Distribution of Eigenvalues for a Sum of Kronecker Products in a Randomized Channel Model

Question

I am working on analyzing a communication channel model where the channel matrix is constructed as a sum of Kronecker products of vectors that represent the effect of path delays and angles on the signal. The model is intended to simulate an OFDM system with multiple transmit and receive antennas, incorporating the effects of multipath propagation with varying delays and angles.

In my model, the channel matrix for each path $l$ is represented as $h_l = a(\theta_l) \otimes b(\tau_l)$, where:

• $a(\theta_l)$ is an $M$-dimensional vector representing the angle of arrival or departure with components $[1, e^{-j 2\pi \frac{d_s}{\lambda_d} \sin(\theta_l)}, \ldots, e^{-j 2\pi \frac{d_s}{\lambda_d} (M-1) \sin(\theta_l)}]^T$. Here, $d_s$ is the antenna element spacing, and $\lambda_d$ is the wavelength of the signal.
• $b(\tau_l)$ is an $N$-dimensional vector representing the path delay with components $[1, e^{-j 2\pi f_s \tau_l}, \ldots, e^{-j 2\pi f_s (N-1) \tau_l}]^T$, with $f_s$ being the subcarrier spacing in an OFDM system.

The parameters $\theta_l$ (angle) and $\tau_l$ (delay) for each path are assumed to be random variables following a normal distribution. The total channel matrix $H$ is then the sum of these individual path contributions, i.e., $H = \sum_{l=1}^L h_l$, where $L$ is the number of paths.

Given the random nature of $\theta_l$ and $\tau_l$, I am interested in understanding the distribution of the eigenvalues of the covariance matrix $\Sigma_H = H H^H$, especially considering the influence of the number of paths $L$ and the randomness introduced by $\theta_l$ and $\tau_l$.

Here is the Matlab code snippet that I used to simulate this scenario:

% Parameters initialization
N = 32; % Number of subcarriers
M = 32; % Number of antennas
L = 10; % Number of paths
fs = 30e3; % OFDM subcarrier spacing
tau_max = 7e-6; % Maximum path delay

% Random path delays and angles
tau = rand(1, L) * tau_max;
theta = rand(1, L) * pi - pi / 2;

% Generate a(theta_l) and b(tau_l)
a = zeros(M, L);
b = zeros(N, L);
for l = 1:L
    a(:, l) = exp(-1j * 2 * pi * (0:M-1)' * sin(theta(l))) / sqrt(M);
    b(:, l) = exp(-1j * 2 * pi * (0:N-1)' * fs * tau(l)) / sqrt(N);
end

% Channel matrix construction
H = zeros(M * N, 1);
for l = 1:L
    H = H + kron(b(:, l), a(:, l)); % Kronecker product and sum
end

% Eigenvalue analysis
Sigma_H = H * H';
eigenvalues = eig(Sigma_H);

My questions are:

1. Given the randomness in $\theta_l$ and $\tau_l$, how can we characterize the distribution of the eigenvalues of $\Sigma_H$?
2. Are there any known results or methods that can predict the eigenvalue distribution for this kind of randomized channel model, especially in relation to the parameters $L$, $N$, and $M$?
3. Is there an effective way to simulate or approximate the eigenvalue distribution for large $L$, $N$, and $M$?

I appreciate any insights or references to relevant literature that could help in understanding the eigenvalue distribution in this context.