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U. Glaeser

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TABLE 26.2 Error Measurements for Fig. 26.5<br />

Filter RMS Error Maximal Error in Passband RMS Error in Stopband<br />

(a) Least-squares 0.0361 0.1793 0.0348<br />

(b) Chebyshev 0.0657 0.0923 0.0660<br />

(c) Constrained least-squares 0.0562 0.0959 0.0571<br />

(d) Least-squares stopband 0.0584 0.0958 0.0374<br />

The norm optimization problem differs quite a bit for the FIR and IIR cases:<br />

FIR filter: The problem is formed on a linear vector space and has been well studied. The optimal<br />

solution is unique by convexity. Many available design methods are not only elegant, but are also<br />

computationally efficient and have guaranteed convergence.<br />

IIR filter: Although the IIR filter design problem does not have the same nice properties as the FIR<br />

filter design problem, optimizing the norm is relatively easy. One iterative approach to IIR filter<br />

design relies on a sub-procedure similar to the method for FIR filter design.<br />

26.4 Conventional Design Methods<br />

Although many filter design papers have been published in the 35 years of DSP, only a handful of filter<br />

design methods are widely used. Some of the older conventional methods can design filters with excellent<br />

magnitude response using a very simple procedure, but the variety of possible filter specifications and error<br />

norms are usually limited. More recent methods offer general design capabilities for both magnitude and<br />

phase approximation, but are based on numerical optimization.<br />

IIR Filters from Analog Filters<br />

Originally digital filters were derived from analog filters [14] because analog filter design techniques had<br />

been studied for a long time and the design usually involved algebraic formulas that were simple to carry<br />

out. The two main design methods are impulse-invariance and the bilinear transformation.<br />

Impulse Invariance<br />

The design is carried out by starting with an already designed analog filter that is bandlimited. Let ha(t) denote the impulse response of the analog filter. Then the impulse response of the digital filter is obtained<br />

by sampling, i.e., by setting h[n] = Tdha(nTd); however, no analog filter is truly bandlimited, so the actual<br />

© 2001 by CRC Press LLC<br />

Note: These are the passband and stopband errors in bandpass filters designed by a different norm problem.<br />

Error<br />

Error<br />

0.2<br />

0.1<br />

(a) Least-squares error<br />

0<br />

0 0.2π 0.4π 0.6π 0.8π π<br />

Frequency (ω)<br />

0.2<br />

0.1<br />

(c) Constrained least squares error<br />

0<br />

0 0.2π 0.4π 0.6π 0.8π π<br />

Frequency (ω)<br />

0<br />

0 0.2π 0.4π 0.6π 0.8π π<br />

Frequency (ω)<br />

FIGURE 26.5 Different error norms. The four filters were designed to approximate the same bandpass filter of order<br />

25 with four different norms. The filter in (c) was designed by minimizing the least-squares norm under the constraint<br />

that the maximal error be smaller than 0.0959. Note that the filter in (c) can also be designed by minimizing the<br />

unconstrained combined norm problem with the norm weighting α = 0.4. The filter can also be designed so that both<br />

the distortion in the pass band and the power of the stopband error are small. The filter in (d) was designed by<br />

optimizing the combination of the Chebyshev error norm of the passband plus the least-squares norm of the stopband.<br />

Error<br />

Error<br />

0.2<br />

0.1<br />

0.2<br />

0.1<br />

(b) Chebyshev error<br />

(d) Least-squares stopband error<br />

0<br />

0 0.2π 0.4π 0.6π 0.8π π<br />

Frequency (ω)

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