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

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

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

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

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