Signals and Systems - Electrical Engineering

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

The time invariance is shown by letting the input bev[n]=x[n−N],n≥N, and zero otherwise.

The corresponding output according to Equation (8.30) is

∑n

k= 0

bakv[n−k]=

∑n

k= 0

bakx[n−N−k]

=

n∑−N

k= 0

bakx[n−N−k]+

∑n

k=n−N+ 1

bakx[n−N−k]=y[n−N]

since the summation

∑n

k=n−N+ 1

bakx[n−N−k]= 0

given thatx[−N]=···=x[−1]=0 is assumed. Thus, the system represented by the above differ-

ence equation is linear and time invariant. As in the continuous-time case, however, if the initial

conditiony[−1] is not zero, or ifx[n]6=0 forn<0, the system characterized by the difference

equation is not LTI. n

nExample 8.22

Autoregressive moving average filter:The recursive system represented by the first-order difference

equation

y[n]=0.5y[n−1]+x[n]+x[n−1] n≥0,y[−1]

is called theautoregressive moving averagegiven that it is the combination of the two systems

discussed before. Consider two cases:

n Let the initial condition bey[−1]=−2, and the input bex[n]=u[n] first and thenx[n]=

2 u[n].

n Let the initial condition bey[−1]=0, and the input bex[n]=u[n] first and thenx[n]= 2 u[n].

Determine in each of these cases if the system is linear.

Find the steady-state response—that is,

lim

n→∞

y[n]