How to implement the given maximum likelihood detection equation in MATLAB ?

Hello all, I am trying to implement the following expression in MATLAB but not getting it properly.

where is a L dimensional set having its cardinality as , .

I am having the values of and but I am not getting how do we proceed with in MATLAB.

Any help in this regard will be highly appreciated.

[charu shree]

Can we say that in this question the task is to find the and which minimizes the given function ?

Please note that I am taking help of the following MATLAB central question: https://in.mathworks.com/matlabcentral/answers/1623125-calculate-argmin-for-the-function-in-matlab?s_tid=prof_contriblnk

[charu shree]

Any hint or suggestion will help me in understanding above expression...

[charu shree]

So can we say that we cannot implement such complicated expression in MATLAB ?

[Torsten]

We can say that we don't understand why both H and X are unknowns in your problem.

And X is a complex matrix having dimension N_t x L ?

What matrix norm is to be chosen for minimization ? Y-H*X is a (N_r x L) complex matrix, I guess.

And you say "I am having the values Y,H and X but I am not getting how do we proceed with argmin in MATLAB". If you have the values, you already solved the problem ?

[charu shree]

Thank you so much sir for your resposne.

1) Sir, X is a matrix having dimension N_t x L but its not complex in nature. In fact each column of X is a vector having 1 at only two positions and 0 every where.

2) We consider Frobenius norm for minimization.

3) Yes, Y-H*X is complex matrix.

4) H and X values are random in nature so we have to estimate them.

[Torsten]

Define

N_t*L + 2*N_r*Nt optimization variables.

The first N_t*L variables are the X_ij defined as integer variables with the constraints

sum_i X_ij = 2 for all j , 0 <= X_ij <= 1, for 1<=j<=L

and the second 2*N_r*Nt variables are the real and imaginary parts of the H_ij = H_ij^(1) + 1i* H_ij^(2).

Then use "ga" to solve this nonlinear optimization problem.

I'm still not convinced that the H_ij are really unknown solution variables.