#### Weighted digital filter design

Many methods for digital filter design support spectral weighting functions
that can be used to focus in on the least-damped modes in the frequency
response. One is the *weighted equation-error* method which is
available in the matlab `invfreqz()` function (§8.6.4).
Figure 8.13 illustrates use of it. For simplicity, only one
frequency-response peak plus noise is shown in this synthetic example.
First, the peak center-frequency is measured using a quadratically
interpolating peak finder operating on the dB spectral magnitude. This is
used to set the spectral weighting function. Next, `invfreqz()` is
called to design a two-pole filter having a frequency response that
approximates the measured data as closely as possible. The weighting
function is also shown in Fig.8.13, renormalized to
overlay on the scale of the plot. Finally, the amplitude response of the
two-pole filter designed by the equation-error method is shown overlaid in
the figure. Note that the agreement is quite good near the peak which is
what matters most. The interpolated peak frequency measured initially in
the nonparametric spectral magnitude data can be used to fine-tune the
pole-angles of the designed filter, thus rendering the equation-error
method a technique for measuring only the peak bandwidth in this case.
There are of course many, many techniques in the signal processing
literature for measuring spectral peaks.

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Linear prediction

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Amplitude response peak measurement