Proving System is Time Invariant (T.I.) or Not

[SpaceDomain]

Homework Statement

Prove that the system is either T.I. or is not T.I.

Homework Equations

y(n) = x(n)*h(n)

x(n) is the input signal
y(n) is the output signal
h(n) is the system

The Attempt at a Solution

Inputing x(n-n0) into the system I get out:
as the output x(n-n0)*h(n)

Since y(n-n0) = x(n-n0)*h(n-n0) != x(n-n0)*h(n) the system is not T.I.

I think I am doing this wrong.

 

[reddvoid]

Inputing x(n-n0) into the system I get out:
as the output x(n-n0)*h(n)

how can you get this output for x(n-n0) input ?

 

[SpaceDomain]

how can you get this output for x(n-n0) input ?

So should it be that an input of x(n-n0) results in x(n-n0)*h(n-n0)?

 

[reddvoid]

no,
you are telling
x(n) is input of the system
y(n) is out put of the system
h(n) is its transfer function

in-order to check whether the system is TV or TIV we need the relation between input and output.
for example if you tell y(t) output =sin times input x(t)
then we can check whether its TV or TIV
or we need the impulse response function h(t) to decide whether it is TV or TIV

you have not given anything
for every system y(t) is equal to x(t)*h(t)
but their response depends on the function h(t) which is different for different systems

P.S.
I hope you understood

 

[DeeNos]

Your approach is correct, but your conclusion is not entirely accurate. Let's break it down step by step.

First, let's define what it means for a system to be time-invariant (T.I.). A system is T.I. if a time shift in the input signal results in the same time shift in the output signal. Mathematically, this can be represented as:

x(n) --> h(n) --> y(n) (original input x(n) results in output y(n))
x(n-n0) --> h(n) --> y(n-n0) (input shifted by n0 results in output shifted by n0)

Now, let's apply this to your solution. You correctly input x(n-n0) into the system and get the output x(n-n0)*h(n). However, your conclusion that y(n-n0) = x(n-n0)*h(n-n0) is not entirely accurate.

If the system is T.I., then the output should be x(n-n0)*h(n-n0), but if the system is not T.I., then the output will not be the same. Therefore, your conclusion should be that the system is not T.I. because the output does not match the expected output x(n-n0)*h(n-n0).

In summary, your approach is correct, but your conclusion should state that the system is not T.I. because the output does not match the expected output when the input is shifted.