Is the system memory-less?

Question

I got a Question on digital signal processing. Is

h(n)= (2 n+5)u(n) 

memory-less? I think that the because of u(n), the system is memory-less. Am I right?

Answer 1

The short answer: No, assuming the provided h(n) represents the usual system impulse response.

A memoryless system is characterized with outputs that do not depend on past inputs (and commonly also do not depend on future input). In terms of the impulse response it means that the response h(n) at a time n>0 should not depend on the input at time 0, namely the impulse itself. In other words you should have h(n) = 0 for n>0 (but h(n) can be non-zero at time n=0).

In your case, the output starts to grow after the impulse is applied, so those outputs at time n>0 clearly depend on the then past impulse input. So, the system is not memoryless.

P.S.: the presence of u(n) indicates that the system's output does not depends on future inputs (e.g. the output at time -1 does not depend on the then future impulse which the system has not seen).