0
$\begingroup$

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.

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
5
  • $\begingroup$ Is generalized eigenvalue problem is the same of generalized eigenvalue decomposition? $\endgroup$ Commented Sep 17, 2018 at 10:56
  • $\begingroup$ @Eng.Badr: The generalized EV problem can be solved by the generalized EV decomposition. $\endgroup$
    – Matt L.
    Commented Sep 17, 2018 at 10:59
  • $\begingroup$ Thank you so much for your help .. that was really helpful $\endgroup$ Commented Sep 17, 2018 at 11:02
  • $\begingroup$ @Eng.Badr: You're welcome! $\endgroup$
    – Matt L.
    Commented Sep 17, 2018 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? $\endgroup$ Commented Oct 25, 2018 at 4:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.