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Introduction to Stochastic Processes

Module by: Behnaam Aazhang

Summary: Describes signals that cannot be precisely characterized.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Definitions, distributions, and stationarity

Definition 1: Stochastic Process
Given a sample space, a stochastic process is an indexed collection of random variables defined for each ωΩ ω Ω .
t,t: X t ω t t X t ω (1)

Example 1

Received signal at an antenna as in Figure 1.

Figure 1
Figure 1 (Figure3-1.png)

For a given tt, X t ω X t ω is a random variable with a distribution

First-order distribution

F X t b=Pr X t b=Pr{ωΩ| X t ωb} F X t b X t b ω Ω X t ω b (2)

Definition 2: First-order stationary process
If F X t b F X t b is not a function of time then X t X t is called a first-order stationary process.

Second-order distribution

F X t 1 , X t 2 b 1 b 2 =Pr X t 1 b 1 X t 2 b 2 F X t 1 , X t 2 b 1 b 2 X t 1 b 1 X t 2 b 2 (3)
for all t 1 t 1 , t 2 t 2 , b 1 b 1 , b 2 b 2

Nth-order distribution

F X t 1 , X t 2 , , X t N b 1 b 2 b N =Pr X t 1 b 1 X t N b N F X t 1 , X t 2 , , X t N b 1 b 2 b N X t 1 b 1 X t N b N (4)

NNth-order stationary : A random process is stationary of order NN if

F X t 1 , X t 2 , , X t N b 1 b 2 b N = F X t 1 + T , X t 2 + T , , X t N + T b 1 b 2 b N F X t 1 , X t 2 , , X t N b 1 b 2 b N F X t 1 + T , X t 2 + T , , X t N + T b 1 b 2 b N (5)

Strictly stationary : A process is strictly stationary if it is NNth order stationary for all NN.

Example 2

X t =cos2π f 0 t+Θω X t 2 f 0 t Θ ω where f 0 f 0 is the deterministic carrier frequency and Θω : Ω Θ ω : Ω is a random variable defined over -ππ and is assumed to be a uniform random variable; i.e., f Θ θ=12πifθ-ππ0otherwise f Θ θ 1 2 θ 0

F X t b=Pr X t b=Prcos2π f 0 t+Θb F X t b X t b 2 f 0 t Θ b (6)
F X t b=Pr-π2π f 0 t+Θ-arccosb+Prarccosb2π f 0 t+Θπ F X t b 2 f 0 t Θ b b 2 f 0 t Θ (7)
F X t b=-π2π f 0 t-arccosb2π f 0 t12πdθ+arccosb2π f 0 tπ2π f 0 t12πdθ=2π2arccosb12π F X t b θ 2 f 0 t b 2 f 0 t 1 2 θ b 2 f 0 t 2 f 0 t 1 2 2 2 b 1 2 (8)
f X t x=ddx11πarccosx=1π1x2if|x|10otherwise f X t x x 1 1 x 1 1 x 2 x 1 0 (9)
This process is stationary of order 1.

Figure 2
Figure 2 (Figure3-3a.png)

The second order stationarity can be determined by first considering conditional densities and the joint density. Recall that

X t =cos2π f 0 t+Θ X t 2 f 0 t Θ (10)
Then the relevant step is to find
Pr X t 2 b 2 | X t 1 = x 1 X t 1 x 1 X t 2 b 2 (11)
Note that
X t 1 = x 1 =cos2π f 0 t+ΘΘ=arccos x 1 2π f 0 t X t 1 x 1 2 f 0 t Θ Θ x 1 2 f 0 t (12)
X t 2 =cos2π f 0 t 2 +arccos x 1 2π f 0 t 1 =cos2π f 0 t 2 t 1 +arccos x 1 X t 2 2 f 0 t 2 x 1 2 f 0 t 1 2 f 0 t 2 t 1 x 1 (13)

Figure 3
Figure 3 (Figure3-3b.png)

F X t 2 , X t 1 b 2 b 1 =- b 1 f X t 1 x 1 Pr X t 2 b 2 | X t 1 = x 1 d x 1 F X t 2 , X t 1 b 2 b 1 x 1 b 1 f X t 1 x 1 X t 1 x 1 X t 2 b 2 (14)
Note that this is only a function of t 2 t 1 t 2 t 1 .

Example 3

Every TT seconds, a fair coin is tossed. If heads, then X t =1 X t 1 for nTt<n+1T n T t n 1 T . If tails, then X t =-1 X t -1 for nTt<n+1T n T t n 1 T .

Figure 4
Figure 4 (Figure3-4.png)

p X t x=12ifx=112ifx=-1 p X t x 1 2 x 1 1 2 x -1 (15)
for all t t . X t X t is stationary of order 1.

Second order probability mass function

p X t 1 X t 2 x 1 x 2 = p X t 2 | X t 1 x 2 | x 1 p X t 1 x 1 p X t 1 X t 2 x 1 x 2 p X t 2 | X t 1 x 2 | x 1 p X t 1 x 1 (16)

The conditional pmf

p X t 2 | X t 1 x 2 | x 1 =0if x 2 x 1 1if x 2 = x 1 p X t 2 | X t 1 x 2 | x 1 0 x 2 x 1 1 x 2 x 1 (17)
when nT t 1 <n+1T n T t 1 n 1 T and nT t 2 <n+1T n T t 2 n 1 T for some nn.
p X t 2 | X t 1 x 2 | x 1 = p X t 2 x 2 p X t 2 | X t 1 x 2 | x 1 p X t 2 x 2 (18)
for all x 1 x 1 and for all x 2 x 2 when nT t 1 <n+1T n T t 1 n 1 T and mT t 2 <m+1T m T t 2 m 1 T with nm n m
p X t 2 X t 1 x 2 x 1 =0if x 2 x 1 for  nT t 1 , t 2 <n+1T p X t 1 x 1 if x 2 = x 1 for  nT t 1 , t 2 <n+1T p X t 1 x 1 p X t 2 x 2 if nm for  nT t 1 <n+1TmT t 2 <m+1T p X t 2 X t 1 x 2 x 1 0 x 2 x 1 for  n T t 1 , t 2 n 1 T p X t 1 x 1 x 2 x 1 for  n T t 1 , t 2 n 1 T p X t 1 x 1 p X t 2 x 2 n m for  n T t 1 n 1 T m T t 2 m 1 T (19)

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