Cross-Entropy-based MIMO Signal Detection
5.1 Generic Cross-Entropy Method
5.1.1 Importance Sampling
The Cross Entropy (CE) method attempts to solve an optimization problem by relating it to a rare event simulation problem [9]. Usually, the sample size for estimat-ing a rare event probability is very large. Important Samplestimat-ing (IS) is a well-known technique used to reduce the variance by simulating a system under a different set of parameters (reference parameters) or under a different probability distribution. With this technique, the rare event is much more likely to occur so that the sample size of the rare event simulation can be reduced. In conventional IS, the optimal reference parameters are difficult to obtain. The CE method provides a simple and fast proce-dure to estimate the optimal reference parameters used in IS. More specifically, the CE method is an iterative Monte-Carlo based approach to find the IS density, i.e. the im-portance distribution, which is closest to the optimal important distribution in the Kullback-Leibler sense. The Kullback-Leibler distance D(g, h) is a distance measure of
two different distributions g(x) and h(x) which defines as follows:
D(g, h) = Z
g(x) lng(x)
h(x)µ(dx). (5.1)
A distance metric must satisfy the three rules below:
(1) The distance is non-negative.
(2) Symmetric property: The distance between two points is the same while measuring from either direction.
(3) Triangle inequality: Considering a triangle formed by three points, the sum of any two edges is larger than the third edge.
Therefore, the K-L distance is not a true distance metric since it is not symmetric and does not satisfy the triangle inequality.
5.1.2 A Generalized CE Method for Optimization
In this subsection, we give a brief description to the relationship between IS and the CE method for optimization problem. The CE method attempts to solve the following optimization function
arg max
ω∈Ω S(ω) (5.2)
where Ω is the domain of variable ω and S is the score function of ω defined on Ω.
Applying IS to this problem, we find another set of parameter, e.g. v, instead of ω. To find the optimal importance distribution within a class of densities f (ω; v), we adapts the parameter v iteratively so that the Kullback-Leibler distance (i.e. the cross entropy) between the associated density and the optimal importance distribution is minimized.
In general, a generic CE method can be described by the following steps :
1. Generate samples according to the importance distribution determined at the pre-vious iteration.
2. Calculate the scores to the generated samples according to a specific score function.
3. Update the importance distribution by the samples with comparatively better scores.
4. Repeat the above steps until the stopping criterion is reached.
At the very beginning, we give a initial distribution to the importance distribution and generate a set of samples depending on it. Then we compute the scores for every sample individually. For an optimization problem, the value of the objective function are usually regarded as the score for each sample. The samples with better scores are called elite samples and the set composed of elite samples is defined as the elite set.
We choose those elite samples to update the importance distribution. The updated importance distribution is a linear combination of the original importance distribution and the distribution determined by the elite samples. Again, new samples are generated according to the updated importance distribution and the same steps mentioned above are repeated. This procedure is processed iteratively until the stopping criterion is reached.
5.2 Cross-Entropy-based MIMO Signal Detection
5.2.1 A CE-based MIMO detection algorithm
To apply the CE method to a MIMO system, we first define a score function
S(x) = ||y − Hx||2 (5.3)
under the assumption of perfect channel estimation for solving the following optimization problem,
arg min
x∈ANTM
||y − Hx||. (5.4)
Following the procedures of the CE method, the importance distribution of x with relatively smaller scores are estimated by minimizing the distance between the initial distribution and the optimal importance distribution. The estimated transmit signal ˆx is the symbol that is the most likely to occur according to the distribution or the sample with the smallest score during the whole process. Intuitively, an estimated transmit signal vector ˆx is regarded as a sample and the score will be calculated for the sample.
However, there are MNT possible candidates if we treat a vector as a sample unit. Thus a large sample size may be required to cover a wider search region so that the computing complexity would inevitably increase. In order to avoid this problem, the importance distribution of every element in a transmit signal x is estimated separatively.
Let f(k)(xi) denote the importance distribution of the ith element for i = 1, · · · , NT, where the superscript k is the index of iteration. U samples, xki,u for u = 1, · · · , U , are generated at kth iteration in accordance with f(k)(xi) for the ith element. To cal-culate the scores for these samples, a vector set {xku}Uu=1 is constructed where xku = [xk1,u, · · · , xki,u, · · · , xkNT,u]T represents the uth sample vector at kth iteration.
Given a specific quantile probability ρ, there are infinite number of thresholds such that the probability of the scores less or equal to these thresholds are larger or equal to ρ. To select elite samples, we choose the threshold at kth iteration γk satisfying
γk= arg min
γ P (S(Z) ≤ γ) ≥ ρ for Z ∈ {xku}Uu=1 (5.5) And the elite samples are those whose scores satisfies S(xku) ≤ γk. The distributions of elite samples are calculated as
where α is the weighting factor and 0 ≤ α < 1. The updated importance distributions are linear combinations of the original importance distributions and the distribution of the elite samples.
The procedure described above is repeated iteratively until the stopping criterion is met. For example, the pre-defined number of iterations is reached or the importance distributions converges. This algorithm is listed as shown in Table 5.1.
Step 1 : Initialize the importance distributions f(k)(xi) with uniform distribution for i = 1, · · · , NT, respectively. And set k = 0.
Step 2 : Generate U samples xki,u from fk(xi) for u = 1, · · · , U.
Construct the set {xku}Uu=1 where xku = [xk1,u, · · · , xki,u, · · · , xkNT,u]T. Step 3 : Calculate the set of scores {S(xku)}Uu=1 according to equation (5.3).
Step 4 : Set a quantile parameter ρ such that there is a γk satisfying equation (5.5).
Step 5 : Calculate the distribution of elite samples in accordance with equation (5.6).
Step 6 : Update the importance distributions according to equation (5.7).
Step 7 : Stop at iteration k = K if the pre-defined stopping criterion is met; otherwise, let k = k + 1 and go back to Step 2.
Table 5.1: The cross-entropy-based MIMO detection algorithm.
5.2.2 Weighting Factors
In equation (3.6), the weighting factor α effects the exploration and exploitation ability. Exploration is an ability to explore more different regions in the search space to find the global optimum. On the contrary, exploitation is an ability to concentrate the search around the a specific region in order to refine a candidate solution. If α is
larger, the component of updated distribution depends on more information from elite samples. Therefore, it is more likely to exploit than to explore and the convergence speed is faster. However, it is also much more possible to trap in one area since the elite samples generated at first several iterations may lead to a local minimum. If α is small, it takes more iterations to converge but has a higher probability to locate the global optimum.
Usually, to balance the exploration and exploitation, the weighting of the original importance distribution, i.e. the importance distribution at the previous iteration, is about twice that of the distribution of elite samples from some experiences of simulations.
Therefore, α is chosen to be about 0.3 in our work. For other optimization problems, it depends.
5.2.3 Simulation Result of the CE-based Detection Method
Figure 5.1 shows the BER performance of ML detection and the CE-based detection method under a 4 × 4 MIMO system using 4-QAM.
As shown in the figure, our simulation indicates that the resulting BER performance is close to that of ML detector at low SNR region. However, there exists an error floor when SNR is larger than about 10dB since the estimated importance distributions do not converge uniformly, i.e., some of the importance distributions do converge but not all the NT importance distributions.