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Problems 101
P3.16 Foralinear, shift-invariant system described by the difference equation
M∑
N∑
y(n) = b mx (n − m) − a l y (n − l)
m=0
l=1
the frequency-response function is given by
H(e jω )=
∑ M
m=0 bme−jωm
1+ ∑ N
l=1 a le −jωl
Write a MATLAB function freqresp to implement this relation. The format of this
function should be
function [H] = freqresp(b,a,w)
% Frequency response function from difference equation
% [H] = freqresp(b,a,w)
% H = frequency response array evaluated at w frequencies
% b = numerator coefficient array
% a = denominator coefficient array (a(1)=1)
% w = frequency location array
P3.17 Determine H(e jω ), and plot its magnitude and phase for each of the following systems:
∑
1. y(n) = 1 4
x(n − m)
5 m=0
2. y(n) =x(n) − x(n − 2)+0.95y(n − 1) − 0.9025y(n − 2)
3. y(n) =x(n) − x(n − 1) + x(n − 2)+0.95y(n − 1) − 0.9025y(n − 2)
4. y(n) =x(n) − 1.7678x(n − 1) + 1.5625x(n − 2)+1.1314y(n − 1) − 0.64y(n − 2)
5. y(n) =x(n) − ∑ 5
l=1 (0.5)l y (n − l)
P3.18 A linear, shift-invariant system is described by the difference equation
y(n) =
3∑
x (n − 2m) −
m=0
3∑
(0.81) l y (n − 2l)
Determine the steady-state response of the system to the following inputs:
1. x(n) =5+10(−1) n
2. x(n) =1+cos(0.5πn + π/2)
3. x(n) =2sin(πn/4) + 3 cos (3πn/4)
4. x(n) = ∑ 5
(k +1)cos (πkn/4)
k=0
5. x(n) =cos (πn)
In each case, generate x(n), 0 ≤ n ≤ 200, and process it through the filter function to
obtain y(n). Compare your y(n) with the steady-state responses in each case.
P3.19 An analog signal x a (t) =sin (1000πt) issampled using the following sampling intervals.
In each case, plot the spectrum of the resulting discrete-time signal.
l=1
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