Editing Adaptive equalizer (section)

An '''adaptive equalizer''' is an [[equalizer (communications)|equalizer]] that automatically adapts to time-varying properties of the [[communication channel]].<ref>S. Haykin. (1996). Adaptive Filter Theory. (3rd edition). Prentice Hall.</ref> It is frequently used with coherent modulations such as [[phase-shift keying]], mitigating the effects of [[multipath propagation]] and [[Fading|Doppler spreading]]. 

Adaptive equalizers are a subclass of adaptive filters. The central idea is altering the filter's coefficients to optimize a filter characteristic. For example, in case of [[Wiener filter#Finite impulse response Wiener filter for discrete series|linear discrete-time filters]], the following equation can be used:<ref>{{cite book |last=Haykin |first=Simon S. |title=Adaptive filter theory |publisher=Pearson Education India |date=2008 |page=118}}</ref>
:<math> \mathbf{w}_{opt} = \mathbf{R}^{-1}\mathbf{p}</math>
where <math>\mathbf{w}_{opt}</math> is the vector of the filter's coefficients, <math>\mathbf{R}</math> is the received signal covariance matrix and <math>\mathbf{p}</math> is the cross-correlation vector between the tap-input vector and the desired response. In practice, the last quantities are not known and, if necessary, must be estimated during the equalization procedure either explicitly or implicitly.
 
Many adaptation strategies exist. They include, e.g.:

* [[Least mean squares filter]] (LMS) Note that the receiver does not have access to the transmitted signal <math>x</math> when it is not in training mode. If the probability that the equalizer makes a mistake is sufficiently small, the symbol decisions <math>d(n)</math> made by the equalizer may be substituted for <math>x</math>.<ref>[http://cnx.org/content/m10481/latest/ Tutorial on the LMS algorithm]</ref>
* [[Stochastic gradient descent]] (SG)
* [[Recursive least squares filter]] (RLS)

{|style="margin: 0 auto;"
| [[File:SG RLS LMS chan inv.png|thumb|400px|The mean square error performance of [[Least mean squares filter|LMS]], [[Stochastic gradient descent|SG]] and [[Recursive least squares filter|RLS]] in dependence of training symbols. Parameter <math>\mu</math> denotes step size, and <math>\lambda</math> means forgetting factor.]]
|[[File:SG RLS LMS chan var.png|thumb|400px|The mean square error performance of [[Least mean squares filter|LMS]], [[Stochastic gradient descent|SG]] and [[Recursive least squares filter|RLS]] in dependence of training symbols in case of changed during the training procedure channel. Signal power is 1 W, noise power is 0.01 W.]]
|}

A well-known example is the [[decision feedback equalizer]],<ref>[https://web.archive.org/web/20071216142936/http://cnx.org/content/m15524/latest/ Decision Feedback Equalizer]</ref><ref>{{cite web|url=http://signal-integrity.blogs.keysight.com/2012/decision-feedback-equalizer-beauty-is-in-the-eye/|title=For Decision Feedback Equalizers, Beauty is in the Eye|last=Warwick|first=Colin|date=March 28, 2012|website=Signal Integrity|publisher=Agilent Technologies}}</ref> a filter that uses feedback of detected [[Modulation|symbols]] in addition to conventional equalization of future symbols.<ref>{{cite web|url=http://literature.cdn.keysight.com/litweb/pdf/5989-3777EN.pdf|title=Equalization: The Correction and Analysis of Degraded Signals|last=Stevens|first=Ransom|website=Keysight.com}}</ref> Some systems use predefined training sequences to provide reference points for the adaptation process.