Statistical signal processing

Statistical signal processing is an area of signal processing dealing with signals and their statistical properties (e.g., mean, covariance, etc.). It is primarily covered by the field of electrical and computer engineering, although important applications exist in almost all scientific fields.

Statistical signal processing is founded on the principle that signals are not deterministic functions. Rather, signals are modeled as functions consisting of both deterministic and stochastic components. A simple example and also a common model of many statistical systems is a signal $y(t)$ that consists of a deterministic part $x(t)$ added to white Gaussian noise $w(t)$:

$y(t)=x(t)+w(t)\,$

where

$w(t)\sim {\mathcal {N}}(0,\sigma ^{2})$

White noise simply means that the noise process is completely uncorrelated. As a result, its autocorrelation function is an impulse:

$R_{ww}(\tau )=\sigma ^{2}\delta (\tau )\,$

where

$\delta (\tau )\,$ is the Dirac delta function.

Given information about a statistical system and the random variable from which it is derived, we can increase our knowledge of the output signal; conversely, given the statistical properties of the output signal, we can infer the properties of the underlying random variable.

These statistical techniques are developed in the fields of estimation theory, detection theory, and numerous related fields that rely on statistical information to maximize their efficiency.

See also

• Bayes theorem
• Bayes estimator

• Wiener filter
• Kalman filter

• Thomas Bayes
• Norbert Wiener
• Rudolph Kalman