Response of an LTI System

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 $h[n]$ is the output when the input is a unit impulse $\delta [n]$.
2. 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 $h[n-m]$.

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