caes lezione 2 - processi stocastici -...

OutlineIntroduction

Stochastic Processes

CAESLezione 2 -

Processi Stocastici

Prof. Michele Scarpiniti

Dip. NFOCOM - “Sapienza” Universita di Roma

http://ispac.ing.uniroma1.it/scarpiniti/index.htm

[email protected]

Roma, 04 Marzo 2010

M. Scarpiniti CAES Lezione 2 - Processi Stocastici 1 / 46

http://ispac.ing.uniroma1.it/scarpiniti/index.htm

OutlineIntroduction


1 IntroductionBasic DefinitionMean and Moments

2 Stochastic ProcessesStochastic ProcessesRandom sequences and discrete-time LTI systems


OutlineIntroduction


Basic DefinitionMean and Moments

Introduction

Introduction


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Introduction

The deterministic signals are defined through precise mathematical ruleswhich are able to exactly predict the sample values. The random signals orstochastic processes, instead, are not exactly predictable, but they arestatistically characterized by functions that can describe their behavior inmean.

A random variable (RV) X(ς) is a function, with domain onS = [ς1, ς2, . . .] defined as universal set of experimental results, whichassigns a number x to every occurrence ς ∈ S .The result ςk is defined event.The predictability of the event ςk is defined by a non-negativefunction p(ςk), k = 1, 2, . . ., named probability function of the eventςk . This function satisfy the two following properties:

1 p[X(ς) = −∞] = 0 and p[X(ς) = +∞] = 1;2 0 ≤ p(·) ≤ 1.

A random variable is: a discrete RV if it can assume a set of discretevalues xk ; a continuous RV if it can assume all possible values.


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Cumulative Density Functions

For simplicity a RV X(ς) is denoted as X.The probability of the event X ≤ x , denoted as p(X ≤ x) is a function ofthe x variable. This function is a measure of the probability, it is denotedby FX(x) and it is named as probability distribution function or cumulativedensity function (cdf):

FX(x) = p(X ≤ x), for −∞ < x <∞

Properties of the cdf:

1 0 ≤ FX(x) ≤ 1;

2 FX(−∞) = 0;

3 FX(+∞) = 1;

4 FX(x1) ≤ FX(x2) if x1 < x2.


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Probability Density Functions

Another very meaningful function is the probability density function (pdf),defined as follows:

fX(x) =dFX(x)

dx

Properties of the pdf:

1∫ +∞−∞ fX(x)dx = 1;

2 FX(x) =∫ x−∞ fX(ν)dν.

We have to note that fX(x) does not represent a probability: to obtain theprobability of the event x ≤ X ≤ x + ∆x we have to multiply the pdf forthe interval ∆x :

fX(x)∆x ≈ ∆xFX(x) ≡ FX(x + ∆x)− FX(x) = p(x ≤ X ≤ x + ∆x).


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Cumulative and Probability Density Functions

The following figure shows some cdf and the respective pdf.


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Mean and Moments

The expected value or mean of a RV X is defined by

µx = E {x} =

∫ ∞−∞

xfX(x)dx

The moment of order m is defined by

r(m)x = E {xm} =

∫ ∞−∞

xmfX(x)dx

The central moment of order m is defined by

c(m)x = E {(x − µ)m} =

∫ ∞−∞

(x − µ)mfX(x)dx


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Variance

The variance σ2x of a RV X is defined as the central moment of order

2:

σ2x = c

(2)x = E

{(x − µ)2

}=

∫ ∞−∞

(x − µ)2fX(x)dx

The positive constant σx =√σ2

x is defined as standard deviation ofthe RV X.

The variance indicates as the values of the RV X are distributed aroundthe mean value µx, as show in the figure below.


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Skewness and Kurtosis

The skewness k(3)x of a RV X is a function of the central moment of order 3:

k(3)x = E

{[x − µσx

]3}

=1

σ3x

c(3)x

The skewness represents a degree of the pdf asymmetry.

The kurtosis k(4)x of a RV X is a function of the central moment of order 4:

k(4)x = E

{[x − µσx

]4}− 3 =

1

σ4x

c(4)x − 3

The term −3 is introduced in order to obtain k(4)x = 0 for a Gaussian RV.

The kurtosis represents the peakiness of the pdf.


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Chebyshev’s inequality

Given a RV X with mean value µ and standard deviation σx, then

p {|x − µ| ≥ kσx} ≤1

k2, k > 0.

This inequality is known as Chebyshev’s inequality and states that a RVdeviates k times from its mean value with probability lower than 1/k2.The Chebyshev’s inequality is independent from the shape of the fx(x).


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The Characteristic Function

The Characteristic Function or CF Φx(s) is defined as the Laplace transform ofthe probability density function (pdf) fx(x):

Φx(s) =

∫ ∞−∞

fx(x)e−sxdx = E{

e−sx}

(1)

where s is the complex-valued Laplace variable and cannot be interpreted as afrequency.The Taylor series expansion of the CF gives

Φx(s) ≡ E{

e−sx}

= E

{1 + sx +

(sx)2

2!+ · · · +

(sx)m

m!+ · · ·

}= 1 + sµ +

s2

2!r (2)x + · · · +

sm

m!r (m)x + · · ·

So the moments can be interpreted as the coefficients of the Taylor seriesexpansion:

r (m)x =

dmΦx(s)

dsmfor m = 1, 2, . . .

For this reason the CF is also called moments generating function.


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The Second Characteristic Function

The Second Characteristic Function Ψx(s) is defined as the natural logarithm ofthe characteristic function Φx(s):

Ψx(s) = ln Φx(s) = ln E{

e−sx}

(2)

The Taylor series expansion of the second CF gives some coefficients each of

them can be interpreted as cumulant κ(m)x , which is a statistical descriptor.

κ(m)x =

dmΨx(s)

dsmm = 1, 2, . . .

For this reason the second CF is also called cumulants generating function. Thefirst five cumulants are:

κ(1)x = r

(1)x = µ = 0

κ(2)x = r

(2)x = σ2

x

κ(3)x = c

(3)x

κ(4)x = c

(4)x − 3σ4

x

κ(5)x = c

(5)x − 10c

(3)x σ2

x


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The Uniform Distribution

A RV X, defined in the interval [a, b], has a uniform distribution if its pdfis

fx (x) =

{1

b−a a ≤ x ≤ b

0 elsewhere

The cdf is

Fx (x) =

x∫−∞

fx (v) dv =

0 x < a

x−ab−a a ≤ x ≤ b

1 x > b

while the CF is

Φx(s) =esb − esa

s(b − a)

Its mean and variance are respectively:

µ =a + b

2and σ2

x =(b − a)2

12


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The Normal Distribution

A RV X, with mean µ and standard deviation σx, has a normal distributionif its pdf is

fx(x) =1√

2πσ2x

e− 1

2σ2x

(x−µ)2

with a CFΦx(s) = eµs− 1

2σ2

x s2

Usually it is denoted by N(µ, σ2x), because the Gaussian RV is completely

determined by its mean µ and variance σ2x .

The moments are evaluated by

c(m)x = E {|x − µ|m} =

{1 · 3 · 5 · . . . · (m − 1)σm

x for m even0 for m odd


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Central Limit Theorem

An important result is the central limit theorem (CLT) stating that:

Theorem

Given n independent RVs xi , consider the sum

x = x1 + x2 + . . .+ xn

This is a RV with mean µ = µ1 + µ2 + . . .+ µn and varianceσ2

x = σ2x1

+ σ2x2

+ . . .+ σ2xn

. Then the distribution fx(x) of x approaches a normaldistribution with the same mean and variance, as n increases:

fx(x) =1√

2πσ2x

e− (x−µ)2

2σ2x

This theorem can be stated as a limit: if z = (x− µ) /σx then

fz(z) →n→∞

1√2π

e−z2/2.


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Vectors of Random Variables

A vector of random variables or random vector is defined as a row or columnvector of random variables. For example X(ς) = [X0(ς),X1(ς), . . . ,XN(ς)]T . Wecan generalize to random vectors the definition of mean, variance and moments.

The mean of a random vector is the vector:

µ = E {x} = [E {x1} ,E {x2} , . . . ,E {xN}]T = [µ1, µ2, . . . , µN ]T

The second order statistic is characterized by the auto-covariance matrix,define as

Cx = E{

(x− µ)(x− µ)T}

It is a symmetric matrix: Cx = CTx .

The auto-correlation matrix is defined as

Rx = E{

xxH}

It is a symmetric matrix: Rx = RTx .


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Vectors of Random Variables

If the random vector X has zero mean µ = [0, 0, . . . , 0]T , then

Cx = Rx .

The auto-correlation matrix Rx of a vector X of RVs is alway non-negativedefined. In fact for every vector w = [w0,w1, . . . ,wM ]T , we have

wT Rxw ≥ 0

that is the quadratic form wT Rxw is non-negative.

For a complex-valued random vector X ∈ CN , we have the same formulassubstituting the transpose operator (·)T with the hermitian operator (·)H .


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Conditional distributions

Conditional probability is the probability of some event X, given theoccurrence of some other event Y and it is written as

p(X|Y).

It is described by the conditional cumulative density function Fx(x|y) andby the conditional probability density function fx(x|y). These terms areevaluated as

Fx(x|y) =Fxy(x,y)Fy(y)

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

In addition it is valid the following

Theorem (Bayes)

p(Y|X) = p(X|Y)p(Y)

p(X)


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Statistical independence

Given two events X and Y, we say that they are statistical independenceif

p (X|Y) = p (X)p (Y|X) = p (Y)

and converselyFx (X|Y) = Fx (X)fx (X|Y) = fx (X)

A consequence of the statistical independence is that the jointdistribution is factorized in the products of its marginal ones, as aconsequence of the Bayes’theorem:

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


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Stochastic ProcessesRandom sequences and discrete-time LTI systems




OutlineIntroduction




A stochastic process (SP) x(t, ς) is a family of bi-dimensional functions inthe variable t and ς. The variable ς is defined by the set of allexperimental results while the variable t represents the time variable,usually t ∈ R or t ∈ Z. If t is real we have a continuous-time stochasticprocess x(t), else, if t ∈ Z, it is a discrete-time stochastic process x[n]:

x[n] = [x1[n], x2[n], ..., xN [n]]


OutlineIntroduction



Statistical representation of stochastic processes

The determination of the statistical representation of a stochastic processcan be done as for the random variables. In fact, for a fixed temporalindex n, the process is a simple random variable. Hence a stochasticprocess is characterized by:

the joint cdf of order k :

Fx(x1, . . . , xk ; n1, . . . , nk) = p (x[n1] ≤ x1, . . . , x[nk ] ≤ xk)

the joint pdf of order k :

fx(x1, . . . , xk ; n1, . . . , nk) =∂Fx(x1, . . . , xk ; n1, . . . , nk)

∂x1∂x2 . . . ∂xk


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Ensemble means

For a stochastic process is possible to define the ensemble means:

mean value:µn = E {x [n]}

It usually depends on the time instant n.autocorrelation:

r [n,m] = E {x [n]x∗[m]}

In addition r [n] = E{|x [n]|2

}is the mean power of the process.

auto-covariance:

c[n,m] = E {(x [n]− µn)(x [m]− µm)∗} = r [n,m]− µnµ∗m

If the SP has zero mean, for every n,m, then r [n,m] = c[n,m].high order moments and high order central moments:

r (m)[n1, . . . , nm] = E {x [n1] · x [n2] · . . . · x [nm]} ,c(m)[n1, . . . , nm] = E {(x [n1]− µn1)(x [n2]− µn2) · · · (x [nm]− µnm)} .


OutlineIntroduction



Ensemble means

In particular the following moments are very important:

variance:

σ2xn = E

{|x [n]− µn|2

}= E

{|x [n]|2

}− |µn|2

The quantity σxn is defined as standard deviation of the SP andmeasures the dispersion of the observation x [n] around its mean value

µn. If the PS has a zero mean then σ2xn = r [n, n] = E

{|x [n]|2

}.

Given two SP x [n] and y [n], we can define the cross-correlation andcross-covariance:

rxy [n,m] = E {x [n]y∗[m]}cxy [n,m] = E {(x [n]− µxn) (y [m]− µym)∗} = rxy [n,m]− µxnµ

∗ym

The normalized cross-correlation is defined as

rxy [n,m] =cxy [n,m]

σxnσym.


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Statistical Independence

Fundamental is the general concept of independence.

A stochastic process is said to be independent if its joint pdf can befactorized into the product of its marginal distribution:

fx(x1, . . . , xk ; n1, . . . , nk) = f1(x1; n1) ·f2(x2; n2) · . . . ·fk(xk ; nk), ∀k , ni

that is: x[n] is a SP formed by the independent RVs x1[n], x2[n], . . ..

Alternatively, for two independent sequences x [n] and y [n] followsthat:

E {x [n] · y [n]} = E {x [n]} · E {y [n]}

If all the sequences of the SP are independent and are characterizedby the same probability density function f1(x1; n1) = . . . = fk(xk ; nk),the the SP is called independent and identically distributed process ori.i.d.


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Statistical Independence

A stochastic process is said to be uncorrelated if

c[n,m] = E {(x [n]− µn)(x [m]− µm)∗} = σ2xnδ[n −m]

The SPs x [n] and y [n] are said to be uncorrelated if

cxy [n,m] = E{

(x [n]− µxn) (y [m]− µym)∗}

= 0

that isrxy [n,m] = µxnµ

∗ym

The SPs x [n] and y [n] are said to be orthogonal if

rxy [n,m] = 0.

Remark

If two SPs x [n] and y [n] are independent the they are also uncorrelated, but thereverse is not always true. (It is true for Gaussian SPs)


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Stationary stochastic process

Usually a SP is stationary if the statistic of x [n] is equal to the statistic ofx [n − k]. To be more precise:

Definition

The SP x [n] is strict-sense stationary (SSS) or stationary of order N, if

fx(x1, . . . , xN ; n1, . . . , nN) = fx(x1, . . . , xN ; n1 − k , . . . , nN − k), ∀k

that is its pdf is invariant to a time translation.

Definition

The SP x [n] is wide-sense stationary (WSS) if

E {x [n]} = E {x [n + k]} = µ, ∀n, k

that is its statistic of first order does not depend on time.


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Stationary stochastic process

As a corollary of the definition of the stationarity we can give the following:

Definition

The SP x [n] is wide-sense periodic (WSP) if

E {x [n]} = E {x [n + N]} = µ, ∀n

that is its statistic of first order is invariant to a translation of N samples.

Definition

The SP x [n] is wide-sense cyclostationary (WSC) if

E {x [n]} = E {x [n + N]}r [m, n] = r [m + N, n + N]

∀m, n

that is its statistic of first and second order are invariant to a translation of Nsamples.


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Auto-correlation and auto-covariance sequence for a WSSSP

The auto-correlation sequence for a WSS SP is defined as

r [m, n] = E {x [n]x∗[m]} = r [n −m]

Let us pose k = n −m, defined as correlation delay or lag , then thecorrelation is written as

r [k] = E {x [n]x∗[n − k]} = E {x [n + k]x∗[n]}

usually named auto-correlation function (acf).

The auto-covariance sequence for a WSS SP is defined as

c[k] = E {(x [n]− µ)(x [n − k]− µ)∗} = r [k]− µ2

for a zero-mean SP r [k] = c[k].


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Properties of the auto-correlation function

The auto-correlation function has some fundamental properties:

1 it has conjugate-symmetry with respect the delay:

r∗[−k] = r [k]

2 it is non-negative defined

M∑k=1

M∑m=1

αk r [k −m]α∗k ≥ 0

this property is a necessary and sufficient condition so that r [k] is anacf.

3 the term at k = 0 is that of maximum amplitude

E{

x2[n]}

= r [0] ≥ |r [k]| , ∀n, k


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Cross-correlation and cross-covariance sequence for twoWSS SPs

The cross-correlation sequence for two WSS SPs x [n] and y [n], isdefined as

rxy [k] = E {x [n]y∗[n − k]} = E {x [n + k]y∗[n]}

usually known as the cross-correlation function (ccf).

The cross-covariance sequence for two WSS SPs x [n] and y [n], isdefined as

cxy [k] = E {(x [n]− µx) (y [n − k]− µy )∗} = rxy [k]− µxµ∗y

for a zero-mean SP rxy [k] = cxy [k].


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Ergodic processes

A stochastic process is said to be ergodic if the ensemble means coincidewith the temporal means.For a discrete sequence x [n] the temporal mean 〈x [n]〉 is defined as

µ = 〈x [n]〉 = limN→∞

1

N

N−1∑n=0

x [n]

So, if the SP is ergodic then we can substitute the expectation operatorE {·} with the operator 〈·〉, hence

µ = 〈x [n]〉 = E {x [n]}

and〈x [n]x∗[n − k]〉 = E {x [n]x∗[n − k]}

and so on.


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Ergodic processes

The ergodicity is a property assuring that we are able to construct thestatistics of order N with only one realization of the SP. This fact is veryimportant in real situations of signal processing where we usually have atour disposal only one realization. Then from time means we can obtain theensemble means: i.e, from a power measure we can obtain the variance.

Two ergodic SP x [n] and y [n] are also jointly ergodic if

E {x [n]y∗[n − k]} = 〈x [n]y∗[n − k]〉

Remark

If a SP is ergodic then it is also WSS, but the reverse is not always true.


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Auto-correlation matrix of sequences

Let us consider a M × 1 random vector xn derived from the SP X [n]:

xn∆=[

x [n] x [n − 1] · · · x [n −M + 1]]T

Then the mean value is defined as

µxn =[µxn µxn−1 · · · µxn−M+1

]TThe auto-correlation matrix is defined as

Rxn = E (xnxHn ) =

rx [n, n] · · · rx [n, n −M + 1]...

. . ....

rx [n −M + 1, n] · · · rx [n −M + 1, n −M + 1]

because rx [n− i , n− j ] = r∗x [n− j , n− i ] for 0 ≤ i , j ≤ M − 1 the matrix Rxn

is hermitian.


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Auto-correlation matrix of sequences

If the stochastic process is stationary then the auto-correlation function does notdepend on the time index n. So, defining k = j − i we have

rx [n − i , n − j ] = rx [j − i ] = rx [k]

With this position we obtain

Rx = E (xnxHn ) =

rx [0] rx [1] rx [2] · · · rx [M − 1]r∗x [1] rx [0] rx [1] · · · rx [M − 2]r∗x [2] r∗x [1] rx [0] · · · rx [M − 3]

......

.... . .

...r∗x [M − 1] r∗x [M − 2] r∗x [M − 3] · · · rx [0]

The auto-correlation matrix Rx of a stationary SP is Hermitian, Toeplitz and

non-negative defined.


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Properties of the auto-correlation matrix

The eigenvalues and eigenvectors of the auto-correlation matrix Rx satisfy some properties:1 The eigenvalues λi of Rx are real and non-negative, and

λi =qH

i Rqi

qHi qi

(Rayleighratio)

2 The eigenvectors qi of Rx are orthogonal for distinct λi

qHi qj = 0, for i 6= j

3 Rx is diagonalized byR = QΛQH

where Q = [ q0 q1 · · · qM−1 ], Λ = diag(λ0, λ1, . . . , λM−1) and QHQ = I.4 Rx has the following spectral representation:

R =

M−1∑i=0

λi qi qHi =

M−1∑i=0

λi Pi

where Pi = qi qHi is defined spectral projection.

5 The trace of Rx satisfies

tr[R] =

M−1∑i=0

λi ⇒1

M

M−1∑i=0

λi = rxx [0] = σ2x


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Random sequences and discrete-time LTI systems

Let us consider a stable discrete-time LTI system characterized by the impulseresponse h[n] with a random sequence WSS x [n] as input andy [n] =

∑∞l=−∞ h[l ]x [n − l ] as output.

It could be useful to know the auto- and cross-correlation between input andoutput of a such system.

ryx [k] = h[k] ∗ rxx [k];

ryy [k] = rhh[k] ∗ rxx [k].

The output pdf py(y) is related to the input pdf px(x) from the following

py(y) =px(x)

|det(J)|where J is the Jacobian of the transformation.M. Scarpiniti CAES Lezione 2 - Processi Stocastici 38 / 46

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Power Spectral Density (PSD)

The power spectral density or PSD is defined as the DTFT of theauto-correlation sequence

Rxx(e jω) =∞∑

k=−∞

rxx [k]e−jωk

It represents a frequency distribution measure of the mean power of a SP.

The cross power spectral density or CPSD is defined as the DTFT of thecross-correlation sequence

Rxy (e jω) =∞∑

k=−∞

rxy [k]e−jωk

Its magnitude represents the frequency of the SP x [n] associates to those ofthe SP y [n], while its phase indicates the phase delay of x [n] with respect toy [n] for each frequency.


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Coherence Function and MSC

The normalized cross-spectrum or coherence function is defined as

γxy (e jω) =Rxy (e jω)√

Rxx(e jω)√

Ryy (e jω)

The magnitude square coherence (MSC) is the magnitude square of thecoherence function

Cxy (e jω) ≡∣∣γxy (e jω)

∣∣2 =

∣∣Rxy (e jω)∣∣2

Rxx(e jω)Ryy (e jω)

The MSC is a kind of correlation in the frequency domain. In particular ifx [n] = y [n] the γxy (e jω) = 1 (maximum correlation). On the contrary if x [n]and y [n] are not correlated then γxy (e jω) = 0. We can deduce that

0 ≤∣∣γxy (e jω)

∣∣ ≤ 1


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Summery

Definitions of the principal definitions of correlation functions are listed below:DefinitionsExpectation µx = E {x [n]}Auto-correlation rxx = E {x [n]x∗[n − k]}Auto-covariance cxx = E {(x [n]− µx) (x∗[n − k]− µx)}Cross-correlation rxy = E {x [n]y∗[n − k]}Cross-covariance cxy = E {(x [n]− µx) (y∗[n − k]− µy )}Power spectral density (PSD) Rxx(e jω) =

∑∞k=−∞ rxx [k]e−jωk

Magnitude square coherence (MSC)∣∣γxy (e jω)

∣∣2 =∣∣Rxy (e jω)

∣∣2/[Ryy (e jω)Rxx(e jω)]

Properties of the principal definitions of correlation functions are listed below:Propertiesrxx [k] is non-negative Rxx(e jω) ≥ 0 and realrxx [k] = r∗xx [−k] Rxx(e jω) = Rxx(e−jω) and x [n] ∈ R|rxx [k]| ≤ rxx [0] Rxx(z) = R∗xx(1/z∗)∣∣cx [k]

/σ2

x

∣∣ ≤ 1 Rxx(z) = Rxx(z−1) and x [n] ∈ Rrxy [k] = r∗yx [−k] Rxy (z) = R∗xy (1/z∗)

|rxy [k]| ≤√

rxx [0]ryy [0] ≤ 12

(rxx [0] + ryy [0]) 0 ≤∣∣γxy (e jω)

∣∣ ≤ 1


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Spectral representation of random sequences

Let us remember the following property on the Z-transform

Z {h[n]} = H(z)⇔ Z {h∗[−n]}=H∗(1/z∗)

Then for previous relations we have

Ryx(z) = H(z)Rxx(z)Ryy (z) = H(z)H∗(1/z∗)Rxx(z)

In particular, considering the DTFT

Ryx(e jω) = H(e jω)Rxx(e jω)

Ryy (e jω) =∣∣H(e jω)

∣∣2 Rxx(e jω)

andRyx(e jω) = R∗xy (e jω)


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Properties on the second order moments

Definitions and properties of second order moments of random sequences for

discrete-time LTI systems, are listed in the following table.Time domain Z-domain Frequency-domain

y [n] = h[n] ∗ x[n]ryx [k] = h[k] ∗ x[k]rxy [k] = h∗[−k] ∗ rxx [k]ryy [k] = h[k] ∗ rxy [k]ryy [k] = h∗[−k] ∗ h[k] ∗ rxx [k]

Y (z) = H(z)X (z)Ryx (z) = H(z)Rxx (z)Rxy (z) = H∗(1

/z∗)Rxx (z)

Ryy (z) = H(z)Rxy (z)Ryy (z) = H(z)H∗(1

/z∗)Rxx (z)

Y (ejω) = H(ejω)X (ejω)

Ryx (ejω) = H(ejω)Rx (ejω)

Rxy (ejω) = H∗(ejω)Rxx (ejω)

Ryy (ejω) = H(ejω)Rxy (ejω)

Ryy (ejω) =∣∣∣H(ejω)

∣∣∣2 Rxx (ejω)


OutlineIntroduction



White noise

The white noise (WN) is a zero-mean sequence with an auto-correlation function

r [k] = σ2xδ[k]

where δ[k] is the impulse function. Then its power spectral density is

Rxx(e jω) = σ2x

so a white noise has its spectrum constant for every frequency.The output of a LTI system when its input is a WN has the following PSD

Ryy (e jω) =∣∣H(e jω)

∣∣2 σ2x

An important WN is that with a Gaussian pdf, known as white Gaussian noise(WGN) x [n] with zero mean and variance σ2

x , denoted by x [n] ∼ N(0, σ2x). It ha

the propertyr [m − n] = E (x [n]x [m]) = 0, for m 6= n


OutlineIntroduction



White noise

The white noise can have different pdf, not only Gaussian; for example a Uniformpdf or a Cauchy pdf, ecc. The figure shows this WNs.

Remark

Differently from the Uniform distribution, the support of the Gaussian or Cauchydistribution is not limited. Then these distributions can assume any amplitudevalue, even if with low probability.


OutlineIntroduction



References

A. Papoulis.Probability, Random Variables and Stochastic Processes.III ed., McGraw-Hill, 1991.

D.G. Manolakis, V.K. Ingle, S.M. KogonStatistical and Adaptive Signal Processing.Artech House, Norwood, MA, 2005.

T. KaylathLinear Systems.Prentice Hall, Englewood Cliffs, NJ, 1980.

R.A. FisherOn the mathematical foundations of theoretical statistics.Philosophical Transactions of the Royal Society, A, 222: 309-368, 1922.


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