1.6 Basic definitions in stochastic processes

A random or stochastic process is the assignment of a function of a real variable to each sample point $$ in a sample space. Thus, the process $X(, t)$ can be considered a function of two variables. For each $$ , 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 $$ , 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 $t$ . $$(X(t))=m_X(t)=\int_{()} \,d x$$∞ ∞ 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. $$R_X(t_1, t_2)=\int_{()} \,d x_2$$∞ ∞ 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 stationaryprocesses depend only on $\left|t_1-t_2\right|$ . In this case, correlation functions are really functions of a single variable (the time difference) and areusually written as $R_X()$ where $=t_1-t_2$ . Related to the correlation function is the covariance function $K_X()$ , which equals the correlation function minus the square of the mean. $$K_X()=R_X()-m_X^2$$ 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 correlationfunction. $${}_X()=\int_{()} \,d$$∞ ∞ R X 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. $$(X(t))=0$$ $$R_X()=\frac{N_0}{2}()$$ The power spectrum of white noise is constant for all frequencies, equaling $\frac{N_0}{2}$ which is known as the spectral height .

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 $${}_Y()=\left|H(i)\right|^2{}_X()$$ where $H(i)$ is the filter's transfer function.