Signals and Systems - Electrical Engineering

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

Solution

This is a nonrecursive system that uses a present sample,x[n], and two past values,x[n−1] and

x[n−2], of the input to get an average,y[n], at everyn. Thus, its name,moving-average filter.

Linearity: If we let the input beax 1 [n]+bx 2 [n], and assume that{yi[n],i=1, 2}are the correspond-

ing outputs to{xi[n],i=1, 2}, the filter output is

1

3

[(ax 1 [n]+bx 2 [n])+(ax 1 [n−1]+bx 2 [n−1])+(ax 1 [n−2]+bx 2 [n−2])]=ay 1 [n]+by 2 [n]

That is, the system is linear.

Time invariance: If the input isx 1 [n]=x[n−N], the corresponding output to it is

1

3

(x 1 [n]+x 1 [n−1]+x 1 [n−2])=

1

3

(x[n−N]+x[n−N−1]+x[n−N−2])

=y[n−N]

That is, the system is time invariant. n

nExample 8.21

Autoregressive discrete filter:The recursive discrete-time system represented by the first-order

difference equation (with initial conditiony[−1])

y[n]=ay[n−1]+bx[n] n≥0, y[−1]

is also called anautoregressive(AR) filter. “Autoregressive” refers to the feedback in the output—that

is, the present value of the outputy[n] depends on its previous valuey[n−1].

Find recursively the solution of the difference equation and determine under what conditions the

system represented by this difference equation is linear and time invariant.

Solution

Let’s first discuss why the initial condition isy[−1]. The initial condition is the value needed to

computey[0], which according to the difference equation

y[0]=ay[−1]+bx[0]

isy[−1] sincex[0] is known.

Assume that the initial condition isy[−1]=0, and that the input isx[n]=0 forn<0 (i.e.,

the system is not energized forn<0). The solution of the difference equation when the input

x[n] is not defined can be found by a repetitive substitution of the input–output relationship.

Thus, replacingy[n−1]=ay[n−2]+bx[n−1] in the difference equation, and then replacing