Invertibility

Table of Contents

### Definition

Suppose $h[n]$ or $h(t)$ is the impulse response.

If an LTI System is invertible it has an LTI Inverse.

The LTI System with impulse response $h[n]$ or $h(t)$ is invertible if $\exists$ an $h_1[n]$ or $h_1(t)$ such that:

(1)
\begin{align} h[n]*h_1[n]=\delta[n] \end{align}
(2)
\begin{align} h(t)*h_1(t)=\delta(t) \end{align}

### Examples

1. $h(t)=\delta(t-t_0)$
$y(t)=x(t)*\delta(t-t_0)=x(t-t_0)$

(3)
\begin{align} \delta(t-t_0)*\delta(t+t_0)=\delta(t) \nonumber \end{align}
(4)
\begin{align} \delta(t-t_0)*\delta(t)=\delta(t-t_0) \nonumber \end{align}
(5)
\begin{align} \delta(t-t_0)*\delta(t+t_0)=\delta(t+t_0-t_0)=\delta(t) \nonumber \end{align}

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1. $h[n]=\big(\frac{1}{2}\big)^n u[n]$
Is there an $h_1[n]$ such that $h[n]*h_1[n]=\delta[n]$?

$h_{eq}[2]=1\frac{1}{4}-\frac{1}{2}\frac{1}{2}+h_1[2]1=0 \Rightarrow h_1[2]=0$

$h_1[n]=\delta[n]-\frac{1}{2}\delta[n-1]$ is the inverse of $h[n]=\big(\frac{1}{2}\big)^n u[n]$

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