Linear Time-Invariant System

Impulse Response

The impulse response $h(t)$ is the inverse Laplace transform of the Transfer Function. $h(t)=L^{-1}\{H(s)\}$

Theorem

The Zero-State Response of a linear DE is the convolution of the input (i.e. forcing term) with the system’s Impulse Response.

Analogies

The impulse response is the system’s response to the Unit Impulse Function. The step response is the system’s response to the Unit Step Function.

I wrote this mathematically, looks like this (see LTI for more definitions) $h(t)=y(t)=(h\ast \delta )(t)$

Motivation

For a linear DE with constant coefficients, the Zero-State Response is $Y(s)=H(s)F(s)$

where

• $H(s)$ is the Transfer Function of the DE
• $F(s)$ is the Laplace transform of the forcing term

Why is it called the impulse response?

Because it’s actually when the forcing term $f(t)$ is set to the Impulse Function.

If the forcing term is the delta impulse function (i.e. $f(t)=\delta (t)$), then $F(s)=1$ and thus $Y(s)=H(s)\cdot 1=H(s)$

Here, we call $h(t)=L^{-1}\{H(s)\}$ the impulse response of the DE.

For more general LTI systems, we might not be able to model them as differential equations. However, we can still find the impulse response!

Theorem 1

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