What Does The Following System Do?

Question

I have a system $y(t) = 0.5 \int^\infty _{-\infty} x(T)[d(t-T) - d(t+T) dT] $

Where d(x) is the Dirac Delta function (couldn't find the LaTEX representation - a little rusty there, so an edit to clarify with it would be appreciated!).

I'm curious as to what the system actually does with the input x(t), and what the plain English explanation of the rule it applies is. Clearly, there are elements of a summation (integral), scaling (factor of 0.5) and limiting (difference between delta terms and their product with the input.) Due to the Dirac Deltas, I was wondering if it was possibly some sort of identity system modification, but I am not sure.

Any pointers would be greatly appreciated!

Answer 1

Convolving with a delta function essentially gives you back the input. So the output should end up being.

$$ y(t) = 0.5 [x(t) - x(-t)] $$

So, the output basically appears to be the difference between the input and the time reversed input.

Edit

Another way to state it is that it creates an anti-symmetric signal from the input. You could imagine that if it were $x(t)+x(-t)$ that would create a symmetric signal.

Answer 2

In general,

$$\int_{-\infty}^{\infty} f(x) \delta(x-x') \, dx = f(x'). $$