Unit 1 PPT .d ocx.pdf ppt presentation

Engineering

1. DigitalCommunication
UnitNo.1
RandomProcessesandNoise
By

2. Prof.S.D.Shipne

3. Contents
•Introduction
•Mathematicaldefinitionofarandomprocess.
•Stationaryprocesses.
•Mean,CorrelationandCovarianceFunctions
Prof.S.D.Shipne

4. •Ergodicprocess
•TransmissionofarandomprocessthroughaLTIfilter
•Powerspectraldensity(PSD)
•Gaussianprocess
•NarrowbandNoise
•In-phaseandQuadraturecomponentsofnoise
Prof.S.D.Shipne

5. References
•T1:DigitalCommunicationSystems,
bySimonHaykin4thEd.
•R3:ModernDigital&Analog
CommunicationSystems,byB.P.Lathi4th
Ed.
Prof.S.D.Shipne

6. Objectives
•Tounderstandrandomsignalsandrandomprocess.
•Todefineandcalculatemean,correlation
andcovariancefunctionsofrandomprocess.
Prof.S.D.Shipne

7. •Todescribeanddefinevarioustypesofrandomprocess
e.g.ErgodicandGaussianprocessetc.
•Tounderstandtransmissionofarandom
processthroughalinearfilter.
•Tounderstandconceptofnarrow-bandnoiseand
studyitsin-phaseandquadraturecomponents.
Prof.S.D.Shipne

8. ImportanceofRandomProcesses
•Randomvariablesandprocessestalkabout
quantitiesandsignalswhichareunknownin
advance
•Thedatasentthroughacommunicationsystem
Prof.S.D.Shipne

9. ismodeledasrandomvariable
•Thenoise,interference,andfadingintroducedby
thechannelcanallbemodeledasrandomprocesses
•Eventhemeasureofperformance(Probabilityof
BitError)isexpressedintermsofaprobability
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10. Deterministicandrandom
Processes:
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11. •bothcontinuousfunctionsoftime(usually)
•Deterministicprocesses:
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12. physicalprocessisrepresentedby
explicitmathematicalrelation
•Randomprocesses:
resultofalargenumberofseparatecauses.
Describedinprobabilistictermsandbyproperties
whichareaverages
Prof.S.D.Shipne

13. RandomProcesses:significance
•Inpracticalproblems,wedealwithtimevarying
waveformswhosevalueatatimeisrandominnature.
Randomnessorunpredictabilityisafundamentalproperty
ofinformation.
Forexample,
Prof.S.D.Shipne

14. •Thespeechwaveformrecordedbyamicrophone,
•Thesignalreceivedbycommunicationreceiver
•Thetemperatureofacertaincityatnoonor
•Thedailyrecordofstock-marketdatarepresents
randomvariablesthatchangewithtime.
•Howdowecharacterizesuchdata?Suchdata
arecharacterizedasrandomorstochastic
Prof.S.D.Shipne

15. processes.
Prof.S.D.Shipne

16. RandomProcesses:significancecontd.
•Arandomvariablemapseachsamplepointin
thesamplespacetoapointintherealline.
•Arandomprocessmapseachsamplepointto
Prof.S.D.Shipne

17. awaveform.
•Thusarandomprocessisafunctionofthesample
point‘s’andindexvariable’t’andmaybewrittenas
X(t,s)
forfixedt=t0X(t0,s)isarandomvariable.
Prof.S.D.Shipne

18. e.g.Temperaturerecordsfortheday
X(t,ξ
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19. X(t,ξ
n
k
X(t,ξ
2
X(t,ξ
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20. 1
)
)
X(t,ξ)
??????
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21. ??????
)
??????
)
0
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22. t
1
t
2
t
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23. Ensembleandsamplefunction
•Thecollectionofallpossiblewaveformis
knownasEnsemble.(samplespaceinrandom
variable)
•Eachwaveforminthecollectionisa
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24. samplefunction(samplepoint)
•Amplitudesofallthesamplefunctionsat
t=t0isensemblestatistics.
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25. Classificationofrandomprocesses
•StationaryandNonstationary
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26. •Wide-SenseorWeaklyStationary
•Ergodic
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27. StrictlyStationaryrandomprocess:
•Ensembleaveragesdonotvarywithtime
•Thestatisticalcharacterizationofthe
processistimeinvariant.
•ThePDFsobtainedatanyinstantsmustbe
Prof.S.D.Shipne

28. identical.
•TheAutocorrelationfunctionmustbe
R
x
(t
1,
t
2
)
=
R
x(t
2
Prof.S.D.Shipne

29. −t
1)
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30. WidesenseStationaryrandomprocess:
•Meanvalueisconstant
•Autocorrelationfunctionisindependentof
theshifts
Prof.S.D.Shipne

31. R
x
(t
1,
t
2
)
=
R
x(t
2
−t
1)=R
x(τ)
Prof.S.D.Shipne

32. •Allstationaryprocessesare
wide-sensestationarybutconverseis
nottrue.
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33. Stationarity
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34. Ergodicrandomprocess:
•Ensembleaveragesareequaltotimeaveragesof
anysamplefunction.
T2
1
[]
Prof.S.D.Shipne

35. 1
T2
Eμ
X
(T)
=x(t)
=lim
T→∞
T
∫
x(t)d
t
−T2
=lim
T→∞
T
∫
μ
X
Prof.S.D.Shipne

36. dt
−T2
=μ
X
•stationaryprocessinwhichaveragesfromasingle
recordarethesameasthoseobtainedfrom
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37. averagingovertheensemble
•Moststationaryrandomprocessescanbe
treatedasErgodic
Prof.S.D.Shipne

38. TerminologyDescribingRandomProcesses
•Astationaryrandomprocesshasstatistical
propertieswhichdonotchangeatalltime
•Awidesensestationary(WSS)processhasa
meanandautocorrelationfunctionwhichdonot
Prof.S.D.Shipne

39. changewithtime
•ArandomprocessisErgodicifthetimeaverage
alwaysconvergestothestatisticalaverage
•Unlessspecified,wewillassumethatall
randomprocessesareWSSandErgodic
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40. Ergodic
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41. Mean,Correlationandcovariance
•Meanvalue:
function
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42. x(t)
x
μ
X(t)=
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43. E[X (t)]=
time,t
T
∞
∫
xf
X(t)
(x)dx
−∞
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44. Mean,Correlationandcovariance
function
Prof.S.D.Shipne

45. •
Meanvalueofastationaryrandom
processisa constant.
Prof.S.D.Shipne

46. μ
X(t)=μ
X
•Autocorrelationfunctionofa
randomprocessX(t)isgivenas
R
X
(t
1,
t
2)
=
E[X
Prof.S.D.Shipne

47. (t
1)X
(t
2)]
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49. 20

50. Prof.S.D.Shipne
21

51. •Autocorrelation:
x(t)
τ
time,t
T
•Theauto-correlation,orautocovariance,
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22

52. describesthegeneraldependencyofx(t)with
itsvalueatashorttimelater,x(t+τ)
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23

53. Auto-correlationproperties
Symmetry
2. PowerofW.S.S.process
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24

54. 3. Maximumvalue
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25

55. Autocorrelationfunctionofslowlyand
rapidlyfluctuatingrandomprocesses
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26

56. Correlationanddecorrelationtimeconcept
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27

57. Mean,Correlationandcovariance
function
•
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28

58. Autocovariancefunctionofastationary
randomprocessX(t)isgivenas
C
X
(t
1,
t
2)
=
μ
2
E[(X
(t
1)−μ
X
)(X
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29

59. (t
2)−μ
X
)]
=R
X
(t
2
−t
1)−
X
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30

60. ProbableQuestionsonthistopic
•Numericalbasedoncalculationof
expectationofagivenrandom
process.
•PropertiesofAutocorrelation
•Findmean,autocorrelationand
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31

61. covarianceofagivenrandomprocess.
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32

62. Prof.S.D.Shipne
33

63. R
x
y
(t
1
,t
2
)
=
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34

64. x
(
t
)
y
(
t
∞∞
=
∫∫
)
x
1y
2f
xy
(x
1,y
2)
dx
1dy
2
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35

65. 12
−∞−∞
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36

66. JointlyStationaryProperties
•Pro
pert
ies
R
xy
(τ)=
R
xy
(−τ)
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37

67. (τ)
(τ)
≤
≤
1
⎡R
x
R
x(0)R
y(0)
(0)+R
y
(0)
⎤
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38

68. 2
⎣ ⎦
•Uncorrelated:
•O
r
t
h
o
g
o
n
a
l
:
R
xy
(τ)=x(t)y(t
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39

69. +τ)
=xy
R
xy(τ)=0
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40

70. 30

71. Ifτ=0thencross-correlationresultsinzeroi.e.therandomvariables
areorthogonal
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31

72. Prof.S.D.Shipne
32

73. Spectraldensity
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33

74. Transmissionofarandomprocessthrougha
X(t)
w.s.s
Y(t)
⎥
Linearfilter
μ
Y(t)=
E[Y(t)]
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34

75. ⎡
=E
⎢
∞
∫
h(τ)X(t
−τ)dτ
⎤
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35

76. Themeanof
theoutput
⎣−∞
∞
=
∫
h(τ
)E[X(t
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36

77. ⎦−τ)]dτ
randomprocessY(t)isgivenas −∞
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37

78. =μ
X
∞
∫
h(τ
)dτ=μ
X
H(0)
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38

79. −∞
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39

80. Pointstoremember
•MeanofrandomprocessY(t)producedatthe
outputofaLTIsysteminresponsetoinput
randomprocessX(t)equalstothemeanof
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40

81. X(t)multipliedbythedcresponseofthe
system.
•Auto-correlationfunctionofthe
randomprocessY(t)isaconstant.
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41

82. Filteringofrandomsignals
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42

84. 37

85. Gaussianprocess
•AstochasticprocessissaidtobeanormalorGaussianprocessif
itsjointprobabilitydistributionisnormal.
•AGaussianprocessisstrictlyandweaklystationarybecause
thenormaldistributionisuniquelycharacterizedbyitsfirsttwo
moments.
1 ⎡(y
−μ)
2
⎤1
2π
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38

86. ⎡y
2
⎤
f
Y(y)=
exp
⎢
−
Y
⎥
=
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39

87. exp
⎢
−
⎥
⎣
2σ
2
⎦
2πσ
Y
Y
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40

88. ⎣
2
⎦
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41

89. •Centrallimittheorem
–Thesumofalargenumberofindependentand
identicallydistributedrandomvariablesgetting
closertoGaussiandistribution
•ThermalnoisecanbecloselymodeledbyGaussian
process
•ThetheoremstatesthattheprobabilitydistributionofN
randomvariablesapproachesanormalizedGaussian
distributioninthelimitasNapproachesinfinity.
Prof.S.D.Shipne
42

90. GaussianRandomProcesses
•Gaussianrandomprocesseshavesomespecialproperties
-IfaGaussianrandomprocessiswide-sensestationary,
thenitisalsostationary
-IftheinputtoalinearsystemisaGaussianrandom
process,thentheoutputisalsoaGaussianrandom
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43

91. process
-iftherandomvariableX(t1),X(t2),…..X(tn)obtained
bysamplingaGaussianprocessX(t)attimest1,t2,……tn
areuncorrelatedthentheserandomvariablesare
statisticallyindependent.
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44

92. Noise
•Unwantedwavesthattendtodisturb
thetransmissionandprocessingof
signalsincommunicationsystems.
Typesofnoise
∞
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45

93. •Shot
noise
asa
rando
m
proce
ss
X(t)=
∑
h(t
−τ
k)
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46

94. currentpulsegeneratedat
timeinstant
k=−∞
(λt)
k
−λt
Poisson
distribution= 0
e
0
k!
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47

95. SamplefunctionofPoissoncountingprocessandautocovaianceplot
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48

96. WHITENOISE
•Theprimaryspectralcharacteristicofthermalnoiseisthatits
powerspectraldensityisthesameforallfrequenciesofinterestin
mostcommunicationsystems
•Athermalnoisesourceemanatesanequalamountofnoisepowerper
unitbandwidthatallfrequencies—fromdctoabout1012Hz.
•PowerspectraldensityG(f)
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42

97. •Autocorrelationfunctionofwhitenoiseis
•TheaveragepowerPofwh
itenoiseifinfinite
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42

98. WhiteNoise
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99. NarrowbandNoiseRepresentation
•Thenoiseprocessappearingattheoutputofa
narrowbandfilteriscallednarrowbandnoise.
•Withthespectralcomponentsofnarrow
±
b
f
candnoise
Prof.S.D.Shipne

100. concentratedaboutsomemidband
f
frequency
c
•Wefindthatasamplefunctionn(t)ofsuchaprocess
appearssomewhatsimilartoasinewaveoffrequency
•Representationsofnarrowbandnoise
–
Prof.S.D.Shipne

101. RepresentationofNarrowbandNoiseinTermsofIn-Phaseand
QuadratureComponents
•Consideranarrowbandnoisen(t)ofbandwidth2Bcenteredon
frequency
f
c,asillustratedinFigure
Prof.S.D.Shipne

102. •Wemayrepresentn(t)inthecanonical(standard)form:
n(t)=n
I(t)cos(2πf
ct)−n
Q(t)sin(2πf
ct)
n
I(t) n(t)n
Q(t)
com
ponentof.
where, is
n(
i
t
n
)
-phasecomponentof
and
isquadrature
Prof.S.D.Shipne

103. Prof.S.D.Shipne

104. PSDofnarrow-bandnoise
46

105. RepresentationofNarrowbandNoiseinTermsof
EnvelopeandPhaseComponents
47

106. Propertiesofnarrow-bandnoise
•Thein-phaseandquadraturecomponents
ofnarrow-bandnoisehavezeromean.
•Ifthenarrow-bandnoiseisGaussianthenitsin-
phasecomponentandquadraturecomponentare
jointlyGaussian.
•Ifthenarrow-bandnoiseiswidesense
stationary(w.s.s.),thenitsin-phasecomponent

107. andquadraturecomponentarejointlyw.s.s.
•PSDsofthesequadraturecomponentsaresame
asthatofPSDoforiginalnarrow-bandnoise.
•Thequadraturecomponentshavesamevarianceas
thenarrow-bandnoise
48

108. ThankYou