0.4 Lecture 5:the laplace transform method of solution

9/23/99 (T.F. Weiss)

Lecture #5: The Laplace transform method of solution

Motivation:

• Continue to describe methods for representing signals as superpositions of complex exponential functions
• Develop efficient methods for analyzing LTI systems

Outline:

• Review of last lecture

• Laplace transform of the family of singularity functions

• More on the region of convergence

• Analysis of networks with the Laplace transform — the impedance method

• Finding inverse transforms — partial fraction expansion

• Conclusion

• Historical perspective — Oliver Heaviside

Review

• The Laplace transform represents a time function as a superposition of complex exponentials.

• A time function is related uniquely to a Laplace transform if the ROC is specified.

• If the Laplace transform of a sum of causal and anti-causal exponential time functions exists, its ROC is a strip in the s-plane parallel to the jω-axis.

Laplace transforms of singularity functions

Unit impulse function

$L\left\{\delta (t)\right\}=\sum _{-\infty }^{\infty }\delta (t)\text{.}{e}^{-\text{st}}\text{dt}$

Recall the definition of the unit impulse

$\sum _{=\infty }^{\infty }\delta (t)f(t)\text{dt}=f(0)$

Hence,

$L\{\delta (t)\}=1$

for all values of s. The region of convergence is the entire s plane.

Unit impulse function delayed — use of properties

The Laplace transform of an impulse located at t = 0 is

$L\{\delta (t)\}=1$

Using the delay property, $x(t)\stackrel{L}{\iff }X(s)$

$x(t-T)\stackrel{L}{\iff }X(s)e^{-s/T}$

the Laplace transform of the delayed impulse is

$L\{\delta (t-T)\}=e^{-\text{sT}}$

and the region of convergence is the whole s plane.

Two-minute miniquiz problem

Problem 5-1

Find the Laplace transform including the ROC for

$x(t)=e^{-2(t-4)}u(t-4)$

Two-minute miniquiz solution

Problem 5-1

We use the Laplace transform of the causal exponential time function and time delay property to solve this problem.

$e^{-\mathrm{2t}}u(t)\stackrel{L}{\iff }\frac{1}{s+2}\begin{array}{cc}& \end{array}\text{for}\begin{array}{cc}& \end{array}\sigma >-2$

$e^{-2(t-4)}u(t-4)\stackrel{L}{\iff }\frac{1}{s+2}{e}^{-\mathrm{4s}}\begin{array}{cc}& \end{array}\text{for}\begin{array}{cc}& \end{array}\sigma >-2$

Singularity functions and their relatives

The Laplace transform of a unit impulse is

$\delta (t)\stackrel{L}{\iff }1\begin{array}{cc}& \end{array}\text{for}\begin{array}{cc}& \end{array}\text{all}\begin{array}{cc}& \end{array}s$

and from the Laplace transform of a causal exponential with $\alpha$ = 0 we have the Laplace transform of a causal step function

$u(t)\stackrel{L}{\iff }\frac{1}{s}\begin{array}{cc}& \end{array}\text{for}\begin{array}{cc}& \end{array}\sigma >0$

Note this fits together with the time differentiation property

$\frac{\text{dx}(t)}{\text{dt}}\stackrel{L}{\iff }\text{sX}(s)$

since in a generalized function sense

$\delta (t)=\frac{\text{du}(t)}{\text{dt}}\stackrel{L}{\iff }L\{\delta (t)\}=s(\frac{1}{s})=1$

Singularity functions and their relatives, cont’d

We use the multiplication by t property

$\text{tx}(t)\stackrel{L}{\iff }-\frac{\text{dX}(s)}{\text{ds}}$

to obtain

$\text{tu}(t)\stackrel{L}{\iff }-\frac{d}{\text{ds}}(\frac{1}{s})=\frac{1}{s^2}\text{for}\sigma >0$

and use it again to obtain

$t^{2}u(t)\stackrel{L}{\iff }-\frac{d}{\text{ds}}(\frac{1}{s^2})=\frac{1}{s^3}\text{for}\sigma >0$

which implies that by induction

or

$\begin{array}{}{t}^{n}u(t)\stackrel{L}{\iff }\frac{n!}{s^{n+1}}\text{for}\sigma >0\\ \end{array}$

$\frac{t^{n-1}}{(n-1)!}u(t)\stackrel{L}{\iff }\frac{1}{s^n}\text{for}\sigma >0$

Summary of singularity functions and their relatives

Wild and crazy singularity functions

Since taking the derivative of a time function corresponds to multiplying the Laplace transform by s we can contemplate the derivative of the unit impulse called the unit doublet.

$\frac{\mathrm{d\delta }(t)}{\text{dt}}=\delta (t)\stackrel{L}{\iff }s$

This process can be continued by taking successive derivatives of the impulse to form the unit triplet which has Laplace transform $s^2$ , unit quadruplet, etc. In general, the nth derivative of the unit impulse has a Laplace transform $s^n$ . We shall consider the usefulness of these higher order singularity functions later!