بسم ا الرمحن الرحيم

LecturerAhmed H. Hadi

Training package inConvolution, and Analogue Correlation

For students of second class

Ministry of Higher Education and Scientific ResearchFoundation of Technical Education

Technical College / Al-Najaf

1 / B –Rationale :-

Convolution is a mathematical operation whichis used to express the input/output relationshipin a linear time invariant system.

The correlation to measure of match or similaritybetween one signal and time delayed version ofanother signal

/ Over view1/ Over view1

1 / A –Target population :-For students of second class inCommunications Techniques Engineering Department

1 / D –Objectives:-Convolution to express the input/output relationship in a lineartime invariant system.

Correlation to measure of match or similarity between one signaland time delayed version of another signal

1 / C –Central Idea :-

Convolution Integral (Direct convolution)Graphical Convolution MethodsAnalogue correlation

2/ Pre test :-2/ Pre test :-

The convolution of two square pulse is

(a) unit step function (b) square pulse

(c) triangular waveform (d) sinusoidal waveform

The convolution of two square pulse is

(a) unit step function (b) square pulse

(c) triangular waveform (d) sinusoidal waveform

Multiple Choice Questions With Answer

١١ Convolution of a function g(t) with (t-t0) is equal to

(a) g(t0) (b) g(t-t0)

(c) g(t+t0) (d) 0

Convolution of a function g(t) with (t-t0) is equal to

(a) g(t0) (b) g(t-t0)

(c) g(t+t0) (d) 0

٢٢

3/ Performance Objectives :-3/ Performance Objectives :-

• Commonly used in engineering, science, math

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Convolution Integral

Convolution IntegralConvolution properties

– Commutative: f1(t) * f2(t) = f2(t) * f1(t)

– Distributive: f1(t) * [f2(t) + f3(t)] = f1(t) * f2(t) + f1(t) * f3(t)

– Associative: f1(t) * [f2(t) * f3(t)] = [f1(t) * f2(t)] * f3(t)

– Shift: If f1(t) * f2(t) = c(t), thenf1(t) * f2(t - T) = f1(t - T) * f2(t) = c(t - T).

– Convolution with impulse, f(t) * δ(t) = f(t)

– Convolution with shifted impulse, f(t) * δ(t-T) = f(t-T)

Graphical Convolution Methods

•From the convolution integral, convolution isequivalent to

Rotating one of the functions about the y axisShifting it by tMultiplying this flipped, shifted function with theother functionCalculating the area under this productAssigning this value to f1(t) * f2(t) at t

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3

t2

f(t)2

-2 + t 2 + t

g(t-t)

*

2

2

t

f(t)

-2 2

3

t

g(t)

Graphical Convolution Example• Convolve the following two functions:

• Replace t withtin f(t) and g(t)• Choose to flip and slide g(t) since it is simpler and symmetric• Functions overlap like this:

62

326

2

232

23)2(3

222

0

22

0

t

tt

d

tt

3

t2

f(t)2

-2 + t 2 + t

g(t-t)

3

t2

f(t)2

-2 + t 2 + t

g(t-t)

Graphical Convolution Example

• Convolution can be divided into 5 partsI. t < -2

• Two functions do not overlap• Area under the product of the

functions is zeroII. -2 t < 0

• Part of g(t) overlaps part of f(t)• Area under the product of the

functions is

Graphical Convolution Example

III. 0 t < 2• Here, g(t) completely overlaps f(t)• Area under the product is just

IV. 2 t < 4• Part of g(t) and f(t) overlap• Calculated similarly to -2 t < 0

V. t 4• g(t) and f(t) do not overlap• Area under their product is zero

622

323

2

0

22

0

d

3

t2

f(t)2

-2 + t 2 + t

g(t-t)

3

t2

f(t)2

-2 + t 2 + t

g(t-t)

Graphical Convolution Example

• Result of convolution (5 intervals of interest):

4for0

42for24122

320for6

02for62

32for0

)(*)()(

2

2

t

ttt

t

tt

t

tgtfty

t

y(t)

0 2 4-2

6

Analogue correlationAnalogue correlation

Analogue correlation is like analogue convolution except that wedo not reverse one of the signals before we integrate. Using Rxy(t)to mean the cross−correlation of x(t) with y(t), the correlationintegral is:

Analogue correlation is like analogue convolution except that wedo not reverse one of the signals before we integrate. Using Rxy(t)to mean the cross−correlation of x(t) with y(t), the correlationintegral is:

As usual, t is the real−time variable. The time variable t representsthe relative displacement of x(t) from y(t) at which we aretesting for similarity.For an integration in progress, t is a constant, and is the lateralshift of x(t) for this comparison. But x(t) is not reversed in time.

As usual, t is the real−time variable. The time variable t representsthe relative displacement of x(t) from y(t) at which we aretesting for similarity.For an integration in progress, t is a constant, and is the lateralshift of x(t) for this comparison. But x(t) is not reversed in time.

To illustrate, we will use the two signals,To illustrate, we will use the two signals,

The sum of areas becomes (2 1 2) =4, plus (1 3 2) =6, a total of 10,which is Rxy(4). After several other similar calculations, we get the resultshown here (Fig. below ). Careful checking of this example should resultin a good understanding of analogue cross−correlation.

The sum of areas becomes (2 1 2) =4, plus (1 3 2) =6, a total of 10,which is Rxy(4). After several other similar calculations, we get the resultshown here (Fig. below ). Careful checking of this example should resultin a good understanding of analogue cross−correlation.

Analogue auto−correlation is no different, except that we use thesame signal twice over. This (Fig below ) is a plot of R xx(t), andit has the expected peak at t = 0, and the expected evensymmetry as well. Also, the information that R xx(t) gives usaboutx(t) is the same as in the digital case.

Analogue auto−correlation is no different, except that we use thesame signal twice over. This (Fig below ) is a plot of R xx(t), andit has the expected peak at t = 0, and the expected evensymmetry as well. Also, the information that R xx(t) gives usaboutx(t) is the same as in the digital case.

١١

٢٢

List two Convolution properties

Quiz /Quiz /

Define the convolutionDefine the convolution

Commutative: f1(t) * f2(t) = f2(t) * f1(t)

Distributive: f1(t) * [f2(t) + f3(t)] = f1(t) * f2(t) + f1(t) * f3(t)

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-/ Post test :5

Multiple Choice Questions With Answer

1.Convolution of a function with a unit impulse functionis

(a) zero (b) function itself

(c) impulse function (d) infinite

Multiple Choice Questions With Answer

1.Convolution of a function with a unit impulse functionis

(a) zero (b) function itself

(c) impulse function (d) infinite

2. Define analogue correlationAnalogue correlation is like analogue convolution except that we donot reverse one of the signals before we integrate. Using Rxy(t) tomean the cross−correlation of x(t) with y(t), the correlation integralis:

2. Define analogue correlationAnalogue correlation is like analogue convolution except that we donot reverse one of the signals before we integrate. Using Rxy(t) tomean the cross−correlation of x(t) with y(t), the correlation integralis:

References:References:

11

2

T. R. Ganesh Babu, and G. Srinivasan:“ Communication Theory and systems”, 2006.

Sanjay Sharma: “Communication Systems(Analog and Digital) ”.Sanjay Sharma: “Communication Systems(Analog and Digital) ”.

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