The Adaptive Reduced Breadth-First Search algorithm

Lai et al. in [84] have examined the possibility of the hybrid tree search. A two stage search was proposed. In the first stage a full-blown breadth-first (FBF) search is performed for NBF levels followed by the sequential DFS tree traversal for every subtree.

After the FBF search the symbol vectors sN_N

BF representing the nodes on level NBF are

sorted based on the path metric M (sN_N

BF) of the symbol vectors. The sorted symbol

vectors are denoted as sN <0>_N

BF , s

N <1> NBF , · · · , s

N <|Ω|NBF>

NBF , where the lowest path metric is

achieved by sN <0>_N

BF . For every expanded node s

N <j>

NBF a subtree of depth N − NBF is

associated, referred to as subtree j. Every subtree is traversed using the DFS similar as in the SD algorithm. The traversal starts on the subtree associated to the best path metric node. Whenever the traversal of subtree j is completed, subtree j + 1 is selected. The DFS search is terminated if M (sN <j+1>_N

BF ) > d

2 _{or all the subtrees have been searched. At} this point the FBF-DFS algorithm finds the ML solution, the only difference compared to the SD algorithm is that there is a sorting on level NBF and the DFS is performed in

the ascending order of the path metrics.

In [84] the FBF-DFS algorithm was extended and further computational complexity reduction was achieved in the Adapting Reduced BF-DFS (ARBF-DFS) algorithm. The extensions of the ARBF-DFS algorithm are: (i) the channel dependent FBF level N_BF and (ii) the reliability index based tree pruning.

In the ARBF-DFS algorithm N_BF is allowed to vary indepently 1 ≤ N_BF ≤ N_BF,max, where Nmax denotes the maximum number of levels allowed for the BF stage. If the

channel condition number is favorable N_BF is closer to one, thus, N_BF becomes a random variable. The worst case memory requirement is proportional to ∼ |Ω|NBF,max_.

The second enhancement was the introduction of the reliability index. The aim of the reliability index is to help in the pruning of the tree, thus, the DFS stage is not

4.7. NON-MAXIMUM LIKELIHOOD TREE-SEARCH BASED DETECTORS

performed on every subtree resulting in complexity reduction. In order to give a deeper insight, the partial ML solution is defined as sk_{M L} = (s_k, sk+1, · · · , sN). The reliability

index is defined as P (sN <j+1>_k = sk_{M L}|nk, R), namely the probability of sN <j+1>_k being

the correct path given the noise realization n_kand R. A straightforward way to determine when to stop the FBF stage is to define a threshold T and evaluate if

P (sN <1>_k = sk_{M L}|n_k, R) > T. (4.48)

Furthermore L<j>_k is define as follows:

L<j>_k = ln P (s N <1> k = skM L|nk, R) P (sN <j>_k = sk_{M L}|nk, R) ! = M (s N <j> k ) − M (sN <1>k ) σ2 (4.49)

where the final result is achieved by using the path metric formula and the Gaussian probability density function. Suming the probabilities leads to a different form of the reliability index as follows:

|Ω|k X j=1 P (sN <j>_k = sk_{M L}|nk, R) = 1 P (sN <1>_k = sk_{M L}|n_k, R) = 1 P|Ω|k j=1e −L<j> k (4.50)

The computation of Eq. 4.50 could be very expensive if k is large, thus, it can be simplified by computing only the first two terms of the sum. In this case the approximation of Eq. 4.50 becomes P (sN <1>_k = sk_{M L}|n_k, R) = 1 P|Ω|k j=1e −L<j> k ≈ 1 1 + e−L<2>k (4.51)

By substituting Eq. 4.49 in 4.51 and rewriting Eq. 4.48 the resulting inequality is

M (sN <2>_k ) > M (sN <1>_k ) + σ2ln

 _T

1 − T



. (4.52)

This result can be further generalized, namely every partial symbol vector on levels 1 < k < N_BF whose path metric exceed the best path metric with threshold σ2_ln T

1−T 

are pruned.

The introduced pruning reduces the average floating point operations by 2-5 times compared to the SD algorithm. The BER performance is highly influenced by the chosen threshold T . BER simulations with T = 0.9 achieved near-ML performance.

4.7. NON-MAXIMUM LIKELIHOOD TREE-SEARCH BASED DETECTORS

4.7.2.2 The Fixed-Complexity Sphere Detector algorithm

The Fixed-Complexity Sphere Detector (FSD) was introduced by Barbero et al. in [56]. The FSD algorithm tree traversal is built on a hybrid scheme as well. However, several important differences are introduced compared to the ARBF-DFS algorithm. The main features of the FSD: (i) a channel independent tree traversal process consisting of full BFS on several levels followed by a single DFS path based on the SE enumeration and (ii) a novel channel matrix ordering method.

A conjecture was presented about the number of nodes that have to be visited in an uncoded MIMO system in order to achieve near-ML performance. The result is that two stages are defined: (i) in the FBF stage the maximum number of nodes are considered for

NBF levels and (ii) in the single path DFS stage only one child node is further expanded

for NSElevels and the selected child is based on the SE enumeration. The levels sum of the

two stages N_BF+N_SE = N equals the depth of the tree. The above configuration implies that the number of predefined paths is fixed and is equal |P | = |Ω|NBF_·1NSE_{. In [85] it was}

shown that FSD achieves the same diversity as the ML detection if N_BF ≥√M − 1. The

advantage of the predefined paths is the resulting rigid structure that enables the parallel implementation of the FSD algorithm and has the same computational complexity for every SNR value.

Another important part of this algorithm is the channel matrix ordering method. It determines the order of the symbols detection based on the stages of the algorithm. The aim is to detect symbols with the highest post-detection noise amplification in the FBF stage and symbols with the smallest post-detection noise amplification in the DFS stage. The motivation behind this strategy is to keep all the paths where the error probability is high and after that it is enough to follow only a single path from every expanded node because the probability of making a bad decision with the SE enumeration is low. The symbol sk to be detected is selected according to

k =        arg max j k(H_i†)_jk2 _{, if i is a FBF level} arg min j k(H_i†)jk2 , if i is a DF level

where (H_i†)_j denotes the j-th row of H_i† and the calculation of matrix H_i† is detailed in Alg. 1 Sec. 4.4.2.

The FSD achieved near-ML BER performance for smaller systems, however, the dif- ference of the BER is getting bigger as the number of antennas are increasing or channel