How do I perform a proper FFT of a gated segment of a time domain waveform with a logmag amplitude?

I have frequency domain data in I/Q to start. I can successfully transform with IFFT to a logmag time domain waveform of 4499 points. The freq domain data is 512 points with freq step of 10.9375Mhz (5.6Ghz of BW). The waveform is 0-91.25nS in length. After IFFT using a Matlab script in other software, I gate a section out for further processing. This is usually around 51 points and 1nS in length. I'm trying to take this gated waveform and FFT back to freq domain with proper amplitudes and frequency. I'm new to data processing and especially Matlab so need help.

Should FFT be same number of points as IFFT? How do I ensure proper frequency distribution with only a 51 point waveform?

Thanks for any help...

[Paul]

Hi Todd,

Do the original 512 points in the frequency domain cover the frequencies from 0 to Fs? Or do those 512 points only cover the "one-sided" transform from 0 to Fs/2?

Starting from the 512 points in the frequency domain, how do you apply ifft to get to 4499 points in the time domain? Presumably some sort of zero-padding is involved. Can you explain, or better yet show in code, exactly how that's done starting from the original 512 points?

Once you have the 4499 points in the time domain, how are the 51 points selected?

What is the expectation after applying fft to those 51 points as compared to the original 512 points?

Also, I'm not tracking the sampling. The original 512 points are spaced at 10.9375 MHz? Which waveform spans 0-91.25ns? The 4599 points? Or the 51?

[Todd]

Paul,

The 512 points in freq domain are complex values simply put into the Matlab function Y=ifft(X,A). X being the complex array and A desired ifft points. I'm using the Matlab script function only, nothing else is done to the data.

Then a section of data is gated depending on what I want in time. For example, I would select the data from the new time domain waveform from 62nS to 63nS. I now have a new time domain waveform of 1nS in length "gated" from the 4499 point waveform.

My expectation is to take this new 1nS waveform and put it back into freq domain spanning my original 5.6Ghz of BW.

This is where I'm stuck. I don't have to use 4499, I can use the next power of 2 or any number as long as I meet nyquist...just playing around with it.

To answer your last, yes, the 10.9375Mhz freq step is from the original 512 points of freq domain data. The 4499 point ifft spans 0-91.25nS.

Not so much worried about accuracy of these numbers at the moment, that can be fixed easily. The proper procedure accuracy is what I'm after. I can do this on the hardware (Keysight ENA) but want to be able to post process myself when necessary.

Thank you my friend,

[Paul]

The numbers still aren't working for me. If the original time domain data has a sampling frequency of

Fs = 10.9375e6; % Hz

Then the sampling period is

Ts = 1/Fs

Ts = 9.1429e-08

So each of the original 512 time-domain points were spaced by 91.43 ns.

How do we go from that to 4499 time domain points from 0-91.25 ns, i.e., all sampled within one sampling period of the original time-domain data?

[Todd]

Sorry, guess I'm not being clear enough.

The original data is not time domain sampled at 10.9375e6. Original data is freq domain stepped at 10.9375e6 Hz. 5.6Ghz of BW, 512 points or "steps". So, original complex data is 512 points of data in freq domain. Each data point is stepped in frequency by 10.9375e6 Hz through 5.6Ghz of BW. 5.6Ghz/512=10.9375e6 Hz. So 10.9375e6 Hz is the original freq step in freq domain, not a sampling rate.

With these settings on the hardware, max time domain range is 91.25nS with a time domain transform. I'm using a kaiser window for the transform but that really doesn't matter here.....

When I ifft (with a Matlab script of 4499 points) the original 512 point complex data I can scale the time domain plot to ge a really accurate repeat of what the hardware shows me when I do a time domain transform on it. My problem is like I said, take a gated portion of my ifft data and putting it back to freq domain.

Does this help?

[Paul]

Very much. Don't know how I missed that 10.9375e6 is the resolution, not the sampling frequency. My mistake.

I'll try to come back to this later today.

After you select the 51 point subset of the 4499 points, do you care about the phase and amplitude in the frequency domain? Or only the amplitude?

[Todd]

Not your fault. I'm not the best communicator. I tend to talk like everyone knows what I'm doing!

Only interested in amplitude at the moment.

Thank you!

[Todd]

Also, thinking about this and talking with a colleague......

I think my problem may be letting Matlab pad my fft instead of me doing it. Without padding in the proper locations my gated waveform maybe somewhat meaningless.....

[Paul]

If X is the 512-point data in the frequency domain, then doing

ifft(X,4499)

is not correct. In the frequency domain, the input to ifft has to be zero-padded in the middle of the array to return an interpolated signal in the time domain.

I'll come back later with a concrete example.

In the meantime, see if this Answer provides any insight.

[Todd]

I think you are correct if you want to track phase. If magnitude is your only concern, trail end padding is sufficient.

I'm no expert, so verify this or correct me I'm wrong.

Your other answer (link you sent) was VERY informative. Thanks so much!

Answer 1

[Paul]

Generate a 512-point reference signal in the time domain

Ts0 = 0.01;
Fs0 = 1/Ts0;
N0 = 512;
n0 = 0:N0-1;
t0 = n0*Ts0;
y0 = sin(2*pi*10*t0);

512-point FFT and the assoicated frequency vector; plot the magnitude; peaks are at expecte locations

Y0 = fft(y0);
f0 = n0/N0*Fs0;
figure
plot(f0,abs(Y0))
xline([10,Fs0-10],'r')

Zero-pad the FFT to 4500 points. I think we want to ensure that the number point zeros in the padding is even to maintain conjugate symmetry.

N1 = 4500;
npad = (N1 - N0)/2;

Zero-pad in the middle of the array. We can accomplish this by fftshift-ing the the transform, zero-pad with the same number of points on each side, and ifftshift back

Y1 = ifftshift([zeros(1,npad),fftshift(Y0),zeros(1,npad)]);

Two extra steps needed to ensure the zero-padded transform is conjugate symmetric

Y1(2 + N0/2 - 1)   = Y1(end - N0/2 + 1)/2;
Y1(end - N0/2 + 1) = Y1(2 + N0/2 - 1);
isequal(Y1(2:end),conj(Y1(end:-1:2)))

ans = logical
   1

The associated sampling frequency is scaled from the original after the padding

Fs1 = Fs0*N1/N0;
n1 = 0:N1-1;
f1 = n1/N1*Fs1;

Peaks are in the expected locations

figure
plot(f1,abs(Y1))
xline([10,Fs1-10],'r')

Back to the time domain from the the padded transform. Scale the output to account for padded transform, which wasn't itself rescaled

y1 = ifft(Y1,'symmetric')*N1/N0;
Ts1 = 1/Fs1;
t1 = n1*Ts1;

The resulting signal interpolates the original signal

figure
plot(t0,y0,'o',t1,y1,'-'),grid

Take 51 points from the interpolated time domain signal

N2 = 51;
y2 = y1(1:N2);
n2 = 0:N2-1;
Ts2 = Ts1;
figure
plot(n2*Ts2,y2)

Those points don't even cover a full cycle

Fs2 = Fs1;

Transform and plot

Y2 = fft(y2);
f2 = n2/N2*Fs2;
figure
plot(f2,abs(Y2)),grid
xline([10,Fs2-10],'r')

Do the same with more points, peaks in the expected locations

N2 = 251;
y2 = y1(1:N2);
n2 = 0:N2-1;
Y2 = fft(y2);
f2 = n2/N2*Fs2;
figure
plot(f2,abs(Y2)),grid
xline([10,Fs2-10],'r')

[Todd]

Thanks so much. Looks good. My actual waveform is not cyclic so 51 pts is enough in my case.

Much appreciate the example!!

[Paul]

Whether or not it's cyclic (and keep in mind that this analysis is based on representing your actual waveform as a sum of cyclics), I was trying to show what happens wrt frequency resolution. We can see that

syms Fs0 N0
df0 = Fs0/N0;
syms N1
Fs1 = Fs0*N1/N0;
df1 = Fs1/N1;
syms N2
Fs2 = Fs1;
df2 = Fs2/N2

df2 = 

df2/df0

ans = 

The frequency step for the N2=51 data wil be ~90x (4500/51) larger than for the original data.

[Todd]

Yes, totally agree.