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I have found the particular definitions.

Causality means that the output of the system does not depend on future inputs, but only on past input.

Memory-less - does not depend on previous values of either input or output.

I am looking for example of systems which are either non-casual or memoryless but not both.

Also i want to check whether I understand it well.

$ T(x[n]) = x[n − n_{0}] $

$T(x[n]) = ax[n] + bx[n − 3]$

Are those an examples of non-casual and memory-less systems?

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A causal system does not need to know the future in order to compute its output. A memoryless system computes the output only from the current input. A memoryless system is always causal (as it doesn't depend on future input values), but a causal system doesn't need to be memoryless (because it may depend on past input or output values).

The system $$y[n]=x[n]+2x[n+1]$$ is non-causal because it needs to look into the future (by $1$ sample) to compute its output. The system $$y[n]=3\big(x[n]\big)^2$$ is memoryless (and necessarily causal) because it only needs the current input sample $x[n]$ to compute the output.

The systems in your question are both causal and have memory (if $n_0>0$).

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A memoryless system's output is determined by the current input value only, hence, every memoryless system must also be causal (a system is causal if its output does not depend on the future input values). Therefore, every memorlyess system is also causal.

The converse in general is not true, however: a causal system can have memory (depend on past input), or can be memoryless (depend on current input).

Also note that a noncausal system cannot be memorlyess: as it must depend on future input values...

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  • $\begingroup$ The OP's definition of a memoryless system requires that outputs not depend upon past inputs or outputs, but says nothing about whether they may rely upon future outputs. I would think that time-reversing a system that neither causal nor memoryless would yield a system that is both. $\endgroup$
    – supercat
    Commented Nov 25, 2018 at 23:09
  • $\begingroup$ @supercat OP's definition of memoryless system is (then) incomplete... anyway I've put my answer wrt. to the correct definition. $\endgroup$
    – Fat32
    Commented Nov 25, 2018 at 23:26
  • $\begingroup$ It might be good to mention that your use of "memoryless" differs from that of the OP. $\endgroup$
    – supercat
    Commented Nov 26, 2018 at 1:31

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