MIMO achievability bound for the Rayleigh fading channel

Achievability bound is based on the expectation over codewords. This package consists of two modules:

cwd_sample is responsible for single codewords set processing
fbl is responsible for parallel optimization of t-error event probailities
parallel implements a shared memory-based parallelism. Note that passing large matrices in parallel pool is time-consuming. Shared memory is based on multiprocessing.Array. This method requires serialization and deserialization routines to be implemented. Serialization converts conmple-valued matrices to 1D numpy array with real and imaginary parts stored separately.
simulation is responsible for multiple test runnning, saving, and postprocessing

Usage

See main.py for more details. To start the simulation, please specify a dictionary with system parameters that must include:

Blocklength n
The number of information bits k
The number of active users Ka
The number of antennas at the receiver L
The SNR range specified by snr_db_min, snr_db_max, and snr_db_step

Requirements

Numerical python numpy
Scientific python scipy

Performance issues

The choice of optimization method

For the projection-based bound, the function to be optimizaed is a log-determinant of some parametric comination of matrices. One can use the following formula to obtain gradients with respect to parameters to be optimized:

$$\frac{d}{dx}\log\left|A\right| = \text{tr}\left(A^{-1}\frac{dA}{dx}\right).$$

In practice, the matrix inversion in the gradient evaluation procedure becomes a bottleneck of the optimization. As a result, we use a gradient-free methods (Nelder-Mead).

Scipy multithreading

scipy optimization routines are multithread and there is no way to control the number of threads from python. The best way is to run multiple parallel simulations (specify n_workers parameter in the parallel.py) and to disable scipy multithreading:

export MKL_NUM_THREADS=1
export NUMEXPR_NUM_THREADS=1
export OMP_NUM_THREADS=1

References

[1] Kirill Andreev, Daria Ustinova, Alexey Frolov, Unsourced Random Access with the MIMO Receiver: Projection Decoding Analysis