Signals and Systems - Electrical Engineering

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

W = 0:0.001:Wmax; % frequency range for plotting

H = freqs(b, a, W); Hm = abs(H); Ha = unwrap(angle(H)) % magnitude (Hm) and phase (Ha)

N = length(W); alpha1 = alphamax∗ones(1, N); alpha2 = alphamin∗ones(1, N); % loss specs

subplot(221)

plot(W, Hm); grid; axis([0 Wmax 0 1.1∗max(Hm)])

subplot(222)

plot(W, Ha); grid; axis([0 Wmax 1.1∗min(Ha) 1.1∗max(Ha)])

subplot(223)

splane(b, a)

subplot(224)

plot(W,− 20 ∗log10(abs(H))); hold on

plot(W, alpha1, ’r’, W, alpha2, ’r’); grid; axis([0 max(W)−0.1 100])

hold off

nExample 6.11

To illustrate the use ofanalogfilconsider the design of low-pass filters using the Chebyshev2 and

the Elliptic design methods. The specifications for the designs are

α( 0 )=0, αmax=0.1, αmin=60 dB

p=10, s=15 rad/sec

We wish to find the coefficients of the designed filters, plot their magnitude and phase, and plot

the loss function for each of the filters and verify that the specifications have been met. The results

are shown in Figure 6.26.

0 5 10 15 20 25

0

0.2

0.4

0.6

0.8

1

Ω

|H

(Ω

)|

− 4 − 2 0 2

− 20

− 10

0

10

20

σ

jΩ

0 5 10 15 20 25

− 10

− 8

− 6

− 4

− 2

0

Ω

<H

(Ω

)

0 5 10 15 20 25

0

20

40

60

80

100

Ω

α(

Ω

)

0 5 10 15 20 25

0

0.2

0.4

0.6

0.8

1

Ω

0 5 10 15 20 25

− 10

− 8

− 6

− 4

− 2

0

Ω

|H

(Ω

)|

<H

(Ω

)

− 15 − 10 − 5 0

− 50

0

50

σ

0 5 10 15 20 25

0

20

40

60

80

100

Ω

jΩ α(Ω

)

(a) (b)

FIGURE 6.26

(a) Elliptic and (b) Chebyshev2 low-pass filter designs using analogfil function. Clockwise: magnitude, phase,

loss function, and poles and zeros are shown for each design.