Contrast functions

Marc Castella

We give here is a brief description of the different contrast functions proposed in the toolbox. We denote by $y(n)$ the output of the separating filter. When calling [Source,Contribution,W] = Deflation(x,Param,FreqCycl);, the following contrast functions have been implemented depending on the value of Param.Method:

• Case of stationnary sources:
  • If Param.Method='kurtosisRealValued', the contrast:

    $$y(n) \mapsto \vert\mathrm{Cum}\left\{y(n),y(n),y(n),y(n)\right\}\vert^2$$

    
    
    is maximized under the constraint $\mathbb{E}\{y(n)^2\}=1$. This contrast function is valid for real-valued stationnary sources.
  • If Param.Method='kurtosis', the contrast:

    $$y(n) \mapsto \vert\mathrm{Cum}\left\{y(n),y(n)^*,y(n),y(n)^*\right\}\vert^2$$

    
    
    is maximized under the constraint $\mathbb{E}\{\vert y(n)\vert^2\}=1$. This contrast function is valid for both real-valued and complex-valued stationnary sources.
  • If Param.Method='quadratic', the contrast:

    $$y(n) \mapsto \vert\mathrm{Cum}\left\{y(n),y(n)^*,z(n),z(n)^*\right\}\vert$$

    
    
    is maximized under the constraint $\mathbb{E}\{\vert y(n)\vert^2\}=1$, where: $z(n)$ is a fixed ``reference'' signal. This contrast function is valid for both real-valued and complex-valued stationnary sources. See the references on quadratic contrasts for more information.
• Case of cyclo-stationnary sources: If Param.Method='cyclostat', then the contrast:

  $$y(n) \mapsto \langle \vert\mathrm{Cum}\left\{y(n),y(n)^*,y(n),y(n)^*\right\}\vert \rangle$$

  
  
  is maximized under the constraint $\langle\mathbb{E}\{\vert y(n)\vert^2\}\rangle=1$ (where ( $\langle.\rangle$ is the time average operator). This contrast function is valid for both real-valued and complex-valued which are cyclo-stationnary.