# circular complex random vectors

Is a vector which components are circular (aka proper) complex random variables also circular complex?

Below I summarized my attempt to solve this problem.

I think the answer is no, but the description of partially polarized thermal light in terms of its coherency matrix (e.g. Goodman 1985) suggests otherwise. Let me elaborate on that. In statistical optics, partially polarized light (or rather its electric field) is defined as a vector field which components are random phasor sums (Goodman) $$\begin{split}\bar{E}(t,\bar{x})&=u_{x}(t,\bar{x})\bar{e}_{x}+u_{y}(t,\bar{x})\bar{e}_{y}\\ u_{x}(t,\bar{x})&=\Psi_{x} \exp[{i(\bar{k}\cdot\bar{x}-\omega t)}]\\ u_{y}(t,\bar{x})&=\Psi_{y} \exp[{i(\bar{k}\cdot\bar{x}-\omega t)}]\end{split}$$ where $\Psi_{x}$ and $\Psi_{y}$ are random phasor sums, which are circular complex random variables which means that $E(\Psi_r^2)=E(\Psi_i^2)$ and $E(\Psi_r\Psi_i)=0$ where subscripts $r$ and $i$ denote real and imaginary part. It is easily verified that $u_x=a+bi$ and $u_y=c+di$ at a particular time $t$ are also circular complex, which means that $E(aa)=E(bb)$, $E(cc)=E(dd)$ and $E(ab)=E(cd)=0$. If we define a vector $U=(a,c,b,d)$ then its covariance matrix is given by $$\begin{split}C=&E((U-E(U))\cdot (U-E(U))^T)\\ =& E(U\cdot U^T)\\ =&\begin{bmatrix} C^{(rr)}&C^{(ri)}\\ C^{(ir)}&C^{(ii)} \end{bmatrix}\\ C^{(rr)}=&\begin{bmatrix} E(aa)&E(ac)\\ E(ac)&E(cc) \end{bmatrix}\quad C^{(ri)}=\begin{bmatrix} E(ab)&E(ad)\\ E(bc)&E(cd) \end{bmatrix}\\ C^{(ir)}=&\begin{bmatrix} E(ab)&E(bc)\\ E(ad)&E(cd) \end{bmatrix}\quad C^{(ii)}=\begin{bmatrix} E(bb)&E(bd)\\ E(bd)&E(dd) \end{bmatrix} \end{split}$$ where $E(U)=\bar{0}$ follows from random phasor properties. If $U$ would be a complex random vector then $C^{(rr)}=C^{(ii)}$ and $C^{(ri)}=-C^{(ir)}$ should hold, which can be written as the following 6 constraints on $C$ $$\begin{split} E(aa)=&E(bb)\\ E(cc)=&E(dd)\\ E(ab)=&-E(ab)=0\\ E(cd)=&-E(cd)=0\\ E(ac)=&E(bd)\\ E(ad)=&-E(bc) \end{split}$$ The first 4 constraints follow directly from $u_x$ and $u_y$ being circular complex (see above). However the last 2 conditions do not (or at least I don't see it). Therefore a vector with circular complex components is not necessarily circular complex.

Now why do I think these last 2 conditions should hold as well (and therefore $U$ should be circular complex)? Because instead of using the covariance matrix $C$, one uses the coherency matrix $J$ to describe partially polarized light $$\begin{split} J&=E(V\cdot \overline{V^{T}}) \quad V=\begin{bmatrix}u_{x}\\u_{y}\end{bmatrix}\\ &=\begin{bmatrix} E(aa)+E(bb)&E(ac)+E(bd)+i(E(bc)-E(ad))\\ E(ac)+E(bd)-i(E(bc)-E(ad))&E(cc)+E(dd)\\ \end{bmatrix}& \end{split}$$ If $J$ describes first order statistics then one should be able to reconstruct $C$ form $J$ which is impossible because $J$ has 4 degrees of freedom while $C$ has 6 (actually 10 but taking into account the first 4 circularity constraints, which we know to be true, gives 6). If the extra two circularity constraints hold then $$J=\begin{bmatrix} 2E(aa)&2E(ac)+2E(bc)i\\ 2E(ac)-2E(bc)i&2E(cc)\\ \end{bmatrix}$$ and we can reconstruct $C$ from $J$. So either the two extra circularity constraints follow from $u_x$ and $u_y$ being circular and I just don't see it or the coherency matrix $J$ doesn't describe the first order statistics of partially polarized light completely. A third option would be that there are two other constraints on $C$, but I wouldn't know which ones that should be.

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