Chapter 1 Random Processes “Probabilities” is considered an important background to communication study

Chapter 1 Random Processes

“Probabilities” is considered an important background to communication study.

© Po-Ning [email protected] 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.


1.1 Examples of Stochastic Models

Channel noise and interference Source of information, such as voice


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.



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)?



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!


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.


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.


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.


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).


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


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


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.



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.



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



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

)(

)(



It is not necessarily true that If

then the pdf of X exists and equals fX(x).

,)(

)(x

xFxf X

X


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


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


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.


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}.


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 [email protected]

X(t,s1)

X(t,s2)

X(t,sn)

Chapter 1-23

Define or view a random process via its inherited probability system


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.


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


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.


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


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



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:





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!




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 )()]([)( )(


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


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


1.4 Autocorrelation

Conceptually, autocorrelation function = “power correlation” between two time instances t1 and t2.


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 [email protected] Background 38

Autocovariance function CX(t1, t2) is given by:


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


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


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).


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


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.



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



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+).



If RX() decreases faster, the correlation decreases faster.


1.4 Decorrelation time

Some researchers define the decorrelation time as:


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.



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)



Then

.0

)2sin()2sin(2

)2sin(2

)2cos(2

2

1)2cos(

)]2cos([)(

tftfA

tfA

dtfA

dtfA

tfAEt

cc

c

c

c

cX



.)(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.




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



I0 I1 I2 I3 I4I1I2I3I4I5I6



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)



Then by assuming that td is uniformly distributed over [0, T), we obtain:

0

)]([0

)]([][

)()(

dn

dn

n

dn

nX

tnTtpEA

tnTtpEIEA

tnTtpIAEt



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


.for 0][][][ Since mnIEIEIIE mnmn

Chapter 1-58


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


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


otherwise,0

)0 and ||0(or )||0(,

)()()()(

2121

2

21

2

21

dd

dn

dd

tTttTtttA

tnTtptnTtpAttXtXE


otherwise,0

||0,||

1

otherwise,0

||0,1

)()()()(

21212

21||

2

2121 21

TttT

ttA

TttdtT

AttXtXEEtXtXET

tt dd




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


1.4 Cross-Correlation

If X(t) and Y(t) are jointly weakly stationary, then

)()(

)()(

)(),(

,

,

,,

utRutR

utRutR

utut

YXY

YXX

YXYX RR


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



),())(2sin(2

1

2

))(2sin()2)(2sin(),(

)]2cos()2[sin()]()([

)]2sin()()2cos()([

)]()([),(,

utRutf

tufutfEutR

tfufEuXtXE

ufuXtftXE

uXtXEutR

Xc

ccX

cc

cc

QIXX QI



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


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.


1.5 Ergodic Process

The answer is negative! An additional ergodicity assumption is

necessary for time average converging to the ensemble average X.


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


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


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)(


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);(


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.


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 )()()]([


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.


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


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


A1.3 Characteristic Function

Characteristic function is indeed the inverse Fourier transform of the pdf.


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|)(|


Example of LTI filter: Mobile Radio Channel

Transmitter Receiver

),( 11

),( 22

),( 33 .)( where

,)()(

)()()()(3

1

332211

ii

iii

h

tsh

tstststY

)(tX )(tY


Example of LTI filter: Mobile Radio Channel

Transmitter Receiver

dtXhtY )()()(

)(tX )(tY


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 |)(|


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 )(


1.6 Autocorrelation

122121

122121

222111

),()()(

)()()()(

)()()()(

)]()([),(

ddutRhh

dduXtXEhh

duXhdtXhE

uYtYEutR

X

Y

122121 )()()()( then

WSS, )( If

ddRhhR

tX

XY


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.


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


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 |)(|


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 !


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


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



Replication property For every continuous point of g(t),

Constant spectrum

dtgtg )()()(

.1)2exp()0()2exp()(

dtftjtdtftjt



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



More properties Multiplication convention

ba

baatbtat

if,0

if),()()(

)(

)()(

)()()()(

ba

dtatba

dtbaatdtbtat


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


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


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 )()(



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/


(Continue from the previous slide.)



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


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


-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


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



A2.1 Impulse Response of LTI Filter

Impulse response = Filter response to Dirac delta function (application of replication property)

h()(t) h(t)

).()()()()()( thdthdtxhty


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


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


A2.1 Fourier Analysis

Remember to self-study Tables A6.2 and A6.3.


1.7 Power Spectral Density

LTIh()

Deterministic x(t)

dtxhty )()()(

LTIh()

WSS X(t)

dtXhtY )()()(



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



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.



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


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



E.g., for a WSS process,

E.g., for a deterministic function,

T

TT

T

TTX

dttxtxT

dttxtxET

R

)()(2

1lim

)]()([2

1lim)(

)]()([)( tXtXERX



E.g., for a cyclostationary process,

).( of periodonary cyclostati theis where

,)]()([2

1)(

tXT

dttXtXET

RT

TX



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



For a WSS process, For a deterministic process,

).()( fSfS XX

).()(2

1lim)( *

2 fxfxT

fS TT

X



Relation of time-average PSDs



1.7 Power Spectral Density under WSS input

For a WSS filter 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



The unit of E[Y2(t)] is, e.g., Watt. So the unit of SY(f) is therefore Watt per Hz.


1.7 Operational Meaning of PSD

Example. Assume h() is real, and |H(f)| is given by:


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.


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()()(



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]



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



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.



Therefore, by passing through a proper filter,

for any fc.

0)]([2)( 2 tYEfS cX



Property 4: PSD is an even function, i.e.,

Proof.

).()( fSfS XX

)(

)()( ,)(

,)(

)()(

2

2

)(2

fS

RRdsesR

sdsesR

deRfS

X

XXfsj

X

fsjX

fjXX



Property 5: PSD is real.

Proof.

Then with Property 4,

Thus, SX(f) is real.

)()(real )( * fSfSR XXX

).()()( * fSfSfS XXX


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


Example 1.5(continue from Example 1.2) Signal with Random Phase


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


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





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).




Example

The ESD of a rectangular pulse of amplitude A and duration T is given by

)(sinc)( 2222

0

2 fTTAdtAefET

ftjg


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


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



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


(Notably, we only have the values of x(t) for t in [T, T).)



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


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


1.7 Cross Spectral Density

Property

)()( ,, XYYX RR

)(

)2exp()(

)2exp()(

)2exp()()(

,

,

,

,,

fS

dfjR

dfjR

dfjRfS

YX

YX

XY

XYXY


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


).()()()()(

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.


Example 1.9

Determine the CSD of output processes induced by separately passing jointly WSS inputs through a pair of stable LTI filters.



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



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



For a Gaussian random vector, “uncorrelation” implies “independence.”

n

iiXX

nn

xfxfXX

XX

i

122

11

)()(},{Cov0

0},{Cov



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


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


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.)


1.8 Properties of Gaussian processes

Property 3. A WSS Gaussian process is also strictly stationary.

(An immediate consequence of Property 2.)


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.


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


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


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


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


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


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


1.9 Thermal Noise

Model of a noisy resistor

222 ][][ RIEVE TNTN


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


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.


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


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


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!


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


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


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



If w(t) is white Gaussian, then the pdf of N is uniquely determined by the first and second moments.


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.


1.10 Narrowband Noise

The analysis on narrowband noise will be covered in subsequent sections.


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.


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).


)(sinc)( 22 fΤTAfSX

. amplitude and duration of pulse

r rectangulaa is )( where),()(

AT

tptnTtpIAtX dn

n


The null-to-null bandwidth is 1/T.



The null-to-null bandwidth in this case is 2B.


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


2

1)(sinc

)0(

)(2

22

TA

fΤTA

S

fS

X

X


The 3-dB bandwidth is 0.32/T.

A2.2 3-dB Bandwidth

T

32.0


A2.2 Root-Mean-Square Bandwidth

rms bandwidth

Disadvantage: Sometimes,

even if

2/12

)(

)(

dffS

dffSfB

X

X

rms

dffSf X )(2

.)(

dffSX


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


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.


A2.2 Noise Equivalent Bandwidth


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


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)()()(



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



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


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.



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


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



).( 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



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



Hence, Hilbert Transform is basically a 90 degree phase shifter.



dt

gtg

dt

gtg

)(ˆ1)(

)(1)(ˆ

PairTransformHilbert

1

) h)tg )ˆ tg

1

) h)ˆ tg )tg

1



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



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


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


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



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


)2exp())()(( tfjtjgtg cQI

)2exp( tfj c))()(( tjgtg QI



Canonical transmitter)2sin()()2cos()()( tftgtftgtg cQcI



Canonical receiver


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


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


Now, is the filter output y(t) also a bandpass signal?


)( fX

)( fH

)( fY



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


There is an additional multiplicative constant 2 at the output!



Conclusion:

)(~ h

)(~ tx

dtxhty )(~)(

~

2

1)(~



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


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 )()()(


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


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


))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)



)()(

)]()([)()(2

1

))]()(())()([(2

1

)](~

)(~

[2

1)(

,

,,

*~

IQI

QIIQQI

NNN

NNNNNN

QIQI

N

jRR

RRjRR

tjNtNtjNtNE

tNtNER


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)



(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


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



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:


)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


)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


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


).( to turn weNext, , QI NNR

)2sin()2cos(),(4

)2sin()()2cos()(4

)]()([),(,

uftfutR

ufuNtftNE

uVtVEutR

ccN

cc

QIVV QI

)2sin())(2cos()(4),(, tftfRttR ccNVV QI


)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



)()()(

.)(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


Finally,


Example 1.12 Ideal Band-Pass Filtered White Noise


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


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



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



R and are therefore independent.

.0for 2

1)(

.0for 2

exp)(2

2

2

f

rrr

rfRRayleigh distribution.

Normalized Rayleigh distribution with 2 = 1.


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


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



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



,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.


1.13 Normalized Rician distribution

,2

exp)( 0

22

avIav

vvfV


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


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


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)()(


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


1.14 Computer Experiments: Flat-Fading Channel

Model of a multi-path channelpaths. Assume N

N

kkck tfA

1

)2cos(

)2cos( tfA c


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


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


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


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


1.14 Experiment 2

N = 10,000

The experimental curve is obtained by averaging 100 histograms.


1.14 Experiment 2

).102cos( Input 6 t

Output channel Rayleigh

ingcorrespond The


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


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


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

Documents

Chapter 1 Random Processes “Probabilities” is considered an important background to communication study