FFT and PSD - normalize values

Question

I have noticed many posts regarding frequency spectrum and many answers to the same questions. How to normalize the values of FFT and PSD from a periodic signal?   Most of us when we refeer to "normalize the frequency domain" we mean that it is required to obtain the frequency domain of the signals where the amplitude of the signal, or the time window, do not affect the obtained plot.   For example in my case I use a periodic sine signal with a frequency of 160HZ, and observed that the amplitude of the foundamental frequency in both figures changes according to the length of the signal. So the questions are: How to obtain a FFT and PSD where the amplitude values are not affected by: The time window or the number of points. The peak amplitude of the signal. I mean for example if the signal amplitude is reduced by the half, still getting the corresponding peak value, or even better the corresponding percentage. Notice that even the amplitud of the signal is 15. The obtained values in FFT and PSD are not 15.   For FFT the value varies between 10 and 14 depending on the time window, but never the 15 that it should be, even it the time window is very high.   For PSD the amplitude value increases as a function of the time window. From 2 up to 800.   Also notice that PSD shows the foundamental frequency at 160 Hz but in the one-side FFT varies 145 - 160 Hz. How to solve it? Why is this happening?     Do you have any comments about how to fix my code in order to get the same amplitude for PSD and FFT despite of the time window. Thanks   %% R is the signal data points extracted from the timetable Datos03 %% tiempo is the signal time points. %% L is the number of data points (Length of the signal) %Time domain plot(Datos3.Time,Datos3.VarName1) R=Datos3.VarName1; tiempo=seconds(Datos3.Time); % FFT Fs=10240; % Sampling frequency T=1/Fs; % Sampling period a=size(tiempo) % Number of points L = a(1,1) % Length of signal Y = fft(R); P2 = abs(Y/L); P1 = P2(1:L/2+1); P1(2:end-1) = 2*P1(2:end-1); f = Fs*(0:(L/2))/L; plot(f,P1) xlim([0 400]) title('FFT') %PSD [SignalSpectrum,SignalFrequencies] = periodogram(R, ... [],[],Fs); plot(SignalFrequencies,SignalSpectrum) xlim([0 400]) title('PSD')

Prashant Kumar · Accepted Answer

Normalizing the FFT and PSD values in MATLAB to ensure that the amplitude values are not affected by the time window or the number of points requires some adjustments. Here are the steps to achieve consistent amplitude values:


FFT Normalization: Normalize the FFT by dividing by the total number of points and adjusting for the peak amplitude.


PSD Normalization: Normalize the PSD by adjusting for the signal's length and amplitude.


Here's how you can modify your MATLAB code:

matlab


%% R is the signal data points extracted from the timetable Datos03
%% tiempo is the signal time points.
%% L is the number of data points (Length of the signal)
    
% Time domain
plot(Datos3.Time, Datos3.VarName1)
R = Datos3.VarName1;
tiempo = seconds(Datos3.Time);

% FFT
Fs = 10240;           % Sampling frequency
T = 1/Fs;             % Sampling period       
L = length(tiempo);   % Length of signal

Y = fft(R);
P2 = abs(Y/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
f = Fs*(0:(L/2))/L;

% Normalization for FFT
P1 = P1 / max(P1) * max(abs(R)); % Normalize to the peak amplitude of the signal

% Plot FFT
figure;
plot(f, P1)
xlim([0 400])
title('FFT')
xlabel('Frequency (Hz)')
ylabel('Amplitude')

% PSD
[SignalSpectrum, SignalFrequencies] = periodogram(R, [], [], Fs);

% Normalization for PSD
SignalSpectrum = SignalSpectrum / max(SignalSpectrum) * max(abs(R))^2 / L; % Normalize to the peak amplitude of the signal and adjust for signal length

% Plot PSD
figure;
plot(SignalFrequencies, SignalSpectrum)
xlim([0 400])
title('PSD')
xlabel('Frequency (Hz)')
ylabel('Power/Frequency (dB/Hz)')




Explanation:


FFT Normalization: After computing the FFT, normalize the amplitude by dividing by the total number of points and scaling it to match the peak amplitude of the original signal.


PSD Normalization: After computing the PSD using the periodogram function, normalize the power spectral density by adjusting for the signal's length and scaling it to match the peak power of the original signal.


These adjustments should help you achieve consistent amplitude values in both the FFT and PSD plots, regardless of the time window or the number of points. This way, the amplitude values will accurately represent the signal's characteristics

FFT and PSD - normalize values

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