1
$\begingroup$

Is the following system time invariant and/or causal?

$$y(t)=\sum_{k=-\infty}^{+\infty}x(t)\delta(t-kT)$$

I think that it isn't time invariant and it is causal, and my thought is based on the fact that this is actually a sampling system that samples on $T-$intervals the input function. Is this correct and if yes, how can the time-variance be formally proven?

$\endgroup$
1
$\begingroup$

I agree with you on the system being a causal system since at a point in time $t$, the output is only made out of the input signal at the same point in time, weighted with an impulse (or not).

You can prove its time-variance by showing that a time-shifted input signal yields another output signal than the original input signal, for example:

  • $x(t) = \delta(t-\frac{T}{2})$ evaluates to $y(t)=0$ while
  • $x(t) = \delta(t)$ does not.
$\endgroup$
  • 1
    $\begingroup$ Your signals $x(t)$ are not functions of $t$ as you wrote them. $\endgroup$ – Tendero Jan 31 '18 at 13:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.