Basic Definitions in Stochastic Processes 1.2 2003/05/16 2003/08/04 17:12:20.379 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Kevin Duh kevinduh@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu A random or stochastic process is the assignment of a function of a real variable to each sample point ω in a sample space. Thus, the process X ω t can be considered a function of two variables. For each ω, the time function must be well-behaved and may or may not look random to the eye. Each time function of the process is called a sample function and must be defined over the entire domain of interest. For each t, we have a function of ω, which is precisely the definition of a random variable. Hence the amplitude of a random process is a random variable. The amplitude distribution of a process refers to the probability density function of the amplitude: p X t x . By examining the process's amplitude at several instants, the joint amplitude distribution can also be defined. For the purposes of this module, a process is said to be stationary when the joint amplitude distribution depends on the differences between the selected time instants. The expected value or mean of a process is the expected value of the amplitude at each t. X t m X t x x p X t x For the most part, we take the mean to be zero. The correlation function is the first-order joint moment between the process's amplitudes at two times. R X t 1 t 2 x 2 x 1 x 1 x 2 p X t 1 X t 2 x 1 x 2 Since the joint distribution for stationary processes depends only on the time difference, correlation functions of stationary processes depend only on t 1 t 2 . In this case, correlation functions are really functions of a single variable (the time difference) and are usually written as R X τ where τ t 1 t 2 . Related to the correlation function is the covariance function K X τ , which equals the correlation function minus the square of the mean. K X τ R X τ m X 2 The variance of the process equals the covariance function evaluated as the origin. The power spectrum of a stationary process is the Fourier Transform of the correlation function. 𝒮 X ω τ R X τ ω τ A particularly important example of a random process is white noise. The process X t is said to be white if it has zero mean and a correlation function proportional to an impulse. X t 0 R X τ N 0 2 δ τ The power spectrum of white noise is constant for all frequencies, equaling N 0 2 which is known as the spectral height. The curious reader can track down why the spectral height of white noise has the fraction one-half in it. This definition is the convention. When a stationary process X t is passed through a stable linear, time-invariant filter, the resulting output Y t is also a stationary process having power density spectrum 𝒮 Y ω H ω 2 𝒮 X ω where H ω is the filter's transfer function.