How to rectify a signal and determine the envelope spectra for the FFT? - Envelope analysis

Hello folks, I am trying to determine the harmful frequencies of my signal. Therefore, I need to do a so called " envelope analysis" for my signal.

I have measured two signal from my sensor. My sensor is located at the tire. Both signals have a shape like sinus. First, I need to rectify the signal. This is done by the following:

XY  = sqrt(X.^2+Y.^2);
XY  = XY - mean(XY);

Now, I need to determine the envelope curve of my signal. This is done by the following code:

[up1,~] = envelope(XY,30,'analytic');

After that, I need to compute the FFT in order to calculate the harmful frequencies. This is done by the following:

FFTq(up1);

function [f,P] = FFTq(data)
Fs = 1000;                  % Sampling frequency                    
T  = 1/Fs;                  % Sampling period       
L  = length(data); % Length of signal in ms
t  = (0:L-1)*T;             % Time vector
f  = Fs*(0:(L/2))/L;        % frequency

FFT = fft(data);
P2  = abs(FFT/L);
P  = P2(1:L/2+1);
end

My Matlab program is compiling without any errors. However, the result is not how I expected it. The literature says, that computing the envelope spectra via FFT will provide a better result than computing the FFT. This not the case here. Therefor, I was thinking that my implementation could have some mistakes. Can someone double check my code?

Kind regards

[Mathieu NOE]

hello

at first look , your code seem fine

could you show a few picture of the signal , then rectified, the enveloppe , the fft output ?

or simply share the data so we can check everything ?

[Hishan Faizer]

Hello, thank you for your respond.

Plots and the data are attached.

[Mathieu NOE]

hello

I have a first problem

what you say and what you do don't match

you say : First, I need to rectify the signal. This is done by the following:

XY = sqrt(X.^2+Y.^2);

XY = XY - mean(XY);

this is not rectification; rectification of X is abs(X)

what you have done is taking the amplitude of two sinus with 90° phase difference

so your XY would be a constant (sinus amplitude) if there were not some measurement noise.

are you sure that it's the way you want to proceed ?

the fft of XY will not show anything but the noise signal . Anything that is related to the X and Y sinus frequency is lost.

see the difference with the fft of X alone.

FYI , this is my code so far

load('Data_Signal.mat');
Fs = 1000;

XY  = sqrt(X.^2+Y.^2);
% XY  = XY - mean(XY);

figure(1);plot(t,X,'b',t,Y,'r',t,XY,'k');
legend('X','Y','XY');


% FFT analysis
NFFT = 256;    % 
OVERLAP = 0.75;

[freq, sensor_spectrumX] = myfft_peak(X,Fs,NFFT,OVERLAP);
[freq, sensor_spectrumXY] = myfft_peak(XY,Fs,NFFT,OVERLAP);

figure(2),plot(freq,20*log10(sensor_spectrumX),'b',freq,20*log10(sensor_spectrumXY),'r');grid
title(['Averaged FFT Spectrum  / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(freq(2)-freq(1)) ' Hz ']);
xlabel('Frequency (Hz)');ylabel('Amplitude (dB)');
legend('X','XY','Location','northwest');



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù


function  [freq_vector,fft_spectrum] = myfft_peak(signal, Fs, nfft, Overlap)
    %FFT peak spectrum of signal  (example sinus amplitude 1   = 0 dB after fft).
    %   signal - input signal, 
    %   Fs - Sampling frequency (Hz).
    %   nfft - FFT window size
    %   Overlap - buffer overlap % (between 0 and 0.95)

    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;
    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)';
        end
    fft_spectrum = fft_spectrum(select);
    freq_vector = (select - 1)*Fs/nfft;
  end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[Hishan Faizer]

Hello, thank you for your comment. The purpose of creating one signal is to analyse the noise signal. I want to know if my bearing has some damage or not. The provided data is form a damaged bearing. Therefore the FFT should show some peaks additionally to the frequency 3.9 Hz. Any ideas, how to solve this problem?

[Mathieu NOE]

ok

so that was my first impression, you are doing bearing damage detection using rectification / envelop / fft

but to the best of my knowledge , you do that with only one direction signal

again, I'm a bit sceptical about doing the XY rms sum ... but maybe

I'll upgrade my code to add the rectification + envelope nefore the fft

I assumed that those publication (attached ) are well known from you

[Hishan Faizer]

That would be awesome, if you could upadate your code. I could try out which option (X, Y or XY) will deliever better results. Thank you for efforts!

[Hishan Faizer]

As I took a closer look to the first paper ( MFPT-Bearing-Envelope-Analysis.pdf ) the author is using some Matlab functions like "envhetsv" which is not defined in the paper. I also googled this function but couldnt find a explanation.

[Mathieu NOE]

you're right this function does not officially exist , I guess it's a home made function designed by the author; too bad he does not publish this code

BTW, are you aware of this function : envspectrum

es = envspectrum(x,fs) returns the envelope spectrum of a signal x sampled at a rate fs. If x is a matrix, then the function computes the envelope spectrum independently for each column and returns the result in the corresponding column of es.

this should do all the hard work for you !

[Hishan Faizer]

Thank you for suggesting the function envspectrum. I already tried envspectrum and it does not deliver good results as the FFT or the Welch Method (Example 2 MFPT-Bearing-Envelope-Analysis.pdf ) does. I am supprised by this, because computing the harmful frequnecies with the envelope signal should deliver better results than with the FFT according to most of the ressources/papers I found.

Results from the function envspectrum:

Results from the function psde:

As you can see, the PSDE function is showing me three peaks (90 Hz, 180 Hz and 270 Hz) whereas the envelope function does only show one peak at 90 Hz.