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Consider we have to design a signal $ \mathbf{x} \in \mathbb{C}^{1_{} \times N_{\mathrm{}}} $ that needs to have the lowest correlation with an unknown signal $\mathbf{y} \in \mathbb{C}^{1_{} \times N_{\mathrm{}}}: $ \begin{equation} \begin{array}{cc} \text { minimize } & E\left[\mathbf{x}_{} \mathbf{y}^{H}_{}\right] \\ \text { subject to } & \mathbf{x}_{} \mathbf{x}^{H}_{}=1 \end{array} \end{equation}

Is it optimum to make $\mathbf{x}$ to be Gaussian? I am trying to prove that based on Entropy of $\mathbf{x}$.

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  • $\begingroup$ no, it wouldn't be more optimal than any other choice. $\endgroup$
    – Marcus Müller
    Feb 18 '21 at 17:22
  • $\begingroup$ The application of the expectation operator $E[\cdot]$, you will have to tell us your stochastic model of $\mathbf y$, otherwise no statement can be made. Also, $\mathbf x\mathbf y^H$ is not lower-bounded, you sure you don't want to minimize $E\left[\left\lvert \mathbf x \mathbf y^H\right\rvert\right]$? $\endgroup$
    – Marcus Müller
    Feb 18 '21 at 17:49
  • $\begingroup$ I think you are right. I need to lower bound it. Assume $\mathbf{y}$ is colored Gaussian. $\endgroup$
    – Am Ki
    Feb 18 '21 at 22:45
  • $\begingroup$ That's a central statement, please specify that in your question instead of just in the comments. I still don't understand what exactly you're trying to optimize, if it's not $E[\mathbf x \mathbf y^H]$. Does the coloredness of the noise relate different entries of $\mathbf y$ to each other, or different $\mathbf y$? I think we'll make the quickest progress if you write down, as formally correct as possible, what you actually want to optimize, and what exactly your stochastic model is. $\endgroup$
    – Marcus Müller
    Feb 19 '21 at 9:06

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