Maximum likelihood sequence estimation: Difference between revisions

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{{Short description|Algorithm for analyzing noisy data streams}}

'''Maximum likelihood sequence estimation''' ('''MLSE''') is a [[mathematical algorithm]] tothat extractextracts useful data out offrom a [[noisy data]] stream.

==Theory==

For an optimized detector for digital signals the priority is not to reconstruct the transmitter signal, but it should do a best estimation of the transmitted data with the least possible number of errors. The receiver emulates the distorted channel. All possible transmitted data streams are fed into this distorted channel model. The receiver compares the time response with the actual received signal and determines the most likely signal.

The receiver emulates the distorted channel. All possible transmitted data streams are fed into this distorted channel model. The receiver compares the time response with the actual received signal and determines the most likely signal.

In cases that are most computationally straightforward, [[root mean square deviation]] can be used as the decision criterion<ref>G. Bosco, P. Poggiolini, and M. Visintin, "Performance Analysis of MLSE Receivers Based on the Square-Root Metric," J. Lightwave Technol. 26, 2098&ndash;2109 (2008)</ref> for the lowest error probability.

==Background==

Suppose that there is an underlying signal {''x''(''t'')}, of which an observed signal {''r''(''t'')} is available. The observed signal ''r'' is related to ''x'' via a transformation that may be nonlinear and may involve attenuation, and would usually involve the incorporation of [[random noise]]. The [[statistical parameter]]s of this transformation are assumed to be known. The problem to be solved is to use the observations {''r''(''t'')} to create a good estimate of {''x''(''t'')}.

Maximum likelihood sequence estimation is formally the application of [[maximum likelihood]] to this problem. That is, the estimate of {''x''(''t'')} is defined to be a sequence of values which maximize the functional

Suppose that there is an underlying signal {''x''(''t'')}, of which an observed signal {''r''(''t'')} is available. The observed signal ''r'' is related to ''x'' via a transformation that may be nonlinear and may involve attenuation, and would usually involve the incorporation of [[random noise]]. The [[statistical parameter]]s of this transformation are assumed known. The problem to be solved is to use the observations {''r''(''t'')} to create a good estimate of {''x''(''t'')}.

:<math>PL(x)=p(x|r\mid x),</math>

where ''p''(''xr''&nbsp;|&nbsp;''rx'') denotes the conditional joint probability density function of the underlyingobserved series {''xr''(''t'')} given that the observedunderlying series has taken the values {''rx''(''t'')}. [[Bayes' theorem]] implies that

MaximumIn likelihoodcontrast, sequencethe related method of maximum a posteriori estimation is formally the application of the [[maximum likelihooda posteriori]] to(MAP) thisestimation problemapproach. ThatThis is more complex than maximum likelihood sequence estimation and requires a known distribution (in [[Bayesian inference|Bayesian terms]], a [[prior distribution]]) for the underlying signal. In this case the estimate of {''x''(''t'')} is defined to be a sequence of values which maximisemaximize the functional

:<math>LP(x)=p(r|x\mid r),</math>

where ''p''(''rx''&nbsp;|&nbsp;''xr'') denotes the conditional joint probability density function of the observedunderlying series {''rx''(''t'')} given that the underlyingobserved series has taken the values {''xr''(''t'')}. [[Bayes' theorem]] implies that

:<math>P(x)=p(x|\mid r)=\frac{p(r|\mid x)p(x)}{p(r)}.</math>

In cases where the contribution of random noise is additive and has a [[multivariate normal distribution]], the problem of maximum likelihood sequence estimation can be reduced to that of a [[least squares]] minimisationminimization.

In contrast, the related mathod of maximum a posteriori estimation is formally the application of the [[Maximum a posteriori]] (MAP) estimation approach. This is more complex than maximum likelihood sequence estimation and requires a known distribution (in [[Bayesian inference|Bayesian terms]], a [[prior distribution]]) for the underlying signal. In this case the estimate of {''x''(''t'')} is defined to be sequence of values which maximise the functional

:<math>P(x)=p(x|r),</math>

where ''p''(''x''|''r'') denotes the conditional joint probability density function of the underlying series {''x''(''t'')} given that the observed series has taken the values {''r''(''t'')}. [[Bayes' theorem]] implies that

:<math>P(x)=p(x|r)=\frac{p(r|x)p(x)}{p(r)}.</math>

==See also==

In cases where the contribution of random noise is additive and has a [[multivariate normal distribution]], the problem of maximum likelihood sequence estimation can be reduced to that of a [[least squares]] minimisation.

* [[Maximum-likelihood estimation]]

* [[Partial-response maximum-likelihood]]

{{morefootnotes}}

==References==

{{More footnotes|date=September 2010}}

<references />

{{reflist}}

== Further reading ==

* {{citeCite book|title=Wireless Communications|author=Andrea Goldsmith|chapter=Maximum Likelihood Sequence Estimation|pages=362&ndash;364|publisher=Cambridge University Press|dateyear=2005|isbn=0521837162|isnb13=9780521837163}}

* {{citeCite book|pages=319&ndash;321|title=Fundamentals of DSL Technology|authorauthor1=Philip Golden, |author2=Hervé Dedieu, and |author3=Krista S. Jacobsen |name-list-style=amp |publisher=CRC Press|dateyear=2006|isbn=0849319137|isnb13=9780849319136}}

* Crivelli, D. E.; Carrer, H. S., Hueda, M. R. (2005) [http://www.scielo.org.ar/pdf/laar/v35n2/v35n2a04.pdf "Performance evaluation of maximum likelihood sequence estimation receivers in lightwave systems with optical amplifiers"], ''Latin American Applied Research'', 35 (2), 95&ndash;98.

* Katz, G., Sadot, D., Mahlab, U., and Levy, A.(2008) "Channel estimators for maximum-likelihood sequence estimation in direct-detection optical communications", ''Optical Engineering'' 47 (4), 045003. {{DOIdoi|10.1117/1.2904827}}

==External links==

* {{citeCite web|title=Maximum-Likelihood Sequence Estimation of Nonlinear Channels in High-Speed Optical Fiber Systems|authorauthor1=W. Sauer-Greff, |author2=A. Dittrich, |author3=M. Lorang, and |author4=M. Siegrist|name-list-style=amp|url=http://wwwold.ftw.at/ftw/events/telekommunikationsforum/SS2001/ss01docs/010406a.pdf|format=PDF|publisher=The Telecommunications Research Center Vienna|date=2001-04-16|access-date=2010-09-02|archive-url=https://web.archive.org/web/20120311181405/http://wwwold.ftw.at/ftw/events/telekommunikationsforum/SS2001/ss01docs/010406a.pdf|archive-date=2012-03-11|url-status=dead}}

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[[Category:Telecommunications techniques]]

[[Category:Error detection and correction]]

[[Category:Signal estimation]]

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