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Discrete-Time Stationary Stochastic Processes

Eric Dubois

School of Electrical Engineering and Computer ScienceUniversity of Ottawa

September 2012

Eric Dubois (EECS) Discrete-Time Stationary Stochastic Processes September 2012 1 / 26

Definitions and Assumptions

We consider only discrete-time signals with a uniform sampling rate.Continuous-time source signals are assumed to have been sampledabove the Nyquist rate of twice the highest frequency in the signal.The sampling instants are nT for n ∈ Z.

An observed signal is denoted u(n) with either u(n) ∈ C or u(n) ∈ Ras appropriate.

The observed signal is assumed to be a realization of a stochasticprocess, which is a collection of random variables U(n) for n ∈ Z.

We can characterize a stochastic process by all the joint probabilitydensities of the form

p(u(n1), u(n2), . . . , u(nK ))

at K arbitrary time instants, and for all K .


Stationarity

The stochastic process is strictly stationary if

p(u(n1), u(n2), . . . , u(nK )) = p(u(n1 + m), u(n2 + m), . . . , u(nK + m))

for all K , all {n1, n2, . . . , nK} and for all m.


First and Second Order Moments – Definitions

Mean function

µ(n) = E[U(n)] =

∫ ∞−∞

upun(u) du

Autocorrelation function

r ′(n, n − k) = E[U(n)U∗(n − k)] =

∫ ∞−∞

∫ ∞−∞

uv∗pun,un−k(u, v) dudv

Autocovariance function

c ′(n, n − k) = E[(U(n)− µ(n))(U(n − k)− µ(n − k))∗]

= r ′(n, n − k)− µ(n)µ∗(n − k)

Autocorrelation and autocovariance are the same for zero-meanprocesses.


Moments for stationary processes

If the stochastic process is strictly stationary, the expressions for themoments are simplified.

For the mean function, pun(u) is independent of n, so

µ(n) = µ for all n

For the autocorrelation and autocovariance functions, pun,un−k(u, v) is

independent of n, and so r ′(n, n − k) depends only on k . Similarly forc ′(n, n − k). The we define

r(k) = r ′(n, n − k) independent of n

c(k) = c ′(n, n − k)


Moments for stationary processes (2)

Note that

r(0) = E[|U(n)|2]

c(0) = E[|U(n)− µ|2] = σ2u

r(−k) = E[U(n)U∗(n + k)] = E[U(n − k)U∗(n)]

= r∗(k)

A stochastic process for which µ(n) = µ and r ′(n, n − k) = r(k)independently of n is called wide-sense stationary.


Ergodicity

Expectation is an average over an ensemble of realizations of arandom process.

Often we want to estimate the properties or parameters of astochastic process by looking at a single realization over a long time.

This works if the process is ergodic

Define the following time average for a particular realization

µ̂(N) =1

N

N−1∑=0

u(n)

µ̂(N) is a random variable, with a mean and a variance.

E[µ̂(N)] =1

N

N−1∑=0

E[U(n)] = µ


Ergodicity (2)

The variance of the time average is

var(µ̂(N)) = E[|µ̂(N)− µ|2]

We say that the process u(n) is mean-ergodic in the mean-squaresense if

limN→∞

var(µ̂(N)) = 0

From the definition, we find that

var(µ̂(N)) =1

N2

N−1∑n=0

N−1∑k=0

c(n − k)

Noting that n − k takes on values from −(N − 1) to N − 1, and that0 occurs N times, ±1 occurs N − 1 times, etc., this can be written

var(µ̂(N)) =1

N

N−1∑`=−(N−1)

(1− |`|

N

)c(`)


Ergodicity (3)

Thus, we need

limN→∞

1

N

N−1∑`=−(N−1)

(1− |`|

N

)c(`) = 0

i.e., the process is asymptotically incorrelated.

Similarly, the process is correlation uncorrelated if

limN→∞

var(r̂(k,N)) = 0

where

r̂(k ,N) =1

N

N−1∑n=0

u(n)u∗(n − k), 0 ≤ k ≤ N − 1

Note thatE[̂r(k ,N)] = r(k), for all N


Summary of Correlation Functions and Spectral Densities

Autocorrelation

ru(k) = E[U(n)U∗(n − k)] = r∗u (−k)

Su(e jω) =∞∑

k=−∞ru(k)e−jωk − π ≤ ω ≤ π

Su(z) =∞∑

k=−∞ru(k)z−k a < |z | < b

Cross-correlation

ruy (k) = E[U(n)Y ∗(n − k)] = r∗yu(−k)

Suy (e jω) =∞∑

k=−∞ruy (k)e−jωk − π ≤ ω ≤ π

Suy(z) =∞∑

k=−∞ruy (k)z−k z ∈ ROC


Some z-transform Properties

If g(k) = f (−k), then G (z) = F (1z ), and G (e jω) = F (e−jω).

If g(k) = f ∗(k), then G (z) = F ∗(z∗) and G (e jω) = F ∗(e−jω).

For autocorrelations and cross correlations

Su(z) = S∗u (1

z∗) Su(e jω) = S∗u (e jω) real

Suy (z) = S∗yu(1

z∗) Suy (e jω) = S∗yu(e jω)


Linear Filtering of Stationary Processes

y(n) = h(n) ∗ u(n)

ry (n) = ru(n) ∗ h(n) ∗ h∗(−n)

Sy (z) = Su(z)H(z)H∗( 1z∗ )

S(e jω) = Su(e jω)H(e jω)H∗(e jω) = Su(e jω)|H(e jω)|2

ryu(n) = h(n) ∗ ru(n) (prove it).

Syu(z) = H(z)Su(z)

Syu(e jω) = H(e jω)Su(e jω)


General Linear ProcessA general linear process is filtered zero-mean white noise

u(n) =∞∑k=0

g∗(k)v(n − k)

∞∑n=0

|g(n)|2 <∞.

Condition is required for the process to have a finite variance:

σ2u = E

[ ∞∑k=0

g∗(k)v(n − k)∞∑`=0

g(`)v∗(n − `)

]

=∞∑k=0

∞∑`=0

g∗(k)g(`) rv (`− k)︸︷︷︸σ2v δ(`−k)

= σ2v

∞∑k=0

|g(k)|2


General Linear Process (2)

The filter is a causal, linear shift-invariant filter HG (z) where

HG (z) =∞∑k=0

g∗(k)z−k is analytic for |z | > 1.

The power density spectrum of the process is

Su(ω) = σ2v |HG (e jω)|2.

By the Cauchy-Schwartz inequality

|c(k)| = |〈U(n) | U(n − k)〉|

≤√‖U(n)‖2‖U(n − k)‖2

= σ2u <∞


Modeling power of the general linear process

Theorem: Let S(ω) be the PDS of a WSS random process. If∫ π

−πln(S(ω)) dω > −∞ (∗)

then there exists a unique sequence g(0), g(1), g(2), . . . with g(0)real and positive, and with

∞∑n=0

|g(n)|2 <∞

such that HG (z) has no zeros in |z | > 1 and such that

S(ω) = |HG (e jω)|2


Modeling power of the general linear process (2)

Note that ln(S(ω)) < S(ω) implies that

1

2π

∫ π

−πln(S(ω)) dω <

1

2π

∫ π

−πS(ω) dω

= σ2u <∞

Condition (*) cannot be satisfied for a bandlimited process.


Stricter condition on the power density spectrum

A stricter condition on the PDS is

ln(S(z)) is analytic for ρ < |z | < 1

ρfor some ρ < 1 (+)

If (+) is satisfied, then

ln(S(z)) =∞∑

n=−∞d(n)z−n for ρ < |z | < 1

ρ

Thus

S(z) = exp

( ∞∑n=−∞

d(n)z−n

)

= ed(0) exp

( −1∑n=−∞

d(n)z−n

)exp

( ∞∑n=1

d(n)z−n

)


Stricter condition on the power density spectrum (2)

Since ln(S(ω)) is real, d(−n) = d∗(n). Let

HG (z) = exp

(d(0)

2+∞∑n=1

d(n)z−n

)

Then

HG (z)H∗G (1

z∗) = ed(0) exp

( ∞∑n=1

d(n)z−n

)exp

( ∞∑n=1

d∗(n)zn

)= S(z)

Thus |HG (e jω)|2 = S(e jω)


Stricter condition on the power density spectrum (2)

We can observe that

d(0)

2+∞∑n=1

d(n)z−n is analytic for |z | > ρ

HG (z) is analytic for |z | > ρ

HG (z) =∞∑n=0

g∗(n)z−n a right-sided sequence

Similarly

1

HG (z)= exp

(−d(0)

2−∞∑n=1

d(n)z−n

)is analytic for |z | > ρ.

Thus HG (z) is a minimum phase filter.


General linear process analysis-synthesis

Sv (ω) =1

|HG (e jω)|2Su(ω) = 1

Thus v is white noise, and

u(n) =∞∑k=0

g∗(k)v(n − k)

u is a general linear process.


Autoregressive (AR) processes – review

The autoregressive AR(M) process is a general linear process with

HG (z) =1∑M

k=0 a∗kz−k

=1

HA(z)a0 = 1

We write equivalently

HA(z) = 1−M∑k=1

w∗k z−k wk = −ak

The process can be viewed as the output of the difference equation

u(n) =M∑k=1

w∗k u(n − k) + v(n)

where v(n) is a zero-mean white noise process with variance σ2v .


Autoregressive (AR) processes – PDS and autocorrelation

The power density spectrum of the process is given by

Su(ω) =σ2v∣∣∣1−∑M

k=1 w∗k e−jωk

∣∣∣2The autocorrelation function satisfies the difference equation

M∑k=1

wk ru(k − `) = r∗u (`) ` = 1, 2 . . . ,M


Autoregressive (AR) processes – Yule-Walker Equations

In matrix forms, these equations, known as the Yule-Walkerequations, can be written

ru(0) ru(1) · · · ru(M − 1)r∗u (1) ru(0) · · · ru(M − 2)

......

...r∗u (M − 1) r∗u (M − 2) · · · ru(0)

w1

w2...

wM

=

r∗u (1)r∗u (2)

...r∗u (M)

Ruw = ru

The correlation matrix is a positive definite Hermitian matrix.

The variance of the white noise satisfies

σ2v = σ2u −M∑k=1

wk ru(k)


Autoregressive AR(1) process

The AR(1) process has the following form

HG (z) =1

1− w∗1 z−1 |w1| < 1

The Yule-Walker equation is

ru(0)w1 = r∗u (1)

If ru(0) = σ2u, then ru(1) = σ2uw∗1 .

The autocorrelation function is

r(`) = (w∗1 )`σ2u

r(−`) = w `1σ

2u ` ≥ 0

The variance of the white noise is given by

σ2v = σ2u(1− |w1|2)


AR(4) Process

u(n) = 2.7607u(n−1)−3.8106u(n−2)+2.6535u(n−3)−0.9238u(n−4)+v(n)


AR(4) Process (2)


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