Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 318

[B, A] [ lp2lp([1],[1 1],103.92);
[b,a] [ bilinear(B,A,fs)
%b ¼ [0.3660 0.3660] numerator coefficients of the digital filter from MATLAB
%a ¼ [1 -0.2679] denominator coefficients of the digital filter from MATLAB
[hz, f] ¼ freqz([0.3660 0.3660],[1 -0.2679],512,fs); % Frequency response
phi ¼ 180*unwrap(angle(hz))/pi;
subplot(2,1,1), plot(f, abs(hz)),grid;
axis([0 fs/2 0 1]);
xlabel(

Frequency (Hz) ); ylabel( Magnitude Response )

' ' ' '

subplot(2,1,2), plot(f, phi); grid;
axis([0 fs/2 -100 0]);
xlabel( Frequency (Hz) ); ylabel( Phase (degrees) )

' ' ' '

8.3 DIGITAL BUTTERWORTH AND CHEBYSHEV FILTER DESIGNS
In this section, we design various types of digital Butterworth and Chebyshev filters using the BLT
design method developed in the previous section.

8.3.1 Lowpass Prototype Function and Its Order
As described in Section 8.2 , BLT design requires obtaining the analog filter with prewarped frequency
specifications. These analog filter design requirements include the ripple specification at the passband
frequency edge, the attenuation specification at the stopband frequency edge, and the type of lowpass
prototype (which we shall discuss) and its order.
Table 8.3 lists the Butterworth prototype functions with 3 dB passband ripple specification. Tables
8.4 and 8.5 contain the Chebyshev prototype functions (type I) with 1 dB and 0.5 dB passband ripple
specifications, respectively. Other lowpass prototypes with different ripple specifications and orders
can be computed using the methods described in Appendix C.
In this section, we will focus on the Chebyshev type I filter. The Chebyshev type II filter design can
be found in Proakis and Manolakis (1996) and Porat (1997).
The magnitude response function of the Butterworth lowpass prototype with order n is
shown in Figure 8.13 , where the magnitude response jH P ðvÞj versus the normalized frequency v is
given by

1
1 þ ε

jH P ðvÞj ¼ p (8.25)

2 2 n

v

With the given passband ripple A P dB at the normalized passband frequency edge v p ¼ 1, and the
stopband attenuation A s dB at the normalized stopband frequency edge v s , the following two equations
must be satisfied to determine the prototype filter order:

1
1 þ ε

A P dB ¼ 20 $ log 10 p (8.26)

2