Statistical Digial Signal Processing

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.