IIR filter design - Continuous and Discrete Time Signals and Systems - page 735

Fig. 16.8. Magnitude response
of the DT highpass filter
designed in Example 16.5.

0

−20

−40

−60

W

− p −0.75 p −0.5 p −0.25 p 0 0.25 p 0.5 p 0.75 p p

Figure 16.8 shows the amplitude gain response of the designed filter. We observe
that the pass-band and stop-band specifications are both satisfied.

Example 16.6
Example 15.6 designed a bandpass FIR filter with the following specifications:

(i) pass-band edge frequencies, Ω p1 = 0 . 375 π and Ω p2 = 0 . 5 π radians/s;
(ii) stop-band edge frequencies, Ω s1 = 0 . 25 π and Ω s2 = 0 . 625 π radians/s;
(iii) stop-band attenuations, δ s1 > 50 dB and δ s2 > 50 dB.

Design an IIR filter with the same specifications.

Solution
Choosing k = 1 (sampling interval T = 2), step 1 transforms the pass-band and
stop-band corner frequencies into the CT frequency domain:

pass-band corner frequency I ω p1 = tan(0 . 5 Ω p1 ) = tan(0 . 1875 π ) = 0 . 6682 radians/s;

= tan(0 . 5 Ω p2 ) = tan(0 . 25 π ) = 1 radian/s; pass-band corner frequency II ω p2

= tan(0 . 5 Ω s1 ) = tan(0 . 125 π ) = 0 . 4142 radians/s; stop-band corner frequency I ω s1

ω s2 = tan(0 . 5 Ω s2 ) = tan(0 . 3125 π ) = 1 . 4966 radians/s .

stop-band corner frequency II

Step 2 designs an analog filter for the aforementioned specifications. We can
either use the analytical techniques developed in Chapter 7 or use the M ATLAB
program. In the following, we calculate the analog elliptic filter for the given
specifications using M ATLAB . Since the pass-band ripple is not specified, we
assume that it is given by 0.03 dB. The M ATLAB code is given by

>> wp = [0.6682 1]; ws = [0.4142 1.4966];
>> Rp = 0.03; Rs = 50;
>> [N, wn] = ellipord(wp,ws,Rp,Rs,'s');
>> [nums,denums] = ellip(N,Rp,Rs,wn,'s');

which results in an eighth-order elliptic filter with the following transfer
function:
H ( s ) = 0.001(3 . 164 s 8 + 30 . 27 s 6 + 57 . 02 s 4 + 13 . 51 s 2 + 0 . 6308)

s 8 + 0 . 7555 s 7 + 3 . 07 s 6 + 1 . 634 s 5 + 3 . 229 s 4 + 1 . 092 s 3 + 1 . 371 s 2 + 0 . 2254 s + 0 . 1994 .
Step 3 derives the z-transfer function of the digital filter using the bilinear trans-
formation. This is achieved by using the bilinear function in M ATLAB .