## Welch's Method

*Welch's method* [296] (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 th windowed, zero-padded frame from the signal by

(7.26) |

where is defined as the window

*hop size*, and let denote the number of available frames. Then the periodogram of the th block is given by

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

In other words, it's just an average of periodograms across time. When 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.

### Welch Autocorrelation Estimate

Since
which is *circular* (or
cyclic) correlation, we must use *zero padding* in each FFT in
order to be able to compute the acyclic autocorrelation function as
the inverse DFT of the Welch PSD estimate. There is no need to
arrange the zero padding in zero-phase form, since all phase
information is discarded when the magnitude squared operation is
performed in the frequency domain.

The Welch autocorrelation estimate is *biased*. That is, as
discussed in §6.6, it converges as
to the true
autocorrelation
weighted by
(a Bartlett window). The
bias can be removed by simply dividing it out, as in
(6.15).

### Resolution versus Stability

A fundamental trade-off exists in Welch's method between
*spectral resolution* and *statistical stability*.
As discussed in §5.4.1, we wish to maximize the block size
in order to maximize *spectral resolution*. On the other
hand, more blocks (larger
) gives *more averaging* and hence
greater spectral stability.
A typical default choice is
, where
denotes the number of available data
samples.

**Next Section:**

Welch's Method with Windows

**Previous Section:**

The Periodogram