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.

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
[264] that
-sample segments of sinusoids at this
frequency-spacing are exactly orthogonal. (
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
sits atop
the first *local minimum* from the main-lobe of the other
. Since the derivative of both
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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University