Near ML detection using Dijkstra's algorithm with bounded list size over MIMO channels
aa r X i v : . [ c s . I T ] F e b Near ML detection using Dijkstra’s algorithm with bounded listsize over MIMO channels
Atsushi OKAWADO, Ryutaroh MATSUMOTOand Tomohiko UYEMATSUDept. of Communications and Integrated SystemsTokyo Institute of Technology, 152-8550 JapanFebruary 13, 2008
Abstract
We propose Dijkstra’s algorithm with boundedlist size after QR decomposition for decreasingthe computational complexity of near maximum-likelihood (ML) detection of signals over multiple-input-multiple-output (MIMO) channels. After that,we compare the performances of proposed algorithm,QR decomposition M-algorithm (QRD-MLD), andits improvement. When the list size is set to achievethe almost same symbol error rate (SER) as theQRD-MLD, the proposed algorithm has smaller av-erage computational complexity.
The channel capacity of multiple-input-multiple-output (MIMO) channels linearly increases with thenumber of antennas [1, 2]. Maximum-likelihood (ML)detection provides the minimum error rate. However,the computational complexity of the simple ML de-tection algorithm grows exponentially with the num-ber of transmit antennas. Thus, we need an effi-cient algorithm that achieves similar error rate to theML detection. The QR decomposition M-algorithm(QRD-MLD) [5, 6] and sphere decoding (SD) [3] arepossibly the most promising algorithms. In [10], toreduce the computational complexity, Dijkstra’s al-gorithm is applied to SD which achieves same errorrate as ML detection. Both the QRD-MLD and Di- jkstra’s algorithm are tree search based algorithms.Dijkstra’s algorithm uses the list of unlimited size tokeep detection candidates. However, the computa-tional complexities of the QRD-MLD and Dijkstra’salgorithm are still high. To reduce the computa-tional complexity, we propose Dijkstra’s algorithmwith bounded list size. When proposed algorithm’slist size is set to achieve the almost same symbol er-ror rate (SER) as the QRD-MLD, the computationalcomplexity of proposed algorithm is lower than theQRD-MLD.This paper is organized as follows. In Section 2,we introduce the system model of MIMO channels.In Section 3, we review the QRD-MLD and its im-provement, then propose Dijkstra’s algorithm withbounded list size. In Section 4, we show the com-parison between the computational complexity of theQRD-MLDs and proposed algorithm by computersimulations. Finally, we give the conclusion in sec-tion 5.
We consider the uncoded system with t transmit an-tennas and r receive antennas, and we assume r ≥ t .We assume that the noise at each receive antenna isthe additive white Gaussian noise (AWGN). Let x be a t × S , H an r × t fading matrix whose ( k, j ) entry is a complex1ading coefficient between j -th transmit antenna and k -th receive antenna, z an r × y an r × y = Hx + z . (1)We assume that the receiver knows the channel stateinformation H perfectly.In this case, the ML detection of the transmittedsignal over the channel (1) can be formulated as find-ing ˆ x ml = arg min x ∈ S t || y − Hx || . (2) In this section, we propose the new near ML detec-tion algorithm. First, to calculate (2) efficiently, weexplain how to find the ML signal by tree search al-gorithm in Section 3.1. Then, we review near MLdetection algorithms called QRD-MLD [5, 6] and itsimprovement [8] in Section 3.2. Finally, we proposeDijkstra’s algorithm with bounded list size in Section3.3.
To calculate (2) efficiently, we compute a QR decom-position of H and obtain an upper triangular matrix R and a unitary matrix Q with H = QR . Since Q isunitary, || y − Hx || = || Q ∗ y − Q ∗ Hx || = || Q ∗ y − Rx || . (3)Let ξ = Q ∗ y = ( ξ , · · · , ξ r ) T . The ML detection prob-lem (2) can be reformulated as findingˆ x ml = arg min x ∈ S t || ξ − Rx || = arg min x ∈ S t t X j =1 | ξ j − t X i = j R j,i x i | + r X k = t | ξ k | = arg min x ∈ S t t X j =1 | ξ j − t X i = j R j,i x i | . (4) The second equality above follows as the second termin the second equation is irrelevant to x .To calculate (4) efficiently, we consider a weighteddirected graph as follows. The decisions on x i con-struct a tree where nodes at k -th depth are corre-spond to the candidate of x t − k +1 [4], and the rootnode is placed at depth 0. Then, the metric value,which is the weight of branch, between a node ˆ x i thathas ˆ x t , · · · , ˆ x i +1 (ˆ x k ∈ S , i + 1 ≤ k ≤ t ) as ancestornodes from the root node to its parent node is definedby m i = | ξ i − R i,i ˆ x i − t X j = i +1 R i,j ˆ x j | . The distance of each node from the root node, whichis called the accumulated metric value in this paper,is equal to the sum of the metric values of branchesfrom the root node to the node itself. The accumu-lated metric value from the root node to the bottomnode whose depth is t is t X i =1 m i = t X j =1 | ξ j − t X i = j R j,i ˆ x i | . (5)Because ˆ x that makes (5) minimum is equal to ˆ x ml of (4), the shortest path from the root node to thebottom node corresponds to the ML signal [4]. The QRD-MLD [5, 6], which is a breadth-first treesearch based algorithm, finds a near ML signal. TheQRD-MLD keeps only M nodes at each depth withthe smallest accumulated metric values [7], instead oftesting all the candidate in S t according to (4). Ateach depth, only M nodes make their child nodes.We call a node that makes its child node detectionnode in this paper.An improvement to QRD-MLD proposed in [8] re-duces the number of detection nodes from the originalQRD-MLD. This improved QRD-MLD has thresholdvalue at each depth. The depth i ’s threshold value∆ i is defined by∆ i = E i,min + Xφ , (6)2here E i,min is the smallest accumulated metricvalue of the node at i -th depth in the nodes whoseparent node is a detection node. X is a fixed constantnumber, and φ is the noise variance. At each depth,select the nodes that have smaller accumulated met-ric value than threshold value ∆ i . If the number ofselected nodes is more than M , only M nodes withsmallest accumulated metric values are selected.Note that both algorithms do not always find theML signal. For small to medium M values, the com-plexity is substantially lower than the simple ML de-tection algorithm. However, the final result is nolonger guaranteed to be the ML signal. Dijkstra’s algorithm is an efficient algorithm to findthe shortest path from a point to a destination in aweighted graph [9]. Dijkstra’s algorithm uses the listof unlimited size to keep candidate nodes. If we useDijkstra’s algorithm to find the shortest path fromthe root to one of nodes at the bottom depth, wecan get the node with minimum || y − H ˆ x || among allnodes at the bottom depth and it corresponds to theML estimate [10]. However, this algorithm still hashigh computational complexity. To reduce the com-putational complxity, we propose a modified versionof Dijkstra’s algorithm whose list keeps only L nodeswith the smallest accumulated metric values in thelist.We show Dijkstra’s algorithm with bounded listsize.1. Create an empty list for nodes.2. Insert all nodes at the first level into the list.3. Select the node A having smallest accumulatedmetric value in the list and remove it from thelist. If the depth of A is t , then output the nodeA and its ancestor nodes as the ML signal andfinish this algorithm.4. Insert all A’s child nodes into the list.5. Arrange the nodes in the list according to the ac-cumulated metric value by the quick sort. If the SE R SNR at single receive antenna [dB]QRD-MLD:X=2Proposed:L=5original QRD-MLDProposed:L=16ML
Figure 1: (4 ×
4) symbol error ratelist has more than L nodes, select the L nodeswith the smallest accumulated metric values inthe list, and discard other nodes from the list.6. Go back to Step 3.The node whose child nodes are inserted into thelist is called detection node in this paper. Because thediscarded nodes, which are decided at Step 5, andtheir descendant nodes are not examined, the pro-posed algorithm dose not examine all the candidatein S t according to (4). Thus, the proposed algorithmdose not always find the ML signal.When we use LDPC codes [12] or turbo codes [13]after detection, we have to compute N most likelysignals [11]. Such signals can be computed by thisalgorithm’s modification that is finished after out-put N signals with the smallest accumulated metricvalue. In this section, we compare the computational com-plexity, the number of detection nodes and the num-ber of comparisons of real numbers among the pro-posed algorithm and the QRD-MLDs. Throughoutthe simulations, we consider the following systemmodel.3 a v e r age c o m pu t a t i ona l c o m p l e x i t y SNR at single receive antenna [dB]original QRD-MLDQRD-MLD:X=2Proposed:L=16Proposed:L=5
Figure 2: (4 ×
4) average computational complexity m a x i m u m c o m pu t a t i ona l c o m p l e x i t y SNR at single receive antenna [dB]Proposed:L=16original QRD-MLDQRD-MLD:X=2Proposed:L=5
Figure 3: (4 ×
4) maximum computational complexity • We do two simulations. In the first simulation,the number of transmit antennas t = 4, and thenumber of receive antennas r = 4. In the secondsimulation, the number of transmit antennas t =6, and the number of receive antennas r = 6. • The signal constellation at each transmit an-tenna is 16-QAM and all signals are drawn ac-cording to the uniform i.i.d. distribution. • The fading coefficients obey the CN (0 ,
1) distri-bution, and the receiver knows it perfectly. • The noise at each recieve antenna obeys the a v e r age nu m be r o f de t e c t i on node s SNR at single receive antenna [dB]original QRD-MLDQRD-MLD:X=2Proposed:L=16Proposed:L=5
Figure 4: (4 ×
4) average number of detection nodes m a x i m u m nu m be r o f de t e c t i on node s SNR at single receive antenna [dB]Proposed:L=16original QRD-MLDQRD-MLD:X=2Proposed:L=5
Figure 5: (4 ×
4) maximum number of detection nodes CN (0 , φ ) distribution. φ is caluculated by φ = tE s × ( − SNR/ , where E s is the average sym-bol energy. • We transmit 100000 signals, which is 400000symbols if the number of transmit antennas is4 and 600000 symbols if the number of transmitantennas is 6, and every 100 signals, change thefading matrix.If M = 16 is used and the signal constellation is16-QAM, QRD-MLD has symbol error rate (SER)near to the ML detection [7]. So, we use M = 16.In QRD-MLD’s improvement, we use X = 2 in (6)4 a v e r age nu m be r o f c o m pa r i s on s o f r ea l nu m be r s SNR at single receive antenna [dB]original QRD-MLDQRD-MLD:X=2Proposed:L=16Proposed:L=5
Figure 6: (4 ×
4) average number of comparisons ofreal numbers m a x i m u m nu m be r o f c o m pa r i s on s o f r ea l nu m be r s SNR at single receive antenna [dB]Proposed:L=16original QRD-MLDQRD-MLD:X=2Proposed:L=5
Figure 7: (4 ×
4) maximum number of comparisonsof real numbersas used in [8]. In order for the proposed algorithmto have the similar SER to QRD-MLD and its im-provement, we use two versions of proposed algorithmwhose list sizes are L = 16 and L = 5. Figures 1 and8 show that the proposed algorithm with L = 16,the original QRD-MLD and the ML algorithm havealmost the same SER throughout this simulations.The proposed algorithm with L = 5 and QRD-MLD’simprovement also have similar SER throughout thissimulations. SE R SNR at single receive antenna [dB]QRD-MLD:X=2Proposed:L=5original QRD-MLDProposed:L=16ML
Figure 8: (6 ×
6) symbol error rateWe count the number of multiplications and divi-sions of complex numbers as the computational com-plexity. Since the part of QR decomposition is thecommon part of all compared algorithms, we do notinclude that part in comparison of complexity.In QRD-MLDs, we use the quick sort to arrangethe nodes and decide M nodes with the smallest ac-cumulated metric value at each depth.Because the QRD-MLD keeps M nodes at eachdepth, the number of detection nodes and the com-putational complexity are completely determined by M . However, in the proposed algorithm and QRD-MLD’s improvement, the number of detection nodesand the computational complexity are not fixed.Figures 1–7 are the results of first simulation whosenumber of transmit antennas and receive antennasare 4. Figures 8–14 are the results of second simula-tion whose number of transmit antennas and receiveantennas are 6.At first, we discuss the result of first simulation.According to Figures 2, 4 and 6, the propose algo-rithm with L = 16 reduece the average computationalcomplexity, average number of detection nodes andaverage number of comparisons of real numbers fromthe original QRD-MLD. Moreover, in the case of highSNR, although the proposed algorithm with L = 16has much smaller SER than QRD-MLD’s improve-ment according to Figure 1, the average computa-5 a v e r age c o m pu t a t i ona l c o m p l e x i t y SNR at single receive antenna [dB]original QRD-MLDQRD-MLD:X=2Proposed:L=16Proposed:L=5
Figure 9: (6 ×
6) average computational complexity m a x i m u m c o m pu t a t i ona l c o m p l e x i t y SNR at single receive antenna [dB]Proposed:L=16original QRD-MLDQRD-MLD:X=2Proposed:L=5
Figure 10: (6 ×
6) maximum computational complex-itytional complexity of proposed algorithm with L = 16is almost the same as QRD-MLD’s improvement. Inthe case of low SNR, the average computational com-plexity, average number of detection nodes and aver-age number of comparisons of real numbers of theproposed algorithm with L = 5 are lower than QRD-MLD’s improvement. In the case of high SNR, the av-erage computational complexity, average number ofdetection nodes and average number of comparisonsof real numbers of proposed algorithm with L = 5are almost same as QRD-MLD’s improvement whilethe proposed algorithm has smaller SER according a v e r age nu m be r o f de t e c t i on node s SNR at single receive antenna [dB]original QRD-MLDQRD-MLD:X=2Proposed:L=16Proposed:L=5
Figure 11: (6 ×
6) average number of detection nodes m a x i m u m nu m be r o f de t e c t i on node s SNR at single receive antenna [dB]Proposed:L=16original QRD-MLDQRD-MLD:X=2Proposed:L=5
Figure 12: (6 ×
6) maximum number of detectionnodesto Figure 1. According to Figures 3, 5 and 7, inthe case of low SNR, maximum computational com-plexity, maximum number of detection nodes and themaximum number of comparisons of real numbers ofthe proposed algorithm with L = 16 are higher thanQRD-MLDs. However, because the average compu-tational complexity, the average number of detectionnodes and the average number of comparisons of realnumber of the proposed algorithm with L = 16 arelower than QRD-MLDs, we find that the proposedalgorithm rarely gets high computational complex-ity, large number of detection nodes or large number6 a v e r age nu m be r o f c o m pa r i s on s o f r ea l nu m be r s SNR at single receive antenna [dB]original QRD-MLDQRD-MLD:X=2Proposed:L=16Proposed:L=5
Figure 13: (6 ×
6) average number of comparisons ofreal numbers m a x i m u m nu m be r o f c o m pa r i s on s o f r ea l nu m be r s SNR at single receive antenna [dB]Proposed:L=16original QRD-MLDQRD-MLD:X=2Proposed:L=5
Figure 14: (6 ×
6) maximum number of comparisonsof real numbersof comparisons of real numbers.According to Figure 8–14, which is the result ofsecond simulation, the characteristic of proposed al-gorithm dose not change with the number of anten-nas.
In this paper, we propose a near ML detection al-gorithm. When the list size is adjusted so that theproposed algorithm has the almost same symbol er- ror rate (SER) as the original QRD-MLD, the aver-age of the computational complexity and the numberof detection nodes are reduced. When the list sizeis adjusted so that the proposed algorithm has thealmost same symbol error rate (SER) as the QRD-MLD’s improvement, in the case of low SNR, boththe average computational complexity and averagenumber of detection nodes are reduced and in thecase of high SNR, the computational complexity andaverage number of detection nodes of proposed algo-rithm is almost same as QRD-MLD’s improvementwhile SER of the proposed algorithm becomes smallerthan QRD-MLD’s improvement.
Acknowledgment
We would like to thank Prof. Kiyomichi Araki fordrawing our attention to the reference [6]. This re-search is partly supported by the International Com-munications Foundation.