Simple Sufficient Condition for Peak Resolution
Recall from §5.4 that the frequency-domain image of a
sinusoid ``through a window'' is the window transform scaled by the
sinusoid's amplitude and shifted so that the main lobe is centered
about the sinusoid's frequency. A spectrum analysis of two sinusoids
summed together is therefore, by linearity of the Fourier transform,
the sum of two overlapping window transforms, as shown in
Fig.5.12 for the rectangular window. A simple
sufficient requirement for resolving two sinusoidal
peaks spaced
Hz apart is to choose a window length long
enough so that the main lobes are clearly separated when the
sinusoidal frequencies are separated by
Hz. For example, we
may require that the main lobes of any Blackman-Harris window meet at
the first zero crossings in the worst case (narrowest frequency
separation); this is shown in Fig.5.12 for the rectangular-window.
To obtain the separation shown in Fig.5.12, we must have
Hz, where
is the main-lobe width in Hz, and
is the minimum sinusoidal frequency separation in Hz.
For members of the
-term Blackman-Harris window family,
can
be expressed as
, as indicated by
Table 5.1. In normalized radian frequency units, i.e.,
radians per sample, we have
. For comparison, Table 5.2 lists minimum effective
values of
for each window (denoted
) given by an
empirically verified sharper lower bound on the value needed for
accurate peak-frequency measurement [1], as discussed
further in §5.5.4 below.
|
We make the main-lobe width
smaller by increasing the window
length
. Specifically, requiring
Hz implies
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(6.26) |
or
Thus, to resolve the frequencies













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Periodic Signals
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Other Definitions of Main Lobe Width