LTI system $y(t)=x(t-T)$ with or without memory

Question

How can I verify if the LTI system ($T$ is a real number, $y(t)$ output, $x(t)$ input):

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

is with or without memory? I know that, if $y(t)$ depends only on $x(t)$, the LTI system will be without memory. In the above example, does it depend on $t-T$ or only on $t$?

Answer 1

From Signals and Systems/Time Domain Analysis -- Memory

A system is said to have memory if the output from the system is dependent on past inputs (or future inputs) to the system. A system is called memoryless if the output is only dependent on the current input.

According to some textbooks, the notion of memory is stricter, and requires causality of the system. To that respect:

• if $T\ge 0$, the system is causal. If $T=0$, it only use the current time (memoryless). If $T>0$, the output need a memory of the past of input $x$
• if $T< 0$, the system is non-causal. The output needed a memory of the future of input $x$. You can either say "non applicable", or still term it "memory".

References (especially the first, with many practical exemples):

• Lecture III: Systems and their properties by Maxim Raginsky
• $y(n)=x(-n)$, causal or not, memory or memoryless?