Statistical signal processing

Statistical signal processing is an area of applied mathematics, more specifically signal processing that treats signals as stochastic processes, dealing with their statistical properties (e.g., mean, covariance, etc.). Because of it very broad range of application Statistical signal processing is taught at the graduate level in Electrical Engineering, Applied Mathematics/Pure Mathematics/Statistics, or even Physics departments around the world, although important applications exist in almost all scientific fields. University of Cambridge for example offers research within this field in all the departments just mentioned.

In many areas 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 noise which can be modeled in many situations as 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.

For example, the Computation of Average Transients (CAT) is used routinely in FT-NMR spectroscopy (nuclear magnetic resonance) to improve the signal-noise ratio of nmr spectra. The signal is measured repeatedly n times and then averaged.

${\bar {y}}={\frac {1}{n}}\sum _{i}y(t)_{i}=x(t)+{\frac {1}{n}}\sum _{i}w(t)_{i}$

Assuming that the noise is white and that its variance is constant in time it follows by error propagation that

$\sigma ({\bar {y}})={\frac {1}{\sqrt {n}}}\sigma$

Thus, if 10,000 measurements are averaged the signal to noise ratio is increased by a factor of 100, enabling the measurement of ¹³C NMR spectra at natural abundance (1.1%) of ¹³C.

See also

• Bayes theorem
• Bayes estimator

• Wiener filter
• Kalman filter
• Sequential Monte Carlo Methods
• Particle filter

• Thomas Bayes
• Norbert Wiener
• Rudolph Kalman

Further reading

• Scharf, Louis L. (1991). Statistical signal processing: detection, estimation, and time series analysis. Boston: Addison–Wesley. ISBN 0-201-19038-9. OCLC 61160161.
• Kay, Steven M. (1993). Fundamentals of Statistical Signal Processing. Upper Saddle River, New Jersey: Prentice Hall. ISBN 0-13-345711-7. OCLC 26504848.