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The Frequency Domain Representation
of LTI Systems 79
1
Input sequence
0.5
x(n)
0
−0.5
−1
0 10 20 30 40 50 60 70 80 90 100
n
Output sequence
5
3.42
4.092
y(n)
0
−5
0 10 20 30 40 50 60 70 80 90 100
n
FIGURE 3.8 Plots in Example 3.15
respectively. Now the array H(e jω k
)in(3.23) can be computed using a ./
operation. This procedure can be implemented in a MATLAB function to
determine the frequency response function, given {b m } and {a l } arrays.
We will explore this in Example 3.16 and in Problem P3.16.
□ EXAMPLE 3.16 A 3rd-order lowpass filter is described by the difference equation
y(n) =0.0181x(n)+0.0543x(n − 1)+0.0543x(n − 2)+0.0181x(n − 3)
+1.76y(n − 1) − 1.1829y(n − 2)+0.2781y(n − 3)
Plot the magnitude and the phase response of this filter, and verify that it is a
lowpass filter.
Solution
We will implement this procedure in MATLAB and then plot the filter
responses.
>> b = [0.0181, 0.0543, 0.0543, 0.0181]; % filter coefficient array b
>> a = [1.0000, -1.7600, 1.1829, -0.2781]; % filter coefficient array a
>> m = 0:length(b)-1; l = 0:length(a)-1; % index arrays m and l
>> K = 500; k = 0:1:K; % index array k for frequencies
>> w = pi*k/K; % [0, pi] axis divided into 501 points.
>> num = b * exp(-j*m’*w); % Numerator calculations
>> den = a * exp(-j*l’*w); % Denominator calculations
>> H = num ./ den; % Frequency response
>> magH = abs(H); angH = angle(H); % mag and phase responses
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