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Stationary and Nonstationary Random Processes

Module by: Michael Haag. E-mail the author

Summary: The module discusses the concept of stationarity in random processes and describes the various types. Also, a review of distribution and density functions is provided to aid in the understanding of stationarity.

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Introduction

From the definition of a random process, we know that all random processes are composed of random variables, each at its own unique point in time. Because of this, random processes have all the properties of random variables, such as mean, correlation, variances, etc.. When dealing with groups of signals or sequences it will be important for us to be able to show whether of not these statistical properties hold true for the entire random process. To do this, the concept of stationary processes has been developed. The general definition of a stationary process is:

Definition 1: stationary process
a random process where all of its statistical properties do not vary with time
Processes whose statistical properties do change are referred to as nonstationary.

Understanding the basic idea of stationarity will help you to be able to follow the more concrete and mathematical definition to follow. Also, we will look at various levels of stationarity used to describe the various types of stationarity characteristics a random process can have.

Distribution and Density Functions

In order to properly define what it means to be stationary from a mathematical standpoint, one needs to be somewhat familiar with the concepts of distribution and density functions. If you can remember your statistics then feel free to skip this section!

Recall that when dealing with a single random variable, the probability distribution function is a simply tool used to identify the probability that our observed random variable will be less than or equal to a given number. More precisely, let XX be our random variable, and let xx be our given value; from this we can define the distribution function as

F x x=PrXx F x x Pr X x
(1)
This same idea can be applied to instances where we have multiple random variables as well. There may be situations where we want to look at the probability of event X and Y both occurring. For example, below is an example of a second-order joint distribution function.
F x xy=PrXxYy F x x y Pr X x Y y
(2)

While the distribution function provides us with a full view of our variable or processes probability, it is not always the most useful for calculations. Often times we will want to look at its derivative, the probability density function (pdf). We define the the pdf as

f x x=dd x F x x f x x x F x x
(3)
f x xdx=Prx<Xx+dx f x x dx Pr x X x dx
(4)
Equation 4 reveals some of the physical significance of the density function. This equations tells us the probability that our random variable falls within a given interval can be approximated by f x xdx f x x dx . From the pdf, we can now use our knowledge of integrals to evaluate probabilities from the above approximation. Again we can also define a joint density function which will include multiple random variables just as was done for the distribution function. The density function is used for a variety of calcutions, such as finding the expected value or proving a random variable is stationary, to name a few.

note:

The above examples explain the distribution and density functions in terms of a single random variable, XX. When we are dealing with signals and random processes, remember that we will have a set of random variables where a different random variable will occur at each time instance of the random process, X t k X t k . In other words, the distribution and density function will also need to take into account the choice of time.

Stationarity

Below we will now look at a more in depth and mathematical definition of a stationary process. As was mentioned previously, various levels of stationarity exist and we will look at the most common types.

First-Order Stationary Process

A random process is classified as first-order stationary if its first-order probability density function remains equal regardless of any shift in time to its time origin. If we let x t 1 x t 1 represent a given value at time t 1 t 1 , then we define a first-order stationary as one that satisifies the following equation:

f x x t 1 = f x x t 1 + τ f x x t 1 f x x t 1 + τ
(5)
The physical significance of this equation is that our density function, f x x t 1 f x x t 1 , is completely independent of t 1 t 1 and thus any time shift, ττ.

The most important result of this statement, and the identifying characteristic of any first-order stationary process, is the fact that the mean is a constant, independent of any time shift. Below we show the results for a random process, XX, that is a discrete-time signal, xn x n .

X-= m x n=Exn=constant (independent of n) X m x n E x n constant (independent of n)
(6)

Second-Order and Strict-Sense Stationary Process

A random process is classified as second-order stationary if its second-order probability density function does not vary over any time shift applied to both values. In other words, for values x t 1 x t 1 and x t 2 x t 2 then we will have the following be equal for an arbitrary time shift ττ.

f x x t 1 x t 2 = f x x t 1 + τ x t 2 + τ f x x t 1 x t 2 f x x t 1 + τ x t 2 + τ
(7)
From this equation we see that the absolute time does not affect our functions, rather it only really depends on the time difference betwee the two variables. Looked at another way, this equation can be described as
PrX t 1 x 1 X t 2 x 2 =PrX t 1 +τ x 1 X t 2 +τ x 2 Pr X t 1 x 1 X t 2 x 2 Pr X t 1 τ x 1 X t 2 τ x 2
(8)

These random proceses are often referred to as strict sense stationary (SSS) when all of the distribution functions of the process are unchanged regardless of the time shift applied to them.

For a second-order stationary process, we need to look at the autocorrelation function to see its most important property. Since we have already stated that a second-order stationary process depends only on the time difference, then all of these types of processes have the following property:

R xx tt+τ=EXtXt+τ= R xx τ R xx t t τ E X t X t τ R xx τ
(9)

Wide-Sense Stationary Process

As you begin to work with random processes, it will become evident that the strict requirements of a SSS process is more than is often necessary in order to adequately approximate our calculations on random processes. We define a final type of stationarity, referred to as wide-sense stationary (WSS), to have slightly more relaxed requirments but ones that are still enough to provide us with adequate results. In order to be WSS a random process only needs to meet the following two requirements.

  1. X-=Exn=constant X E x n constant
  2. EXtXt+τ= R xx τ E X t X t τ R xx τ
Note that a second-order (or SSS) stationary process will always be WSS; however, the reverse will not always hold true.

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