0.12 Lecture 13:the bilateral laplace transform

Lecture #13:

THE BILATERAL LAPLACE TRANSFORM

Motivation: Method for representing signals as superpositions of complex exponential functions, which leads to simpler solving process.

Outline:

• Review of last lecture
• The bilateral Laplace transform

• Definition
• Properties of the Laplace transform
• Transforms of simple time functions
• The region of convergence

• Conclusion

I. REVIEW OF LAST LECTURE

Solve linear differential equation for a causal exponential input

$$ $\sum _{n=0}^N{a}_n\frac{d^{n}y(t)}{{\text{dt}}^n}=\sum _{m=0}^M{b}_m\frac{d^{m}x(t)}{{\text{dt}}^m}\begin{array}{cc}& \end{array}\text{for}\begin{array}{cc}& \end{array}x(t)={\text{Xe}}^{\text{st}}u(t)$

Solve homogeneous equation for t>0

$\sum _{n=0}^N{a}_n\frac{d^{n}y(t)}{{\text{dt}}^n}=0\begin{array}{cc}& \end{array}\text{by}\begin{array}{cc}& \end{array}\text{assuming}\begin{array}{cc}& \end{array}{y}_h(t)={\text{Ae}}^{\mathrm{\lambda t}}$

Solve characteristic polynomial

$\sum _{n=0}^N{a}_n{\lambda }^n=0\begin{array}{cc}& \end{array}\text{for}\begin{array}{cc}& \end{array}\lambda$

Solve for a particular solution for t>0

$\sum _{n=0}^N{a}_n\frac{d^n{y}_p(t)}{{\text{dt}}^n}=\sum _{m=0}^M{b}_m\frac{d^{m}x(t)}{{\text{dt}}^m}\begin{array}{cc}& \end{array}\text{for}\begin{array}{cc}& \end{array}x(t)={\text{Xe}}^{\text{st}}$

Assuming $y_p(t)={\text{Ye}}^{\text{st}}n$ and solving for Y yields

$Y=H(s)X=\frac{\sum _{m=0}^M{b}_m{s}^m}{\sum _{n=0}^N{b}_n{s}^n}X$

$H(s)=\frac{V}{I}=\frac{1}{C}(\frac{s}{s^2+\frac{1}{\text{RC}}s+\frac{1}{\text{RC}}})$

Logic for an analysis method for CT LTI systems

• H(s) characterizes system  compute H(s) efficiently.
• In steady state, response to ${\text{Xe}}^{\text{st}}$ is $H(s){\text{Xe}}^{\text{st}}$ .
• Represent arbitrary x(t) as superpositions of ${\text{Xe}}^{\text{st}}$ on s.
• Compute response y(t) as superpositions of $H(s){\text{Xe}}^{\text{st}}$ on s.

II. THE BILATERAL LAPLACE TRANSFORM DEFINITION

1/ Analysis formula

The bilateral Laplace transform is defined by the analysis formula

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

X(s) is defined for regions in s — called the region of conver­gence (ROC) — for which the integral exists.

2/ Synthesis formula

The inverse transform is defined by the synthesis formula

$x(t)=\frac{1}{\mathrm{2\pi j}}{\int }_{\sigma -j\infty }^{\sigma +j\infty }X(t)e^{\text{st}}\text{ds}$

Since s is a complex quantity, X(s) is a complex function of a complex variable, and $\sigma$ is in the ROC.

• The synthesis formula involves integration in the complex s domain. We shall not perform this integration in this subject. The synthesis formula will be used only to prove theorems and not to compute time functions directly.
• The synthesis formula makes apparent that x(t) is synthe­sized by a superposition of an uncountably infinite number of eternal complex exponentials $e^{\text{st}}$ each for a different value of s and each of infinitesimal magnitude X(s)ds.

3/ Relation to unilateral Laplace transform

The difference between the unilateral and the bilateral Laplace transform is in the lower limit of integration, i.e.,

$\text{Bilateral}\Rightarrow X(s)={\int }_{-\infty }^{\infty }x(t)e^{-\text{st}}\text{dt}$

$\text{Unilateral}\Rightarrow X(s)={\int }_0^{\infty }x(t)e^{-\text{st}}\text{dt}$

• The unilateral Laplace transform is restricted to causal time functions, and takes initial conditions into account in a sys­tematic, automatic manner both in the solution of differential equations and in the analysis of systems.
• The bilateral Laplace transform can represent both causal and non-causal time functions. Initial conditions are ac­counted by including additional inputs.

4/ Approach

Databases of time functions and their Laplace transforms are developed as follows:

• Determine the Laplace transforms of simple time functions directly,

• Use the Laplace transform properties to extend the database of transform pairs.

Notation

We shall use two useful notations — L {x(t)} signifies the Laplace transform of x(t) and a Laplace transform pair is indicated by