Is $x(t/2)$ a causal/memoryless system?

Question

Basically, is any system $$y(t) =x(\alpha t)$$ a causal system? Where $0<\alpha<1$ (a fraction) and $t>0$.

I know that a causal system depends only on current or previous inputs, however, does multiplying the variable $t$ by a fraction make the system causal or non-causal?

Answer 1

Give the definition following wikipedia:

Defintion. Let system $y:F([0;\infty),S)\to F([0;\infty),S)$, where $S$ is some space and $F([0,\infty),S)$ if the space of functions from $[0,\infty)$ to $S$. The system $y$ is said to be is causal iff for every two input signals $x_1,x_2\in F([0,\infty);S)$ such that there exist $t_0\in [0,\infty)$ so that $$ x_1(t)=x_2(t),\ \text{for every }t\le t_0,\ \ \text{it holds that }\ y(x_1(t))=y(x_2(t)),\text{ for every }t\le t_0, $$

Then your system is causal, which can be proved by \begin{align*} y(x_1(t)):=x_1(\alpha t) = x_2(\alpha t) =:y(x_2(t)), \text{ for every }t\le t_0, \end{align*} as $\alpha t\le t$ for every $\alpha\in[0,1]$.

I would recommend checking causality of other systems like $$y(x(t)):=x(\alpha^{-1}t),\quad y(x(t)):=\int_0^{\alpha t}x(s)ds+ x(1),\quad y(x(t)):=\sup_{s\in [0,t]}|x(s)|\quad y(x(t)):= x(\sin(2^{-1}\pi t)),\quad y(x(t)):= \sin(2^{-1}\pi x( t)). $$

Answer 2

If you forget for a minute the $t<0$ constraint: It is not casual since for any t<0 it will depend on the future.
It is not memoryless since at any given t is depends on the past.

Now if $t>0$ then $t/2<t$ so it is casual but again not memoryless