% Problem 9.37

% Impulse response of 3rd order Butterworth filter

figure(1)t = −15:0.001:15;wc = 1;h = wc*exp(−wc*t)+(2/sqrt(3))*wc*exp(−wc*t/2).*cos(sqrt(3)*wc*t/2 − 150*pi/180);h(t<0) = 0;subplot(211)plot(t,h)gridxlabel(’t (seconds)’);ylabel(’h(t)’);title(’Problem 9.37’);

% Impulse response of ideal filtersubplot(212)hi = (1/pi)*sinc(t/pi);plot(t,hi)gridxlabel(’t (seconds)’);ylabel(’h_i(t)’);print −dpsc2 p9_37.ps

% Problem 9.38

figure(2)t = 0:0.001:4;x = (t >= 0) − (t >= 1);subplot(311)plot(t,x)axis([0 3 −1 2])gridxlabel(’t (seconds)’);ylabel(’x(t)’);

%(a) wc = 2 piwc = 2*pi;gt = (1 − exp(−wc*t) + (2/sqrt(3))*exp(−wc*t/2).*cos(sqrt(3)*wc*t/2+pi/2)).*(t >= 0);gtm1 = (1 − exp(−wc*(t−1)) + (2/sqrt(3))*exp(−wc*(t−1)/2).*cos(sqrt(3)*wc*(t−1)/2+pi/2)).*(t >= 1);y = gt − gtm1;subplot(312)plot(t,y);gridxlabel(’t (seconds)’);ylabel(’y_a(t)’);axis([0 3 −1 2])

%(b) wc = 4 piwc = 4*pi;t = 0:0.001:4;gt = (1 − exp(−wc*t) + (2/sqrt(3))*exp(−wc*t/2).*cos(sqrt(3)*wc*t/2+pi/2)).*(t >= 0);gtm1 = (1 − exp(−wc*(t−1)) + (2/sqrt(3))*exp(−wc*(t−1)/2).*cos(sqrt(3)*wc*(t−1)/2+pi/2)).*(t >= 1);y = gt − gtm1;subplot(313)plot(t,y);axis([0 3 −1 2])

Feb 28, 05 19:42 Page 1/3hw06.mgridxlabel(’t (seconds)’);ylabel(’y_b(t)’);print −dpsc2 p9_38.ps

% Problem 9.45

% (a)figure(3)b = conv([1 0 150],conv([1 0 150],[1 0 150]));a = conv([1 5 150],[1 5 325 750 22500]);w = 0:0.01:25;H = freqs(b,a,w);subplot(211)plot(w,abs(H))axis([0 25 0 1.2])gridxlabel(’\omega (rad/sec)’);ylabel(’|H(\omega)|’);title(’Problem 9.45 (a)’)subplot(212)plot(w,unwrap(angle(H))*180/pi)gridxlabel(’\omega (rad/sec)’);ylabel(’\angle H(\omega)’);print −dpsc2 p9_45_a.ps

% (c)figure(4)clft=0:0.01:5;x = sin(5*t) + sin(12*t);y = lsim(b,a,x,t);plot(t,y)hold onyss = sin(5*t − 23*pi/180) + 0.01*sin(12*t − 259*pi/180);plot(t,yss,’r’);legend(’y(t)’,’y_{ss}(t)’)gridxlabel(’t (seconds)’);ylabel(’output’)title(’Probelm 9.45 (c)’)print −dpsc2 p9_45_c.ps

% Problem 9.47

N = 4;[z,p,k] = buttap(N);[b,a] = zp2tf(z,p,k);[b1,a1]=lp2lp(b,a,10);w = 0:0.001:40;H = freqs(b1,a1,w);figure(5)subplot(211)plot(w,abs(H));gridxlabel(’\omega (rad/sec)’);ylabel(’|H(\omega)|’);title(’Problem 9.47’)

subplot(212)plot(w,unwrap(angle(H))*180/pi);

Feb 28, 05 19:42 Page 2/3hw06.m

Printed by Bill Rison

Monday February 28, 2005 1/2hw06.m

gridxlabel(’\omega (rad/sec)’);ylabel(’\angle H(\omega)’);print −dpsc2 p9_47_freq_resp.ps

figure(6)clf% (a)t = 0:0.01:5;x = sin(5*t);y = lsim(b1,a1,x,t);subplot(411)plot(t,x)hold onplot(t,y,’r’)gridxlabel(’t (seconds)’);ylabel(’y_a(t)’);title(’Problem 9.47’);legend(’x(t)’,’y_a(t)’)

%(b)x = sin(25*t);y = lsim(b1,a1,x,t);subplot(412)plot(t,x)hold onplot(t,y,’r’)plot(t,y)gridxlabel(’t (seconds)’);ylabel(’y_b(t)’);legend(’x(t)’,’y_b(t)’)

%(c)x = sin(5*t) + sin(25*t);y = lsim(b1,a1,x,t);subplot(413)plot(t,x)hold onplot(t,y,’r’)gridxlabel(’t (seconds)’);ylabel(’y_c(t)’);legend(’x(t)’,’y_c(t)’)

%(d)t = 0:0.05:10;x = rand(201,1);y = lsim(b1,a1,x,t);subplot(414)plot(t,x)hold onplot(t,y,’r’)gridxlabel(’t (seconds)’);ylabel(’y_d(t)’);legend(’x(t)’,’y_d(t)’)print −dpsc2 p9_47_abcd.ps

Feb 28, 05 19:42 Page 3/3hw06.m

Printed by Bill Rison

Monday February 28, 2005 2/2hw06.m

−15 −10 −5 0 5 10 15−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t (seconds)

h(t)

Problem 9.37

−15 −10 −5 0 5 10 15−0.1

0

0.1

0.2

0.3

0.4

t (seconds)

h i(t)

0 0.5 1 1.5 2 2.5 3−1

0

1

2

t (seconds)

x(t)

0 0.5 1 1.5 2 2.5 3−1

0

1

2

t (seconds)

y a(t)

0 0.5 1 1.5 2 2.5 3−1

0

1

2

t (seconds)

y b(t)

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

ω (rad/sec)

|H(ω

)|

Problem 9.45 (a)

0 5 10 15 20 25−350

−300

−250

−200

−150

−100

−50

0

ω (rad/sec)

∠ H

(ω)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

t (seconds)

outp

ut

Probelm 9.45 (c)

y(t)y

ss(t)

0 5 10 15 20 25 30 35 400

0.2

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0.6

0.8

1

1.2

1.4

ω (rad/sec)

|H(ω

)|

Problem 9.47

0 5 10 15 20 25 30 35 40−350

−300

−250

−200

−150

−100

−50

0

ω (rad/sec)

∠ H

(ω)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

t (seconds)

y a(t)

Problem 9.47

x(t)y

a(t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1

0

1

t (seconds)

y b(t)

x(t)y

b(t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

t (seconds)

y c(t)

x(t)y

c(t)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

t (seconds)

y d(t)

x(t)y

d(t)