摘要：傅里葉變換的核心在于，“任何連續周期信號可以由一組適當的正弦曲線組合而成”，在這個基礎上對信號的中特定頻率的正弦波進行分解或者重組，基于頻率方面分析波形。

1、傅里葉變換的意義

近似周期性的方波（橙色），可采用6組正弦波（藍色）合成，這是傅里葉的基礎。對數字信號處理或者工程數學有一定基礎，就明白傅里葉變換的價值。一般情況下的信號或者波形隨時間變化，稱為時域信號，時域（Time domain）是描述數學函數或物理信號對時間的關系。而以頻率和幅度表示信號稱為頻域，頻域（frequency domain）是描述信號在頻率方面特性時用到的一種坐標系。其數學理論上暫且不理，針對嵌入式系統開發，只探討其物理意義或者應用場景。

時域和頻域是信號的基本性質，時域的表示較為形象與直觀，比較符合一般認知，而頻域分析則更為簡練，剖析問題更為深刻和方便。

例如下圖的左側時域信號，其可以分解為2路正弦波的疊加效果，右側為頻域信號，表示2路正弦波的頻率和幅度。傅里葉變化可簡單理解為求解一段信號或波形，由哪些正弦波組成，也可以反向推導多路正弦波合并后的效果。基于動畫形式表現如下：

傅里葉變換是一種信號分析方法，讓我們對信號的構成和特點進行深入的、定量的研究。把信號通過頻譜的方式進行準確的、定量的描述。將原來難以處理的時域信號轉換成了易于分析的頻域信號，即傅里葉變換的核心是從時域到頻域的變換。

2、變換方式

數字信號屬于離散值，對應的稱為離散傅里葉變換（DFT），是傅里葉變換在時域和頻域上都呈現離散的形式，將時域信號的采樣變換為在離散時間傅里葉變換（DTFT）頻域的采樣。在形式上，變換兩端（時域和頻域上）的序列是有限長的，而實際上這兩組序列都應當被認為是離散周期信號的主值序列。即使對有限長的離散信號作DFT，也應當將其看作經過周期延拓成為周期信號再作變換。

在實際應用中通常采用快速傅里葉變換以高效計算效率，快速傅里葉變換 FFT（Fast Fourier Transformation）是離散傅里葉變換 DFT（Discrete Fourier Transform）的快速算法。采用這種算法能使計算機計算離散傅里葉變換所需要的乘法次數大為減少，特別是被變換的抽樣點數N越多，FFT算法計算量的節省就越顯著。同理，從頻域到時域的變換，稱為逆變換，快速傅里葉逆變換 IFFT和離散傅里葉逆變換IDFT。

3、應用

一般嵌入式系統使用快速傅里葉變換是分析某段信號中混合的雜波干擾，或者剔除某個頻段后再逆變換。有些高端示波器可以對信號快速傅里葉變換 FFT，直接顯示高頻雜波的頻率或者較大幅度干擾源的頻率，以便由針對性的檢查電路竄擾。本文主要是講解使用fft/ifft進行干擾信號的過濾。

1、python生成正弦波，以及混合波形，提取數值作為c語言FFT/IFFT的數據源?

2、先進行FFT，輸出幅頻數據，導入Excel看看效果?

3、對數據進行簡單過濾

4、過濾后的數據進行IFFT運算，再導入Excel看還原的效果?

5、關注微信公眾號 ?嵌入式系統

3.1 基于python生成數據源

#?This?is?a?Python?script.import?numpy?as?np import?matplotlib.pyplot?as?plt def?sin_wave(A,?f,?fs,?phi,?t):''':params?A:????振幅:params?f:????信號頻率:params?fs:???采樣頻率:params?phi:??相位:params?t:????時間長度'''#?若時間序列長度為?t=1s,#?采樣頻率?fs=1000?Hz,?則采樣時間間隔?Ts=1/fs=0.001s#?采樣點個數為?n=t/Ts=1/0.001=1000,?即有1000個點,每個點間隔為?TsTs?=?1/fsn?=?t?/?Tsn?=?np.arange(n)y?=?A*np.sin(2*np.pi*f*n*Ts?+?phi*(np.pi/180))return?yfs?=?360*40 my_sin1?=?sin_wave(100,?100,?fs=fs,?phi=0,?t=0.08) my_sin2?=?sin_wave(20,?500,?fs=fs,?phi=0,?t=0.08) my_sin3?=?sin_wave(10,?1100,?fs=fs,?phi=0,?t=0.08)x?=?np.arange(0,?0.08,?1/fs) plt.xlabel('x-t?(samRate=14400)') plt.ylabel('y-A') plt.grid() plt.plot(x,?my_sin1,?'k--',label="f=100") plt.plot(x,?my_sin2,?'b--',label="f=500") plt.plot(x,?my_sin3,?'g--',label="f=1100") plt.plot(x,?my_sin1+my_sin2+my_sin3,?'r',label="mix") plt.legend() plt.show()np.savetxt("sin.txt",my_sin1+my_sin2+my_sin3,?fmt='%.06f')def?print_name(name):print(f'Hi,?{name}')if?__name__?==?'__main__':print_name('fft?test')

生成的波形圖如下：三種頻率的sin以及合成后的紅色曲線，變形的正弦波；同時也將數據存入文件sin.txt備用。

3.2 基于C驗證算法

將前面生成的數據使用c語言的FFT/IFFT進行處理，輸出結果，并導入Excels生成圖表展示效果。

#include?<math.h> #include?<stdio.h> #include?"string.h"typedef?struct {float?real;float?imag; }?complex_t;#ifndef?PI #define?PI?????????????(3.14159265) #endif#define?TYPE_FFT_E?????float????/*?Type?is?the?same?with?complex_t?member?*/typedef?complex_t?TYPE_FFT;??/*?Define?complex_t?in?Config.h?*/ typedef?unsigned?int??????????????????????uint32_t;#define?????SAMPLE_NODES????(1024) complex_t???fft_buff[SAMPLE_NODES];//python生成的3個sin混合的波形數組 float?my_sin_wave_table[]?= 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};//fft算法來自開源?https://github.com/xiahouzuoxin/fftconst?float?sin_tb[]?=????//?精度(PI?PI/2?PI/4?PI/8?PI/16?...?PI/(2^k)) {0.000000,?1.000000,?0.707107,?0.382683,?0.195090,?0.098017,0.049068,?0.024541,?0.012272,?0.006136,?0.003068,?0.001534,0.000767,?0.000383,?0.000192,?0.000096,?0.000048,?0.000024,0.000012,?0.000006,?0.000003};const?float?cos_tb[]?=????//?精度(PI?PI/2?PI/4?PI/8?PI/16?...?PI/(2^k)) {-1.000000,?0.000000,?0.707107,?0.923880,?0.980785,?0.995185,0.998795,?0.999699,?0.999925,?0.999981,?0.999995,?0.999999,1.000000,?1.000000,?1.000000,?1.000000,?1.000000,?1.000000,1.000000,?1.000000,?1.000000};int?ones_32(uint32_t?n) {unsigned?int?c?=?0?;for(c?=?0;?n;?++c){n?&=?(n?-?1)?;}return?c?; }uint32_t?floor_log2_32(uint32_t?fft_buff) {fft_buff?|=?(fft_buff?>>?1);fft_buff?|=?(fft_buff?>>?2);fft_buff?|=?(fft_buff?>>?4);fft_buff?|=?(fft_buff?>>?8);fft_buff?|=?(fft_buff?>>?16);return?(ones_32(fft_buff?>>?1)); }/**?FFT?Algorithm*?===?Inputs?===*?fft_buff?:?complex?numbers*?N?:?nodes?of?FFT.?@N?should?be?power?of?2,?that?is?2^(*)*?===?Output?===*?the?@fft_buff?contains?the?result?of?FFT?algorithm,?so?the?original?data*?in?@fft_buff?is?destroyed,?please?store?them?before?using?FFT.*/ int?fft(TYPE_FFT?*fft_buff,?uint32_t?N) {int?i,?j,?l,?k,?ip;static?uint32_t?M?=?0;static?int?le,?le2;static?TYPE_FFT_E?sR,?sI,?tR,?tI,?uR,?uI;M?=?floor_log2_32(N);/**?bit?reversal?sorting*/l?=?N?>>?1;j?=?l;ip?=?N?-?2;for(i?=?1;?i?<=?ip;?i++){if(i?<?j){tR?=?fft_buff[j].real;tI?=?fft_buff[j].imag;fft_buff[j].real?=?fft_buff[i].real;fft_buff[j].imag?=?fft_buff[i].imag;fft_buff[i].real?=?tR;fft_buff[i].imag?=?tI;}k?=?l;while(k?<=?j){j?=?j?-?k;k?=?k?>>?1;}j?=?j?+?k;}/**?For?Loops*/for(l?=?1;?l?<=?M;?l++)??/*?loop?for?ceil{log2(N)}?*/{le??=?(int)(1?<<?l);le2?=?(int)(le?>>?1);uR?=?1;uI?=?0;k?=?floor_log2_32(le2);sR?=?cos_tb[k];sI?=?-sin_tb[k];for(j?=?1;?j?<=?le2;?j++)??/*?loop?for?each?sub?DFT?*/{for(i?=?j?-?1;?i?<?N;?i?+=?le)?/*?loop?for?each?butterfly?*/{ip?=?i?+?le2;tR?=?fft_buff[ip].real?*?uR?-?fft_buff[ip].imag?*?uI;tI?=?fft_buff[ip].real?*?uI?+?fft_buff[ip].imag?*?uR;fft_buff[ip].real?=?fft_buff[i].real?-?tR;fft_buff[ip].imag?=?fft_buff[i].imag?-?tI;fft_buff[i].real?+=?tR;fft_buff[i].imag?+=?tI;}??/*?Next?i?*/tR?=?uR;uR?=?tR?*?sR?-?uI?*?sI;uI?=?tR?*?sI?+?uI?*?sR;}?/*?Next?j?*/}?/*?Next?l?*/return?0; }/**?Inverse?FFT?Algorithm*?===?Inputs?===*?fft_buff?:?complex?numbers*?N?:?nodes?of?FFT.?@N?should?be?power?of?2,?that?is?2^(*)*?===?Output?===*?the?@fft_buff?contains?the?result?of?FFT?algorithm,?so?the?original?data*?in?@fft_buff?is?destroyed,?please?store?them?before?using?FFT.*/int?ifft(TYPE_FFT?*fft_buff,?uint32_t?N) {int?k?=?0;for(k?=?0;?k?<=?N?-?1;?k++){fft_buff[k].imag?=?-fft_buff[k].imag;}fft(fft_buff,?N);????/*?using?FFT?*/for(k?=?0;?k?<=?N?-?1;?k++){fft_buff[k].real?=?fft_buff[k].real?/?N;fft_buff[k].imag?=?-fft_buff[k].imag?/?N;}return?0; }static?void?import_data(void) {int?i;for(i?=?0;?i?<?SAMPLE_NODES;?i++){fft_buff[i].real?=?my_sin_wave_table[i];//取前1024個數進行fft變換fft_buff[i].imag??=?0.0f;} }int?main(int?argc,?char?*argv[]) {int?i;int?f;//頻率float?a;//幅度int?fd;float?t;printf("FFT\r\n");import_data();fft(fft_buff,?SAMPLE_NODES);//fft后的結果在fft_buff//將其實部與虛部處理，輸出頻點與幅度值，導入Excel看效果//因為是周期性質，取前半部分即可//數據采樣頻率是360*40，均分到SAMPLE_NODES，則對應頻點間隔是?360*40/SAMPLE_NODESfd=360*40/SAMPLE_NODES;for(i?=?0;?i?<?SAMPLE_NODES?/?2;?i++){f?=?i?*fd;a?=?(double)sqrt(fft_buff[i].real?*?fft_buff[i].real?+?fft_buff[i].imag?*?fft_buff[i].imag)/?(SAMPLE_NODES?/?2);//轉換幅度//printf("%d,%f\n",?f,?a);//>>導入excel查看幅頻圖效果}//過濾高頻部分//將幅度小于某個值的，以30為例過濾for(i?=?0;?i?<?SAMPLE_NODES;?i++){a?=?(double)sqrt(fft_buff[i].real?*?fft_buff[i].real?+?fft_buff[i].imag?*?fft_buff[i].imag)/?(SAMPLE_NODES?/?2);if(a<30){fft_buff[i].real?=?0;fft_buff[i].imag?=?0;}}//再進行逆運算還原ifft(fft_buff,?SAMPLE_NODES);for(i?=?0;?i?<?SAMPLE_NODES;?i++){t=1.0/(360*40)*i;//結合采樣頻率步進，方便查看波形效果//printf("%f,%f\n",?t,fft_buff[i].real);//>>導入excel查看還原后的sin效果}return?0; }

3.3 幅頻圖

混合波形使用FFT后的幅頻圖：看圖可知對應的頻點和幅度，與原始混合波形的3種正弦波吻合。

柵欄效應：其中頻點與采樣頻率和FFT轉換的采樣數有關，需要結合實際定義，因為數字離散信號，頻點并不是連續的，如上大概相差14，因為數據點數N為1024點，原始信號的采樣頻率為fs=14400Hz，則頻率間隔fs/N約等于14Hz。這樣看頻譜圖只有fs/N倍數的，例如沒有410Hz，只能看到相鄰頻率406Hz和420Hz處頻譜泄漏的數據，相當于透過柵欄觀賞風景，只能看到頻譜的一部分，而其它頻率點看不見，因此很可能部分有用的頻率成分被漏掉，這種現象被稱為柵欄效應。

3.4 逆變換還原

FFT后的幅頻數據過濾其中2個頻點的小幅度波形，還原后的效果如下圖：過濾還原后的波形，與原始圖，前面python生成的黑色虛線吻合。

4、應用

傅里葉變換在數字信號領域中應用廣泛，在嵌入式系統中，一般用來分析ADC高速采集的數據、濾波處理，或者對音頻進行簡單分析。但傅里葉變換算法內存消耗有點大，低端設備可能無法運行。

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