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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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Output of a memoryless system depends only on the current input value and therefore every memoryless system is also causal; since a causal system's output cannot depend on the future input values.

The converse in general is not true; causal systems can be memoryless as well as can exhibit memory (if their outputs depend on the past input values in addition to current input sample).

Also note that a noncausal system cannot be memorlyess is it must depend on future values (and hence exhibit memory) to be noncausal.

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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 Nov 25 '18 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 Nov 25 '18 at 23:26
  • $\begingroup$ It might be good to mention that your use of "memoryless" differs from that of the OP. $\endgroup$ – supercat Nov 26 '18 at 1:31

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