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Solutions of the Difference Equations 131
while the second part as
Y ZI (z) =H(z)X IC (z)
where X IC (z) can be thought of as an equivalent initial-condition input
that generates the same output Y ZI as generated by the initial
conditions. In this example x IC (n) is
x IC (n) ={1, −2}
↑
Now taking the inverse z-transform of each part of (4.27), we write the
complete response as
[ ( ) n ( ) n 1 1 1
y(n) = − 2 + 8 [ ( ) n 1
u(n) + 3 − 2]
u(n)
3 4 2 3]
2
} {{ } } {{ }
Zero-state response
Zero-input response
From this example, it is clear that each part of the complete solution
is, in general, a different function and emphasizes a different aspect of
system analysis.
4.5.1 MATLAB IMPLEMENTATION
In Chapter 2 we used the filter function to solve the difference equation,
given its coefficients and an input. This function can also be used to find
the complete response when initial conditions are given. In this form the
filter function is invoked by
y = filter(b,a,x,xic)
where xic is an equivalent initial-condition input array. To find the complete
response in Example 4.14, we will use the MATLAB script
>> n = [0:7]; x = (1/4).^n; xic = [1, -2];
>> format long; y1 = filter(b,a,x,xic)
y1 =
Columns 1 through 4
2.00000000000000 1.25000000000000 0.93750000000000 0.79687500000000
Columns 5 through 8
0.73046875000000 0.69824218750000 0.68237304687500 0.67449951171875
>> y2 = (1/3)*(1/4).^n+(1/2).^n+(2/3)*ones(1,8) % MATLAB Check
y2 =
Columns 1 through 4
2.00000000000000 1.25000000000000 0.93750000000000 0.79687500000000
Columns 5 through 8
0.73046875000000 0.69824218750000 0.68237304687500 0.67449951171875
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