I have a major problem to solve and implement my solution and I believe that I lack in theory. My problem is about RF microwave engineering but I believe it is not essential to explain in detail and I can only explain the mathematics to find a path to solve my problem with your help.
I have sets of equations that usually can be written as : $b_{in}=f(\Gamma_1,\Gamma_2,\Gamma_{out},a_{in})$ that have these assumptions:
$\Gamma_1$ and $\Gamma_2$: are real and between -1 and +1. $a_{in}$: is assumed to be real and positive for simplicity. $\Gamma_{out}$: is a complex stochastic variable within the unit circle.
As an example, one function is:
$b_{in}=\frac{{\left(\Gamma_1 \,a_{\textrm{in}} +\Gamma_2 \,a_{\textrm{in}} +2\,\Gamma_1 \,\Gamma_2 \,a_{\textrm{in}} \right)}\,\Gamma_{\textrm{out}} +\Gamma_1 \,\Gamma_2 \,a_{\textrm{in}} -a_{\textrm{in}} }{{\left(1-\Gamma_1 \,\Gamma_2 \right)}\,\Gamma_{\textrm{out}} +\Gamma_1 +\Gamma_2 +2}$
My goal is to find the proper $\Gamma_1$ and $\Gamma_2$ to minimize the effect of $\Gamma_{out}$ on $b_{in}$ in a symbolic form.
I have an idea that I should take the derivative of $b_{in}$ with respect to $\Gamma_{out}$:
$\frac{d\,b_{in}}{d\,\Gamma_{out}}$
Then, assume that it is zero and solve the results to find $\Gamma_1$ and $\Gamma_2$.
I am using MATLAB (symbolic equations) to find the solutions or sets of solutions of $\Gamma_1$ and $\Gamma_2$, but Matlab returns "Empty".
Am I on the right track? What do you suggest?