Understanding Time-Invariant and Variant Systems: Examples and Solutions

[steven-ka]

I would like to have some assistance with two examples of checking times in/variant systems.

1) y(t)=x(-t)

2) y(t)=x(t^2)

I would like to know what's wrong with the following solution of mine(especially the second one):

1) y(t)=x(-t)

x1(t)=x1(t-t0) => y1(t)=x(-t-t0)

y2(t)=y2(t-t0)=> y2(t)=x(-(t-t0))=x(-t+t0)

y1(t)=!y2(t) => time variant system.

2) y(t)=x(t^2)

x1(t)=x1(t-t0)=> y1(t)=x(t^2-t0)

y2(t)=y2(t-t0)=> y2(t)=x((t-t0)^2)

y1(t)=!y2(t)=> time variant system.

I'll appreciate any helpful comment :) thanks.

 

[haruspex]

I agree with your answers but I am baffled by the proofs.
I assume you are defining x1 as the tIme shifted input and y1 as the corresponding time shifted output. So surely you should write x1(t)=x(t-t0), y1(t)=x1(-t), etc?

 

[steven-ka]

Yeah yea exactly, so do you think its true?

 

[haruspex]

Yes, but surely the point is to come up with a working proof.