Maximum likelihood sequence estimation (MLSE) is a

mathematical algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing c ...

to extract useful data out of a

noisy data Noisy data are data that is corrupted, distorted, or has a low signal-to-noise ratio. Improper procedures (or improperly-documented procedures) to subtract out the noise in data can lead to a false sense of accuracy or false conclusions. Noisy d ...

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. In cases that are most computationally straightforward,

root mean square deviation The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents ...

can be used as the decision criterionG. Bosco, P. Poggiolini, and M. Visintin, "Performance Analysis of MLSE Receivers Based on the Square-Root Metric," J. Lightwave Technol. 26, 2098–2109 (2008) for the lowest error probability.

Background

Suppose that there is an underlying signal , of which an observed signal 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 In electronics, noise is an unwanted disturbance in an electrical signal. Noise generated by electronic devices varies greatly as it is produced by several different effects. In particular, noise is inherent in physics, and central to the ...

. The

statistical parameter In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population ex ...

s of this transformation are assumed known. The problem to be solved is to use the observations to create a good estimate of . Maximum likelihood sequence estimation is formally the application of

maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed sta ...

to this problem. That is, the estimate of is defined to be sequence of values which maximize the functional :

L(x)=p(r\mid x),

where ''p''(''r'' , ''x'') denotes the conditional joint probability density function of the observed series given that the underlying series has the values . In contrast, the related method of maximum a posteriori estimation is formally the application of the

maximum a posteriori In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the ...

(MAP) estimation approach. This is more complex than maximum likelihood sequence estimation and requires a known distribution (in Bayesian terms, a

prior distribution In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken int ...

) for the underlying signal. In this case the estimate of is defined to be sequence of values which maximize the functional :

P(x)=p(x\mid r),

where ''p''(''x'' , ''r'') denotes the conditional joint probability density function of the underlying series given that the observed series has taken the values .

Bayes' theorem In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exa ...

implies that :

P(x)=p(x\mid r)=\frac.

In cases where the contribution of random noise is additive and has a

multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...

, the problem of maximum likelihood sequence estimation can be reduced to that of a

least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the r ...

minimization.

References

External links

* {{DEFAULTSORT:Maximum Likelihood Sequence Estimation Telecommunications techniques Error detection and correction Signal estimation

Theory

Background

See also

References

Further reading

External links