Signals and Systems - Electrical Engineering

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

FIGURE 8.9

(a) Cascade and

(b) parallel

connections of LTI

systems with impulse

responsesh 1 [n]and

h 2 [n]. Equivalent

systems on the right.

Notice the

interchange of

systems in the

cascade connection.

(a)

(b)

h 2 [n]

h 1 [n]

x[n] + h 1 [n] + h 2 [n]

y[n] x[n] y[n]

x[n] h 1 [n] h 2 [n]

h 2 [n] h 1 [n]

(h 1 ∗h 2 ) [n]

x[n]

y[n]

y[n]

x[n] y[n]

nExample 8.23

Consider a moving-averaging filter where the input isx[n] and the output isy[n]:

y[n]=

1

3

(x[n]+x[n−1]+x[n−2])

Find the impulse responseh[n] of this filter. Then,

(a) Letx[n]=u[n]. Find the output of the filtery[n] using the input–output relation and the

convolution sum.

(b) If the input of the filter isx[n]=Acos( 2 πn/N)u[n], determine the values ofAandN, so that

the steady-state response of the filter is zero. Explain. Use MATLAB to verify your results.

Solution

(a) If the input isx[n]=δ[n], the output of the filter isy[n]=h[n], or the impulse response of the

system. No initial conditions are needed. We thus have that

h[n]=

1

3

(δ[n]+δ[n−1]+δ[n−2])

so thath[0]= 1 /3 asδ[0]=1 butδ[−1]=δ[−2]=0; likewise,h[1]=h[2]= 1 /3 so that the

coefficients of the filter equal the impulse response of the filter atn=0, 1, and 2.