# Linear Time-Invariant System (LTI)

This is a Linear System that is a Time-Invariant.

See Impulse Response first for a motivation. You can’t always write $f(t)$ as a differential equation.

Theorem 1

The response of an LTI system is the convolution of the input with the system’s Impulse Response.

Theorem 4: LTI response to an exponential

If $S:f→y$ is a LTI then for any $s∈C$ $e_{st}→H(s)e_{st}$

- $H(s)$ is the LT of the system’s impulse response and is called the system’s transfer function

Note this means that complex exponentials are the eigenfunctions of LTIs and the transfer function tells you the eigenvalues of the system! (like with DEs).