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Page 1: Application of Convolution Theorem

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ISSN No: 2456

InternationalResearch

Application of Convolution Theorem

Associate Professor, Yogananda College of Engineering & Technology, Jammu

ABSTRACT Generally it has been noticed that differential equation is solved typically. The Laplace transformation makes it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is usesolving the time domain function by converting it into frequency domain. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. In this paper we will discuss how to foconvolution theorem holds the Commutative property, Associative Property and Distributive Property.

Keywords: Laplace transformation, Inverse Laplace transformation, Convolution theorem

INTRODUCTION:

Laplace transformation is a mathematical tool which is used in the solving of differential equations by converting it from one form into another form. Generally it is effective in solving linear differential equation either ordinary or partial. It reduces ordinary differential equation into algebraic equation. Ordinary linear differential equation with constant coefficient and variable coefficient can be easily solved by the Laplace transformation method without finding the generally solution and the arbconstant. It is used in solving physical problems. this involving integral and ordinary differential equation with constant and variable coefficient.

It is also used to convert the signal system in frequency domain for solving it on a simple and easy way. It has some applications in nearly all engineering disciplines, like System Modeling, Analysis of Electrical Circuit, Digital Signal Processing, Nucle

@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 4 | May-Jun 2018

ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume

International Journal of Trend in Scientific Research and Development (IJTSRD)

International Open Access Journal

Application of Convolution Theorem

Dr. Dinesh Verma Yogananda College of Engineering & Technology, Jammu

Generally it has been noticed that differential equation solved typically. The Laplace transformation makes

it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is used in solving the time domain function by converting it into frequency domain. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. In this paper we will discuss how to follow convolution theorem holds the Commutative property, Associative Property and Distributive Property.

Laplace transformation, Inverse Laplace

Laplace transformation is a mathematical tool which is used in the solving of differential equations by converting it from one form into another form. Generally it is effective in solving linear differential equation either ordinary or partial. It reduces an ordinary differential equation into algebraic equation. Ordinary linear differential equation with constant coefficient and variable coefficient can be easily solved by the Laplace transformation method without finding the generally solution and the arbitrary constant. It is used in solving physical problems. this involving integral and ordinary differential equation

It is also used to convert the signal system in frequency domain for solving it on a simple and easy way. It has some applications in nearly all engineering disciplines, like System Modeling, Analysis of Electrical Circuit, Digital Signal Processing, Nuclear

Physics, Process Controls, Applications in Probability, Applications in Physics, Applications in Power Systems Load Frequency Control etc.

DEFINITION

Let F (t) is a well defined function of t for all t The Laplace transformation of F (t), denotedor L {F (t)}, is defined as

L {F (t)} =∫ 𝑒 𝐹(𝑑

Provided that the integral exists, i.e. convergent. If the integral is convergent for some value ofLaplace transformation of F (t) exists otherwise not. Where 𝑝 the parameter which may be real or complex number and L is is the Laplace transformation operator.

The Laplace transformation of F (t) i.e.

∫ 𝑒 𝐹(𝑑)𝑑𝑑 exists for 𝑝>a, if

F (t) is continuous andlim β†’

should however, be keep in mind that above condition are sufficient and not necessary.

Inverse Laplace Transformation

Definition:

If be the Laplace Transformation of a function F(t),then F(t) is called the Inverse Laplace transformation of the function f(p) and is written as 𝐹(𝑑) = 𝐿 {𝑓(𝑝)} , π‘Šβ„Žπ‘’π‘Ÿπ‘’ 𝐿

π‘–π‘›π‘£π‘’π‘Ÿπ‘ π‘’ πΏπ‘Žπ‘π‘™π‘Žπ‘π‘’ π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘“π‘œπ‘Ÿπ‘šπ‘Žπ‘‘π‘–π‘œπ‘›

General Property of inverse Laplace transformation,

Jun 2018 Page: 981

6470 | www.ijtsrd.com | Volume - 2 | Issue – 4

Scientific (IJTSRD)

International Open Access Journal

Yogananda College of Engineering & Technology, Jammu, India

Physics, Process Controls, Applications in Probability, Applications in Physics, Applications in Power Systems Load Frequency Control etc.

Let F (t) is a well defined function of t for all t β‰₯ 0. The Laplace transformation of F (t), denoted by f (𝑝)

(𝑑)𝑑𝑑 = 𝑓(𝑝)

Provided that the integral exists, i.e. convergent. If the integral is convergent for some value of 𝑝 , then the Laplace transformation of F (t) exists otherwise not.

the parameter which may be real or complex number and L is is the Laplace transformation

The Laplace transformation of F (t) i.e. >a, if

{𝑒 𝐹(𝑑)} is finite. It should however, be keep in mind that above condition are sufficient and not necessary.

Inverse Laplace Transformation

If be the Laplace Transformation of a function F(t),then F(t) is called the Inverse Laplace

he function f(p) and is written as 𝑖𝑠 π‘π‘Žπ‘™π‘™π‘’π‘‘ π‘‘β„Žπ‘’

π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘“π‘œπ‘Ÿπ‘šπ‘Žπ‘‘π‘–π‘œπ‘›.

General Property of inverse Laplace transformation,

Page 2: Application of Convolution Theorem

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470

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(1) π‘³π’Šπ’π’†π’‚π’“π’•π’š π‘·π’“π’π’‘π’†π’“π’•π’š:

𝐼𝑓 𝑐 π‘Žπ‘›π‘‘ 𝑐 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘π‘  π‘Žπ‘›π‘‘ 𝑖𝑓

𝐿 {𝑓(𝑝)} = 𝐹(𝑑)π‘Žπ‘›π‘‘πΏ{𝐺(𝑑)} = 𝑔(𝑝)

π‘‘β„Žπ‘’π‘›,

𝐿 [𝑐 𝑓(𝑝) + 𝑐 𝑔(𝑝)]= 𝑐 𝐿 [𝑓(𝑝)] + 𝑐 𝐿 [𝑔(𝑝)]

(2)πΉπ‘–π‘Ÿπ‘ π‘‘ π‘†β„Žπ‘–π‘“π‘‘π‘–π‘›π‘” π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦:

𝑖𝑓𝐿 {𝑓(𝑝)} = 𝐹(𝑑), π‘‘β„Žπ‘’π‘›

𝐿 {𝑓(𝑝 βˆ’ π‘Ž)} = 𝑒 𝐹(𝑑), (3)πΆβ„Žπ‘Žπ‘›π‘”π‘’ π‘œπ‘“ π‘†π‘π‘Žπ‘™π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦:

𝑖𝑓𝐿 {𝑓(𝑝)} = 𝐹(𝑑), π‘‘β„Žπ‘’π‘›

𝐿 {𝑓(π‘Žπ‘)} =1

π‘ŽπΉ

𝑑

π‘Ž,

(4)πΌπ‘›π‘£π‘’π‘Ÿπ‘ π‘’ πΏπ‘Žπ‘π‘™π‘Žπ‘π‘’ π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘“π‘œπ‘Ÿπ‘šπ‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“

π‘‘π‘’π‘Ÿπ‘–π‘£π‘Žπ‘‘π‘–π‘£π‘’:

𝑖𝑓𝐿 {𝑓(𝑝)} = 𝐹(𝑑), π‘‘β„Žπ‘’π‘›

𝐿 βˆ’{ ( )}

= 𝑑𝐹(𝑑)

𝐿 βˆ’{ ( )}

= (βˆ’1) 𝑑 𝐹(𝑑),

π‘€β„Žπ‘Ÿπ‘’π‘Ÿ 𝑛 = 123 … … …

(5)πΌπ‘›π‘£π‘’π‘Ÿπ‘ π‘’ πΏπ‘Žπ‘π‘™π‘Žπ‘π‘’ π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘“π‘œπ‘Ÿπ‘šπ‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“

π‘–π‘›π‘‘π‘’π‘”π‘Ÿπ‘Žπ‘™:

𝑖𝑓𝐿 {𝑓(𝑝)} = 𝐹(𝑑), π‘‘β„Žπ‘’π‘›

𝐿 𝑓(𝑝)𝑑𝑝 = 𝐹(𝑑)

𝑑

(6)πΌπ‘›π‘£π‘’π‘Ÿπ‘ π‘’ πΏπ‘Žπ‘π‘™π‘Žπ‘π‘’ π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘“π‘œπ‘Ÿπ‘šπ‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“

π‘‘π‘–π‘£π‘–π‘ π‘–π‘œπ‘› 𝑏𝑦 𝑝:

𝑖𝑓𝐿 {𝑓(𝑝)} = 𝐹(𝑑), π‘‘β„Žπ‘’π‘›

𝐿𝐹(𝑝)

𝑝 = 𝑓(𝑒)𝑑𝑒

The convolution of two given functions acting an essential role in a number of physical applications. It is usually convenient to determination a Laplace transformation into the product of two transformations

when the inverse transform of both transforms are known. Convolution is used to get inverse Laplace transformation in solving differential equations and integral equations.

If 𝐻 (𝑑)π‘Žπ‘›π‘‘ 𝐻 (𝑑) 𝑏𝑒 π‘‘π‘€π‘œ π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘π‘™π‘Žπ‘ π‘  𝐴

π‘Žπ‘›π‘‘ π‘–π‘“β„Ž (𝑠) = 𝐿{𝐻 (𝑑)} , β„Ž (𝑠) = 𝐿{𝐻 (𝑑)}.

π‘‡β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘π‘œπ‘›π‘£π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘‘β„Žπ‘’π‘ π‘’ π‘‘π‘€π‘œ

π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘  𝐻 (𝑑)π‘Žπ‘›π‘‘ 𝐻 (𝑑) ,

𝑑 > 0 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 π‘‘β„Žπ‘’ π‘–π‘›π‘‘π‘’π‘”π‘Ÿπ‘Žπ‘™

{𝐻 βˆ— 𝐻 }(𝑑) = 𝐻 (𝑑)𝐻 (𝑑 βˆ’ 𝑦)𝑑𝑦

π‘œπ‘Ÿ 𝐻 (𝑑 βˆ’ 𝑦)𝐻 (𝑦)𝑑𝑦

Which of course exists if 𝐻 (𝑑)π‘Žπ‘›π‘‘ 𝐻 (𝑑) are piecewise continuous .the relation is called the convolution or falting of 𝐻 (𝑑)π‘Žπ‘›π‘‘ 𝐻 (𝑑).

Proof of convolution theorem:

By the definition of Laplace Transformation 𝐿( 𝐻 βˆ—

𝐻 )(𝑑) = ∫ 𝑒 {(𝐻 βˆ— 𝐻 )𝑑}𝑑𝑑

= 𝑒 𝐻 (𝑑)𝐻 (𝑑 βˆ’ 𝑦)𝑑𝑦 𝑑𝑑

Where the double integral is taken over the infinite region in the first quadrant deceitful linking the limit𝑦 = 0 π‘‘π‘œ 𝑦 = 𝑑.

Now order of integral are changing

𝑒 𝐻 (𝑦)𝑑𝑦 𝑒 ( )𝐻 (𝑑 βˆ’ 𝑦)𝑑𝑑

𝑑 βˆ’ 𝑦 = 𝑒 β‡’ 𝑑𝑑 = 𝑑𝑒

π‘€β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ 𝑑 𝑖𝑠 𝑦 π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ 𝑒

𝑖𝑠 0 π‘Žπ‘›π‘‘ π‘€β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ 𝑑 𝑖𝑠 ∞ π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’

π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ 𝑒 𝑖𝑠 ∞.

π‘π‘œπ‘€ π‘“π‘Ÿπ‘œπ‘š π‘Žπ‘π‘œπ‘£π‘’

𝑒 𝐻 (𝑦)𝑑𝑦 𝑒 𝐻 (𝑒)𝑑𝑒

β„Ž (𝑝)β„Ž (𝑝)

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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470

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Hence

𝐿( 𝐻 βˆ— 𝐻 )(𝑑) = β„Ž (𝑝)β„Ž (𝑝)

Properties of convolution theorem: convolution theorem holds the following Properties:

(1) Commutative Property: property states that there is no alter in result from side to side the numbers in an appearance are exchange. Commutative property holds for addition and multiplication but not for subtraction and division.

π΄π‘‘π‘‘π‘–π‘‘π‘–π‘œπ‘› 𝐻 + 𝐻 = 𝐻 + 𝐻

π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘› 𝐻 βˆ’π» β‰  𝐻 βˆ’ 𝐻

π‘€π‘’π‘™π‘‘π‘–π‘π‘™π‘’π‘π‘Žπ‘‘π‘–π‘œπ‘› 𝐻 βˆ— 𝐻 = 𝐻 βˆ— 𝐻

π·π‘–π‘£π‘–π‘ π‘–π‘œπ‘› 𝐻 Γ· 𝐻 β‰  𝐻 Γ· 𝐻

Now 𝑀𝑒 𝑀𝑖𝑙𝑙 π‘π‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘’ πΆπ‘œπ‘šπ‘šπ‘’π‘‘π‘Žπ‘‘π‘–π‘£π‘’

π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘“π‘œπ‘Ÿ π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’π‘π‘Žπ‘‘π‘–π‘œπ‘›

{𝐻 βˆ— 𝐻 } = {𝐻 βˆ— 𝐻 }

𝑷𝒓𝒐𝒐𝒇 (𝟏):

By the definition of convolution theorem

𝐻 βˆ— 𝐻 = ∫ 𝐻 (𝑑)𝐻 (𝑑 βˆ’ 𝑦)𝑑𝑦

𝐿𝑒𝑑 𝑑 βˆ’ 𝑦 = 𝑒 β‡’ 𝑑𝑑 = βˆ’π‘‘π‘’,

π‘€β„Žπ‘’π‘› the π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ 𝑦 𝑖𝑠 0 π‘‘β„Žπ‘’π‘› 𝑒 𝑖𝑠 𝑦

π‘Žπ‘›π‘‘ π‘€β„Žπ‘’π‘› 𝑦 𝑖𝑠 𝑑 π‘‘β„Žπ‘’π‘› 𝑒 𝑖𝑠 0.

𝐻 βˆ— 𝐻 = ∫ 𝐻 (𝑑 βˆ’ 𝑒)𝐻 (𝑒)𝑑𝑒

𝐻 βˆ— 𝐻 = ∫ 𝐻 (𝑑 βˆ’ 𝑒)𝐻 (𝑒)𝑑𝑒

𝐻 βˆ— 𝐻 = ∫ 𝐻 (𝑒)𝐻 (𝑑 βˆ’ 𝑒)𝑑𝑒

𝐻 βˆ— 𝐻 = 𝐻 βˆ— 𝐻

The Convolution of 𝐻 π‘Žπ‘›π‘‘ 𝐻 follow the commutative property.

(2) Associative Property: Associative Property states that the order of grouping the numbers does not matter. This law holds for addition and multiplication but not for subtraction and division.

π΄π‘‘π‘‘π‘–π‘‘π‘–π‘œπ‘› 𝐻 +( 𝐻 + 𝐻 ) = (𝐻 + 𝐻 ) + 𝐻

π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘› 𝐻 βˆ’( 𝐻 βˆ’ 𝐻 ) = (𝐻 βˆ’ 𝐻 ) βˆ’ 𝐻 π‘€π‘’π‘™π‘‘π‘–π‘π‘™π‘’π‘π‘Žπ‘‘π‘–π‘œπ‘› 𝐻 βˆ— ( 𝐻 βˆ— 𝐻 ) = (𝐻 βˆ— 𝐻 ) βˆ— 𝐻

π·π‘–π‘£π‘–π‘ π‘–π‘œπ‘› 𝐻 Γ· ( 𝐻 Γ· 𝐻 ) = (𝐻 Γ· 𝐻 ) Γ· 𝐻

Now we will see that how to convolution theorem follow the Associative property for multiplication

𝐻 βˆ— ( 𝐻 βˆ— 𝐻 ) = (𝐻 βˆ— 𝐻 ) βˆ— 𝐻

𝑙𝑒𝑑 𝐻 βˆ— 𝐻 = 𝐻

π‘π‘œπ‘€, 𝐻 = 𝐻 βˆ— 𝐻 = 𝐻 (𝑦)𝐻 (𝑑 βˆ’ 𝑦)𝑑𝑦

From above the commutative property

𝐻 βˆ— 𝐻 = 𝐻 βˆ— 𝐻

π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ 𝐻 βˆ— 𝐻 = 𝐻 βˆ— 𝐻

π‘†π‘œ, 𝐻 βˆ— 𝐻 = 𝐻 (𝑦)𝐻 (𝑑 βˆ’ 𝑦)𝑑𝑦

Hence , 𝐻 βˆ— 𝐻 = 𝐻 (𝑧)𝐻 (𝑑 βˆ’ 𝑧)𝑑𝑧

𝐻 βˆ— 𝐻 = 𝐻 (𝑧) 𝐻 (𝑦)𝐻 (𝑑 βˆ’ 𝑧

βˆ’ 𝑦)𝑑𝑦 𝑑𝑧

Change of order of integration

𝐻 βˆ— 𝐻 = ∫ 𝐻 (𝑧) ∫ 𝐻 (𝑦)𝐻 (𝑑 βˆ’ 𝑧 βˆ’

𝑦)𝑑𝑧 𝑑𝑦 = 𝐻 βˆ— (𝐻 βˆ— 𝐻 )

𝐻𝑒𝑛𝑐𝑒, 𝐻 βˆ— ( 𝐻 βˆ— 𝐻 ) = (𝐻 βˆ— 𝐻 ) βˆ— 𝐻

Distributive Property: The property with respect to addition is used to eliminate the bracket in an expression. The distributive property states that each term inside the bracket should be multiplied with the term outside. The property is very useful while simplifying the expressions and solving the complicated equations.

Distributive property over addition

𝐻 βˆ— (𝐻 + 𝐻 ) = 𝐻 βˆ— 𝐻 + 𝐻 βˆ— 𝐻

Here the terms which are inside the bracket (𝐻 π‘Žπ‘›π‘‘ 𝐻 ) are multiplied with the external terms (while is 𝐻 ).

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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470

@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 4 | May-Jun 2018 Page: 984

Now we will see that how to convolution theorem follow the Distributive property for multiplication

𝐻 βˆ— (𝐻 + 𝐻 ) = ∫ 𝐻 (𝑦)[𝐻 (𝑑 βˆ’ 𝑦) + 𝐻 (𝑑 βˆ’

𝑦)]𝑑𝑦

= ∫ 𝐻 (𝑦)𝐻 (𝑑 βˆ’ 𝑦)𝑑𝑦 + ∫ 𝐻 (𝑦)𝐻 (𝑑 βˆ’

𝑦)𝑑𝑦

𝐻 βˆ— 𝐻 + 𝐻 βˆ— 𝐻

𝐻𝑒𝑛𝑐𝑒 , 𝐻 βˆ— (𝐻 + 𝐻 ) = 𝐻 βˆ— 𝐻 + 𝐻 βˆ— 𝐻

CONCLUSION:

In this paper we have discussed the Applications of convolution theorem of Laplace transformation i.e. how to follow the convolution theorem holds the Commutative property, Associative Property and Distributive Property. The primary use of Laplace transformation is converting a time domain functions into frequency domain function. Here, Some Property of inverse Laplace transformations like, linearity property, First shifting property, Change of scale property, Inverse Laplace transformation of derivative etc. has been discussed.

REFERENCES:

1. B. V. Ramana, Higher Engineering Mathematics. 2. Dr. B. S. Grewal, Higher Engineering

Mathematics. 3. Dr. S. K. Pundir, Engineering Mathematics with

gate tutor. 4. Erwin Kreyszig, Advanced Engineering

Mathematics, Wiley, 1998. 5. J.L. Schiff, The Laplace Transform: Theory and

Applications, Springer Science and Business Media (1999).

6. Advanced engineering mathematics seventh edition, peter v. Oneil.

7. H. K. Dass, ’’Higher engineering Mathematics’’ S. Chand and company limited, New Delhi


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