Introduction

### Introduction

• What does it mean?
• What is its significance for LTI Systems?

The impulse response completely characterizes an LTI System. We can find the output $H(x[n])$ for any $x[n]$ if $h[n]$ is known.

### Examples

Any input signal $x[n]$ can be written as:

(1)
\begin{align} x[n]=\sum_{k=-\infty}^{\infty}x[k]\delta[n-k] \end{align}

where $x[k]$ is the value taken by the signal $x[n]$ at $n=k$

(2)
\begin{align} y[n]=H(x[n])=\sum_{k=-\infty}^{\infty}x[k]h[n-k] \end{align}

The above equation is known as the Convolution Sum and is written as:

(3)
$$y[n]=x[n]*h[n]$$