Response of an LTI System

Learn how an LTI system responds to a discrete-time signal built with scaled and shifted unit impulses.

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In this lesson, we learn how to determine the output response of a system for any arbitrary input.

Finding the output

To find the answer to this question, the following two results are needed:

  1. System impulse response h[n]h[n] is the output when the input is a unit impulse δ[n]\delta [n].
  2. Any signal can be decomposed as a set of scaled and shifted unit impulses as:

x[n]=+x[1]δ[n+1]+x[0]δ[n]+x[1]δ[n1]+=mx[m]δ[nm] \begin{align*} x[n]&= \cdots + x[-1]\delta[n+1] + \nonumber \\ &\quad\quad\quad\quad\quad x[0]\delta[n] + x[1]\delta[n-1] + \cdots \nonumber \\ &= \sum _mx[m] \delta[n-m] \end{align*}

The algorithm

From here, we can follow the steps below:

  • For a time-invariant system, the response to a delayed unit impulse δ[nm]\delta[n-m] becomes h[nm]h[n-m].

δ[nm]h[nm]\delta[n-m] \qquad \rightarrow \qquad h[n-m]

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