Page 416 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS 403

Note that the sum in Eq. (7.118) has no terms for n = 0 and that the first term is cA"-'b
for n > 0. The second term on the right-hand side of Eq. (7.118) is equal to d for n = 0
and zero otherwise. Thus, we conclude that

(6) From the result from Prob. 7.29 we have

and &-I,, = (+)"-' - '(')"-I n2l
4
4
Thus, by Eq. (7.117) h[n] is

which is the same result obtained in Prob. 4.32(b).

7.31. Use the state space method to solve the difference equation [Prob. 4.38(6)]

3y[n] - 4y[n - 11 +y[n - 21 =x[n] (7.119)
with x[n] = (;)"u[n] and y[- 11 = 1, y[-2]= 2.
Rewriting Eq. (7.119), we have

y[n] - $y[n - 11 + +y[n - 21 = ix[n]
Let q,[n] = y[n - 21 and q,[n] = y[n - 11. Then

d n + 11 =+I
sz[n + 11 = - fql[n] + ;q2[n] + ix[n]

~[nl= -fql[nl + ;q2[n] + +[nl
In matrix form

and

Then, by Eq. (7.25)