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Weighted digital filter design

Figure 8.13: Illustration of one way to determine the parameters of a least-damped resonant mode. In this example, the sampling rate is 8 kHz, mode center-frequency 1 kHz, mode 3dB-bandwidth 200 Hz, rectangular weighting interval 200 Hz, and relative weighting in the passband is 100, where 1 means no weighting.

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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-10-15 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University