I have a system given,

y(t) = ay(t-1) + x(t)- 2x(t-2).

As I understand a system is time invariant if

x(t) -> y(t)

and

x(t-T) -> y(t-T).

What is confusing me is the delayed output that this system depends on. Is this a time invariant system? Thanks.

 

I figured splitting the equation into two time intervals, t > 1 and t < 1.

At t < 1, I ignored the delayed output ay(t-1) in the system and solved for y(t) in terms of x only. I then delayed both input and output signals and found equivalence.

Then for t > 1, I added ay(t-1) back into the input. I replaced y(t-1) with what I found earlier with all time parameters delayed by 1. I was then able to find equivalence again from delaying the inputs and the outputs which confirmed TI.

Can anyone confirm that this is correct?

 

You are making it far more complicated than it is.

This might help you:

http://en.wikipedia.org/wiki/Time-invariant_system