# Linear Time Invariant System

### Linear Time Invariant Systems (LTI Systems)

A linear time invariant system is one that is linear (superposition and proportionality apply) and one that doesn't change with time. For example a circuit with fixed capacitors, resistors, and inductors having an input and an output is linear and time invariant. If a capacitor changed value with time, then it would not be time invariant.

### Eigenfunctions and Eigenvalues of an LTI Systems

It is an interesting exercise to show that ${\displaystyle e^{j\omega t}}$ are eigenfunctions of any LTI system. The eigenvalues are ${\displaystyle H(\omega )}$.

LTI System
Input Output Reason
${\displaystyle \delta (t)}$ ${\displaystyle h(t)}$ Given
${\displaystyle \delta (t-\lambda )}$ ${\displaystyle h(t-\lambda )}$ Time invariance
${\displaystyle x(\lambda )\delta (t-\lambda )}$ ${\displaystyle x(\lambda )h(t-\lambda )}$ Proportionality
${\displaystyle x(t)=\int _{-\infty }^{\infty }x(\lambda )\delta (t-\lambda )d\lambda }$ ${\displaystyle \int _{-\infty }^{\infty }x(\lambda )h(t-\lambda )d\lambda =y(t)}$ Superposition
${\displaystyle e^{j\omega t}}$ ${\displaystyle \int _{-\infty }^{\infty }e^{j\omega \lambda }h(t-\lambda )d\lambda =\int _{-\infty }^{\infty }e^{j\omega (t-\lambda )}h(\lambda )d\lambda =e^{j\omega t}\int _{-\infty }^{\infty }e^{-j\omega \lambda }h(\lambda )d\lambda =e^{j\omega t}H(\omega )}$ Applying the line above to ${\displaystyle e^{j\omega t}}$

Note that the last line is obtained by doing a change of variables, then recognizing the Fourier Transform.