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.