Page 59 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

-2-10 1 2 3 4 n -2-1 0 1 2 3 4 n
Fig. 1-38

(d) Let y ,[n] be the response to xl[n] = x[n - no]. Then

-
y ,[n] = ~{x[n no]) = nx[n - n,]
But ~ [ n
-no] = (n -n,)x[n -no] #~,(n]
Hence, the system is not time-invariant.
(el Let x[n] = u[nl. Then y[nl= nu[nl. Thus, the bounded unit step sequence produces an
output sequence that grows without bound (Fig. 1-38) and the system is not BIB0 stable.

139. A system has the input-output relation given by

where k,, is a positive integer. Is the system time-invariant?

Hence, the system is not time-invariant unless k, = 1. Note that the system described by Eq.
(1.114) is called a compressor. It creates the output sequence by selecting every koth sample of
the input sequence. Thus, it is obvious that this system is time-varying.

1.40. Consider the system whose input-output relation is given by the linear equation

y=ax+b (1.115)

where x and y are the input and output of the system, respectively, and a and b are
constants. Is this system linear?
If b + 0, then the system is not linear because x = 0 implies y = b # 0. If b = 0, then the
system is linear.

1.41. The system represented by T in Fig. 1-39 is known to be time-invariant. When the
inputs to the system are xl[n], x,[n], and x,[n], the outputs of the system are yl[n],
y,[n], and y,[n] as shown. Determine whether the system is linear.
From Fig. 1-39 it is seen that

x3[n] =xl[n] +x,[n - 21