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4 Measurement of Biosignals and Analog Signal Processing

[h1,w1] = freqs(b1,a1,4096);

% inverse Tschebyscheff filter

[z2,p2,k2] = cheby2(n,30,fg,'s');

[b2,a2] = zp2tf(z2,p2,k2);

[h2,w2] = freqs(b2,a2,4096);

% Cauer filter

[ze,pe,ke] = ellip(n,3,30,fg,'s');

[be,ae] = zp2tf(ze,pe,ke);

[he,we] = freqs(be,ae,4096);

% graphical representation

plot(wb,abs(hb))

hold on

plot(w1,abs(h1))

plot(w2,abs(h2))

plot(we,abs(he))

grid on

xlabel('f_g / Hz')

ylabel('|A(f_g)|')

legend('Butterworth','Tschebyscheff','inverse Tschebyscheff','Cauer')

4.5.1.1 General Procedure for Filter Design

When designing selective filters for the magnitude frequency response, specifica-

tions are only made for the magnitude frequency response |A(f)|. Requirements for

the phase frequency response and the group delay are not defined. In biosignal pro-

cessing, however, the group delay in the passband should be as flat as possible in

order to avoid signal distortion.

To simplify the filter synthesis, first only normalised lowpass filters are designed,

which have a dimensionless frequency normalised to a reference frequency fB. Then,

by frequency transformation, a filter can be created which meets the desired require-

ments in terms of passband and stopband. The values of the tolerance range of the

magnitude transfer function remain unchanged. Only the cut-off frequencies are

changed. The frequency transformation can be done by the denormalisation. Not

only the reconversion from a normalised low pass to a non-normalised low pass, but

also a conversion to a high pass, band pass or band stop can be done (cf. Figure 4.36).

The conversion must take place in such a way that afterwards a transfer function

A(p) (realisable with analogue components) arises again. It can be represented in

fractional-rational form with p := σ + jω and ω := 2πf like