Chapter 5: Laplace Transform

Introduction

The Laplace transform generalizes the Fourier transform to handle a broader class of signals, including growing exponentials. It is essential for analyzing linear systems and solving differential equations.

Definition

Definition: Bilateral Laplace Transform

The bilateral Laplace transform of $x(t)$ is:

$$X(s)=\mathcal{L}\{x(t)\}={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}x(t)e^{-st}dt$$

where $s=\sigma +j\omega$ is the complex frequency.

Definition: Unilateral Laplace Transform

For causal signals and systems, the unilateral Laplace transform is:

$$X(s)={\int }_{0^{-}}^{+\mathrm{\infty }}x(t)e^{-st}dt$$

Region of Convergence (ROC)

Definition: Region of Convergence

The ROC is the set of values of $s$ for which the Laplace transform integral converges. The ROC is always a vertical strip in the $s$-plane.

Proposition: ROC Properties

1. The ROC does not contain any poles
2. For right-sided signals: ROC is to the right of the rightmost pole
3. For left-sided signals: ROC is to the left of the leftmost pole
4. For two-sided signals: ROC is a vertical strip between poles
5. For causal and stable systems: ROC includes the $j\omega$ axis

Common Transform Pairs

Signal $x(t)$	Laplace Transform $X(s)$	ROC
$\delta (t)$	$1$	All $s$
$u(t)$	$\frac{1}{s}$	$\text{Re}(s)>0$
$e^{-at}u(t)$	$\frac{1}{s+a}$	$\text{Re}(s)>-a$
$t^{n}u(t)$	$\frac{n!}{s^{n+1}}$	$\text{Re}(s)>0$
$t{e}^{-at}u(t)$	$\frac{1}{(s+a{)}^2}$	$\text{Re}(s)>-a$
$\cos ({\omega }_{0}t)u(t)$	$\frac{s}{s^2+{\omega }_0^2}$	$\text{Re}(s)>0$
$\sin ({\omega }_{0}t)u(t)$	$\frac{{\omega }_0}{s^2+{\omega }_0^2}$	$\text{Re}(s)>0$
$e^{-at}\cos ({\omega }_{0}t)u(t)$	$\frac{s+a}{(s+a{)}^2+{\omega }_0^2}$	$\text{Re}(s)>-a$

Properties

Theorem: Laplace Transform Properties

Let $x(t)\leftrightarrow X(s)$ with ROC $R_x$.

Linearity:

$$ax(t)+by(t)\leftrightarrow aX(s)+bY(s)$$

Time Shift:

$$x(t-t_0)u(t-t_0)\leftrightarrow e^{-s{t}_0}X(s)$$

Frequency Shift:

$$e^{s_{0}t}x(t)\leftrightarrow X(s-s_0)$$

Time Scaling:

$$x(at)\leftrightarrow \frac{1}{|a|}X\left(\frac{s}{a}\right)$$

Differentiation in Time:

$$\frac{dx(t)}{dt}\leftrightarrow sX(s)-x(0^{-})$$

Integration:

$${\int }_{0^{-}}^{t}x(\tau )d\tau \leftrightarrow \frac{X(s)}{s}$$

Convolution:

$$x(t)\ast h(t)\leftrightarrow X(s)H(s)$$

Transfer Function

Definition: Transfer Function

The transfer function $H(s)$ of an LTI system is the Laplace transform of its impulse response:

$$H(s)=\mathcal{L}\{h(t)\}$$

For a system described by:

$$\sum _{k=0}^N{a}_k\frac{d^{k}y(t)}{d{t}^k}=\sum _{k=0}^M{b}_k\frac{d^{k}x(t)}{d{t}^k}$$

The transfer function is:

$$H(s)=\frac{Y(s)}{X(s)}=\frac{\sum _{k=0}^M{b}_k{s}^k}{\sum _{k=0}^N{a}_k{s}^k}$$

Poles and Zeros

Definition: Poles and Zeros

For a rational transfer function $H(s)=\frac{N(s)}{D(s)}$:

• Zeros: roots of $N(s)=0$
• Poles: roots of $D(s)=0$

Theorem: Stability from Poles

A causal LTI system is BIBO stable if and only if all poles of $H(s)$ are in the left half-plane:

$$\text{Re}(p_i)<0\text{for all poles }{p}_i$$

Inverse Laplace Transform

Definition: Partial Fraction Expansion

For a rational $X(s)$, use partial fraction expansion:

$$X(s)=\frac{N(s)}{D(s)}=\sum _i\frac{A_i}{s-p_i}$$

Then invert term by term:

$$x(t)=\sum _i{A}_i{e}^{p_{i}t}u(t)$$

for causal signals (ROC to the right of all poles).

Example: Second-Order System

Consider $H(s)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$ where $\zeta$ is the damping ratio.

• $\zeta >1$: overdamped (two real poles)
• $\zeta =1$: critically damped (repeated real pole)
• $0<\zeta <1$: underdamped (complex conjugate poles)
• $\zeta =0$: undamped (purely imaginary poles)

For the underdamped case, the impulse response is:

$$h(t)=\frac{{\omega }_n}{\sqrt{1-{\zeta }^2}}{e}^{-\zeta {\omega }_{n}t}\sin ({\omega }_n\sqrt{1-{\zeta }^2}t)u(t)$$