# Signals: Time Invariance

1. Sep 30, 2012

### jegues

1. The problem statement, all variables and given/known data

Prove whether or not,

$$y(t) = \frac{1}{2}\left( x(t) - x(-t) \right)$$

Is time invariant or not

2. Relevant equations

3. The attempt at a solution

Shifting the output by -T results in,

$$y(t-T) = \frac{1}{2}\left( x(t-T) - x(-(t-T)) \right)$$

$$y(t-T) = \frac{1}{2}\left( x(t-T) - x(-t+T) \right)$$

Shifting the input by -T results in,

$$\frac{1}{2}\left( x(t-T) - x(-t-T) \right)$$

Since the last two lines are not the same they are not time invariant.

I feel like this is wrong because,

$$x(-t-T)$$

shifts the input to the left while the other input (i.e. x(t)) is shifted to the right.

What is the correct procedure to prove whether or not this is time invariant or not?