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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$$∞
When a stationary process $X(t)$ is passed through a stable linear, time-invariant filter, the resulting output $Y(t)$ is also a stationary process having power density spectrum $${}_{Y}()=\left|H(i)\right|^{2}{}_{X}()$$ where $H(i)$ is the filter's transfer function.
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