## Welch's Method with Windows

As usual, the purpose of the window function (Chapter 3) is to reduce side-lobe level in the spectral density estimate, at the expense of frequency resolution, exactly as in the case of sinusoidal spectrum analysis.

When using a non-rectangular analysis window, the window hop-size
cannot exceed half the frame length
without introducing a
non-uniform sensitivity to the data
over time. In the
rectangular window case, we can set
and have non-overlapping
windows. For Hamming, Hanning, and any other generalized Hamming
window, one normally sees
for odd-length windows. For the
Blackman window,
is typical. In general, the hop size
is chosen so that the analysis window
*overlaps and adds
to a constant* at that hop size. This consideration is explored more
fully in Chapter 8. An equivalent parameter
used by most matlab implementations is the *overlap* parameter
.

Note that the number of blocks averaged in (6.27) increases as decreases. If denotes the total number of signal samples available, then the number of complete blocks available for averaging may be computed as

(7.28) |

where the

*floor function*denotes the largest integer less than or equal to .

###

Matlab for Welch's Method

Octave and the Matlab Signal Processing Toolbox have a `pwelch`
function. Note that these functions also provide *confidence intervals*
(not covered here).
Matlab for generating the data shown in Fig.6.1 (employing Welch's method)
appears in Fig.6.2.

M = 32; Ks=[1 8 32 128] nkases = length(Ks); for kase = 1:4 K = Ks(kase); N = M*K; Nfft = 2*M; % zero pad for acyclic autocorrelation Sv = zeros(Nfft,1); % PSD "accumulator" for m=1:K v = randn(M,1); % noise sample V = fft(v,Nfft); Vms = abs(V).^2; % same as conj(V) .* V Sv = Sv + Vms; % sum scaled periodograms end Sv = Sv/K; % average of all scaled periodograms rv = ifft(Sv); % Average Bartlett-windowed sample autocor. rvup = [rv(Nfft-M+1:Nfft)',rv(1:M)']; % unpack FFT rvup = rvup/M; % Normalize for no bias at lag 0 end |

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Filtered White Noise

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Welch's Method