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138 Chapter 4 THE z-TRANSFORM
2. Determine the difference equation representation of the overall system that relates the
output y(n) tothe input x(n).
3. Is this system causal? BIBO stable? Explain clearly to receive full credit.
4. Determine the frequency response H(e jω )ofthe overall system.
5. Using MATLAB, provide a plot of this frequency response over 0 ≤ ω ≤ π.
P4.17 For the linear and time-invariant systems described by the following system functions,
determine (i) the impulse response representation, (ii) the difference equation
representation, (iii) the pole-zero plot, and (iv) the output y(n) ifthe input is
x(n) =3cos(πn/3)u(n).
1. H(z) =(z +1)/(z − 0.5), causal system
2. H(z) =(1+z −1 + z −2 )/(1 + 0.5z −1 − 0.25z −2 ), stable system
3. H(z) =(z 2 − 1)/(z − 3) 2 ,anticausal system
z
4. H(z) =
z − 0.25 + 1 − 0.5z−1 , stable system
1+2z−1 5. H(z) =(1+z −1 + z −2 ) 2
P4.18 For the linear, causal, and time-invariant systems described by the following difference
equations, determine (i) the impulse response representation, (ii) the system function
representation, (iii) the pole-zero plot, and (iv) the output y(n) ifthe input is
x(n) =2(0.9) n u(n).
1. y(n) =[x(n)+2x(n − 1) + x(n − 3)] /4
2. y(n) =x(n)+0.5x(n − 1) − 0.5y(n − 1) + 0.25y(n − 2)
3. y(n) =2x(n)+0.9y(n − 1)
4. y(n) =−0.45x(n) − 0.4x(n − 1) + x(n − 2) + 0.4y(n − 1) + 0.45y(n − 2)
5. y(n) = ∑ 4
m=0 (0.8)m x(n − m) − ∑ 4
l=1 (0.9)l y(n − l)
P4.19 The output sequence y(n) inProblem P4.18 is the total response. For each of the systems
given in Problem P4.18, separate y(n) into (i) the homogeneous part, (ii) the particular
part, (iii) the transient response, and (iv) the steady-state response.
P4.20 A stable system has four zeros and four poles as given here:
zeros: ± 1, ±j1
Poles: ± 0.9, ±j0.9
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