6.4 MIMO detector with RSS-MCMC and FST-MCMC Methods
6.4.3 Force State Transitions (FST) MCMC Method
The most significant difference between the RSS-MCMC and conventional MCMC detectors is that the statistically significant samples can be drawn in an interfer- ence reduced system rather than over the entire signal space. In the RSS-MCMC detector, interference from reliable bits is removed in (6.29), which results a MIMO system with less interference, whereby the Gibbs sampler performance can be im- proved. If all the bits are reliable, the RSS-MCMC detector is an interference canceler. On the other hand, if all the bits are unreliable, the RSS-MCMC detec- tor is the same as conventional MCMC detector. Otherwise, if the bits are partially reliable, the RSS-MCMC detector is the hybrid conventional MCMC detector and interference canceler.
6.4.3
Force State Transitions (FST) MCMC Method
Gibbs Sampler with bits fipping Deinterleaver Decoder Decoded information bits Interleaver BPSK/QPSK/ 16QAM Mapping Extrinsic LLR computation Reliable/ Unreliable Signal preprocessing MIMO detector with FST-MCMC
Figure 6.3: Iterative MIMO spatial multiplexing receiver with FST-MCMC detec- tor
Fig. 6.3 shows the Iterative MIMO spatial multiplexing receiver with FST- MCMC detector, which consists of a pre-processor, Gibbs sampler with bit flipping, and the extrinsic LLR computation module. The pre-processor and the extrinsic LLR computation modules are the same as the RSS-MCMC detector. Hence, in this section, the Gibbs sampler with bit flipping is presented for the FST-MCMC detector.
As discussed before, the problem of Gibbs sampler at high SNR is that the LLRs of the coded bits may have large values so that the transition probability in the underlying Markov chain may becomes very small. Hence, the Markov chain would
6.4 MIMO detector with RSS-MCMC and FST-MCMC Methods 155 stay in the same state no matter how many samples are drawn. In this section, we develop a forced state transitions MCMC method. In this method, we force the Markov chain to move by manually changing some coded bits which stay the same for a long period of time. In the MCMC methods, the event that the Markov chain changes state occurs when a certain bit dk is to be drawn by the Gibbs
sampler will differ from the previous sample, i.e. d(kn) 6=d(kn−1). The Gibbs sampler draws the sample according to the conditional probability P(dk|Y,X\(nk), λe2), which is directly related to the a posterior LLR in a non-linear manner. A small value of LLR indicates that the coded bit is more likely to change state, while a large value of the LLR indicates that the coded bits will stay the same. Obviously, the coded bits that remain the same are normally associated with large LLR values.
In Section 6.4.1, it has been shown that the LLRs can be assumed as Gaussian distributed with conditional mean µ and conditional variance σ2
η. The decision
error would occur at the tail of the Gaussian distribution, which falls into two cases. In the first case, the LLR value is small and the sign of the LLR is flipped. The Gibbs sampler will take care of this case because the coded bit associated with such small LLR value will likely to be changed in the next sample so that the Markov chain can visit more states. In the second case, the LLR value is large and sign of the LLR is flipped. In this case, the coded bit associated with such large LLR value will less likely to be changed in the next sample so that the Markov chain would be trapped in the current state and never move forward. Hence, the FST-MCMC method is going to flip these “ill conditioned” coded bits to force the Markov chain to move to next state.
Similar to the RSS-MCMC method, the FST-MCMC method will first partition the full signal set into the reliable signal set and unreliable signal set. The reliable signal signal set will keep as it is. For the unreliable signal set, the FST-MCMC method will check the following two criteria:
1. The coded bit has not been changed for the last m samples. 2. Its ML metric and the a priori metric is not consistent.
Then the FST-MCMC method will flip the bit once these two conditions are true, and continue with the Gibbs sampler. The FST-MCMC method is summarized as follows:
1. Partition X into XR and XU.
3. for n= 1 to N
for k= 0 to I−1
draw d(kn) fromP(dk|d(0n). . . , dk(n+1−1), . . . , d(In−−11),YIC, λe
2) If (dk∈XU)
check the criterion 1. check the criterion 2. If (both criteria are true)
flip the coded bit d(kn).
4. Compute the extrinsic LLRs for all coded bits.
The FST-MCMC is different from the RSS-MCMC method in such a way that we keep the reliable bits while try to flip the unreliable bits as long as they are trapped in the same state and would never move forward for a long period of time.