Adaptive equalizer

<p>An adaptive equalizer is an <a href="/facts/Equalizer_(communications)/44Mc0WXw">equalizer</a> that automatically adapts to time-varying properties of the <a href="/facts/Communication_channel/MtlxEShV">communication channel</a>. It is frequently used with coherent modulations such as <a href="/facts/Phase-shift_keying/TFnpYe64">phase-shift keying</a>, mitigating the effects of <a href="/facts/Multipath_propagation/dJFXvAbW">multipath propagation</a> and <a href="/facts/Fading/RsDwBbdz">Doppler spreading</a>. 
</p><p>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 <a href="/facts/Wiener_filter/oGWdBTYa">linear discrete-time filters</a>, the following equation can be used:
</p>

w
          
          
            o
            p
            t
          
        
        =
        
          
            R
          
          
            −
            1
          
        
        
          p
        
      
    
    {\displaystyle \mathbf {w} _{opt}=\mathbf {R} ^{-1}\mathbf {p} }

<p>where 
  
    
      
        
          
            w
          
          
            o
            p
            t
          
        
      
    
    {\displaystyle \mathbf {w} _{opt}}
  
 is the vector of the filter's coefficients, 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbf {R} }
  
 is the received signal covariance matrix and 
  
    
      
        
          p
        
      
    
    {\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.
</p><p>Many adaptation strategies exist. They include, e.g.:
</p>
<ul><li><a href="/facts/Least_mean_squares_filter/1L0XBsyT">Least mean squares filter</a> (LMS) Note that the receiver does not have access to the transmitted signal 
  
    
      
        x
      
    
    {\displaystyle x}
  
 when it is not in training mode. If the probability that the equalizer makes a mistake is sufficiently small, the symbol decisions 
  
    
      
        d
        (
        n
        )
      
    
    {\displaystyle d(n)}
  
 made by the equalizer may be substituted for 
  
    
      
        x
      
    
    {\displaystyle x}
  
.</li>
<li><a href="/facts/Stochastic_gradient_descent/HbcaYqQP">Stochastic gradient descent</a> (SG)</li>
<li><a href="/facts/Recursive_least_squares_filter/K3y6q7db">Recursive least squares filter</a> (RLS)</li></ul>
<table><tbody><tr><td></td><td></td></tr></tbody></table>
<p>A well-known example is the decision feedback equalizer, a filter that uses feedback of detected <a href="/facts/Modulation/aAXPjINv">symbols</a> in addition to conventional equalization of future symbols. Some systems use predefined training sequences to provide reference points for the adaptation process.
</p>

Adaptive equalizer open-in-new

Adaptive equalizer