Corrupt the signal with zero-mean white noise with a variance of 4.

Do i have to corrupt the signal with zero-mean white noise with a variance of 4. What does it mean and why its necessary?

[dpb]

This seems to parallel another Q? along same lines of yesterday or day before -- except it used a std deviation of 2 which would be the same thing as.

There, there seemed no discernible justification for "why"; it would have swamped the actual data shown by a factor of about 30X for no apparent reason whatsoever.

Would have to see the context for the Q? to have any idea unless it's simply a pedgogical exercise to illustrate that averaging during calculation of PSD by FFT can reduce random noise or some such---there's really no reason to add noise to a signal just to do so...

Answer 1

[William Rose]

@KYRIAKOS PIPIGKAS,

As @dpb said, it would help to know the context of the statement.

Adding random noise to a signal can be used to demonstrate that a signal that looks random in the time domain may reveal its deterministic nature in the frequency domain.

a=2;                    %signal amplitude
b=4;                    %noise variance
t=1:100;                %time
x=a*cos(2*pi*t/10);     %signal without noise
xn=x+sqrt(b)*randn(size(x));  %signal with noise added
Xn=fft(xn);             %FFT of xn               
f=0:.01:.5;             %vector of frequencies, up to Nyquist
%plot the results
subplot(211); plot(t,xn,'-rx'); 
xlabel('Time'); title('Time Domain');
subplot(212); plot(f,abs(Xn(1:51)),'-rx'); 
xlabel('Frequency'); title('Frequency Domain');

The sinusoidal nature of x(t) is not obvious in the time domain plot. The frequency domain plot shows a noticable spike at 0.1 Hz, the sine wave frequency. See above.

[dpb]

a=2
b=4;
t=1:1000;                       %time  (10X the length of original -- dpb)
x=a*cos(2*pi*t/10);             %signal without noise
xn=x+sqrt(b)*randn(size(x));    %signal with noise added
xn=reshape(xn,100,[]);          % N traces of original length -- dpb
Xn=abs(fft(xn));                %abs(FFT) of xn               
f=0:.01:.5;                     %vector of frequencies, up to Nyquist
%plot the results
subplot(211); plot(t,xn(:),'-rx'); 
xlabel('Time'); title('Time Domain');
subplot(212); plot(f,Xn(1:51,:));      % show the individual PSDs -- DPB
hold on
plot(f,mean(Xn(1:51,:),2),'k-','linewidth',2);  % and the result of averaging
xlabel('Frequency'); title('Frequency Domain');

That illustrates the power of averaging out random noise.

Other than such a pedgogical purpose, however, I can think of no reason to introduce the noise into a measured signal -- it can be used on an input signal for exciting a DUT, maybe, but why contaminate a measurement that worked hard to get without being corrupted is beyond my ken...

[Image Analyst]

@dpb I think it's probably a question on his homework assignment.

[dpb]

I don't understand the Q? as you're asking it about "add[ing] the step time" -- the time is simply the sample rate.

The fundamental problem is for modal analysis you need an input and a response to that input at multiple locations and it appears you have above only a single 3-axis acceleration trace. Where's the reference input?

Looks like you need to go back to fundamentals of modal analysis and figure out just what it is that you're trying to do here...

I'm 80 years old (almost, anyways... :) ) and so my points of reference go way back, but the best summary of modal analysis I've ever seen is the old HP (Hewlet Packard from when it was still the original measurements/instruments company) <application note>. It develops the basic theoretical concepts and outlines the measurements necessary to be able to do modal analysis and then how to manipulate the measured data to produce the results.

[William Rose]

@dpb, Very impressive!

Answer 2

[William Rose]

@KYRIAKOS PIPIGKAS, I am sorry to hear in your comment that you have been struggling with this prblem for a long time. You said "trying to compute mode shapes and frequency of this data". I assume that you intend to identify modes of vibraiotn by their appearance in the frequency domain. You said someone proposed adding random noise to the signal to aid in this approach. Adding random noise to the measured acceleraiton signals will not help you identify modes in the acceleraiton signal.

Adding random noise to the input to the system, if that is possible, can assist in identifying vibration modes, because uncorrelated noise contains power at all frequencies. Adding power at all frequencies to the input can reveal resonance frequencies in the output that might be difficult to detect otherwise. In your case, the output is the measured acceleration

Like @dpb, I am not sure what you mean by "i am finding it difficult to add the step time in the next data of acceleration". I notice that column 1 in your data file is "step time". It looks fine to me. Perhaps you are not sure how to use the time data to determine the frequencies of the FFT. The frequencies of the FFT are given by f=df*(0:N-1), where df=1/(N*dT) is the spacing of the samples in the frequency domain, N is the total number of points in the time domain, and dT is the time step size.

%Read data from file
data =table2array(readtable('METRISEIS.txt')) ;
step_time=data(:,1);
acc_x=data(:,2);
acc_y=data(:,3); 
acc_z=data(:,4); 
%Add noise to data
sd=2;                %standard deviation of added noise
acc_xn= acc_x + 2*randn(size(step_time));
acc_yn= acc_y + 2*randn(size(step_time));
acc_zn= acc_z + 2*randn(size(step_time));
%Compute the FFTs of the data
N=length(step_time);                    %number of data points
dT=(step_time(end)-step_time(1))/(N-1); %time step size
df=1/(N*dT);                            %spacing of frequencies
f=(0:N-1)*df;                           %vector of frequencies
AX = fft(acc_x); 
AY = fft(acc_y); 
AZ = fft(acc_z);
AXn = fft(acc_xn); 
AYn = fft(acc_yn); 
AZn = fft(acc_zn);
%Plot results, up to the Nyquist frequency
figure;
subplot(3,2,1); plot(f(1:N/2),abs(AX(1:N/2)),'-r'); ylabel('AX'); title('No Added Noise')
subplot(3,2,3); plot(f(1:N/2),abs(AY(1:N/2)),'-g'); ylabel('AY')
subplot(3,2,5); plot(f(1:N/2),abs(AZ(1:N/2)),'-b'); ylabel('AZ'); xlabel('Frequency')
subplot(3,2,2); plot(f(1:N/2),abs(AXn(1:N/2)),'-r'); ylabel('AXn'); title('With Added Noise')
subplot(3,2,4); plot(f(1:N/2),abs(AYn(1:N/2)),'-g'); ylabel('AYn')
subplot(3,2,6); plot(f(1:N/2),abs(AZn(1:N/2)),'-b'); ylabel('AZn'); xlabel('Frequency')

The code above is based on your code but with a few changes. I compute the amplitude spectrum for the signal without noise and with noise. As you can see, adding noise obscures the details in the amplitude spectra.