Page 89 - DSP Integrated Circuits

74 Chapter 3 Digital Signal Processing

3.10.1 Magnitude Function
The magnitude function is related to the frequency response according to

mT
where \H(eJ ) I is the magnitude response and <&(GfT) is the phase response. The
frequency response, which is the Fourier transform of the impulse response, is a
0
rational function in eJ -^. The frequency response describes how the magnitude
and phase of a sinusoidal signal are modified by the system.

EXAMPLE 3.4
Use MATLAB™ or any other standard filter design program to design a digital
Cauer (elliptic) filter meeting the following specification:

Lowpass IIR filter with
Passband ripple: < 1 dB Stopband attenuation: > 20 dB
Passband edge: 2 kHz Stopband edge: 2.5 kHz
Sample frequency: 10 kHz
The following MATLAB-program

fcnorm = 2/5; % The band edges in MATLAB are normalized with
fsnorm = 2.5/5; % respect to fsample/2, e.g., fcnorm = 2fc/fsample
Amax = 1;
Amin = 20;
[N,Wn] = ellipord(fcnorm, fsnorm, Amax, Amin);
N
[Num, Den] = ellip(N, Amax, Amin, Wn);
[Z, P, K] = tf2zp(Num, Den)

yields a filter that has

Filter degree: N = 3
Passband ripple: A max = 1.0 dB
Stopband attenuation: A mi n = 20.0 dB

Figure 3.12 shows the magnitude response of a third-order lowpass filter of
Cauer type. These types of filters will be discussed further in Chapter 4. The mag-
nitude function can be shown in either linear or logarithmic (dB) scales, although
the latter is more common. The attenuation, phase, and group delay responses are
shown in Figures. 3.13, 3.14, and 3.15, respectively.