Impulse Response

For an LTI system:

$$\begin{array}{rl}\dot{x}& =Ax+Bu\\ y& =Cx\end{array}$$

where:

• $x(t)$ is the state at time $t$
• $u(t)$ is the input at time $t$
• $y(t)$ is the output at time $t$

Consider the input

$$u(t)=\delta (t-\tau )$$

which is an unit impulse applied at $t=\tau$.

Given that the system is LTI and has zero input condition $x(0)=0$, and we have

$$y(t)=h(t-\tau )$$

then $h(t)$ is the impulse response of this system.

Using impulse response with sifting

If we know $h$, how can we find the system’s response to other arbitrary inputs?

Recall the sifting property of the $\delta$-function: for any function $f$ which is “well-behaved” at $t=\tau$,

$${\int }_{-\infty }^{\infty }f(t)\delta (t-\tau )dt=f(\tau )$$

Any reasonably regular function can be represented as an integral of impulses.

Then, by the sifting property we can write a general input $u(t)$ as

$$u(t)={\int }_{-\infty }^{\infty }u(\tau )\delta (t-\tau )d\tau$$

Then, with the superposition principle, the response of a linear system to a sum (or integral) of inputs is the sum (or integral) of the individual responses to these inputs:

$$u(t)={\int }_{-\infty }^{\infty }u(\tau )\delta (t-\tau )d\tau ⟶y(t)={\int }_{-\infty }^{\infty }u(\tau )\underset{\text{response to }\delta (t-\tau )}{\underbrace{h(t-\tau )}}d\tau$$

where $h(t-\tau )$ is the response to $\delta (t-\tau )$. ==Thus, the integral that defines $y(t)$ is a convolution of $u$ and $h$:==

$$\begin{array}{rl}y(t)& =u(t)\ast h(t)\\ & =h(t)\ast u(t)\\ & ={\int }_{-\infty }^{\infty }u(\tau )h(t-\tau )dt\end{array}$$

This is even better in Laplace Transform form:

$$Y(s)=H(s)U(s)$$

where

$$H(s)={\int }_{-\infty }^{\infty }h(\tau )e^{-s\tau }d\tau$$

Transfer Functions

Since the transfer function $G(s)$ of an LTI system is the ratio of the output Laplace transform $Y(s)$ and the input Laplace transform $U(s)$ such that $G(s)=\frac{Y(s)}{U(s)}$, if we let

$$U(s)=1\text{i.e.,}u(t)=\delta (t)$$

then

$$Y(s)=G(s)$$

or

$$y(t)={\mathcal{L}}^{-1}[G(s)]=g(t)$$

Hence, $g(t)$ is the impulse response of the system.