(PDF) notes-Digital Communication Lecture-1.ppt – DOKUMEN.TIPS

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Digital CommunicationLecture-1, Prof. Dr.

Habibullah JamalUnder Graduate, Spring 2008

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Text: Digital Communications: Fundamentals and
Applications, By “Bernard Sklar”, Prentice

Hall, 2nd

 ed, 2001. Probability and andom !ignals “or
#lectrical #ngineers, $eon %arcia

References:

 Digital Communications, Fourt& #dition, ‘.%. Proa(is,
)c%ra* Hill, 2000.  

Course Books

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Course Outline Review of Probability

Signal and Spectra (Chapter ! “or#atting and $a%e band
&odulation (Chapter 2! $a%e band ‘e#odulation’etection (Chapter
)! Channel Coding (Chapter *, + and 8! $and pa%% &odulation and
‘e#od’etect

(Chapter -! Spread Spectru# .echni/ue% (Chapter 2!

Synchroniation (Chapter 0! Source Coding (Chapter )! “ading
Channel% (Chapter 1!

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Toda!s “oal

Review of $a%ic Probability

‘igital Co##unication $a%ic

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Communication

&ain purpo%e of co##unication i% to tran%fer infor#ation

fro# a %ource to a recipient via a channel or #ediu#

$a%ic bloc diagra# of a co##unication %y%te#3

Source Transmitter Receiver  

Recipient

Channel

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Brief Descri#tion

Source: analog or digital

Transmitter: tran%ducer, a#plifier, #odulator, o%cillator,
power

a#p, antenna

Channel: eg cable, optical fibre, free %pace

Receiver: antenna, a#plifier, de#odulator, o%cillator,
powera#plifier, tran%ducer 

Recipient: eg per%on, (loud! %peaer, co#puter 

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Types of information

4oice, data, video, #u%ic, e#ail etc

Types of communication systems

Public Switched .elephone 5etwor (voice,fa6,#ode#!

Satellite %y%te#%

Radio,.4 broadca%ting

Cellular phone%

Co#puter networ% (75%, 95%, 975%!

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$nformation %e#resentation

Co##unication %y%te# convert% infor#ation into
electricalelectro#agneticoptical %ignal% appropriate for the
tran%#i%%ion#ediu#

 nalog %y%te#% convert analog #e%%age into %ignal% that
canpropagate through the channel

‘igital %y%te#% convert bit%(digit%, %y#bol%! into %ignal%

Co#puter% naturally generate infor#ation a% character%bit%
&o%t infor#ation can be converted into bit%  nalog %ignal%
converted to bit% by %a#pling and /uantiing

(‘ conver%ion!

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&h digital’

‘igital techni/ue% need to di%tingui%h between di%crete
%y#bol%allowing regeneration ver%u% a#plification

Good proce%%ing techni/ue% are available for digital %ignal%,
%ucha% #ediu#

‘ata co#pre%%ion (or %ource coding! :rror Correction (or channel
coding!(‘ conver%ion! :/ualiation Security

:a%y to #i6 %ignal% and data u%ing digital techni/ue%

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$a%ic ‘igital Co##unication .ran%for#ation% “or#attingSource
Coding .ran%for#% %ource info into digital %y#bol% (digitiation!
Select% co#patible wavefor#% (#atching function! ;ntroduce%
redundancy which facilitate% accurate decoding

de%pite error%

It is essential for reliable communication

&odulation’e#odulation &odulation i% the proce%% of
#odifying the info %ignal to

facilitate tran%#i%%ion ‘e#odulation rever%e% the proce%% of
#odulation ;t

involve% the detection and retrieval of the info %ignal .ype%
Coherent3 Re/uire% a reference info for detection 5oncoherent3 ‘oe%
not re/uire reference pha%e infor#ation

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Basic Digital Communication

Transformations  Coding’ecoding

.ran%lating info bit% to tran%#itter data %y#bol%

.echni/ue% u%ed to enhance info %ignal %o that they arele%%
vulnerable to channel i#pair#ent (eg noi%e, fading, 

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Performance (etrics

 nalog Co##unication Sy%te#% &etric i% fidelity3 want
S5R typically u%ed a% perfor#ance #etric

‘igital Co##unication Sy%te#% &etric% are data rate (R bp%!
and probability of bit error

Sy#bol% already nown at the receiver 

9ithout noi%edi%tortion%ync proble#, we will never

#ae bit error%

ˆ ( ) ( )m t m t  ≈

( )ˆ( )b P p b b= ≠

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(ain Points

.ran%#itter% #odulate analog #e%%age% or bit% in ca%e of a
‘CSfor tran%#i%%ion over a channel

Receiver% recreate %ignal% or bit% fro# received %ignal
(#itigate

channel effect%!

Perfor#ance #etric for analog %y%te#% i% fidelity, for digital
it i%

the bit rate and error probability

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&h Digital Communications’  :a%y to regenerate the
di%torted %ignal Regenerative repeater% along the tran%#i%%ion path
can

detect a digital %ignal and retran%#it a new, clean (noi%efree!
%ignal

.he%e repeater% prevent accu#ulation of noi%e along thepath

.hi% i% not po%%ible with analog co##unication%y%te#% .wo=%tate
%ignal repre%entation

.he input to a digital %y%te# i% in the for# of a%e/uence of
bit% (binary or &>ary! ;##unity to di%tortion and
interference ‘igital co##unication i% rugged in the %en%e that it
i% #ore

i##une to channel noi%e and di%tortion

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&h Digital Communications’  ?ardware i% #ore
fle6ible

‘igital hardware i#ple#entation i% fle6ible and per#it%the u%e
of #icroproce%%or%, #ini=proce%%or%, digital%witching and 47S;

Shorter de%ign and production cycle

7ow co%t .he u%e of 7S; and 47S; in the de%ign of co#ponent%

and %y%te#% have re%ulted in lower co%t :a%ier and #ore
efficient to #ultiple6 %everal digital

%ignal% ‘igital #ultiple6ing techni/ue% @ .i#e A Code
‘ivi%ion

&ultiple cce%% = are ea%ier to i#ple#ent than
analogtechni/ue% %uch a% “re/uency ‘ivi%ion &ultiple cce%%

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&h Digital Communications’  Can co#bine different
%ignal type% @ data, voice, te6t, etc

‘ata co##unication in co#puter% i% digital in naturewherea%
voice co##unication between people i% analog innature

.he two type% of co##unication are difficult to co#bine overthe
%a#e #ediu# in the analog do#ain

U%ing digital techni/ue%, it i% po%%ible to co#bineboth for#at
for tran%#i%%ion through a co##on#ediu#

:ncryption and privacy techni/ue% are ea%ier toi#ple#ent

$etter overall perfor#ance ‘igital co##unication i% inherently
#ore efficient thananalog in realiing the e6change of S5R for
bandwidth

‘igital %ignal% can be coded to yield e6tre#ely low rate%
andhigh fidelity a% well a% privacy

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&h Digital Communications’ 

‘i%advantage%

Re/uire% reliable B%ynchroniation

Re/uire% ‘ conver%ion% at high rate

Re/uire% larger bandwidth

5ongraceful degradation

Perfor#ance Criteria

Probability of error or $it :rror Rate

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Goals in Communication SystemDesign

.o #a6i#ie tran%#i%%ion rate,R 

.o #a6i#ie %y%te# utiliation, U 

.o #ini#ie bit error rate, P e

.o #ini#ie re/uired %y%te#% bandwidth, W 

.o #ini#ie %y%te# co#ple6ity, C  x 

.o #ini#ie re/uired power, E b /N o

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Comparative Analysis of Analog and

Digital Communication

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Digital Signal Nomenclature

Information Source ‘i%crete output value% eg Deyboard

 nalog %ignal %ource eg output of a #icrophone

Character  &e#ber of an alphanu#eric%y#bol ( to E, 0 to
F!

Character% can be #apped into a %e/uence of binary digit%

u%ing one of the %tandardied code% %uch a% ASCII: American
Standard Code for Information Interchange

EBCDIC: Etended Binary Coded Decimal Interchange Code

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Digital Signal Nomenclature

Digital !essage &e%%age% con%tructed fro# a finite nu#ber of
%y#bol% eg, printed

language con%i%t% of 2* letter%, 0 nu#ber%, B%pace and
%everal

punctuation #ar% ?ence a te6t i% a digital #e%%age con%tructed
fro#

about 10 %y#bol%

&or%e=coded telegraph #e%%age i% a digital #e%%age
con%tructed fro#

two %y#bol% B!ar” and BSpace

! # ary   digital #e%%age con%tructed with M %y#bol%

Digital $aveform Current or voltage wavefor# that repre%ent% a
digital %y#bol

Bit Rate  ctual rate at which infor#ation i% tran%#itted
per %econd

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Digital Signal Nomenclature

Baud Rate

Refer% to the rate at which the %ignaling ele#ent% are

tran%#itted, ie nu#ber of %ignaling ele#ent% per

%econd

Bit Error Rate

.he probability that one of the bit% i% in error or %i#ply

the probability of error 

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1.2 Classification Of Signals1. Deterministic and %andom
)ignals 

  %ignal i% deterministic  #ean% that there i% no
uncertainty with

re%pect to it% value at any ti#e

‘eter#ini%tic wavefor#% are #odeled by e6plicit #athe#atical

e6pre%%ion%, e6a#ple3

  %ignal i% random #ean% that there i% %o#e degree
of

uncertainty before the %ignal actually occur%

Rando# wavefor#% Rando# proce%%e% when e6a#ined over a

long period #ay e6hibit certain regularitie% that can be
de%cribed

in ter#% of probabilitie% and %tati%tical average%

x(t) = Cos(!”t)

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*. Periodic and +on-#eriodic )ignals

  %ignal 6(t ! i% called periodic in time if
there e6i%t% a con%tant

T 0  H 0 %uch that

(2!

t denote% ti#e

T 0  i% the period
of x (t !.

“x(t) = x(t # T ) for $ % t %∞ ∞

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. nalog and Discrete )ignals

 n analog signal x (t ! i% a continuou% function
of ti#e that i%, x (t !

i% uni/uely defined for all t 

  discrete signal x (kT ! i% one that e6i%t% only
at di%crete ti#e% it

i% characteried by a %e/uence of nu#ber% defined for eachti#e,
kT, where

k i% an integer

T i% a fi6ed ti#e interval

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. /nerg and Po0er )ignals

.he perfor#ance of a co##unication %y%te# depend% on thereceived
%ignal energ! higher energy %ignal% are detected #orereliably (with
fewer error%! than are lower energy %ignal%

 x (t ! i% cla%%ified a% an energ signal if, and
only if, it ha% nonero

but finite energy (0 I ”  x  I J! for all
ti#e, where3 

(+!

 n energy %ignal ha% finite energy but #ero average
po$er.

Signal% that are both deter#ini%tic and non=periodic
arecla%%ified a% energy %ignal%

T&’

‘ ‘

xT & ‘

= x (t) dt = x (t) dtlimT 

∞

→∞   − −∞∫ ∫ 

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%o$er i% the rate at which energy i% delivered

  %ignal i% defined a% a power %ignal if, and only if, it
ha% finite

but nonero power (0 I %  x  I J! for all
ti#e, where

(8!

Power %ignal ha% finite average power but infinite energ.

 % a general rule, periodic %ignal% and rando# %ignal%
are

cla%%ified a% power %ignal%

. /nerg and Po0er )ignals

T&’

‘

xT & ‘

! = x (t) dt

Tlim

T →∞   −∫ 

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&irac delta function ‘ (t ! or i#pul%e function i%
an ab%tractionKan

infinitely large a#plitude pul%e, with ero pul%e width, and
unity

weight (area under the pul%e!, concentrated at the point where
it%

argu#ent i% ero

 (F!

(0!

  (!

Sifting or Sa#pling Property

(2!

. The 2nit $m#ulse 3unction

(t) dt = !

(t) = ” for t “

(t) is *ounded at t “

δ 

δ 

δ 

∞

−∞

≠=

∫ 

” “( ) (t$t )dt = x(t ) x t   δ ∞

−∞∫ 

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1.3 Spectral Density

.he spectral densit of a %ignal characterie% the di%tribution
of

the %ignalL% energy or power in the fre/uency do#ain

.hi% concept i% particularly i#portant when con%idering
filtering in

co##unication %y%te#% while evaluating the %ignal and noi%e
atthe filter output

.he energy %pectral den%ity (:S’! or the power %pectral
den%ity

(PS’! i% u%ed in the evaluation

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1. /nerg )#ectral Densit 4/)D5

:nergy %pectral den%ity de%cribe% the %ignal energy per unit

bandwidth #ea%ured in

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*. Po0er )#ectral Densit 4P)D5

.he po$er spectral densit (PS’! function (6(f ! of the
periodic

%ignal x (t ! i% a real, even, and nonnegative
function of fre/uencythat give% the di%tribution of the power
of x (t ! in the fre/uency

do#ain

PS’ i% repre%ented a%3

  (8!

9herea% the average power of a periodic %ignal 6(t! i%

repre%ented a%3

  (+!

U%ing PS’, the average nor#alied power of a real=valued%ignal i%
repre%ented a%3

  (F!

‘x n “

n=$

– (f ) = +C + ( )  f nf  δ ∞

∞

−

∑”

“

&’

‘ ‘

x nn=$” & ‘

! x (t) dt +C +

T 

T T 

∞

∞−

= = ∑∫ 

x x x

“

– (f) df ‘ – (f) df  ∞ ∞

−∞

= =∫ ∫ 

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1.4 Autocorrelation1. utocorrelation of an /nerg )ignal

Correlation i% a #atching proce%% autocorrelation refer% to
the

#atching of a %ignal with a delayed ver%ion of it%elf

 utocorrelation function of a real=valued energy
%ignal x (t ! i%

defined a%3

(2!

.he autocorrelation function R6(.! provide% a #ea%ure of how

clo%ely the %ignal #atche% a copy of it%elf a% the copy i%
%hifted. unit% in ti#e

R6(.! i% not a function of ti#e it i% only a function of the
ti#e

difference . between the wavefor# and it% %hifted copy

xR ( ) = x(t) x (t # ) dt for $ % %τ τ τ ∞

−∞

∞ ∞∫ 

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1. utocorrelation of an /nerg )ignal

.he autocorrelation function of a real=valued energ %ignal
ha%

the following propertie%3

%y##etrical in about ero

  #a6i#u# value occur% at the origin

autocorrelation and :S’ for# a

“ourier tran%for# pair, a% de%ignated

by the double=headed arrow%

  value at the origin i% e/ual to

the energy of the %ignal

x xR ( ) =R ($ )τ τ 

x xR ( ) R (“) for allτ τ ≤

x xR ( ) (f)τ ψ ↔

‘

xR (“) x (t) dt

∞

−∞

= ∫ 

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*. utocorrelation of a Po0er )ignal

 utocorrelation function of a real=valued power
%ignal x (t ! i%

defined a%3

(22!

9hen the power %ignal x (t ! i% periodic with
period T 0, the

autocorrelation function can be e6pre%%ed a%

(2)!

& ‘

xT & ‘

!R ( ) x(t) x (t # ) dt for $ % %lim

T 

T T 

τ τ τ →∞   −

= ∞ ∞∫ 

“

“

& ‘

x

” & ‘

!R ( ) x(t) x (t # ) dt for $ % %

T 

T T 

τ τ τ −

= ∞ ∞∫ 

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*. utocorrelation of a Po0er )ignal

.he autocorrelation function of a real=valued periodic
%ignal ha%

the following propertie% %i#ilar to tho%e of an energy
%ignal3

%y##etrical in about ero

#a6i#u# value occur% at the origin

  autocorrelation and PS’ for# a

“ourier tran%for# pair 

value at the origin i% e/ual to theaverage power of the
%ignal

x xR ( ) =R ($ )τ τ 

x xR ( ) R (“) for allτ τ ≤

x xR ( ) (f)Gτ    ↔

“

“

T & ‘

‘x

” T & ‘

!R (“) x (t)dtT −

= ∫ 

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1.5 Random Signals1. %andom 6ariables

 ll u%eful #e%%age %ignal% appear rando# that i%, the
receiverdoe% not now, a priori, which of the po%%ible wavefor# have
been%ent

7et a random variable ) ( *! repre%ent the functional
relation%hip

between a rando# event * and a real nu#ber

.he +cumulative distribution function
–  ) ( x ! of the rando# variable )
i% given by

(2-!

 nother u%eful function relating to the rando#
variable ) i% the probabilit densit function (pdf!

  (21!

( ) ( ) X  F x P X x= ≤

( )

( )  X 

 X 

dF x

 P x dx=

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1.1 /nsemble 7erages

Te first moment of a

 probabilit distribution of arandom variable ) is
calledmean value m ) , or expectedvalue of a rando#
variable )

Te second moment of a probabilit distribution is
temean/square value of ) 

Central moments are the#o#ent% of the
differencebetween ) and m )  and the%econd
central #o#ent i% the

variance of ) 4ariance i% e/ual to the

difference between the #ean=%/uare value and the %/uare ofthe
#ean

/ 0 ( ) X X m E X x p x dx

∞

−∞= = ∫ ‘ ‘/ 0 ( ) X  E X x p x dx

∞

−∞

= ∫ 

‘

‘

var( ) /( ) 0

( ) ( )

 X 

 X X 

 X E X m

 x m p x dx

∞

−∞

= −

= −∫ 

‘ ‘var( ) / 0 / 0 X E X E X = −

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2. Random Processes

  rando# proce%% ) ( *, t ! can be
viewed a% a function of two

variable%3 an event * and time.

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1.5.2.1 Statistical Averages of aRandom Process   rando#
proce%% who%e di%tribution function% are continuou% can

be de%cribed %tati%tically with a probability den%ity function
(pdf!

  partial de%cription con%i%ting of the #ean and
autocorrelation

function are often ade/uate for the need% of co##unication

%y%te#%

&ean of the rando# proce%% ) (t ! 3

()0!

 utocorrelation function of the rando#
proce%% ) (t !()!

/ ( )0 ( ) ( )k k X X k  

 E X t xp x dx m t 

∞

−∞

= =∫ 

! ‘ ! ‘( 1 ) / ( ) ( )0 X  R t t E X t X
t =

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1.5.5. Noise in CommunicationSystems

.he ter# noise refer% to un$anted electrical %ignal% that
arealway% pre%ent in electrical %y%te#% eg %par=plug ignitionnoi%e,
%witching tran%ient%, and other radiating
electro#agnetic%ignal%

Can de%cribe ther#al noi%e a% a ero=#ean (aussian
rando#proce%%

  Gau%%ian proce%% n(t ! i% a rando# function who%e
a#plitude atany arbitrary ti#e t i% %tati%tically characteried
by the Gau%%ian

probability den%ity function

(-0!

‘! !( ) exp

”

n p n

σ σ π 

 = − ÷  

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Noise in Communication Systems

.he normali#ed or standardi#ed (aussian densit function of a

ero=#ean proce%% i% obtained by a%%u#ing unit variance

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1.5.5.1 White Noise

.he pri#ary %pectral characteri%tic of ther#al noi%e i% that
it%power %pectral den%ity i% te same for all fre/uencie% of
intere%tin #o%t co##unication %y%te#%

Power %pectral den%ity (n(f !

  (-2!

 utocorrelation function of white noi%e i%

(-)!

.he average power % n of white noi%e i% infinite

(–!

“( ) &

‘n

 N G f watts hertz  =

! “( ) / ( )0 ( )’

n n

 N  R G f  τ δ τ −= ℑ =

“( )’

 N  p n df  

∞

−∞

= = ∞∫ 

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.he effect on the detection proce%% of a channel with
additive$ite (aussian noise (9G5! i% that the noi%e affect%
each

tran%#itted %y#bol independentl.

Such a channel i% called a memorless cannel.

.he ter# Badditive #ean% that the noi%e i% %i#ply
%uperi#po%ed

or added to the %ignal

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1.6 Signal ransmission t!roug!

“inear Systems

  %y%te# can be characteried e/ually well in the ti#e
do#ain

or the fre/uency do#ain, techni/ue% will be developed in
both

do#ain%

.he %y%te# i% a%%u#ed to be linear and ti#e invariant

;t i% al%o a%%u#ed that there i% no %tored energy in the
%y%te#

at the ti#e the input i% applied

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1.6.1. Impulse Response

.he linear ti#e invariant %y%te# or networ i% characteried in
theti#e do#ain by an i#pul%e re%pon%e (t !,to an input unit
i#pul%e

δ(t(-1!

.he re%pon%e of the networ to an arbitrary input %ignal x
(t !i%found by the convolution of x (t !with (t !

(-*!

.he %y%te# i% a%%u#ed to be causal,which #ean% that there canbe
no output prior to the ti#e, t N0,when the input i% applied

.he convolution integral can be e6pre%%ed a%3 

(-+a!

( ) ( ) ( ) ( ) y t h t when x t t δ = =

( ) ( ) ( ) ( ) ( ) y t x t h t x h t d τ τ
τ 

∞

−∞= ∗ = −∫ 

“

( ) ( ) ( ) y t x h t d τ τ τ  

∞

= −∫ 

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1.6.2. Freuency !rans”er Function

.he fre/uency=do#ain output %ignal 0 (f !i% obtained by taingthe
“ourier tran%for#

(-8!

-requenc transfer function or the frequenc response i%
defineda%3

  (-F!

  (10!

.he pha%e re%pon%e i% defined a%3

  (1!

( ) ( ) ( )Y f X f H f    =

( )

( )( )

( )

( ) ( )   j f  

Y f   H f  

 X f  

 H f H f e   θ 

=

=

! 2m/ ( )0( ) tanRe/ ( )0

 H f    f  

 H f  θ    −=

d d

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1.6.2.1. Random Processes and#inear Systems

;f a rando# proce%% for#% the input to a ti#e=

invariant linear %y%te#,the output will al%o be a

rando# proce%%

.he input power %pectral den%ity ( )  (f !and
the

output power %pectral den%ity (  (f !are related

a%3

(1)!

‘( ) ( ) ( )Y X G f G f H f    =

1 6 $ i i l i i

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1.6.$. Distortionless !ransmissionWhat is the required behavior
of an idealtransmission line?

.he output %ignal fro# an ideal tran%#i%%ion line #ay have
%o#eti#e delay and different a#plitude than the input

;t #u%t have no di%tortionKit #u%t have the %a#e %hape a%
theinput

“or ideal di%tortionle%% tran%#i%%ion3

(1-!

(11!

(1*!

Output %ignal in ti#e do#ain

Output %ignal in fre/uency do#ain

Sy%te# .ran%fer “unction

“( ) ( ) y t Kx t t = −

“

‘

( ) ( )  j ft 

Y f KX f e  π −

=

“‘( )  j ft 

 H f Ke  π −=

Wh t i th i d b h i f id l

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What is the required behavior of an idealtransmission line? .he
overall %y%te# re%pon%e #u%t have a con%tant #agnitude

re%pon%e .he pha%e %hift #u%t be linear with fre/uency  ll
of the %ignalL% fre/uency co#ponent% #u%t al%o arrive with

identical ti#e delay in order to add up correctly .i#e delay
t 0 i% related to the pha%e %hift θ and the
radian

fre/uency ω  N 2πf by3  t 0 (%econd%! N
θ (radian%! 2πf +radians1seconds (1+a!

 nother characteri%tic often u%ed to #ea%ure delay
di%tortion of a%ignal i% called envelope dela or group dela2

  (1+b!! ( )( )

‘

d f    f  

df  

θ τ 

π = −

1 6 $ 1 Id l Fil

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1.6.$.1. Ideal Filters

“or the ideal low=pa%% filter  tran%fer function with
bandwidth W f  N

f u hert can be written a%3

“igure (b! ;deal low=pa%% filter 

%&'()*

$here

 %&'(+*

 %&’,-*

( )( ) ( )   j f   H f H f e 
θ −=

! + +( )

” + +

u

u

 for f f   H f  

 for f f  

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.he i#pul%e re%pon%e of the ideal low=pa%% filter 3

“

“

!

‘

‘ ‘

‘ ( )

“

“

“

( ) / ( )0

( )

sin ‘ ( )’

‘ ( )

‘ sin ‘ ( )

u

u

u

u

  j ft 

  f  

  j ft    j ft 

  f  

  f  

  j f t t 

  f  

uu

u

u u

h t H f    

 H f e df  

e e df    

e df  

  f t t   f  

  f t t 

  f n f t t 

π 

π    π 

π 

π 

π 

−

∞

−∞

−

−

−

−

= ℑ

=

=

=

−=−

= −

∫ 

∫ 

∫ 

Id l Filt

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Ideal Filters

“or the ideal band=pa%% filter  

tran%fer function

“or the ideal high=pa%% filter  

tran%fer function

“igure (a! ;deal band=pa%% filter  “igure (c! ;deal
high=pa%% filter 

1 6 $ 2 R li &l Filt

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1.6.$.2. Reali%a&le Filters

.he %i#ple%t e6a#ple of a realiable low=pa%% filter an RC
filter 

  *)!

.igure &’&/

( )

‘! !( )

! ‘ ! (‘ )

  j f   H f e  j
f     f  

θ 

π    π −= =+ ℜ   + ℜ£   £

R li &l Filt

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Reali%a&le Filters

Phase characteristic of RC filter

“igure )

R li bl Filt

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Realizable Filters

.here are %everal u%eful appro6i#ation% to the ideal
low=pa%%filter characteri%tic and one of the%e i% the 3utter$ort
filter 

  (*1!

Butterworth ltersare popular because

they are the bestapproximation to the

ideal, in the sense ofmaximal fatness inthe lter passband.

‘

!( ) !

! ( & )n

n

u

 H f n f f  

= ≥+

1 ‘ ( d idt* +” Di it l D t

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n easy way to translate thespectrum of a low!pass or
basebandsi”nal x#t$ to a hi”her frequency isto multiply or
heterodyne thebaseband si”nal with a carrier wavecos
%πf ct

xc#t$ is called a double!sideband

#&’B$ modulated si”nal

 xc#t$ ( x#t$ cos %πf ct   #).*+$

rom the frequency shiftin”theorem

 -c#f$ ( )% /-#f!f c$ 0 -#f0f c$
1 #).*)$

2enerally the carrier wavefrequency is much hi”her than
thebandwidth of the baseband si”nal

f c 33 f m and therefore W&’B (
%f m

1.’. (and)idt* +” Digital Data 1.’.1 Baseband versus
Bandass

1 ‘ 2 (and idt* Dilemma

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.heore#% ofco##unication and

infor#ation theory are

ba%ed on the

a%%u#ption of strictl

bandlimited channel%

.he #athe#atical

de%cription of a real

%ignal doe% not per#it

the %ignal to be %trictlyduration li#ited and

%trictly bandli#ited

1.’.2 (and)idt* Dilemma

1 ‘ 2 (and)idt* Dilemma

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1.’.2 (and)idt* Dilemma

 ll bandwidth criteria have in co##on the atte#pt to
%pecify a

#ea%ure of the width, W, of a nonnegative real=valued
%pectral

den%ity defined for all fre/uencie% f I J

.he %ingle=%ided power %pectral den%ity for a %ingle
heterodyned

pul%e x c (t ! tae% the analytical for#3

#).*4$

‘

sin ( )( )

( )

 x

  f f T G f T 

  f f T 

π  

π  

−=   −

!i”erent Band#idth Criteria

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!i”erent Band#idth Criteria

(a! ?alf=power bandwidth

(b! :/uivalent rectangularor noi%e e/uivalentbandwidth

(c! 5ull=to=null bandwidth

(d! “ractional powercontain#entbandwidth

(e! $ounded power%pectral den%ity

(f! b%olute bandwidth