# Welch's Power Spectral Density - Time-Averaging Explanation

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.

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.