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The Laplace Transform• Previous basis functions: 1, x, cosx, sinx, exp(jwt).• New basis function for the LT => complex

exponential functions• LT provides a broader characteristics of CT signals

and CT LTI systems• Two types of LT

– Unilateral (one-sided): good for solving differential equations with initial conditions.

– Bilateral LT (two-sided): good for looking at the system characteristics such as stability, causality, and frequency response

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I. Laplace Transform• So far, signals represented using superpositions of complex sinusoids,

exp(jωt)• Now, let’s consider complex exponentials as basis

exp(st)=exp(s+jω)t=exp(st)cos(ωt)+jexp(st)sin(ωt)• Why? LT can be used to analyze a larger class of CT problems involving

signals that are not absolutely integrable• Remember, the FT does not exist for signals that are not absolutely

integrable• The properties of LT are very much similar to those of FT• Two types of LT

– Unilateral, or one sided: good for solving differential equations with initial conditions

– Bilateral, or two sided: good for looking at the system characteristics such as stability, causality, and frequency responses

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Laplace Transform

sine dampledlly exponentia}Im{cosine dampedlly exponentia}Re{

),sin()cos(

=

=

+=+=

st

st

ttst

ee

jstjetee wsww ss

Real and imaginary parts of the complex exponential est, where s = s + jw.

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• Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics.

• He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems.

• In statistics, the so-called Bayesian interpretation of probability was mainly developed by Laplace.

• He formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him.

• He restated and developed the nebular hypothesis of the origin of the solar system and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.

• He is remembered as one of the greatest scientists of all time, sometimes referred to as a French Newton or Newton of France, with a phenomenal natural mathematical faculty superior to any of his contemporaries.

Who is Laplace?

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Transfer Function

òò

òò

¥

¥-

-

¥

¥-

-

¥

¥-

-

¥

¥-

=\

=

=

-=

*==

=

tt

tt

tt

ttt

t

t

t

dehsH

dehe

deh

dtxh

txthtxHty

thetx

s

sst

ts

st

)()(

)(

)(

)()(

)()()}({)(

)(function response impulsean and )(Given

)(

Transfer function

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Bilateral Laplace Transform

ò

ò¥+¥-

¥¥-

-

=

=

jj

st

st

dsesXj

tx

dtetxsX

ssp

)(21)(

)()(

)()( sXtx «

• Signal x(t) is expressed as a weighted superposition of complex exponentials exp(st)

• In practice, we usually do not evaluate this integral directly (it requires a lecture on contour integration)

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Convergence

¥<ò¥¥-

- dtetx st |)(|

• We must have the above as a necessary condition for convergence of the LT

• The range of s for which the LT converges is termed the region of convergence (ROC)

• LT exists for signals that do not have a FT: i.e., within a certain range of s, we can ensure that x(t)exp(-st) is absolutely integrable.

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Example• FT of x(t)=exp(t)u(t) does not exist. Why?• If s>1, x(t)exp(- st)=exp((1-s)t)u(t) is absolutely integrable.• So the LT exists.

The Laplace transform applies to more general signals than the Fourier transform does. (a) Signal for which the Fourier transform does not exist.

(b) Attenuating factor associated with Laplace transform. (c) The modified signal x(t)e-st is absolutely integrable for s > 1.

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s-plane• Represent the complex frequency s graphically in terms of a complex

plane = s-plane• Horizontal axis represents the real part• Vertical axis represents the imaginary of s• If s=0, then X(jω)=X(s)|s=0. That is replace jw by s.

Right side of the s-plane

Left side of the s-plane

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Poles and Zeros

The s-plane. The horizontal axis is Re{s} and the vertical axis is Im{s}. Zeros are depicted at s = –1 and s = –4 ± 2j, and poles are depicted at

s = –3, s = 2 ± 3j, and s = 4.

ÕÕ

=

=

-

-= N

k k

Mk kM

ds

csbsX

1

1

)(

)()(

Zeros (“o”)

Poles“x”

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II. Unilateral LT• Since most of signals and systems we deal are causal signals and

systems, use unilateral LT is good enough; for noncausal signals and systems, use bilateral LT.

• Thus ULT is limited to causal signals: that is signals are zeros for time t<0

• In practice, LT means ULT• Unilateral LT is used to analyze the behavior of the causal system

described by a differential equations with initial conditions• Unilateral LT

ò¥ --= 0 )()( dtetxsX st

• 0- implies that we do not include discontinuites and impulses that occur at t=0

Anti-causal (noncausal) signals are not covered in this class.

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III. Some Properties of Unilateral LT• Linearity• Scaling• Time Shift• S-domain Shift• Convolution• Differentiation in the s-domain• Differentiation in the Time Domain

– General form of the differentiation property• Integration Property• Initial and Final Value Theorem

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Examples

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Examples

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Examples

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ExamplesSolving differential equations

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Examples

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ExamplesTransfer functions

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ExamplesThree Types of the Systems

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Some Signals and Their ROCs

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Bilateral LT: Causality

The relationship between the locations of poles and the impulse response in a causal system. (a) A pole in the left half of the s-plane corresponds to an exponentially decaying impulse response. (b) A pole in the right half of the s-plane corresponds to an exponentially increasing impulse response. The system is unstable in this case.

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Bilateral LT: Stability

The relationship between the locations of poles and the impulse response in a stable system. (a) A pole in the left half of the s-plane corresponds to a right-sided impulse

response. (b) A pole in the right half of the s-plane corresponds to an left-sided impulse response. In this case, the system is noncausal.

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Bilateral LT: Both Stable & Causal

A system that is both stable and causal must have a transfer function with all of its poles in the left half of the s-plane, as shown here.

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Inverse Systems