Near ML detection using Dijkstras algorithm with bounded list size over MIMO channels

Near ML detection using Dijkstra’ s algorithm with b ounded list size o v er MIMO c hannels A tsushi OKA W ADO, Ry utaroh MA TSUMOTO and T omohiko UYE MA TSU Dept. of Comm unications and Integrated Systems T oky o Institute of T ec hnology , 152-8550 Japa n F ebruary 13, 2008 Abstract W e prop ose Dijkstra’s algorithm with b ounded list size after QR decomp osition for decreasing the computational complexity of near maximum- likelihoo d (ML) detection of signals ov er m ultiple- input-m ultiple-output (MIMO) c hannels. After that, we compare the p erfor mances of prop osed algo rithm, QR decomp osition M-alg o rithm (QRD-MLD), and its improv ement. When the list size is set to achiev e the almost sa me symbol error rate (SER) as the QRD-MLD, the pr op osed algorithm has smaller av- erage co mputational co mplexity . 1 In tro duction The c hannel capacity of mu ltiple-input-multiple- output (MIMO) channels linear ly increases with the nu mber of antennas [1, 2]. Maximum-lik eliho o d (ML) detection provides the minimum error rate. How ever, the computational complexity of the simple ML de- tection alg orithm grows exp onentially with the num - ber of tra nsmit antennas. Thu s, we need an effi- cient algorithm that achiev es s imila r err or rate to the ML detection. The Q R decomp osition M-a lgorithm (QRD-MLD) [5, 6] and spher e deco ding (SD) [3] are po ssibly the mo st promising a lgorithms. In [10], to reduce the computational complexity , Dijkstra ’s al- gorithm is applied to SD which achiev es same error rate a s ML detection. Both the Q RD-MLD and Di- jkstra’s alg o rithm a re tr ee search based alg orithms. Dijkstra’s a lgorithm uses the lis t of unlimited size to keep detection candidates. Ho wev er, the computa- tional complexities of the QRD-MLD and Dijkstra’s algorithm a re still high. T o reduce the co mputa- tional complexity , we prop os e Dijkstra ’s a lgorithm with bounded list size . When prop osed alg orithm’s list size is se t to a chiev e the a lmost sa me s y m b ol er- ror rate (SE R) as the QRD-MLD, the computatio nal complexity of pro p o sed algorithm is low er than the QRD-MLD. This paper is org anized as follows. In Section 2, we intro duce the system mo del o f MIMO channels. In Section 3, w e review the QRD-MLD and its im- prov ement, then prop ose Dijkstra’s algo rithm with bo unded list size. In Section 4, we show the com- parison betw een the computationa l complexity of the QRD-MLDs and prop osed a lgorithm by c omputer simulations. Fina lly , we give the co nclusion in sec- tion 5. 2 System mo del W e co nsider the unco ded s ystem with t transmit an- tennas and r receive a nt ennas, and we assume r ≥ t . W e assume that the noise a t each r eceive antenna is the additive white Gauss ian noise (A W GN). L et x be a t × 1 vector consisting of c o mplex en velopes of transmitted sig nals with the signal co ns tellation S , H an r × t fading ma trix whose ( k , j ) entry is a complex 1 fading co e fficient b etw een j - th tr ansmit a ntenna and k -th receive antenna, z an r × 1 c o mplex vector whose comp onent is no is e a t each r eceive antenna, a nd y an r × 1 co mplex vector who se comp onent is the received signal comp onent at each receive antenna. The mo del of this channel is written as y = Hx + z . (1) W e a ssume that the receiver knows the channel state information H p erfectly . In this ca se, the ML detection of the tra nsmitted signal over the channel (1) ca n b e formulated as find- ing ˆ x ml = ar g min x ∈ S t || y − Hx || 2 . (2) 3 Near ML detection algorithm In this section, we pro po se the new near ML detec- tion algorithm. First, to calcula te (2 ) efficiently , we explain how to find the ML signal b y tree search al- gorithm in Sectio n 3 .1. Then, we rev iew near ML detection alg orithms ca lled QRD-MLD [5, 6] a nd its improv ement [8] in Section 3.2. Finally , w e pro p o se Dijkstra’s a lgorithm with b ounded list size in Section 3.3. 3.1 QR decomp osition T o ca lculate (2) efficiently , we compute a QR deco m- po sition of H and obtain an upper triangula r matrix R and a unitary matrix Q with H = QR . Since Q is unitary , || y − Hx || 2 = || Q ∗ y − Q ∗ Hx || 2 = || Q ∗ y − Rx || 2 . (3) Let ξ = Q ∗ y = ( ξ 1 , · · · , ξ r ) T . The ML detection prob- lem (2 ) ca n b e refor mulated as finding ˆ x ml = arg min x ∈ S t || ξ − Rx || 2 = arg min x ∈ S t    t X j =1 | ξ j − t X i = j R j,i x i | 2 + r X k = t | ξ k | 2    = arg min x ∈ S t    t X j =1 | ξ j − t X i = j R j,i x i | 2    . (4) The s econd equa lity ab ov e follows as the second term in the se c ond equatio n is irr elev an t to x . T o calculate (4) efficiently , we consider a weighted directed graph as follows. The decisions on x i con- struct a tree where no des at k -th depth are corr e- sp ond to the candidate of x t − k +1 [4], and the ro ot no de is placed at depth 0. Then, the metric v alue, which is the w eight o f branch, b et ween a no de ˆ x i that has ˆ x t , · · · , ˆ x i +1 ( ˆ x k ∈ S , i + 1 ≤ k ≤ t ) as ances to r no des from the ro ot no de to its parent no de is defined by m i = | ξ i − R i,i ˆ x i − t X j = i +1 R i,j ˆ x j | 2 . The distance of each no de fro m the ro o t no de, which is called the accum ulated metric v alue in this pap er, is equal to the sum of the metric v alues of branches from the ro ot node to the node itself. The a c c um u- lated metric v alue fro m the ro ot no de to the b ottom no de 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 | 2 . (5) Because ˆ x tha t ma kes (5) minimum is e qual to ˆ x ml of (4), the shortest path from the r o ot no de to the bo ttom no de co rresp onds to the ML signal [4 ]. 3.2 QRD-MLD The QRD-MLD [5 , 6], which is a br eadth-first tree search based algor ithm, finds a near ML s ig nal. The QRD-MLD keeps only M no des at each depth with the sma llest accumulated metric v alues [7], instea d of testing all the ca ndidate in S t according to (4). At each depth, only M nodes make their c hild no des. W e ca ll a no de that ma kes its child no de detectio n no de in this pa pe r . An improv ement to Q RD-MLD pr op osed in [8] re- duces the n umber of detection no des from the o riginal QRD-MLD. This improv ed QRD-MLD has threshold v alue a t each de pth. The depth i ’s threshold v alue ∆ i is defined by ∆ i = E i,min + X φ 2 , (6) 2 where E i,min is the smallest accumulated metric v alue of the no de at i -th depth in the no des whose parent no de is a detection no de. X is a fixed co nstant nu mber, and φ 2 is the noise v ariance. At each depth, select the no des that have smaller a ccumulated met- ric v alue than threshold v alue ∆ i . If the n umber of selected no des is more than M , only M no des with smallest a ccumulated metric v alues are selected. Note that both algor ithms do not alw ays find the ML signal. F or s mall to medium M v alues, the c om- plexity is substantially low er than the simple ML de- tection algo rithm. How ev er, the final r esult is no longer g ua ranteed to b e the ML s ignal. 3.3 Prop osed a lgorithm: Dijkstra’s algorithm with b ounded list size Dijkstra’s algor ithm is an efficient a lgorithm to find the shortest path fr o m a point to a destination in a weigh ted gra ph [9]. Dijkstra’s a lg orithm uses the list of unlimited size to keep candidate no des. If w e use Dijkstra’s algorithm to find the shortest path from the r o ot to one of no des at the b ottom depth, we can get the no de with minimum || y − H ˆ x || 2 among all no des a t the b ottom depth and it co rresp onds to the ML estimate [10]. How ev er, this algo rithm still has high computational co mplexity . T o reduce the co m- putational co mplxit y , we prop ose a modified version of Dijkstra ’s algor ithm whos e list keeps only L no des with the smallest accumulated metric v alues in the list. W e show Dijkstra ’s alg o rithm with b ounded list size. 1. Create an empty list for no des. 2. Insert a ll no de s at the first level int o the list. 3. Select the no de A having smallest accumulated metric v alue in the list and remov e it fro m the list. If the depth of A is t , then output the no de A and its ancestor no des as the ML signal a nd finish this a lgorithm. 4. Insert a ll A’s child no des into the list. 5. Arrange the no de s in the list ac c ording to the ac- cum ulated metric v alue by the quick sor t. If the 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 0 5 10 15 20 25 30 SER SNR at single receive antenna [dB] QRD-MLD:X=2 Proposed:L=5 original QRD-MLD Proposed:L=16 ML Figure 1 : (4 × 4) s ymbol er ror rate list has more than L no des , select the L node s with the smallest a ccumulated metric v a lues in the lis t, a nd disc ard other no des from the list. 6. Go back to Step 3. The no de w ho se child no des are inserted into the list is called detection no de in this pap e r . Bec ause the discarded no des, which a re decided a t Step 5, and their descenda n t no des a re not exa mined, the pro- po sed algo rithm do se not examine all the candidate in S t according to (4). Thus, the prop os ed alg orithm dose no t a lwa ys find the ML signal. When we use LDPC co des [12] or turb o co de s [13] after detection, we have to co mpute N mo st likely signals [11]. Such signals can b e computed by this algorithm’s mo dification that is finished after out- put N signa ls with the smallest accumulated metric v alue. 4 Computer sim ulation In this sectio n, we compare the computationa l com- plexity , the num be r of detection no des and the num- ber of comparis o ns of real num b ers amo ng the pro- po sed algorithm and the QRD-MLDs. Throug hout the simulations, w e consider the following s ystem mo del. 3 0 500 1000 1500 2000 2500 0 5 10 15 20 25 30 average computational complexity SNR at single receive antenna [dB] original QRD-MLD QRD-MLD:X=2 Proposed:L=16 Proposed:L=5 Figure 2 : (4 × 4) av era ge co mputational co mplexity 0 1000 2000 3000 4000 5000 0 5 10 15 20 25 30 maximum computational complexity SNR at single receive antenna [dB] Proposed:L=16 original QRD-MLD QRD-MLD:X=2 Proposed:L=5 Figure 3: (4 × 4) maximum computationa l complexity • W e do tw o simulations. In the first simulation, the num b er of trans mit a ntennas t = 4, and the nu mber of receive antennas r = 4. In the s econd simulation, the num be r of transmit antennas t = 6, and the num b er o f r eceive antennas r = 6. • The signal constella tion at each transmit a n- tenna is 16- Q AM and all signals ar e drawn ac- cording to the uniform i.i.d. distr ibutio n. • The fading co efficients ob ey the C N (0 , 1) distri- bution, a nd the receiver knows it p erfectly . • The noise a t each recieve antenna o b e y s the 0 5 10 15 20 25 30 35 40 45 50 55 0 5 10 15 20 25 30 average number of detection nodes SNR at single receive antenna [dB] original QRD-MLD QRD-MLD:X=2 Proposed:L=16 Proposed:L=5 Figure 4 : (4 × 4) average num be r of detection no des 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 maximum number of detection nodes SNR at single receive antenna [dB] Proposed:L=16 original QRD-MLD QRD-MLD:X=2 Proposed:L=5 Figure 5: (4 × 4) max im um n umber o f detection no des C N (0 , φ ) distribution. φ is caluculated by φ 2 = tE s × 10 ( − S N R/ 10) , where E s is the average sym- bo l energy . • W e transmit 100 000 sig nals, which is 400 000 symbols if the num b er of transmit antennas is 4 and 600 000 sy m b ols if the num b er of tra nsmit antennas is 6, a nd every 100 signals, change the fading matrix. If M = 1 6 is used and the signal co nstellation is 16-QAM, QRD-MLD has symbol er ror ra te (SER) near to the ML detection [7]. So, we us e M = 1 6. In QRD-MLD’s improv ement, we use X = 2 in (6) 4 0 1000 2000 3000 4000 5000 6000 7000 8000 0 5 10 15 20 25 30 average number of comparisons of real numbers SNR at single receive antenna [dB] original QRD-MLD QRD-MLD:X=2 Proposed:L=16 Proposed:L=5 Figure 6: (4 × 4 ) a verage num be r of comparis ons o f real num b ers 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 5 10 15 20 25 30 maximum number of comparisons of real numbers SNR at single receive antenna [dB] Proposed:L=16 original QRD-MLD QRD-MLD:X=2 Proposed:L=5 Figure 7: (4 × 4) maximum nu mber of co mpa risons of real num b e rs as used in [8]. In order for the prop osed alg orithm to hav e the similar SE R to Q RD-MLD and its im- prov ement, we use t wo versions of prop os ed algorithm whose list sizes are L = 16 and L = 5. Fig ures 1 and 8 show that the prop osed alg orithm with L = 1 6, the original QRD-MLD and the ML a lgorithm hav e almost the same SER througho ut this simulations. The pro p o sed algorithm with L = 5 and QRD-MLD’s improv ement also hav e s imila r SER throug hout this simulations. 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 0 5 10 15 20 25 30 SER SNR at single receive antenna [dB] QRD-MLD:X=2 Proposed:L=5 original QRD-MLD Proposed:L=16 ML Figure 8 : (6 × 6) s ymbol er ror rate W e count the n umber of m ultiplications and divi- sions of co mplex num b ers as the computational com- plexity . Since the part of QR decompo s ition is the common par t of all c ompared alg o rithms, we do not include tha t pa rt in compar ison of complexity . In QRD-MLDs, we use the quick so rt to arra nge the no des and decide M no des with the smallest a c - cum ulated metric v a lue a t each depth. Because the QRD-MLD keeps M no des at e ach depth, the n umber o f detection no de s and the com- putational co mplexity are completely determined b y M . How ever, in the prop osed a lgorithm and QRD- MLD’s improvemen t, the num be r of detection no des and the computational complexity are not fixed. Figures 1–7 a re the results of first simulation w ho se nu mber of transmit antennas and receive antennas are 4. Figures 8– 14 are the r esults of second simula- tion whose num b er of transmit antennas and receive antennas are 6. A t first, we discuss the res ult of first s im ulation. According to Figures 2, 4 and 6, the prop ose alg o- rithm with L = 1 6 reduece the av erag e computational complexity , av er age num b er of detectio n no des a nd av erage nu mber of compar isons of real nu mbers from the orig ina l QRD-MLD. Mor eov er, in the ca se of high SNR, although the pro p o s ed algo rithm with L = 16 has muc h s ma ller SER than QRD-MLD’s improv e- men t ac cording to Figure 1 , the average computa- 5 0 1000 2000 3000 4000 5000 6000 0 5 10 15 20 25 30 average computational complexity SNR at single receive antenna [dB] original QRD-MLD QRD-MLD:X=2 Proposed:L=16 Proposed:L=5 Figure 9 : (6 × 6) av era ge co mputational co mplexity 0 3000 6000 9000 12000 0 5 10 15 20 25 30 maximum computational complexity SNR at single receive antenna [dB] Proposed:L=16 original QRD-MLD QRD-MLD:X=2 Proposed:L=5 Figure 10: (6 × 6) ma ximum computational co mplex- it y tional complexity of pro po sed algo rithm with L = 1 6 is almost the s ame as Q RD-MLD’s improv ement. In the ca se of low SNR, the av erage computational co m- plexity , average num b er of detection no des and aver- age num b er o f co mparisons of real num b ers o f the prop osed a lg orithm with L = 5 are lower than QRD- MLD’s improv ement. In the case of high SNR, the av- erage computational complexity , average num b er of detection nodes and av era ge num b er of co mparisons of r eal n umbers of prop osed alg orithm with L = 5 are almos t same as QRD-MLD’s improv ement while the prop ose d algo rithm ha s smaller SER accor ding 0 10 20 30 40 50 60 70 80 90 0 5 10 15 20 25 30 average number of detection nodes SNR at single receive antenna [dB] original QRD-MLD QRD-MLD:X=2 Proposed:L=16 Proposed:L=5 Figure 11 : (6 × 6 ) average num b er o f detection no des 0 50 100 150 0 5 10 15 20 25 30 maximum number of detection nodes SNR at single receive antenna [dB] Proposed:L=16 original QRD-MLD QRD-MLD:X=2 Proposed:L=5 Figure 12 : (6 × 6) maximum num b er o f detection no des to Fig ure 1. According to Figure s 3, 5 and 7, in the ca se of low SNR, maximum computational co m- plexity , maximum nu mber of detection no des and the maximum num b er of compar isons of real num b er s of the prop os e d algo rithm with L = 16 ar e higher tha n QRD-MLDs. How ever, b e cause the average compu- tational complexity , the average n umber of detection no des and the av erage num b er of co mparisons of re al nu mber of the prop ose d algorithm with L = 16 are low er than QRD-MLDs, we find tha t the pro po sed algorithm ra rely g e ts high co mputational complex- it y , larg e num b er o f detectio n no des or large num be r 6 0 3000 6000 9000 12000 15000 0 5 10 15 20 25 30 average number of comparisons of real numbers SNR at single receive antenna [dB] original QRD-MLD QRD-MLD:X=2 Proposed:L=16 Proposed:L=5 Figure 13: (6 × 6) av erag e num b er o f compar isons of real num b ers 0 5000 10000 15000 0 5 10 15 20 25 30 maximum number of comparisons of real numbers SNR at single receive antenna [dB] Proposed:L=16 original QRD-MLD QRD-MLD:X=2 Proposed:L=5 Figure 14: (6 × 6) maximum num b e r of comparis ons of real num b e rs of compariso ns of re al num b ers . According to Figure 8–1 4, which is the r e sult of second sim ulation, the c haracter istic of propo s ed al- gorithm dose not c hange with the num b er of anten- nas. 5 Conclusion In this pap er , we prop os e a near ML detection a l- gorithm. When the list s ize is adjusted so that the prop osed algorithm has the almost same symbol er- ror rate (SER) as the or ig inal QRD-MLD, the aver- age of the computational c o mplexity and the n umber of detec tio n no des are reduced. When the list size is adjusted so that the pro p o s ed algo rithm has the almost same symbo l erro r rate (SE R) as the QRD- MLD’s impr ovemen t, in the case of low SNR, b oth the average computational complexity and av era ge nu mber of detec tio n no des a re reduced and in the case of high SNR, the co mputational complexity and av erage num b er of detection no des of prop osed algo - rithm is a lmost sa me a s QRD-MLD’s improv ement while SER of the prop osed algor ithm b ecomes smaller than QRD-MLD’s improv ement. Ac kno wledgmen t W e would like to thank Prof. Kiyomichi Araki for drawing our atten tion to the refer ence [6]. This re - search is pa rtly supp orted by the International Com- m unications F o undation. References [1] E. T elatar , ” Capacity of multi-an tenna Gaussian channels,” Eur o p. T rans. T elecommun., vol.10, pp.585–5 95, Nov. 1 999. [2] G. J. 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