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is the maximum absolute weighted error. These equations may be written in matrix form in terms of the unknowns a(O), . . . a ( L ) and E as follows:
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Given the extremal frequencies, these equations may be solved for a(O), . . . , a ( L ) and c . To find the extremal frequencies, there is an efficient iterative procedure known as the Parks-McClellan algorithm, which involves the following steps:
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1. Guess an initial set of extremal frequencies. 2. Find r by solving Eq. (9.5). The value of c has been shown to be
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FILTER DESIGN
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Evaluate the weighted error function over the set . by interpolating between the extremal frequencies F using the Lagrange interpolation formula. Select a new sel of extremal frequencies by choosing the L error function is maximum.
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+ 2 frequencies for which the interpolated
If the extremal frequencies have changed, repeat the iteration from step 2.
A design formula that may be used to estimate the equiripple filter order for a low-pass filter with a transition width Af , passband ripple 6,. and stopband ripple 6 , is
EXAMPLE 9 3 3 Suppose thal we would like to design an equiripple low-pass filter with a passband cutoff frequency .. w,, = 0 . 3 ~a. stopband cutoff frequency o,= 0 . 3 5 ~a, passband ripple of 6, = 0.01, and a stopband ripple of 6, = 0.001. Estimating the filter using Eq. (9.h),we find
Because we want the ripple in Ihe stopband to be I0 times smaller than the ripple in the passband, the error must be weighted usmg the weighting function
Using the Parks-McClellan algorilhm to design the filter. we obtain a filter with the frequency response magnitude shown below.
9.4 IIR FILTER DESIGN There are two general approaches used to design IIR digital filters. The most common is to design an analog IIR filter and then map it into an equivalent digital filter because the art of analog filter design is highly advanced. Therefore, it is prudent to consider optimal ways for mapping these filters into the discrete-time domain. Furthermore, because there are powerful design procedures that facilitate the design of analog filters, this approach
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FILTER DESIGN
to IIR filter design is relatively simple. The second approach to design IIR digital filters is to use an algorithmic design procedure, which generally requires the use of a computer to solve a set of linear or nonlinear equations. These methods may be used to design digital filters with arbitrary frequency response characteristics for which no analog filter prototype exists or to design filters when other types of constraints are imposed on the design. In this section, we consider the approach of mapping analog filters into digital filters. Initially, the focus will be on the design of digital low-pass filters from analog low-pass filters. Techniques for transforming these designs into more general frequency selective filters will then be discussed.
9.4.1 Analog Low-Pass Filter Prototypes
To design an IIR digital low-pass filter from an analog low-pass filter, we must first know how to design an analog low-pass filter. Historically, most analog filter approximation methods were developed for the design of passive systems having a gain less than or equal to 1. Therefore, a typical set of specifications for these filters is as shown in Fig. 9-5(a),with the passband specifications having the form
((I)
Specifications in terms of
and 6,.
( I ) Specifications in terms of e and A.
Fig. 9-5. Tivo different conventions for specifying the passband and stopband deviations for an analog
low-pass filter. Another convention that is commonly used is to describe the passband and stopband constraints in terms of the parameters E and A as illustrated in Fig. 9-5(h). Two auxiliary parameters of interest are the dist.riminatior7 factor,
and the selectivity factor
k ="
The three most commonly used analog low-pass filters are the Butterworth, Chebyshev, and elliptic filters. These filters are described below.