chapter 2

2/9/2014

Communication Systems 1

Communication Systems

Instructor: Engr. Dr. Sarmad Ullah Khan

Assistant ProfessorAssistant ProfessorElectrical Engineering Department

CECOS University of IT and Emerging [email protected]

Chapter 2

Dr. Sarmad Ullah Khan

Signals and Signal Space

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Outlines

• Signals and systems• Size of signal


Size of signal• Classification of signals• Signal operations• The unit impulse function• Signals versus Vectors• Correlation

O th l i l• Orthogonal signals• Trigonometric Fourier Series• Exponential Fourier Series

3

Outlines





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Signal and system

• SignalSignal is a set of Information / Data


Signal is a set of Information / Data

For example, Telephone, Telegraph, Stock Market

Price

Signals are function of independent variable time

In electric charge distribution over surface signalIn electric charge distribution over surface, signal

is function of space rather than time

5

Signal and system

• SystemSystem process the received signal


System process the received signal

System might modify or extract information from

signal

For example, Anti aircraft missile launcher may

want to know the future location of targetwant to know the future location of target

Anti aircraft missile launcher gets information

from radar (INPUT)

Radar provides target past location and velocity 6

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Signal and system

• SystemAnti aircraft missile launcher calculates the future


Anti aircraft missile launcher calculates the future

location (OUTPUT) using the received

information

• Definition

System gets a set of signals as INPUT and yield aSystem gets a set of signals as INPUT and yield a

set of signals as OUTPUT after proper processing

System might be a physical device or it might be

an algorithm 7

Signal and system


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Outlines





9

Size of Signal • Size of an entity is a quantity that shows

largeness or strength of entity


g g y

• It is a single value/number measure

• How signal (amplitude and duration) can berepresented by a single number measure?

• For example, person’s width ‘r’ and height ‘h’

• To be more precise, single value measure of aperson is its volume

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Size of Signal • Signal Energy Area under a signal g(t) is its SIZE


g g( )

Signal Size takes two values “Amplitude” and“Duration”

This measuring approach is defective for largesignals having positive and negative portions

11

Size of Signal • Signal Energy Positive portion is cancelled by negative portion


p y g presulting in small signal

This can be solve by calculating area under g2(t)

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Size of Signal • Signal Energy If signal is complex signal, then


g p g ,

h ibl h i d | ( )| Other possible approach is area under |g(t)|

Energy measure is more desirable

13

Size of Signal • Signal Power Signal energy must be finite for it to be


g gymeaningful

Necessary conditions to make it finite Amplitude goes to zero as time approaches infinity

Signal must converge

IfIf Amplitude does not go to zero

Then Energy is infinite

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Size of Signal


Does this mean that a 60 hertz sine wave feeding intoyour headphones is as strong as the 60 hertz sine wavecoming out of your outlet? Obviously not. This iswhat leads us to the idea of signal power. 15

Size of Signal • Signal Power To make energy finite and meaningful, time


gy g ,average of signal energy is taken into consideration

For comple signals For complex signals

Signal power is time average of signal amplitude 16

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Size of Signal


• Square root of signal power is the Root MeanSquare (RMS) value of signal 17

Size of Signal • Are all energy signals also power signals?• No. In fact, any signal with finite energy will have


zero power.

• Are all power signals also energy signals?• No, any signal with non-zero power will have infinite

energy.

• Are all signals either energy or power signals?• No. Any infinite-duration, increasing-magnitude

function will not be either. (e.g. f(t)=t is neither)

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Size of Signal

• Remark:


• The terms energy and power are not used intheir conventional sense as electrical energyor power, but only as a measure for the signalsize.

19

Example 2.1• Determine the suitable measures of the signals given

below:


• The signal (a) amp 0 as t infinity Therefore• The signal (a) amp. 0 as t infinity .Therefore, the suitable measure for this signal is its energy, given by

gE 8444)2()(= 0

0

1

22 =+=+= ∞

−

−

∞

∞−

dtedtdttg t

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Example 2.1The signal in the fig. Below does not --- 0 as t . However it is periodic, therefore its power exits. ∞


21

Example 2.2


(a)

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Example 2.2


Remarks:A sinusoid of amplitude C has power of regardless of its frequency and phase .23

Example 2.2


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Example 2.2


We can extent this result to a sum of any number of sinusoids with distinct frequencies.

25

Example 2.2


Recall that ThereforeThereforeThe rms value is

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Outlines





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Classification of signals

• Continuous-time and discrete-time signalsA l d di it l i l


• Analog and digital signals• Periodic and Aperiodic signals• Energy and power signals• Deterministic and random signals• Causal vs Non-causal signals• Causal vs. Non-causal signals• Right Sided and Left Sided Signals• Even and Odd Signals

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• Continuous-time Signal


A signal that is specified for every value of time t e.g. Audio and video signals

29


• Discrete-time signal A i l h i ifi d f di l f


A signal that is specified for discrete value of time t = nT, e.g. Stock Market daily average

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• Analog signal A l i l d i i i l


Analog signal and continuous-time signal are two different signals

Analog signal whose amplitude can have any value over a continuous range

Analog continuous time signal x(t) Analog discrete time signal x[n]31


• Digital signal Di i l i l d di i i l


Digital signal and discrete-time signal are two different signals

Discrete signal whose amplitude can have only a finite number of value

Digital continuous time signal Digital discrete time signal 32

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• Periodic signal A i l ( ) i id b i di if h i


A signal g(t) is said to be periodic if there exist a positive constant T0, such that

g(t) = g(t+T0) for all t

Oth i i di i l Otherwise aperiodic signal

33


• Properties of Periodic Signal P i di i l i


Periodic signal must start at time t = - Periodic signal shifted by integral multiple of T0

remains unchanged A periodic signal g(t) can be generated by

periodic extension of any segment of g(t) withduration T0duration T0

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• Energy Signal A i l i h fi i i ll d i l


A signal with finite energy is called energy signal

• Power Signal A signal with finite power is called power signal

35


• Remarks:


A signal with finite energy has zero power.

A signal can be either energy signal or power signal, notboth.

Every signal in daily life is energy signal, NOT powersignal

Power signal in practice is not possible because of infiniteduration and infinite energy

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• Deterministic Signal:


A signal whose physical description is know completely,either mathematically or graphically is called deterministicsignal

• Random Signal:

A signal which is known in terms of probabilisticdescription such as mean value, mean squared value anddistribution

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• Casual Signal:


A signal which is zero prior to zero time

Signal amplitude A=0 for T = -t

• Non Casual Signal:

A signal which is zero after zero time

Signal amplitude A=0 for T = +t

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• Right sided and Left sided Signal:A i ht id d i l i f t < T d l ft id d i l


A right-sided signal is zero for t < T and a left-sided signalis zero for t > T where T can be positive or negative.

39


• Even and Odd Signal:E i l ( ) d dd i l ( ) d fi d


Even signals xe(t) and odd signals xo(t) are defined as

xe(t) = xe(−t) and xo(−t) = −xo(t).

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• Even and Odd Signal:If h i l i i i d f i If h


If the signal is even, it is composed of cosine waves. If thesignal is odd, it is composed out of sine waves. If thesignal is neither even nor odd, it is composed of both sineand cosine waves.

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Outlines





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Signal Operation


Time Shifting

A signal g(t) is said to be time shifted if g(t) isdelayed or advanced by time T

If signal g(t) is delayed by time T, then

43

If signal g(t) is advanced by time T, then

Signal Operation

Time Scaling


The compression or expansion of signal g(t) intime is known as Time Scaling

Compression

g(t)=g(at)

Expansion

g(t)=g(t/a)

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Signal Operation

Time Inversion


In time inversion, signal g(t) is multiplied by afactor a = -1 in time domain

g(t) = g(at) if a=1

Time inverted signal

g(t) = g(at) if a=-1

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Signal Operation

• Example: 2.4

F th i l (t) h i th fi B l k t h


• For the signal g(t), shown in the fig. Below , sketch g(-t)

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Outlines





47

Unit Impulse Signal

• The Dirac delta function or unit impulse or oftenreferred to as the delta function is the function that


referred to as the delta function, is the function thatdefines the idea of a unit impulse in continuous-time

• It is infinitesimally narrow, infinitely tall, yetintegrates to one

• simplest way to visualize this as a rectangular pulsep y g pfrom a -D/2 to a +D/2 with a height of 1/D

• The impulse function is often written as

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Unit Impulse Signal


=0 for al t ≠ 0

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Unit Impulse Signal

• Multiplication of a Function by an Impulse


If a function g(t) is multiplied by impulse function we getimpulse value of g(t)

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Unit Impulse Signal


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Unit Impulse Signal


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Unit Impulse Signal

• Unit Step Function u(t)


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Outlines





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Signals versus Vectors

• A Vector can be represented as a sum of itscomponents


components

• A Signal can also be represented as a sum of its• A Signal can also be represented as a sum of itscomponents

55


• A Signal defined over a finite number of time instantscan be written as a Vector


can be written as a Vector• Consider a signal g(t) defined over interval [a,b]• Uniformly divide interval [a,b] in N points

T1 = a, T2 = a+ϵ, T3 = a+2ϵ, …. TN = a+(N-1)ϵ

• WhereStep Sizeϵ =

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• Signal vector g can be written a N-dimensional vector


• Signal vector g grows as N increases

g = [g(t1) g(t2) … g(tN)]

• Signal transforms into continuous time signal g(t)

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• Signal transforms into continuous time signal g(t)


• Continuous time signals are straightforwardgeneralization of finite dimension vectors

• Vector properties can be applied to signals

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• A vector is represented by bold-face type


• Specified by its magnitude and its direction

• For example, Vector x have magnitude | x | andVector g have magnitude | g |

• Inner product (dot or scalar) of two real valuedvectors ‘g’ and ‘x’ is

59


• Component of a Vector


Consider two vectors ‘x’ and ‘g’

‘cx’ (projection) is component of ‘g’ along ‘x’

What is the mathematical significance of a vector alonganother vector?

g = cx + eg

However, this is not a unique way of vector decomposition

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Oth t ‘ ’ i


Other ways to express ‘g’ is

g is represented in terms of x plus another vectorwhich is called the error vector e

61




If we approximate

e = g – cx

Geometrically component of g along x is

Hence

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Based on definition of inner product, multiply both side by|x|

63




If g and x are orthogonal, then

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• Component of Signal


Vector component and orthogonality can be extended tocontinuous time signals

Consider approximating a real signal g(t) in terms ofanother real signal x(t)

And

65




As energy is one possible measure of signal size.

To minimize the effect of error signal we need to minimize its size-----which is its energy over the interval [t1 , t2]

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But ‘e’ is a function of ‘c’, not ‘t’, hence

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69




Two signals g(t) and x(t) are said to be orthogonal if thereis zero contribution from one signal to other signal

For N dimensional vectors ‘g’ and ‘x’

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For the square signal g(t), find the component of g(t) of the form sint or in other words approximate g(t) in terms of sint

Example 2.5


sint or in other words approximate g(t) in terms of sint

tctg sin)( ≅ π20 ≥≤ t

71

ttx sin)( = and

Example 2.5


)(

From equation for signals

dttxtg

Ec

t

tx=2

1

)()(1

72

πππ

π π

π

π 4sinsin

1sin)(

1

0

22

=

−+== tdttdttdttgc

o

ttg sin4

)(π

≅

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• Orthogonality in Complex Signals


For complex function g(t), its approximation by anothercomplex function x(t) over finite interval

‘c’ and ‘e’ are complex functions

73




Energy of a complex signal x(t) over finite interval

Choose ‘c’ such that it reduces ‘Ee’

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We know that

After certain manipulations

( )( ) ∗∗∗∗ ++=++=+ uvvuvuvuvuvu222

222 2222

1

2

1

2

1

)()(1

)()(1

)( ∗∗ −+−=t

tx

x

t

tx

t

t

e dttxtgE

EcdttxtgE

dttgE

dttxtgE

ct

tx

)()(1 2

1

∗= 75




So, two complex functions are orthogonal over an interval,if

0)()( 21

2

1

=∗ dttxtxt

t

0)()( 21

2

1

= ∗ dttxtxt

t

or

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• Energy of the Sum of Orthogonal Signals


Sum of the two orthogonal vectors is equal to the sum ofthe lengths of the squared of two vectors. z = x+y then

Sum of the energy of two orthogonal signals is equal to the

222yxz +=

gy g g qsum of the energy of the two signals. If x(t) and y(t) areorthogonal signals over the interval, and if

z(t) = x(t)+ y(t) then

yxz EEE += 77

Outlines





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Correlation of Signals

• Correlation addresses the question: “to what degree issignal A similar to signal B”


signal A similar to signal B

• Two vectors ‘g’ and ‘x’ are similar if ‘g’ has a largecomponent along ‘x’

• If ‘c’ has a large value, then the two vectors will besimilar

‘ ’ ld b id d h i i f• ‘c’ could be considered the quantitative measure ofsimilarity between ‘g’ and ‘x’But such a measure could be defective. The amount of similarity should be independent of the lengths of g and x 79


• Doubling g should not change the similarity between gand x


and x

• Similarity between the vectors is indicated by angle

However:

Doubling g doubles the value of c

Doubling x halves the value of cc is faulty measure for similarity

y y gbetween the vectors.

• The smaller the angle , the largest is the similarity, andvice versa

• Thus, a suitable measure would be , given by80

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• Where


• This similarity measure is known as correlationco-efficient

Independent of the lengths of g and x

• And

81


• Same arguments for defining a similarity index(correlation co efficient) for signals


(correlation co-efficient) for signals

• Consider signals over the entire time interval

• To establish a similarity index independent ofenergies (sizes) of g(t) and x(t), normalize c bynormalizing the two signals to have unit energiesg g g

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• Best Friends


• Opposite personalities (Enemies)

• Complete Strangers

83

Example 2.6

Find the correlation co-efficient between the pulse x(t) and the pulses

nc6,5,4,3,2,1,)( == itgi


p i

84

5)(5

0

5

0

2 === dtdttxEx5

1=gE

155

1 5

0

=×

= dtcndttxtgEE

cxg

n ∞

∞−

= )()(1

Similarly

Maximum possible similarity

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Example 2.6 (cont…)Dr. Sarmad Ullah Khan

5)(5

0

5

0

2 === dtdttxEx25.1

2 =gE

85

1)5.0(525.1

1 5

0

=×

= dtcndttxtg

EEc

xg

n ∞

∞−

= )()(1

Maximum possible similarity……independent of amplitude

Example 2.6 (cont…)Dr. Sarmad Ullah Khan

5)(55

2 === dtdttxEx 51

=gESimilarly

86

00 1g

1)1)(1(55

1 5

0

−=−×

= dtcndttxtgEE

cxg

n ∞

∞−

= )()(1

y

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Example 2.6(cont…)Dr. Sarmad Ullah Khan

)1(2

1)( 22

2

aTT

atT

at ea

dtedteE −−− −===

5)(5

0

5

0

2 === dtdttxEx

87

200 a

5

1=a 5=T

1617.24 =gE961.0

1617.25

1 5

0

5 =×

= −

dtect

n

Here

Reaching Maximum similarity

Outlines





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Orthogonal Signal Sets

• A signal can be represented as a sum of orthogonalset of signals


set of signals

• Orthogonal set of signals form a basis for specificsignal space

• For example, a vector is represented as a sum oforthogonal set of vectors

I f di f• It forms a coordinate system for vector space

89


• Orthogonal Vector SpaceC t i i d ib d b th t ll th l


Cartesian space is described by three mutually orthogonalvectors x1, x2, and x3

If a three dimensional vector g is approximated by twoorthogonal vectors x1 and x2, then

And

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• Orthogonal Vector SpaceIf th di i l t i t d b th


If a three dimensional vector g is represented by threeorthogonal vectors x1 , x2 and x3, then

And e = 0 in this case

x1,x2 and x3 is complete set of orthogonal space, No x4 exist

91


• Orthogonal Vector SpaceTh th l t ll d b i t


These orthogonal vectors are called basis vectors

Complete set of vectors is called complete orthogonal basisof a vector

A set of vector {xi} is mutually{ i} y

orthogonal if

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• Orthogonal Signal SpaceLik t th lit f i l t (t) (t)


Like vector, orthogonality of signal set x1(t), x2(t), …..xN(t) over time interval [t1, t2] is defined as

If all signal energies are equal En = 1 then set is normalizedand is called an orthogonal set

An orthogonal set can be normalized by dividing xN(t) by

93


• Orthogonal Signal SpaceN i l (t) [t t ] b t f N th l


Now signal g(t) over [t1, t2] by a set of N-orthogonalsignals x1(t), x2(t), ….. xN(t) is

Energy of error signal e(t) can be minimized if

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Outlines





95

Trigonometric Fourier Series

• Like vector, signal can be represented as a sum of itsorthogonal signal (Basis signals)


g g ( g )• There are number of such basis signals e.g. trigonometric

function, exponential function, Walsh function, Besselfunction, Legendre polynomial, Laguerre functions,Jaccobi polynomial

• Consider a periodic signal of period T0

• Consider a signal set• Consider a signal set

{1+Cos w0t+Cos 2w0t+……Cos nw0t…. Sin w0t+Sin 2w0t……Sin nw0t….}

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• nw0 is called the nth harmonic of sinusoid of angularfrequency w where n is an integer


frequency w0 where n is an integer

• A sinusoid of frequency w0 is called the fundamentaltone/anchor of the set

97


• This set is orthogonal over any interval of duration because:

oo wT π2=


because:

=2

0coscos

oTo

oo Ttdtmwtnwomn

mn

≠=≠

mn≠

=0

sinsin oo Ttdtmwtnwomn ≠=

2

oTo

oo T

0cossin = tdtmwtnwTo

oofor all n and m

and

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• The trigonometric set is a complete set.


• Each signal g(t) can be described by a trigonometric Fourier series over the interval To :

...2sinsin 21 +++ twbtwb oo

...2coscos)( 21 +++= twatwaatg ooooTttt +≤≤ 11

∞

=

++=1

sincos)(n

onono tnwbtnwaatg oTttt +≤≤ 11

or

on T

wπ2=

99


• We determine the Fourier co-efficient as:


+

+

=o

o

Tt

t o

o

Tt

tn

tdtnw

tdtnwtgC

1

1

1

1

2cos

cos)(

doTt

+1

)(1

,......3,2,1=n 100

dttgT

ato=1

)(1

0

tdtnwtgT

a o

Tt

ton

o

cos)(2 1

1

+

=

tdtnwtgT

b o

Tt

ton

o

sin)(2 1

1

+

=

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Compact Trigonometric Fourier Series

• Consider trigonometric Fourier series


• It contains sine and cosine terms of the samefrequency. We can represents the above equation in ai l t f th f i th

...2sinsin 21 +++ twbtwb oo

...2coscos)( 21 +++= twatwaatg ooo oTttt +≤≤ 11

single term of the same frequency using thetrigonometry identity

)cos(sincos nononon tnwCtnwbtnwa θ+=+

22nnn baC +=

−= −

n

nn a

b1tanθ

oo aC =

oTttt +≤≤ 11 101∞

=

++=1

0 )cos()(n

non tnwCCtg θ

Example 2.7

Find the compact trigonometric Fourier series for the following function


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Example 2.7Solution:We are required to represent g(t) by the trigonometric Fourier series over the interval andπ≤≤ t0 π=T


series over the interval and π≤≤ t0 πoT

sec22 radT

wo

o == π

T i i f f F i i

103

Trigonometric form of Fourier series:

??,?,0 nn baa

ntbntaatg nn

no 2sin2cos)(1

++= ∞

=

π≤≤ t0

Example 2.7

5001 2 ==

−dtea

tπ


50.00

0 == dteaπ

+==

−

20

2

161

2504.02cos

2

ndtntea

t

n

π

π

+==

−

20

2

161

8504.02sin

2

n

nntdteb

t

n

π

π

22nnn

oo

baC

aC

+=

=

104

0

Compact Fourier series is given by

)cos()(1

0 non

n tnwCCtg θ++= ∞

=π≤≤ t0

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Example 2.7

2644

504.0

2n

aC oo ==


( ) )161

2(504.0

)161(

64

161

4504.0

22222

22

nn

n

nbaC nnn

+=

++

+=+=

( ) nna

b

n

nn 4tan4tantan 11 −=−=

−= −−θ

( )4t22

50405040)( 1−∞

tt π≤≤ t0

105

( )

.......)42.868cos(063.0)24.856cos(084.0

)87.824cos(25.1)96.752cos(244.0504.0

4tan2cos161

504.0504.0)( 1

12

+−+−+−+−+=

−+

+==

oo

oo

n

tt

tt

nntn

tg π≤≤ t0

π≤≤ t0

Example 2.7

n 0 1 2 3 4 5 6 7


Cn 0.504 0.244 0.125 0.084 0.063 0.054 0.042 0.063

Өn 0 -75.96 -82.87 -85.24 -86.42 -87.14 -87.61 -87.95

l d d h f f h

106

Amplitudes and phases for first seven harmonics

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• Periodicity of the trigonometric Fourier series


The co-efficient of the of the Fourier series are calculatedfor the interval [t1, t1+T0]

∞

=

∞

=

+++=+

++=

100

1

])([cos()(

)cos()(

nnono

nnono

TtnwCCTt

tnwCCt

θφ

θφ for all t

)(

)cos(

)2cos(

1

1

t

nwtCC

nnwtCC

no

n

no

no

n

no

φ

θ

θπ

=

++=

+++=

∞

=

∞

=

for all t

107


• Periodicity of the trigonometric Fourier series


tdtnwtgT

a o

Ton

o

2

cos)(2 =

dttgT

aTo

o

o

)(1 =

n= 1,2,3,……

tdtnwtgT

b o

Ton

o

sin)(2 = n= 1,2,3,……

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• Fourier Spectrum


Consider the compact Fourier series

This equation can represents a periodic signal g(t) of

f i

)cos()(1

0 non

n tnwCCtg θ++= ∞

=

d )(frequencies:

Amplitudes:

Phases:

oooo nwwwwdc ,.....,3,2,),(0

nCCCCC ,......,3210 ,,,

nθθθθ ,.....,,,0 321

109


• Fourier SpectrumF d i d i ti f )(


nc vs w (Amplitude spectrum)

wvsθ (phase spectrum)

Frequency domain description of )( tφ

Time domain description of )( tφ

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Example 2.8


Find the compact Fourier series for the periodic square wave w(t) shown in figure and sketch amplitude and phase spectrum( ) g p p p

F i i

111

∞

=

++=1

sincos)(n

onono tnwbtnwaatw

Fourier series:

2

11 4

4

0 == dtT

a

o

o

T

To

W(t)=1 only over (-To/4, To/4) andw(t)=0 over the remaining segmentdttgT

aoTt

to+

=1

1

)(1

0

Example 2.8


== 2

sin2

cos2 4 π

πn

ndttnw

Ta

oT

on

− 24

πnToTo

−

=

π

π

n

n2

2

0

...15,11,7,3

...13,9,5,1

=

=

−

n

n

evenn

112

πn ,,,

0sin2 4

4

== ntdtT

b

o

o

T

Ton 0= nb

All the sine terms are zero

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Example 2.8


+−+−+= ....7cos

7

15cos

5

13cos

3

1cos

2

2

1)( twtwtwtwtw ooooπ 7532 π

The series is already in compact form as there are no sine terms

Except the alternating harmonics have negative amplitudes

The negative sign can be accommodated by a phase of radians asπ)cos(cos π−=− xx

113

Series can be expressed as:

++−++−++= ....9cos

9

1)7cos(

7

15cos

5

1)3cos(

3

1cos

2

2

1)( twtwtwtwtwtw ooooo ππ

π

Example 2.8


2

1=oC

=πn

C n 2

0

oddn

evenn

−

−

−=

πθ

0n

for all n 3,5,7,11,15,…..

for all n = 3,5,7,11,15,…..≠

We could plot amplitude and phase spectra using these values….

In this special case if we allow Cn to take negative values we do not need a phase of to account for sign.π−

114

Means all phases are zero, so only amplitude spectrum is enough

2/9/2014


Example 2.8


Consider figure

)5.0)((2)( −= twtwo

115

+−+−= ....7cos

7

15cos

5

13cos

3

1cos

4)( twtwtwtwtw ooooπ

Outlines





116

2/9/2014


Exponential Fourier Series

• According to Euler’s theorem, each Sin function canbe represented as a sum of ejwt and e-jwt


be represented as a sum of ejwt and e jwt

• Also a set of exponentials ejnwt is orthogonal over anytime interval T=2*pi/w

A i l (t) l b t d• A signal g(t) can also be represented as anexponential Fourier series over an interval T0

117



• Where the coefficient Dn can be calculated as

• Exponential Fourier series is an another form oftrigonometric Fourier series

118

2/9/2014



• Exponential Fourier series is an another form oftrigonometric Fourier series


trigonometric Fourier series

119


• The compact trigonometric Fourier series of aperiodic signal g(t) is given by


periodic signal g(t) is given by

120

2/9/2014



• Exponential Fourier Spectra


To draw Dn we need to find its spectra

As Dn is a complex quantity having real and imaginaryvalue, thus we need two plots (real and imaginary parts ORamplitude and angle of Dn)

Amplitude and phase is prefered because of its closeti ith th di t fconnection with the corresponding components of

trigonometric Fourier series

We plot |Dn| vs ω and ∟Dn vs ω

121




Comparing exponential with trigonometric Fourierspectrum yields

For real periodic signal, Dn and D n are conjugate, thusp g , n -n j g ,

122

2/9/2014





123




sec22 radT

wo

o == π

∞

−∞=

=n

ntjn eDt 2)(ϕ

dteedtetT

D ntjtntj

To

no

2

0

22 1)(

1 −−− ==π

πϕ

== +−

π)2

2

1(1

dtetn

π=oT

124

π 0

nj 41

504.0

+=

2/9/2014





sec22 radT

wo

o == π

π=oTntj

n

enj

t 2

41

1504.0)(

∞

−∞= +=ϕ

and

+

++

++

++

=111

...121

1

81

1

41

11

504.0

642 tjtjtj ej

ej

ej

Dn are complex

Dn and D-n are conjugates

125

+

++

−+−

−−− ...

121

1

81

1

41

1 642 tjtjtj ej

ej

ej




sec22 radT

wo

o == π

π=oTnnn CDD2

1== −

nnD θ=< nnD θ−=< −and

thusnj

nn eDD θ= njnn eDD θ−

− =

126o

o

j

j

o

ej

D

ej

D

D

96.751

96.751

122.041

504.0

122.041

504.0

504.0

−−

−

−

=

+

=

=

o

o

D

D

96.75

96.75

1

1

=<

−=<

−

2/9/2014





sec22 radT

wo

o == π

π=oT

o

o

j

j

ej

D

ej

D

87.822

87.822

625.081

504.0

625.081

504.0

−−

−

−

=

+

=

o

o

D

D

87.82

87.82

1

1

=<

−=<

−

127

And so on….




128

chapter 2

Engineering

system signal signal

size of signal size

systems size of signal

signal gt

signal space

small signal

size of signal remark

average of signal energy