Signals and Systems - Electrical Engineering

402 C H A P T E R 6: Application to Control and Communications

− 8 − 6 − 4 − 2 0 2

− 5

0

5

Butterworth

σ

σ

− 4 − 2 0 2

− 5

0

5

Chebyshev

jΩ

jΩ

0 10 20 30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ω

|H

(j

Ω

)|

Butterworth

Chebyshev

0 5 10 15 20

−5

0

5

t

y(

t)

0 5 10 15 20

−5

0

5

t

x(

t)

cos(10t)−2cos(5t)+4sin(20t)

FIGURE 6.24

Comparison of filtering of an analog signalx(t)using a low-pass Butterworth and Chebyshev filter with the same

half-power frequency.

wp = whp/cosh(acosh(1/epsi)/N) % recomputing wp to get same whp

[N1, wn] = cheb1ord(wp, ws, alphamax, alphamin, ’s’);

[b1, a1] = cheby1(N1, alphamax, wn, ’s’);

% frequency responses

W = 0:0.01:30;

Hm = abs(freqs(b, a, W));

Hm1 = abs(freqs(b1, a1, W));

% generation of frequency response from coefficients

n = N:−1:0; n1 = N1:−1:0;

U = ( j∗w).ˆn; U1 = ( j∗w).ˆn1

num = b∗conj(U’); den = a∗conj(U’);

num1 = b1∗conj(U1’); den1 = a1∗conj(U1’)

H = num/den; % Butterworth LPF

H1 = num1/den1; % Chebyshev LPF

% output of filter

Y = X∗H;

Y1 = X∗H1;

y = ifourier(Y, t)

y1 = ifourier(Y1, t) n

6.5.4 Frequency Transformations

As indicated before, the design of an analog filter is typically done by transforming the frequency of

a normalized prototype low-pass filter. The frequency transformations were developed by Professor

Ronald Foster [72] using the properties of reactance functions. The frequency transformations for the