff1b\u7531\u6b64\u53ef\u7372\u5f97\u632f\u5e45\u983b\u8b5c \u2013 \u5373\u6b63\u983b\u7387\u8655\u7684\u5be6\u6578\u975e\u8ca0\u632f\u5e45\uff08\u8207\u76f8\u4f4d\uff09\u96c6\u5408\uff0c\u800c\u5be6\u52d9\u4e0a\u7e6a\u88fd\u7684\u6b63\u662f\u6b64\u55ae\u908a\u632f\u5e45\u983b\u8b5c\u3002\u5305\u542b\u8ca0\u983b\u7387\u7684\u96d9\u908a\u8907\u6578\u5f62\u5f0f\u8207\u55ae\u908a\u632f\u5e45\/\u76f8\u4f4d\u5f62\u5f0f\u70ba\u540c\u4e00\u983b\u8b5c\u7684\u7b49\u6548\u8868\u793a \u2013 \u5c0d\u65bc\u8a0a\u865f\u5206\u6790\u800c\u8a00\uff0c\u901a\u5e38\u4f7f\u7528\u55ae\u908a\u632f\u5e45\u983b\u8b5c\u66f4\u70ba\u65b9\u4fbf\u3002<\/p>\n 3. \u5728\u6642\u9593 T \u5167\u6e2c\u91cf\u7684\u8a0a\u865f\u50c5\u5728\u6642\u9593 T \u5167\u88ab\u78ba\u5b9a\u3002\u5728\u6211\u5011\u958b\u59cb\u6e2c\u91cf\u8a0a\u865f\u4e4b\u524d\u767c\u751f\u4e86\u4ec0\u9ebc\uff0c\u4ee5\u53ca\u4e4b\u5f8c\u6703\u767c\u751f\u4ec0\u9ebc\uff0c\u79d1\u5b78\u4e0a\u4e26\u4e0d\u77e5\u9053\u3002\u800c\u5728\u6211\u5011\u7684\u60c5\u6cc1\u4e0b\uff0c\u9019\u4e5f\u4e0d\u91cd\u8981\u3002\u6709\u9650\u6642\u9593\u8a0a\u865f\u7684 FFT \u7d66\u51fa\u4e86\u5176\u300c\u771f\u5be6\u300d\u983b\u8b5c\uff0c\u610f\u5373\u5728\u7279\u5b9a\u689d\u4ef6\u4e0b\uff0c\u5b83\u5141\u8a31\u8a08\u7b97\u5176\u5206\u91cf\u7684\u632f\u5e45\u8207\u983b\u7387\u3002<\/p>\n <\/p>","protected":false},"excerpt":{"rendered":" Application of the Fourier Transform to the Analysis of Vibration Signals Andrei Shelkovenko. One of the developers and founder of Vibromera. The translation of the article may contain inaccuracies. Fourier transform and signal spectrum In many cases the task of obtaining (calculating) the spectrum of a signal is as follows. […]<\/p>\n","protected":false},"author":2,"featured_media":2161,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ai_generated_summary":"","footnotes":""},"categories":[4],"tags":[],"class_list":["post-2138","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-example"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/posts\/2138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/comments?post=2138"}],"version-history":[{"count":2,"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/posts\/2138\/revisions"}],"predecessor-version":[{"id":102137,"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/posts\/2138\/revisions\/102137"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/media\/2161"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/media?parent=2138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/categories?post=2138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/zh-tw\/wp-json\/wp\/v2\/tags?post=2138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\u5716](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US779.webp\")
\n\u5f9e\u6578\u5b78\u89d2\u5ea6\u4f86\u770b – \u9019\u53e5\u8a71\u6709\u591a\u5c11\u8655\u932f\u8aa4\uff1f <\/span><\/i><\/em><\/p>\n![\"\u5716](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US1459.png\")
![\"\u5716](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US1564.webp\")
![\"\u5716](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US1861.png\")
![\"\u5716](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US1916.webp\")
\u9023\u7e8c\u51fd\u6578\u53ca\u5176\u5085\u7acb\u8449\u7d1a\u6578\u8868\u793a<\/h2>\n
![\"\u9023\u7e8c\u51fd\u6578\u53ca\u5176\u5085\u7acb\u8449\u7d1a\u6578\u8868\u793a\"](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US2409.png\")
\nT – \u51fd\u6578\u5b9a\u7fa9\u7684\u5340\u9593\uff08\u8a0a\u865f\u7684\u6301\u7e8c\u6642\u9593\uff09
\nAk- \u7b2c k \u500b\u8ae7\u6ce2\u5206\u91cf\u7684\u632f\u5e45\uff0c
\n\u03b8k- \u7b2c k \u500b\u8ae7\u6ce2\u5206\u91cf\u7684\u521d\u59cb\u76f8\u4f4d
\n\u300c\u5c07\u51fd\u6578\u8868\u793a\u70ba\u7d1a\u6578\u4e4b\u548c\u300d\u662f\u4ec0\u9ebc\u610f\u601d\uff1f\u9019\u610f\u5473\u8457\uff0c\u901a\u904e\u5728\u6bcf\u500b\u9ede\u4e0a\u5c07\u5085\u7acb\u8449\u7d1a\u6578\u7684\u8ae7\u6ce2\u5206\u91cf\u503c\u76f8\u52a0\uff0c\u6211\u5011\u5f97\u5230\u8a72\u9ede\u4e0a\u6211\u5011\u51fd\u6578\u7684\u503c\u3002
\n\uff08\u66f4\u56b4\u683c\u5730\u8aaa\uff0c\u7d1a\u6578\u8207\u51fd\u6578 f(x) \u7684\u5747\u65b9\u504f\u5dee\u5c07\u8da8\u65bc\u96f6\uff0c\u4f46\u5118\u7ba1\u5b58\u5728\u5747\u65b9\u6536\u6582\uff0c\u4e00\u822c\u800c\u8a00\uff0c\u51fd\u6578\u7684\u5085\u7acb\u8449\u7d1a\u6578\u4e0d\u4e00\u5b9a\u9010\u9ede\u6536\u6582\u65bc\u8a72\u51fd\u6578\u3002\uff09
\n\u6b64\u7d1a\u6578\u4e5f\u53ef\u4ee5\u5beb\u6210\u4ee5\u4e0b\u5f62\u5f0f\uff1a<\/p>\n![\"(2),\"](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US3314.png\")
![\"Fourier](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US3326.webp\")
\uff0c\u7b2c k \u500b\u8907\u632f\u5e45\u3002<\/p>\n![\"](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US3362.png\")
![\"Formula](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US3464.png\")
<\/p>\n![\"Fourier](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US3470.png\")
<\/p>\n
\n<\/span><\/b><\/strong>\u8a0a\u865f\u983b\u8b5c\u5206\u6790\u7684\u6578\u5b78\u57fa\u790e\u662f\u5085\u7acb\u8449\u8b8a\u63db\u3002<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>\u5085\u7acb\u8449\u8b8a\u63db\u5141\u8a31\u5c07\u5b9a\u7fa9\u5728\u5340\u9593 {0, T} \u4e0a\u7684\u9023\u7e8c\u51fd\u6578 f(x)\uff08\u8a0a\u865f\uff09\u8868\u793a\u70ba\u7121\u7aae\u591a\u500b\uff08\u7121\u7aae\u7d1a\u6578\uff09\u5177\u6709\u78ba\u5b9a\u632f\u5e45\u548c\u76f8\u4f4d\u7684\u4e09\u89d2\u51fd\u6578\uff08\u6b63\u5f26\u548c\/\u6216\u9918\u5f26\uff09\u4e4b\u548c\uff0c\u9019\u4e9b\u4e09\u89d2\u51fd\u6578\u540c\u6a23\u8003\u616e\u5728\u5340\u9593 {0, T} \u4e0a\u3002\u6b64\u985e\u7d1a\u6578\u7a31\u70ba\u5085\u7acb\u8449\u7d1a\u6578\u3002<\/span><\/b><\/strong><\/p>\n![\"Figure](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US5209.png\")
\nThe spectrum of this function is discrete, i.e. it is represented as an infinite series of harmonic components – a Fourier series.
\nIn fact, Fourier series defines some periodic function, which coincides with our function on the interval {0, T}, but for us this periodicity is not essential.<\/p>\n![\"Fig.](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US6155.png\")
\n\u82e5\u8981\u5c07\u89e3\u6790\u5ea6\u63d0\u9ad8 2 \u500d\u81f3 0.5 Hz\uff0c\u5247\u5fc5\u9808\u5c07\u6e2c\u91cf\u6642\u9577\u63d0\u9ad8 2 \u500d\u81f3 2 \u79d2\u300210 \u79d2\u7684\u8a0a\u865f\u53ef\u5206\u89e3\u70ba\u983b\u7387\u89e3\u6790\u5ea6\u70ba 0.1 Hz \u7684\u8ae7\u6ce2\u5206\u91cf\uff08\u983b\u8b5c\uff09\u3002\u6c92\u6709\u5176\u4ed6\u65b9\u6cd5\u53ef\u63d0\u9ad8\u983b\u7387\u89e3\u6790\u5ea6\u3002\u60a8\u53ef\u4ee5\u900f\u904e\u6211\u5011\u7684 FFT \u89e3\u6790\u5ea6\u8a08\u7b97\u5668<\/a>.<\/p>\n\u96e2\u6563\u8a0a\u865f\u8207\u96e2\u6563\u5085\u7acb\u8449\u8b8a\u63db<\/h2>\n
\n(Fig.9 Schematic of the measuring channel)<\/p>\n![\"Fig.10](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US8285.webp\")
![\"Fig.](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US9777.webp\")
![\"Discrete](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US10917.png\")
<\/p>\n![\"Discrete](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US10958.png\")
<\/p>\n
\nThe value s shows the number of full harmonic oscillations per period T (signal measurement duration). The discrete Fourier transform is used to find the amplitudes and phases of harmonics numerically, i.e. “on the computer”.<\/p>\n![\"Fig.12.](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US11706.png\")
![\"Fig.13](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US13102.png\")
![\"\u5716](\"https:\/\/vibromera.eu\/wp-content\/uploads\/2023\/04\/\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435-\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f-\u0424\u0443\u0440\u044c\u0435-\u0434\u043b\u044f-\u0430\u043d\u0430\u043b\u0438\u0437\u0430-\u0432\u0438\u0431\u0440\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432-en-US13237.png\")
Some results<\/h2>\n