chapter 2 introduction to signals students

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EE 23353 Analog Communications

Chapter 2: Introduction To Signals

Dr. Rami A. Wahsheh

Communications Engineering Department

Ch. 2: Introduction To SignalsReview of signals and linear systems: Chapters 2&3

2.1Size of a signal.g

2.2Classification of signals.

2.3Some useful signal operations

2.4Unit impulse function.

2.5Signals and vectors.

2

g

2.6Signal comparison: correlation.

2.7Signal representation by orthogonal signal set.

2.8Trigonometric Fourier series.

2.9Exponential Fourier series.

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Definitions

Signal: is a single-valued function of time, whichmeans that to each assigned instant of time (themeans that to each assigned instant of time (theindependent variable) there is one unique value ofthe function (the dependent variable). This valuemay be a real number, in which case we have real-valued signal, or it may be complex, and then wecan speak of a complex-valued signal. In eithercase the independent variable (time) is real-valued

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case the independent variable (time) is real valued.

Definitions

System: is an entity that processes a set of signals(inputs) to yield another set of signals (outputs)(inputs) to yield another set of signals (outputs).

SystemSignal Response

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2.1 Size of a Signal

How can a signal that exists over a certain timeand interval with varying amplitude be measured byand interval with varying amplitude be measured byone number that will indicate the signal size orstrength?

Signal Energy: the area under the signal |g(t)|2,g(t) is a complex valued signal.

5

A necessary condition for the energy to be finite isthat the signal amplitude goes to zero as |t| goesto ∞. Otherwise the integral will not converge.


Signal Power: if the amplitude of g(t) does not goto zero as |t| goes to ∞ then the signal energy isto zero as |t| goes to , then the signal energy isinfinite.

The measure of the signal size in such a case wouldbe: the time average of the energy (if it exists:periodic or has a statistical regularity), which isthe average power Pg (i.e., the mean of the signal).

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The root mean square (rms) value of g(t) is thesquare root of Pg.

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Units of Energy and Power

Depend on the nature of the signal g(t).

f ( ) l lIf g(t) is a voltage signal

The energy has units of volts squared seconds

The power has units of volts squared.

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Example 2.1 Page 17

Determine the suitable measures of the followingsignal:signal:

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Example 2.1 Page 17


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Solution: g(t) = 2e-t/2 as |t| goes to ∞ g(t) goes to 0

844)(44)1

1(4422)( 0

0

0

1 0

22/22

eeedtedtdttgE tt

g

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Example 2.1 Page 17


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Example 2.1 Page 17


Solution: g(t) is periodic as |t| goes to ∞ g(t) doest t 0 th f th i t f thi

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not go to 0 therefore the power exists for thisperiodic signal. Averaging it over one period

3

1

3

1)

3

1

3

1(

2

1

32

1

2

1)(

11

1

31

1

22/

2/

2

g

T

T

g

Prms

tdttdttg

TP

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Example 2.2 Page 18

Determine the power and the rms value of:

tj

o

oDetgc

wheretCtCtgb

tCtga

)()

)cos()cos()()

)cos()()

21222111

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Example 2.2 Page 18Determine the power and the rms value of:

)cos()( o

S l ti

tCtg

2/

2/

2/

222/

2/

2 )(cos1

lim)(1

lim

2

T

T

T

oT

T

TT

g

oo

dttCT

dttgT

P

SignalPeriodicTPeriod

Solution

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2/

2/

22/

2/

2

2/

2/

2

)22cos(2

lim2

lim

)22cos(12

1lim

T

T

oT

T

TT

g

T

T

oT

g

dttT

Cdt

T

CP

dttC

TP

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Example 2.2 Page 18

Determine the power and the rms value of:

02

)22cos(2

lim2

lim

2

2

2/

2/

22/

2/

2

CP

TasC

P

dttT

Cdt

T

CP

g

T

T

oT

T

TT

g

Cancellation of +ve and –veareas of Sinusoid

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2

2C

rms

Pg

Example 2.2 Page 18

Determine the power and the rms value of:htCtCt )()()(

dttCdttCP

dttCtCT

P

Solution

wheretCtCtg

TT

T

TT

g

2/22

2/22

2/

2/

2222111

21222111

)(cos1

lim)(cos1

lim

)cos()cos(1

lim

)cos()cos()(

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dtttCCT

dttCT

dttCT

P

T

TT

TT

TT

g

2/

2/

221121

2/

222

2/

111

cos()cos(21

lim

)(coslim)(coslim

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Example 2.2 Page 18

)(cos1

lim)(cos1

lim2/

2222

2

2/

1122

1 dttCT

dttCT

PT

T

T

Tg

022

cos()cos(21

lim

22

22

21

2/

2/

221121

2/2/

TasCC

P

dtttCCT

TT

g

T

TT

TT

TT

Cancellation of +ve and –veareas of Sinusoid

17

2

222

21

22

21

CCrms

CCPg

Example 2.2 Page 18

Determine the power and the rms value of:tj oDetg )(

The signal is complex.

DP

DdtDT

dtDeT

PT

o

Ttj

og

oo

o 2

0

2

0

2 11

18

DPrms g

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2.2 Classification of Signals

1. Continuous-time and discrete-time signals2. Analog and digital signals3. Periodic and aperiodic signals4. Energy and power signals5 Deterministic and probabilistic (random) signals

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5. Deterministic and probabilistic (random) signals6. Causal vs. Non-causal Signals

1. Continuous-Time and Discrete-Time Signals

•Qualify the nature of a signal along the time axis.

•A Continuous time signal: is specified for everyvalue of time t. Example: a telephone or videocamera

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•A discrete time signal: is specified only atdiscrete values of t. Example: monthly sales of acompany.

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2. Analog and Digital SignalsAn analog signal:

1. Is a continuous function of time with continuousamplitude.

2. Its amplitude can take an infinite number ofvalues.

A digital signal:

1 Is a discrete function of time and amplitude

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1. Is a discrete function of time and amplitude(i.e., the amplitude can only have a finite set ofvalues).

2. Its amplitude can take a finite number ofvalues.

Analog and Digital Signals

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3. Periodic and Aperiodic Signals

Periodic Signal: if it satisfies the following equation

tallforTtgtg o )()(

Where To is the period of g(t). The signal muststart at -∞ and continue forever.

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Aperiodic Signal: if it is not periodic.

4. Energy and Power Signals

Energy Signal: if it has a finite energy. (zeropower) power)

dttgtg 2)()(

Power Signal: if it has a finite and nonzero power.(infinite energy)

2/

2)(lim0T

Tdttg

24

2/T

T

•A signal can not both be an energy and a powersignal. If it is one it can not be the other.

•There are signals that are neither energy norpower signals. The ramp signal is such an example.

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5. Deterministic and Random Signals

A deterministic signal: can be modeled as acompletely specified function of time In othercompletely specified function of time. In otherwords, there is no uncertainty about its value atany time. For example, the sinusoid signalAcos(2πfct+θ) is deterministic if A, fc and θ areknown constants.

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A random (or stochastic) signal: cannot becompletely specified as a function of time.

5. Deterministic and Random Signals

• Deterministic signal: is a signal whose physicaldescription (i.e., mathematical or graphicalform) is known completelyform) is known completely.

• Random signal: is when the signal is known onlyin terms of probabilistic description, such as– Distribution– Mean value (i.e., the average or expected

l )

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value)– Squared mean value (i.e., the expected

value of the squared error)– Standard deviation (i.e., the square root

of the variance)

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6. Causal vs. Non-causal Signals• A causal signal is zero for t < 0 and a non-

causal signal is zero for t > 0 or

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Causal Signal Non-causal Signal

2.3 Some Useful Signal Operations

Time Shifting

T lTime Scaling

Time Inversion

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Time Shifting:

( T)•g(t-T) representsg(t) time-shifted byT seconds.

•If T is +ve, theshift is to the right(delay)

)(tg

)( Ttg

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(delay).

•If T is –ve, theshift is to the left(advance).

)( Ttg


Time Scaling:

Th•The compression orexpansion of a signalin time.

•g(2t) represents asignal that iscompressed in time

Compression

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compressed in timeby a factor of 2.

•Therefore, whateverhappens in g(t) at talso happens at timet/2 for Φ(t)

Expansion

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Time Inversion:

l f•Is a special case of timescaling with a=-1.

•Φ(t)=g(-t): Mirror imageof g(t) about the verticalaxis.

N M f

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•Note: Mirror image ofg(t) about the horizontalaxis is –g(t).

2.3 Example Page 26

Sketch g(3t) for the signal that is shown in a.

g(t): the value oft at 6,12,15,24

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g(3t): is g(t) compressedby 3. Therefore,

t at 2,4,5,8

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Homework #1 •Solve the following problems: Due to one weekfrom today

•2.1-1

•2.1-5

•2.1-8

•2.2-1

33

•2.3-1

•2.3-2

•2.3-3

2.4 Example Page 27

For the signal g(t) shown below, sketch g(-t):TimeInversionInversion

-1&-5 in g(t) aremapped into 1&5 ing(-t).

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2.4Unit Impulse Function

•One can visualize a unit impulse as a tall, narrowrectangular pulse of unit area

)(t

rectangular pulse of unit area.

•A unit impulse is regarded as a rectangular pulsewith a width that is very small, and a height thatis very large

1)(

00)(

dtt

tt

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1)( dtt

Multiplication of a Function by an Impulse

Impulse only exists at t=0)()0()()( ttt Impulse only exists at t 0

That is why the functiononly exists at t=0

)()0()()( ttt

)()()()( TtTTtt

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Sampling Property of the Unit Impulse Function

)0()()0()()(

dttdttt

This means the area under the product of afunction with an impulse δ(t) is equal to the value ofthat function at the instant where the unit impulseis located.

)()()()()(

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This property is known as the sampling or siftingproperty of the unit impulse function.

)()()( TdtTtt

Unit Step Function u(t)

00

01)(

t

ttu

)()(

00

tdt

tdu

t

•If one wants a signal to start at t=0, thenmultiply the signal by u(t).

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•e-at starts at t=-∞.

•To make it start at t=0,just multiply it by u(t).

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2.5 Signals and Vectors•Analogy between Signals and Vectors

• A vector can be represented as a sum of itspcomponents in a variety of ways.

• A Signal can also be represented as a sum of itscomponents in a variety of ways.

•Component of a vector:

• Specified by its magnitude and its direction

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• Specified by its magnitude and its direction.

• All vectors are denoted by boldface type.

Component of a VectorExample:

• Vector x of magnitude |x| and Vector g ofg gmagnitude |g|

• Let the component of g along x be cx

• Geometrically this component is the projection of gon x

• The component can be obtained by drawing a

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• The component can be obtained by drawing aperpendicular from the tip of g on x and expressedin terms of x as: g = cx + e

• e is the error vector.

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Component of a Vector•Other ways to express g in terms of x:

g=c1x+e1=c2x+e2

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Component of a Vector•The dot (inner or scalar) product of two vectors gand x is defined as:and x is defined as:

g.x=|g| |x| cosθ

Where θ is the angle between vectors g and x.

•The length of the vector x is |x| where:

|x|2=x x

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|x| x.x

•The length of the component g along x is:

c|x| = |g| cosθ

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Component of a Vector• Multiplying both sides by |x| yields

| |2 | | | | θc|x|2 =|g| |x| cosθ= g.x

c= (g.x)/(x.x) = (1/|x|2) g.x

• If g.x = 0 then

1.g and x are perpendicular (orthogonal)

2 h l d 0

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2.g has a zero component along x and c=0

Component of a Signal•Approximating a real signal g(t) in terms ofanother real signal x(t) over an interval [t1 t2]another real signal x(t) over an interval [t1,t2]

•The error e(t) in the approximation is given by:

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Component of a Signal• Energy is one possible measure of a signal size.

• To minimize the error signal we need to minimizegits size (its energy Ee over the interval [t1,t2]

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• Ee is minimum for some choice of c (not t)

Component of a Signal

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Component of a Signal1.Remarkable similarity

between the behavior ofvectors and signals Thevectors and signals. Thearea under the product oftwo signals corresponds tothe inner product of twovectors.

2.The energy of the signal is

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gy gthe inner product of a signalwith itself, and correspondsto the vector length squared(which is the inner productof the vector with itself).

Component of a Signal

dt

2

)()(1

• cx(t) is the projection of g(t) on x(t)

• If c = 0, then g(t) and x(t) are orthogonal over the interval [t1 t2]

dttxtgE

ctx1

)()(

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the interval [t1,t2]

0)()(2

1

dttxtgt

t

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Example 2.5 Page 33Find the component of g(t) of the form of sin (t) sothat the energy of the error signal is minimum.Where:Where: 20sin)( ttctg

49

Example 2.5 Page 33ttx

t

sin)(2

222

tdttgdttxtgE

c

dttdttxE

o

t

tx

t

x

sin)(1

)()(1

)(sin)(

2

2

0

22

2

1

1

50ttg

tdttdtc

sin4

)(

4sinsin

1

0

2

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Orthogonality in Complex SignalsFor complex functions of t over an interval [t1,t2]

tetcxtg )()()(

dttxE

tcxtg

tetcxtg

t

t

x

22

1

)(

)()(

)()()(

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dttcxtgE

tcxtgtet

t

e

22

1

)()(

)()()(

Orthogonality in Complex Signals

uvvuvuvuvuvu222

Recall that:

dtttE

dttxtgE

dttgE

t

t

tx

t

t

e

)()(1

)()(1

)(

2

2

1

2

1

2

22

Independentof c

To minimizeEe, the 3rd

52

dttxtgE

cTherefore

dttxtgE

Ec

t

tx

tx

x

)()(1

)()(

2

1

1

e,term shouldbe zero

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Orthogonality in Complex SignalsTwo complex functions x1(t) and x2(t) are orthogonalover an interval [t1,t2], if

0)()(0)()( 2121

2

1

2

1

dttxtxordttxtxt

t

t

t

When the two functions are real, then

0)()(2

dt

53

0)()(

0)()(

2

1

1

21

dttxtg

dttxtx

t

t

t

Energy of the Sum of Orthogonal Signals

Sum of two orthogonal vectors (x and y) is equal tothe sum of the lengths squared of the two vectors.If z=x+y thenIf z=x+y then

222yxz

The energy of the sum of two orthogonal signals isequal to the sum of the energies of the two signals.

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If x(t) and y(t) are orthogonal over an interval [t1,t2]and z(t)=x(t)+y(t), then

yxz EEE

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2.6 Signal Comparison: Correlation

Two vectors g and x are similar if g has a largelcomponent along x.

This means that c= (g.x)/(x.x) = (1/|x|2) g.x is large, thenthe two vectors g and x are similar.

Is c a quantitative measure of the similarity?

Such a measure would be defective The amount of

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Such a measure would be defective. The amount ofsimilarity should be independent of the lengths of g andx.

Signal Comparison: CorrelationDoubling the length of g should not change the

similarity between g and x:• Doubling g doubles the value of c• Doubling x halves the value of cThen c is a faulty measure of the similarity

Similarity between two vectors is indicated by the

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angle θ between the vectors.A suitable measure would be:

xg

xgcn

.cos

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Signal Comparison: Correlation

xg

xgcn

.cos

• The larger the cosθ, larger is the similarity.• cn is independent of the lengths of g and x.• cn is called Correlation Coefficient.• The magnitude of cn is never greater than unity

If 1 th th t t li d

xg

11 nc

57

• If cn=1 then, the two vectors are aligned • If cn=-1 then, the two vectors are aligned in opposite

direction• If cn=0 then, the two vectors are orthogonal

Signal Comparison: CorrelationConsider signals over the entire time interval -∞ to ∞

To make c independent of the energies of g(t) and x(t),p g gall what we need to do is to normalize the two signals tohave unit energies.

dttxtgEE

cxg

n

)()(1

11 nc

58

Correlation between two signals is a measure of thedegree of similarity between the two signals

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Example 2.6 Page 37Find the correlation coefficient cn between the pulse x(t)and the pulses gi(t), i=1,2,3,4,5 and 6

dttxtgEE

cxg

n

)()(1

59


55 1

dttxtgEE

cxg

n

)()(1

60

5)(

5)(

5

0

5

0

21

5

0

5

0

2

dtdttgE

dtdttxE

g

x

1.15.5

1

)()(1

5

0

1

dtc

dttxtgEE

cxg

n

MaximumSimilarity

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dttxtgEE

cxg

n

)()(1

61


55 1

dttxtgEE

cxg

n

)()(1

62

25.1)5.0()(

5)(

5

0

25

0

22

5

0

5

0

2

dtdttgE

dtdttxE

g

x

1)5.0(5).25.1(

1

)()(1

5

0

2

dtc

dttxtgEE

cxg

n

MaximumSimilarity

c is independent of theamplitude of signals

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dttxtgEE

cxg

n

)()(1

63


dttxtgEE

cxg

n

)()(1

55 1

64

MaximumDissimilarity

g(t) = -x(t)

5)(

5)(

5

0

5

0

23

5

0

5

0

2

dtdttgE

dtdttxE

g

x

1)1(5.5

1

)()(1

5

0

3

dtc

dttxtgEE

cxg

n

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dttxtgEE

cxg

n

)()(1

65


dttxtgEE

cxg

n

)()(1

5)(55

2 dtdttxE 1

661617.2)1(2

5

2

5)(

5)(

2

5

0

5

25

0

5

25

0

254

00

e

edtedteE

dtdttxE

ttt

g

x

961.0)1617.2(5

1

)()(1

5

0

54

dtec

dttxtgEE

c

t

xg

n

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dttxtgEE

cxg

n

)()(1

67


dttxtgEE

cxg

n

)()(1

5)(55

2 dtdttE 1

685.0)1(

2

1

2

1)(

5)(

10

5

0

25

0

25

0

25

00

2

e

edtedteE

dtdttxE

tttg

x

628.0)5.0(5

1

)()(1

5

0

5

dtec

dttxtgEE

c

t

xg

n

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dttxtgEE

cxg

n

)()(1

69


dttxtgEE

cxg

n

)()(1

1

70

5.22sin

5)(

5

0

26

5

0

5

0

2

dttE

dtdttxE

g

x

02sin)5.2(5

1

)()(1

5

0

6

dttc

dttxtgEE

cxg

n

g6(t) is orthogonal to x(t)

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Orthogonal Signal Space

Orthogonality of a signal set x1(t), x2(t), …, xN(t) over theinterval [t1, t2] as,

If the energies En=1 for all n, then the set is normalizedand is called an orthonormal set

71

and is called an orthonormal set.


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Trigonometric Fourier SeriesConsider a signal set: ,....sin....2sin,sin,....cos........2cos,cos,1 tnwtwtwtnwtwtw oooooo

• A sinusoid function with frequency nωo is called thenth harmonic of the sinusoid of frequency ωo when n isan integer.

• A sinusoid of frequency ωo is called the fundamental

• This set is orthogonal over any interval of duration

74

• This set is orthogonal over any interval of durationTo=2П/ωo because:

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Trigonometric Fourier Series

And

The trigonometric set is a complete set. Each signal g(t)can be described by a trigonometric Fourier series overany interval To :

75

any interval To :

Trigonometric Fourier SeriesWe determine the Fourier coefficients a0, an, and bn:

76

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Compact Trigonometric Fourier Series

The trigonometric Fourier series contains sine and cosineterms of the same frequency. We can represents theabove equation in a single term of the same frequencyusing the trigonometry identity

77


The trigonometric Fourier series contains sine and cosineterms of the same frequency. We can represents theabove equation in a single term of the same frequencyusing the trigonometry identity

78

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79

Webpagehttp://en.wikipedia.org/wiki/File:Fourier_Series.svg

80

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Webpagehttp://upload.wikimedia.org/wikipedia/commons/e/e8/Periodic_identity_function.gif

81

Example 2.7

82

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Example 2.7

83

??,?,0 nn baa

Example 2.7

84

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Example 2.7Compact Fourier series is given by

85

Example 2.7

86

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Example 2.7

87

Periodicity of the Trigonometric Fourier Series

We now show that the trigonometric Fourier series is aperiodic function of period To (The period of thef d t l)fundamental).

88

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Periodicity of the Trigonometric Fourier Series

89

Fourier SpectrumThe compact trigonometric Fourier series

g(t) is expressed as a sum of sinusoids of

Frequencies: 0, wo , 2wo, …, nwo, …

Amplitudes: Co, C1, …, Cn, …

90

p o 1 n

Phases: 0, θ1, θ2, …, θn, …

Cn vs. w (Amplitude Spectrum)

θn vs. w (Phase Spectrum)

The two plots together are the frequency spectra of g(t)

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Fourier Spectrum

• A signal has a dual identity: the time domain identityd th f d i id tit (F i t )and the frequency domain identity (Fourier spectra).

• The two identities complement each other. Takentogether, they provide a better understanding of asignal.

91

Example 2.8• Find the compact Fourier series for the periodic

square wave w(t) shown in figure and sketch amplituded h tand phase spectrum.

Fourier series:

92

Fourier series:

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Example 2.8

93

Example 2.8

94

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Example 2.8

The series is already in compact form as there are nosine terms. Except the alternating harmonics havenegative amplitudes.

95

g p

The negative sign can be accommodated by a phase ofradians as

Example 2.8

The series is already in compact form as there are nosine terms. Except the alternating harmonics havenegative amplitudes.

96

g p

The negative sign can be accommodated by a phase ofradians as

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Example 2.8

97

Example 2.8

98

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Example 2.8We could plot amplitude and phase spectra using thesevalues.

In this special case if we allow Cn to take negative valueswe do not need a phase of –П to account for sign.

Means all phases are zero, so only amplitude spectrum isenough.

99

Example 2.8Bipolar square wave wo(t): is basically w(t) minus its dccomponent

100

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Example 2.8Bipolar square wave wo(t): is basically w(t) minus its dccomponent

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Example 2.9Find the trigonometric Fourier series and sketch thecorresponding spectra for the periodic impulse trainδ (t)δTo(t)

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Example 2.9The trigonometric Fourier series for δTo(t)

??,?,0 nn baa

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Example 2.9The trigonometric Fourier series for δTo(t)

??,?,0 nn baa

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Example 2.9

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Exponential Fourier SeriesThe set of exponentials

Orthogonal over any interval of duration To=2П/ωoOrthogonal over any interval of duration To 2П/ωo

A signal g(t) can be expressed over an interval ofduration To seconds as an exponential Fourier series

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Exponential Fourier SeriesEach sinusoid of frequency ω can be expressed as thesum of two exponentials ejωt and e-jωt (using Euler’sf l )formula)

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Exponential Fourier Series

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Example 2.10Find the exponential Fourier series for the followingsignal

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Example 2.10

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Example 2.10

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Exponential Fourier Spectra

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Exponential Fourier Spectra

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From Example 2.10

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From Example 2.10

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From Example 2.10

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Exponential Fourier Spectra1. The spectra exist for

positive as well as negativel f (th f )values of ω (the frequency).

2. The amplitude spectrum isan even function of ω andthe angle spectrum is an oddfunction of ω.

3 Th i l ti

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3. There is a close connectionbetween these spectra andthe spectra of thecorresponding trigonometricFourier Series for Ψ(t).

Exponential Fourier SpectraTrigonometric Fourier SeriesExponential Fourier Series

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Example 2.11Find the exponential Fourier series for the periodicsquare wave ω(t) shown below:

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Example 2.11

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Example 2.11

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Example 2.12Find the exponential Fourier series and sketch thecorresponding spectra for the impulse train δTo(t) shownb lbelow:

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Example 2.12

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Example 2.12

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Parseval’s Theorem• A periodic signal g(t) is a power signal.

• The power Pg of g(t) is equal to the power of itsThe power Pg of g(t) is equal to the power of itsFourier series.

• The power of the Fourier Series is equal to the sum ofthe powers of its Fourier components.

• For the trigonometric Fourier series

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Parseval’s TheoremFor the exponential Fourier series

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Homework #2 •Solve the following problems: Due to one weekfrom today

•2.3-4

•2.4-1

•2.4-2

•2.6-1

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•2.8-4

Problem 2.5-2

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Problem 2.5-2

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Problem 2.8-3

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Problem 2.8-3

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Problem 2.8-3

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Problem 2.8-3

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chapter 2 introduction to signals students

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