4.5 Design of Analogue Filters

|

125

Fig. 4.35: Approximations of the ideal magnitude frequency response for a 4th order low pass fil-

ter with a cut-off frequency of 200 Hz using Butterworth, Chebyshev and Cauer filters: The gain

in the passband AD is 1. The passband and stopband tolerances are each 0.3 (corresponding to a

passband tolerance of 3 dB), the passband frequency fD is 200 Hz and the stopband frequency fS is

approximately 210 Hz.

magnitude frequency response in the tolerance range should be flat or wavy. Depend-

ing on the specification, these distinctions are made:

–

a flat course in the passband and stopband (power- or Butterworth-filter),

–

only a wavy course in the passband (Chebyshev-filter),

–

only a wavy course in the stopband (inverse Chebyshev-Filter) or

–

a wavy course in both the passband/stopband (elliptical filter or Cauer filter).

Figure 4.35 shows an example of a low-pass filter with a cut-off frequency of 200 Hz

approximated to the 4th order by the above filter types, and Listing 4.5.1 shows the

associated Matlab-script.

Listing 4.5.1: Matlab example of calculating the frequency response of different low-pass filters.

n = 4;

% filter order

fg = 200;

% filter cut-off frequency

% Butterworth Filter

[zb,pb,kb] = butter(n,fg,'s');

% filter coefficients

[bb,ab] = zp2tf(zb,pb,kb);

% transfer function

[hb,wb] = freqs(bb,ab,4096);

% frequency response

% Tschebyscheff filter

[z1,p1,k1] = cheby1(n,3,fg,'s');

[b1,a1] = zp2tf(z1,p1,k1);