Anadaptive equalizer is anequalizer that automatically adapts to time-varying properties of thecommunication channel.^[1] It is frequently used with coherent modulations such asphase-shift keying, mitigating the effects ofmultipath propagation andDoppler 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 oflinear discrete-time filters, the following equation can be used:^[2]

${\displaystyle {\mathbf{w}}_{opt}={\mathbf{R}}^{-1}\mathbf{p}}$

where${\displaystyle {\mathbf{w}}_{opt}}$ is the vector of the filter's coefficients,${\displaystyle \mathbf{R}}$ is the received signal covariance matrix and${\displaystyle \mathbf{p}}$ 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${\displaystyle x}$ when it is not in training mode. If the probability that the equalizer makes a mistake is sufficiently small, the symbol decisions${\displaystyle d(n)}$ made by the equalizer may be substituted for${\displaystyle x}$.^[3]
• Stochastic gradient descent (SG)
• Recursive least squares filter (RLS)

A well-known example is thedecision feedback equalizer,^[4][5] a filter that uses feedback of detectedsymbols in addition to conventional equalization of future symbols.^[6] Some systems use predefined training sequences to provide reference points for the adaptation process.

• Equalizer
• Intersymbol interference

• Data transmission
• Digital signal processing