Blind equalization

<h2 id="problem-statement">Problem statement</h2>
<h3>Noiseless model</h3>
<p>Assuming a <a href="/facts/LTI_system_theory/kEmj0QDC">linear time invariant</a> channel with impulse response 
  
    
      
        {
        h
        [
        n
        ]
        
          }
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
      
    
    {\displaystyle \{h[n]\}_{n=-\infty }^{\infty }}
  
, the noiseless model relates the received signal 
  
    
      
        r
        [
        k
        ]
      
    
    {\displaystyle r[k]}
  
 to the transmitted signal 
  
    
      
        s
        [
        k
        ]
      
    
    {\displaystyle s[k]}
  
 via
</p>

r
        [
        k
        ]
        =
        
          ∑
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
        h
        [
        n
        ]
        s
        [
        k
        −
        n
        ]
      
    
    {\displaystyle r[k]=\sum _{n=-\infty }^{\infty }h[n]s[k-n]}

<p>The blind equalization problem can now be formulated as follows; Given the received signal 
  
    
      
        r
        [
        k
        ]
      
    
    {\displaystyle r[k]}
  
, find a filter 
  
    
      
        w
        [
        k
        ]
      
    
    {\displaystyle w[k]}
  
, called an equalization filter, such that
</p>

s
              ^
            
          
        
        [
        k
        ]
        =
        
          ∑
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
        w
        [
        n
        ]
        r
        [
        k
        −
        n
        ]
      
    
    {\displaystyle {\hat {s}}[k]=\sum _{n=-\infty }^{\infty }w[n]r[k-n]}

<p>where 
  
    
      
        
          
            
              s
              ^
            
          
        
      
    
    {\displaystyle {\hat {s}}}
  
 is an estimation of 
  
    
      
        s
      
    
    {\displaystyle s}
  
.
The solution 
  
    
      
        
          
            
              s
              ^
            
          
        
      
    
    {\displaystyle {\hat {s}}}
  
 to the blind equalization problem is not unique. In fact, it may be determined only up to a signed scale factor and an arbitrary time delay. That is, if 
  
    
      
        {
        
          
            
              s
              ~
            
          
        
        [
        n
        ]
        ,
        
          
            
              h
              ~
            
          
        
        [
        n
        ]
        }
      
    
    {\displaystyle \{{\tilde {s}}[n],{\tilde {h}}[n]\}}
  
 are estimates of the transmitted signal and channel impulse response, respectively, then 
  
    
      
        {
        c
        
          
            
              s
              ~
            
          
        
        [
        n
        +
        d
        ]
        ,
        
          
            
              h
              ~
            
          
        
        [
        n
        −
        d
        ]
        
          /
        
        c
        }
      
    
    {\displaystyle \{c{\tilde {s}}[n+d],{\tilde {h}}[n-d]/c\}}
  
 give rise to the same received signal 
  
    
      
        r
      
    
    {\displaystyle r}
  
 for any real scale factor 
  
    
      
        c
      
    
    {\displaystyle c}
  
 and integral time delay 
  
    
      
        d
      
    
    {\displaystyle d}
  
. In fact, by symmetry, the roles of 
  
    
      
        s
      
    
    {\displaystyle s}
  
 and 
  
    
      
        h
      
    
    {\displaystyle h}
  
 are Interchangeable.
</p>
<h3>Noisy model</h3>
<p>In the noisy model, an additional term, 
  
    
      
        n
        [
        k
        ]
      
    
    {\displaystyle n[k]}
  
, representing additive noise, is included. The model is therefore
</p>

r
        [
        k
        ]
        =
        
          ∑
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
        h
        [
        n
        ]
        s
        [
        k
        −
        n
        ]
        +
        n
        [
        k
        ]
      
    
    {\displaystyle r[k]=\sum _{n=-\infty }^{\infty }h[n]s[k-n]+n[k]}

<h2 id="algorithms">Algorithms</h2>
<p>Many algorithms for the solution of the blind equalization problem have been suggested over the years.
However, as one usually has access to only a finite number of samples from the received signal 
  
    
      
        r
        (
        t
        )
      
    
    {\displaystyle r(t)}
  
, further restrictions must be imposed over the above models to render the blind equalization problem tractable.
One such assumption, common to all algorithms described below is to assume that the channel has <a href="/facts/Finite_impulse_response/5u4Gaqcb">finite impulse response</a>, 
  
    
      
        {
        h
        [
        n
        ]
        
          }
          
            n
            =
            −
            N
          
          
            N
          
        
      
    
    {\displaystyle \{h[n]\}_{n=-N}^{N}}
  
, where 
  
    
      
        N
      
    
    {\displaystyle N}
  
 is an arbitrary natural number.
</p><p>This assumption may be justified on physical grounds, since the energy of any real signal must be finite, and therefore its impulse response must tend to zero. Thus it may be assumed that all coefficients beyond a certain point are negligibly small.
</p>
<h3>Minimum phase</h3>
<p>If the channel impulse response is assumed to be <a href="/facts/Minimum_phase/a9WcqXXc">minimum phase</a>, the problem becomes trivial.
</p>
<h3>Bussgang methods</h3>
<p>Bussgang methods make use of the <a href="/facts/Least_mean_squares_filter/1L0XBsyT">Least mean squares filter</a> algorithm
</p>

w
          
            n
            +
            1
          
        
        [
        k
        ]
        =
        
          w
          
            n
          
        
        [
        k
        ]
        +
        μ
        
        
          e
          
            ∗
          
        
        [
        n
        ]
        r
        [
        n
        −
        k
        ]
        ,
        k
        =
        −
        N
        ,
        .
        .
        .
        N
      
    
    {\displaystyle w_{n+1}[k]=w_{n}[k]+\mu \,e^{*}[n]r[n-k],k=-N,...N}

e
        [
        n
        ]
        =
        
          g
        
        (
        
          
            
              s
              ^
            
          
        
        [
        n
        ]
        )
        −
        
          
            
              s
              ^
            
          
        
        [
        n
        ]
      
    
    {\displaystyle e[n]=\mathbf {g} ({\hat {s}}[n])-{\hat {s}}[n]}

<p>where 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 is an appropriate positive adaptation step and 
  
    
      
        
          g
        
      
    
    {\displaystyle \mathbf {g} }
  
 is a suitable nonlinear function.
</p>
<h3>Polyspectra techniques</h3>
<p>Polyspectra techniques utilize <a href="/facts/Higher-order_statistics/pdPLtQBA">higher order statistics</a> in order to compute the equalizer.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Independent_component_analysis/gsdVaDFR">Independent component analysis</a></li>
<li><a href="/facts/Principal_components_analysis/pCbVTqdR">Principal components analysis</a></li>
<li><a href="/facts/Blind_deconvolution/E6mgQQ3a">Blind deconvolution</a></li>
<li><a href="/facts/Linear_predictive_coding/2Bl1xdSN">Linear predictive coding</a></li></ul>

<p>[1] C. RICHARD JOHNSON, JR., et. el., "Blind Equalization Using the Constant Modulus Criterion: A Review", PROCEEDINGS OF THE IEEE, VOL. 86, NO. 10, OCTOBER 1998.
</p>

Blind equalization open-in-new

Blind equalization