For a discrete LTI system, the impulse response is u[n]. What is

SOLUTION

Concept:

The step response of a discrete – LTI system is the convolution of the unit step with the impulse response i.e.

s(n) = u(n) * h(n)

s(n) = Step Response. It is the response of the LTI system to a step input u(n).

According to the convolution property, the input and the impulse response are interchangeable, i.e. we can write:

s(n) = h(n) * u(n)

\(s\left( n \right) = \mathop \sum \limits_{k = - \infty }^\infty h\left( k \right) \cdot u\left( {n - k} \right)\)

\(u\left( {n - k} \right) = \left\{ {\begin{array}{*{20}{c}} {0\;for\;n - k < 0\;\;}\\ {1\;for\;\;n - k \ge 0\;} \end{array}} \right.\)

\(\therefore s\left( n \right) = \mathop \sum \limits_{k = - \infty }^n h\left( k \right)\)

We conclude that the step response of a  discrete LTI system is the running sum of its impulse response given

h(n) = u(n)

\(s\left( n \right) = \mathop \sum \limits_{k = - \infty }^n u\left( k \right) = nu\left( n \right)\)