# Blind Estimation of Signal Parameters for Zero mean Normal distribution

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!

• You mention that you attempted the problem, can you include those steps in your question? That will help – Engineer May 30 at 22:37
• @Engineer Thank you for having a look, I have added the steps which I tried. – Andrew Smith May 31 at 11:09
• @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. – mark leeds May 31 at 15:30
• @markleeds Unfortunately, the variance of s is unknown, thanks for your time! – Andrew Smith May 31 at 15:56
• Hi Andrew:: Ok but maybe it can give you some more equations since the variance of $K s = K ^2 s ^ 2$. – mark leeds May 31 at 21:26