Choi–Williams distribution function

<h2 id="mathematical-definition">Mathematical definition</h2>
<p>The definition of the <a href="/facts/Cone-shape_distribution_function/PV1UkRlq">cone-shape distribution function</a> is shown as follows:
</p>

C
          
            x
          
        
        (
        t
        ,
        f
        )
        =
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          A
          
            x
          
        
        (
        η
        ,
        τ
        )
        Φ
        (
        η
        ,
        τ
        )
        exp
        ⁡
        (
        j
        2
        π
        (
        η
        t
        −
        τ
        f
        )
        )
        
        d
        η
        
        d
        τ
        ,
      
    
    {\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,}

<p>where
</p>

A
          
            x
          
        
        (
        η
        ,
        τ
        )
        =
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        x
        (
        t
        +
        τ
        
          /
        
        2
        )
        
          x
          
            ∗
          
        
        (
        t
        −
        τ
        
          /
        
        2
        )
        
          e
          
            −
            j
            2
            π
            t
            η
          
        
        
        d
        t
        ,
      
    
    {\displaystyle A_{x}(\eta ,\tau )=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi t\eta }\,dt,}

<p>and the kernel function is:
</p>

Φ
        
          (
          
            η
            ,
            τ
          
          )
        
        =
        exp
        ⁡
        
          [
          
            −
            α
            
              
                (
                
                  η
                  τ
                
                )
              
              
                2
              
            
          
          ]
        
        .
      
    
    {\displaystyle \Phi \left(\eta ,\tau \right)=\exp \left[-\alpha \left(\eta \tau \right)^{2}\right].}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cone-shape_distribution_function/PV1UkRlq">Cone-shape distribution function</a></li>
<li><a href="/facts/Wigner_distribution_function/l5lUlUDV">Wigner distribution function</a></li>
<li><a href="/facts/Ambiguity_function/5uTy0kn4">Ambiguity function</a></li>
<li><a href="/facts/Short-time_Fourier_transform/XHo1t2m6">Short-time Fourier transform</a></li></ul>

<ul><li><a href="http://djj.ee.ntu.edu.tw/TFW.htm">Time frequency analysis and wavelet transform class notes</a>, Jian-Jiun Ding, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.</li>
<li>S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.</li>
<li>H. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE. Trans. Acoustics, Speech, Signal Processing, vol. 37, no. 6, pp. 862–871, June 1989.</li>
<li>Y. Zhao, L. E. Atlas, and R. J. Marks, “The use of cone-shape kernels for generalized time-frequency representations of nonstationary signals,” IEEE Trans. Acoustics, Speech, Signal Processing, vol. 38, no. 7, pp. 1084–1091, July 1990.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” Digital Signal Processing, vol. 19, no. 1, pp. 153-183, January 2009. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Choi–Williams distribution function open-in-new

Choi–Williams distribution function