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In that paper https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/1687-1499-2012-72 .. Equation 3,

$J(w) = w^HBw/w^HCw$ ..

$B$ and $C$ are matrices, $w$ the filter coefficients vector we need to design.

The author is supposed to maximize that cost function $J(w)$. So, he used the generalized eigenvalue decomposition of the two equations $(B,C)$ then he set the eigenvector correspondent to maximum eigenvalue as equalizer filter $w$.

My question, Is my understanding for that idea correct? I mean we can use generalized eigenvalue decomposition for that purpose maximize/minimize, which means if set setting the eigenvector correspondent to maximum eigenvalue as equalizer filter $w$ will maximize, and setting the eigenvector correspondent to minimum eigenvalue as equalizer filter $w$ will minimize.

thank you.

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The generalized eigenvalue problem is given by

$$Bw=\lambda Cw\tag{1}$$

where $\lambda$ is the generalized eigenvalue of the matrices $B$ and $C$. Multiplying $(1)$ from the left with $w^H$ (with $^H$ denoting the Hermitian conjugate) and dividing both sides by $w^HCw$ (assuming that this term is non-zero), we obtain

$$\frac{w^HBw}{w^HCw}=J(w)=\lambda\tag{2}$$

This shows that the value of the objective function $J(w)$ (which is a generalized Rayleigh quotient) equals the generalized eigenvalue $\lambda$. Consequently, $J(w)$ is maximized by the eigenvector corresponding to the maximum eigenvalue, and it is minimized by the eigenvector corresponding to the minimum eigenvalue.

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  • $\begingroup$ Is generalized eigenvalue problem is the same of generalized eigenvalue decomposition? $\endgroup$ – New_student Sep 17 '18 at 10:56
  • $\begingroup$ @Eng.Badr: The generalized EV problem can be solved by the generalized EV decomposition. $\endgroup$ – Matt L. Sep 17 '18 at 10:59
  • $\begingroup$ Thank you so much for your help .. that was really helpful $\endgroup$ – New_student Sep 17 '18 at 11:02
  • $\begingroup$ @Eng.Badr: You're welcome! $\endgroup$ – Matt L. Sep 17 '18 at 11:03
  • $\begingroup$ I add a related question here dsp.stackexchange.com/questions/52833/… Could you please check if you have an idea about? $\endgroup$ – New_student Oct 25 '18 at 4:18

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