How to solve equations for the MMSE receiver?

Question

Given the following equations for the achievable rate of the Minimum Mean-Squared Error Receiver [1]:

$$\mu = \frac{1}{K-1} \sum_{i=1, i \neq k}^{K}{\frac{1}{Mpd_{i}\left(1 - \frac{K-1}{M}+ \frac{K-1}{M}\mu \right) + 1}}$$

and

$$\kappa \left(1 + \sum_{i=1, i \neq k}^{K}{\frac{pd_{i}}{ \left( Mpd_{i}\left(1 - \frac{K-1}{M}+ \frac{K-1}{M}\mu \right) + 1 \right)^2} } \right) = \sum_{i=1, i \neq k}^{K}{\frac{pd_{i}\mu + 1}{ \left( Mpd_{i}\left(1 - \frac{K-1}{M}+ \frac{K-1}{M}\mu \right) + 1 \right)^2} }$$

I've got to find, $\mu$ and $\kappa$. I'm trying to solve these equations with matlab, firstly I tried with solve and then vpasolve, however, when $d$ is a vector with different numbers (i.e., random) it seems matlab never finds a solution. For the case when $d$ is a vector with all values equal to 1 for example, it finds the solution quickly.

This is the script I'm using to find the solution: https://pastebin.com/AWDwyR0U

Am I doing something wrong? Is there a faster way to find a solution to these equations?

Reference:

[1] Hien Quoc Ngo, Erik G. Larsson, and Thomas L. Marzetta, "Energy and Spectral Efficiency of Very Large Multiuser MIMO Systems", IEEE Transactions on Communications, vol. 61, no. 4, April 2013.

Answer 1

I doubt there is general closed-form solution. For the first equation, you essentially want to solve $$\mu=\sum_k \frac{1}{a_k+b_k \mu}$$ for $\mu$ which equals to search for the zeros of a $k+1$th order polynomial, which is impossible for $k>3$. So, I fear you need to do a numeric solution. Once you have $\mu$ you can go for a simple calculation of $\kappa$.

Regarding a numerical solution, here's a plot of $$\mu-\sum_k \frac{1}{a_k+b_k \mu}$$ which should become $0$ for the solution:

N = 7
d = np.random.rand(N)
K = N+1
M = 16
p = 1

f = lambda mu: mu - 1/(K-1)*sum(1/(M*p*d_i*(1-(K-1)/M+(K-1)/M*mu)+1) for d_i in d)

mu = np.linspace(-5,5,10000)
plt.grid(True)

plt.plot(mu, f(mu))
plt.ylim((-0.5,0.5))

So, there are multiple solutions. Maybe you need to pick to positive one?

Answer 2

I've found a way to solve numerically. First we change the equations to

$$\mu - \frac{1}{K-1} \sum_{i=1, i \neq k}^{K}{\frac{1}{Mpd_{i}\left(1 - \frac{K-1}{M}+ \frac{K-1}{M}\mu \right) + 1}} = 0$$

and

$$\kappa \left(1 + \sum_{i=1, i \neq k}^{K}{\frac{pd_{i}}{ \left( Mpd_{i}\left(1 - \frac{K-1}{M}+ \frac{K-1}{M}\mu \right) + 1 \right)^2} } \right) - \sum_{i=1, i \neq k}^{K}{\frac{pd_{i}\mu + 1}{ \left( Mpd_{i}\left(1 - \frac{K-1}{M}+ \frac{K-1}{M}\mu \right) + 1 \right)^2} } = 0$$

Next, by using fzero from Matlab we find $\mu$, finally, using $\mu$ and fzero again we can find $\kappa$.