Is $y(n) = x(nT)$ causal,time invariant and linear?

Question

If $T>0$ is the sampling period of a device which produces one output signal $Yn$ from a continuous input signal $x(t)$, and n is any integer, would it be causal,time invariant and/or linear?

My guess would be all three as:

Causality: only depends on current n

Time invariant: a time offset of t yields the same output

$y(n-t) = x((n-t)T)$

Linear as:

$y_{n1} = x_1(nT)$, $y_{n2} = x_2(nT)$ and

$yn = [x_1(nT)+x_2(nT)] = y_{n1} + y_{n2}$

However I'm not too sure about linearity as it is not specified if the sampling rate actually changes. Any thoughts?

Answer 1

Your proof of time invariance is incorrect.

Take $x'(t) = x(t-\tau)$. Then

$$y'(n) = x'(nT) = x(nT-\tau) \neq y(n-\tau)$$

There is no such thing as a time invariant transformation from continuous to discrete time.