Signals and Systems - Electrical Engineering

554 C H A P T E R 9: The Z-Transform

To solve the difference equation witha=0.8,x[n]=u[n]−u[n−11], and zero initial condition

y[−1]=0, using MATLAB, we use the following script.

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% Example 9.16

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N = 100; n = 0:N - 1; x = [ones(1,10) zeros(1,N - 10)];

den=[1 -0.8]; num = [1 0];

y = filter(num, den,x)

The functionfilterrequires that the initial conditions are zero. The results are shown in Figure 9.11.

Let us now find the impulse responseh[n] of the system. For that, letx[n]=δ[n] andy[−1]=0,

then we havey[n]=h[n] orY(z)=H(z). Thus, we have

H(z)=

1

1 −az−^1

so that h[n]=anu[n]

0 5 10 15 20 25 30

0

1

2

3

4

5

x[

n]

0 5 10 15 20 25 30

0

2

4

6

(a)

n

(b)

n

y[

n]

FIGURE 9.11

(a) Solution of the first-order difference equation with (b) input.