Question

Assume you are given an LTI system with a step response s(n), which is defined as the output of the system when excited with (input) a step function u(n)? Can you derive the expression for the output y(n) in terms of s(n) and an arbitrary input x(n)?

Answer 1

`\boldsymbol{y(n)=x(n)* s(n) - x(n)* s(n-1)}`

Step-by-step explanation:

Given:

`s(n)` is the step of response with LTI system.

and, It response h(n) then relates to phase response as,

`h(n)=s(n)-s(n-1)`

Therefore, for the arbitrary
`x(n)` input, the output is
`x(n)` converted to
`h(n)`.

`y(n)=x(n)* h(n)`

So, put the value of
`h(n)` that is
`s(n)-s(n-1)`.

`y(n)=x(n)* (s(n)-s(n-1))`

Finally, open the bracket and multiply
`x(n)` in the equation.

`y(n)=x(n)* s(n) - x(n)* s(n-1).`