# On FFT, interpolating signal vs extending signal in time

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:

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)

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.

• 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. Oct 23, 2023 at 12:08

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).