# 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 $e^{j\omega t}$ are eigenfunctions of any LTI system. The eigenvalues are $H(\omega)$.

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

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