Statistical signal processing

Statistical signal processing is an area of signal processing that treats signals as stochastic processes, dealing with their statistical properties (e.g., mean, covariance, etc.). Traditionally it is taught at the graduate level in electrical engineering departments around the world, although important applications exist in almost all scientific fields.

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.

v • d • e Digital signal processing

Theory	Discrete frequency \| Nyquist–Shannon sampling theorem \| estimation theory \| detection theory

Sub-fields	audio signal processing \| control engineering \| digital image processing \| speech processing \| statistical signal processing

Techniques	Discrete Fourier transform (DFT) \| Discrete-time Fourier transform (DTFT) \| Impulse invariance \| bilinear transform \| Z-transform, advanced Z-transform

Sampling	oversampling \| undersampling \| downsampling \| upsampling \| aliasing \| anti-aliasing filter \| sampling rate \| Nyquist rate/frequency

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