Chapter 2

Signals and Spectra

(All sections, except Section 8, are covered.)

Physically Realizable Waveform

1. Non zero over finite duration (finite energy)

2. Non zero over finite frequency range (physical limitation of media)

3. Continuous in time (finite bandwidth)4. Finite peak value (physical limitation of

equipment)5. Real valued (must be observable)

• Power Signal: finite power, infinite energy

• Energy Signal: finite energy, non-zero power over limited time

• All physical signals are energy signals. Nothing can have infinite power. However, mathematically it is more convenient to deal with power signals. We will use power signals to approximate the behavior of energy signals over the time intervals of interest.

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power inSystem

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signal, s(t)System

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The phasor is a complex number that carries the amplitude and phase angle information of a sinusoidal function. It does not include the angular frequency.

Euler’s identify :

)cos()(

cos

sincos

)(

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Given

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Fourier Transform

Signal is a measurable, physical quantity which carries information. In time, it is quantified as w(t). Sometimes it is convenient to view through its frequency components.

Fourier Transform (FT) is a mathematical tool to identify the presence of frequency component for any wave form.

(Hz).frequency theis and of FT thedenotes where

)()( 2

f

dtetww(t)fW ftj

F

F

W(f)w(t)

W(f)w(t)

dfefWfWtw ftj

FT. under thepair matcheduniquely a thusare and

. of FT inverse thedenotes where

)()()( ,Conversely 2

1-

1-

F

F

πfjπfj

edteeW(f)

t

tew(t)

πf)t-(πft-j-t

t

21

1

21

0 0

0

:Example

0

212

0

Note: It is in general difficult to evaluate the FT integrationsfor arbitrary functions. There are certain well known functionsused in the FT along with the properties of the FT.

Properties of Fourier Transform

1. If w(t) is real, W(-f) = W*(f).

2. Linearity:

a1w1(t)+a2w2(t) a1W1(f) + a2 W2(f)

3. Time delay: w(t – T) = W(f) e-j2fT

4. Frequency Translation:

w(t) ej2fot W(f – fo)

5. Convolution: w1(t) w2(t) W1(f)W2(f)

6. Multiplication: w1(t)w2(t) W1(f) W2(f)Note:* is complex conjugate. is convolution integral.X

X

X

Parseval’s Theorem

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) :energy normalized Total

hertz)per joules (unit )( :(ESD)density spectralEnergy

).( ofenergy theis

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function allfor )0()()( satisfies ),( function, delta Dirac

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xx x

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Unit Step Function

x). δ dx

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x

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x

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x

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t

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20

21

function,r Rectangula

More Commonly Used Functions

Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-

0

Spectrum of Sine Wave

Figure 2–8 Waveform and spectrum of a switched sinusoid.Spectrum of Truncated Sine Wave

2

0

21

0

21

20

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21

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)(

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etw

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t

Example: Double Exponential

Convolution

dtwwtwtwtw

)()()()()( 21213

Supposed t is fixed at an arbitrary value.Within the integration, w2(t-) is a “horizontally flipped about=0, and move to the right by t version” of w2(). Now, multiply w1() with w2(t-) for each point of .Then, integrate over - < < . The result is w3(t) for this fixed value of t.Repeat this process for all values of t, - < t < .

)()()(

).()(

2

1

)(

Example

213

2

1

twtwtw

tuetw

T

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T

t

Power Spectrum Density

)0()()(

)()( :Theorem Khintchine-Wiener

)()(1

lim)()()( :ationAutocorrel

)()(power average normalized

)()( wherehertzper watts)(

lim)(

22

2

2

2

2

w

w

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. and ),( ),( Find

elsewhere0

11Given

:Example

1

1

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Orthogonal Series Representation

l.orthonorma

mutually is,...3,2,1 , allfor 1 If . 0

)(*

if interval on the orthogonalmutually is,...3,2,1 , functions, ofset A .Definition

0

1 ith function w deltaKronecker is .Definition

.0)(*

if interval on the orthogonal are and functions, Two .Definition

space.ector abstract van in elements as noise and signals ngrepresenti of method a is

series Orthogonal series. orthogonal of use theis noise and signalsrepresent way toconvenientA

functions. almathematic form closedby drepresente be todifficult are noise and signals general,In

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n

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l.orthonormamutually is ,...3,2,1 ,1

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Therefore,

. 1

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interval over the orthogonal are ,...3,2,1 , that Show Example.

2

2

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Examples of Orthogonal Functions

Sinusoids Polynomials Square Waves

).assumption goverlappin-non (by the 0)(*)(

)otherwise. 0 and, if 1 (because )()(*)(

)(*)(

)(*)(

)( )( )(*)(

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21

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2221

22

2121

21

2121

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)(* )(* )(*(t)

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22

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321 ),( from )( of Generation n ,,,nttw

later. see will WeYes.

y?similar wa ain )( from 321 , find way toa thereIs n

tw,,,na

Fourier Series (pages 71 – 78 not covered)

space. vector real ain seriesFourier of formknown well theis This

.0 allfor sin) 2

and

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0) 1

wheresin cos ely,Alternativ

. 1

where energy, finite with waveperiodic aFor Theorem.

.)( of period theis and

2

2 where,...3,2,1 ,set basis orthogonal a has SeriesFourier Complex

Series.Fourier complex aby drepresente becan energy finite with function periodAny

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0

0

10

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022

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. real, is If 1.

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: theoremsParseval' of Proof

2

*

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0

*

2*2

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00

2

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ommn

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dt w(t)w*(t)T

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n

(t)

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.frequency at outputs theand inputs theof angles phase theand magnitudes therelates

odd. is andeven is Then, function. real a is system, physical aFor

function.) responsefrequency thecalled also is( )(

ly equivalentor

)()( :)(Function Transfer

)()( )(arbitrary For

).(Causality 0for ,0 systems, physicalFor

)()( if )()(

.completely system thedescribes

, function, response impulse thedelay), timeartificial(without systemsinvariant elinear timFor

H(f)πft H(f)Ay(t)πft A x(t)

fH(f)

H(f)H(f)h(t)

H(f) X(f)

fYH(f)X(f)H(f)Y(f)

fHthfH

dthxthx(t)x(t), y(t)

t h(t)

ttxthty

h(t)

Impulse Response

. than)less 3(or of half a is ofpower the,At

.350log10log10

becausefrequency 3 thecalled is . if 5.0)(

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1

1)( :FunctionTransfer Power

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00

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equation. above theof sidesboth of TransformFourier theTake

)()()(

)()( and )( )()(

tageoutput vol :(t)

ageinput volt :(t)

Filter :Example

2

2

x(t) dBy(t)f

.)(f G

-dBffffG

πRCf

f

fH(f)fG

RC

t

te

th

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. offunction linear a is .2

. allfor constant is )( .1

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)(

)()(

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)()(

if less,distortion be tosaid is systemion Communicat

2

2

fH(f)

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c

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versa. viceand limited, timebecannot waveformdbandlimiteA Theorem.

.for 0

if seconds toed timelimity)(absolutel be tosaid is A waveform :.Definition

.for 0

if herz todbandlimite y)(absolutel be tosaid is A waveform .Definition

Ttw(t)

Tw(t)

BfW(f)

Bw(t)

Note: Sections 2.7 and 2.9 will be covered briefly. Section 2.8 will not be covered.

functions. Sinc of sumcertain a toequivalent are signals dBandlimite :tionInterpreta

frequency.Nyquist theasknown is

.2 :error without hertz) (to waveformdbandlimite at reconstruc To

)./( ,2 and hertz todbandlimite is If

value.minimumcertain asatisfy tohas Note.

./

/sin)( where

/

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by interval over the drepresente bemay signal)energy finite (i.e., waveformphysicalAny

.321 samples thefrom tedreconstruc completely becan then smooth,ly sufficient is If

)frequency. sampling : period. sampling :( Usually,in time.

points, ofset discrete aat signal evaluating of process a is Sampling .Definition

min

min

14321

s

s

sns

s

ss

sssn

ss

ssn

n

sss

s

f

BfB

fnwaBfBw(t)

f

dtfntf

fntftwfa

fntf

fntfatw

t-

, ...,,), nw(tw(t) w(t)

fT.f

nnTt, ....,t,t,tt

w(t)

Theorem Sampling

Sampling

w(t)

t1 t5t4t3t2

t

.321 samples thefrom tedreconstruc completely becan then smooth,ly sufficient is If

system.ion communicat digital through signals analog ofon tranmissi thegoverns

which principle lfundamenta a is This 321 samples thefrom tedreconstruc

compltely becan 2

1 of intervalsat known is and hertz, todbandlimite is If

, ...,,), nw(tw(t) w(t)

.,...,,), nw(nT

, w(t)B

Tw(t)Bw(t)

n

s

s

)(1

)(1

)(1

1)(

1)(

)()(

).( of series sampled impulse : ).()(Let

2

2

2

ss

ss

tfjn

s

tfjn

ss

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ss

sss

sss

nffWT

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eT

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s

s

s

F

F

w(t)

t1 t5t4t3t2

t

ws(t)

t1 t5t4t3t2

t

w(t)

(t-nTs)

ws(t)sampler

Low Pass FilterIdeal cutoff at B

w(t)

If fs < 2B, the sampling rate is insufficient, i.e., there aren’t enough samplesto reconstruct the original waveform. Aliasing or spectral folding. The original waveform cannot be reconstructed without distortion.

Dimensionality Theorem

For a bandlimited waveform with bandwidth B hertz, if the waveform can be completely specified (i.e., later reconstructed by an ideal low pass filter) by N=2BTo samples during a time period of To,then N is the dimension of the wave form.

Conversely, to estimate the bandwidth of a waveform, find a numberN such that N=2BTo is the minimum number of samples needed to reconstructthe waveform during a time period To. Then B follows.

As To , any approximation goes to zero.

A slightly modified version of this theorem is the Bandpass Dimensionality Theorem: Any bandpass waveform (with bandwidth B) can be determined by N=2BTo samples taken during a period of To.

Data Rate Theorem (Corollary to Dimensionality Theorem)

The maximum number of independent quantities which can be transmitted by a bandlimited channel (B hertz) during a time period of To is N=2BTo.

Definition. The baud rate of a digital communication system is the rate of symbols or quantities transmitted per second.

From the Data Rate Theorem, the maximum baud rate of a system with a bandlimited channel (B hertz) is 2B symbols / second.

Definition. The data rate (or bit rate), R, of a system is the baud rate times the information content per symbol (H): R= 2BH bits / second

Suppose a source transmits one of M equally likely symbols. The information

content of each symbol: H = log 2 (1/probablity of each symbol) = log2 M R= 2Blog2 M Data rate is (also known as the Channel Capacity) is determined by (1) channelbandwidth and (2) channel SNR.

bandwidth FCC 7.

bandwidthPower 6.

bandwidth spectrum Bounded 5.

bandwidth null toNull 4.

bandwidth Equivalent 3.

s)definition useful (Less

.over of value

maximum theof 2

1 ,any For : bandwidth power) half(or 3 2.

.0 ,or For : bandwidth Absolute 1.

terms)used(Commonly

12

2

2

*2*112

1212

fffH(f)

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W(f) f ff ff-f

Bandwidth of sDefinitionDifferent