Signals and Systems - Electrical Engineering

492 C H A P T E R 8: Discrete-Time Signals and Systems

FIGURE 8.10

Convolution sums for

a moving-averaging

filter with input

x 1 [n]=u[n]and

x 2 [n]=cos( 2 πn/ 3 )u[n].

0 5 10 15 0 5 10 15

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

n

0 5 10 15

n

n

− 1

−0.5

0

0.5

1

0 5 10 15

−0.1

0

0.1

0.2

0.3

0.4

n

x^1

[n

]

x^2

[n

]

y^2

[n

]

y^1

[n

]

Notice that each of the input sequences has two zeros at the beginning so that the response can

be found atn≥−2. Also, when the input is of infinite support, like whenx[n]=u[n], we can

only approximate it as a finite sequence in MATLAB, and as such the final values of the convolu-

tion obtained fromconvare not correct and should not be considered. In this case, the final two

values of the convolution results are not correct and are not considered. The results are shown in

Figure 8.10. n

nExample 8.24

Consider an autoregressive system represented by a first-order difference equation

y[n]=0.5y[n−1]+x[n] n≥ 0

Find the impulse responseh[n] of the system and then compute the response of the system to

x[n]=u[n]−u[n−3] using the convolution sum. Verify results with MATLAB.

Solution

The impulse responseh[n] can be found recursively. Lettingx[n]=δ[n],y[n]=h[n], and initial

conditiony[−1]=h[−1]=0, we have

h[0]=0.5h[−1]+δ[0]= 1

h[1]=0.5h[0]+δ[1]=0.5

h[2]=0.5h[1]+δ[2]=0.5^2

h[3]=0.5h[2]+δ[3]=0.5^3

···