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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
A First Course in Digital CommunicationsHa H. Nguyen and E. Shwedyk
February 2009
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Analog and Digital Amplitude Modulations
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
What is Digital Communication?
x(t)
(a)
0
(c)
Ts0
t
(b)
(d)
Ts
0
x(t)
x(t) x(t)
t
t
t
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Why Digital Communications?
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Why Digital Communications?
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Ch I d Ch P b b l R d V bl R d P
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Regenerative Repeater in Digital Communications
Propagation distance
Original pulse Some distortion DegradedSeverely
degraded Regenerated
21 3 4 5
Digital communications: Transmitted signals belong to afinite set of waveforms
The distorted signal can be
recovered to its ideal shape, hence removing all the noise.Analog communications: Transmitted signals are analogwaveforms, which can take infinite variety of shapes Oncethe analog signal is distorted, the distortion cannot be
removed.A First Course in Digital Communications 6/75
Ch 1 I d i Ch 3 P b bili R d V i bl R d P
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Block Diagram of a Communication System
Source
(User)Transmitter Channel Receiver
Sink
(User)
Synchronization
Transmitter Receiver
Source
Encoder
Channel
EncoderModulator
De-
modulator
Channel
Decoder
Source
Decoder
(a)
(b)
Note: Synchronization block is only present in a digital system.A First Course in Digital Communications 7/75
Ch te 1 I t d ti Ch te 3 P b bilit R d V i bles R d P esses
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Digital vs. AnalogAdvantages:
Digital signals are much easier to be regenerated.Digital circuits are less subject to distortion and interference.
Digital circuits are more reliable and can be produced at alower cost than analog circuits.
It is more flexible to implement digital hardware than analoghardware.
Digital signals are beneficial from digital signal processing(DSP) techniques.
Disadvantages:
Heavy signal processing.
Synchronization is crucial.
Larger transmission bandwidth.
Non-graceful degradation.A First Course in Digital Communications 8/75
Chapter 1: Introduction Chapter 3: Probability Random Variables Random Processes
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
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 variable evolving intime.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
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 by s and each lies in , i.e., .A sample space can be discrete or continuous.
Events are subsets of the sample space for which measures oftheir occurrences, called probabilities, can be defined ordetermined.
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Chapter 1: Introduction Chapter 3: Probability Random Variables Random Processes
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Example of Throwing a Fair Die
Various events can be defined: the outcome is even number ofdots, the outcome is smaller than 4 dots, the outcome is morethan 3 dots, etc.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
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 % scaleprobability 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 12/75
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p p y, ,
Important Properties of the Probability Measure
1. P(Ec) = 1 P(E), where Ec denotes the complement of E.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. If E1 E2 then P(E1) P(E2). This says that if event E1is contained in E2 then occurrence of E1 means E2 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 of E1 hasoccurred changes the probability of E2 occurring.
If it was P(E2) before, it now becomes P(E2|E1), theprobability of E2 occurring given that event E1 has occurred.
This conditional probability is given by
P(E2|E1) =
P(E2 E1)P(E1)
, if P(E1) = 00, otherwise
.
If P(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)ni=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 mapping from 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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Random Variable in the Example of Throwing a Fair Die
R
52 3 41 6
There could be many other random variables defined to describethe outcome of this random experiment!
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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) if x1 x2 (this is
because event x() x1 is contained in event x() x2).3. Fx() = 0 and Fx(+) = 1 (x() is the empty set,
hence an impossible event, while x() is the wholesample space, i.e., a certain event).
4. P(a < x b) = Fx(b) Fx(a).
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Typical Plots of cdf I
A random variable can be discrete, continuous or mixed.
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.
fx(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 has pi 0 and i
pi = 1.
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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 withprobabilities p 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 of n independent Bernoulli 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 as N(, 2).Most important and frequently encountered random variablein communications.
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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 of g(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, or moments, play an important role in the
characterization of the random variable.The expected value(also called the mean value, first moment)of the random variable x is defined as
mx = E{x}
xfx(x)dx,
where E denotes 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 nth central moment of 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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(b) Hi t d df fit
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1 0.5 0 0.5 10
0.5
1
1.5
2
2.5
3
3.5
4
x (volts)
fx(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
2ea|x| (Laplacian)
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
G i Di ib i (U i i )
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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
fx(x)
x=1
x=2
x=5
Range (kx) k = 1 k = 2 k = 3 k = 4P(mx kx < x mx kx) 0.683 0.955 0.997 0.999Error probability 103 104 106 108
Distance from the mean 3.09 3.72 4.75 5.61
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E l
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Example
The noise voltage in an electric circuit can be modeled as aGaussian random variable with zero mean and variance 2.
(a) Show that the probability that the noise voltage exceeds somelevel A can be expressed as Q A .
(b) Given that the noise voltage is negative, express in terms ofthe Q function the probability that the noise voltage exceedsA, where A < 0.
(c) Using the MATLAB function Q.m available from the courses
website, evaluate the expressions found in (a) and (b) for2 = 108 and A = 104.
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S l i
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Solution
fx(x) = 12
e (x
)2
22 (=0)= 12
e x2
22 .
The Q-function is defined as:
Q(t) =1
2
t
e2
2 d
The probability that the noise voltage exceeds some level A is
P(x > A) =
A
fx(x)dx
= 12
Ae
x222 dx
(= x )=(d=dx
)
12
A
e22 d
= Q
A
.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Gi th i lt i ti th b bilit th t th i
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Given the noise voltage is negative, the probability that the noisevoltage exceeds some level A < 0 is:
P(x
> A|x
< 0) =
P(x > A , x < 0)
P(x < 0) =
P(A < x < 0)
P(x < 0)
=QA
Q(0)Q(0)
=QA
121/2
= 2Q
A
1.
0x
( )f xx
22
1
0x
( | 0)f x
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With 2 = 108 and A = 104,
P(x >
104) = Q
104
108 = Q(
1) = 0.8413
P(x > 104|x < 0) = 2Q(1) 1 = 0.6827.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Example
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Example
The Matlab function randn defines a Gaussian random variable of
zero mean and unit variance. Generate a vector of L = 106
samples according to a zero-mean Gaussian random variable x withvariance 2 = 108. This can be done as follows:x=sigma*randn(1,L).
(a) Based on the sample vector x, verify the mean and variance of
random variable x.
(b) Based on the sample vector x, compute the probability thatx > A, where A = 104. Compare the result with thatfound in Question 2-(c).
(c) Using the Matlab command hist, plot the histogram ofsample vector x with 100 bins. Next obtain and plot the pdffrom the histogram. Then also plot on the same figure thetheoretical Gaussian pdf (note the value of the variance forthe pdf). Do the histogram pdf and theoretical pdf fit well?
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Solution
L=10^6; % Length of the noise vector
sigma=sqrt(10^(-8)); % Standard deviation of the noiseA=-10^(-4);
x=sigma*randn(1,L);
%[2] (a) Verify mean and variance:
mean_x=sum(x)/L; % this is the same as mean(x)
variance_x=sum((x-mean_x).^2)/L; % this is the same as var(x);
% mean_x = -3.7696e-008
% variance_x = 1.0028e-008
%[3] (b) Compute P(x>A)P=length(find(x>A))/L;
% P = 0.8410
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
%[5] ( ) Hi d G i df fi
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%[5] (c) Histogram and Gaussian pdf fit
N_bins=100;
[y,x_center]=hist(x,100); % This bins the elements of x into N_
% and returns the number of elements in each con
% while the bin centers are stored in x_center.
dx=(max(x)-min(x))/N_bins; % This gives the width of each bin.
hist_pdf= (y/L)/dx; % This approximate the pdf to be a constant
pl(1)=plot(x_center,hist_pdf,color,blue,linestyle,--,li
hold on;
x0=[-5:0.001:5]*sigma; % this specifies the range of random vari
true_pdf=1/(sqrt(2*pi)*sigma)*exp(-x0.^2/(2*sigma^2));
pl(2)=plot(x0,true_pdf,color,r,linestyle,-,linewidth,1
xlabel({\itx},FontName,Times New Roman,FontSize,16);
ylabel({\itf}_{\bf x}({\itx}),FontName,Times New Roman,Fo
legend(pl,pdf from histogram,true pdf,+1);
set(gca,FontSize,16,XGrid,on,YGrid,on,GridLineStyle,
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5 0 5
x 104
0
500
1000
1500
2000
2500
3000
3500
4000
x
fx
(x)
pdf from histogram
true pdf
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Multiple Random Variables I
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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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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Multiple Random Variables II
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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
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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
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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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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Multiple Random Variables V
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Multiple Random Variables V
Let 2x 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 of linear dependence between two
random variables.It can be shown that |x,y| 1.x,y = 1 implies an increasing/decreasing linear relationship.If x,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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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Examples of Uncorrelated Dependent Random Variables
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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 variables y = x and z = x2 are
uncorrelated 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 butdependent.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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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Example 1
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p
x {1, 0, 1} with probabilities {1/4, 1/2, 1/4} y = x
3
{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!
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Jointly Gaussian Distribution (Bivariate)
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y ( )
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
,
where mx, 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) randomvariables 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.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Joint pdf and Contours for x = y = 1 and x,y = 0
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p x y x,y
32
10
12 3
2
0
2
0
0.05
0.1
0.15
x
x,y
=0
y
fx,y
(x,y)
x
y
2 1 0 1 22.5
2
1
0
1
2
2.5
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Joint pdf and Contours for x = y = 1 and x,y = 0.3
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p y ,y
32
10
12
3
2
0
2
0
0.05
0.1
0.15
x
x,y
=0.30
y
fx,y
(x,y)
a crosssection
x
y
2 1 0 1 22.5
2
1
0
1
2
2.5
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Joint pdf and Contours for x = y = 1 and x,y = 0.7
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y ,y
32
10
1 23
2
0
2
0
0.05
0.1
0.15
0.2
a crosssection
x
x,y
=0.70
y
fx,y
(x,y)
x
y
2 1 0 1 22.5
2
1
0
1
2
2.5
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Joint pdf and Contours for x = y = 1 and x,y = 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
fx,y
(x,y)
x
y
2 1 0 1 22.5
2
1
0
1
2
2.5
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
MultivariateGaussian pdf
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Definex = [x1,x2, . . . ,xn], a vector of the means
m = [m1, m2, . . . , mn], and the n n covariance matrix Cwith Ci,j = cov(xi,xj) = E{(xi mi)(xj mj)}.The random variables {xi}ni=1 are jointly Gaussian if:
fx1,x2,...,xn(x1, x2, . . . , xn) =1
(2)n det(C)
exp
1
2(x m)C1(x m)
.
If C is diagonal (i.e., the random variables {xi}ni=1 are alluncorrelated), the joint pdf is a product of the marginal pdfs:Uncorrelatedness implies statistical independent for multipleGaussian random variables.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Random Processes I
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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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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Random Processes II
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Ensemble: The set of possible time functions that one sees.
Denote this set by x(t), where the time functions x1(t, 1),x2(t, 2), x3(t, 3), . . . are specific members of the ensemble.
At any time instant, t = tk, we have random variable x(tk).
At any two time instants, say t1 and t2, we have two different
random variables x(t1) and x(t2).Any relationship between them is described by the joint pdffx(t1),x(t2)(x1, x2; t1, t2).
A complete description of the random process is determined by
the joint pdf fx(t1),x(t2),...,x(tN)(x1, x2, . . . , xN; t1, t2, . . . , tN).The most important joint pdfs are the first-order pdffx(t)(x; t) and the second-order pdf fx(t1)x(t2)(x1, x2; t1, t2).
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Examples of Random Processes I
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(a) Thermal noise
0 t
(b) Uniform phase
t0
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Examples of Random Processes II
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t0
(c) Rayleigh fading process
t
+V
V
(d) Binary random data
0
Tb
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Classification of Random Processes
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Based on whether its statistics change with time: the processis non-stationary or stationary.
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 58/75
Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Statistical Averages or Joint Moments
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Consider N random 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 value and (auto)correlation.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Mean Value or the First Moment
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The mean value of the process at time t is
mx(t) = E{x(t)} =
xfx(t)(x; t)dx.
The average is across the ensemble and 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 of tor a constant:
mx = E{x(t)} =
xfx(x)dx.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Mean-Squared Value or the Second Moment
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This is defined as
MSVx(t) = E{x2(t)} =
x2fx(t)(x; t)dx (non-stationary),
MSVx = E{x2
(t)} =
x2
fx(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).
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Correlation
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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.
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Properties of the Autocorrelation Function
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1. Rx() = Rx(). It is an even function of because thesame 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. If mx = 0 then Rx() will have a constant component equalto m2x.
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.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Power Spectral Density of a Random Process (I)
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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
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Power Spectral Density of a Random Process (II)
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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
TTx2T(t)dt =
1
2T
|XT(f)|2 df (watts).
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Power Spectral Density of a Random Process (III)
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Find the average value ofP:
E{P} = E
1
2T
T
Tx2T(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
T
T Ex
2
T(t)
dt
=
limT
E|XT(f)|2
2T
df (watts).
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Power Spectral Density of a Random Process (IV)
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Finally,
Sx(f) = limT
E|XT(f)|2
2T
(watts/Hz),
is the power spectral density of the process.
It can be shown that the power spectral density and theautocorrelation function are a Fourier transform pair:
Rx() Sx(f) =
=Rx()ej2fd.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
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))
}=
xnfx(x)dx
= limT
1
2T
TT
[xk(t, k)]ndt, n, k.
For an ergodic process: To measure various statistical
averages, 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.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
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 statistically
independent. This process is not ergodic, but strictlystationary.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
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().
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
Thermal Noise in Communication Systems
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A natural noise source is thermal noise, whose amplitude
statistics 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 of 1012 sec).
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
(a) Power Spectral Density, Sw
(f)
N0/2
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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
()
0
N0/2()
White noise
Thermal noise
White noise
Thermal noise
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
The noise PSD is approximately flat over the frequency rangeof 0 to 10 GHz let the spectrum be flat from 0 to :
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Sw(f) =N0
2(watts/Hz),
where N0 = 4kG is a constant.
Noise that has a uniform spectrum over the entire frequencyrange is referred to as white noise
The autocorrelation of white noise isRw() =
N02
() (watts).
Since Rw() = 0 for = 0, any two different samples ofwhite noise, no matter how close in time they are taken, areuncorrelated.
Since the noise samples of white noise are uncorrelated, if thenoise is both white and Gaussian (for example, thermal noise)then the noise samples are also independent.
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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
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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Chapter 1: Introduction Chapter 3: Probability, Random Variables, Random Processes
R 1
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H(f) =R
R +j2f L=
1
1 +j2fL/R.
Sy(f) =N02
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
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