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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)?

Respuesta :

Limosa

Answer:

[tex]\boldsymbol{y(n)=x(n)\times s(n) - x(n)\times s(n-1)}[/tex]

Explanation:

Given:

[tex]s(n)[/tex] is the step of response with LTI system.

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

              [tex]h(n)=s(n)-s(n-1)[/tex]

Therefore, for the arbitrary [tex]x(n)[/tex] input, the output is [tex]x(n)[/tex] converted to [tex]h(n)[/tex].

              [tex]y(n)=x(n)\times h(n)[/tex]

So, put the value of [tex]h(n)[/tex] that is [tex]s(n)-s(n-1)[/tex].

             [tex]y(n)=x(n)\times (s(n)-s(n-1))[/tex]

Finally, open the bracket and multiply [tex]x(n)[/tex] in the equation.

             [tex]y(n)=x(n)\times s(n) - x(n)\times s(n-1).[/tex]

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