find a minima in a symbolic equations that have multiple symbolic variables and a stochastic variable

Dear MATLAB representative,

I have a major problem to solve and implement my solution and I believe that I lack in theory or even implementation.

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 :

that have these assumptions:

and : are real and between -1 and +1

: is assumed to be real and positive for simplicity.

: is a complex stochastic variable within the circle one.

as an example, one of the functions is:

My goal is to find the proper and to minimize the effect of on in a symbolic form.

I have an idea that I should take the derivative of with respect to :

Then, assume that it is zero and solve the results to find and .

I am using MATLAB R2019 (symbolic equations) to find the solutions or sets of solutions of and , but Matlab returns "Empty".

Am I on the right track? What do you suggest?

p.s. I attached my code, just in case if it is needed.

[Night Elf]

I assumed is complex. Does it influence ?

Answer 1

[Sulaymon Eshkabilov]

I think the problem in equation diff11 that you are trying to solve it for two unknowns.

[Night Elf]

What problem?

I have bounderies so I guess there should be a minima?

Answer 2

[Walter Roberson]

The conditions are contradictory.

assumeAlso(-1<Gamma_2<1)
assumeAlso(-1<Gamma_1<1)

forces Gamma_1 and Gamma_2 to never be +1 or -1 exactly (only between) but the other equations force either Gamma_1 or Gamma_2 to be -1 exactly.

[Night Elf]

Thanks. I just commented the bounderies (, and also rewrote the code). The problem is solved.

Just one more question. Based on the MATLAB manual, if I do not specify the type of symbol, MATLAB assumes that the variable can be complex.

1. Does it mean that is considered as a complex variable? Should I check that the is analytic? checking conj( )?
2. My main question emereges from the point that I want to normalize my sensitivity and then integrate the result over the complex area circle 0.5 (). and I relaized I do not know how to handle dGamma_out

is the reflection factor that physically should be within the complex circle one. However, I want to do my analysis with the condition of .

I would be more than grateful, if you I can have your opinion and hints on this matter.

[Walter Roberson]

if I do not specify the type of symbol, MATLAB assumes that the variable can be complex.

Correct. MATLAB does assume that complex values are possible for the variable, but it does not assume that the variable will definitely have a non-zero real component. For example if it assumed that X was definitely real (and not NaN) then

x<=0 | x > 0

would have to be true, but without the assumption of real,

>> simplify(x<=0 | x > 0)
ans =
in(x, 'real')

This is a conditional test -- that is, the test is true if and only if x is real. It does not assume that x is real-valued; it does not assume that x is complex-valued. There are cases in which it might be able to prove that an expression is real-valued, and if so then it will make analytic simplifications as appropriate. x^2 >= 0 is, for example, not assumed to be definitely true, but if at some point it can prove the expression x is real valued then it will happy simplify that x^2>=0 to true.

[Night Elf]

thanks for your answer :).

I am almost convinced that I should rewrite my variable in a complex form and starts from the begining then.

Thanks again.