The principle of inverse Fourier transform (IFFT) in signal processing

In signal processing , It is often necessary to convert the signal to the frequency domain for analysis , Sometimes it will return from the frequency domain to the time domain , be used FFT and IFFT function .

FFT Transformation is to transform the signal from time domain to frequency domain , It seems convenient and easy to convert complex signals in time domain to frequency domain . But sometimes it is also necessary to convert the frequency domain signal to the time domain , So you need to use IFFT Transformation .

Actually IFFT One of the calculation principles of is Frequency domain （ Note that the frequency domain is complex ） Data is taken as conjugate complex （ The imaginary part is reversed ）, And then we can move on FFT Transformation , In this way, the frequency domain signal is converted to the time domain . because FFT The result of the transformation is a complex number , So from the frequency domain FFT The result of transformation is also plural , At this time, just take The real part of a complex number , It is the original time domain signal （ At the same time, the imaginary part of this complex number is already very small , Just omit ）.

Be careful ：FFT The next sequence is the order of conjugate symmetry , When clearing the data of a certain range of frequencies , The frequency data that is conjugate and symmetrical with it must also be clear , Don't forget , Don't be too clear , So there will be no plural .

stay MATLAB in , By calculation MATLAB Self contained IFFT Functions and FFT The result is recalculated by inverting the imaginary part FFT. Two methods show ： Consistent result .

%%  Make the sine wave FFT Calculation ,50% Overlap . verification IFFT The calculation of —— Compare the built-in functions IFFT function , And for FFT The result is conjugate , Proceed again FFT The difference of calculation .
clear;clf;
Fs=24000;t = 0:1/Fs:0.07;y=sin(200*2*pi*t+50);      % 24k At sampling rate , Generate 200Hz、0.07 Second sine wave 
i=(floor(length(y)/512)-1)*2;                       %  Because the frame calculation , Here we calculate the number of cycles . Because 50% Overlap , So it needs to be multiplied by 2
yfft=zeros(1,length(y));            %  Initialization function 
yifft=zeros(1,length(y));           %  Initialization function 
Windows=hamming(512);               %  Take the window function 
for a =0:i
    StartPoint=1+256*a;            
    y0(1:512)=y(StartPoint:(StartPoint+512-1)); % Take what needs to be handled every time 512 Data 
    y0=y0.*Windows';                %  Add windows 
    TempData=fft(y0);               %  do FFT Calculation 
    Yifft=ifft(TempData);           % MATLAB Bring their own IFFT function 
    Yfft=conj(TempData);            %  Take the conjugate complex number 
    Yfft=fft(Yfft);                 %  The empty part takes the reverse FFT Calculation 
    Yfft=(real(Yfft)/512)*0.98;     %  Take out the real part , And divide it by N. multiply 0.98 In order to make a comparison 
    yfft(StartPoint:StartPoint+512-1)=yfft(StartPoint:StartPoint+512-1)+Yfft;       % restructuring 
    yifft(StartPoint:StartPoint+512-1)=yifft(StartPoint:StartPoint+512-1)+Yifft;    % restructuring 
end
%%  Make a picture 
figure(1);
plot(y,'*');            hold on
plot(yfft,'r');         hold on ;
plot(yifft,'k');        grid on
legend(' Original signal time domain ',' The empty part takes the reverse fft Time domain ','Ifft Time domain ');
xlabel(' Time domain （t）');
ylabel(' amplitude ');