### 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.

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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Simple Sufficient Condition for Peak Resolution

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Frequency Resolution