Hello Mingcheng,
Yes, your approach of computing the expectation, 
) by generating multiple instances of 
, accumulating them, and then dividing by the number of instances N is a Monte Carlo method for estimating expectations. This method can work well, especially if the distribution of 
and 
is known and you can sample from it effectively. However, it is important to ensure that your sample size N is large enough to get a good estimate of the expectation.
) by generating multiple instances of , accumulating them, and then dividing by the number of instances N is a Monte Carlo method for estimating expectations. This method can work well, especially if the distribution of and is known and you can sample from it effectively. However, it is important to ensure that your sample size N is large enough to get a good estimate of the expectation.Here is a basic MATLAB implementation of this approach:
% Define dimensions
M = 3; % Size of H
L = 2; % Size of G
N = 1000; % Number of samples
% Initialize the accumulator
accumulator = zeros(L, L);
for i = 1:N
% Generate H and G for each sample
% Assuming H is from a complex normal distribution
H = randn(M, M) + 1i*randn(M, M);
% Assuming G is from a real normal distribution
G = randn(M, L);
% Compute G^H H^H H G
tempMatrix = G' * H' * H * G; % G^H is the conjugate transpose of G
% Accumulate the results
accumulator = accumulator + tempMatrix;
end
% Estimate the expectation
expectation = accumulator / N;
% Display the result
disp('Estimated expectation of G^H H^H H G:');
disp(expectation);
Notes:
- Distribution of
and 
: Adjust the code to generate 
and 
according to their actual distributions in your specific problem. - Convergence: For more accurate results, you might need to increase N, especially if the variance of
is large. - Parallelization: For large N, consider parallelizing the for-loop to speed up the computation, using MATLAB's "parfor" function from Parallel Computing Toolbox if available (https://www.mathworks.com/help/parallel-computing/parfor.html).
This Monte Carlo approach is straightforward but keep in mind that its accuracy depends on the number of samples N and the variance of the quantity you are estimating.
I hope this helps!
