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

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

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