Stochastic Processes - part 1

Let ξ denote the random outcome of an experiment. To ev-
ery such outcome suppose a waveform X(t, ξ) is assigned.
The collection of such waveforms form a stochastic process.
The set of {ξk} and the time index t can be continuous or
discrete (countably infinite or finite) as well. For fixed ξk ∈ S
(the set of all experimental outcomes), X(t, ξ) is a specific
time function. For fixed t, X1 = X(t1, ξi) is a random vari-
able.
AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.1/105

The ensemble of all such realizations X(t, ξ) over time
represents the stochastic process X(t). For example
X(t) = a cos(ω0t + φ), φ ∼ U(0, 2π)
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) repre-
sents a random variable.

Its distribution function is given by
FX(x, t) = P{X(t) ≤ x}
Notice that FX(x, t) depends on t, since for a different t, we
obtain a different random variable. Further
fX(x, t) =
dFX(x, t)
dx

FX(x1, x2, t1, t2) = P{X(t1) ≤ x1, X(t2) ≤ x2}
fX(x1, x2, t1, t2) =
∂2
FX(x1, x2, t1, t2)
∂x1∂x2
represents the 2nd order density function of the process
X(t).

Similarly fX(x1, x2, · · · xn, t1, t2 · · · , tn)represents the nth or-
der density function of the process X(t). Complete specifica-
tion of the stochastic process X(t) requires the knowledge
of fX(x1, x2, · · · xn, t1, t2 · · · , tn) for all ti, i = 1, 2, · · · , n
and for all n, an almost impossible task in reality.

µ(t) = E{X(t)} =
Z +∞
−∞
xfX(x, t)dx
In general, the mean of a process can depend on the time
index t.
RXX(t1, t2)=E{X(t1)X∗
(t2)}=
Z Z
x1x∗
2fX(x1, x2, t1, t2)dx1dx2
It represents the interrelationship between the random vari-
ables X1 = X(t1) and X2 = X(t2) generated from the pro-
cess X(t).

1. RXX(t1, t2) = R∗
XX(t2, t1) = [E{X(t2)X∗
(t1)}]∗
2. RXX(t, t) = E{|X(t)|2
} > 0.
3.
n
P
i=1
n
P
j=1
aia∗
j RXX(ti, tj) ≥ 0.
CXX(t1, t2) = RXX(t1, t2) − µX(t1)µ∗
X(t2)

Example:
z =
Z T
−T
X(t)dt
E[|z|2
] =
Z T
−T
Z T
−T
E{X(t1)X∗
(t2)}dt1dt2
=
Z T
−T
Z T
−T
RXX(t1, t2)dt1dt2

Example:
X(t) = a cos(ω0t + ϕ), ϕ ∼ U(0, 2π).
µX(t) = E{X(t)} = aE{cos(ω0t + ϕ)}
= a cos ω0tE{cos ϕ} − a sin ω0tE{sin ϕ} = 0,
E{cos ϕ} = 1
2π
Z 2π
0
cos ϕdϕ = 0 = E{sin ϕ}.
RXX(t1, t2) = a2
E{cos(ω0t1 + ϕ) cos(ω0t2 + ϕ)}
= a2
2
E{cos ω0(t1 − t2) + cos(ω0(t1 + t2) + 2ϕ)}
= a2
2
cos ω0(t1 − t2).

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 proper-
ties 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(ti) and X(ti + c) are the
same for any c.

Strict-Sense Stationary (S.S.S)
if
fX(x1, · · · xn, t1, · · · , tn) ≡ fX(x1, · · · xn, t1 +c, · · · , tn +c) (1)
for any c, where the left side represents the joint density
function of the random variables

X1 = X(t1), X2 = X(t2), · · · , Xn = X(tn) and the right side
corresponds to the joint density function of the random
variables
X′
1 = X(t1 + c), X′
2 = X(t2 + c), · · · , X′
n = X(tn + c).
A process X(t) is said to be strict-sense stationary if (1) is
true for all ti, i = 1, 2, · · · , n, n = 1, 2, · · · , for any c.

For a first-order strict sense stationary process,
fX(x, t) ≡ fX(x, t + c)
for c = −t,
fX(x, t) = fX(x) (2)
the first-order density of X(t) is independent of t. In that
case
E[X(t)] =
Z +∞
−∞
xf(x)dx = µ, a constant.

Similarly, for a 2nd-order strict-sense stationary process
fX(x1, x2, t1, t2) ≡ fX(x1, x2, t1 + c, t2 + c)
For c = −t2:
fX(x1, x2, t1, t2) ≡ fX(x1, x2, t1 − t2) (3)

the second order density function of a strict sense
stationary process depends only on the difference of the
time indices t2 − t2 = τ. In that case the autocorrelation
function is given by
E{X(t1)X∗
(t2)} =
Z Z
x1x∗
2fX(x1, x2, τ = t1 − t2)dx1dx2
= RXX(t1 − t2) = RXX(τ) = R∗
XX(−τ),
the E{X(t1)X∗
(t2)} of a 2nd SSS depends only on τ.

The conditions for 1st and 2nd order SSS are in equations
(2), (3), these are usually hard to verify. For this reason we
resort to wide-sense stationarity WSS.

WSS:
1. E{X(t)} = µ
2. E{X(t1)X∗
(t2)} = RXX(|t2 − t1|)
SSS always implies WSS, but the converse is not true, but,
the only exception are Gaussian processes.

This follows, since if X(t) is a Gaussian process, then by
definition
X1 = X(t1), X2 = X(t2), · · · , Xn = X(tn)
are jointly Gaussian random variables for any t1, t2 · · · , tn
whose joint characteristic function is given by

φX(ω1, ω2, · · · , ωn) = e
j
n
P
k=1
µ(tk)ωk−0.5
n
P
l=1
n
P
k=1
CXX (tl,tk)ωlωk
CXX(ti, tk) is the element of covariance matrix, for WSS
process CXX(ti, tk) = CXX(ti − tk)
φX(ω1, ω2, · · · , ωn) = e
j
n
P
k=1
µωk−0.5
n
P
l=1
n
P
k=1
CXX (tl−tk)ωlωk
(4)

and hence if the set of time indices are shifted by a
constant c to generate a new set of jointly Gaussian
random variables X′
2 = X(t2 + c), · · · ,X′
n = X(tn + c) then
their joint characteristic function is identical to (4). Thus the
set of random variables {Xi}n
i=1 and {X′
i}n
i=1 have the
same JPDF for all n and all c, establishing the SSS of
Gaussian processes from it WSS.
For Gaussian processes: WSS⇒SSS

Why GPs?
1. Easy prior assumption
2. Regression
3. Classification
4. Optimization
5. Unimodal

N-dim Gaussian
e−0.5(x−µ)H C−1(x−µ)
p
(2π)N |C|

N-dim Gaussian
C=[.5,-.15;-.15,.2]; mu=[1 -1];
[X1,X2] = meshgrid(linspace(-2,2,100)’,
linspace(-2,2,100)’);
X = [X1(:) X2(:)];
p = mvnpdf(X, mu, C);
surf(X1,X2,reshape(p,100,100));
contour(X1,X2,reshape(p,100,100));

−2
−1
0
1
2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
2D Gaussian PDF, surface level plot
X
2
X1
PDF
Figure 1: Multivariate Gaussian PDF.

2D Gaussian PDF, contour plot
X1
X
2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2

C = V DV H
, V = [V1 V2]
C =

0.5 −.15
−0.15 0.2

V =

0.31625 −.9487
−0.9487 0.3162

D =

0.05 0
0 0.55


.
. σ2
2
V2
σ
1
2
V
1
X1
X
2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2

For LTI sytems:
Y (t) =
Z +∞
−∞
h(t − τ)X(τ)dτ
=
Z +∞
−∞
h(τ)X(t − τ)dτ
X(t) =
Z +∞
−∞
X(τ)δ(t − τ)dτ

Y (t) = L{X(t)} = L{
Z +∞
−∞
X(τ)δ(t − τ)dτ}
=
Z +∞
−∞
L{X(τ)δ(t − τ)dτ}
=
Z +∞
−∞
X(τ)L{δ(t − τ)}dτ
=
Z +∞
−∞
X(τ)h(t − τ)dτ =
Z +∞
−∞
h(τ)X(t − τ)dτ

Output statistics:
µY (t) = E{Y (t)} =
Z +∞
−∞
E{X(τ)h(t − τ)dτ}
=
Z +∞
−∞
µX(τ)h(t − τ)dτ = µX(t) ∗ h(t).

RXY (t1, t2) = E{X(t1)Y ∗
(t2)}
= E{X(t1)
Z +∞
−∞
X∗
(t2 − α)h∗
(α)dα}
=
Z +∞
−∞
E{X(t1)X∗
(t2 − α)}h∗
(α)dα
=
Z +∞
−∞
RXX(t1, t2 − α)h∗
(α)dα
= RXX(t1, t2) ∗ h∗
(t2)

RY Y (t1, t2) = E{Y (t1)Y ∗
(t2)}
= E{
Z +∞
−∞
X(t1 − β)h(β)dβY ∗
(t2)}
=
Z +∞
−∞
E{X(t1 − β)Y ∗
(t2)}h(β)dβ
=
Z +∞
−∞
RXY (t1 − β, t2)h(β)dβ
= RXY (t1, t2) ∗ h(t1),

RY Y (t1, t2) = RXX(t1, t2) ∗ h∗
(t2) ∗ h(t1).

If X(t) is WSS:
µY (t) = µX
Z +∞
−∞
h(τ)dτ = µXc, a constant .
RXY (t1, t2) =
Z +∞
−∞
RXX(t1 − t2 + α)h∗
(α)dα
= RXX(τ) ∗ h∗
(−τ) = RXY (τ), τ = t1 − t2

RY Y (t1, t2) =
Z +∞
−∞
RXY (t1 − β − t2)h(β)dβ, τ = t1 − t2
= RXY (τ) ∗ h(τ) = RY Y (τ)
RY Y (τ) = RXX(τ) ∗ h∗
(−τ) ∗ h(τ).

Example: If X(t) is WSS:
z =
Z T
−T
X(t)dt.
Y (t) = X(t) ∗ h(t), h(t) = u(t + T) − u(t − T)
RY Y (τ) = RXX(τ) ∗ h(τ) ∗ h(−τ)
h(τ) ∗ h(−τ) = (2T − |τ|), |τ| ≤ 2T

Y (t) = g{X(t)}
SSS input SSS output
Memoryless
system

Y (t) = g{X(t)}
WSS input Can’t say about
stationarity
in any sense
Memoryless
system

Y (t) = g{X(t)}
Gaussian input Not Gaussian
Memoryless
system

If X(t) is a zero mean stationary Gaussian process, and
Y (t) = g{X(t)}, where g(·) represents a nonlinear
memoryless device, then
RXY (τ) = ηRXX(τ), η = E{g′
(X)}.
Proof:
RXY (τ) = E{X(t)Y (t − τ)} = E[X(t)g{X(t − τ)}]
=
Z Z
x1g(x2)fX1X2 (x1, x2)dx1dx2, (5)

where X1 = X(t) and X2 = X(t − τ) are jointly Gaussian
RVs.
fX1X2 (x1, x2) = 1
2π
√
|A|
e−x∗A−1x/2
X = (X1, X2)T
, x = (x1, x2)T
A = E{X X∗
} =

RXX(0) RXX(τ)
RXX(τ) RXX(0)

= LL∗

where L is an upper triangular factor matrix with positive
diagonal entries.
L =

ℓ11 ℓ12
0 ℓ22

.
Consider the transformation:
Z = L−1
X = (Z1, Z2)T
, z = L−1
x = (z1, z2)T
so that
E{Z Z∗
} = L−1
E{XX∗
}L∗−1
= L−1
AL∗−1
= I

and hence Z1, Z2 are zero mean independent Gaussian
random variables. Also
x = Lz ⇒ x1 = ℓ11z1 + ℓ12z2, x2 = ℓ22z2
and hence,
x∗
A−1
x = z∗
L∗
A−1
Lz = z∗
z = z2
1 + z2
2
then the Jacobian of transformation
|J| = |L−1
| = |A|−1/2
.

RXY (τ) =
Z +∞
−∞
Z +∞
−∞
(ℓ11z1 + ℓ12z2)g(ℓ22z2) 1
|J|
1
2π|A|1/2 e−z2
1/2
e−z2
2/2
= ℓ11
Z +∞
−∞
Z +∞
−∞
z1g(ℓ22z2)fz1 (z1)fz2 (z2)dz1dz2
+ ℓ12
Z +∞
−∞
Z +∞
−∞
z2g(ℓ22z2)fz1 (z1)fz2 (z2)dz1dz2
= ℓ11
Z +∞
−∞
z1fz1 (z1)dz1
Z +∞
−∞
g(ℓ22z2)fz2 (z2)dz2

+ ℓ12
Z +∞
−∞
z2g(ℓ22z2) fz2 (z2)
| {z }
1
√
2π
e−z2
2/2
dz2
= ℓ12
ℓ2
22
Z +∞
−∞
ug(u) 1
√
2π
e−u2/2ℓ2
22 du
where u = ℓ22z2

This gives
RXY (τ) = ℓ12ℓ22
R +∞
−∞
g(u) u
ℓ2
22
fu(u)
z }| {
1
√
2πℓ2
22
e−u2/2ℓ2
22
| {z }
−
dfu(u)
du
=−f′
u(u)
du
= −RXX(τ)
R +∞
−∞
g(u)f′
u(u)du,
Since A = LL∗
gives ℓ12ℓ22 = RXX(τ). Hence,

RXY (τ) = RXX(τ){−g(u)fu(u)|+∞
−∞ +
R +∞
−∞
g′
(u)fu(u)du}
= RXX(τ)E{g′
(X)} = ηRXX(τ),
This is the desired result, where η = E{g′
(X)}. 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.

WSS input WSS output
LTI system
h(t)

SSS input SSS output
LTI system
h(t)

Gaussian input Gaussian output
LTI system
h(t)

White noise W(t) is said to be a white noise process if
RWW (t1, t2) = q(t1)δ(t2 − t1)
W(t) is said to be WSS white noise if E{W(t)} = constant,
and
RWW (t1, t2) = qδ(t1 − t2) = qδ(τ).

If W(t) is also a Gaussian process (white Gaussian pro-
cess), then all of its samples are independent random vari-
ables(Why?)

White noise Colored noise
W(t) N(t) = h(t) ∗ W(t)
LTI system
h(t)

E[N(t)] = µW
Z +∞
−∞
h(τ)dτ = a constant
RNN (τ) = qδ(τ) ∗ h∗
(−τ) ∗ h(τ)
= qh∗
(−τ) ∗ h(τ) = qρ(τ)
ρ(τ) = h(τ) ∗ h∗
(−τ) =
Z +∞
−∞
h(α)h∗
(α + τ)dα.

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!

♦ ♦
♦

X(t)
t
Figure 4: Upcrossing (♦), Downcrossing ()

Upcrossings and Downcrossings of a stationary Gaussian pro-
cess: Consider a zero mean stationary Gaussian process
X(t) with autocorrelation function RXX(τ). An upcrossing
over the mean value occurs whenever the realization X(t)
passes through zero with positive slope. Let ρ∆t represent
the probability of such an upcrossing in the interval (t, t+∆t)
We wish to determine ρ.

Since X(t) is a stationary Gaussian process, its derivative
process X′
(t) is also zero mean stationary Gaussian with
autocorrelation function
RXX(τ) =
Z ∞
−∞
SXX(ω)ejωτ dω
2π
R
′′
XX(τ) =
Z ∞
−∞
(jω)2
SXX(ω)ejωτ dω
2π
SX′X′ (ω) = |jω|2
SXX(ω) ⇒
RX′X′ (τ) = −R
′′
XX(τ)

Further X(t) and X′
(t) and are jointly Gaussian stationary
processes, and since
RXX′ (τ) = −
dRXX(τ)
dτ
,
RXX′ (−τ) = −
dRXX(−τ)
d(−τ)
=
dRXX(τ)
dτ
= −RXX′ (τ)
which for τ = 0 gives
RXX′ (0) = 0 ⇒ E[X(t)X′
(t)] = 0

the jointly Gaussian zero mean random variables
X1 = X(t), X2 = X′
(t)
are uncorrelated hence independent with variances
σ2
1 = RXX(0) and σ2
2 = RX′X′ (0) = −R′′
XX(0) 0
fX1X2 (x1, x2) = fX(x1)fX(x2) =
1
2πσ1σ2
e
−
x2
1
2σ2
1
+
x2
1
2σ2
2
!
.
To determine ρ the probability of upcrossing rate,

we argue as follows: In an interval (t, t + ∆t) the realization
moves from X(t) = X1 to
X(t + ∆t) = X(t) + X′
(t)∆t = X1 + X2∆t, and hence the
realization intersects with the zero level somewhere in that
interval if
X1 0, X2 0, and X(t + ∆t) = X1 + X2∆t 0
i.e., X1 −X2∆t Hence the probability of upcrossing in
(t, t + ∆t) is given by

ρ∆t =
Z ∞
x2=0
Z 0
x1=−x2∆t
fX1X2 (x1, x2)dx1dx2
=
Z ∞
0
fX2 (x2)dx2
Z ∞
−x2∆t
fX1 (x1)dx1. (6)
Differentiating both sides of 6 with respect to ∆t we get
ρ =
Z ∞
0
fX2 (x2)x2fX1 (−x2∆t)dx2
and letting ∆t → 0, we have

c
ρ =
Z ∞
0
x2fX(x2)fX(0)dx2 =
1
p
2πRXX(0)
Z ∞
0
x2fX(x2)dx2
=
1
p
2πRXX(0)
1
2
(σ2
p
2/π) =
1
2π
s
−R′′
XX(0)
RXX(0)
There is an equal probability for downcrossings, and hence
the total probability for crossing the zero line in an interval
(t, t + ∆t) equals ρ0∆t where

ρ0 =
1
π
q
−R′′
XX(0)/RXX(0) 0.

A discrete time stochastic process Xn = X(nT) is a
sequence of random variables. The mean, autocorrelation
and auto-covariance functions of a discrete-time process
are gives by
µn = E{X(nT)}
R(n1, n2) = E{X(n1T)X∗
(n2T)}
C(n1, n2) = R(n1, n2) − µn1 µ∗
n2

WSS:
E{X(nT)} = µ, a constant
E[X{(k + n)T}X∗
{(k)T}] = R(n) = rn = r∗
−n

The positive-definite property of the autocorrelation
sequence can be expressed in terms of certain
Hermitian-Toeplitz matrices as follows:
Theorem: A sequence {rn} |∞
−∞ forms an autocorrelation se-
quence of a wide sense stationary stochastic process if and
only if every Hermitian-Toeplitz matrix Tn given by

Tn =





r0 r1 r2 · · · rn
r∗
1 r0 r1 · · · rn−1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
r∗
n r∗
n−1 · · · r∗
1 r0





= T∗
n
is non-negative (positive) definite for n = 0, 1, 2 · · · , ∞

a∗
Tna =
n
X
i=0
n
X
k=0
aia∗
krk−i, ∀~
a
a∗
Tna =
n
X
i=0
n
X
k=0
aia∗
kE{X(kT)X∗
(iT)}
= E




2



≥ 0.

Systems and discrete WSS processes:
RXY (n) = RXX(n) ∗ h∗
(−n)
RY Y (n) = RXY (n) ∗ h(n)
RY Y (n) = RXX(n) ∗ h∗
(−n) ∗ h(n).

ARMA processes: input: W(n), output: X(n)
X(n) = −
p
X
k=1
akX(n − k) +
q
X
k=0
bkW(n − k),
X(z)
p
X
k=0
akz−k
= W(z)
q
X
k=0
bkz−k
, a0 ≡ 1

H(z) =
∞
X
k=0
h(k)z−k
=
X(z)
W(z)
=
b0 + b1z−1
+ b2z−2
+ · · · + bqz−q
1 + a1z−1 + a2z−2 + · · · + apz−p
=
B(z)
A(z)
X(n) =
∞
X
k=0
h(n − k)W(k)

the transfer function H(z) is rational with p poles and q zeros
that determine the model order of the underlying system.
X(n) undergoes regression over p of its previous values and
at the same time a moving average based on W(n), W(n −
1), · · · , W(n − q) 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 σ2
W so that
RWW (n) = σ2
W δ(n).
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 we have an AR(p) process (all-pole process),
and if p = 0, then an MA(q)

AR(1) process: An AR(1) process has the form
X(n) = aX(n − 1) + W(n)
H(z) =
1
1 − az−1
=
∞
X
n=0
an
z−n
h(n) = an
, |a| 1

RXX(n) = σ2
W δ(n) ∗ {a−n
} ∗ {an
} = σ2
W
∞
X
k=0
a|n|+k
ak
= σ2
W
a|n|
1 − a2

The normalized autocorrelation sequence:
ρX(n) =
RXX(n)
RXX(0)
= a|n|
, |n| ≥ 0. (7)

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
Y (n) = X(n) + V (n), X(n) ∼ AR(1), V (n) ∼ White noise (8)

RY Y (n) = RXX(n) + RV V (n) = RXX(n) + σ2
V δ(n)
= σ2
W
a|n|
1 − a2
+ σ2
V δ(n)

ρY (n) =
RY Y (n)
RY Y (0)
=

1, n = 0
ca|n|
, n = ±1, ±2, · · ·
(9)
c =
σ2
W
σ2
W + σ2
V (1 − a2)
1.

Eqs. (7) and (8) 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 re-
duced by a constant factor compared to the original process
{X(n)}. From (8), 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 observa-
tions.

AR(2) Process: An AR(2) process has the form, W(n) is a
zero-mean white process
X(n) = a1X(n − 1) + a2X(n − 2) + W(n)
H(z) =
∞
X
n=0
h(n)z−n
=
1
1 − a1z−1 − a2z−2
=
b1
1 − λ1z−1
+
b2
1 − λ2z−1
h(0) = 1, h(1) = a1, h(n) = a1h(n − 1) + a2h(n − 2), n ≥ 2

h(n) = b1λn
1 + b2λn
2 , n ≥ 0
b1 + b2 = 1, b1λ1 + b2λ2 = a1.
λ1 + λ2 = a1, λ1λ2 = −a2,
Stability condition:
|λ1| 1, |λ2| 1.

RXX(n) = E{X(n + m)X∗
(m)}
= E{[a1X(n + m − 1) + a2X(n + m − 2)]X∗
(m)}
+ E{W(n + m)X∗
(m)}
= a1RXX(n − 1) + a2RXX(n − 2)
Because W(n) and X(n) are uncorrelated
E{W(n + m)X∗
(m)} = 0
ρX(n) =
RXX(n)
RXX(0)
= a1ρX(n − 1) + a2ρX(n − 2).

RXX(n) = RWW (n) ∗ h∗
(−n) ∗ h(n) = σ2
W h∗
(−n) ∗ h(n)
= σ2
W
∞
X
k=0
h∗
(n + k)h(k)
= σ2
W

|b1|2
(λ∗
1)n
1 − |λ1|2
+
b∗
1b2(λ∗
1)n
1 − λ∗
1λ2
+
b1b∗
2(λ∗
2)n
1 − λ1λ∗
2
+
|b2|2
(λ∗
2)n
1 − |λ2|2


The normalized autocorrelation function:
ρX(n) =
RXX(n)
RXX(0)
= c1λ∗n
1 + c2λ∗n
2
where c1 and c2 are the appropriate constant.

ARMA(p, q) system has only p + q + 1 independent
coefficients, and hence its impulse response sequence
{hk} also must exhibit a similar dependence among them.
An old result due to Kronecker (1881) states that the nec-
essary and sufficient condition for H(z) =
P∞
k=0 h(n)z−n
to
represent a rational system (ARMA) is that

det Hn = 0, n ≥ N (for all sufficiently large n),
where
Hn =





h0 h1 h2 · · · hn
h1 h2 h3 · · · hn+1
.
.
.
hn hn+1 hn+2 · · · h2n





. (10)

In the case of rational systems for all sufficiently large n, the
Hankel matrices Hn in (10) all have the same rank.

The necessary part easily follows from
∞
X
k=0
h(n)z−n
=
Pq
k=0 bkz−k
Pp
k=0 akz−k
by cross multiplying and equating coefficients of like
powers of z−n
for n = 0, 1, 2, · · · .
We have

b0 = h0
b1 = h0a1 + h1
.
.
.
bq = h0aq + h1aq−1 + · · · + hm
0 = h0aq+i + h1aq+i−1 + · · · + hq+i−1a1 + hq+i, i ≥ 1.
For systems with q ≤ p−1, letting i = p−q, p−q+1, · · · , 2p−q
in the last equation we get

h0ap + h1ap−1 + · · · + hp−1a1 + hp = 0
.
.
.
hpap + hp+1ap−1 + · · · + h2p−1a1 + h2p = 0
which gives det(Hp) = 0, and similarly for i = p − q + 1, · · · .

h0ap+1 + h1ap + · · · + hp+1 = 0
h1ap+1 + h2ap + · · · + hp+2 = 0
.
.
.
hp+1ap+1 + hp+2ap + · · · + h2p+2 = 0, †

the ARMA (p, q) model, the input white noise process W(n)
is uncorrelated with its own past sample values as well as
the past values of the system output. This gives
E{W(n)W∗
(n − k)} = 0, k ≥ 1
E{W(n)X∗
(n − k)} = 0, k ≥ 1.

ri = E{x(n)x∗
(n − i)}
= −
p
X
k=1
ak{x(n − k)x∗
(n − i)} +
q
X
k=0
bk{w(n − k)w∗
(n − i)}
= −
p
X
k=1
akri−k +
q
X
k=0
bk{w(n − k)x∗
(n − i)}

Stochastic Processes - part 1

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