Page 112 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS 101

In Fig. 2-29 the output of the first (from the right) unit delay element is y[n - 11 and the
output of the second (from the right) unit delay element is y[n - 21. Thus, from Fig. 2-29 we
see that
y[n]=a,y[n-l]+a,y[n-2]+x[n] (2.144)
or y[n] - a,y[n - 1] - a,y[n - 21 =x[n] (2.145)

which is the required second-order linear difference equation.
Note that, in general, the order of a discrete-time LTI system consisting of the interconnec-
tion of unit delay elements and scalar multipliers is equal to the number of unit delay elements
in the system.

2.41. Consider the discrete-time system in Fig. 2-30. Write a difference equation that relates
the output y[n] and the input x[n].

Unit
delay
I
qb - 11
Fig. 2-30

Let the input to the unit delay element be q[nl. Then from Fig. 2-30 we see that

q[n] = 2q[n - 11 + x[n] ( 2.146a)
~[n] s[nI+ 3q[n - 1 1 (2.1466)
=
Solving Eqs. (2.146a) and (2.146b) for q[n] and q[n - 11 in terms of x[n] and y[n], we obtain
9[n]= fy[n] + $+I (2.147a)

q[n - 11 = iy[n] - ix[n] (2.147b)
Changing n to (n - 1) in Eq. (2.147a), we have
q[n - 11 = fy[n - 11 + ix[n - 11 (2.147~)

Thus, equating Eq. (2.1476) and (Eq. (2.147c), we have
;y[n]- fx[n] = fy[n- 11 + +x[n- 11
Multiplying both sides of the above equation by 5 and rearranging terms, we obtain
y[n] - 2y[n - 11 =x[n] + 3x[n - 11 (2.148)

which is the required difference equation.

2.42. Consider a discrete-time system whose input x[n] and output y[n] are related by

y[n] -ay[n - 11 =x[n] (2.149)