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Problems 139
It is also known that the frequency response function H(e jω )evaluated at ω = π/4 isequal
to 1, i.e.,
H(e jπ/4 )=1
1. Determine the system function H(z), and indicate its region of convergence.
2. Determine the difference equation representation.
3. Determine the steady-state response y ss(n) ifthe input is x(n) =cos(πn/4)u(n).
4. Determine the transient response y tr(n) ifthe input is x(n) =cos(πn/4)u(n).
P4.21 A digital filter is described by the frequency response function
( )
H(e jω ω
)=[1+2cos(ω)+3cos(2ω)] cos e −j5ω/2
2
1. Determine the difference equation representation.
2. Using the freqz function, plot the magnitude and phase of the frequency response of the
filter. Note the magnitude and phase at ω = π/2 and at ω = π.
3. Generate 200 samples of the signal x(n) =sin(πn/2) + 5 cos(πn), and process through
the filter to obtain y(n). Compare the steady-state portion of y(n) tox(n). How are the
amplitudes and phases of two sinusoids affected by the filter?
P4.22 Repeat Problem 4.21 for the following filter
H(e jω )=
1+e −j4ω
1 − 0.8145e −j4ω
P4.23 Solve the following difference equation for y(n) using the one-sided z-transform approach.
y(n) =0.81y(n − 2) + x(n) − x(n − 1), n≥ 0; y(−1) =2,y(−2) =2
x(n) =(0.7) n u(n +1)
Generate the first 20 samples of y(n) using MATLAB, and compare them with your answer.
P4.24 Solve the difference equation for y(n), n≥ 0
y(n) − 0.4y(n − 1) − 0.45y(n − 2) = 0.45x(n)+0.4x(n − 1) − x(n − 2)
driven by the input x(n) = [ 2+ ( 1
2) n ] u(n) and subject to
y(−1) =0,y(−2) =3; x(−1) = x(−2) = 2
Decompose the solution y(n) into (i) transient response, (ii) steady-state response, (iii)
zero-input response, and (iv) zero-state response.
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