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Other Definitions of Main Lobe Width

Our simple definition of main-lobe band-width (distance between zero-crossings) works pretty well for windows in the Blackman-Harris family, which includes the first six entries in Table 5.1. (See §3.3 for more about the Blackman-Harris window family.) However, some windows have smooth transforms, such as the Hann-Poisson (Fig.3.21), or infinite-duration Gaussian window (§3.11). In particular, the true Gaussian window has a Gaussian Fourier transform, and therefore no zero crossings at all in either the time or frequency domains. In such cases, the main-lobe width is often defined using the second central moment.6.5

A practical engineering definition of main-lobe width is the minimum distance about the center such that the window-transform magnitude does not exceed the specified side-lobe level anywhere outside this interval. Such a definition always gives a smaller main-lobe width than does a definition based on zero crossings.

Figure 5.11: Lowpass filter design parameters.

In filter-design terminology, regarding the window as an FIR filter and its transform as a lowpass-filter frequency response [263], as depicted in Fig.5.11, we can say that the side lobes are everything in the stop band, while the main lobe is everything in the pass band plus the transition band of the frequency response. The pass band may be defined as some small interval about the midpoint of the main lobe. The wider the interval chosen, the larger the ``ripple'' in the pass band. The pass band can even be regarded as having zero width, in which case the main lobe consists entirely of transition band. This formulation is quite useful when designing customized windows by means of FIR filter design software, such as in Matlab or Octave (see §4.5.1, §4.10, and §3.13).

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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