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| {{Nofootnotes|article|date=March 2009}}'''Statistical signal processing''' is an area of [[Applied Mathematics]] and [[Signal Processing]] that treats signals as [[stochastic process]]es, dealing with their statistical properties (e.g., [[mean]], [[covariance]], etc.). Because of its very broad range of application Statistical signal processing is taught at the graduate level in either [[Electrical Engineering]], [[Applied Mathematics]], [[Pure Mathematics]]/[[Statistics]], or even [[Biomedical Engineering]] and [[Physics]] departments around the world, although important applications exist in almost all scientific fields.
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| 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 <math>y(t)</math> that consists of a deterministic part <math>x(t)</math> added to noise which can be modeled in many situations as white [[Gaussian noise]] <math>w(t)</math>:
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| :<math>y(t) = x(t) + w(t) \, </math>
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| where
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| :<math>w(t) \sim \mathcal{N}(0,\sigma^2)</math>
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| [[White noise]] simply means that the noise process is completely uncorrelated. As a result, its [[autocorrelation]] function is an [[Dirac delta function|impulse]]:
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| :<math> R_{ww}(\tau) = \sigma^2 \delta(\tau) \, </math>
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| where
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| :<math>\delta(\tau) \, </math> is the [[Dirac delta function]].
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| 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.
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| 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.
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| : <math>\bar y = \frac{1}{n} \sum_i y(t)_i=x(t)+ \frac{1}{n} \sum_i w(t)_i</math>
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| Assuming that the noise is white and that its variance is constant in time it follows by [[error propagation]] that
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| :<math>\sigma(\bar y)= \frac{1}{\sqrt{n}}\sigma</math>
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| Thus, if 10,000 measurements are averaged the signal to noise ratio is increased by a factor of 100, enabling the measurement of [[carbon|<sup>13</sup>C]] NMR spectra at natural abundance (1.1%) of <sup>13</sup>C.
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| == See also ==
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| * [[Wiener filter]]
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| * [[Kalman filter]]
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| * [[Particle filter]]
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| == Further reading ==
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| * {{cite book |first=Louis L. |last=Scharf |title=Statistical signal processing: detection, estimation, and time series analysis |publisher=[[Addison–Wesley]] |location=[[Boston]] |year=1991 |pages= |isbn=0-201-19038-9 |oclc=61160161}}
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| * {{cite book|last=P Stoica|first=R Moses|title=SPECTRAL ANALYSIS OF SIGNALS|year=2005; Chinese Edition, 2007|publisher=Prentice Hall|location=NJ|url=http://user.it.uu.se/%7Eps/SAS-new.pdf}}
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| * {{cite book |first=Steven M. |last=Kay |title=Fundamentals of Statistical Signal Processing |publisher=[[Prentice Hall]] |location=[[Upper Saddle River, New Jersey]] |year=1993 |pages= |isbn=0-13-345711-7 |oclc=26504848}}
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| * Kainam Thomas Wong[http://www.eie.polyu.edu.hk/~enktwong/]: Statistical Signal Processing lecture notes at the University of Waterloo, Canada.
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| * [[Ali H. Sayed]], Adaptive Filters, Wiley, NJ, 2008, ISBN 978-0-470-25388-5.
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| * [[Thomas Kailath]], [[Ali H. Sayed]], and [[Babak Hassibi]], Linear Estimation, Prentice-Hall, NJ, 2000, ISBN 978-0-13-022464-4.
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| {{DSP}}
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| {{Signal-processing-stub}}
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| {{stat-stub}}
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| [[Category:Signal processing]]
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| [[Category:Time series analysis]]
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