Autocorrelation technique

<h2 id="derivation">Derivation</h2>
<p>The <a href="/facts/Autocorrelation/dm4MqK0A">autocorrelation</a> of lag 1 can be expressed using the inverse Fourier transform of the power spectrum 
  
    
      
        S
        (
        ω
        )
      
    
    {\displaystyle S(\omega )}
  
:
</p>

R
        (
        1
        )
        =
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            −
            π
          
          
            π
          
        
        S
        (
        ω
        )
        
          e
          
            i
            
            ω
          
        
        d
        ω
        .
      
    
    {\displaystyle R(1)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }S(\omega )e^{i\,\omega }d\omega .}

<p>If we model the power spectrum as a single frequency 
  
    
      
        S
        (
        ω
        )
         
        
          
            
              
                =
              
              
                
                  d
                  e
                  f
                
              
            
          
        
         
        δ
        (
        ω
        −
        
          ω
          
            0
          
        
        )
      
    
    {\displaystyle S(\omega )\ {\stackrel {\mathrm {def} }{=}}\ \delta (\omega -\omega _{0})}
  
, this becomes:
</p>

R
        (
        1
        )
        =
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            −
            π
          
          
            π
          
        
        δ
        (
        ω
        −
        
          ω
          
            0
          
        
        )
        
          e
          
            i
            
            ω
          
        
        d
        ω
      
    
    {\displaystyle R(1)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\delta (\omega -\omega _{0})e^{i\,\omega }d\omega }

R
        (
        1
        )
        =
        
          
            1
            
              2
              π
            
          
        
        
          e
          
            i
            
            
              ω
              
                0
              
            
          
        
      
    
    {\displaystyle R(1)={\frac {1}{2\pi }}e^{i\,\omega _{0}}}

<p>where it is apparent that the phase of 
  
    
      
        R
        (
        1
        )
      
    
    {\displaystyle R(1)}
  
 equals the signal frequency.
</p>
<h2 id="implementation">Implementation</h2>
<p>The mean frequency is calculated based on the <a href="/facts/Autocorrelation/dm4MqK0A">autocorrelation</a> with lag one, evaluated over a signal consisting of N samples:
</p>

ω
        =
        ∠
        
          R
          
            N
          
        
        (
        1
        )
        =
        
          tan
          
            −
            1
          
        
        ⁡
        
          
            
              
                im
              
              {
              
                R
                
                  N
                
              
              (
              1
              )
              }
            
            
              
                re
              
              {
              
                R
                
                  N
                
              
              (
              1
              )
              }
            
          
        
        .
      
    
    {\displaystyle \omega =\angle R_{N}(1)=\tan ^{-1}{\frac {{\text{im}}\{R_{N}(1)\}}{{\text{re}}\{R_{N}(1)\}}}.}

<p>The spectral variance is calculated as follows:
</p>

var
        
        {
        ω
        }
        =
        
          
            2
            N
          
        
        
          (
          
            1
            −
            
              
                
                  
                    |
                  
                  
                    R
                    
                      N
                    
                  
                  (
                  1
                  )
                  
                    |
                  
                
                
                  
                    R
                    
                      N
                    
                  
                  (
                  0
                  )
                
              
            
          
          )
        
        .
      
    
    {\displaystyle {\text{var}}\{\omega \}={\frac {2}{N}}\left(1-{\frac {|R_{N}(1)|}{R_{N}(0)}}\right).}

<h2 id="applications">Applications</h2>
<ul><li>Estimation of blood velocity and turbulence in <i>color flow imaging</i> used in <a href="/facts/Medical_ultrasonography/Ju5BFuuV">medical ultrasonography</a>.</li>
<li>Estimation of target velocity in <a href="/facts/Pulse-doppler_radar/Ibbq5nJS">pulse-doppler radar</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://ieeexplore.ieee.org/xpl/abs_free.jsp?arNumber=1054886">A covariance approach to spectral moment estimation</a>[<i>dead link</i>], Miller et al., IEEE Transactions on Information Theory. </li>
<li>Doppler Radar Meteorological Observations <a href="https://web.archive.org/web/20060708085413/http://www.ofcm.gov/fmh11/fmh11partb/2005pdf/fmh-11B-2005.pdf">Doppler Radar Theory</a>. Autocorrelation technique described on p.2-11</li>
<li><a href="http://bme.elektro.dtu.dk/31545/documents/kasai_et_al_1985.pdf">Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique</a>, by Chihiro Kasai, Koroku Namekawa, Akira Koyano, and Ryozo Omoto, IEEE Transactions on Sonics and Ultrasonics, Vol. SU-32, No.3, May 1985.</li></ul>

Autocorrelation technique open-in-new

Autocorrelation technique