Phy_Detect_EP

This detection method is proposed in Expectation Propagation Detection for High-Order High-Dimensional MIMO Systems by Javier Céspedes. It has three process: cavity distribution, Bayesian estimation and iterative update.

Cespedes, J., Olmos, P. M., Sánchez-Fernández, M., & Perez-Cruz, F. (2014). Expectation propagation detection for high-order high-dimensional MIMO systems. IEEE Transactions on Communications, 62(8), 2840-2849.

Also, the initial version of this code is from Alva Kosasih.

How to use

All EP codes are uniform in matlab and python as a class of EP. This class contains the whole process. This section will illustrate the methods of this class following the process of detection.

• EP(the parameter names follow the notations in Expectation Propagation Detection for High-Order High-Dimensional MIMO Systems)
  @constellation: the constellation, a vector
  @beta: the percentage for taking values from the previous iteration
  @epsilon: the default minimal variance
  @l: the maximal iteration
  @early_stop: whether stop early
  @early_stop_min_diff: the elemental minimal (element-wise) difference in mean and variance
  @batch_size(optional) : the batch size (only used in python).
  

  % matlab
  M = 16;
  constellation = qammod([0: M - 1], M, "UnitAveragePower", true);
  ep = EP(constellation, "beta", 0.1, "epsilon", 1e-13, "early_stop_min_diff", 1e-4);
  

  # python
  ep = EP(constellation, beta=0.1, epsilon=1e-13, early_stop_min_diff=1e-4); # no batch
  ep = EP(constellation, beta=0.1, epsilon=1e-13, early_stop_min_diff=1e-4, batch_size=10); # using batch
  

• detect
  @y: the received signal, a vector of [(batch_size), rx_num], [(batch_size), rx_num, 1] or [(batch_size), 1, rx_num]
  @H: the channel matrix, a matrix of [(batch_size), rx_num, tx_num]
  @No: the noise (linear) power
  @sym_map: false by default. If true, the output will be mapped to the constellation
  

  # matlab
  syms = ep.detect(y, H, No); # no hard mapping
  syms = ep.detect(y, H, No, sym_map=True); # hard mapping
  

  # python
  syms = ep.detect(y, H, No); # no hard mapping
  syms = ep.detect(y, H, No, sym_map=True); # hard mapping
  

Samples

Before running any sample code, please make sure you are at the root path of this repository. Also, Matlab codes require running init in the command window first to load directories.

• Alva: this is the original code from Alva Kosasih.
• Tests
  • Tests/test_case_01: test the difference between the mature code from and newly proposed EP object (no early stop).
  • Tests/test_case_02: draw the SER vs SNR between the mature code from and newly proposed EP object (early stop).

Further Reading

The division of two Gaussian distribution

Now we set and we are going to calculate X to see the distribution

Therefore, we can conclude that the the new mean and new variance is given as below (ignoring the constant part because this is just the likelihood of two Xs are same).

Cavity distribution

Cavity Marginal

Now we are going to use the 2 Gaussian Distribution Division knowledge to calculate

Now, we follow the Math symbols in the paper, suppose ( is the diagonal of the matrix ) and

Here, we need pay attention that the mean of p(x) is instead of . The proof is give below where we follow the exponential family.

Here, we can see that the mean is

Now, we are going to find out in Lth iteration

The Pair Update

From the step computing the mean and the variance of prediction , we can use those 2 results to calculate ; that is, for (L+1)th iteration,

Using 2 Gaussian Distribution Division knowledge, for the variance,

For the mean,