Short Notes Theory of Random Variables & Processes - Communication System

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TheoryofRandomVariables&Processes
Random variables and processes are fundamental to analyzing noise, signals, and system per-
formance in communication systems. This section covers key concepts, properties, and appli-
cations.
1. RandomVariables
Arandomvariable(RV)mapsoutcomesofarandomexperimenttorealnumbers,characterized
byitsprobabilitydistribution.
• Types:
– Discrete RV: Takes countable values, described by a probability mass function
(PMF),P(X = x
i
).
– Continuous RV: Takes values in a continuum, described by a probability density
function(PDF),f
X
(x),whereP(a= X= b) =
?
b
a
f
X
(x)dx.
• KeyMetrics:
– ExpectedValue(Mean): E[X] =
?
x
i
P(X = x
i
)(discrete)orE[X] =
?
8
-8
xf
X
(x)dx
(continuous).
– Variance: Var(X) = E[(X-E[X])
2
] = E[X
2
]-(E[X])
2
.
– CumulativeDistributionFunction(CDF):F
X
(x) = P(X= x).
• CommonDistributions:
– Gaussian: f
X
(x) =
1
v
2ps
2
e
-
(x-µ)
2
2s
2
,meanµ,variances
2
.
– Uniform: f
X
(x) =
1
b-a
,forx? [a,b].
– Exponential: f
X
(x) = ?e
-?x
,forx= 0,mean
1
?
.
2. RandomProcesses
A random process is a collection of RVs indexed by time, X(t), modeling signals or noise in
communicationsystems.
• Classi?cation:
– Stationary: Statisticalproperties(mean,variance)aretime-invariant.
– Wide-Sense Stationary (WSS):Constantmeanandautocorrelationdependsonlyon
timedifference,R
X
(t) = E[X(t)X(t+t)].
– Ergodic: Timeaveragesequalensembleaverages.
• KeyProperties:
– Autocorrelation: R
X
(t
1
,t
2
) = E[X(t
1
)X(t
2
)]. ForWSS,R
X
(t).
– Power Spectral Density (PSD): Fourier transform of autocorrelation, S
X
(f) =
?
8
-8
R
X
(t)e
-j2pft
dt,describespowerdistributionacrossfrequencies.
– Mean: µ
X
(t) = E[X(t)].
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Page 2


TheoryofRandomVariables&Processes
Random variables and processes are fundamental to analyzing noise, signals, and system per-
formance in communication systems. This section covers key concepts, properties, and appli-
cations.
1. RandomVariables
Arandomvariable(RV)mapsoutcomesofarandomexperimenttorealnumbers,characterized
byitsprobabilitydistribution.
• Types:
– Discrete RV: Takes countable values, described by a probability mass function
(PMF),P(X = x
i
).
– Continuous RV: Takes values in a continuum, described by a probability density
function(PDF),f
X
(x),whereP(a= X= b) =
?
b
a
f
X
(x)dx.
• KeyMetrics:
– ExpectedValue(Mean): E[X] =
?
x
i
P(X = x
i
)(discrete)orE[X] =
?
8
-8
xf
X
(x)dx
(continuous).
– Variance: Var(X) = E[(X-E[X])
2
] = E[X
2
]-(E[X])
2
.
– CumulativeDistributionFunction(CDF):F
X
(x) = P(X= x).
• CommonDistributions:
– Gaussian: f
X
(x) =
1
v
2ps
2
e
-
(x-µ)
2
2s
2
,meanµ,variances
2
.
– Uniform: f
X
(x) =
1
b-a
,forx? [a,b].
– Exponential: f
X
(x) = ?e
-?x
,forx= 0,mean
1
?
.
2. RandomProcesses
A random process is a collection of RVs indexed by time, X(t), modeling signals or noise in
communicationsystems.
• Classi?cation:
– Stationary: Statisticalproperties(mean,variance)aretime-invariant.
– Wide-Sense Stationary (WSS):Constantmeanandautocorrelationdependsonlyon
timedifference,R
X
(t) = E[X(t)X(t+t)].
– Ergodic: Timeaveragesequalensembleaverages.
• KeyProperties:
– Autocorrelation: R
X
(t
1
,t
2
) = E[X(t
1
)X(t
2
)]. ForWSS,R
X
(t).
– Power Spectral Density (PSD): Fourier transform of autocorrelation, S
X
(f) =
?
8
-8
R
X
(t)e
-j2pft
dt,describespowerdistributionacrossfrequencies.
– Mean: µ
X
(t) = E[X(t)].
1
• Examples:
– WhiteNoise: ConstantPSD,S
X
(f) = N
0
/2,zeromean,uncorrelatedsamples.
– GaussianProcess: All?nite-dimensionaldistributionsareGaussian.
3. ApplicationsinCommunicationSystems
• SignalAnalysis: Randomprocessesmodelmodulatedsignals,noise,andinterference.
• NoiseCharacterization: White Gaussian noise (WGN) is used to analyze receiver per-
formance.
• SystemDesign: PSDhelpsdesign?lterstooptimizesignal-to-noiseratio(SNR).
• PerformanceMetrics: Biterrorrate(BER)dependsonRVdistributions(e.g.,Gaussian
noise).
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