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Tighter Bounds for Minimum Window Length

[This section, adapted from [1], is relatively advanced and may be skipped without loss of continuity.]

Figures 5.14(a) through 5.14(d) show four possible definitions of main-lobe separation that could be considered for purposes of resolving closely spaced sinusoidal peaks.

Figure 5.14: Four alternative main-lobe displacement rules for resolving closely spaced sinusoidal peaks. For peak-frequency measurements based on a few samples at the main-lobe center, (a) is suboptimal, (b) is nearly optimal [1], (c) is analogous to a filter-design specification based on side-lobe level, and (d) is overly conservative, but sufficient and simple to compute for Blackman-Harris windows.
\begin{figure*}\centering
\subfigure[Minimum orthogonal separation]{
\epsfxsize =4.0in
\epsfysize =1.0in \epsfbox{eps/SpecResCases1.eps}
}
\subfigure[Separation to first stationary-point \cite{AbeAndSmith04}]{
\epsfxsize =4.0in
\epsfysize =1.0in \epsfbox{eps/SpecResCases2.eps}
}
\subfigure[Separation to where main-lobe level equals side-lobe level]{
\epsfxsize =4.0in
\epsfysize =1.0in \epsfbox{eps/SpecResCases3.eps}
}
\subfigure[Full main-lobe separation]{
\epsfxsize =4.0in
\epsfysize =1.0in \epsfbox{eps/SpecResCases4.eps}
}\end{figure*}

In Fig.5.14(a), the main lobes sit atop each other's first zero crossing. We may call this the ``minimum orthogonal separation,'' so named because we know from Discrete Fourier Transform theory [263] that $ M$ -sample segments of sinusoids at this frequency-spacing are exactly orthogonal. ($ M$ is the rectangular-window length as before.) At this spacing, the peak of each main lobe is unchanged by the ``interfering'' window transform. However, the slope and higher derivatives at each peak are modified by the presence of the interfering window transform. In practice, we must work over a discrete frequency axis, and we do not, in general, sample exactly at each main-lobe peak. Instead, we usually determine an interpolated peak location based on samples near the true peak location. For example, quadratic interpolation, which is commonly used, requires at least three samples about each peak (as discussed in §5.7 below), and it is therefore sensitive to a nonzero slope at the peak. Thus, while minimum-orthogonal spacing is ideal in the limit as the sampling density along the frequency axis approaches infinity, it is not ideal in practice, even when we know the peak frequency-spacing exactly.6.8

Figure 5.14(b) shows the ``zero-error stationary point'' frequency spacing. In this case, the main-lobe peak of one $ \hbox{asinc}$ sits atop the first local minimum from the main-lobe of the other $ \hbox{asinc}$ . Since the derivative of both $ \hbox{asinc}$ functions is zero at both peak frequencies at this spacing, the peaks do not ``sit on a slope'' which would cause the peak locations to be biased away from the sinusoidal frequencies. We may say that peak-frequency estimates based on samples about the peak will be unbiased, to first order, at this spacing. This minimum spacing, which is easy to compute for Blackman-Harris windows, turns out to be very close to the optimal minimum spacing [1].

Figure 5.14(c) shows the minimum frequency spacing which naturally matches side-lobe level. That is, the main lobes are pulled apart until the main-lobe level equals the worst-case side-lobe level. This spacing is usually not easy to compute, and it is best matched with the Chebyshev window (see §3.10). Note that it is just a little wider than the stationary-point spacing discussed in the previous paragraph.

For ease of comparison, Fig.5.14(d) shows once again the simple, sufficient rule (''full main-lobe separation'') discussed in §5.5.2 above. While overly conservative, it is easily computed for many window types (any window with a known main-lobe width), and so it remains a useful rule-of-thumb for determining minimum window length given the minimum expected frequency spacing.

A table of minimum window lengths for the Kaiser window, as a function of frequency spacing, is given in §3.9.


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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