# Time-variance and causality of $y(t)=\sum_{k=-\infty}^{+\infty}x(t)\delta(t-kT)$

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?

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