communication fundamentals1 properties of delta function delta function is a particular class of...

Communication Fundamentals 1

Properties of Delta Function

• Delta function is a particular class of functions which plays a significant role in signal analysis.

• They have simple mathematical form but they are either not finite everywhere or they do not have finite derivatives of all orders everywhere. They are also known as singularity functions.

• Unit impulse function or Dirac delta function is a singularity function of great importance. This function has the property

• for any f(t) continuous at for finite t0.

b

a

btatfdttttf

elsewhere0

)()()( 00

0

0tt


• The impulse function selects or sifts out a particular value of the function f(t), namely, the value at t=t0, in the integration process.

• If f(t) = 1, then the above equation becomes

• Therefore (t) has unit area.

• Also

• The symmetry properties of delta function stipulates that

• Time scaling property --

1)( 0 b

adttt

00 allfor 0)( tttt

)()( tt )(

1)( t

aatt


• Multiplication by a time function

• Relationship with the Unit step function, which is given by

• For f(t)=1, we have

• Thus the derivative of the unit step function yields a delta function

)()()()( 000 tttftttf

0

00 0

1)(

tt

ttttu

t

b

a

t

ttudt

ttutt

ttdtdtf

)()(

or)(0

1)()()(

00

00

000


Fourier Transform

• We consider an aperiodic function f(t) as shown

• We wish to represent this function as a sum o fexponential functions over the entire interval . For this purpose, we construct a new periodic function , with period T so that the function f(t) is forced to repeat itself completely every T seconds. The original function can be obtained by letting

,)(tfT

T


• The new function fT(t) is a periodic function and consequently can be represented by an exponential Fourier Series

• we define Using these definitions we obtain

TdtetfT

FeFtfT

T

tjnTn

n

tjnnT /2;)(

1,)( 0

2/

2/

00

nnn TFFn )(;0

;)()(,)(1

)(2/

2/

T

T

tjTn

n

tjnT dtetfFeF

Ttf nn


• The spacing between adjacent lines in the line spectrum of fT(t) is Using this relation for T we obtain the alternate form

• Now as T becomes very large, becomes smaller and the spectrum becomes denser. As , the discrete lines in the spectrum merge and the frequency spectrum becomes continuous. Thus,

T/2

n

tjnT

neFtf

2)()(

T

n

tjn

TT

T

neFtf LimLim

)(2

1)(


• becomes

• similarly

• Symbolically

• The complex Fourier series coefficients can be evaluated in terms of the Fourier Transform

deFtf tj)(

2

1)(

dtetfF tj )()(

FtftfF 1)(;)(

0

1

n

n FT

F


Spectral Density Function

• The area under the spectral density function F() gas the dimensions voltage. Each point on the F() curve contributes nothing to the representation of f(t). It is the area that contributes. But each point does indicate the relative weighting of each frequency component. The contribution of a given frequency band to the representation of f(t) may be found by integrating to find the desired area.

• A periodic waveform has its amplitude components at discrete frequencies. At each of these discrete frequencies there is some definite contribution. To portray the amplitude components of a periodic waveform on a spectral-density graph requires a representation with area


• equal to the respective amplitude components yet occupying zero frequency width. This can be done by representing each amplitude component of the periodic function by an impulse function. The area of the impulse is equal to the amplitude component and the position of the impulse is determined by the particular discrete frequency.

• Summarizing, a signal of finite energy can be described by a continuous spectral density function. This spectral density function is found by taking the Fourier transform of the signal.


• e.g. Find the Fourier transform of a gate function defined as

• We have

2/0

2/1/

t

ttrect

2/

2/sin/

/)(

2/2/

2/

2/

jee

dtedtetrectF

jj

tjtj


Fourier Transform Involving Impulse Functions

• The Fourier transform of a unit impulse is

• The phase spectrum of the time-shifted impulse is linear with a slope that is proportional to the time shift.

0)()(

1)()(

00

0

tjtj

jtj

edtetttt

edtett


Complex Exponentials

• We would expect that the spectral density of

• as shown

0

01-

001

2

Thus, .2

1

have wesidesboth of ansformFourier tr Taking2

12

1

0

0

0

tj

tj

tj

tj

e

e

e

de

0at edconcentrat be will0 tje


Sinusoids

• The sinusoidal signals can be written in terms of the complex exponentials using Euler’s identities

tt 00 sin and cos

j

eet

eet

tjtj

tjtj

/

2

1

2

1sin

2

1

2

1cos

00

0

00

0

00

00


Signum Function and the Unit Step

• The Signum function, sgn(t), changes sign when its argument is zero

• The signum function has an average value of zero and is piecewise continuous, but not absolutely integrable. To make it absolutely integrable we multiply sgn(t) by

• and then take the limit as

01

00

01

)sgn(

t

t

t

t

tt

tae

0a teLimt ta

asgn)sgn(

0


• Interchanging the operations of taking the limit and integrating we have

• The unit step function can be expressed as

• Thus

ja

j

dtedte

dtetet

a

tjatja

a

tjta

a

22lim

lim

sgnlimsgn

220

0

00

0

ttu sgn2

1

2

1

j

tu1


Periodic Functions

• A periodic function, of period T, can be expressed as

• Taking the Fourier transform, we find

TeFtfn

tjnnT /2 where 0

0

nn

tjn

nn

n

tjnnT

nF

eFeFtf

02

00


Properties of Fourier TransformLinearity

• This follows directly from the integral definition of Fourier transform

Complex Conjugate• For any complex signal we have

• If f(t) is real, then

22112211 FaFatfatfa

**

**

** toDue .

Fdtetfdtetftf

Ftf

tjtj

FFtftf ** and


Symmetry

• Any signal can be expressed as a sum of an even function and an odd function

Duality• Duality exists between time and frequency domain as

shown below

0cos2

sincos

:Proof .imaginary) (and and (real)

tdttf

tdttfjtdttfdtetftf

FtfFtf

e

eetj

ee

ooee

ftFFtf 2 then ,)(


• The proof can be done by interchanging t and in the Fourier transform integral.

tFdtetFf

dtetFfdeftF

deFtfdtetfF

tj

tjtj

tjtj

-

-

-

2 Thus

2

1 ;

get we and t inginterchang2

1 ;


• e.g. It is given that find

• Let

•

2/Satrect 2/tSa

rectrecttF

tSatFSaF

22

2/2/


Coordinate Scaling• The expansion or compression of a time waveform affects

the spectral density of the waveform. For a real-valued scaling constant and any pulse signal f(t),

Ftf

FF

dxexfxftftx

dtetftfFtf

xj

tj

1simply or

0for 1

and;0for 1

/ have wefor

and 1

/


• If is positive and greater than unity, f(t) is compressed, and its spectral density is expanded in frequency by 1/ . The magnitude of the spectral density also changes -- an effect necessary to maintain energy balance between the two domains. If > 0 but less than unity, f(t) is an expanded version of f(t) and its spectral density is compressed. When < 0, f(t) is reversed in time compared to f(t) and is expanded or compressed depending whether | | is greater than or less than unity.


Time Shifting

Frequency Shifting

dxexfedxexfxf

ttxdtettfttf

eFttf

xjtjtxj

tj

tj

00

0

get wethenlet ;

:Proof

000

0

0

)(

0

000

0

:Proof

;

Fdtetfdteetfetf

Fetf

tjtjtjtj

tj


Differentiation and Integration• If df/dt is absolutely integrable, then

• The corresponding integration property is

deFjdeFdt

dtf

dt

d

deFtf

Fjtfdt

d

tjtj

tj

2

1

2

1

;2

1 :Proof

;


• Consider the function g(t) defined as

• Let g(t) have Fourier transform G(). Now

• However, for g(t) to have a transform G() must exist. One condition is that This means

• which is equivalent to F(0) = 0. If then g(t) is no longer an energy function and the transform will include an impulse function

''

tdttftg

FGjtfdt

tdg thatsee weabove From.

0lim

tgt

0dttf

00 F

01

'' FFj

dttft


Time Convolution• There are two ways of characterizing a system --

frequency response and impulse response. The two can be related using the principle of convolution.

• For the test signal , the system impulse response is defined as where is the delay or age variable. If the system is time-invariant, h(t,) takes the special form . The input signal f(t) may be expressed in terms of impulse functions by

• If we define

ttf ,tht T

thtT

dtfdtftf

dττtδτftg T


• From integration theory, we can rewrite this as

• Using the principle of superposition, we move the system operator inside the summation. Also, the f(n) are the weights (areas) of the impulse functions and are constants for each impulse. Therefore we have

• Therefore we have

• This is a key result in signal analysis for it links the input to the output by means of an integral operation.

nnn tfTtg

0lim

nn tftg T0

lim dthftg ,


• The equation reduces to

• This is known as the convolution integral.

• An important property of the Fourier transform is that it reduces the convolution integral to an algebraic product.

• Proof

thtfdthftg

HFhf

dtedthfhf tj


• Changing the order of integration and integrating with respect to t first yields

HFdHefhf

Heth

ddtethfhf

j

j

tj

have weThus.

have weproperty,delay timeusing


Frequency Convolution

• A dual to the preceding property can be established

2121

2211

2Then

,

ffFF

FtfFtf


Some Convolution Relationships

• The convolution integral

holds as long as the system is linear, time-invariant, and causal. Thus h(t) = 0 for all t < 0 and there is no contribution to the integration for (t-) < 0.

• Often the input, f(t), also satisfies f(t) = 0 for t < 0.

• Properties of Convolution• Commutative Law --

• Distributive Law --

• Associative Law --

dthftg

1221 ffff 3121321 fffffff

321321 ffffff


Convolution involving Singularity Functions

• The unit step response is the indefinite integral of the unit impulse response as shown

• This provides a technique for determining the impulse response of a system in the laboratory.

• Convolution with the unit impulse function gives

t

dxxhhu

txdthdthuthtu

then

Let . 0

000 ttfdttfttf


• Example: Find as shown:

•

thtfhf , for the,

20

20sin

00

22sinsin

2sin

2,sin

t

ttA

t

tg

tutAtutA

dttuAhftg

ttthttuAtf


Graphical Interpretation of Convolution

• The graphical interpretation of convolution permits to understand visually the results of the more abstract mathematical operations. For instance

• The required operations are as listed below: Replace t by in f1(t) giving f1() Replace t by (- ) in f2(). This folds the function f2() about the

vertical axis passing through the origin of the axis. Translate the entire frame of reference of f2(- ) by an amount

t. Thus the amount of translation, t, is the difference

dtffff

221


between the moving frame of reference and the fixed frame of reference and the fixed frame of reference. The origin in the moving is at = t, the origin in the fixed frame is at = 0. The function in the moving frame represents f2(t- ). The function in the fixed frame represents f1(t).

At any given relative shift between the frames of reference, e.g. t0, we must find the area under the product of the two functions

0

21021tt

tftftff


This procedure is to be repeated for different values of t=t0 by successively progressing the movable frame and finding the values of the convolution integral at those values of t.

If the amount of shift of the movable frame is along the negative axis, t is negative. If the shift is along the positive axis, t is positive.


• Example:

• Find the convolution of a rectangular pulse and a triangular pulse

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