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I am just branching out to taking the power spectrum of short-term audio frames ($20$ ms) in order to extract useful audio features. I have been reading about Welsh's method which states that after computing the squared magnitude of the result, the individual periodograms are time-averaged which reduced the variance of the individual power measurements.

I was hoping somebody could explain what and how time-averaging is achieved? Is it done by looking at the individual squared bins within a given frame, or does it compare a previous frame to a present frame like a Spectral Flux? Also, I take it that a periodogram refers to the periodicity of a given audio frame? Thanks.

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Periodogram averaging means that the squared DFTs of consecutive frames are averaged (per frequency bin). If $k$ denotes the frequency index and $l$ denotes the frame index, the averaged periodogram is computed as

$$\overline{P}[k]=\frac{1}{L}\sum_{l=0}^{L-1}P[k,l]\tag{1}$$

where $P[k,l]$ is the $k^{th}$ element of the squared magnitude of the DFT of the frame with index $l$. In (1) the averaging is done over $L$ frames. This is only useful if the signal can be assumed to be stationary during these $L$ time frames.

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  • $\begingroup$ Thanks, so basically after windowing is performed on a particular frame, we square each DFT bin, sum their magnitude's, and divide by the total number of DFT bins in that frame. Then just do the same for the next frame, and so on? So I take it a periodogram pertains to frame in the context of the power spectrum? $\endgroup$ – user1574598 May 11 '15 at 21:28

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