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Welch's Method

Welch's method [297] (also called the periodogram method) for estimating power spectra is carried out by dividing the time signal into successive blocks, forming the periodogram for each block, and averaging.

Denote the $ m$ th windowed, zero-padded frame from the signal $ x$ by

$\displaystyle x_m(n)\isdef w(n)x(n+mR), \quad n=0,1,\ldots,M-1,\; m=0,1,\ldots,K-1,$ (7.26)

where $ R$ is defined as the window hop size, and let $ K$ denote the number of available frames. Then the periodogram of the $ m$ th block is given by

$\displaystyle P_{x_m,M}(\omega_k)
= \frac{1}{M}\left\vert\hbox{\sc FFT}_{N,k}(x_m)\right\vert^2
\isdef \frac{1}{M}\left\vert\sum_{n=0}^{N-1} x_m(n) e^{-j2\pi nk/N}\right\vert^2

as before, and the Welch estimate of the power spectral density is given by

$\displaystyle {\hat S}_x^W(\omega_k) \isdef \frac{1}{K}\sum_{m=0}^{K-1}P_{x_m,M}(\omega_k). \protect$ (7.27)

In other words, it's just an average of periodograms across time. When $ w(n)$ is the rectangular window, the periodograms are formed from non-overlapping successive blocks of data. For other window types, the analysis frames typically overlap, as discussed further in §6.13 below.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University