# signal construction based on cross correlation

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 $$\mathbf{y} \in \mathbb{C}^{1_{} \times N_{\mathrm{}}}:$$ $$$$\begin{array}{cc} \text { minimize } & E\left[\mathbf{x}_{} \mathbf{y}^{H}_{}\right] \\ \text { subject to } & \mathbf{x}_{} \mathbf{x}^{H}_{}=1 \end{array}$$$$

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

• no, it wouldn't be more optimal than any other choice. Feb 18 '21 at 17:22
• 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 $E\left[\left\lvert \mathbf x \mathbf y^H\right\rvert\right]$? Feb 18 '21 at 17:49
• I think you are right. I need to lower bound it. Assume $\mathbf{y}$ is colored Gaussian. Feb 18 '21 at 22:45
• 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. Feb 19 '21 at 9:06