Don´t know how to perform IFFT considering my frequency.

Hi! I have a frequency domain data which I want to transform into time domain. I have the following file where the first column is the frequency in range [0 Nyquist frequency], second column is real component of the response and the 3rd colums is the imaginary part.

I know I have to use "ifft" function but I just don´t understand how to define it properly. I know the result should be time response doman with maximum value around 600-800 but when I launch the script I obtain peak values of 30. As far as I see there should be some way to input my frequency as the parameter. The help section includes this:

X = ifft(Y,n) returns the n-point inverse Fourier transform of Y by padding Y with trailing zeros to length n.

But I don´t understand what is the correlation between parameter "n" and my frequency, how do I know how many points to solve?

Part of my code as well as the attached data, keep in mind I have one sided spectra:

Mreaction=readmatrix("Mreaction_damped0.xlsx");
freq=Mreaction(:,1); % frequency of my response spectrum
hMoment=Mreaction(:,2)+i*Mreaction(:,3); % obtain complex number for the response moment
myTdata=ifft(hMoment,"symmetric"); % I´ve also tried with 1000 (and some others) points but the results just don´t 
% make any sense to me
plot(myTdata) % Plot without time vector since I struggle to define it

Thanks in advance!

Answer 1

[Matt J]

But I don´t understand what is the correlation between parameter "n" and my frequency, how do I know how many points to solve?

The key relationship involving n is,

n=1/(dF*dT)

where dT and dF are the time and frequency sampling intervals . The choice of n depends on what dF you have (in your case dF=0.03) and what dT you want. Also, n must be an integer.

When using fft(x,n) you are telling the code to append zeros to x making it length n. However, in applications to time-frequency analysis (like yours) I find it is often better just to manually pad x to the length that you want, because the padding has to be done in a very specific way in conjunction with fftshift.

I illustrate all this below:

Mreaction=readmatrix('https://www.mathworks.com/matlabcentral/answers/uploaded_files/1129380/Mreaction_damped0.xlsx');

dT0=0.01; %maximum desired time sampling interval
dF=Mreaction(2)-Mreaction(1); %given frequency sampling interval

hMoment=complex( Mreaction(:,2) , Mreaction(:,3)  );
n0=numel(hMoment);

n=max( ceil(1/dT0/dF) , n0 ); %ensures dT<=dT0=0.01
dT=1/dF/n; %actual time sampling interval

nAxis=( (0:n-1)-ceil((n-1)/2) );  %normalized axis
freqAxis=dF*nAxis;%frequency axis
timeAxis=dT*nAxis;%time axis


leftside=flip(conj(hMoment));
hMoment=[leftside(1:end-1); hMoment]; %two sided spectrum

hMoment(end+1:n)=0;
hMoment=fftshift(circshift(hMoment,1-n0));  %zero padding circulantly is complicated

myTdata=fftshift(real( ifft(  ifftshift(hMoment)) ))/dT;

figure
plot(freqAxis,abs(hMoment));
xlabel Frequency

figure
plot(timeAxis, myTdata) 
xlabel Time

[Andriy Voshchenko]

Hi Matt, thank you for your answer. I´m still trying to comprehend it since I don´t really understand the purpose and the reason of setting

dT0=0.01;

Also, please check my comment for the answer below sinse I tried to clarify my problem there, maybe the ifft approach is not the best option?

Meanwhile, I´ll be studying your solution since, like I said before, there are things that are new for me and I don´t fully understand. Thanks again!

[Matt J]

I don´t really understand the purpose and the reason of setting dT0=0.01

You need to decide how finely spaced the reconstructed time samples of myTdata are going to be. dT<0.01 was just an example selection.

Also, please check my comment for the answer below sinse I tried to clarify my problem there, maybe the ifft approach is not the best option?

I have read it, but there is nothing it adds to the answer I originally gave you, I'm afraid. Moreover, since both Paul and I get the same time signal as a result, I am even more confident that we are both processing the spectrum correctly, given what you've told us so far.

If the time signal our code gives you isn't what you expect, then there's something you haven't told us about how the spectrum is produced. If the point, however, is to reconstruct the exact same time signal as you started with, it should be a simple matter of just reversing each step in the process you followed to generate the spectrum.

[Andriy Voshchenko]

You need to decide how finely spaced the reconstructed time samples of myTdata are going to be dT<0.01 was just an example selection.

I see, thanks for the clarification.

Also, I have no doubt the signal is obtained correctly, the only thing left is the scaling as I´ve already mentioned. You see Paul gets something in range [-20 15] and in your case it rises to 600 and I don´t see how I can make sure the scaling is done correctly.

The previous comment with time series and its FFT is just an example that ilustrates that for the given time series which reaches 2000 [units] at its maximas the FFT provides basically the same spectra that I´m trying to invert with ifft, therefore when I obtain something like 15 or even 600 with your method I still don´t see why they are so different.

I hope I explained this last point clear enough.

[Matt J]

There's no way to account for the scaling fully without seeing the code you used to to obtain the spectrum.

My code assumes that the given spectrum was correctly scaled to approximate the continuous Fourier transform. You would have needed to multiply the FFT by dT in order to ensure this.

Assuming you did so, then dividing the IFFT by dT is the appropriate way to invert. You can see in my code that I have done this, whereas Paul has not.

If you didn't scale the forward FFT, then Paul's scaling is probably correct and your expectations are probably wrong.

[Andriy Voshchenko]

I see, as I stated in comment below the best way seems to be using another techniques instead of ifft. Thank you both for the contribution!

Answer 2

[Paul]

Hi Andriy

I would attack the problem this way, assuming throughout the original time-domain signal is real and the freq vector in the file is in Hz.

Read in the data

%Mreaction=readmatrix("Mreaction_damped0.xlsx");
Mreaction = readmatrix('https://www.mathworks.com/matlabcentral/answers/uploaded_files/1129380/Mreaction_damped0.xlsx');
freq      = Mreaction(:,1); % frequency of my response spectrum
hMoment   = Mreaction(:,2) + 1i*Mreaction(:,3); % obtain complex number for the response moment

Check the first point in the freq vector

freq(1)

ans = 0.0300

It's not zero like it should be, so I'll synthesize a point at f = 0.

freq = [0;freq];
hMoment = [real(hMoment(1));hMoment];

The last point in the one-sided spectrum has a signficanct imaginary part

hMoment(end)

ans = 0.5224 + 0.7874i

If this point truly corresponded to the Nyquist frequency, then it would be real and N, the length of the time-domain signal, would be even. But it's not, so it looks like N must be odd.

N = 2*numel(freq) + 1;

The indices for the signal and the difference between successive frequency points are

n = 0:(N-1);
dF  = 0.03; % assuming this is Hz?

Therefore, the sampling frequency and sampling period are

Fs = N*dF;
Ts = 1/Fs;

The time domain signal is then

myTdata = ifft(hMoment,N,"symmetric");
plot(n*Ts,myTdata)
xlabel('Time (sec)')

I know that you were expecting the signal to have a maximum value in the 600-800 range. The appropriate scaling, if any, would depend on how the original frequency domain data was collected and scaled in the first place.

[Andriy Voshchenko]

Hi Paul, thank you for you answer but the problem is still how do I scale the results? The spectrums I´m trying to process are the output of the simulation software therefore I´ve already specified all the awaliable data from there.

Initially I have the same time series as input, I perform 2 simulations, one in time domain and one in frequency domain (applying fft, windowing etc.). Therefore, I obtain time response and frequency response, the time response is something like this

And if I perform the FFT of the image above I obtain the following spectrum

Now this spectrum is very similar to what I have in the original problem posted. What I am pretending (and failing) to do is the opposite, obtain the time response out of my spectra (the max value is all I want).

I thought the ifft is the best approach for this but maybe there is a better way? I´ve also tried the approach where I perform the sum of all tha values of the spectra ^2 and then square root it but I don´t know if this approach is valid since well, it depends on the number of points consideres (I was using the same frequencies as the load spectra for it)

Any idea of what else I cant try would be appreciated.

P.S. If you think the data would help I can attach the files later since I don´t have access to the data right now.

[Paul]

As already stated, the appropriate scaling of the output of ifft() depends very much (entirely?) on how the frequency domain data was developed in the first place. Was it based on an FFT algorithm? If so, what scaling was assumed on that FFT? 1? 1/N? As @Matt J hypothesized, maybe the data was scaled to be consistent with the Continuous Time Fourier Transform, in which as a divide by Ts is in order as he showed, and seems to yield a result consistent with your expectations.

I agree with @Matt J that there really is not any way to know for sure without knowing exactly the process used to generate the frequency domaing data in the .xlsx file.

[Andriy Voshchenko]

Oh I see, I think in this case in particular it would be better to avoid using ifft then since I don´t have enough data and only the output spectras. Thank you both for the contribution!