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450 Chapter 8 IIR FILTER DESIGN
and
or
e −jω′ = ∣ ∣G(e −jω ) ∣ ∣ e j̸ G(e −jω )
−ω ′ = ̸ G(e −jω )
The general form of the function G(·) that satisfies these requirements is
a rational function of the all-pass type given by
Z −1 = G ( z −1) = ±
n∏
k=1
z −1 − α k
1 − α k z −1
where |α k | < 1 for stability and to satisfy requirement 3.
Now by choosing an appropriate order n and the coefficients {α k },we
can obtain a variety of mappings. The most widely used transformations
are given in Table 8.2. We will now illustrate the use of this table for
designing a highpass digital filter.
□ EXAMPLE 8.25 In Example 8.22 we designed a Chebyshev-I lowpass filter with specifications
ω ′ p =0.2π,
ω ′ s =0.3π,
R p =1dB
A s =15dB
and determined its system function
H LP (Z) =
0.001836(1 + Z −1 ) 4
(1 − 1.4996Z −1 +0.8482Z −2 )(1 − 1.5548Z −1 +0.6493Z −2 )
Design a highpass filter with these tolerances but with passband beginning at
ω p =0.6π.
Solution
We want to transform the given lowpass filter into a highpass filter such that
the cutoff frequency ω ′ p =0.2π is mapped onto the cutoff frequency ω p =0.6π.
From Table 8.2
cos[(0.2π +0.6π)/2]
α = − = −0.38197 (8.70)
cos[(0.2π − 0.6π)/2]
Hence
H LP (z) =H(Z)| Z=−
z −1 −0.38197
1−0.38197z −1
0.02426(1 − z −1 ) 4
=
(1+0.5661z −1 +0.7657z −2 )(1+1.0416z −1 +0.4019z −2 )
which is the desired filter. The frequency response plots of the lowpass filter
and the new highpass filter are shown in Figure 8.31.
□
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