What is the correct way to scale results when taking the Fast Fourier Transform (FFT) and/or the Inverse Fast Fourier Transform (IFFT)?
MATLAB: Scaling the FFT and the IFFT
fftifftMATLABscalescaling
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Hello fgb
Here is a revised answer that attempts to better address the issue. I believe there are a couple of things going on here. In your code,
df = 2*Fs/L
should not have the factor of 2. Also, if the length of the x array is not a power of 2, then the procedure
NFFT = nextpow2(L)Y ~ fft(x,NFFT)
changes the length of the x array that gets fft'd. So both uses of L afterwards have to be changed, and df has to become
df = Fs/NFFT.
For a transient response the transform and the energies in the time and frequency domains are
Y = fft(x)*dt % dt is the same as T
ex = sum(conj(x).*x)*dt % dt is the same as T
eY = sum(conj(Y).*Y)*df
The first two you have. The frequency domain calculation eY agrees with ex, and it uses the entire frequency array. All frequency components have identical energy content. Each energy is the abs value squared of the amplitude. Life is good. The issue is with using positive frequencies only.
When you fft a real quantity and double the positive positive frequency components Y(f), you are going to a representation (just at one frequency for simplicity)
x = 2 |Y(f)| cos(2pi f t + phase angle)
But the energy content is different, because the average value of cos(x)^2 is 1/2. So if somebody has handed you a positive-frequency-absolute-value spectrum like the one that you have after this step
YA(1)=YA(1)/2; %Correction for the fact that DC doesn't occur twice
then you also have to make sure that point N/2+1 has been included (as you did), and that that point is not doubled. So there needs to have been a step
YA(end)=YA(end)/2; %Correction because nyquist point doesn't occur twice
Then the expression
eY_new = (YA(1)^2 + (1/2)*sum(YA(2:end-1).^2) + YA(end)^2)*df;
works. But it's not straightforward.
Stuff like this is why I try to stay away from positive-frequencies-only as much as possible for real calculations. It's not bad for illustrative plotting purposes.
hello Anna
look at my fft function at the end of my code
clc%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% load signal
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% data
data = importdata('SFC5S_nov25_ST_1_1.txt');dt = 1e-6; % 1 micro seconds
signal = data(:,1);samples = length(signal);Fs = 1/dt; % sampling frequency (Hz)
%% decimate (if needed)
% NB : decim = 1 will do nothing (output = input)
decim = 1;if decim>1 signal = decimate(signal,decim); Fs = Fs/decim;end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FFT parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%NFFT = 512; %
OVERLAP = 0.95;% spectrogram dB scale
spectrogram_dB_scale = 80; % dB range scale (means , the lowest displayed level is XX dB below the max level)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% options
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if you are dealing with acoustics, you may wish to have A weighted
% spectrums
% option_w = 0 : linear spectrum (no weighting dB (L) )
% option_w = 1 : A weighted spectrum (dB (A) )
option_w = 0;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 1 : averaged FFT spectrum
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%[freq, sensor_spectrum] = myfft_peak(signal,Fs,NFFT,OVERLAP);% convert to dB scale (ref = 1)
sensor_spectrum_dB = 20*log10(sensor_spectrum);% apply A weigthing if needed
if option_w == 1 pondA_dB = pondA_function(freq); sensor_spectrum_dB = sensor_spectrum_dB+pondA_dB; my_ylabel = ('Amplitude (dB (A))');else my_ylabel = ('Amplitude (dB (L))');endfigure(1),plot(freq,sensor_spectrum_dB,'b');gridtitle(['Averaged FFT Spectrum / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(freq(2)-freq(1)) ' Hz ']);xlabel('Frequency (Hz)');ylabel(my_ylabel);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 2 : time / frequency analysis : spectrogram demo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%[sg,fsg,tsg] = specgram(signal,NFFT,Fs,hanning(NFFT),floor(NFFT*OVERLAP)); % FFT normalisation and conversion amplitude from linear to dB (peak)
sg_dBpeak = 20*log10(abs(sg))+20*log10(2/length(fsg)); % NB : X=fft(x.*hanning(N))*4/N; % hanning only
% apply A weigthing if neededif option_w == 1 pondA_dB = pondA_function(fsg); sg_dBpeak = sg_dBpeak+(pondA_dB*ones(1,size(sg_dBpeak,2))); my_title = ('Spectrogram (dB (A))');else my_title = ('Spectrogram (dB (L))');end% saturation of the dB range :
% saturation_dB = 60; % dB range scale (means , the lowest displayed level is XX dB below the max level)
min_disp_dB = round(max(max(sg_dBpeak))) - spectrogram_dB_scale;sg_dBpeak(sg_dBpeak<min_disp_dB) = min_disp_dB;% plots spectrogram
figure(2);imagesc(tsg,fsg,sg_dBpeak);colormap('jet');axis('xy');colorbar('vert');gridtitle([my_title ' / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(fsg(2)-fsg(1)) ' Hz ']);xlabel('Time (s)');ylabel('Frequency (Hz)');function pondA_dB = pondA_function(f) % dB (A) weighting curve
n = ((12200^2*f.^4)./((f.^2+20.6^2).*(f.^2+12200^2).*sqrt(f.^2+107.7^2).*sqrt(f.^2+737.9^2))); r = ((12200^2*1000.^4)./((1000.^2+20.6^2).*(1000.^2+12200^2).*sqrt(1000.^2+107.7^2).*sqrt(1000.^2+737.9^2))) * ones(size(f)); pondA = n./r; pondA_dB = 20*log10(pondA(:));endfunction [freq_vector,fft_spectrum] = myfft_peak(signal, Fs, nfft, Overlap)% FFT peak spectrum of signal (example sinus amplitude 1 = 0 dB after fft).
% Linear averaging
% signal - input signal,
% Fs - Sampling frequency (Hz).
% nfft - FFT window size
% Overlap - buffer overlap % (between 0 and 0.95)
signal = signal(:);samples = length(signal);% fill signal with zeros if its length is lower than nfft
if samples<nfft s_tmp = zeros(nfft,1); s_tmp((1:samples)) = signal; signal = s_tmp; samples = nfft;end% window : hanning
window = hanning(nfft);window = window(:);% compute fft with overlap
offset = fix((1-Overlap)*nfft); spectnum = 1+ fix((samples-nfft)/offset); % Number of windows
% % for info is equivalent to :
% noverlap = Overlap*nfft;
% spectnum = fix((samples-noverlap)/(nfft-noverlap)); % Number of windows
% main loop
fft_spectrum = 0; for i=1:spectnum start = (i-1)*offset; sw = signal((1+start):(start+nfft)).*window; fft_spectrum = fft_spectrum + (abs(fft(sw))*4/nfft); % X=fft(x.*hanning(N))*4/N; % hanning only
end fft_spectrum = fft_spectrum/spectnum; % to do linear averaging scaling
% one sidded fft spectrum % Select first half
if rem(nfft,2) % nfft odd
select = (1:(nfft+1)/2)'; else select = (1:nfft/2+1)'; endfft_spectrum = fft_spectrum(select);freq_vector = (select - 1)*Fs/nfft;end
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