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Optimized Windows

We close this chapter with a general discussion of optimal windows in a wider sense. We generally desire

$\displaystyle W(\omega) \approx \delta(\omega),$ (4.59)

but the nature of this approximation is typically determined by characteristics of audio perception. Best results are usually obtained by formulating this as an FIR filter design problem (see Chapter 4). In general, both time-domain and frequency-domain specifications are needed. (Recall the potentially problematic impulses in the Dolph-Chebyshev window shown in Fig.3.33 when its length was long and ripple level was high). Equivalently, both magnitude and phase specifications are necessary in the frequency domain.

A window transform can generally be regarded as the frequency response of a lowpass filter having a stop band corresponding to the side lobes and a pass band corresponding to the main lobe (or central section of the main lobe). Optimal lowpass filters require a transition region from the pass band to the stop band. For spectrum analysis windows, it is natural to define the entire main lobe as ``transition region.'' That is, the pass-band width is zero. Alternatively, the pass-band could be allowed to have a finite width, allowing some amount of ``ripple'' in the pass band; in this case, the pass-band ripple will normally be maximum at the main-lobe midpoint ( $ W(0)= 1+\delta$ , say), and at the pass-band edges ( $ W(\epsilon) = W(-\epsilon) = 1-\delta$ ). By embedding the window design problem within the more general problem of FIR digital filter design, a plethora of optimal design techniques can be brought to bear [204,258,14,176,218].

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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