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Let $\mathbb{s}$ be zero mean with a normal distribution. Let one of the observed set of signals be given by $K\mathbb{s}$ and another by $L\mathbb{s^*}$, where $K$ and $L$ both are constants, and $\mathbb{s^*}$ is defined by complex conjugate of $\mathbb{s}$. Also the values of $KL$, $\mid K\mid$, $\mid L\mid$ are known, along with the known relation that $\mid K\mid^2 + \mid L\mid^2=1$.

Is there a way to estimate parameters $K$ and $L$? I tried solving non-linear set of equations, splitting real and imaginary terms of products $K\mathbb{s}$, $L\mathbb{s^*}$, however the system of equations seems to be underdetermined.

The set of equations by separating real and imaginary parts, I tried to solve were as follows: $$ K_{re}s_{re}-K_{im}s_{im} = A\\ K_{re}s_{im}+K_{im}s_{re} = B\\ L_{re}s_{re}+L_{im}s_{im} = C\\ -L_{re}s_{im}+L_{im}s_{re} = D\\ K_{re}L_{re} - K_{im}L_{im} = E\\ K_{re}L_{im} + K_{im}L_{re} = F $$ where $A$ & $B$ are the real & imaginary parts of $K\mathbb{s}$ respectively, $C$ & $D$ are the real & imaginary parts of $K\mathbb{s}^*$ respetively and $E$ & $F$ are the real & imaginary parts of $KL$ respectively.

After substitutions, I had following set of equations: $$ s_{re}=\frac{CL_{re}+DL_{im}}{|L|^2}=\frac{AK_{re}+BK_{im}}{|K|^2}\\ s_{im}=\frac{CL_{im}-DL_{re}}{|L|^2}=\frac{BK_{re}-AK_{im}}{|K|^2} $$ and got the relations between real & imaginary parts of $L$ and $K$ as follows: $$ L_{re}=\frac{FK_{im}+EK_{re}}{|K|^2}\\ L_{im}=\frac{FK_{re}-EK_{im}}{|K|^2} $$ However, I could not find a way forward to estimate $K$ and $L$, as further substitutions could not give me unique solutions for the parameters. I hope someone has a better idea to attack the problem. Thanks in advance!

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  • $\begingroup$ You mention that you attempted the problem, can you include those steps in your question? That will help $\endgroup$ – Engineer May 30 at 22:37
  • $\begingroup$ @Engineer Thank you for having a look, I have added the steps which I tried. $\endgroup$ – Andrew Smith May 31 at 11:09
  • $\begingroup$ @Andrew: Hi. I don't have time to think about it a lot but you may have variance relations also because Ks is normal with zero mean and variance K^2. This of course assumes that the variance of s = 1. If it is also unknown, then disregard my noise and no pun intended. $\endgroup$ – mark leeds May 31 at 15:30
  • $\begingroup$ @markleeds Unfortunately, the variance of s is unknown, thanks for your time! $\endgroup$ – Andrew Smith May 31 at 15:56
  • $\begingroup$ Hi Andrew:: Ok but maybe it can give you some more equations since the variance of $K s = K ^2 s ^ 2 $. $\endgroup$ – mark leeds May 31 at 21:26

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