Properties of Systems Memoryless, Causal, time invariant, linear


by itsdinamo
Tags: signals and systems
itsdinamo
itsdinamo is offline
#1
Aug20-11, 07:43 PM
P: 1
Hi, I need a hand with my reasoning on the following question.
I have answered all the questions, but not too sure weather they are correct.
Please guide me or point me on the right direction if they are not correct.
Regards

1. A system defined by the following equation y(t)=[x(t-2)]^2
Is the system Memoryless, Causal, Time Invariant, linear

The attempt at a solution

Memoryless ?

y(t)=x(t-2) has memory, the output is equal to the input two seconds ago.
y(t)=[x(t)]^2 = memoryless, the output at time t0 depends only on the input value at time t0.

Now, y(t)=[x(t-2)]^2 can be thought as y(t)=[x(t-2)*x(t-2)] right?, if so, my best guess would be that the system y(t)=[x(t-2)]^2 has memory.

Causal?
by definition; A system is causal if the output at any time t0 is dependent on the input only for t<=t0.
y(t)=[x(t-2)]^2 = causal

Time Invariant?
by definition if
x(t)--->y(t), then
x(t-t0)--->y(t-t0)
so;
x(t-t0)--->[x((t-t0)-2)]^2
y(t-t0)=[x((t-t0)-2)]^2,
Therefore, y(t)=[x(t-2)]^2 is time invariant.

linear ?
A system is linear if it is homogeneous and additive.
Homogeneous Property
if x(t)--->y(t), then
ax(t)--->ay(t) a= alpha
Additive Property
if x1(t)--->y1(t), and x2(t)--->y2(t), then
x1(t)+x2(t)--->y1(t)+y2(t),

x(t)--->y(t)=[x(t-2)]^2
multiply input by alpha a, then
ax(t)--->[ax(t-2)]^2
ay(t)--->a[(x(t-2))^2]
system y(t)=[x(t-2)]^2 is not homogeneous, thus not linear.
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