Complementary code keying

<h2 id="mathematical-description">Mathematical description</h2>
<p>The CCK modulation used by 802.11b transmits data in symbols of eight <a href="/facts/Direct-sequence_spread_spectrum/l3u1HVTd">chips</a>, where each chip is a complex <a href="/facts/QPSK/TFnpYe64">QPSK</a> bit-pair at a chip rate of 11Mchip/s. In 5.5 Mbit/s and 11 Mbit/s modes respectively 4 and 8 bits are modulated onto the eight chips of the symbol c0,...,c7, where
</p>

c
        
        =
        (
        
          c
          
            0
          
        
        ,
        …
        ,
        
          c
          
            7
          
        
        )
        =
        
          (
          
            
              e
              
                j
                (
                
                  ϕ
                  
                    1
                  
                
                +
                
                  ϕ
                  
                    2
                  
                
                +
                
                  ϕ
                  
                    3
                  
                
                +
                
                  ϕ
                  
                    4
                  
                
                )
              
            
            ,
            
              e
              
                j
                (
                
                  ϕ
                  
                    1
                  
                
                +
                
                  ϕ
                  
                    3
                  
                
                +
                
                  ϕ
                  
                    4
                  
                
                )
              
            
            ,
            
              e
              
                j
                (
                
                  ϕ
                  
                    1
                  
                
                +
                
                  ϕ
                  
                    2
                  
                
                +
                
                  ϕ
                  
                    4
                  
                
                )
              
            
            ,
            −
            
              e
              
                j
                (
                
                  ϕ
                  
                    1
                  
                
                +
                
                  ϕ
                  
                    4
                  
                
                )
              
            
            ,
            
              e
              
                j
                (
                
                  ϕ
                  
                    1
                  
                
                +
                
                  ϕ
                  
                    2
                  
                
                +
                
                  ϕ
                  
                    3
                  
                
                )
              
            
            ,
            
              e
              
                j
                (
                
                  ϕ
                  
                    1
                  
                
                +
                
                  ϕ
                  
                    3
                  
                
                )
              
            
            ,
            −
            
              e
              
                j
                (
                
                  ϕ
                  
                    1
                  
                
                +
                
                  ϕ
                  
                    2
                  
                
                )
              
            
            ,
            
              e
              
                j
                
                  ϕ
                  
                    1
                  
                
              
            
          
          )
        
      
    
    {\displaystyle \mathbf {c} =(c_{0},\ldots ,c_{7})=\left(e^{j(\phi _{1}+\phi _{2}+\phi _{3}+\phi _{4})},e^{j(\phi _{1}+\phi _{3}+\phi _{4})},e^{j(\phi _{1}+\phi _{2}+\phi _{4})},-e^{j(\phi _{1}+\phi _{4})},e^{j(\phi _{1}+\phi _{2}+\phi _{3})},e^{j(\phi _{1}+\phi _{3})},-e^{j(\phi _{1}+\phi _{2})},e^{j\phi _{1}}\right)}

<p>and 
  
    
      
        
          ϕ
          
            1
          
        
        ,
        …
        ,
        
          ϕ
          
            4
          
        
      
    
    {\displaystyle \phi _{1},\ldots ,\phi _{4}}
  
 are determined by the bits being modulated.
</p><p>In other words, the phase change 
  
    
      
        
          ϕ
          
            1
          
        
      
    
    {\displaystyle \phi _{1}}
  
 is applied to every chip, 
  
    
      
        
          ϕ
          
            2
          
        
      
    
    {\displaystyle \phi _{2}}
  
 is applied to all even code chips (starting with 
  
    
      
        
          c
          
            0
          
        
      
    
    {\displaystyle c_{0}}
  
), 
  
    
      
        
          ϕ
          
            3
          
        
      
    
    {\displaystyle \phi _{3}}
  
 is applied to the first two of every four chips, and 
  
    
      
        
          ϕ
          
            4
          
        
      
    
    {\displaystyle \phi _{4}}
  
 is applied to the first four of the eight chips. Therefore, it can also be viewed as a form of generalized <a href="/facts/Hadamard_transform/JFD5uojA">Hadamard transform</a> encoding.
</p>

<ul><li>IEEE Std 802.11b-1999, §18.4.6.5</li>
<li>Van Nee, Richard; Awater, Geert; Morikura, Masahiro; Takanashi, Hitoshi; Webster, Mark; Halford, Karen (December 1999). <a href="http://www.jorianvannee.nl/wifi">"New High Rate Wireless LAN Standards"</a>. <i>IEEE Communications Magazine</i>.</li></ul>

Complementary code keying open-in-new

Complementary code keying