Determining Time Invariance in Signal Statements: Examples and Solutions

[jegues]

Homework Statement

I just have a general question about what one of my professors had written on the board today in class.

He was writing down examples where we had to determine whether the given statement was time invariant or not.

One example was written as follows,

$$x(-t) = y(t)$$

$$\text{Input: }x(t-T)$$

My attempt at solving this would be,

$$x(-(t-T)) = x(T-t) = y(t-T)$$

Thus a leftward shift in the input causes a rightward shift in the output. Therefore not time invariant.

The solution my professor proposed was something like the following

$$x(-t -T) = x(-(t+T)) = y(t-T)$$

I can't quite remember what he wrote on the RHS of the equation but it was something like that.

Am I misinterpreting what he means by,

$$\text{Input: }x(t-T)$$

Is he just saying, check if its time invariant when the input is shifted to the right by T?

Another example he gave was,

$$x(at) = y(t)$$

$$\text{Input: }x(t-T)$$

My attempt at the solution would be,

$$x(a(t-T)) = x(at-aT) = y(t-T)$$

A shift by aT ≠ a shift by T therefore not time invariant.

His solution,

$$x(at-T) = x(a(t-\frac{T}{a}))$$

Therefore not invariant.

Am I misinterpreting what he is saying what when he writes,

$$\text{Input: x(t-T)}$$

?

Should it be written that way in the first place? Is what he is writing and what he wants from us inconsistent?

Thanks again!

 

[jegues]

Bump, anyone!?