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Comparison to Optimal Chebyshev FIR Filter

Let's now compare the window-method design using the Kaiser window to the optimal equiripple FIR filter design given by the Remez multiple exchange algorithm.

Note, by the way, that many filter-design software functions, such as firpm have special modes for designing Hilbert-transform filters [223].

It turns out that the Remez exchange algorithm has convergence problems for filters larger than a few hundred taps. Therefore, the FIR filter length $ M=257$ was chosen above to be small enough to work out in this comparison. However, keep in mind that for very large filter orders, the Remez exchange method may not be an option. There are more recently developed methods for optimal Chebyshev FIR filter design, using ``convex optimization'' techniques, that may continue to work at very high orders [217,22,153]. The fast nonparametric methods discussed above (frequency sampling, window method) will work fine at extremely high orders.

The following Matlab command will try to design the FIR Hilbert-transform filter of the desired length with the desired transition bands:

 hri = firpm(M-1, [f1,f2]/fn, [1,1], [1], 'Hilbert');
Instead, however, we will use a more robust method [227] which uses the Remez exchange algorithm to design a lowpass filter, followed by modulation of the lowpass impulse-response by a complex sinusoid at frequency $ f_s/4$ in order to frequency-shift the lowpass to the single-sideband filter we seek:
tic; % remember the current time
hrm = firpm(M-1, [0,(f2-fs/4)/fn,0.5,1], [1,1,0,0], [1,10]);
dt = toc; % design time dt can be minutes
hr = hrm .* j .^ [0:M-1]; % modulate lowpass to single-sideband
The weighting [1,10] in the call to firpm above says ``make the pass-band ripple $ 10$ times that of the stop-band.'' For steady-state audio spectra, pass-band ripple can be as high as $ 0.1$ dB or more without audible consequences.5.11 The result is shown in Fig.4.16 (full amplitude response) and Fig.4.17 (zoom-in on the dc transition band). By symmetry the high-frequency transition region is identical (but flipped):

Figure 4.16: Frequency response of the optimal Chebyshev FIR filter designed by the Remez exchange algorithm.
\includegraphics[width=0.8\twidth]{eps/OptimalHilbertFR}

Figure 4.17: Transition region of the optimal Chebyshev frequency response.
\includegraphics[width=0.8\twidth]{eps/OptimalHilbertZoomedFR}

In this case we did not normalize the peak amplitude response to 0 dB because it has a ripple peak of about 1 dB in the pass-band. Figure 4.18 shows a zoom-in on the pass-band ripple.

Figure 4.18: Pass-Band ripple for optimal Chebyshev frequency response.
\includegraphics[width=0.8\twidth]{eps/OptimalHilbertZoomedPB}


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2014-06-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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