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Welch's Method with Windows

As usual, the purpose of the window function $ w(n)$ (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 $ R$ cannot exceed half the frame length $ M/2$ without introducing a non-uniform sensitivity to the data $ x(n)$ over time. In the rectangular window case, we can set $ R=M$ and have non-overlapping windows. For Hamming, Hanning, and any other generalized Hamming window, one normally sees $ R=(M-1)/2$ for odd-length windows. For the Blackman window, $ R\approx M/3$ is typical. In general, the hop size $ R$ is chosen so that the analysis window $ w$ 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 $ M-R$ .

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

$\displaystyle K = \left\lfloor \frac{N_x-M}{R}\right\rfloor + 1,$ (7.28)

where the floor function $ \left\lfloor x\right\rfloor $ denotes the largest integer less than or equal to $ x$ .

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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