random-signal.md

Statistical Digial Signal Processing

Random Process

A random process can be defined as an ensemble of real or complex functions of two variables $$ {X(t, \zeta)} $$. Variable $$ \zeta $$ is an element of sample space. If the variable is discrete then the random process is a collection, if it is continuous then it is continuous.

Random Signal

For each $$\zeta$$ there is unique ensemble from all possible realizations. The realization is called as random signal which is denoted by $$x_i(t)$$ or $$x(t)$$

Random Variable

For each time $$t_i$$ the process becomes only a random variable $$X(t_i)$$ or $$X_i$$. The behaviour of $$X_i$$ is described by it's probability distribution $$P(x_i,t_i)$$ or its probability density $$p(x_i,t_i)$$

The probability that a random variable $$X_n$$ takes a value $$\infty \leq x$$ is described by the probability distribution function

$$ P_{X_n}(x_n, n) = Probability[X_n \leq x_n] $$

where $$X_n$$ is a random variable and $$x_n$$ is a particular value of $$X_n$$

For continuous

References

Ramesh Babu 2007. Digital Signal Processing, Fourth Edition.