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Stochastic Processes

We have already seen discrete-time random processes, but we will now formalize the concept of random process, including both discrete-time and continuous time.

'Definition $ \qquad $ a stochastic process, or random process, defined on (S,F,P) is a family of random variables {X(t), t ∈ T} indexed by a set T.

Fig 1: The mapping from the sample space to the reals under X$ _j $.

Each waveform is referred to as a sample realization. Note that T can be uncountable, as shown above, or countable.

Note that

• X(t,$ \omega $) (or simply X(t)) is a random process.
• X(t$ _0 $,$ \omega $) is a random variable for fixed t$ _0 $.
• X(t,$ \omega_0 $) is a real-valued function of t for fixed $ \omega_0 $.
• X(t$ _0 $,$ \omega_0 $) is a real number for fixed t$ _0 $ and $ \omega_0 $.

There are four types or random processes we will consider

1. T ⊂ R uncountable, X(t) a discrete random variable for every t ∈ T is a continuous-time discrete random process.
2. T ⊂ R uncountable, X(t) a continuous random variable for every t ∈ T is a continuous time continuous random process.
3. T ⊂ R countable, X(t) a discrete random variable for every t ∈ T is a discrete-time discrete random process.
4. T ⊂ R countable, X(t) a continuous random variable for every t ∈ T is a discrete-time continuous random process.

Example $ \qquad $ if T = N = {1,2,3,...}, then X(t) is a discrete time random process, usually written as X$ _1 $,X$ _2 $

Example $ \qquad $ a binary waveform with random transition times

Fig 2: A binary waveform with random transition times.

Example $ \qquad $ A sinusoid with random frequency

where $ \Omega $ is a random variable.

Probabilistic Description of a Random Process

We can use joint pdfs of pmfs, but often we use the first and second order moments instead.

Definition $ \qquad $ The nth order cdf of X(t) is

and the nth order pdf is

Notation $ \qquad $ for n=1, we have

and for n= 2,

Definition $ \qquad $ The nth order pmf of a discrete random process is

It can be shown that if f$ _{X(t1)...X(tn)} $(x$ _1 $,...x$ _n $) is specified ∀t$ _1 $,...,t$ _n $; ∀n = 1,2,..., then X(t) is a valid random process consistent with a probability space (S,F,P). This result comes from the Kolmogorov existence theorem, which we will not cover.

Now consider the first and second order moments for a random process.

Definition $ \qquad $ The mean of a random process X(t) is

Definition $ \qquad $ The autocorrelation function of a random process X(t) is

Note: R$ _{XX} $(t$ _1 $,t$ _2 $) = R$ _{XX} $(t$ _2 $,t$ _1 $)

Definition $ \qquad $ The autocovariance function of a random process X(t) is

Important property of R$ _{XX} $ and C$ _{XX} $:
R$ _{XX} $ and C$ _{XX} $ are non-negative definite functions, i.e., ∀a$ _1 $,...,a$ _n $ ∈ R and t$ _1 $,...,t$ _n $ ∈ R, and ∀n ∈ N,

Proof $ \qquad $ See the proof of NND property of correlation matrix R$ _X $. Let R$ _{ij} $ = R$ _{XX} $(t$ _i $, t$ _j $).

Two important properties of random processes:

Definition $ \qquad $ A random process W(t) is called a white noise process if C$ _{WW} $(t$ _1 $,t$ _2 $) = 0 ∀t$ _1 $ ≠ t$ _2 $.

This means that ∀t$ _1 $ ≠ t$ _2 $, W(t$ _1 $) and W(t$ _2 $) are uncorrelated.

Definition $ \qquad $ A random process X(t) is called a Gaussian random process if X(t$ _1 $),...,X(t$ _n $) are jointly Gaussian random variables ∀t$ _1 $,...,t$ _n $ for any n ∈ N.

The nth order characteristic function of a Gaussian random process is given by

Stationarity

Intuitive idea: A random process is stationary (is some sense) if its probabilistic description (nth order cdf/pdf/pmf, or mean, autocorrelation, autocovariance functions) does not depend on the time origin.

Fig 3

Does the nth order cdf/pdf/pmf depend on where t=0 is? Do $ \mu_X $(t), R$ _{XX} $(t$ _1 $,t$ _2 $), C$ _{XX} $(t$ _1 $,t$ _2 $)?

Definition $ \qquad $ a random process X(t) is strict sense stationary (SSS), or simply stationary, if

Note that if X(t) is SSS, then

for some pdf f$ _X $(x) and

where $ \tau=t_2+\alpha-(t_1+\alpha)=t_2-t_1 $ and f$ _{X1X2} $ is a second order joint pdf that depends on $ \tau $.

Wide Sense Stationary Random Processes
A random process X(t) is wide sense stationary (WSS) if it satisfies

1. E[X(t)] = $ \mu_X $(t) = $ \mu_X $ ∀t, where $ \mu_X $ ∈ R does not depend on t.
2. R$ _{XX} $(t$ _1 $,t$ _2 $) = R$ _X $(t$ _2 $ - t$ _1 $) = R$ _X $($ \tau $) where $ \tau $ = t$ _2 $ - t$ _1 $, and R$ _X $ is a function mapping R to R.

Interesting properties:

• If X(t) is WSS then
  • E[X$ ^2 $(t)] = R$ _{XX} $(t,t) = R$ _X $(0) (so R$ _X $(0) ≥ 0).
  • C$ _{XX} $(t$ _1 $,t$ _2 $) = R$ _{XX} $(t$ _1 $,t$ _2 $) - $ \mu_X $(t$ _1 $)$ \mu_X $(t$ _2 $) = R$ _X $($ \tau $) - $ \mu_X $$ ^2 $, where $ \tau $ = t$ _2 $ - t$ _1 $, and C$ _X $ is a function mapping R to R.
• if X(t) is SSS, then X(t) is WSS, but the converse is not true in general.
• If X(t) is Gaussian and WSS, then X(t) is SSS.

Proof $ \qquad $ The random variables X(t$ _1 $ + $ \alpha $),...,X(t$ _n $ + $ \alpha $)have characteristic function

This does not depend on $ \alpha $, and hence F$ _{X(t1+\alpha)...X(tn+\alpha)} $ does not depend on $ \alpha $. Thus X(t) is SSS.

References

• M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.

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