Is the System y(t) = (1/2)(x(t) - x(-t)) Time Invariant?

[jegues]

Homework Statement

Prove whether or not,

[tex]y(t) = \frac{1}{2}\left( x(t) - x(-t) \right)[/tex]

Is time invariant or not

Homework Equations

The Attempt at a Solution

Shifting the output by -T results in,

[tex]y(t-T) = \frac{1}{2}\left( x(t-T) - x(-(t-T)) \right) [/tex]

[tex]y(t-T) = \frac{1}{2}\left( x(t-T) - x(-t+T) \right) [/tex]

Shifting the input by -T results in,

[tex]\frac{1}{2}\left( x(t-T) - x(-t-T) \right)[/tex]

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

I feel like this is wrong because,

[tex]x(-t-T)[/tex]

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?

 

[KataKoniK]

Your attempt at a solution is correct. To prove time invariance, we need to show that shifting the input and output by the same amount results in the same output. In this case, we are shifting both the input and output by -T. However, as you have correctly pointed out, the two expressions are not the same, which means that the system is not time invariant.

To further clarify, let's look at the two expressions you have derived:

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

and

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

When we shift the input by -T, we are essentially replacing t with t-T in the input signal. This means that the input signal will be shifted to the right by T units. However, when we shift the output by -T, we are replacing t with t-T in the output signal. This means that the output signal will be shifted to the left by T units. As you can see, these two expressions are not the same, which indicates that the system is not time invariant.

In general, to prove time invariance, we need to show that the shifted output is equal to the output of the shifted input. This can be done by substituting t with t-T in the original expression and showing that it is equal to the shifted output expression. If the two expressions are equal, then the system is time invariant.

I hope this helps clarify the concept of time invariance and how to prove it. Keep up the good work!