Consider a discrete-time system with input
x[n]
and output
y[n]
, respectively. Let * denote the operator taking convolution. (a)
(20%)
Show that this system is linear and time-invariant (LTI) if and only if
y[n]=x[n]^(**)h[n]
. where and
h[n]
stands for the impulse response of the system. (b)
(10%)
Given
h_(1)[n],h_(2)[n],h_(3)[n]
, show that
(h_(1)[n]^(**)h_(2)[n])^(**)h_(3)[n]=h_(1)[n]^(**)(h_(2)[n]^(**)h_(3)[n])
. (c)
(5%)
Based on (a) and (b), show that the serial interconnection of two LTT systems is also LTI.