# Response of an LTI System

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

## We'll cover the following

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:

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

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

## The algorithm

From here, we can follow the steps below:

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

$\delta[n-m] \qquad \rightarrow \qquad h[n-m]$

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