Fig. 14.1

process X(t). (see Fig 14.1). For example

​

​

where is a uniformly distributed random variable in

represents a stochastic process. Stochastic processes are everywhere:

Brownian motion, stock market fluctuations, various queuing systems

all represent stochastic phenomena.

If X(t) is a stochastic process, then for fixed t, X(t) represents

a random variable. Its distribution function is given by

​

​

Notice that depends on t, since for a different t, we obtain

a different random variable. Further

​

​

represents the first-order probability density function of the

process X(t).

(14-1)

(14-2)

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For t = t₁ and t = t₂, X(t) represents two different random variables

X₁ = X(t₁) and X₂ = X(t₂) respectively. Their joint distribution is

given by

​

​

​

and

​

​

​

represents the second-order density function of the process X(t).

Similarly represents the n^th order density

function of the process X(t). Complete specification of the stochastic

process X(t) requires the knowledge of

for all and for all n. (an almost impossible task

in reality).

(14-3)

(14-4)

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Mean of a Stochastic Process:

​

​

represents the mean value of a process X(t). In general, the mean of

a process can depend on the time index t.

​

Autocorrelation function of a process X(t) is defined as

​

​

and it represents the interrelationship between the random variables

X₁ = X(t₁) and X₂ = X(t₂) generated from the process X(t).

​

Properties:

​

​

2.

(14-5)

(14-6)

(14-7)

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(Average instantaneous power)

3. represents a nonnegative definite function, i.e., for any

set of constants

​

Eq. (14-8) follows by noticing that

The function

​

​

represents the autocovariance function of the process X(t).

Example 14.1

Let

​

​

Then

​

​

​

(14-9)

(14-10)

(14-8)

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Similarly

(14-12)

(14-13)

Example 14.2

(14-11)

This gives

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Stationary Stochastic Processes

Stationary processes exhibit statistical properties that are

invariant to shift in the time index. Thus, for example, second-order

stationarity implies that the statistical properties of the pairs

{X(t₁) , X(t₂) } and {X(t₁+c) , X(t₂+c)} are the same for any c.

Similarly first-order stationarity implies that the statistical properties

of X(t_i) and X(t_i+c) are the same for any c.

In strict terms, the statistical properties are governed by the

joint probability density function. Hence a process is n^th-order

Strict-Sense Stationary (S.S.S) if

​

​

for any c, where the left side represents the joint density function of

the random variables and

the right side corresponds to the joint density function of the random

variables

A process X(t) is said to be strict-sense stationary if (14-14) is

true for all

(14-14)

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For a first-order strict sense stationary process,

from (14-14) we have

​

​

for any c. In particular c = – t gives

​

​

i.e., the first-order density of X(t) is independent of t. In that case

​

​

Similarly, for a second-order strict-sense stationary process

we have from (14-14)

​

​

for any c. For c = – t₂ we get

(14-16)

(14-15)

(14-17)

(14-18)

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i.e., the second order density function of a strict sense stationary

process depends only on the difference of the time indices

In that case the autocorrelation function is given by

​

​

​

​

​

i.e., the autocorrelation function of a second order strict-sense

stationary process depends only on the difference of the time

indices

Notice that (14-17) and (14-19) are consequences of the stochastic

process being first and second-order strict sense stationary.

On the other hand, the basic conditions for the first and second order

stationarity – Eqs. (14-16) and (14-18) – are usually difficult to verify.

In that case, we often resort to a looser definition of stationarity,

known as Wide-Sense Stationarity (W.S.S), by making use of

(14-19)

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(14-17) and (14-19) as the necessary conditions. Thus, a process X(t)

is said to be Wide-Sense Stationary if

and

(ii)

i.e., for wide-sense stationary processes, the mean is a constant and

the autocorrelation function depends only on the difference between

the time indices. Notice that (14-20)-(14-21) does not say anything

about the nature of the probability density functions, and instead deal

with the average behavior of the process. Since (14-20)-(14-21)

follow from (14-16) and (14-18), strict-sense stationarity always

implies wide-sense stationarity. However, the converse is not true in

general, the only exception being the Gaussian process.

This follows, since if X(t) is a Gaussian process, then by definition

are jointly Gaussian random

variables for any whose joint characteristic function

is given by

(14-21)

(14-20)

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​

​

where is as defined on (14-9). If X(t) is wide-sense

stationary, then using (14-20)-(14-21) in (14-22) we get

​

​

​

and hence if the set of time indices are shifted by a constant c to

generate a new set of jointly Gaussian random variables

then their joint characteristic

function is identical to (14-23). Thus the set of random variables

and have the same joint probability distribution for all n and

all c, establishing the strict sense stationarity of Gaussian processes

from its wide-sense stationarity.

To summarize if X(t) is a Gaussian process, then

wide-sense stationarity (w.s.s) strict-sense stationarity (s.s.s).

Notice that since the joint p.d.f of Gaussian random variables depends

only on their second order statistics, which is also the basis

(14-22)

(14-23)

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for wide sense stationarity, we obtain strict sense stationarity as well.

From (14-12)-(14-13), (refer to Example 14.2), the process

in (14-11) is wide-sense stationary, but

not strict-sense stationary.

​

Similarly if X(t) is a zero mean wide

sense stationary process in Example 14.1,

then in (14-10) reduces to

​

​

As t₁, t₂ varies from –T to +T, varies

from –2T to + 2T. Moreover is a constant

over the shaded region in Fig 14.2, whose area is given by

​

​

and hence the above integral reduces to

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(14-24)

Systems with Stochastic Inputs

A deterministic system¹ transforms each input waveform into

an output waveform by operating only on the

time variable t. Thus a set of realizations at the input corresponding

to a process X(t) generates a new set of realizations at the

output associated with a new process Y(t).

Our goal is to study the output process statistics in terms of the input

process statistics and the system function.

​

¹A stochastic system on the other hand operates on both the variables t and

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Deterministic Systems

Systems with Memory

Time-Invariant

systems

Linear systems

Linear-Time Invariant

(LTI) systems

Memoryless Systems

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Time-varying

systems

Fig. 14.3

Memoryless Systems:

The output Y(t) in this case depends only on the present value of the

input X(t). i.e.,

(14-25)

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Theorem: If X(t) is a zero mean stationary Gaussian process, and

Y(t) = g[X(t)], where represents a nonlinear memoryless device,

then

​

​

Proof:

​

​

​

​

where are jointly Gaussian random

variables, and hence

(14-26)

(14-27)

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where L is an upper triangular factor matrix with positive diagonal

entries. i.e.,

​

​

Consider the transformation

​

​

so that

​

​

and hence Z₁, Z₂ are zero mean independent Gaussian random

variables. Also

​

​

and hence

​

​

The Jacobaian of the transformation is given by

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​

​

Hence substituting these into (14-27), we obtain

​

​

​

​

​

​

​

​

​

​

​

​

​

​

where This gives

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the desired result, where Thus if the input to

a memoryless device is stationary Gaussian, the cross correlation

function between the input and the output is proportional to the

input autocorrelation function.

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Linear Systems: represents a linear system if

​

​

Let

​

represent the output of a linear system.

Time-Invariant System: represents a time-invariant system if

​

​

i.e., shift in the input results in the same shift in the output also.

If satisfies both (14-28) and (14-30), then it corresponds to

a linear time-invariant (LTI) system.

LTI systems can be uniquely represented in terms of their output to

a delta function

(14-28)

(14-29)

(14-30)

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Eq. (14-31) follows by expressing X(t) as

​

​

and applying (14-28) and (14-30) to Thus

​

​

​

​

​

​

(14-31)

(14-32)

(14-33)

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Output Statistics: Using (14-33), the mean of the output process

is given by

​

​

​

​

Similarly the cross-correlation function between the input and output

processes is given by

​

​

​

​

​

​

​

​

Finally the output autocorrelation function is given by

(14-34)

(14-35)

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or

(14-36)

(14-37)

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Fig. 14.7

In particular if X(t) is wide-sense stationary, then we have

so that from (14-34)

​

​

Also so that (14-35) reduces to

​

​

​

​

Thus X(t) and Y(t) are jointly w.s.s. Further, from (14-36), the output

autocorrelation simplifies to

​

​

​

From (14-37), we obtain

​

​

(14-38)

(14-39)

(14-40)

(14-41)

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From (14-38)-(14-40), the output process is also wide-sense stationary.

This gives rise to the following representation

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Fig. 14.8

White Noise Process:

W(t) is said to be a white noise process if

​

​

i.e., E[W(t₁) W^*(t₂)] = 0 unless t₁ = t₂.

W(t) is said to be wide-sense stationary (w.s.s) white noise

if E[W(t)] = constant, and

​

​

If W(t) is also a Gaussian process (white Gaussian process), then all of

its samples are independent random variables (why?).

​

​

​

​

​

For w.s.s. white noise input W(t), we have

(14-42)

(14-43)

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Fig. 14.9

​

​

​

and

​

​

​

​

where

​

​

​

Thus the output of a white noise process through an LTI system

represents a (colored) noise process.

Note: White noise need not be Gaussian.

“White” and “Gaussian” are two different concepts!

(14-45)

(14-46)

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(14-44)

Upcrossings and Downcrossings of a stationary Gaussian process:

Consider a zero mean stationary Gaussian process X(t) with

autocorrelation function An upcrossing over the mean value

occurs whenever the realization X(t)

passes through zero with

positive slope. Let

represent the probability

of such an upcrossing in

the interval

We wish to determine

​

Since X(t) is a stationary Gaussian process, its derivative process

is also zero mean stationary Gaussian with autocorrelation function

(see (9-101)-(9-106), Text). Further X(t) and

are jointly Gaussian stationary processes, and since (see (9-106), Text)

Fig. 14.10

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we have

​

​

​

which for gives

​

​

i.e., the jointly Gaussian zero mean random variables

​

​

are uncorrelated and hence independent with variances

​

​

respectively. Thus

​

​

​

To determine the probability of upcrossing rate,

(14-48)

(14-47)

(14-49)

(14-50)

(14-51)

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we argue as follows: In an interval the realization moves

from X(t) = X₁ to

and hence the realization intersects with the zero level somewhere

in that interval if

​

​

i.e.,

Hence the probability of upcrossing

in is given by

​

​

​

​

Differentiating both sides of (14-53) with respect to we get

​

​

and letting Eq. (14-54) reduce to

(14-52)

(14-53)

(14-54)

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​

​

​

​

​

[where we have made use of (5-78), Text]. There is an equal

probability for downcrossings, and hence the total probability for

crossing the zero line in an interval equals where

​

​

​

It follows that in a long interval T, there will be approximately

crossings of the mean value. If is large, then the

autocorrelation function decays more rapidly as moves

away from zero, implying a large random variation around the origin

(mean value) for X(t), and the likelihood of zero crossings should

increase with increase in agreeing with (14-56).

(14-55)

(14-56)

Discrete Time Stochastic Processes:

A discrete time stochastic process X_n = X(nT) is a sequence of

random variables. The mean, autocorrelation and auto-covariance

functions of a discrete-time process are gives by

​

​

​

and

​

​

respectively. As before strict sense stationarity and wide-sense

stationarity definitions apply here also.

For example, X(nT) is wide sense stationary if

​

​

and

(14-57)

(14-58)

(14-59)

(14-60)

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(14-61)

i.e., R(n₁, n₂) = R(n₁ – n₂) = R^*(n₂ – n₁). The positive-definite

property of the autocorrelation sequence in (14-8) can be expressed

in terms of certain Hermitian-Toeplitz matrices as follows:

Theorem: A sequence forms an autocorrelation sequence of

a wide sense stationary stochastic process if and only if every

Hermitian-Toeplitz matrix T_n given by

​

​

​

​

​

​

is non-negative (positive) definite for

Proof: Let represent an arbitrary constant vector.

Then from (14-62),

​

​

since the Toeplitz character gives Using (14-61),

Eq. (14-63) reduces to

(14-62)

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(14-63)

​

​

​

​

From (14-64), if X(nT) is a wide sense stationary stochastic process

then T_n is a non-negative definite matrix for every

Similarly the converse also follows from (14-64). (see section 9.4, Text)

​

If X(nT) represents a wide-sense stationary input to a discrete-time

system {h(nT)}, and Y(nT) the system output, then as before the cross

correlation function satisfies

​

and the output autocorrelation function is given by

​

or

​

Thus wide-sense stationarity from input to output is preserved

for discrete-time systems also.

(14-64)

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(14-65)

(14-66)

(14-67)

Auto Regressive Moving Average (ARMA) Processes

​

Consider an input – output representation

​

​

​

where X(n) may be considered as the output of a system {h(n)}

driven by the input W(n).

Z – transform of

(14-68) gives

​

​

​

or

​

​

​

(14-68)

(14-69)

(14-70)

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Fig.14.12

represents the transfer function of the associated system response {h(n)}

in Fig 14.12 so that

​

​

Notice that the transfer function H(z) in (14-70) is rational with p poles

and q zeros that determine the model order of the underlying system.

From (14-68), the output undergoes regression over p of its previous

values and at the same time a moving average based on

of the input over (q + 1) values is added to it, thus

generating an Auto Regressive Moving Average (ARMA (p, q))

process X(n). Generally the input {W(n)} represents a sequence of

uncorrelated random variables of zero mean and constant variance

so that

​

If in addition, {W(n)} is normally distributed then the output {X(n)}

also represents a strict-sense stationary normal process.

If q = 0, then (14-68) represents an AR(p) process (all-pole

process), and if p = 0, then (14-68) represents an MA(q)

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(14-72)

(14-71)

process (all-zero process). Next, we shall discuss AR(1) and AR(2)

processes through explicit calculations.

AR(1) process: An AR(1) process has the form (see (14-68))

​

​

and from (14-70) the corresponding system transfer

​

​

​

provided | a | < 1. Thus

​

​

represents the impulse response of an AR(1) stable system. Using

(14-67) together with (14-72) and (14-75), we get the output

autocorrelation sequence of an AR(1) process to be

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(14-73)

(14-75)

(14-74)

(14-76)

where we have made use of the discrete version of (14-46). The

normalized (in terms of R_XX (0)) output autocorrelation sequence is

given by

​

​

It is instructive to compare an AR(1) model discussed above by

superimposing a random component to it, which may be an error

term associated with observing a first order AR process X(n). Thus

​

​

where X(n) ~ AR(1) as in (14-73), and V(n) is an uncorrelated random

sequence with zero mean and variance that is also uncorrelated

with {W(n)}. From (14-73), (14-78) we obtain the output

autocorrelation of the observed process Y(n) to be

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(14-78)

(14-77)

(14-79)

so that its normalized version is given by

​

​

​

where

​

​

Eqs. (14-77) and (14-80) demonstrate the effect of superimposing

an error sequence on an AR(1) model. For non-zero lags, the

autocorrelation of the observed sequence {Y(n)}is reduced by a constant

factor compared to the original process {X(n)}.

From (14-78), the superimposed

error sequence V(n) only affects

the corresponding term in Y(n)

(term by term). However,

a particular term in the “input sequence”

W(n) affects X(n) and Y(n) as well as

all subsequent observations.

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(14-80)

(14-81)

Fig. 14.13

AR(2) Process: An AR(2) process has the form

​

​

and from (14-70) the corresponding transfer function is given by

​

​

​

so that

​

​

and in term of the poles of the transfer function,

from (14-83) we have

​

​

that represents the impulse response of the system.

From (14-84)-(14-85), we also have

From (14-83),

​

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(14-82)

(14-83)

(14-84)

(14-85)

(14-86)

and H(z) stable implies

Further, using (14-82) the output autocorrelations satisfy the recursion

​

​

​

​

​

and hence their normalized version is given by

​

​

By direct calculation using (14-67), the output autocorrelations are

given by

​

​

​

​

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(14-88)

(14-87)

(14-89)

where we have made use of (14-85). From (14-89), the normalized

output autocorrelations may be expressed as

​

​

where c₁ and c₂ are appropriate constants.

Damped Exponentials: When the second order system in

(14-83)-(14-85) is real and corresponds to a damped exponential

response, the poles are complex conjugate which gives

in (14-83). Thus

​

​

In that case in (14-90) so that the normalized

correlations there reduce to

​

​

But from (14-86)

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(14-90)

(14-91)

(14-92)

(14-93)

and hence which gives

​

​

​

Also from (14-88)

​

​

so that

​

​

​

where the later form is obtained from (14-92) with n = 1. But

in (14-92) gives

​

​

Substituting (14-96) into (14-92) and (14-95) we obtain the normalized

output autocorrelations to be

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(14-94)

(14-95)

(14-96)

(14-105)

(14-106)