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The 'binning' process consists in summing the energies (squared magnitude) within groups of adjacent FFT values. This will give you the total energy in a set of disjoint frequency bands.

A more elaborate form consists in using overlapping triangle filter banks - you compute a weighted sum of the energy in a range of FFT bins to get a number which can be interpreted as the energy measured at the output of a band-pass filter of a given center frequency/width.

Can anybody help in generating such overlapping filter banks as I am new to signal processing and having hard time creating such a filter.

Thanks in advance

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Let's say we have spectrum $X[k]$ and it's power spectrum $P[k]=|X[k]|^2$. Smoothing can we written as

$$S[k] = \sum_n W[n+k]P[n+k]$$

where $W[k]$ is a the smoothing window. For a rectangular window of length $2N+1$ this simply turns into.

$$S[k] = \frac{1}{2N+1}\sum_{n=k-N}^{k+N} P[n+k]$$

An efficient way to calculate this is to simply calculate the integral $Q[k] = \sum_0^kP[k]$ and this simply becomes

$$S[k] = Q[k+N]-Q[k-N]$$

To get a triangular window, you can simply apply rectangular smoothing twice. The effective window of consecutive smoothing operation is the convolution of all individual smoothing windows. The convolution of two rectangles is a triangle.

CAVEAT: I have been intentionally sloppy with the summation indices at the edges of the spectrum. What you want to do for $n<0$ and $n > K$ depends on the specific application. I have also assumed that the window is symmetric.

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  • $\begingroup$ Thank you so much for your response. So, you are saying that I need to convolve my power spectrum twice with the Rectangle window in order to get the triangular filter affect? $\endgroup$ Oct 16, 2022 at 12:49
  • $\begingroup$ Yes. That's correct $\endgroup$
    – Hilmar
    Oct 16, 2022 at 13:26

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