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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 

    ∞−

    = = ∑∫ 

    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 

    τ τ τ →∞   −

    = ∞ ∞∫ 

    "

    "

    & '

    x

    " & '

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

    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