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Second-order Description of Stochastic Processes

Module by: Tuan Do-Hong

Second-order description

Practical and incomplete statistics
Definition 1: Mean
The mean function of a random process X t X t is defined as the expected value of X t X t for all tt's.
μ X t =E X t =∫-∞∞xf X t xdxifcontinuous∑k=-∞∞ x k p X t x k ifdiscrete μ X t X t x x f X t x continuous k x k p X t x k discrete (1)
Definition 2: Autocorrelation
The autocorrelation function of the random process X t X t is defined as
R X t 2 t 1 =E X t 2 X t 1 ¯=∫-∞∞∫-∞∞ x 2 x 1 ¯f X t 2 X t 1 x 2 x 1 d x 1 d x 2 ifcontinuous∑k=-∞∞∑l=-∞∞ x l x k ¯p X t 2 X t 1 x l x k ifdiscrete R X t 2 t 1 X t 2 X t 1 x 2 x 1 x 2 x 1 f X t 2 X t 1 x 2 x 1 continuous k l x l x k p X t 2 X t 1 x l x k discrete (2)
Fact 1 
If X t X t is second-order stationary, then R X t 2 t 1 R X t 2 t 1 only depends on t 2 - t 1 t 2 t 1 .
Proof
R X t 2 t 1 =E X t 2 X t 1 ¯=∫-∞∞∫-∞∞ x 2 x 1 ¯f X t 2 X t 1 x 2 x 1 d x 2 d x 1 R X t 2 t 1 X t 2 X t 1 x 1 x 2 x 2 x 1 f X t 2 X t 1 x 2 x 1 (3)
R X t 2 t 1 =∫-∞∞∫-∞∞ x 2 x 1 ¯f X t 2 - t 1 X 0 x 2 x 1 d x 2 d x 1 = R X t 2 - t 1 0 R X t 2 t 1 x 1 x 2 x 2 x 1 f X t 2 - t 1 X 0 x 2 x 1 R X t 2 t 1 0 (4)
If R X t 2 t 1 R X t 2 t 1 depends on t 2 - t 1 t 2 t 1 only, then we will represent the autocorrelation with only one variable τ= t 2 - t 1 τ t 2 t 1
R X τ= R X t 2 - t 1 = R X t 2 t 1 R X τ R X t 2 t 1 R X t 2 t 1 (5)
    Properties
  1. R X 0≥0 R X 0 0
  2. R X τ= R X -τ¯ R X τ R X τ
  3. | R X τ|≤ R X 0 R X τ R X 0
Example 1 
X t =cos2π f 0 t+Θω X t 2 f 0 t Θ ω and ΘΘ is uniformly distributed between 00 and 2π 2 . The mean function
μ X t=E X t =Ecos2π f 0 t+Θ=∫02πcos2π f 0 t+θ12πdθ=0 μ X t X t 2 f 0 t Θ θ 0 2 2 f 0 t θ 1 2 0 (6)
The autocorrelation function
R X t+τt=E X t + τ X t ¯=Ecos2π f 0 t+τ+Θcos2π f 0 t+Θ=1/2Ecos2π f 0 τ+1/2Ecos2π f 0 2t+τ+2Θ=1/2cos2π f 0 τ+1/2∫02πcos2π f 0 2t+τ+2θ12πdθ=1/2cos2π f 0 τ R X t τ t X t + τ X t 2 f 0 t τ Θ 2 f 0 t Θ 12 2 f 0 τ 12 2 f 0 2 t τ 2 Θ 12 2 f 0 τ 12 θ 0 2 2 f 0 2 t τ 2 θ 1 2 12 2 f 0 τ (7)
Not a function of tt since the second term in the right hand side of the equality in Equation 7 is zero.
Example 2 
Toss a fair coin every TT seconds. Since X t X t is a discrete valued random process, the statistical characteristics can be captured by the pmf and the mean function is written as
μ X t=E X t =1/2×-1+1/2×1=0 μ X t X t 12 -1 12 1 0 (8)
R X t 2 t 1 =∑k∑l x k x l p X t 2 X t 1 x k x l =1×1×1/2+-1×-1×1/2=1 R X t 2 t 1 k k l l x k x l p X t 2 X t 1 x k x l 1 1 12 -1 -1 12 1 (9)
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
R X t 2 t 1 =1×1×1/4+-1×-1×1/4+-1×1×1/4+1×-1×1/4=0 R X t 2 t 1 1 1 14 -1 -1 14 -1 1 14 1 -1 14 0 (10)
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 n≠m n m
R X t 2 t 1 =1ifnT≤ t 1 <n+1T∧nT≤ t 2 <n+1T0otherwise R X t 2 t 1 1 n T t 1 n 1 T n T t 2 n 1 T 0 (11)
A function of t 1 t 1 and t 2 t 2 .
Definition 3: Wide Sense Stationary
A process is said to be wide sense stationary if μ X μ X is constant and R X t 2 t 1 R X t 2 t 1 is only a function of t 2 - t 1 t 2 t 1 .
fact 1 
If X t X t is strictly stationary, then it is wide sense stationary. The converse is not necessarily true.
Definition 4: Autocovariance
Autocovariance of a random process is defined as
C X t 2 t 1 =E X t 2 - μ X t 2 X t 1 - μ X t 1 ¯= R X t 2 t 1 - μ X t 2 μ X t 1 ¯ C X t 2 t 1 X t 2 μ X t 2 X t 1 μ X t 1 R X t 2 t 1 μ X t 2 μ X t 1 (12)
The variance of X t X t is Var X t = C X tt Var X t C X t t
Two processes defined on one experiment (Figure 1).
Figure 1
Definition 5: Crosscorrelation
The crosscorrelation function of a pair of random processes is defined as
R X Y t 2 t 1 =E X t 2 Y t 1 ¯=∫-∞∞∫-∞∞xyf X t 2 Y t 1 xydxdy R X Y t 2 t 1 X t 2 Y t 1 y x x y f X t 2 Y t 1 x y (13)
C X Y t 2 t 1 = R X Y t 2 t 1 - μ X t 2 μ Y t 1 ¯ C X Y t 2 t 1 R X Y t 2 t 1 μ X t 2 μ Y t 1 (14)
Definition 6: Jointly Wide Sense Stationary
The random processes X t X t and Y t Y t are said to be jointly wide sense stationary if R X Y t 2 t 1 R X Y t 2 t 1 is a function of t 2 - t 1 t 2 t 1 only and μ X t μ X t and μ Y t μ Y t are constant.

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