8.3 Discrete-Time Systems 481
the signals in the above two cases and compare the resulting signal with the one we obtain when
finding the square root of the previous two values ofα, 2 and 4, they do not coincide. The additive
condition is not satisfied either, so the system is not linear. n
8.3.1 Recursive and Nonrecursive Discrete-Time Systems.....................
Depending on the relation between the inputx[n]and the outputy[n]two types of discrete-time systems of
interest are:
n Recursive system:
y[n]=−
N∑− 1
k= 1
aky[n−k]+
M∑− 1
m= 0
bmx[n−m] n≥ 0
initial conditionsy[−k],k=1,...,N− 1 (8.28)
This system is also calledinfinite-impulse response(IIR).
n Nonrecursive system:
y[n]=
M∑− 1
m= 0
bmx[n−m] (8.29)
This system is also calledfinite-impulse response(FIR).
The recursive system is analogous to a continuous-time system represented by a differential equation.
For this type of system the discrete-time inputx[n] and the discrete-time outputy[n] are related by an
(N− 1 )th-order difference equation. If such a difference equation is linear, with constant coefficients,
zero initial conditions, and the input is zero forn<0, then it represents a linear and time-invariant
system. For these systems the output at a present timen,y[n], depends or recurs on previous values
of the output{y[n−k],k=1,...,N− 1 }, and thus they are calledrecursive. We will see that these
systems are also calledinfinite-impulse responseorIIRbecause their impulse responses are typically of
infinite length.
On the other hand, if the outputy[n] does not depend on previous values of the output, but only
on weighted and shifted inputs{bmx[n−m],m=0,...,M− 1 }, the system is callednonrecursive. We
will see that the impulse response of nonrecursive systems is of finite length; as such, these systems
are also calledfinite impulse responseorFIR.
nExample 8.20
Moving-average discrete filter:A third-order moving-average filter (also called asmoothersince
it smooths out the input signal) is an FIR filter for which the inputx[n] and the outputy[n] are
related by
y[n]=
1
3
(x[n]+x[n−1]+x[n−2])
Show that this system is linear time invariant.
