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Chapter 1 Random Processes
“Probabilities” is considered an important background to communication study.
© Po-Ning Chen@ece.nctu Chapter 1-2
1.1 Mathematical Models
To model a system mathematically is the basis of its analysis, either analytically or empirically.
Two models are usually considered: Deterministic model
No uncertainty about its time-dependent behavior at any instance of time.
Random or stochastic model Uncertain about its time-dependent behavior at any
instance of time, but certain on the statistical behavior at any instance of
time.
© Po-Ning Chen@ece.nctu Chapter 1-3
1.1 Examples of Stochastic Models
Channel noise and interference Source of information, such as voice
© Po-Ning Chen@ece.nctu Chapter 1-4
A1.1: Relative Frequency
How to determine the probability of “head appearance” for a coin?
Answer: Relative frequency.
Specifically, by carrying out n coin-tossing experiments, the relative frequency of head appearance is equal to Nn(A)/n, where Nn(A) is the number of head appearance in these n random experiments.
© Po-Ning Chen@ece.nctu Chapter 1-5
A1.1: Relative Frequency
Is relative frequency close to the true probability (of head appearance)? It could occur that 4-out-of-10 tossing results are
“head” for a fair coin!
Can one guarantee that the true “head appearance probability” remains unchanged (i.e., time-invariant) in each experiment (performed at different time instance)?
© Po-Ning Chen@ece.nctu Chapter 1-6
A1.1: Relative Frequency
Similarly, the previous question can be extended to “In a communication system, can we estimate the noise by repetitive measurements at consecutive but different time instance?”
Some assumptions on the statistical models are necessary!
© Po-Ning Chen@ece.nctu Chapter 1-7
A1.1: Axioms of Probability Definition of a Probability System (S, F, P) (also
named Probability Space)1. Sample space S
All possible outcomes (sample points) of the experiment
2. Event space F Subset of sample space, which can be probabilistically
measured. F and S F A F implies Ac F. A F and B F implies A∪B F.
3. Probability measure P
A F and B F implies A∩B F.
© Po-Ning Chen@ece.nctu Chapter 1-8
A1.1: Axioms of Probability
3. Probability measure P A probability measure satisfies:
P(S) = 1 and P(EmptySet) = 0 For any A in F, 0 ≦ P(A) 1.≦ For any two mutually exclusive events A and B, P(A ∪
B) = P(A) + P(B)
)()()( .3
1.)(
0 .2
.0)emptySet( and )( .1
BNANBANn
AN
NnSN
nnn
n
nn
These Axioms coincide with the relative-frequency expectation.
© Po-Ning Chen@ece.nctu Chapter 1-9
A1.1: Axioms of Probability
Example. Rolling a dice. S = { } F = {EmpySet, { }, { }, S} P satisfies
P(EmptySet) = 0 P({ }) = 0.4 P({ }) = 0.6 P(S) = 1.
© Po-Ning Chen@ece.nctu Chapter 1-10
A1.1: Properties from Axioms
All the properties are induced from Axioms Example 1. P(Ac) = 1 – P(A).
Proof. Since A and Ac are mutually exclusive events, P(A) + P(Ac) = P(A∪Ac) = P(S) = 1.
Example 2. P(A∪B) = P(A) + P(B) – P(A∩B).
Proof. P(A∪B) = P(A/B) + P(B/A) + P(A∩B) = [P(A/B) + P(A∩B)] + [P(B/A) + P(A∩B)] – P(A∩B) = P(A) + P(B) – P(A∩B).
© Po-Ning Chen@ece.nctu Chapter 1-11
A1.1: Conditional Probability
Definition of conditional probability
Independence of events A knowledge of occurrence of event A tells us no
more about the probability of occurrence of event B than we knew without this knowledge.
Hence, they are statistically independent.
)(
)(
)(
)( )|(
AP
BAP
AN
BANABP
n
n
)()|( BPABP
© Po-Ning Chen@ece.nctu Chapter 1-12
A1.2: Random Variable
A probability system (S, F, P) can be “visualized” (or observed, or recorded) through a real-valued random variable X
1
2
3
4
5
6
After mapping the sample point in the sample space to a real number, the cumulative distribution function (cdf) can be defined as:
})(:{]Pr[)( xsXSsPxXxFX
: SX
© Po-Ning Chen@ece.nctu Chapter 1-13
A1.2: Random Variable Since the event [X ≦ x] has to be probabilistically
measurable for any real number x, the event space F has to consist of all the elements of the form [X ≦ x].
In previous example, the event space F must contain the intersections and unions of the following 6 sets.
{
{
{
{
{
{
}={sS:X(s)≦1}
}={sS:X(s)≦2}
}={sS:X(s)≦3}
}={sS:X(s)≦4}
}={sS:X(s)≦5}
}={sS:X(s)≦6}
Otherwise, the cdf is not well-defined.
© Po-Ning Chen@ece.nctu Chapter 1-14
A1.2: Random Variable
It can be proved that we can construct a well-defined probability system (S, F, P) for any random variable X and its cdf FX. So to define a real-valued random variable by its
cdf is good enough from engineering standpoint. In other words, it is not necessary to mention the
probability system (S, F, P) before defining a random variable.
© Po-Ning Chen@ece.nctu Chapter 1-15
A1.2: Random Variable
It can be proved that any function satisfying:
is a legitimate cdf for some random variable. It suffices to check the above three properties for
FX(x) to well-define a random variable.
.decreasing-Non 3.
.continuous - Right.2
.1)(lim and 0)(lim .1
xGxGxx
© Po-Ning Chen@ece.nctu Chapter 1-16
A1.2: Random Variable
A non-negative function fX(x) satisfies
is called the probability density function (pdf) of random variable X.
If the pdf of X exists, then
x
X dttfxX )()Pr(
x
xFxf X
X
)(
)(
© Po-Ning Chen@ece.nctu Chapter 1-17
A1.2: Random Variable
It is not necessarily true that If
then the pdf of X exists and equals fX(x).
,)(
)(x
xFxf X
X
© Po-Ning Chen@ece.nctu Chapter 1-18
A1.2: Random Vector
We can extend a random variable to a (multi-dimensional) random vector. For two random variables X and Y, the joint cdf is
defined as:
Again, the event [X ≦ x and Y ≦ y] must be probabilistically measurable in some probability system (S, F, P) for any real numbers x and y.
)} )(:{ } )(:({
) and Pr(),(,
ysYSsxsXSsP
yYxXyxF YX
© Po-Ning Chen@ece.nctu Chapter 1-19
A1.2: Random Vector
As continuing from previous example, the event space F must contain the intersections and unions of the following 4 sets.
{
{
{
{
}={sS : X(s)≦1 and Y(s)≦1}
}={sS : X(s)≦2 and Y(s)≦1}
}={sS : X(s)≦1 and Y(s)≦2}
}={sS : X(s)≦2 and Y(s)≦2}
(1,1)
(2,1)
(1,1)
(2,2)
(1,2)
(2,2)
:),( SYX
© Po-Ning Chen@ece.nctu Chapter 1-20
A1.2: Random Vector
If its joint density fX,Y(x,y) exists, then
The conditional density function of Y given that [X = x] is
provided that fX(x) 0.
© Po-Ning Chen@ece.nctu Chapter 1-21
1.2 Random Process
Random process: An extension of multi-dimensional random vectors Representation of 2-dimensional random vector
(X,Y) = (X(1), X(2)) = {X(j), jI}, where the index set I equals {1, 2}.
Representation of m-dimensional random vector {X(j), jI}, where the index set I equals {1, 2,…, m}.
© Po-Ning Chen@ece.nctu Chapter 1-22
1.2 Random Process
How about {X(t), t}? It is no longer a random vector since the index set is
continuous! This is a suitable model for, e.g., a noise because a
noise often exists continuously in time. Its cdf is well-defined through the mapping:
for every t .
StX :)(
© Po-Ning Chen@ece.nctu
X(t,s1)
X(t,s2)
X(t,sn)
Chapter 1-23
Define or view a random process via its inherited probability system
© Po-Ning Chen@ece.nctu Chapter 1-24
1.2 Random Process
X(t, sj) is called a sample function (or a realization) of the random process for sample point sj.
X(t, sj) is deterministic.
© Po-Ning Chen@ece.nctu Chapter 1-25
1.2 Random Process
Notably, with a probability system (S, F, P) over which the random process is defined, any finite-dimensional joint cdf is well-defined. For example,
332211
332211
),( : ),( :),(:
)( and )( and )(Pr
xstXSsxstXSsxstXSsP
xtXxtXxtX
© Po-Ning Chen@ece.nctu Chapter 1-26
1.2 Random Process
Summary A random variable
maps s to a real number X A random vector
maps s to a multi-dimensional real vector. A random process
maps s to a real-valued deterministic function.
s is where the probability is truly defined; yet the image of the mapping is what we can observe and experiment over.
© Po-Ning Chen@ece.nctu Chapter 1-27
1.2 Random Process
Question: Can we map s to two or more real-valued deterministic functions?
Answer: Yes, such as (X(t), Y(t)). Then, we can discuss any finite-dimensional joint
distribution of X(t) and Y(t), such as, the joint distribution of
))(),(),(),(),(( 41321 tYtYtXtXtX
© Po-Ning Chen@ece.nctu Chapter 1-28
1.3 (Strictly) Stationary
For strict stationarity, the statistical property of a random process encountered in real world is often independent of the time at which the observation (or experiment) is initiated.
Mathematically, this can be formulated as that for any t1, t2, …, tk and :
),...,,(
),...,,(
21)(),...,(),(
21)(),...,(),(
21
21
ktXtXtX
ktXtXtX
xxxF
xxxF
k
k
© Po-Ning Chen@ece.nctu Chapter 1-29
1.3 (Strictly) Stationary
Example 1.1. Suppose that any finite-dimensional cdf of a random process X(t) is defined. Find the probability of the joint event.
222111 )( and )( btXabtXaA
),(),(
),(),()(
21)(),(21)(),(
21)(),(21)(),(
2121
2121
aaFbaF
abFbbFAP
tXtXtXtX
tXtXtXtX
Answer:
© Po-Ning Chen@ece.nctu Chapter 1-30
1.3 (Strictly) Stationary
© Po-Ning Chen@ece.nctu Chapter 1-31
1.3 (Strictly) Stationary
Example 1.1. Further assume that X(t) is strictly stationary.
Then, P(A) is also equal to:
),(),(
),(),()(
21)(),(21)(),(
21)(),(21)(),(
2121
2121
aaFbaF
abFbbFAP
tXtXtXtX
tXtXtXtX
Why introducing “stationarity?” With stationarity, we can be certain that the
observations made at different time instances have the same distributions!
For example, X(0), X(T), X(2T), X(3T), ….
Suppose that Pr[X(0) = 0] = Pr[X(0)=1] = ½. Can we guarantee that the relative frequency (i.e., their average) of “1’s appearance” for experiments performed at several different times approach ½ by stationarity? No, we need additional assumption!
© Po-Ning Chen@ece.nctu Chapter 1-32
1.3 (Strictly) Stationary
© Po-Ning Chen@ece.nctu Chapter 1-33
1.4 Mean
The mean of a random process X(t) at time t is equal to:
where fX(t)() is the pdf of X(t) at time t.
If X(t) is stationary, the mean X(t) is independent of t, and is a constant for all t.
dxxfxtXEt tXX )()]([)( )(
© Po-Ning Chen@ece.nctu Chapter 1-34
1.4 Autocorrelation
The autocorrelation function of a real random process X(t) is given by:
If X(t) is stationary, the autocorrelation function RX(t1, t2) is equal to RX(t1t2, 0).
2121)(),(21
2121
),(
)]()([),(
21dxdxxxfxx
tXtXEttR
tXtX
X
© Po-Ning Chen@ece.nctu Chapter 1-35
1.4 Autocorrelation
)(
)0,(
)]0()([
),(
),(
)]()([),(
21
21
21
2121)(),0(21
2121)(),(21
2121
12
21
ttR
ttR
XttXE
dxdxxxfxx
dxdxxxfxx
tXtXEttR
X
X
ttXX
tXtX
X
A short-hand for autocorrelation function of a stationary process
© Po-Ning Chen@ece.nctu Chapter 1-36
1.4 Autocorrelation
Conceptually, autocorrelation function = “power correlation” between two time instances t1 and t2.
© Po-Ning Chen@ece.nctu Chapter 1-37
1.4 Autocovariance
“Variance” is the degree of variation to the standard value (i.e., mean).
)()(),(
)()()()(
)()(),(
)()()()(
)()()()(
)()()()(
)()()()(
)()()()(),(
2121
2121
2121
2121
2121
2121
2121
221121
ttttR
tttt
ttttR
tttXEt
ttXEtXtXE
tttXt
ttXtXtXE
ttXttXEttC
XXX
XXXX
XXX
XXX
X
XXX
X
XXX
© Po-Ning Chen@ece.nctu Background 38
Autocovariance function CX(t1, t2) is given by:
© Po-Ning Chen@ece.nctu Chapter 1-39
1.4 Autocovariance
If X(t) is stationary, the autocovariance function CX(t1, t2) becomes
)(
)0,(
)0,(
)()(),(),(
21
21
2
21
212121
ttC
ttC
ttR
ttttRttC
X
X
XX
XXXX
© Po-Ning Chen@ece.nctu Chapter 1-40
1.4 Wide-Sense Stationary (WSS)
Since in most cases of practical interest, only the first two moments (i.e., X(t) and CX(t1, t2)) are concerned, an alternative definition of stationarity is introduced.
Definition (Wide-Sense Stationarity) A random process X(t) is WSS if
)(),(
constant;)(
2121 ttCttC
t
XX
X
).(),(
constant;)(or
2121 ttRttR
t
XX
X
© Po-Ning Chen@ece.nctu Chapter 1-41
1.4 Wide-Sense Stationary (WSS)
Alternative names for WSS include weakly stationary stationary in the weak sense second-order stationary
If the first two moments of a random process exist (i.e., are finite), then strictly stationary implies weakly stationary (but not vice versa).
© Po-Ning Chen@ece.nctu Chapter 1-42
1.4 Cyclostationarity
Definition (Cyclostationarity) A random process X(t) is cyclostationary if there exists a constant T such that
).,(),(
;)()(
2121 ttCTtTtC
tTt
XX
XX
© Po-Ning Chen@ece.nctu Chapter 1-43
1.4 Properties of autocorrelation function for WSS random process
1. Second Moment: RX(0) = E[X2(t)] > 0.
2. Symmetry: RX() = RX().
1. In concept, autocorrelation function = “power correlation” between two time instances t1 and t2.
2. For a WSS process, this “power correlation” only depends on time difference.
© Po-Ning Chen@ece.nctu Chapter 1-44
1.4 Properties of autocorrelation function for WSS random process
3. Peak at zero: |RX()| ≦ RX(0)
)(2)0(2
)(2)0()0(
)]()([2)()(
)()(022
2
XX
XXX
RR
RRR
tXtXEtXEtXE
tXtXE
Proof:
Hence, )0()()0( XXX RRR with equality holds when
.1)0()(Pr)()(Pr XXtXtX
© Po-Ning Chen@ece.nctu Chapter 1-45
1.4 Properties of autocorrelation function for WSS random process
Operational meaning of autocorrelation function: The “power” correlation of a random process at
seconds apart. The smaller RX() is, the less correlation between
X(t) and X(t+).
© Po-Ning Chen@ece.nctu Chapter 1-46
1.4 Properties of autocorrelation function for WSS random process
If RX() decreases faster, the correlation decreases faster.
© Po-Ning Chen@ece.nctu Chapter 1-47
1.4 Decorrelation time
Some researchers define the decorrelation time as:
© Po-Ning Chen@ece.nctu Chapter 1-48
Example 1.2 Signal with Random Phase
Let X(t) = A cos(2fct + ), where is uniformly distributed over [, ). Application: A local carrier at the receiver side may
have a random “phase difference” with respect to the carrier at the transmitter side.
© Po-Ning Chen@ece.nctu Chapter 1-49
Example 1.2 Signal with Random Phase
ChannelEncoder
…0110Modulator
…,m(t), m(t), m(t), m(t)
m(t)
T
Carrier waveAccos(2fct)
s(t)
w(t)
x(t)
Local carriercos(2fct+)
T
dt0
correlator
yT>< 0
0110…
X(t)=Acos(2fct+)Local carriercos(2fct)
© Po-Ning Chen@ece.nctu Chapter 1-50
Example 1.2 Signal with Random Phase
Then
.0
)2sin()2sin(2
)2sin(2
)2cos(2
2
1)2cos(
)]2cos([)(
tftfA
tfA
dtfA
dtfA
tfAEt
cc
c
c
c
cX
© Po-Ning Chen@ece.nctu Chapter 1-51
Example 1.2 Signal with Random Phase
.)(2cos2
)(2cos)(22cos2
1
2
)2()2(cos
)2()2(cos2
1
2
2
1)2cos()2cos(
)]2cos()2cos([),(
21
2
2121
2
21
21
2
212
2121
ttfA
dttfttfA
dtftf
tftfA
dtftfA
tfAtfAEttR
c
cc
cc
cc
cc
ccX
Hence, X(t) is WSS.
© Po-Ning Chen@ece.nctu Chapter 1-52
Example 1.2 Signal with Random Phase
© Po-Ning Chen@ece.nctu Chapter 1-53
Example 1.3 Random Binary Wave
Let
where …, I2, I1, I0, I1, I2, … are independent, and each Ij is either 1 or 1 with equal probability, and
)()( dn
n tnTtpIAtX
otherwise,0
0,1)(
Tttp
© Po-Ning Chen@ece.nctu Chapter 1-54
Example 1.3 Random Binary Wave
I0 I1 I2 I3 I4I1I2I3I4I5I6
© Po-Ning Chen@ece.nctu Chapter 1-55
Example 1.3 Random Binary Wave
ChannelEncoder
…0110Modulator
m(t)
T
No/Ignorecarrier wave
s(t)
w(t)
x(t)
correlator
yT>< 0
0110…
…,m(t), m(t), m(t), m(t)
X(t) = A p(t−td)
© Po-Ning Chen@ece.nctu Chapter 1-56
Example 1.3 Random Binary Wave
Then by assuming that td is uniformly distributed over [0, T), we obtain:
0
)]([0
)]([][
)()(
dn
dn
n
dn
nX
tnTtpEA
tnTtpEIEA
tnTtpIAEt
© Po-Ning Chen@ece.nctu Chapter 1-57
Example 1.3 Random Binary Wave
A useful probabilistic rule: E[X] = E[E[X|Y]]
dttXtXEEtXtXE )()()]()([ 2121
So we have:
)()(
)()(][
)()(][
]|)()([]|[
)()(
)()(
212
2122
212
212
21
21
dn
d
dn
dn
dn m
dmn
ddn m
ddmn
ddm
mdn
n
d
tnTtptnTtpA
tnTtptnTtpIEA
tmTtptnTtpIIEA
ttmTtptnTtpEtIIEA
ttmTtpIAtnTtpIAE
ttXtXE
© Po-Ning Chen@ece.nctu
.for 0][][][ Since mnIEIEIIE mnmn
Chapter 1-58
© Po-Ning Chen@ece.nctu
1ly equivalentor
0 if 1)(
11
11
T
ttn
T
tt
TtnTttnTtp
dd
dd
1ly equivalentor
0 if 1)(
22
22
T
ttn
T
tt
TtnTttnTtp
dd
dd
integer. an bemust n
Three conditions must be simultaneously satisfied in order to obtain a non-zero
n dd tnTtptnTtp )()( 21
.0 Ttd Notably,
Chapter 1-59
.0 and 0 01 0 and 0
0
111 and 0
:gives 1 withCombining
dd
dd
ddd
dd
tTTT
nt
TtTtT
t
T
tnTt
T
tn
T
t
© Po-Ning Chen@ece.nctu
0 if,0
0 if,1
integer an bemust
1
d
ddd
tn
Ttn
nT
tn
T
t
For an easier understanding, we let t1 = 0 and t2 = 0.
The below two conditions reduce to:
Chapter 1-60
© Po-Ning Chen@ece.nctu Chapter 1-61
otherwise,0
)0 and ||0(or )||0(,
)()()()(
2121
2
21
2
21
dd
dn
dd
tTttTtttA
tnTtptnTtpAttXtXE
Example 1.3 Random Binary Wave
otherwise,0
||0,||
1
otherwise,0
||0,1
)()()()(
21212
21||
2
2121 21
TttT
ttA
TttdtT
AttXtXEEtXtXET
tt dd
© Po-Ning Chen@ece.nctu Chapter 1-62
Example 1.3 Random Binary Wave
© Po-Ning Chen@ece.nctu Chapter 1-63
1.4 Cross-Correlation
The cross-correlation between two processes X(t) and Y(t) is:
Sometimes, their correlation matrix is given instead for convenience:
)]()([),(, uYtXEutR YX
),(),(
),(),(),(
,
,
, utRutR
utRutRut
YXY
YXX
YXR
© Po-Ning Chen@ece.nctu Chapter 1-64
1.4 Cross-Correlation
If X(t) and Y(t) are jointly weakly stationary, then
)()(
)()(
)(),(
,
,
,,
utRutR
utRutR
utut
YXY
YXX
YXYX RR
© Po-Ning Chen@ece.nctu Chapter 1-65
Example 1.4 Quadrature-Modulated Processes
Consider a pair of quadrature decomposition of X(t) as:
where is independent of X(t) and is uniformly distributed over [0, ).
)2sin()()(
2cos)()(
tftXtX
tftXtX
cQ
cI
© Po-Ning Chen@ece.nctu Chapter 1-66
Example 1.4 Quadrature-Modulated Processes
),())(2sin(2
1
2
))(2sin()2)(2sin(),(
)]2cos()2[sin()]()([
)]2sin()()2cos()([
)]()([),(,
utRutf
tufutfEutR
tfufEuXtXE
ufuXtftXE
uXtXEutR
Xc
ccX
cc
cc
QIXX QI
© Po-Ning Chen@ece.nctu Chapter 1-67
Example 1.4 Quadrature-Modulated Processes
Notably, if t = u, i.e., synchronize in two quadrature components, then
which indicates that simultaneous observations of the quadrature-modulated processes are “orthogonal” to each other!
(See Slide 1-78 for a formal definition of orthogonality.)
0),(, ttRQI XX
© Po-Ning Chen@ece.nctu Chapter 1-68
1.5 Ergodic Process
For a random-process-modeled noise (or random-process-modeled source) X(t), how can we know its mean and variance? Answer: Relative frequency. Specifically, by measuring X(t1), X(t2), …, X(tn), and
calculating their average, it is expected that this time average will be close to its mean.
Question is “Will this time average be close to its mean, if X(t) is stationary?” For a stationary process, the mean function X(t) is
independent of time t.
© Po-Ning Chen@ece.nctu Chapter 1-69
1.5 Ergodic Process
The answer is negative! An additional ergodicity assumption is
necessary for time average converging to the ensemble average X.
© Po-Ning Chen@ece.nctu Chapter 1-70
1.5 Time Average versus Ensemble Average
Example. X(t) is stationary. For any t, X(t) is uniformly distributed over {1, 2, 3,
4, 5, 6}. Then ensemble average is equal to:
5.36
16
6
15
6
14
6
13
6
12
6
11
© Po-Ning Chen@ece.nctu Chapter 1-71
1.5 Time Average versus Ensemble Average
We make a series of observations at time 0, T, 2T, …, 10T to obtain 1, 2, 3, 4, 3, 2, 5, 6, 4, 1. (They are deterministic!)
Then, the time average is equal to:
1.310
1465234321
© Po-Ning Chen@ece.nctu Chapter 1-72
1.5 Ergodicity
Definition. A stationary process X(t) is ergodic in the mean if
where
0)(Varlim .2
and 1,)(lim Pr.1
T
T
XT
XXT
T
TX dttX
TT )(
2
1)(
© Po-Ning Chen@ece.nctu Chapter 1-73
1.5 Ergodicity
Definition. A stationary process X(t) is ergodic in the autocorrelation function if
where
0);(Varlim .2
and 1,)();(lim Pr.1
TR
RTR
XT
XXT
T
TX dttXtXT
TR )()(2
1);(
© Po-Ning Chen@ece.nctu Chapter 1-74
1.5 Ergodic Process
Experiments or observations on the same process can only be performed at different time.
“Stationarity” only guarantees that the observations made at different time come from the same distribution.
© Po-Ning Chen@ece.nctu Chapter 1-75
A1.3 Statistical Average
Alternative names of ensemble average Mean Expectation value Sample average. (Recall that sample space consists
of all possible outcomes!)
How about the sample average of a function g( ) of a random variable X ?
dxxfxgXgE X )()()]([
© Po-Ning Chen@ece.nctu Chapter 1-76
A1.3 Statistical Average nth moment of random variable X
The 2nd moment is also named mean-square value. nth central moment of random variable X
The 2nd central moment is also named variance. Square root of the 2nd central moment is also named
standard deviation.
© Po-Ning Chen@ece.nctu Chapter 1-77
A1.3 Joint Moments
The joint moment of X and Y is given by:
When i = j = 1, the joint moment is specifically named correlation.
The correlation of centered random variables is specifically named covariance.
dxdyyxfyxYXE YX
jiki ),(][ ,
YXYX XYEYXEYX ][)])([(],[Cov
© Po-Ning Chen@ece.nctu Chapter 1-78
A1.3 Joint Moments
Two random variables, X and Y, are uncorrelated if, and only if, Cov[X, Y] = 0.
Two random variables, X and Y, are orthogonal if, and only if, E[XY] = 0.
The covariance, normalized by two standard deviations, is named correlation coefficient of X and Y.
YX
YX
],[Cov
© Po-Ning Chen@ece.nctu Chapter 1-79
A1.3 Characteristic Function
Characteristic function is indeed the inverse Fourier transform of the pdf.
© Po-Ning Chen@ece.nctu Chapter 1-80
1.6 Transmission of a Random Process Through a Stable Linear Time-Invariant Filter
Linear Y(t) is a linear function of X(t). Specifically, Y(t) is a weighted sum of X(t).
Time-invariant The weights are time-independent.
Stable Dirichlet’s condition (defined later) and “Stability” implies that the output is an energy function, which has
finite power (second moment), if the input is an energy function.
dh 2|)(|
© Po-Ning Chen@ece.nctu Chapter 1-81
Example of LTI filter: Mobile Radio Channel
Transmitter Receiver
),( 11
),( 22
),( 33 .)( where
,)()(
)()()()(3
1
332211
ii
iii
h
tsh
tstststY
)(tX )(tY
© Po-Ning Chen@ece.nctu Chapter 1-82
Example of LTI filter: Mobile Radio Channel
Transmitter Receiver
dtXhtY )()()(
)(tX )(tY
© Po-Ning Chen@ece.nctu Chapter 1-83
1.6 Transmission of a Random Process Through a Linear Time-Invariant Filter
What are the mean and autocorrelation functions of the LTI filter output Y(t)? Suppose X(t) is stationary and has finite mean. Suppose Then
dhdtXEh
dtXhEtYEt
X
Y
)()]([)(
)()()]([)(
dh |)(|
© Po-Ning Chen@ece.nctu Chapter 1-84
1.6 Zero-Frequency (DC) Response
dh )(1
The mean of the LTI filter output process is equal to the mean of the stationary filter input multiplied by the DC response of the system.
dhXY )(
© Po-Ning Chen@ece.nctu Chapter 1-85
1.6 Autocorrelation
122121
122121
222111
),()()(
)()()()(
)()()()(
)]()([),(
ddutRhh
dduXtXEhh
duXhdtXhE
uYtYEutR
X
Y
122121 )()()()( then
WSS, )( If
ddRhhR
tX
XY
© Po-Ning Chen@ece.nctu Chapter 1-86
1.6 WSS input induces WSS output
From the above derivations, we conclude: For a stable LTI filter, a WSS input induces a WSS
output. In general (not necessarily WSS),
As the above two quantities also relate in “convolution” form, a spectrum analysis is perhaps better in characterizing their relationship.
© Po-Ning Chen@ece.nctu Chapter 1-87
A2.1 Fourier Analysis
Fourier Transform Pair
Fourier Transform G(f) is the frequency spectrum content of a signal g(t). |G(f)| magnitude spectrum arg{G(f)} phase spectrum
dfftjfGtgtg
dtftjtgfGtg
)2exp()()(:)( of Transform Fourier Inverse
)2exp()()(:)( of TransformFourier
© Po-Ning Chen@ece.nctu Chapter 1-88
A2.1 Dirichlet’s Condition
Dirichlet’s condition In every finite interval, g(t) has a finite number of
local maxima and minima, and a finite number of discontinuity points.
Sufficient conditions for the existence of Fourier transform g(t) satisfies Dirichlet’s condition Absolute integrability.
dttg |)(|
© Po-Ning Chen@ece.nctu Chapter 1-89
A2.1 Dirichlet’s Condition
“Existence” means that the Fourier transform pair is valid only for continuity points.
.1||,0
;11,1)(
t
ttg
.1||,0
;11,1)(
t
ttgand
has the same spectrum G(f).
However, the above two functions are not equal at t = 1 and t = −1 !
© Po-Ning Chen@ece.nctu Chapter 1-90
A2.1 Dirac Delta Function
A function that exists only in principle. Define the Dirac Delta Function as a function
(t) satisfies:
(t) can be thought of as a limit of a unit-area pulse function.
.0,0
;0,)(
t
tt and .1)(
dtt
otherwise.,0
;2
1
2
1,)( where,)()(lim n
tn
ntstts nnn
© Po-Ning Chen@ece.nctu Chapter 1-91
A2.1 Properties of Dirac Delta Function
Sifting property If g(t) is continuous at t0, then
The sifting property is not necessarily true if g(t) is discontinuous at t0.
)()()()(
)()()(
0
)2/(1
)2/(10
00
0
0
tgdtntgdtttstg
tgdttttg
nt
ntn
© Po-Ning Chen@ece.nctu Chapter 1-92
A2.1 Properties of Dirac Delta Function
Replication property For every continuous point of g(t),
Constant spectrum
dtgtg )()()(
.1)2exp()0()2exp()(
dtftjtdtftjt
© Po-Ning Chen@ece.nctu Chapter 1-93
A2.1 Properties of Dirac Delta Function
Scaling after integration Although
their integrations are different
Hence, the “multiplicative constant” to Dirac delta function is significant, and shall never be ignored!
0,0
0,)(2)(
t
ttt
.2)(2 while1)(
dttdtt
??? )()()()(
dxxgdxxfxgxf
© Po-Ning Chen@ece.nctu Chapter 1-94
A2.1 Properties of Dirac Delta Function
More properties Multiplication convention
ba
baatbtat
if,0
if),()()(
)(
)()(
)()()()(
ba
dtatba
dtbaatdtbtat
© Po-Ning Chen@ece.nctu Chapter 1-95
A2.1 Fourier Series
The Fourier transform of a periodic function does not exist! E.g., for integer k,
otherwise.,0
;122,1)(
ktktg
0
1
0 2 4 6 8 10
© Po-Ning Chen@ece.nctu Chapter 1-96
A2.1 Fourier Series
Theorem: If gT(t) is a bounded periodic function satisfying Dirichlet condition, then
at every continuity points of gT(t), where T is the smallest real number such that gT(t) = gT(t+T), and
nnT t
T
njctg
2exp)(
2/
2/
2exp)(
1 T
TTn dtt
T
njtg
Tc
© Po-Ning Chen@ece.nctu Chapter 1-97
A2.1 Relation between a Periodic Function and its Generating Function
Define the generating function of a periodic function gT(t) with period T as:
Then
otherwise.,0
;2/2/),()(
TtTtgtg T
m
T mTtgtg )()(
© Po-Ning Chen@ece.nctu Chapter 1-98
A2.1 Relation between a Periodic Function and its Generating Function
Let G(f) be the spectrum of g(t) (which should exist).
Then from Theorem in Slide 1-96,
mTtsdttT
njmTtg
T
dttT
njmTtg
T
dttT
njtg
Tc
m
T
T
T
Tm
T
TTn
,2
exp)(1
2exp)(
1
2exp)(
1
2/
2/
2/
2/
2/
2/
T
nG
T
dssT
njsg
T
dssT
njsg
T
dsmTsT
njsg
Tc
m
mTT
mTT
m
mTT
mTTn
1
2exp)(1
2exp)(
1
)(2
exp)(1
2/
2/
2/
2/
© Po-Ning Chen@ece.nctu Chapter 1-99
(Continue from the previous slide.)
© Po-Ning Chen@ece.nctu Chapter 1-100
A2.1 Relation between a Periodic Function and its Generating Function
This concludes to the Poisson’s sum formula
nT t
T
nj
T
nG
Ttg 2exp
1)(
T
nG
)(tgT
T
1
T
)(tg )( fG1
T/1
© Po-Ning Chen@ece.nctu Chapter 1-101
A2.1 Spectrums through LTI filter
h()x1(t) y1(t)
h()x2(t) y2(t)
A linear filter satisfies the principle of superposition, i.e.,
h()x1(t) + x2(t) y1(t) + y2(t)
excitation response
© Po-Ning Chen@ece.nctu Chapter 1-102
-2 -1 1 2
-1
-0.5
0.5
1
A2.1 Linearity and Convolution
A linear time-invariant filter can be described by convolution integral
Example of a non-linear system
h()x(t) y(t)
)(1.0)()( 3 txtxty
.)()()( dtxhty
© Po-Ning Chen@ece.nctu Chapter 1-103
A2.1 Linearity and Convolution
Time-Convolution = Spectrum-Multiplication
dffjfHh
dfftjfxtxdtxhty
)2exp()()(
)2exp()()( and )()()(
ddtftjtxh
dtftjdtxh
dtftjtyfy
)2exp()()(
)2exp()()(
)2exp()()(
)()(
)2exp()()(
)2exp()()2exp()(
,))(2exp()()()(
fHfx
dfjhfx
ddsfsjsxfjh
tsddtsfjsxhfy
© Po-Ning Chen@ece.nctu Chapter 1-104
© Po-Ning Chen@ece.nctu Chapter 1-105
A2.1 Impulse Response of LTI Filter
Impulse response = Filter response to Dirac delta function (application of replication property)
h()(t) h(t)
).()()()()()( thdthdtxhty
© Po-Ning Chen@ece.nctu Chapter 1-106
A2.1 Frequency Response of LTI Filter
Frequency response = Filter response to a complex exponential input of unit amplitude and frequency f
h()exp(j2ft)
)()2exp(
2exp)(2exp
)(2exp)(
)()()(
fHftj
dfjhftj
dtfjh
dtxhty
© Po-Ning Chen@ece.nctu Chapter 1-107
A2.1 Measures for Frequency Response
response phase)(
response amplitude|)(| where)],(exp[|)(|)(
f
fHfjfHfH
)()(
)(|)(|log)(log
fjf
fjfHfH
response phase)(
gain)( where
f
f
Expression 1
Expression 2
dB |)(|log20
nepers |)(|ln)(
10 fH
fHf
© Po-Ning Chen@ece.nctu Chapter 1-108
A2.1 Fourier Analysis
Remember to self-study Tables A6.2 and A6.3.
© Po-Ning Chen@ece.nctu Chapter 1-109
1.7 Power Spectral Density
LTIh()
Deterministic x(t)
dtxhty )()()(
LTIh()
WSS X(t)
dtXhtY )()()(
© Po-Ning Chen@ece.nctu Chapter 1-110
1.7 Power Spectral Density
How about the spectrum relation between filter input and filter output? An apparent relation is:
LTIH(f)
Deterministic x(f) )()()( fxfHfy
LTIH(f)
X(f) )()()( fXfHfY
© Po-Ning Chen@ece.nctu Chapter 1-111
1.7 Power Spectral Density
This is however not adequate for a random process. For a random process, what concerns us is the
relation between the input statistic and output statistic.
© Po-Ning Chen@ece.nctu Chapter 1-112
1.7 Power Spectral Density
How about the relation of the first two moments between filter input and output? Spectrum relation of mean processes
dth
dtXhEtYEt
X
Y
)()(
)()()]([)(
)()()( fHff XY
© Po-Ning Chen@ece.nctu Chapter 1-113
補充 : Time-Average Autocorrelation Function
For a non-stationary process, we can use the time-average autocorrelation function to define the average power correlation for a given time difference.
It is implicitly assumed that is independent of the location of the integration window. Hence,
T
TTX dttXtXET
R )]()([2
1lim)(
)(XR
2/3
2/)]()([
2
1lim)(
T
TTX dttXtXET
R
© Po-Ning Chen@ece.nctu Chapter 1-114
補充 : Time-Average Autocorrelation Function
E.g., for a WSS process,
E.g., for a deterministic function,
T
TT
T
TTX
dttxtxT
dttxtxET
R
)()(2
1lim
)]()([2
1lim)(
)]()([)( tXtXERX
© Po-Ning Chen@ece.nctu Chapter 1-115
補充 : Time-Average Autocorrelation Function
E.g., for a cyclostationary process,
).( of periodonary cyclostati theis where
,)]()([2
1)(
tXT
dttXtXET
RT
TX
© Po-Ning Chen@ece.nctu Chapter 1-116
補充 : Time-Average Autocorrelation Function
The time-average power spectral density is the Fourier transform of the time-average autocorrelation function.
)]()([
2
1lim
)()(2
1lim
)]()([2
1lim)(
*
2
2
2
2
fXfXET
dedttXtXET
dedttXtXET
fS
TT
fj
TT
fjT
TTX
.||)()( where 2 TttXtX T 1
© Po-Ning Chen@ece.nctu Chapter 1-117
補充 : Time-Average Autocorrelation Function
For a WSS process, For a deterministic process,
).()( fSfS XX
).()(2
1lim)( *
2 fxfxT
fS TT
X
© Po-Ning Chen@ece.nctu Chapter 1-118
1.7 Power Spectral Density
Relation of time-average PSDs
© Po-Ning Chen@ece.nctu Chapter 1-119
© Po-Ning Chen@ece.nctu Chapter 1-120
1.7 Power Spectral Density under WSS input
For a WSS filter input,
© Po-Ning Chen@ece.nctu Chapter 1-121
1.7 Power Spectral Density under WSS input
Observation
E[Y2(t)] is generally viewed as the average power of the WSS filter output process Y(t).
This average power distributes over each spectrum frequency f through SY(f). (Hence, the total average power is equal to the integration of SY(f).)
Thus, SY(f) is named the power spectral density (PSD) of Y(t).
dffSfHdffSRtYE XYY )(|)(|)()0()]([ 22
© Po-Ning Chen@ece.nctu Chapter 1-122
1.7 Power Spectral Density under WSS input
The unit of E[Y2(t)] is, e.g., Watt. So the unit of SY(f) is therefore Watt per Hz.
© Po-Ning Chen@ece.nctu Chapter 1-123
1.7 Operational Meaning of PSD
Example. Assume h() is real, and |H(f)| is given by:
© Po-Ning Chen@ece.nctu Chapter 1-124
1.7 Operational Meaning of PSD
Then
)]()([
)()(
)(|)(|
)()0()]([
2/
2/
2/
2/
2
2
cXcX
ff
ff X
ff
ff X
X
YY
fSfSf
dffSdffS
dffSfH
dffSRtYE
c
c
c
c
The filter passes only those frequency components of the input random process X(t), which lie inside a narrow frequency band of width f centered about the frequency fc and fc.
© Po-Ning Chen@ece.nctu Chapter 1-125
1.7 Properties of PSD
Property 0. Einstein-Wiener-Khintchine relation Relation between autocorrelation function and PSD
of a WSS process X(t)
dffjfSR
dfjRfS
XX
XX
)2exp()()(
)2exp()()(
© Po-Ning Chen@ece.nctu Chapter 1-126
1.7 Properties of PSD
Property 1. Power density at zero frequency
Property 2: Average power
[Second] [Watt] )(
)0([Watt/Hz] )0(
dR
SS
X
XX
[Hz] [Watt/Hz] )([Watt] )]([ 2 dffStXE X
[Watt-Second]
© Po-Ning Chen@ece.nctu Chapter 1-127
1.7 Properties of PSD
Property 3: Non-negativity for WSS processes
Proof: Pass X(t) through a filter with impulse response h() = cos(2fc).
Then H(f) = (1/2)[(f fc) + (f + fc)].
0)( fSX
© Po-Ning Chen@ece.nctu Chapter 1-128
1.7 Properties of PSD
4. Property from )()( since ),(2
1
))()((4
1
)()()()(4
1
)(|)(|
)()]([
2
2
fSfSfS
fSfS
dffSffdffSff
dffSfH
dffStYE
XXcX
cXcX
XcXc
X
Y
As a result,
This step requires that SX(f) is continuous, which is true in general.
© Po-Ning Chen@ece.nctu Chapter 1-129
1.7 Properties of PSD
Therefore, by passing through a proper filter,
for any fc.
0)]([2)( 2 tYEfS cX
© Po-Ning Chen@ece.nctu Chapter 1-130
1.7 Properties of PSD
Property 4: PSD is an even function, i.e.,
Proof.
).()( fSfS XX
)(
)()( ,)(
,)(
)()(
2
2
)(2
fS
RRdsesR
sdsesR
deRfS
X
XXfsj
X
fsjX
fjXX
© Po-Ning Chen@ece.nctu Chapter 1-131
1.7 Properties of PSD
Property 5: PSD is real.
Proof.
Then with Property 4,
Thus, SX(f) is real.
)()(real )( * fSfSR XXX
).()()( * fSfSfS XXX
© Po-Ning Chen@ece.nctu Chapter 1-132
Example 1.5(continue from Example 1.2) Signal with Random Phase
Let X(t) = A cos(2fct + ), where is uniformly distributed over [, ).
.2cos2
)(2
cX fA
R
)()(4
4
4
2cos2
)(
2
)(2)(22
2222
22
cc
ffjffj
fjfjfj
fjcX
ffffA
dedeA
deeeA
defA
fS
cc
cc
© Po-Ning Chen@ece.nctu Chapter 1-133
Example 1.5(continue from Example 1.2) Signal with Random Phase
© Po-Ning Chen@ece.nctu Chapter 1-134
Example 1.6 (continue from Example 1.3) Random Binary Wave
Let
where …, I2, I1, I0, I1, I2, … are independent, and each Ij is either 1 or 1 with equal probability, and
)()( dn
n tnTtpIAtX
otherwise,0
0,1)(
Tttp
© Po-Ning Chen@ece.nctu Chapter 1-135
otherwise,0
||,||
1)(
2 TT
ARX
Example 1.6 (continue from Example 1.3) Random Binary Wave
T
T
fj
T
T
fj
T
T
fj
T
T
fjX
defTj
A
defjT
AefjT
A
deT
AfS
22
2222
22
)sgn(2
2
1)sgn(
1
2
1||1
||1)( duvuvdvu
)(sinc)(sin
)2cos(224
114
4
2222
)sgn(2
)(
22222
2
22
2
2222
2
02
0
222
2
0 2
0
22
22
fTTAfTTf
A
fTTf
A
eeTf
A
eeTf
A
defjdefjfjfTj
A
defTj
AfS
fTjfTj
T
fjTfj
T
fjT fj
T
T
fjX
© Po-Ning Chen@ece.nctu Chapter 1-136
(Continue from the previous slide.)
© Po-Ning Chen@ece.nctu Chapter 1-137
© Po-Ning Chen@ece.nctu Chapter 1-138
1.7 Energy Spectral Density
Energy of a (deterministic) function p(t) is given by Recall that the average power of p(t) is given by
Observe that
dtdfefpdfefp
dttptpdttp
tfjftj*
'22
*2
')'()(
)()(|)(|
.|)(| 2
dttp
.|)(|2
1lim 2
T
TTdttp
T
dffp
dffpfp
dfdffffpfp
dfdfdtefpfp
dtdfefpdfefpdttp
tffj
tfjftj
2
*
*
)'(2*
'2*22
|)(|
)()(
')'()'()(
')'()(
')'()(|)(|
For the same reason as PSD, |p(f)|2 is named energy spectral density (ESD) of p(t).
(Continue from the previous slide.)
© Po-Ning Chen@ece.nctu Chapter 1-139
© Po-Ning Chen@ece.nctu Chapter 1-140
Example
The ESD of a rectangular pulse of amplitude A and duration T is given by
)(sinc)( 2222
0
2 fTTAdtAefET
ftjg
© Po-Ning Chen@ece.nctu Chapter 1-141
Example 1.7 Mixing of a Random Process with a Sinusoidal Process
Let Y(t) = X(t) cos(2fct + ), where is uniformly distributed over [, ), and X(t) is WSS and independent of .
2
))(2cos()(
)]2cos()2[cos()]()([
)]2cos()2cos()()([),(
utfutR
uftfEuXtXE
uftfuXtXEutR
cX
cc
ccY
)()(4
1)( cXcXY ffSffSfS
© Po-Ning Chen@ece.nctu Chapter 1-142
1.7 How to measure PSD?
If X(t) is not only (strictly) stationary but also ergodic, then any (deterministic) observation sample x(t) in [T, T) has:
Similarly,
X
T
TTtXEdttx
T
)]([)(2
1lim
)()()(2
1lim X
T
TTRdttxtx
T
© Po-Ning Chen@ece.nctu Chapter 1-143
1.7 How to measure PSD?
Hence, we may use the time-limited Fourier transform of the time-averaged autocorrelation function:
to approximate the PSD.
T
TTdttxtx
T)()(
2
1lim
)()(2
1
))(2exp()()2exp()(2
1
)2exp()()2exp()(2
1
,))(2exp()()(2
1
)2exp()()(2
1
)2exp()()(2
1
22 fxfxT
dttfjtxdsfsjsxT
dtdsfsjsxftjtxT
tsdtdstsfjsxtxT
dtdfjtxtxT
dfjdttxtxT
TT
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
© Po-Ning Chen@ece.nctu Chapter 1-144
(Notably, we only have the values of x(t) for t in [T, T).)
© Po-Ning Chen@ece.nctu Chapter 1-145
1.7 How to measure PSD?
The estimate
is named the periodogram. To summarize:
)()(2
122 fxfx
T TT
)()(2
1)( Then .3
.)2exp()()( Calculate .2
).,[ durationfor )( Observe .1
22
2
fxfxT
fS
dtftjtxfx
TTtx
TTX
T
TT
© Po-Ning Chen@ece.nctu Chapter 1-146
1.7 Cross Spectral Density
Definition: For two (jointly WSS) random processes, X(t) and Y(t), their cross spectral densities are given by:
).(for
)()( and )]()([),( and
),()( and )]()([),( where
)2exp()()(
)2exp()()(
,,,
,,,
,,
,,
ut
utRRuXtYEutR
utRRuYtXEutR
dfjRfS
dfjRfS
XYXYXY
YXYXYX
XYXY
YXYX
© Po-Ning Chen@ece.nctu Chapter 1-147
1.7 Cross Spectral Density
Property
)()( ,, XYYX RR
)(
)2exp()(
)2exp()(
)2exp()()(
,
,
,
,,
fS
dfjR
dfjR
dfjRfS
YX
YX
XY
XYXY
© Po-Ning Chen@ece.nctu Chapter 1-148
Example 1.8 PSD of Sum Process
Determine the PSD of sum process Z(t) = X(t) + Y(t) of two zero-mean WSS processes X(t) and Y(t). Answer:
).,(),(),(),(
)]()([)]()([)]()([)]()([
))]()())(()([(
)]()([
),(
,, utRutRutRutR
uYtYEuXtYEuYtXEuXtXE
uYuXtYtXE
uZtZE
utR
YXYYXX
Z
© Po-Ning Chen@ece.nctu Chapter 1-149
).()()()()(
Hence,
).()()()()(
thatimplies WSS
,,
,,
fSfSfSfSfS
RRRRR
YXYYXXZ
YXYYXXZ
Q.E.D.
).()()(
)]([)]([)]()([ i.e.,
ed,uncorrelat are )( and )( If
fSfSfS
tYEtXEtYtXE
tYtX
YXZ
The PSD of a sum process of zero-mean uncorrelated processes is equal to the sum of their individual PSDs.
© Po-Ning Chen@ece.nctu Chapter 1-150
Example 1.9
Determine the CSD of output processes induced by separately passing jointly WSS inputs through a pair of stable LTI filters.
© Po-Ning Chen@ece.nctu Chapter 1-151
© Po-Ning Chen@ece.nctu Chapter 1-152
1.8 Gaussian Process
Definition. A random variable is Gaussian distributed, if its pdf has the form
2
2
2 2
)(exp
2
1)(
Y
Y
Y
Y
yyf
© Po-Ning Chen@ece.nctu Chapter 1-153
1.8 Gaussian Process
Definition. An n-dimensional random vector is Gaussian distributed, if its pdf has the form
)()(
2
1exp
|)|2(
1)( 1
2/
xxxf T
nX
matrix. covariance theis },{Cov},{Cov
},{Cov},{Cov
and vector,mean theis ]][],...,[],[[ where
2212
2111
21
nn
Tn
XXXX
XXXX
XEXEXE
© Po-Ning Chen@ece.nctu Chapter 1-154
1.8 Gaussian Process
For a Gaussian random vector, “uncorrelation” implies “independence.”
n
iiXX
nn
xfxfXX
XX
i
122
11
)()(},{Cov0
0},{Cov
© Po-Ning Chen@ece.nctu Chapter 1-155
1.8 Gaussian Process
Definition. A random process X(t) is said to be Gaussian distributed, if for every function g(t), satisfying
is a Gaussian random variable.
T T
X dtduutRugtg0 0
),()()(
T
dttXtgY0
)()(
.),()()(][ Notably,0 0
2 T T
X dtduutRugtgYE
© Po-Ning Chen@ece.nctu Chapter 1-156
1.8 Central Limit Theorem
Theorem (Central Limit Theorem) For a sequence of independent and identically distributed (i.i.d.) random variables X1, X2, X3, …
y
X
XnX
ndx
xy
n
XX
2exp
2
1)()(Prlim
2
1
].[ and ][ where 22jXjX XEXE
© Po-Ning Chen@ece.nctu Chapter 1-157
1.8 Properties of Gaussian processes
Property 1. Output of a stable linear filter is a Gaussian process if the input is a Gaussian process.
(This is self-justified by the definition of Gaussian processes.)
Property 2. A finite number of samples of a Gaussian process forms a multi-dimensional Gaussian vector.
(No proof. Some books use this as the definition of Gaussian processes.)
© Po-Ning Chen@ece.nctu Chapter 1-158
1.8 Properties of Gaussian processes
Property 3. A WSS Gaussian process is also strictly stationary.
(An immediate consequence of Property 2.)
© Po-Ning Chen@ece.nctu Chapter 1-159
1.9 Noise
Noise An unwanted signal that will disturb the
transmission or processing of signals in communication systems.
Types Shot noise Thermal noise … etc.
© Po-Ning Chen@ece.nctu Chapter 1-160
1.9 Shot Noise
A noise arises from the discrete nature of diodes and transistors. E.g., a current pulse is generated every time an
electron is emitted by the cathode.
Mathematical model
k
kShot tptX )()(
duration. finite withshape pulsea is )( and
generated, is pulsea timeof sequence theare }{ where
tpkk
© Po-Ning Chen@ece.nctu Chapter 1-161
1.9 Shot Noise
XShot(t) is called the shot noise.
A more useful model is to count the number of electrons emitted in the time interval (0, t].
}:max{)( tktN k
© Po-Ning Chen@ece.nctu Chapter 1-162
1.9 Shot Noise
N(t) behaves like a Poisson Counting Process. Definition (Poisson counting process) A Poisson
counting process with parameter is a process {N(t), t 0} with N(0) = 0 and stationary independent increments satisfying that for 0 < t1 < t2, N(t2) N(t1) is Poisson distributed with mean (t2 t1). In other words,
)](exp[!
)]([)()(Pr 12
1212 tt
k
ttktNtN
k
© Po-Ning Chen@ece.nctu Chapter 1-163
1.9 Shot Noise
A detailed statistical characterization of the shot-noise process X(t) is in general hard.
Some properties are quoted below. XShot(t) is strictly stationary.
Mean
Autocovariance function
dttp
ShotX )(
dttptpC
ShotX )()()(
Chapter 1-164
1.9 Shot Noise
For your reference
© Po-Ning Chen@ece.nctu Chapter 1-165
1.9 Shot Noise
Example. p(t) is a rectangular pulse of amplitude A and duration T.
ATAdtdttpT
X Shot
0)(
otherwise,0
|||),|()(
2 TTAC
ShotX
© Po-Ning Chen@ece.nctu Chapter 1-166
1.9 Thermal Noise A noise arises from the random motion of
electrons in a conductor. Mathematical model
Thermal noise voltage VTN that appears across the terminals of a resistor, measured in a bandwidth of f Herz, is zero-mean Gaussian distributed with variance ][volts 4][ 22 fkTRVE TN
Kelvin.degrees in re temparatuabsolute theis and
ohms, in resistance theis constant, sBoltzmann'
theis Kelvindegreeper joules 1038.1 where 23
T
R
k
© Po-Ning Chen@ece.nctu Chapter 1-167
1.9 Thermal Noise
Model of a noisy resistor
222 ][][ RIEVE TNTN
© Po-Ning Chen@ece.nctu Chapter 1-168
1.9 White Noise
A noise is white if its PSD equals constant for all frequencies. It is often defined as:
Impracticability The noise has infinite power
2)( 0N
fSW
.2
)()]([ 02
df
NdffStWE W
© Po-Ning Chen@ece.nctu Chapter 1-169
1.9 White Noise
Another impracticability No matter how closely in time two samples are,
they are uncorrelated!
So impractical, why white noise is so popular in the analysis of communication system? There do exist noise sources that have a flat power
spectral density over a range of frequencies that is much larger than the bandwidths of subsequent filters or measurement devices.
© Po-Ning Chen@ece.nctu Chapter 1-170
1.9 White Noise
Some physically measurements have shown that the PSD of (a certain kind of) noise has the form
where k is the Boltzmann’s constant, T is the absolute temperature, and R are the parameters of physical medium.
When f << ,
22
2
)2(
2)(
fkTRfSW
22
)2(
2)( 0
22
2 NkTR
fkTRfSW
© Po-Ning Chen@ece.nctu Chapter 1-171
Example 1.10 Ideal Low-Pass Filtered White Noise
After the filter, the PSD of the zero-mean white noise becomes:
otherwise,0
||,2)(|)(|)(
02 Bf
NfSfHfS WFW
)2(sinc)2exp(2
)( 00 BBNdffj
NR
B
BFW
ed.uncorrelat i.e., ,0)( implies )2/( FWRBk
© Po-Ning Chen@ece.nctu Chapter 1-172
Example 1.10 Ideal Low-Pass Filtered White Noise
So if we sample the noise at rate of 2B times per second, the resultant noise samples are uncorrelated!
© Po-Ning Chen@ece.nctu Chapter 1-173
Example 1.11
ChannelChannelEncoderEncoder
……01100110ModulatorModulator
……,-,-mm((tt), ), mm((tt), ), mm((tt), -), -mm((tt))
mm((tt))
TT
Carrier waveCarrier waveAccos(2Accos(2ffcctt))
ss((tt))
w(t)
xx((tt))
T
dt0
correlator
NN>><< 00
0110…0110…
)cos(22/
carrier Local
ctfT
© Po-Ning Chen@ece.nctu Chapter 1-174
Example 1.11
energy. signal thenomalize carrier to local
the toadded is /2factor scalinga figure, previous theIn T
.1
)4cos(1
)2(cos2
)2cos(2
EnergySignal
0
0
2
0
2
Tc
T
c
T
c
dtT
tf
dttfT
dttfT
© Po-Ning Chen@ece.nctu Chapter 1-175
Example 1.11
T
c dttfT
twN0
)2cos(2
)( Noise
.0)2cos(2
)]([)2cos(2
)(00
T
c
T
cN dttfT
twEdttfT
twE
T
cc
T
T
c
T
cN
dsdtsftfswtwET
dssfT
swdttfT
twE
0 0
00
2
)2cos()2cos()]()([2
)2cos(2
)()2cos(2
)(
.2
)2(cos
)2cos()2cos()(2
2
0
0
20
0 0
02
N
dssfT
N
dsdtsftfstN
TT
c
T
cc
T
N
© Po-Ning Chen@ece.nctu Chapter 1-176
(Continue from the previous slide.)
If w(t) is white Gaussian, then the pdf of N is uniquely determined by the first and second moments.
© Po-Ning Chen@ece.nctu Chapter 1-177
1.10 Narrowband Noise
In general, the receiver of a communication system includes a narrowband filter whose bandwidth is just large enough to pass the modulated component of the received signal.
The noise is therefore also filtered by this narrowband filter.
So the noise’s PSD after being filtered may look like the figures in the next slide.
© Po-Ning Chen@ece.nctu Chapter 1-178
1.10 Narrowband Noise
The analysis on narrowband noise will be covered in subsequent sections.
© Po-Ning Chen@ece.nctu Chapter 1-179
A2.2 Bandwidth
The bandwidth is the width of the frequency range outside which the power is essentially negligible. E.g., the bandwidth of a (strictly) band-limited
signal shown below is B.
© Po-Ning Chen@ece.nctu Chapter 1-180
A2.2 Null-to-Null Bandwidth
Most signals of practical interest are not strictly band-limited. Therefore, there may not be a universally accepted
definition of bandwidth for such signals. In such case, people may use null-to-null
bandwidth. The width of the main spectral lobe that lies inside the
positive frequency region (f 0).
A2.2 Null-to-Null Bandwidth
)(sinc)( 22 fΤTAfSX
. amplitude and duration of pulse
r rectangulaa is )( where),()(
AT
tptnTtpIAtX dn
n
© Po-Ning Chen@ece.nctu Chapter 1-181
The null-to-null bandwidth is 1/T.
© Po-Ning Chen@ece.nctu Chapter 1-182
A2.2 Null-to-Null Bandwidth
The null-to-null bandwidth in this case is 2B.
© Po-Ning Chen@ece.nctu Chapter 1-183
A2.2 3-dB Bandwidth
A 3-dB bandwidth is the displacement between the two (positive) frequencies, at which the magnitude spectrum of the signal reaches its maximum value, and at which the magnitude spectrum of the signal drops to of the peak spectrum value. Drawback: A small 3-dB bandwidth does not
necessarily indicate that most of the power will be confined within a small range. (E.g., the signal may have slowly decreasing tail.)
21
© Po-Ning Chen@ece.nctu Chapter 1-184
2
1)(sinc
)0(
)(2
22
TA
fΤTA
S
fS
X
X
© Po-Ning Chen@ece.nctu Chapter 1-184
The 3-dB bandwidth is 0.32/T.
A2.2 3-dB Bandwidth
T
32.0
© Po-Ning Chen@ece.nctu Chapter 1-185
A2.2 Root-Mean-Square Bandwidth
rms bandwidth
Disadvantage: Sometimes,
even if
2/12
)(
)(
dffS
dffSfB
X
X
rms
dffSf X )(2
.)(
dffSX
© Po-Ning Chen@ece.nctu Chapter 1-186
A2.2 Bandwidth of Deterministic Signals
The previous definitions can also be applied to Deterministic Signals where PSD is replaced by ESD. For example, a deterministic signal with spectrum
G(f) has rms bandwidth:2/1
2
22
|)(|
|)(|
dffG
dffGfBrms
© Po-Ning Chen@ece.nctu Chapter 1-187
A2.2 Noise Equivalent Bandwidth
An important consideration in communication system is the noise power at a linear filter output due to a white noise input. We can characterize the noise-resistant ability of
this filter by its noise equivalent bandwidth. Noise equivalent bandwidth = The bandwidth of an
ideal low-pass filter through which the same output filter noise power is resulted.
© Po-Ning Chen@ece.nctu Chapter 1-188
A2.2 Noise Equivalent Bandwidth
© Po-Ning Chen@ece.nctu Chapter 1-189
A2.2 Noise Equivalent Bandwidth Output noise power for a general linear filter
Output noise power for an ideal low-pass filter of bandwidth B and the same amplitude as the general linear filter at f = 0.
dffH
NdffHfSW
202 |)(|2
|)(|)(
20
202 |)0(||)0(|2
|)(|)( HBNdfHN
dffHfSB
BW
2
2
|)0(|2
|)(|
H
dffHBNE
© Po-Ning Chen@ece.nctu Chapter 1-190
A2.2 Time-Bandwidth Product
Time-Scaling Property of Fourier Transform Reducing the time-scale by a factor of a extends the
bandwidth by a factor of a.
This hints that the product of time- and frequency- parameters should remain constant, which is named the time-bandwidth product or bandwidth-duration product.
a
fG
aatgfGtg
FourierFourier
||
1)()()(
© Po-Ning Chen@ece.nctu Chapter 1-191
A2.2 Time-Bandwidth Product
Since there are various definitions of time-parameter (e.g., duration of a signal) and frequency-parameter (e.g., bandwidth), the time-bandwidth product constant may change for different definitions.
E.g., rms duration and rms bandwidth of a pulse g(t)2/1
2
22
|)(|
|)(|
dttg
dttgtTrms
2/1
2
22
|)(|
|)(|
dffG
dffGfBrms
.4
1 Then
rmsrmsBT
© Po-Ning Chen@ece.nctu Chapter 1-192
A2.2 Time-Bandwidth Product
Example: g(t) = exp(t2). Then G(f) = exp(f2).
Example: g(t) = exp(t|). Then G(f) = 2/(1+42f2).
.2
12/1
2
22
2
2
dte
dtetBT
t
t
rmsrms .4
1 Then
rmsrmsBT
.4
1
2
1
2
1
)41(1
)41(
2/1
222
222
22/1
||2
||22
dff
dff
f
dte
dtetBT
t
t
rmsrms
© Po-Ning Chen@ece.nctu Chapter 1-193
A2.3 Hilbert Transform
Let G(f) be the spectrum of a real function g(t). By convention, denote by u(f) the unit step
function, i.e.,
Put g+(t) to be the function
corresponding to 2u(f)G(f).
0,0
0,2/1
0,1
)(
f
f
f
fu
)( fG )()(2 fGfu
Multiply 2 to unchange the area.
© Po-Ning Chen@ece.nctu Chapter 1-194
A2.3 Hilbert Transform
How to obtain g+(t)?
Answer: Hilbert Transformer.
Proof: Observe that
0,1
0,0
0,1
)sgn( where),sgn(1)(2
f
f
f
fffu
Then by the next slide, we learn that
}0{1
)()(2
tt
jtfuFourierInverse
1
© Po-Ning Chen@ece.nctu Chapter 1-195
By extended Fourier transform,
0,0
0,4
4lim)sgn(
220
rierInverseFou
t
tt
j
ta
tjf
a
}0{)()sgn(1)(2rierInverseFou
tt
jtffu 1
© Po-Ning Chen@ece.nctu Chapter 1-196
A2.3 Hilbert Transform
).( of Transform Hilbert thenamed is )(
)()(ˆ where
),(ˆ)(
)(*}0{1
)(
)(*}0{1
)(
)}({*)}(2{
)()(2)(11
1
tgdt
gtg
tgjtg
tgtt
jtg
tgtt
jt
fGFourierfuFourier
fGfuFouriertg
1
1
© Po-Ning Chen@ece.nctu Chapter 1-197
A2.3 Hilbert Transform
1
) h)tg )ˆ tg
0,1
0,0
0,1
)sgn( where),sgn()(1
)
f
f
f
ffjfHhFourier
0]},2/)([exp{|)(|
0,0
0]},2/)([exp{|)(|
)()sgn()(ˆ
ffGjfG
f
ffGjfG
fGfjfG
© Po-Ning Chen@ece.nctu Chapter 1-198
A2.3 Hilbert Transform
Hence, Hilbert Transform is basically a 90 degree phase shifter.
© Po-Ning Chen@ece.nctu Chapter 1-199
A2.3 Hilbert Transform
dt
gtg
dt
gtg
)(ˆ1)(
)(1)(ˆ
PairTransformHilbert
1
) h)tg )ˆ tg
1
) h)ˆ tg )tg
1
© Po-Ning Chen@ece.nctu Chapter 1-200
A2.3 Hilbert Transform
An important property of Hilbert Transform is that:
slide.)next thein proof the(See
.0)(ˆ)( s,other word In
n.Integratio of sense thein orthogonal are )(ˆ and )(
dttgtg
tgtg
(Examples of Hilbert Transform Pairs can be found in Table A6.4.)
The real and imaginary parts of are orthogonal to each other.
.)()( if ,0
)()()()(
)()()()(
)()()sgn(
)()(ˆ
)()(ˆ
)(ˆ)()(ˆ)(
0
00
0
0
2
2
dffGfG
dffGfGdffGfGj
dffGfGdffGfGj
dffGfGfj
dffGfG
dfdtetgfG
dtdfefGtgdttgtg
ftj
ftj
© Po-Ning Chen@ece.nctu Chapter 1-201
© Po-Ning Chen@ece.nctu Chapter 1-202
A2.4 Complex Representation of Signals and Systems
g+(t) is named the pre-envelope or analytical signal of g(t).
We can similarly define
)(ˆ)()( tgjtgtg
)( fG)())(1(2 fGfu
© Po-Ning Chen@ece.nctu Chapter 1-203
A2.4 Canonical Representation of Band-Pass Signals
Now let G(f) be a narrow-band signal for which 2W << fc. Then we can obtain its pre-
envelope G+(f).
Afterwards, we can shift the pre-envelope to its low-pass signal )()(
~cffGfG
© Po-Ning Chen@ece.nctu Chapter 1-204
A2.4 Canonical Representation of Band-Pass Signal
These steps give the relation between the complex lowpass signal (baseband signal) and the real bandpass signal (passband signal).
Quite often, the real and imaginary parts of complex lowpass signal are respectively denoted by gI(t) and gQ(t).
)2exp()(~Re)(Re)( tfjtgtgtg c
© Po-Ning Chen@ece.nctu Chapter 1-205
A2.4 Canonical Representation of Band-Pass Signal
In terminology,
)( signal pass-band theofcomponent quadrature)(
)( signal pass-band theofcomponent phase-in)(
envelopecomplex )(~envelope-pre)(
tgtg
tgtg
tg
tg
Q
I
This leads to a canonical, or standard, expression for g(t).
)2sin()()2cos()(
)2exp())()((Re)(
tftgtftg
tfjtjgtgtg
cQcI
cQI
© Po-Ning Chen@ece.nctu Chapter 1-206
)2exp())()(( tfjtjgtg cQI
)2exp( tfj c))()(( tjgtg QI
© Po-Ning Chen@ece.nctu Chapter 1-207
A2.4 Canonical Representation of Band-Pass Signal
Canonical transmitter)2sin()()2cos()()( tftgtftgtg cQcI
© Po-Ning Chen@ece.nctu Chapter 1-208
A2.4 Canonical Representation of Band-Pass Signal
Canonical receiver
© Po-Ning Chen@ece.nctu Chapter 1-209
A2.4 More Terminology
)( of phase )(
)(tan)(
)( of envelopeor envelope natural
)()(|)(~||)(|)(
)( signal pass-band theofcomponent quadrature)(
)( signal pass-band theofcomponent phase-in)(
envelopecomplex )(~envelope-pre)(
1
22
tgtg
tgt
tg
tgtgtgtgta
tgtg
tgtg
tg
tg
I
Q
QI
Q
I
© Po-Ning Chen@ece.nctu Chapter 1-210
A2.4 Bandpass System
Consider the case of passing a band-pass signal x(t) through a real LTI filter h() to yield an output y(t).
Can we have a low-pass equivalent system for the bandpass system?
)(h)(tx
dtxhty )()()(
Similar to the previous analysis, we have:
)()()(~)2sin()()2cos()( )(~ Re)( 2
tjxtxtx
tftxtftxetxtx
QI
cQcItfj c
. and ,frequency carrier
theof Hz within tolimited is )( of spectrum The :
cc fWf
Wtx
Assumption
response impulsecomplex )()()(~
)2sin()()2cos()( )(~
Re)( 2
QI
cQcIfj
jhhh
fhfhehh c
. and ,frequency carrier
theof Hz within tolimited is )( of spectrum The :
cc fBf
Bh
Assumption
© Po-Ning Chen@ece.nctu Chapter 1-211
Now, is the filter output y(t) also a bandpass signal?
© Po-Ning Chen@ece.nctu Chapter 1-212
)( fX
)( fH
)( fY
© Po-Ning Chen@ece.nctu Chapter 1-213
A2.4 Bandpass System
Question: Is the following system valid?
The advantage of the above equivalent system is that there is no need to deal with the carrier frequency in the system analysis.
The answer to the question is YES (with some modification)! It will be substantiated in the sequel.
)(~ h
)(~ tx
dtxhty )(~)(
~)(~
).(~
)(~
)(~
show that tosufficesIt fHfXfY
.
)()(~
)()(~
)()(~
and
)()(2)(
)()(2)(
)()(2)(
that Observe
c
c
c
ffHfH
ffXfX
ffYfY
fHfufH
fXfufX
fYfufY
ly,Consequent
)(~
2
)(2
)()(4
)()()(4
)()(2)()(2
)()()(~
)(~
fY
ffY
ffYffu
ffHffXffu
ffHffuffXffu
ffHffXfHfX
c
cc
ccc
cccc
cc
© Po-Ning Chen@ece.nctu Chapter 1-214
There is an additional multiplicative constant 2 at the output!
© Po-Ning Chen@ece.nctu Chapter 1-215
A2.4 Bandpass System
Conclusion:
)(~ h
)(~ tx
dtxhty )(~)(
~
2
1)(~
© Po-Ning Chen@ece.nctu Chapter 1-216
A2.4 Bandpass System
Final note on bandpass system Some books define H+(f) = u(f)H(f) for a filter,
instead of H+(f) = 2u(f)H(f) as for a signal.
As a result of this definition (i.e., H+(f) = u(f)H(f)),
It is justifiable to remove 2 in H+(f) = u(f)H(f), because a filter is used to filter out the signal; hence, it is not necessary to make the total area constant.
dtxhtyehh cfj )(~)(
~)(~ and )(
~ Re2)( 2
© Po-Ning Chen@ece.nctu Chapter 1-217
1.11 Representation of Narrowband Noise in terms of In-Phase and Quadrature Components
In Appendix 2.4, the bandpass system representation is discussed based on deterministic signals.
How about a random process? Can we have a low-pass isomorphism system to a bandpass random process. Take the noise process N(t) as an example.
)(h)(tN
dtNhtY )()()(
© Po-Ning Chen@ece.nctu Chapter 1-218
1.11 Representation of Narrowband Noise in Terms of In-Phase and Quadrature Components
A WSS real-valued zero-mean noise process N(t) is a band-pass process if its PSD SN(f) 0 for |f fc| B and |f + fc| B, and also B < fc. Similar to the analysis for deterministic signals, let
for some joint zero-mean WSS of NI(t) and NQ(t). Notably, the joint WSS of NI(t) and NQ(t)
immediately imply WSS of
)()()(~
)2sin()()2cos()()(
tjNtNtN
tftNtftNtN
QI
cQcI
).(~
tN
© Po-Ning Chen@ece.nctu Chapter 1-219
1.11 PSD of NI(t ) and NQ(t)
First, we note that joint WSS of NI(t ) and NQ(t) and the WSS of N(t) imply:
sequel.) thein proof the(See )()(
)()(
,,
IQQI
QI
NNNN
NN
RR
RR
© Po-Ning Chen@ece.nctu Chapter 1-220
))2(2sin()]()([2
1
))2(2cos()]()([2
1
)2sin()]()([2
1
)2cos()]()([2
1)(
,,
,,
tfRR
tfRR
fRR
fRRR
cNNNN
cNN
cNNNN
cNNN
IQQI
QI
IQQI
QI
(Continue from the previous slide.) These two terms must equal zero, since RN() is not a function of t.
)()(
)()(
,,
IQQI
QI
NNNN
NN
RR
RR
)2sin()()2cos()()( , cNNcNN fRfRRIQI
(Property 1)
(Property 2)
© Po-Ning Chen@ece.nctu Chapter 1-221
© Po-Ning Chen@ece.nctu Chapter 1-222
)()(
)]()([)()(2
1
))]()(())()([(2
1
)](~
)(~
[2
1)(
,
,,
*~
IQI
QIIQQI
NNN
NNNNNN
QIQI
N
jRR
RRjRR
tjNtNtjNtNE
tNtNER
1.11 PSD of NI(t ) and NQ(t)
Some other properties
slide.)next The (Cf.
power. same thehave reasonably
)(~
isomophism lowpass its and
)( that so added is 2
1Factor
)].(~
)(~
[2
1)(
),(~
complex For
)].()([)(
),( realFor
*~
tN
tN
tNtNER
tN
tNtNER
tN
N
N
(Property 3)
© Po-Ning Chen@ece.nctu Chapter 1-223
1.11 PSD of NI(t ) and NQ(t)
(Property 4)
Properties 2 and 3 jointly imply
hold. tofail )()(
)()(2)( of
relation spectrum desired the,2
1factor he without tfact, In
~
cNN
NN
ffSfS
fSfufS
© Po-Ning Chen@ece.nctu Chapter 1-224
1.11 Summary of Spectrum Properties
)()( and )()( .1 ,, IQQIQI NNNNNN RRRR
)2sin()()2cos()()( .2 , cNNcNN fRfRRIQI
)()()( .3 ,~ IQI NNNN
jRRR
.)2exp()(Re)( 4. ~ cNN fjRR
)()( and )()( .6 ,, fSfSfSfSIQQIQI NNNNNN [From 1.]
[From 4. See the next slide.]
)()(2
1)( 7. ~~ cNcNN ffSffSfS
valued.real is )( .5 ~ fSN
).()(By *~~ NN
RR
)0(
)0()0(
Q
I
N
NN
R
RR
valued.real is )( since ,)()(2
1
)()(2
1
)(2
1)(
2
1
)()(2
1
)()(2
1
)(Re
)()(
~~~
*~~
*)(2
~)(2
~
22*~
2~
2*2~
2~
22~
2
fSffSffS
ffSffS
deRdeR
deeReR
deeReR
deeR
deRfS
NcNcN
cNcN
ffj
N
ffj
N
fjfj
N
fj
N
fjfj
N
fj
N
fjfj
N
fjNN
cc
cc
cc
c
© Po-Ning Chen@ece.nctu Chapter 1-225
© Po-Ning Chen@ece.nctu Chapter 1-226
means. zero have they since ed,uncorrelat are )( and )( .8 tNtN QI
.0)]()([)0(
)0()0(
)()(
)()]()([)(
)()(
,
,,
,,
,,
,,
tNtNER
RR
RR
RtNtNER
RR
QINN
NNNN
NNNN
NNQINN
NNNN
QI
QIQI
QIQI
IQQI
IQQI
)()(
)()(2)(
~
cNN
NN
ffSfS
fSfufS
Now, let us examine the desired spectrum properties of:
© Po-Ning Chen@ece.nctu Chapter 1-227
)2cos()2cos(),(4
)]2cos()()2cos()([4
)]()([),(
uftfutR
ufuNtftNE
uVtVEutR
ccN
cc
IIVI
)2sin()(
)2cos()()(
tftN
tftNtN
cQ
cI
)(tN I
)(tNQ
)(tVI
)(tVQ
)2cos(2 tfc
)2sin(2 tfc
WSS.not is )( Notably, tVI
© Po-Ning Chen@ece.nctu Chapter 1-228
)2cos()(2
)2cos()(2))2(2cos(1
lim)(
)]2cos())2(2[cos(1
lim)(
)2cos())(2cos()(42
1lim
),(2
1lim)(
cN
cN
T
T cTN
T
T ccTN
T
T ccNT
T
T VTV
fR
fRdttfT
R
dtftfT
R
dttftfRT
dtttRT
RII
)()()( cNcNV ffSffSfSI
Bf
BffHfSfHfS
II VN ,0
||,1|)(| where),(|)(|)( 22
© Po-Ning Chen@ece.nctu Chapter 1-229
otherwise,0
||),()(
|)(|)]()([)()( 2
BfffSffS
fHffSffSfSfS
cNcN
cNcNNN II
otherwise,0
||),()()(
BfffSffSfS cNcN
NQ
Similarly,
)2sin()(
)2cos()()(
tftN
tftNtN
cQ
cI
)(tN I
)(tNQ
)(tVI
)(tVQ
)2cos(2 tfc
)2sin(2 tfc
© Po-Ning Chen@ece.nctu Chapter 1-230
).( to turn weNext, , QI NNR
)2sin()2cos(),(4
)2sin()()2cos()(4
)]()([),(,
uftfutR
ufuNtftNE
uVtVEutR
ccN
cc
QIVV QI
)2sin())(2cos()(4),(, tftfRttR ccNVV QI
© Po-Ning Chen@ece.nctu Chapter 1-231
)2sin()(2
)]2sin()2(2[sin(1
lim)(
),(2
1lim)( ,,
cN
T
T ccTN
T
T VVTVV
fR
dtftfT
R
dtttRT
RQIQI
)]()([)(, cNcNVV ffSffSjfSQI
2121,21
212121
222111
,
),()()(
)]()([)()(
)()()()(
)()(),(
ddutRhh
dduVtVEhh
duVhdtVhE
uNtNEutR
QI
QI
VVQI
QIQI
QQII
QINN
© Po-Ning Chen@ece.nctu Chapter 1-232
© Po-Ning Chen@ece.nctu Chapter 1-233
)()()(
.)(Let
,)()()(
)()()()(
,
12
21
)(2
,21
21
2
12,21,
12
fSfHfH
u
ddudeuRhh
dddeRhhfS
QI
QI
QIQI
VVQI
ufj
VVQI
fj
VVQINN
otherwise0,
||)],()([
|)(|)]()([
)()(2
,,
BfffSffSj
fHffSffSj
fSfS
cNcN
cNcN
NNNN QIQI
© Po-Ning Chen@ece.nctu Chapter 1-234
Finally,
© Po-Ning Chen@ece.nctu Chapter 1-235
Example 1.12 Ideal Band-Pass Filtered White Noise
© Po-Ning Chen@ece.nctu Chapter 1-236
1.12 Representation of Narrowband Noise in Terms of Envelope and Phase Components
Now we turn to envelope R(t) and phase (t) components of a random process of the form
)](2cos[)(
)2sin()()2cos()()(
ttftR
tftNtftNtN
c
cQcI
)].(/)([tan)( and )()()( where 122 tNtNttNtNtR IQQI
© Po-Ning Chen@ece.nctu Chapter 1-237
1.12 Pdf of R(t) and (t)
Assume that N(t) is a white Gaussian process with two-sided PSD 2 = N0/2.
For convenience, let NI and NQ be snapshot samples of NI(t) and NQ(t).
Then
2
22
2, 2exp
2
1),(
QI
QINN
nnnnf
QI
© Po-Ning Chen@ece.nctu Chapter 1-238
1.12 Pdf of R(t) and (t)
By nI = r cos() and nQ = r sin(),
drdrr
drd
d
dn
d
dndr
dn
dr
dnr
dndnnn
rA
rAQI
QI
nnA
QIQI
QI
),(2
2
2
),(2
2
2),(
2
22
2
2exp
2
1
2exp
2
1
2exp
2
1
.2
exp2
1
2exp
2),( So
2
2
22
2
2,
rrrrrfR
© Po-Ning Chen@ece.nctu Chapter 1-239
1.12 Pdf of R(t) and (t)
R and are therefore independent.
.0for 2
1)(
.0for 2
exp)(2
2
2
f
rrr
rfRRayleigh distribution.
Normalized Rayleigh distribution with 2 = 1.
© Po-Ning Chen@ece.nctu Chapter 1-240
1.13 Sine Wave Plus Narrowband Noise
Now suppose the previous Gaussian white noise is added to a sinusoid of amplitude A.
Then
Uncorrelation for Gaussian nI(t) and nQ(t) implies their independence.
)2sin()()2cos()(
)2sin()2cos()()2cos(
)()2cos()(
cQcI
cQcIc
c
ftxftx
tfntftntfA
tntfAtx
© Po-Ning Chen@ece.nctu Chapter 1-241
1.13 Sine Wave Plus Narrowband Noise This gives the pdf of xI(t) and xQ(t) as:
By xI = r cos() and xQ = r sin(),
2
22
2, 2
)(exp
2
1),(
QI
QIXX
xAxxxf
QI
2
22
2
2
222
2,
2
)cos(2exp
2
1
2
)(sin))cos((exp
2
1),(
ArAr
d
dx
d
dxdr
dx
dr
dxrAr
rfQI
QI
R
© Po-Ning Chen@ece.nctu Chapter 1-242
1.13 Sine Wave Plus Narrowband Noise
Notably, in this case, R and are no longer independent.
We are more interested in the marginal distribution of R.
2
0 2
22
2
2
0,
2
)cos(2exp
2
),()(
dArArr
drfrf RR
© Po-Ning Chen@ece.nctu Chapter 1-243
1.13 Sine Wave Plus Narrowband Noise
,2
exp
2
)cos(2exp
2exp
2)(
202
22
2
2
0 22
22
2
ArI
Arr
dArArr
rfR
order. zero of kindfirst theof
function Besselmodified theis ))cos(exp(2
1)( where
2
00
dxxI
This distribution is named the Rician distribution.
© Po-Ning Chen@ece.nctu Chapter 1-244
1.13 Normalized Rician distribution
,2
exp)( 0
22
avIav
vvfV
© Po-Ning Chen@ece.nctu Chapter 1-245
A3.1 Bessel Functions
Bessel’s equation of order n
Its solution Jn(x) is the Bessel function of the first kind of order n.
0)( 22
2
22 ynx
dx
dyx
dx
ydx
0
2
0
)!(!
4/
2sinexp
2
1
))sin(cos(1
)(
m
m
n
n
n
mnm
xxdjnjx
dnxxJ
© Po-Ning Chen@ece.nctu Chapter 1-246
A3.2 Properties of the Bessel Function
.1)( .7
))sin(exp()exp()( .6 .0)(lim .5
24cos
2)( large, When .4
.!2
)( small, When .3 )(2
)()( .2
)()1()()1()( 1.
2
11
nn
nnnn
n
n
n
nnnn
nn
nn
n
xJ
jxjnxJxJ
nx
xxJx
n
xxJxxJ
x
nxJxJ
xJxJxJ
© Po-Ning Chen@ece.nctu Chapter 1-247
A3.3 Modified Bessel Function
Modified Bessel’s equation of order n
Its solution In(x) is the Modified Bessel function of the first kind of order n.
The modified Bessel function is monotonically increasing in x for all n.
0)( 222
2
22 ynxj
dx
dyx
dx
ydx
dnxxJjxI n
nn )cos())cos(exp(
2
1)()(
© Po-Ning Chen@ece.nctu Chapter 1-248
A3.3 Properties of Modified Bessel Function
))cos(exp()exp()( '.6
.2
)exp()( large, When '.4
.1,0
0,1)(lim '.3
0
xjnxI
x
xxIx
n
nxI
nn
n
nx
© Po-Ning Chen@ece.nctu Chapter 1-249
1.14 Computer Experiments: Flat-Fading Channel
Model of a multi-path channelpaths. Assume N
N
kkck tfA
1
)2cos(
)2cos( tfA c
© Po-Ning Chen@ece.nctu Chapter 1-250
1.14 Computer Experiments: Flat-Fading Channel
)2sin()2cos()2cos()(1
tfYtfYtfAtY cQcI
N
kkck
.)sin( and )cos( where11
N
kkkQ
N
kkkI AYAY
. oft independen is which
,)(~
output induces )(~
Input
t
jYYtYAtX QI
).2,0[over uniform and
)1,1[over uniform and i.i.d., )},{( Assume
k
kkk AA
© Po-Ning Chen@ece.nctu Chapter 1-251
1.14 Experiment 1
By the central limit theorem,
)1,0(6/
)cos()cos(
6/11 Normal
N
AA
N
Y NNI
).6/( varianceand 0 mean
withddistribute Gaussianely approximat is So
N
YI
have we, variablerandom Gaussianany For G
3.]||[
]||[ and 0
]||[
]||[22
4
223
32
1
GE
GE
GE
GE
© Po-Ning Chen@ece.nctu Chapter 1-252
1.14 Experiment 1
We can therefore use 1 and 2 to examine the degree of resemblance to Gaussian for a random variable.
N
10 100 1000 10000
YI
1 0.2443 0.0255 0.0065 0.0003
2 2.1567 2.8759 2.8587 3.0075
YQ
1 0.0874 0.0017 0.0004 0.0000
2 1.9621 2.7109 3.1663 3.0135
40/3)](sin[)](cos[
6/1)](sin[)](cos[
0)]sin([)]cos([
3][/][3
][
][)1(3][
])[(
])[(
][
][~
}{ i.i.d. mean-zerofor /)(
4444
2222
224
222
224
21
2
41
22
4
2
1
AEAE
AEAE
AEAE
N
UEUE
UEN
UENNUNE
UUE
UUE
SE
SE
UaUUS
N
N
N
N
kNNN
0000.39975.29745.2745.2
10000100010010N
© Po-Ning Chen@ece.nctu Chapter 1-253
1.14 Experiment 2
We can re-formulate Y(t) as:
If YI and YQ approach independent Gaussian, then R and respectively approach Rayleigh and uniform distributions.
)./(tan and where
),2cos()(
122IQQI
c
YYYYR
tfRtY
© Po-Ning Chen@ece.nctu Chapter 1-254
1.14 Experiment 2
N = 10,000
The experimental curve is obtained by averaging 100 histograms.
© Po-Ning Chen@ece.nctu Chapter 1-255
1.14 Experiment 2
).102cos( Input 6 t
Output channel Rayleigh
ingcorrespond The
© Po-Ning Chen@ece.nctu Chapter 1-256
1.15 Summary and Discussion Definition of Probability System and Probability
Measure Random variable, vector and process Autocorrelation and crosscorrelation Definition of WSS Why ergodicity?
Time average as a good “estimate” of ensemble average Characteristic function and Fourier transform
Dirichlet’s condition Dirac delta function Fourier series and its relation to Fourier transform
© Po-Ning Chen@ece.nctu Chapter 1-257
1.15 Summary and Discussion Power spectral density and its properties Energy spectral density Cross spectral density Stable LTI filter
Linearity and convolution Narrowband process
Canonical low-pass isomorphism In-phase and quadrature components Hilbert transform Bandpass system
© Po-Ning Chen@ece.nctu Chapter 1-258
1.15 Summary and Discussion
Bandwidth Null to null, 3dB, rms, noise-equivalent Time-bandwidth product
Noise Shot noise, thermal noise and white noise
Gaussian, Rayleigh and Rician Central limit theorem
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