Introduction to Analog And Digital Communications Second Edition Simon Haykin, Michael Moher
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Introduction to Analog And Digital Communications
Second Edition
Simon Haykin, Michael Moher
Chapter 8 Random Signals and Noise
8.1 Probability and Random Variables8.2 Expectation8.3 Transformation of Random Variables8.4 Gaussian Random Variables8.5 The Central Limit Theorem8.6 Random Processes8.7 Correlation of Random Processes8.8 Spectra of Random Signals8.9 Gaussian Processes8.10 Narrowband Noise8.11 Summary and Discussion
3
Random Signals And Noise “random” is used to describe erratic and apparently unpredictable
variations of an observed signal. This randomness or unpredictability is a fundamental property of
information. If the information were predictable, there would be no need to
communicate, because the other end could predict the information before receiving it.
Noise may be defined as any unwanted signal interfering with or distorting the signal being communicated.
It is the purpose of this chapter to introduce the tools that are necessary to analyze information and noise and that are required to understand the signal detection techniques described in the remainder of the book.
4
Lesson 1 : Random events and signals can be modeled as the outcomes of random experiments.
Lesson 2 : Random variables provide a general representation for analyzing, and processing outcomes of random experiments. Statistical properties of random events can be obtained from the expectation of the various functions of these fandom variables; the expectation should be viewed as an operator.
Lesson 3 : Gaussian random variables play a key role in the analysis of random signals because of the central limit theorem and their mathematical tractability.
Lesson 4 : A random process may be viewed as a family of random variables, indexed by a time parameter. Then we can extend the analysis techniques for random variables to study the time-variation of random process.
Lesson 5 : White noise is one of the most important random processes in the study of communication systems, both from a practical viewpoint and for the mathematical tractability of its statistical properties.
Lesson 6 : Narrowband noise can be analyzed in terns of its in-phase and quadrature components, in a manner similar to narrowband communication signals.
5
8.1 Probability and Random Variables
We ask that a random experiment have the following properties:1. On any trial of the experiment, the outcome is unpredictable.2. For a large number of trials of the experiment, the outcomes exhibit statistical
regularity. That is, a definite average pattern of outcomes is observed if the experiment is repeated a large number of times.
Relative-Frequency Approach The relative frequency is a nonnegative real number less than or equal to one.
The experiment exhibits statistical regularity if, for any sequence of n trials, the relative frequency converges to a limit as n becomes large.
probability of event A.
)1.8(10 ≤≤nnA
nnA /
)2.8(lim][P
=
∞→ nnA A
n
6
The Axioms of probability Each possible outcome of the experiment, we associate a point called
the sample point. The totality of all sample point Sanple space The entire sample space S Sure event, the null set null or impossible event; a single sample point
elementary event. A formal definition of probability. A probability system consists of the
triple:1. A sample space S of elementary events (outcomes).2. A class of events that are subsets of S.3. A probabilitry measure P[A] assigned to each event A in the class ,
which has the following properties:
φ
εε
thenclasstheineventsexclusivemutuallytwoofuniontheisBAIfiiiAPii
SPi
,)(1][0)(
1][)(
ε
≤≤=
7
The axioms of probability.
Mathematical form similar to that of axiom (iii). This abstract definition of a probability system is illustrated in
Fig.8.2
)3.8(][P][P][P BABA +=
)4.8(nn
nn
nn BABA +=
8
Fig. 8.1 Back Next
Fig. 8.1
9
Fig. 8.2 Back Next
Fig. 8.2
10
Random Varlables A function whose domain is a sample space and whose range is a set of
real numbers is called a random variable of the experiment. There may be more than one random variable associated with the same
random experiment. The comcept of a random variable is illustrated in Fig.8.3 For a discrete-valued random variable, the probability mass function
describes the probability of each possible value of the random variable. For the coin-tossing experiment, the probability mass function of the associated random variable may be written as
)5.8(otherwise,0
1 , 0 ,
][P21
21
==
== xx
xX
11
Fig. 8.3 Back Next
Fig. 8.3
12
Fig. 8.4 Back Next
Fig. 8.4
13
14
Distribution Functions Closely related to the probability mass function is the probability
distribution funciton.
The distribution funciton has two basic properties1. The distribution function is bounded between zero and one.2. The distribution function is a monotone nondecreasing
function of x; that is,
)7.8(][P)( xXxFX ≤=
)(xFX
)(xFX
2121 if)()( xxxFxF XX ≤≤
15
Probability density function, denoted by
A probability density function has three basic properties:1. Since the distribution function is monotone nondecreasing, it follows
that the density function is nonnegative for all values of x.2. The distribution function may be recovered from the density function
by integration, as shown by
3. Property 2 implies that the total area under the curve of the density function is unity.
)(xFX
)8.8()()( xFx
xf XX ∂∂
=
)9.8()()( ∫ ∞−=
x
XX dssfxF
16
17
Fig. 8.5
18
Fig. 8.5 Back Next
19
Several Random Variables Joint distribution function The probability that the random variable X is less than or equal to a
specified value x and that the random variable Y is less than or equal to a specified value y.
the joint probability density function.
),(, yxF YX
)11.8(],[P),(, yYxXyxF YX ≤≤=
)12.8(),(
),( ,2
, yxyxF
yxf YXYX ∂∂
∂=
),(, yxF YX
)13.8(),()( ,∫∫ ∞−
∞
∞−=
x
YXX ddfxF ηξηξ
)14.8(),()( ,∫∞
∞−= ηη dxfxf YXX
20
The probability density function may be obtained from the joint probability density function by simply integrating over all possible value of the random variable Y.
The probability density functions and marginal densities.
Statistically independent if the outcome of X does cot affect the outcome Y.
)(xFX
),(, yxF YX
)(xFX)(yfY
)15.8(][P][P],[P BYAXBYAX ∈∈=∈∈
)16.8()()(),(, yFxFyxF YXYX =
21
22
Fig. 8.6
23
Fig. 8.6 Back Next
24
Fig. 8.7
25
Fig. 8.7 Back Next
26
Conditional Probability denote the probability mass function of Y given that X has occurred. Conditional probability of Y given X.
Bayes’ rule.
Random variables X and Y that satisfy this condition Statistically indipendent.
]|[P XY
)22.8(][P],[P]|[P
XYXXY =
)23.8(][P]|[P],[P XXYYX =
)24.8(][P]|[P],[P YYXYX =
)25.8(][P
][P]|[P]|[PX
YYXXY =
)26.8(][P]|[P YXY =
)27.8(][P][P],[P YXYX =
27
28
29
Fig. 8.8
30
Fig. 8.8 Back Next
31
8.2 Expectation
Mean Statistical averages expectations For the expected value of a function(.) of the random variable
X. For a discrete random variable X, the mean, , is the weighted sum
of the possible outcomes.
The analogous definition of the expected value is
)]([ XgE
Xµ
)32.8(][P
][E
xXx
X
X
X
==
=
∑µ
)33.8()(][E dxxxfX X∫∞
∞−=
32
estimator The mean value of a random variable is estimated from N observations
If we consider X as a random variable representing observations of the voltage of a random signal, then the mean value of X represents the average voltage or dc offset of the signal.
)34.8(1ˆ1
X ∑=
=N
nnx
Nµ
NnMnn M⋅+⋅⋅⋅+⋅+⋅
= 21Z
21µ̂
∑
∑
=
=
=≈
=
M
i
M
i
i
iZi
Nni
1
1Z
][P
µ̂
33
Variance The variance, By the expectation of the squared distance of each outcome from the mean
value of the distribution.
X2σ
)35.8(][P)(
])[(E )(Var
2
2
2
xXx
XX
XX
X
X
=−=
−=
=
∑ µ
µσ
)36.8()()( 22 dxxfx XXX µσ ∫∞
∞−−=
)37.8()ˆ(1
1ˆ1
22 ∑=
−−
=N
nXnX x
Nµσ
34
Eq.(8.37) is an unbiased estimator of the variance
mean-square value of the random signal Total power of the signal.
)38.8(]P[)ˆ(1
)ˆ(1
1)ˆ()ˆ2()ˆ1(ˆ
1
2
1
2
22
21
22
∑
∑
=
=
=−−
≈
−−
=
−⋅−+⋅⋅⋅+⋅−+⋅−
=
M
iZ
M
n
iZ
MZZZZ
iZiN
NNni
NN
NnMnn
i
µ
µ
µµµσ
][ 2XE
35
36
Covariance The covariance of two random variables, X and Y
The expectation term of Eq. (8.40) is
If the two random variables happen to be independent
We find that the covariance of independent random variables is zero. Zero covariance does not, in general, imply independence.
)39.8()])([(E),(Cov YX YXYX µµ −−=
)40.8(][E),(Cov YXXYYX µµ−=
)41.8(),(][E ,∫ ∫∞
∞−
∞
∞−= dxdyyxxyfXY YX
)42.8( ][E][E
)()(
)()(][E
YX
dyyyfdxxxf
dxdyyfxxyfXY
XX
YX
=
=
=
∫∫∫ ∫
∞
∞−
∞
∞−
∞
∞−
∞
∞−
37
8.3 Transformation of Random Variables
What is the new distribution of the random variable after processing? If it follows that where B is defined by B = aA + b
The set A is gives by
AX ∈ BY ∈
)43.8(][P][P BYAX ∈=∈
]/)(,(/)( abyabBA −−∞=−=
)44.8(
]]/)(,([P ]],([P)(
−
=
−−∞∈=−∞∈=
abyF
abyXyXyF
X
Y
38
If Y = g(X) is a one-to-one transformation of the random variable X to the random variable X to the random variable Y
Functional inverse of g(y)
)45.8())(()( 1 ygFyF XY−=
)(1 yg −
39
Fig. 8.9
40
Fig. 8.9 Back Next
41
42
8.4 Gaussian Random Variables
A Gaussian random variable
A Gaussian random variable has a number of properties that we will state without proof
1. A Gaussian random variable is completely characterized by its mean and variance.
2. A Gaussian random variable plus a constant is another Gaussian random variable with the mean adjusted by the constant.
3. A Gaussian random variable multiplied by a constant is another Gaussian random variable where both the mean and variance are affected by the constant.
4. The sum of two independent Gaussian random variable is also a Gaussian random variable.
5. The weighted sum of N independent Gaussian random variables is a Gaussian random variable.
6. If two Gaussian random variable have zero covariance (uncorrelated), they are also independent.
)48.8(2
)(exp2
1)( 2
2
−−=
XXX
xxxfσµ
σπ
43
For the special case of a Gaussian random variable with a mean of zero,
Normalized Gaussian random variable
Q-funciton The complement of the normalized Gaussian distribution function.
0=Xµ
)49.8(),2/exp(21)( 2 ∞<<∞−−= xxxf X π
)50.8()2/exp(21
)()(
2∫
∫
∞−
∞−
−=
=
x
x
XX
dss
dssfxF
π
)51.8( )(1
)2/exp(21)( 2
xF
dssxQ
X
x
−=
−= ∫∞
πFig. 8.10
Fig. 8.11
44
Fig. 8.10 Back Next
45
Fig. 8.11 Back Next
46
Fig. 8.12
47
Fig. 8.12 Back Next
48
49
8.5 The Central Limit Theorem
The Gaussian distribution is the central limit theorem.1. The with k = 1, 2, 3…., n are statistically indepindint.2. The all have the same probability density function.3. Both the mean and the variance exist for each .
The normalized random variable
A zero-mean Gaussian random variable with unit variance
kXkX
kX
)56.8(1∑=
=n
kkXY
)57.8(][EY
YYZσ−
=
)58.8(2
exp21)(
2
dsszFz
Z ∫ ∞−
−→
π
50
A mathematical statement of the central limit theorem. The normalized distribution of the sum of independent, identically
distributed random variables approaches a Gaussian distribution as the number of random variables increases, regardless of the individual distributions.
Computer Experiment : Sums of random variables We consider the random variable
Where the are independent, uniformly distributed random variables on the interval from -1 to +1. In the computer experiment, we compute 20,000 samples of Z for N = 5, and estimate the corresponding density function by forming a histogram of the results.
The results of this experiment indicate how powerful the central limit theorem is and explain why Gaussian models are ubiquitous in the analysis of random signals in communications and elsewhere.
∑=
=N
iiXZ
1
iX
Fig. 8.13
51
Fig. 8.13 Back Next
52
8.6 Random Process
We combine the concepts of time variation and random variables to introduce the concept of random processes.
Random processes have the following properties:1. Random processes are functions of time.2. Random processes are random in the sense that it is not possible to
predict exactly what waveform will be observed in the future. Suppose that we assign to each sample point s a function of time
with the label
This sample function as
)59.8(),,( TtTstX <<−
)60.8(),()( jj stXtx =
)},(),...,,(),,({)}(),...,(),({ 2121 nkkkknkk stXstXstXtxtxtx =
53
To compare: With a random variable, the outcome of random experiment is mapped
to a real number. With a random process, the outcome of random experiment is mapped
into a waveform that is a function of time. The family of all such random variables, indexed by the time
variable t, forms the random process. Stationary random processes Stationary If a random process is divided into a number of time intervals, the various
sections of the process exhibit essentially the same statistical properties. Otherwise, nonstationary. It is said to be nonstationary.
Fig. 8.14
54
Suppose the same random process is observed at time , and the corresponding distribution function is .
A first-order stationary random process A first-order stationary random process has a distribution function that
is independent of time. Suppose has the density . The mean value
Does not change with time because the distribution function (and hence the density) are time invariant.
τ+1t)()( 1
xF tX τ+
)61.8()()( )()( 11xFxF tXtX =+τ
)()( 1xF tX
)()( 1xf tX
)62.8()()( )()( 11dsssfdsssf tXtXX ∫∫
∞
∞−+
∞
∞−== τµ
55
Suppose a second set of observations are made at times and, and the corresponding joint distribution is
Stationary to the second order. Statistical quantities such as covariance and correlation, which we will
discuss in the following, do not depend upon absolute time. In other words, a random process X(t) is strictly stationary of the joint
distribution of any set of random variables obtained by observing the random process X(t) is invariant with respect to the location of the origin t = 0.
τ+1tτ+2t ),( 21)(),( 21
xxF tXtX ττ ++
)63.8(),(),( 21)(),(21)(),( 2121xxFxxF tXtXtXtX =++ ττ
56
Fig. 8.14 Back Next
57
8.7 Correlation of Random Process
The covariance of the two random variables and is given by
1. The mean of the random process is a constant independent of time: for all t.
2. The autocorrelation of the random process only depends upon the time difference:
Wide-sense stationary or weakly stationary. If a random process has these two properties, then we say it is wide-
sense stationary or weakly stationary
)( 1tX )( 2tX
)64.8()]()([E))(),((Cov )()(2121 21 tXtXtXtXtXtX µµ−=
)65.8()](*)([E),( sXtXstRX =
)66.8()( )](*)([E),(
stRsXtXstR
X
X
−==
XtXE µ=)]([
τττ andtallforRtXtXE X ),()]()([ * =−
58
Properties of the autocorrelation function Property 1 Power of a Wide-Sense Stationary Process The second moment or mean-square value of a real-valued random process
is given by
Property 2 Symmetry The autocorrelation of a real-valued wide-sense stationary process has
even symmetry.
Property 3 Maximum Value The autocorrelation function of a wide-sense stationary random process is
a maximum at the origin.
)67.8()]([E )]()([E)0(
2 tXtXtXRX
=
=
)68.8()( )]()([E )]()([E)(
ττ
ττ
X
X
RtXtX
tXtXR
=+=
+=−
)69.8()(2)0(2 )]()([E2)]([E)]([E
]))()([(E022
2
τττ
τ
XX RRtXtXtXtX
tXtX
±≤−±−+≤
−±≤Fig. 8.15
59
Fig. 8.15 Back Next
60
61
62
Erfodicity Ensemble The expected value and second moment of the random process at time
The time average of a continuous sample function drawn from a real-valued process is given by
Time-autocorrelation of the sample function
ktt =
)70.8()(1)]([E1∑=
=N
jkjk tx
NtX
)71.8()(1)]([E1
22 ∑=
=N
jkjk tx
NtX
)72.8()(21lim][ dttxT
xT
TT ∫−∞→=ε
)73.8()()(21lim)( dttxtxT
RT
TTx ∫−∞→−= ττ
63
When are the time averages of a sample function equal to the ensemble averages of the corresponding random process
Of the statistics of the random process X(t) do not change with time, then we might expect the time averages and ensemble averages to be equivalent.
Ergodic. Random processes for which this equivalence holds are said to be
ergodic. Wide-sense stationary processes are assumed to be ergodic, in
which case time averages and expectations can be used interchangeably.
64
If we assume that the real-valued random process is ergodic
An estimate of the autocorrelation of a real-valued process for is (see Fig. 2.29)
)74.8()()(21lim
)]()([E)(
dttxtxT
tXtXRT
TT
X
∫−∞→−=
−=
τ
ττ
0ττ =lag
)75.8()()(1)(ˆ1
00 ∑=
−=N
nnnX txtx
NR ττ
65
Fig. 8.16
66
Fig. 8.16 Back Next
67
8.8 Spectra of Random Signals
Figure 8.17 shows a plot of the waveform of on the interval -T<t<T.
Fourier transform of the sample function as
Effectively, the Fourier transform has converted a family of random variables X(t) indexed by parameter t to a new family of random variables indexed by parameter f.
The ensemble-averaged value of the new random process asand the corresponding power spectral density of the
random process X(t) as
)(txT
)(txT
)78.8()2exp()()( ∫∞
∞−−= dtftjtxf TT πξ
)( fTΞ2)( fTΞ
[ ]2)( fE TΞ
)79.8(]|)([|E21lim)( 2fT
fS TTX Ε∞→
=
Fig. 8.17
68
Fig. 8.17 Back Next
69
The discrete Fourier transform is defines as
We may estimate the power spectral density of a random process by the following three steps:
1. Partition the sample function x(t) into M sections of length and sample at intervals
2. Perform a DFT on each section of length . Let where m = 0,…,M-1 represent the M DFT outputs, one set for each esction.
3. Average the magnitude squared of each DFT, and then the power spectral density estimate is given by
)80.8(W1
0∑
−
=
=N
n
knnk xξ
sNTsT
sNT { }mNk+ξ
)81.8(1,...,0,W1
||1ˆ
1
0
21
0
1
0
2
∑∑
∑−
=
−
=+
−
=+
−==
=
M
m
N
n
knmNn
M
mmNk
sX
MkxM
MNTkS ξ
70
Properties of the power spectral density Weiner-Khintchine relations
The Weiner-Khintchine relations show that if either the autocorrelation or the power spectral density of the random process is known, the other may be found exactly.
Property 1 Mean-Square Value The mean-square value of a stationary process equals the total area
under the graph of the power spectral density; that is,
∫∞
∞−−= )82.8()2exp()()( ττπτ dfjRfS XX
∫∞
∞−= )83.8()2exp()()( dffjfSR XX τπτ
∫∞
∞−= )84.8()(]|)([|E 2 dffStX X
]|)([|E)0( 2tXRX =
71
Property 2 Nonnegativity The power spectral density of a stationary random process is always
nonnegative; that is,
Property 3 Symmetry The power spectral density of a real random process is an even function
of frequency; that is,
)85.8( allfor ,0)( ffSX ≥
)86.8()()( fSfS XX =−
∫∞
∞−=− ττπτ dfjRfS XX )2exp()()(
)()2exp()()( fSdfjRfS XXX =−=− ∫∞
∞−ττπτ
72
Property 4 Filtered random process If a stationary random process X(t) with spectrum is passed
through a linear filter with frequency response H(f), the spectrum of the stationary output random process Y(t) is given by
)( fSX
)87.8()(|)(|)( 2 fSfHfS XY =
73
74
75
8.9 Gaussian Processes
The joint distribution of N Gaussian random variables may be written as
Represents an N-dimensional vector of Gaussian random variables
is the corresponding vector of indeterminates. is the N-dimensional vector of means is the N-by-N covariance matrix with individual elements gives
by
)90.8(}2/)μx()μx(exp{||)2(
1)x( 12/12/
TNXf −Λ−−
Λ= −
π
)( ,,....2,1 NXXXX =
)( ,,....2,1 Nxxxx =
])[],....,[],[( 21 NXEXEXE=µΛ
),( jiij XXCov=Λ
76
A Gaussian process has the following properties:1. If a Gaussian process is wide-sense stationary, then it is also stationary
in the strict sense.2. If a Gaussian process is applied to a stable linear filter, then the
random process Y(t) produced at the output of the filter is also Gaussian.
3. If integration is defined in the mean-square sense, then we may interchange the order of the operations of integration and expectation with a Gaussian random process.
This first property comes from observing that if a Gaussian process is wide-sense stationary
The second property comes from observing that the filtering operation can be written as
)91.8()()()(0
dssXsthtYt
∫ −=
77
1. The integral of Eq. (8.91) is defined as the mean-square limit of the sums
we observe that the right-hand side is a weighted sum of the Gaussian random variables
2. Recall from the properties of Gaussian random variables that a weighted sum of Gaussian random variables is another Gaussian random variable.
3. If a sequence of Gaussian random variables converges in the mean-square sense, then the result is a Gaussian random variable.
The third property of Gaussian processes implies that if Y(t) is given by Eq.(8.91) then the mean of the output is given by
∑ ∆∆∆−=→∆
is
ssixsithtY )()(lim)(0
).( siX ∆
)(
)]([E)(
)()(E)]([E
0
0
t
dssXsth
dssXsthtY
Y
t
t
µ=
−=
−=
∫
∫
78
8.10 White Noise
The power spectral density of white noise is independent of frequency.
We denote the power spectral density of a white noise process W(t)
The autocorrelation of white noise is given by
White noise has infinite average power and, as such, it is not physically realizable.
White noise has convenient mathematical properties and is therefore useful in system analysis.
)92.8(2
)( 0W
NfS =
)93.8()(2
)( 0W τδτ NR =
Fig. 8.18
79
Fig. 8.18 Back Next
80
Fig. 8.19
81
Fig. 8.19 Back Next
82
Fig. 8.20
83
Fig. 8.20 Back Next
84
8.11 Narrowband Noise
Narrowband noise. The noise process appearing at the output of such a filter
Narrowband noise can be represented mathematically using in-phase and quadrature components
is called the in-phase component of N(t) and is the quadrature component.
)97.8()2sin()()2cos()()( tftNtftNtN cQcI ππ −=
)(tNI )(tNQ
Fig. 8.21
Fig. 8.22
85
Fig. 8.21 Back Next
86
Fig. 8.22 Back Next
87
The in-phase and quadrature components of narrowband noise have the following important properties:
1. The in-phase component and quadrature component of narrowband noise N(t) have zero mean.
2. If the narrowband noise N(t) is Gaussian, then its in-phase and quadrature components are Gaussian.
3. If the narrowband noise N(t) is stationary, then its in-phase and quadrature components are stationary.
4. Both the in-phase component and have the same power spectral density. This power spectral density is related to the power spectral density of the narrowband density by
5. The in-phase component and quadrature componenthave the same variance as the narrowband noise N(t).
)(tNI )(tNQ
)(tNI )(tNQ
)( fSN
)98.8(otherwise ,0
),()()()(
≤≤−++−
==BfBffSffS
fSfS cNcNNN QI
)(tNI )(tNQ
Fig. 8.23
88
Fig. 8.23 Back Next
89
The spectrum of the in-phase component of the narrowband noise is given by
The is-phase component has the same variance (power) as the narrowband noise.
≤≤−
= otherwise ,0
,)( 0 BfBN
fSIN
BNdffSdffSINN 02)()( == ∫∫
∞
∞−
∞
∞−
90
Fig. 8.23
91
Noise –equivalent bandwidth Noise –equivalent bandwidth Power spectral densities , the average output noise power is
The average output noise power is
)102.8( |)(|
2|)(|
)(|)(|
0
20
02
W2
∫
∫
∫
∞
∞
∞−
∞
∞−
=
=
=
dffHN
dfNfH
dffSfHPN
)103.8(|)0(| 20 HBNP NN =
)104.8(|)0(|
|)(|2
0
2
H
dffHBN
∫∞
=
92
noise-equivalent bandwidth for a low-pass filter. The noise-equivalent bandwidth for a band-pass filter, as illustrated in
Fig.8.25. The noise-equivalent bandwidth for a band-pass filter may be defined as
The general result
And the effect of passing white noise through a filter may be separated into two parts: The center-frequency power gain The noise equivalent bandwidth ,representing the frequency selectivity of
the filter.
Fig. 8.24 Fig. 8.25
NB
)105.8(|)(|
|)(|2
0
2
cN fH
dffHB ∫
∞
=
)106.8(|)(| 20 NcN BfHNP =
.)( 2
cfHNB
93
Fig. 8.24 Back Next
94
Fig. 8.25 Back Next
95
96
8.12 Summary and Discussion
We introduced a random experiment as a model for unpredictable phenomena; and the relative frequency definition of probability as a means of assigning a likelihood to the outcome of a random experiment.
Random variables were introduced as a domain is the sample space of the random experiment and whose range is the real numbers.
The concept of expectation and the statistical moments and covariance of random variables.
Gaussian random variables were introduced as a particular important type of random variable in the study of communication systems.
A random process was defined as a family of random variables indexed by time as a parameter.
Gaussian processes and white noise were introduced as important random processes in the analysis of communication systems.
This narrowband noise has in-phase and quadrature components, similar to deterministic signals.
The two subsequent chapters will illustrate the importance of the material presented in this chapter in designing receivers and evaluating communication system performance.
97
Fig. 8.26 Back Next
Fig. 8.26
98
Fig. 8.27 Back Next
Fig. 8.27
99
Fig. 8.28 Back Next
Fig. 8.28
100
Fig. 8.29 Back Next
Fig. 8.29
101
Fig. 8.5 Back Next
Fig. 8.5
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