institute of engineering and management sciences

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INSTITUTE OF ENGINEERING AND MANAGEMENTSCIENCES

REPORT

LAPLACE TRANSFORM AND Z-TRANSFORM

SUBMITTED TO:

Dr, daness bill

SUBMITTED BY:ww.chin chon moon

TABLE OF CONTENTS

Laplace Transform Page No. History 3 Formal Definition 4

Unilateral & Bilateral

Laplace Transforms 5 Domain 6 Flowchart 6 Advantages 7 Inverse Laplace Transform 7 ROC 8 Derivation 9

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Applications 9 Appendix A 10

Z-Transform

History14

Formal Definition 14 ROC 16 Flowchart 21 Relationship To Fourier 21 Transfer Function 22 Zeros and Poles 22 Applications 23 Appendix B 24

References 27

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LAPLACE TRANSFORM

HISTORY

The Laplace transform is named in honor ofmathematician andastronomerPierre-Simon Laplace, who used the transform in hiswork on probability theory.

From 1744, Leonhard Eulerinvestigated integrals of the form:

As solutions of differential equations but did not pursue thematter very far. Joseph Louis Lagrange was an admirer of Euler and,in his work on integrating probability density functions, investigatedexpressions of the form:

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That are some modern historians have interpreted withinmodern Laplace transform theory.

These types of integrals seem first to have attracted Laplace'sattention in 1782 where he was following in the spirit of Euler in usingthe integrals themselves as solutions of equations. However, in 1785,Laplace took the critical step forward when, rather than just look for asolution in the form of an integral; he started to apply the transformsin the sense that was later to become popular. He used an integral ofthe form:

Akin to a Mellin transform, to transform the whole of adifference equation, in order to look for solutions of the transformedequation. He then went on to apply the Laplace transform in the sameway and started to derive some of its properties, beginning toappreciate its potential power.

Laplace also recognized that Joseph Fourier's method ofFourier series for solving the diffusion equation could only apply to alimited region of space as the solutions were periodic. In 1809,Laplace applied his transform to find solutions that diffusedindefinitely in space.

FORMAL DEFINITION:

The two main techniques in signal processing, convolution andFourier analysis, teach that a linear system can be completelyunderstood from its impulse or frequency response. This is a much

generalized approach, since the impulse and frequency responsescan be of nearly any shape or form. In fact, it is too general for manyapplications in science and engineering. Many of the parameters inour universe interact through differential equations. For example, thevoltage across an inductor is proportional to the derivative of thecurrent through the device. Likewise, the force applied to a mass isproportional to the derivative of its velocity. Physics is filled with these

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kinds of relations. The frequency and impulse responses of thesesystems cannot be arbitrary, but must be consistent with the solutionof these differential equations. This means that their impulseresponses can only consist of exponentials and sinusoids. TheLaplace transform is a technique for analyzing these special systemswhen the signals are continuous. The z-transform is a similartechnique used in the discrete case.

The Laplace transform of a functionf(t), defined for all realnumberst 0, is the function F(s), defined by:

The lower limit of 0 is short notation to mean

And assures the inclusion of the entire Dirac delta function (t)at 0 if there is such an impulse in f(t) at 0.

The parameters is in general complex:

This integral transform has a number of properties that make ituseful for analyzing lineardynamic systems. The most significantadvantage is that differentiation and integration become multiplicationand division, respectively; by s. (This is similar to the way thatlogarithms change an operation of multiplication of numbers toaddition of their logarithms.) This changes integral equations anddifferential equations to polynomial equations, which are much easierto solve. Once solved, use of the inverse Laplace transform revertsback to the time domain.

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UNILATERAL AND BILATERAL LAPLACE TRANSFORM

When one says "the Laplace transform" without qualification, theunilateral or one-sided transform is normally intended. The Laplace

transform can be alternatively defined as the bilateral Laplacetransform ortwo-sided Laplace transform by extending the limits ofintegration to be the entire real axis. If that is done the commonunilateral transform simply becomes a special case of the bilateraltransform where the definition of the function being transformed ismultiplied by the Heaviside step function.

The bilateral Laplace transform is defined as follows:

DOMAIN :

Time-Domain

Frequency-Domain

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Relationship between the time domain and the frequency domain. Note the * inthe time domain, denoting convolution

FLOW CHART OF SOLVING IVP BY LAPLACETRANSFORM:

(a) If we have the function g(t), then G(s) = G = {g(t)}.

(b) g(0) is the value of the function g(t) at t= 0.

(c) g'(0), g''(0),... are the values of the derivatives of the function at t=0.

Ifg(t) is continuous and g'(0), g''(0),... are finite, then

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(1)

(2) {g''(t)} = s2G s g(0) g'(0)

ADVANTAGES:

Solving a nonhomogeneous ODE does not require first solvingthe homogeneous ODE

Initial values are automatically taken care of

Complicated inputs r(t) (right sides of linear ODEs) can behandled very efficiently

INVERSE LAPLACE TRANSFORM:

In mathematics, the inverse Laplace transform ofF(s) is thefunction f(t) which has the property

That is the Laplace transform.

It can be proven, that if a function F(s) has the inverse Laplacetransform f(t), i.e. fis a piecewise continuous and exponentiallyrestricted real function fsatisfying the condition

Then f(t) is uniquely determined (considering functions whichdiffer from each other only on a point set having Lebesgue measurezero as the same).

The Laplace transform and the inverse Laplace transformtogether have a number of properties that make them useful foranalysing linear dynamic systems.

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An integral formula for the inverse Laplace transform, called theBromwich integral, the Fourier-Mellin integral, and Mellin'sinverse formula, is given by the line integral:

Where the integration is done along the vertical line Re(s) = inthe complex plane such that is greater than the real part of allsingularities ofF(s). This ensures that the contour path is in theregion of convergence. If all singularities are in the left half-plane,then can be set to zero and the above inverse integral formulaabove becomes identical to the inverse Fourier transform.

In practice, computing the complex integral can be done by using theCauchy residue theorem.

It is named afterHjalmar Mellin (Finland 1854 1933), JosephFourier, and Thomas John I'Anson Bromwich (1875-1929).

Region Of C onvergence (ROC):

The Laplace transform F(s) typically exists for all complexnumbers such that Re{s} > a, where a is a real constant whichdepends on the growth behavior of f(t), whereas the two-sidedtransform is defined in a range a < Re{s} < b. The subset of values ofs for which the Laplace transform exists is called the region ofconvergence (ROC) or the domain of convergence. In the two-sidedcase, it is sometimes called the strip of convergence.

The integral defining the Laplace transform of a function mayfail to exist for various reasons. For example, when the function hasinfinite discontinuities in the interval of integration, or when it

increases so rapidly that e pt

cannot damp it sufficiently forconvergence on the interval to take place. There are no specificconditions that one can check a function against to know in all casesif its Laplace transform can be taken, other than to say the definingintegral converges. It is however possible to give theorems on caseswhere it may or may not be taken.

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Signal Processing

Probability Theory

APPENDIX A

LAPLACE TABLE

ID Function Time domain Laplace s-domain Region of convergenc

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eforcausalsystems

1 ideal delay

1a unit impulse 1

2

delayed nthpowerwith

frequencyshift

2anth power

( for integern )

2a.1

qth power( for real q )

2a.2

unit step

2bdelayed unit

step

2c ramp

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2d

nth powerwith

frequencyshift

2d.1

exponentialdecay

3exponentialapproach

4 sine

5 cosine

6hyperbolic

sine

7hyperbolic

cosine

8Exponentially-decayingsine wave

9

Exponentiall

y-decayingcosine wave

10 nth root

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11natural

logarithm

12

Besselfunction

of the firstkind,

of ordern

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ModifiedBesselfunction

of the first

kind,of ordern

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Besselfunctionof the

second kind,of order 0

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ModifiedBesselfunctionof the

second kind,of order 0

16Error

function

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Z-TRANSFORM

In mathematics and signal processing, the Z-transformconverts a discretetime-domain signal, which is a sequence ofreal orcomplex numbers, into a complex frequency-domain representation.

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It is like a discrete equivalent of the Laplace transform. This similarityis explored in the theory oftime scale calculus.

HISTORY :

The Z-transform was introduced, under this name, by Ragazziniand Zadeh in 1952.

The modified oradvanced Z-transform was later developed byE. I. Jury, and presented in his book Sampled-Data ControlSystems (John Wiley & Sons 1958). The idea contained withinthe Z-transform was previously known as the "generatingfunction method".

FORMAL DEFINITION:

The Z-transform, like many other integral transforms, can bedefined as either a one-sided or two-sided transform.

Bilateral Z-transform

The bilateral or two-sided Z-transform of a discrete-time signal x[n]is the function X(z) defined as

Where n is an integer and z is, in general, a complex number:

z = Aej (OR)z = A(cos + jsin)

Where A is the magnitude of z, and is the complex argument (alsoreferred to as angle or phase) in radians.

Unilateral Z-transform

Alternatively, in cases where x[n] is defined only for n 0, thesingle-sided or unilateral Z-transform is defined as

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In signal processing, this definition is used when the signal is causal.

An important example of the unilateral Z-transform is the probability-generating function, where the component x[n] is the probability that adiscrete random variable takes the value n, and the function X(z) isusually written as X(s), in terms of s = z 1. The properties of Z-transforms (below) have useful interpretations in the context ofprobability theory.

In geophysics, the usual definition for the Z-transform is apolynomial in z as opposed to z 1. This convention is used byRobinson and Treitel and by Kanasewich. The geophysical definitionis

The two definitions are equivalent; however, the difference results ina number of changes. For example, the location of zeros and polesmove from inside the unit circle, using one definition, to outside theunit circle, using the other definition (and vice versa).

Region of C onvergence (ROC):

The region of convergence (ROC) is the set of points in thecomplex plane for which the Z-transform summation converges.

(No ROC):

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Let . Expanding on the interval it becomes

Looking at the sum

Therefore, there are no such values of that satisfy this condition.

(Causal ROC):

ROC shown in blue, the unit circle as a dotted grey circle and

the circle is shown as a dashed black circle. Let

(where uis the Heaviside step function). Expanding

on the interval it becomes

Looking at the sum

The last equality arises from the infinite geometric series and

the equality only holds if which can be rewritten in terms

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of as . Thus, the ROC is . In this case the ROC isthe complex plane with a disc of radius 0.5 at the origin "punchedout".

(Anticausal ROC):

ROC shown in blue, the unit circle as a dotted grey circle and

the circle is shown as a dashed black circle

Let (where uis the Heaviside step function).

Expanding on the interval it becomes

Looking at the sum

Using the infinite geometric series, again, the equality only holds if

which can be rewritten in terms of as . Thus, the

ROC is . In this case the ROC is a disc centered at the originand of radius 0.5.

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What differentiates this example from the previous example isonlythe ROC. This is intentional to demonstrate that the transformresult alone is insufficient.

C onclusion:CAUSAL AND ANTICAUSAL Z Transform clearly show that the Z-

transform of is unique when and only when specifying theROC. Creating the pole-zero plot for the causal and anticausal caseshow that the ROC for either case does not include the pole that is at0.5. This extends to cases with multiple poles: the ROC will nevercontain poles.

the causal system yields an ROC that includes while the

anticausal system yields an ROC that includes .

ROC shown as a blue ring

In systems with multiple poles it is possible to have an ROC that

includes neither nor . The ROC creates a circular

band. For example, have poles at

0.5 and 0.75. The ROC will be , which includesneither the origin nor infinity. Such a system is called a mixed-

causality system as it contains a causal term and an

anticausal term .

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The stability of a system can also be determined by knowing the ROC

alone. If the ROC contains the unit circle (i.e., ) then thesystem is stable. In the above systems the causal system (Example

2) is stable because contains the unit circle.

If you are provided a Z-transform of a system without an ROC (i.e.,

an ambiguous ) you can determine a unique provided youdesire the following:

Stability Causality

If you need stability then the ROC must contain the unit circle. If youneed a causal system then the ROC must contain infinity and the

system function will be a right-sided sequence. If you need ananticausal system then the ROC must contain the origin and thesystem function will be a left-sided sequence. If you need both,stability and causality, all the poles of the system function must beinside the unit circle.

FLOW CHART FOR SOLVING PARTIAL FRACTIONEXPANSION

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EXAMPLE OF PARTIAL FRACTION EXPASION

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Relationship to Fourier:

The Z-transform is a generalization of the discrete-time Fouriertransform (DTFT). The DTFT can be found by evaluating the Z-

transform at or, in other words, evaluated on the unitcircle. In order to determine the frequency response of the system theZ-transform must be evaluated on the unit circle, meaning that the

system's region of convergence must contain the unit circle.Otherwise, the DTFT of the system does not exist.

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Transfer F unction:

Taking the Z-transform of the above equation (using linearityand time-shifting laws) yields

and rearranging results in

Zeros and P oles:

From the fundamental theorem of algebra the numeratorhas Mroots (corresponding to zeros of H) and the denominatorhas N roots(corresponding to poles). Rewriting the transfer function in terms ofpoles and zeros

Where is the zero and is the pole. The zeros andpoles are commonly complex and when plotted on the complex plane(z-plane) it is called the pole-zero plot.

In simple words, zeros are the solutions to the equationobtained by setting the numerator equal to zero, while poles are thesolutions to the equation obtained by setting the denominator equal tozero.

In addition, there may also exist zeros and poles at z= 0 and. If we take these poles and zeros as well as multiple-orderzeros and poles into consideration, the number of zeros and polesare always equal.

By factoring the denominator, partial fraction decomposition canbe used, which can then be transformed back to the time domain.

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Doing so would result in the impulse response and the linear constantcoefficient difference equation of the system.

If such a system is driven by a signal then the output

is . By performing partial fraction decompositionon and then taking the inverse Z-transform the output can

be found. In practice, it is often useful to fractionally decompose

before multiplying that quantity by to generate a form of whichhas terms with easily computable inverse Z-transforms.

APPLICATIONS

Mathematics

Physics

Optics

Electrical Engineering

Control Engineering

Signal Processing

Probability Theory

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APPENDIX B

Table of common Z-transform pairs

Signal,x[n] Z-transform,X(z) ROC

1

2

3

4

5

6

7

8

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9

10

11

1

2

13

14

15

16

17

18

28


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19

20

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REFERENCES

en.wikipedia.org/wiki/Laplace_transform mathworld.wolfram.com/LaplaceTransform.html

www.didaktik.itn.liu.se/thesis/margarita_thesis3.pdf www.stanford.edu/~boyd/ee102/laplace.pdf www.physicsforums.com/ en.wikipedia.org/wiki/Z-transform www.dspguide.com/ math.fullerton.edu fourier.eng.hmc.edu embeddedsystemdesign.blogspot.com dspcan.homestead.com

www.vocw.edu www.csupomona.edu www.intmath.com www.physicsforums.com

CONSULTED BOOKS

Erwin Kreyszig, Advanced Engineering Mathematics, 5th

Edition, Prentice Hall, 1988, Page no.230, 232, 259, 272

Alan V.Oppenheim, Alan S.Willsky, Signals and Systems,Seventh Edition, Prentice Hall, Page no.175

B.P. Lathi, Signal Processing and Linear Systems,OxfordUniversity press,Page no.163
http://www.didaktik.itn.liu.se/thesis/margarita_thesis3.pdfhttp://www.stanford.edu/~boyd/ee102/laplace.pdfhttp://www.physicsforums.com/http://www.dspguide.com/http://www.vocw.edu/http://www.csupomona.edu/http://www.intmath.com/http://www.physicsforums.com/http://www.didaktik.itn.liu.se/thesis/margarita_thesis3.pdfhttp://www.stanford.edu/~boyd/ee102/laplace.pdfhttp://www.physicsforums.com/http://www.dspguide.com/http://www.vocw.edu/http://www.csupomona.edu/http://www.intmath.com/http://www.physicsforums.com/

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