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When we interpolate, then FFT the output will have more bins. When we extend the signal in time, Then FFT output will have more bins too but:

Interpolation increases max bin frequency but time extention not. Interpolation simply scale signal in f domain. But how time extention will effects? Depend on extended signal content.

What if in case of single frequency? Assume reference signl is:

real(ifft([0 zeros(1,1) ones(1,1) zeros(1,6) ones(1,1) zeros(1,1)]));

This frequency response signal have this structure: [0 PrePad 1 PostPad 1 PrePad]. Then in our example we have 1 PrePad and 6 PostPad, Don't worry this is just naming. IFFT of this signal is simple cosine signal. which you can look it by MATLAB|GNU Octave.

Now in case of Interpolation:

Nx Pre pad post pad
1x 1 6
2x 1*2+1 6*2+1
3x 1*3+2 6*3+2
4x 1*4+3 6*4+3
... ... ...

On time extention: (Just replication of the signal appeds to itself)

Mx Pre pad post pad
1x 1 6
2x 1 6*2+1
3x 1 6*3+2
4x 1 6*4+3
... ... (remains the same) ...

This means PrePad is also scales in case of interpolation. Whole f is scaled it's obvious but is there any answer to tell how time extension affect the frequency in better sense? The nature of this question ariesed for me to find IFFT of frequency of LTI system. I want to deform frequency respnse in sense that causes to IFFT give longer duration impulse response. Longer duration impulse response means less aliasing(If possible) and more accurate model of the system.

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  • $\begingroup$ I'm not quite sure what you are asking but A) zero-padding in time is interpolation in frequency and B) zero-padding in frequency is interpolation in time. Most properties of the DFT are completely symmetrical between the domains. Your first example is zero-padding in frequency, which is interpolation in time. $\endgroup$
    – Hilmar
    Oct 23, 2023 at 12:08

2 Answers 2

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It is like you took a vacation to Italy, took a bunch of photos but forgot to take pictures of the colosseum in Rome (didn't capture samples for long enough to see the whole impulse response). There is no amount of editing of your other photos (zero-padding/interpolation) that can be done to capture the colosseum, you just have to go back there and take the picture (capture more samples to begin with).

The frequency resolution is always the capture duration $\frac{f}{N}$, where $f$ is the sampling rate and $N$ is the number of the samples captured. When you are zero-padding, you are adding no useful information to the signal (just zeros) so you can't expect a better resolution in the result. Zero-padding is increasing both $f$ and $N$ equally, which is why resolution does not increase. The way to make your IFFT result give a longer duration impulse response as the output is to have captured more samples to begin with (I'd start with increasing sample rate since you mentioned you see aliasing).

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    $\begingroup$ Excelent analogy, I'll use it next time I have to explain this: by interpolating the photos, you cat get a nice "smooth" vacation video, but the colosseum will not appear. $\endgroup$
    – MBaz
    Oct 23, 2023 at 14:25
  • $\begingroup$ Fantastic analogy. But taking fotoes means time domain sample capturing. But in my case I have Frequency domain. I want to infere the impulse response from it using IFFT. I can achieve frequency response with resolution with my desiered frequency been. $\endgroup$ Nov 1, 2023 at 7:35
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Interpolating in time causes Frequncy domain to scale completly, also the boundaries of frequency domain changes accordingly, since nyquist frequiency will be changed.

On time extentioning, Nyquist doesn't changes, just frequency bins increases. Then if I only increase the resolution of frequency domain in my algorithm which determines the frequency domain. Time extension in impulse response or time domain have to automatically apply.

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