7074 chapter3 slides

8/12/2019 7074 Chapter3 Slides

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Chapter 3: Probability, Random Variables, Random Processes

A First Course in Digital CommunicationsHa H. Nguyen and E. Shwedyk

February 2009

A First Course in Digital Communications 1/58
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Introduction

The main objective of a communication system is the transfer

of information over a channel.Message signal is best modeled by a random signal: Anysignal that conveys information must have some uncertainty init, otherwise its transmission is of no interest.

Two types of imperfections in a communication channel:Deterministic imperfection, such as linear and nonlineardistortions, inter-symbol interference, etc.Nondeterministic imperfection, such as addition of noise,interference, multipath fading, etc.

We are concerned with the methods used to describe andcharacterize a random signal, generally referred to as arandom process(also commonly called stochastic process).

In essence, a random process is a random variableevolving in

time.A First Course in Digital Communications 2/58
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Sample Space and Probability

Random experiment: its outcome, for some reason, cannot bepredicted with certainty.

Examples: throwing a die, flipping a coin and drawing a card

from a deck.Sample space: the set of all possible outcomes, denoted by .Outcomes are denoted bys and each lies in, i.e., .A sample space can be discreteor continuous.

Eventsare subsets of the sample space for which measures oftheir occurrences, called probabilities, can be defined ordetermined.

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Three Axioms of Probability

For a discrete sample space , define a probability measure P on as a set function that assigns nonnegative values to all events,denoted by E, in such that the following conditions are satisfied

Axiom 1: 0

P(E)

1 for all E

(on a % scale

probability ranges from 0 to 100%. Despite popular sportslore, it is impossible to give more than 100%).

Axiom 2: P() = 1 (when an experiment is conducted therehas to be an outcome).

Axiom 3: For mutually exclusive events1

E1, E2, E3, . . .wehave P(

i=1 Ei) =

i=1 P(Ei).

1The events E1, E2, E3,. . . are mutually exclusive ifEi Ej = for all

i= j, where is the null set.A First Course in Digital Communications 4/58
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Important Properties of the Probability Measure

1. P(Ec) = 1 P(E), where Ec denotes the complement ofE.This property implies that P(Ec) + P(E) = 1, i.e., somethinghas to happen.

2. P(

) = 0 (again, something has to happen).

3. P(E1 E2) =P(E1) + P(E2) P(E1 E2). Note that iftwo events E1 and E2 are mutually exclusive thenP(E1 E2) =P(E1) + P(E2), otherwise the nonzerocommon probability P(E1 E2) needs to be subtracted off.

4. IfE1 E2 then P(E1) P(E2). This says that if event E1is contained in E2 then occurrence ofE2 means E1 hasoccurred but the converse is not true.


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Conditional Probability

We observe or are told that event E1 has occurred but are

actually interested in event E2: Knowledge that ofE1 hasoccurred changes the probability ofE2 occurring.

If it was P(E2) before, it now becomes P(E2|E1), theprobability ofE2 occurring given that event E1 has occurred.

This conditional probabilityis given by

P(E2|E1) =

P(E2 E1)P(E1)

, ifP(E1) = 00, otherwise

.

IfP(E2|E1) =P(E2), or P(E2 E1) =P(E1)P(E2), thenE1 and E2 are said to be statistically independent.

Bayes rule

P(E2|E1) = P(E1|E2)P(E2)P(E1)

,


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Total Probability Theorem

The events

{Ei

}ni=1 partition the sample space if:

(i)n

i=1

Ei= (1a)

(ii) Ei Ej = for all 1 i, j n and i =j (1b)If for an event A we have the conditional probabilities{P(A|Ei)}ni=1, P(A) can be obtained as

P(A) =n

i=1

P(Ei)P(A|Ei).

Bayes rule:

P(Ei|A) = P(A|Ei)P(Ei)P(A)

= P(A|Ei)P(Ei)

nj=1 P(A

|Ej)P(Ej)

.


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Random Variables

R

1

4

3

2

4( )x 1( )x 2( )x3( )x

A random variable is a mappingfrom the sample space tothe set of real numbers.

We shall denote random variables by boldface, i.e., x, y, etc.,while individual or specific values of the mapping x aredenoted by x().


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Cumulative Distribution Function (cdf)

cdf gives a complete description of the random variable. It isdefined as:

Fx(x) =P( :x() x) =P(x x).

The cdf has the following properties:1. 0 Fx(x) 1 (this follows from Axiom 1 of the probability

measure).2. Fx(x) is nondecreasing: Fx(x1) Fx(x2) ifx1 x2 (this is

because event x()

x1 is contained in event x()

x2).

3. Fx() = 0andFx(+) = 1(x() is the empty set,hence an impossible event, while x() is the wholesample space, i.e., a certain event).

4. P(a


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Typical Plots of cdf I

A random variable can be discrete, continuousormixed.

0.1

0 x

( )F xx

(a)


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Typical Plots of cdf II

0.1

0 x

( )F xx

(b)

0.1

0 x

( )F xx

(c)


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Probability Density Function (pdf)

The pdf is defined as the derivative of the cdf:

fx(x) =dFx(x)

dx .

It follows that:

P(x1

x

x2) =P(x

x2)

P(x

x1)

=Fx(x2) Fx(x1) = x2

x1

fx(x)dx.

Basic properties of pdf:1. fx(x) 0.2. f

x

(x)dx= 1.3. In general, P(x A) = A

fx(x)dx.

For discrete random variables, it is more common to definethe probability mass function (pmf): pi=P(x=xi).

Note that, for all i, one haspi

0 and

i

pi= 1.


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p y, ,

Bernoulli Random Variable

0x

1 0x

1

p1

1

(1 )p( )p

( )F xx

( )f xx

A discrete random variable that takes two values 1 and 0 withprobabilitiesp and 1 p.Good model for a binary data source whose output is 1 or 0.

Can also be used to model the channel errors.


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Binomial Random Variable

0x

2

05.0

10.0

20.0

15.0

25.0

30.0

4 6

( )f xx

A discrete random variable that gives the number of 1s in asequence ofn independentBernoulli trials.

fx(x) =n

k=0

n

k

pk(1p)nk(xk), where

n

k

=

n!

k!(n

k)!

.


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Uniform Random Variable

0x

ab

1

a b 0x

a b

1

( )F xx( )f xx

A continuous random variable that takes values between aand b with equal probabilities over intervals of equal length.

The phase of a received sinusoidal carrier is usually modeledas a uniform random variable between 0 and 2. Quantizationerror is also typically modeled as uniform.




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Gaussian (or Normal) Random Variable

0x

0x

12

2

1

2

1

( )f xx

( )F xx

A continuous random variable whose pdf is:

fx(x) = 122

exp(x )2

22

,

and 2 are parameters. Usually denoted asN(, 2).Most important and frequently encountered random variable

in communications.A First Course in Digital Communications 16/58

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Functions of A Random Variable

The function y=g(x) is itself a random variable.

From the definition, the cdf ofy can be written as

Fy(y) =P( :g(x()) y).Assume that for all y, the equation g(x) =y has a countable

number of solutions and at each solution point, dg(x)/dxexists and is nonzero. Then the pdf ofy=g(x) is:

fy(y) = i

fx(xi)

dg(x)dx x=xi,

where{xi} are the solutions ofg(x) =y.A linear function of a Gaussian random variable is itself aGaussian random variable.




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Expectation of Random Variables I

Statistical averages, ormoments, play an important role in the

characterization of the random variable.

Theexpected value(also called the mean value, first moment)of the random variable x is defined as

mx=E{x}

xfx(x)dx,

where Edenotes the statistical expectation operator.

In general, the nth moment ofx is defined as

E{xn}

xnfx(x)dx.

For n= 2, E{x2} is known as the mean-squared value of therandom variable.


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Expectation of Random Variables II

The nthcentral momentof the random variable x is:

E{y} =E{(x mx)n} =

(x mx)nfx(x)dx.

When n= 2 the central moment is called the variance,

commonly denoted as 2x:

2x= var(x) =E{(x mx)2} =

(x mx)2fx(x)dx.

The variance provides a measure of the variablesrandomness.

The mean and variance of a random variable give a partialdescription of its pdf.


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Expectation of Random Variables III

Relationship between the variance, the first and secondmoments:

2x=E

{x2

} [E

{x

}]2 =E

{x2

} m2x.

An electrical engineering interpretation: The AC power equalstotal power minus DC power.

The square-root of the variance is known as the standarddeviation, and can be interpreted as the root-mean-squared(RMS) value of the AC component.


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The Gaussian Random Variable

0 0.2 0.4 0.6 0.8 10.8

0.6

0.4

0.2

0

0.2

0.4

0.6

t(sec)

Signalamplitude(volts)

(a) A muscle (emg) signal


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1 0.5 0 0.5 1

0

0.5

1

1.5

2

2.5

3

3.5

4

x(volts)

f x(x)(1/volts)

(b) Histogram and pdf fits

Histogram

Gaussian fit

Laplacian fit

fx(x) = 1

22x

e (xmx)

2

22x (Gaussian)

fx(x) = a

2

ea|x| (Laplacian)


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Gaussian Distribution (Univariate)

15 10 5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

f x(x)

x=1

x

=2

x=5

Range (kx) k = 1 k= 2 k= 3 k = 4P(mx kx


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Multiple Random Variables I

Often encountered when dealing with combined experimentsor repeated trials of a single experiment.

Multiple random variables are basically multidimensionalfunctions defined on a sample space of a combinedexperiment.

Let x and y be the two random variables defined on the samesample space . The joint cumulative distribution function isdefined as

Fx,y(x, y) =P(x

x,y

y).

Similarly, the joint probability density function is:

fx,y(x, y) =2Fx,y(x, y)

xy .


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Multiple Random Variables II

When the joint pdf is integrated over one of the variables, oneobtains the pdf of other variable, called the marginal pdf:

fx,y(x, y)dx=fy(y),

fx,y(x, y)dy=fx(x).

Note that:

fx,y(x, y)dxdy=F(, ) = 1Fx,y(, ) =Fx,y(, y) =Fx,y(x, ) = 0.


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Multiple Random Variables III

The conditional pdf of the random variable y, given that the

value of the random variable x is equal to x, is defined as

fy(y|x) =

fx,y(x, y)

fx(x) , fx(x) = 0

0, otherwise.

Two random variables x and y are statistically independent ifand only if

fy(y|x) =fy(y) or equivalently fx,y(x, y) =fx(x)fy(y).

The joint moment is defined as

E{xjyk} =

xjykfx,y(x, y)dxdy.


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Multiple Random Variables IV

The joint central moment is

E{(xmx)j(ymy)k} =

(xmx)j(ymy)kfx,y(x, y)dxdy

where mx=E{x

}and my =E

{y

}.

The most important moments are

E{xy}

xyfx,y(x, y)dxdy (correlation)

cov{x,y} E{(x mx)(y my)}= E{xy} mxmy (covariance).


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Multiple Random Variables V

Let2x and 2y be the variance ofx and y. The covariance

normalized w.r.t. xy is called the correlation coefficient:

x,y =cov{x,y}

xy.

x,y indicates the degree oflinear dependencebetween two

random variables.

It can be shown that|x,y| 1.x,y = 1 implies an increasing/decreasing linear relationship.Ifx,y = 0, x and y are said to be uncorrelated.

It is easy to verify that ifx and y are independent, thenx,y = 0: Independence implies lack of correlation.

However, lack of correlation (no linear relationship) does notin general imply statistical independence.


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Examples of Uncorrelated Dependent Random Variables

Example 1: Let x be a discrete random variable that takes on

{1, 0, 1} with probabilities{14 , 12 , 14}, respectively. Therandom variables y=x3 and z= x2 are uncorrelated butdependent.

Example 2: Let x be an uniformly random variable over

[1, 1]. Then the random variablesy

=x

andz

=x2

areuncorrelated but dependent.

Example 3: Let x be a Gaussian random variable with zeromean and unit variance (standard normal distribution). Therandom variables y=x and z=

|x

|are uncorrelated but

dependent.Example 4: Let u and v be two random variables (discrete orcontinuous) with the same probability density function. Thenx= u v and y=u+ v are uncorrelated dependent randomvariables.


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Example 1

x

{1, 0, 1

}with probabilities

{1/4, 1/2, 1/4

} y=x3 {1, 0, 1} with probabilities{1/4, 1/2, 1/4} z= x2 {0, 1} with probabilities{1/2, 1/2}my = (1)14 + (0)12 + (1)14 = 0; mz= (0)12 + (1)12 = 12 .The joint pmf (similar to pdf) ofy and z:

0

1

1

1

12

1

4

1

4

y

zP(y= 1, z= 0) = 0P(y= 1, z= 1) =P(x= 1) = 1/4P(y= 0, z= 0) =P(x= 0) = 1/2

P(y= 0, z= 1) = 0

P(y= 1, z= 0) = 0

P(y= 1, z= 1) =P(x= 1) = 1/4Therefore, E{yz} = (1)(1)14 + (0)(0)12 + (1)(1)14 = 0

cov

{y, z

}=E

{yz

} mymz= 0

(0)1/2 = 0!



G ( )
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Jointly Gaussian Distribution (Bivariate)

fx,y(x, y) = 1

2xy

1 2x,yexp

1

2(1 2x,y)

(x mx)22x

2x,y(x mx)(y my)xy

+(y my)2

2y

,

wheremx, my, x, y are the means and variances.

x,y is indeed the correlation coefficient.

Marginal density is Gaussian: fx(x) N(mx, 2x) andfy(y) N(my, 2y).When x,y = 0

fx,y(x, y) =fx(x)fy(y)

random

variables x and y are statistically independent.Uncorrelatedness means that joint Gaussian random variablesare statistically independent. The converse is not true.

Weighted sum of two jointly Gaussian random variables is alsoGaussian.



J i df d C f 1 d
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Joint pdf and Contours for x = y = 1and x,y = 0

32

10

12 3

2

02

0

0.05

0.1

0.15

x

x,y

=0

y

f x,y

(x,y

)

x

y

2 1 0 1 22.5

2

1

0

1

2

2.5



J i df d C f 1 d 0 3


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Joint pdf and Contours for x = y = 1and x,y = 0.3

32

10

12 3

2

0

2

0

0.05

0.1

0.15

x

x,y

=0.30

y

f x,y

(x,y

)

a crosssection

x

y

2 1 0 1 22.5

2

1

0

1

2

2.5



J i df d C f 1 d 0 7
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32

10

12 3

2

0

2

0

0.05

0.1

0.15

0.2

a crosssection

x

x,y

=0.70

y

f x,y

(x,y

)

x

y

2 1 0 1 22.5

2

1

0

1

2

2.5



J i t df d C t f 1 d 0 95
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32

10

12

3

2

0

2

0

0.1

0.2

0.3

0.4

0.5

x

x,y

=0.95

y

f x,y

(x,y

)

x

y

2 1 0 1 22.5

2

1

0

1

2

2.5

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Random Processes I


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Random Processes I

Real number

Timetk

x1(t,

1)

x2(t,

2)

xM

(t,M

)

.

.

.

1

2

M

3

j

x(t,)

t2

t1

.

.

.

.

.

.

. . .

x(tk,)

.

.

.

A mapping from a sample space to a set of time functions.



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Examples of Random Processes I


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Examples of Random Processes I

(a) Thermal noise

0 t

(b) Uniform phase

t0



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Examples of Random Processes II

t0

(c) Rayleigh fading process

t

+V

V

(d) Binary random data

0

Tb



Classification of Random Processes
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Classification of Random Processes

Based on whether its statistics change with time: the processis non-stationaryorstationary.

Different levels of stationarity:

Strictly stationary: the joint pdf of any order is independent ofa shift in time.Nth-order stationarity: the joint pdf does not depend on thetime shift, but depends on time spacings:

fx(t1),x(t2),...x(tN)(x1, x2, . . . , xN; t1, t2, . . . , tN) =

fx(t1+t),x(t2+t),...x(tN+t)(x1, x2, . . . , xN; t1+ t, t2+ t , . . . , tN+ t).

The first- and second-order stationarity:

fx(t1)(x, t1) =fx(t1+t)(x; t1+ t) =fx(t)(x)

fx(t1),x(t2)(x1, x2; t1, t2) =fx(t1+t),x(t2+t)(x1, x2; t1+ t, t2+ t)

=fx(t1),x(t2)(x1, x2; ), =t2 t1.A First Course in Digital Communications 41/58


Statistical Averages or Joint Moments
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Statistical Averages or Joint Moments

Consider Nrandom variables x(t1),x(t2), . . .x(tN). Thejoint moments of these random variables is

E{xk1(t1),xk2(t2), . . .xkN(tN)} = x1=

xN=

xk11 xk22 xkNN fx(t1),x(t2),...x(tN)(x1, x2, . . . , xN; t1, t2, . . . , tN)

dx1dx2 . . . dxN,

for all integers kj 1 and N 1.Shall only consider the first- and second-order moments, i.e.,E{x(t)}, E{x2(t)} and E{x(t1)x(t2)}. They are the meanvalue, mean-squared valueand (auto)correlation.



Mean Value or the First Moment
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Mean Value or the First Moment

The mean value of the process at time t is

mx(t) =E{x(t)} =

xfx(t)(x; t)dx.

The average isacross the ensembleand if the pdf varies withtime then the mean value is a (deterministic) function of time.

If the process is stationary then the mean is independent oftor a constant:

mx =E{x(t)} =

xfx(x)dx.



Mean-Squared Value or the Second Moment
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Mean Squared Value or the Second Moment

This is defined as

MSVx(t) =E{x2(t)} =

x2fx(t)(x; t)dx (non-stationary),

MSVx=E{x2(t)} =

x2fx(x)dx (stationary).

The second central moment (or the variance) is:

2x(t) = E[x(t) mx(t)]

2 =MSVx(t) m2x(t) (non-stationary),

2x = E

[x(t) mx]2

=MSVx m2x (stationary).



Correlation
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Correlation

The autocorrelation function completely describes the power

spectral density of the random process.Defined as the correlation between the two random variablesx1=x(t1) and x2=x(t2):

Rx(t1, t2) =E{x(t1)x(t2)}= x1=

x2=

x1x2fx1,x2(x1, x2; t1, t2)dx1dx2.

For a stationary process:

Rx() =E

{x(t)x(t+ )

}=

x1=

x2=x1x2fx1,x2(x1, x2; )dx1dx2.

Wide-sense stationarity(WSS) process: E{x(t)} =mx forany t, and Rx(t1, t2) =Rx() for =t2

t1.



Properties of the Autocorrelation Function


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Properties of the Autocorrelation Function

1. Rx() =Rx(). It is an even function of because the

same set of product values is averaged across the ensemble,regardless of the direction of translation.

2.|Rx()| Rx(0). The maximum always occurs at = 0,though there maybe other values of for which it is as big.Further Rx(0)is the mean-squared value of the random

process.3. If for some 0 we have Rx(0) =Rx(0), then for all integers

k, Rx(k0) =Rx(0).

4. Ifmx= 0 then Rx() will have a constant component equalto m2

x.

5. Autocorrelation functions cannot have an arbitrary shape.The restriction on the shape arises from the fact that theFourier transform of an autocorrelation function must begreater than or equal to zero, i.e.,F{Rx()} 0.



Power Spectral Density of a Random Process (I)
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p y ( )

Taking the Fourier transform of the random process does not work.

x2(t,

2)

xM

(t,M

)

x1(t,

1)

t

t0

0

0

0

0

0

|X1(f,

1)|

|X2(f,2)|

|XM

(f,M

)|

f

f

f.

.

.

.

.

.

.

.

.

.

.

.

Timedomain ensemble Frequencydomain ensemble

t



Power Spectral Density of a Random Process (II)


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p y ( )

Need to determine how the average power of the process is

distributed in frequency.Define a truncated process:

xT(t) =

x(t), T t T

0, otherwise .

Consider the Fourier transform of this truncated process:

XT(f) =

xT(t)ej2ftdt.

Average the energy over the total time, 2T:

P= 1

2T

TT

x2T(t)dt= 1

2T

|XT(f)|2 df (watts).



Power Spectral Density of a Random Process (III)
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p y ( )

Find the average value ofP:

E{P} =E

1

2T

TT

x2T(t)dt

=E

1

2T

|XT(f)|2 df

.

Take the limit as T :

limT

12T

TT

Ex2T(t)

dt= lim

T1

2T

E|XT(f)|2 df,

It follows that

MSVx = limT

1

2T TT E

x2T(t)

dt

=

limT

E|XT(f)|2

2T

df (watts).



Power Spectral Density of a Random Process (IV)
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p y ( )

Finally,

Sx(f) = limT

E|XT(f)|2

2T

(watts/Hz),

is the power spectral densityof the process.

It can be shown that the power spectral density and theautocorrelation function are a Fourier transform pair:

Rx() Sx(f) =

=Rx()ej2fd.



Time Averaging and Ergodicity
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A process where any member of the ensemble exhibits the

same statistical behavior as that of the whole ensemble.All time averages on a single ensemble member are equal tothe corresponding ensemble average:

E{xn

(t))} =

x

n

fx(x)dx

= limT

1

2T

TT

[xk(t, k)]ndt, n, k.

For an ergodic process: To measure various statisticalaverages, it is sufficient to look at only one realization of theprocess and find the corresponding time average.

For a process to be ergodic it must be stationary. Theconverse is not true.



Examples of Random Processes
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(Example 3.4) x(t) =A cos(2f0t + ), where is arandom variable uniformly distributed on [0, 2]. This processis both stationary and ergodic.

(Example 3.5) x(t) =x, where x is a random variable

uniformly distributed on [A, A], where A >0. This processis WSS, but not ergodic.

(Example 3.6) x(t) =A cos(2f0t + ) where A is azero-mean random variable with variance, 2A, and isuniform in [0, 2]. Furthermore, A and are statisticallyindependent. This process is not ergodic, but strictlystationary.



Random Processes and LTI Systems
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Linear, Time-Invariant

(LTI) System

Input Output

( ) ( )h t H f ( )tx ( )ty

, ( ) ( )m R S f x x x

, ( ) ( )m R S f y y y

, ( )R x y

my = E

{y(t)

}=E

h()x(t

)d =mxH(0)

Sy(f) = |H(f)|2 Sx(f)Ry() = h() h() Rx().



Thermal Noise in Communication Systems
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A natural noise source is thermal noise, whose amplitudestatistics are well modeled to be Gaussian with zero mean.

The autocorrelation and PSD are well modeled as:

Rw() = kGe||/t0

t0

(watts),

Sw(f) = 2kG

1 + (2f t0)2 (watts/Hz).

where k = 1.38 1023 joule/0K is Boltzmanns constant, Gis conductance of the resistor (mhos); is temperature indegrees Kelvin; and t0 is the statistical average of timeintervals between collisions of free electrons in the resistor (onthe order of1012 sec).



(a) Power Spectral Density, Sw

(f)
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15 10 5 0 5 10 150

f(GHz)

Sw

(f)(watts/Hz)

0.1 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.1(picosec)

Rw

()(watts)

(b) Autocorrelation,Rw

()

N0/2

N0/2()

White noise

Thermal noise

White noise

Thermal noise



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Example


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Suppose that a (WSS) white noise process, x(t), of zero-mean andpower spectral density N0/2 is applied to the input of the filter.

(a) Find and sketch the power spectral density andautocorrelation function of the random process y(t) at theoutput of the filter.

(b) What are the mean and variance of the output process y(t)?

L

R( )tx ( )ty




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H(f) = R

R +j2f L=

1

1 +j2fL/R.

Sy(f) =N0

2

1

1 +2LR

2f2

Ry() = N0R4L

e(R/L)||.

0 0

2

0N

(Hz)f (sec)

L

RN

4

0

( ) (watts/Hz)S fy

( ) (watts)R y

http://find/

7074 chapter3 slides

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