Normalization of zero padded signals
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I have a simple question regarding zero padding and normalization. Consider an impulse resonse of a 4 point moving average filter. and its fft zero padded to 1024 points..
x=[1/4 1/4 1/4 1/4]
By Parsevals theorem the two energies are equal as expected. However, the fft without scaling shows the correct frequency response with a gain of 1 at 0 Hz. So why do I always read the FFT should be scaled by the number of samples before zero padding (in this case 4) if I am interested in the magnitude response of the filter?
Matt J on 21 Oct 2022
Edited: Matt J on 21 Oct 2022
So why do I always read the FFT should be scaled by the number of samples before zero padding (in this case 4) if I am interested in the magnitude response of the filter?
The FFT is a tool with many applications, each with its own appropriate scaling.
Scaling by 1/N is done when the FFT is being used to evaluate the Discrete Fourier Series.
When it is being used to approximate the continuous Fourier transform, it is scaled by the time sampling interval 1/Fs.
To achieve Parseval's equality, the fft should be scaled by 1/sqrt(N):
x=[1/4 1/4 1/4 1/4];