SIMPLIFIED ROBUST FIXED-COMPLEXITY SPHERE DECODER

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1 19 European Signal Processing Conference (EUSIPCO 2011) Barcelona, Spain, August 29 - September 2, 2011 SIMPLIFIED ROBUST FIXED-COMPLEXITY SPHERE DECODER Yuehua DING, Yide WANG, Jean-François DIOURIS IREENA, Ecole Polytechnique de l Université de Nantes, yuehua.ding@univ-nantes.fr ABSTRACT A simplified robust fixed-complexity sphere decoder (SRFSD) is proposed in is paper. SRFSD reduces e complexity of robust fixed complexity sphere decoder (RFSD) by choosing a detection order minimizing e upper bound of e power of e interference in single expansion (SE) stage. Theoretical proof is given to support e feasibility of SRFSD. Simulation results show at SRFSD retains e robustness of RFSD, and sharply reduces e complexity of RFSD wi little sacrifice in bit error rate (BER) performance. ROOT 1+j -1-j -1+j 1-j Level 4, n4=4 Level 3, n3=1 1. INTRODUCTION Multiple input multiple output (MIMO) has become one of e most promising technology in wireless communication, but e detection of MIMO system is really a timedemanding process. The maximum lielihood (ML) detection has an unaffordable complexity and e suboptimal algorims wi acceptable complexity usually have significant performance degradation. Various algorims have been proposed to achieve e tradeoff between performance and complexity. The zero-forcing (ZF) [1] detector has an attractive low complexity but wi bad performance. Minimum mean square error (MMSE) [2] technique is superior in performance compared to ZF. However, ere is still a huge gap between MMSE and ML. Even if MMSE and ZF are improved by MMSE ordered successive interference canceling (MMSE-OSIC) [3] and ZF ordered successive interference canceling (ZF-OSIC) [1], respectively, ese gaps still can not be effectively filled. The sphere decoder (SD) [4] does greatly reduce e complexity of ML wiout significant performance sacrifice. The main drawbac of SD lies in its variable complexity and e choice of radius, which deps on e system parameters such as SNR and channel conditions. A lower bound on its average complexity has been shown to be exponential [5]. Fixed-complexity sphere decoder (FSD) [6] can efficiently fix e complexity order of SD wi a quasi-ml performance. However, FSD is not robust to e antenna configurations (e.g. N T > N R ). The principe of FSD is based on e assumption at N T N R. For a MIMO configuration wi N T > N R, is meod is no more effective since e left pseudo inverse of channel matrix doesn t exist. To overcome is problem, robust fixedcomplexity sphere decoder (RFSD), which is robust to all e antenna configurations, is proposed in [7]. Besides its robustness, RFSD has better BER performance an at of FSD. The problem wi RFSD is its computational complexity. The purpose of is paper is to develop furer e wor in [7], and find a simplified RFSD (SRFSD) wi reduced This wor is supported by China Scholarship Council (No ), National natural science foundation of China (No ) and Guangdong natural science foundation (No ). Level 2, n2=1 Level 1, n1=1 Figure 1: FSD(1, 1, 1, 4) search tree (QPSK, N T = 4, p=1) complexity but wiout significant performance degradation. Notation: Capital letters of boldface and lowercase letters of boldface are used for matrices, column vectors, respectively. ( ) H, ( ) T, ( ), ( ), E[ ] and denote operation of Hermitian, transpose, complex conjugate, left pseudo inverse, expectation and Frobenius norm respectively; A(:, ) represents e column of matrixa;i N is e N N identity matrix; 0 represents zero matrix or vector. tr(a) denotes e trace ofa. 2. MIMO COMMUNICATION SYSTEM MODEL The MIMO system model considered in is paper is a V- BLAST [1] system wi N R antennas at e receiver and N T antennas at e transmitter. N R is not necessarily equal or greater an N T. The symbols of transmitted vector x = [x 1,x 2,,x NT ] T are indepently drawn from M-QAM constellation of M points. The received vector y is given by equation (1): y=hx+w (1) where w is an additive white Gaussian noise vector, H = [h 1,h 2,...,h NT ] represents e frequency-flat channel. In e next sections, e following assumptions are made: 1. E[xx H ]=PI NT, 2. E[ww H ]=σ 2 I NR, 3. E[wx H ]= FSD 3. FSD AND RFSD FSD proposed by [6] efficiently fixes e complexity of SD by searching a fixed number of candidate vectors. EURASIP, ISSN

2 FSD follows a search order which is determined by a preprocessing algorim dividing e searching into two stages: full expansion (FE) stage in e first p levels (level N T, N T 1,, N T p+1 ) and single expansion (SE) stage in e remaining N T p levels (level N T p, N T p 1,, 1 ), as shown in Fig. 1. In FE stage, FSD, similar to SD, is usually considered as a tree search: M branches are generated by each node at e first p levels. In SE stage, unlie SD which should calculate all possible children according to e information of its current node, FSD only needs to calculate one child for each node. The preprocessing algorim proposed in [6] can be expressed as follows: Preordering o f FSD index=[] H NT =H f or i=n T : 1 : 1 H i =(H H i H i ) 1 H H i de f ine γ i =[ (H i ) 1 2, (H i ) 2 2,, (H i ) N T 2 ] T i f n i = M l i = arg{max j (γ i )}, j {1, 2,, N T } {index} else l i = arg{min j (γ i )}, j {1, 2,, N T } {index} H i (:,l i )=0, H i 1 =H i,index=[l i, index] where H i is e channel matrix at level i, it preserves i columns of H, wi oer N T i columns zeroed. H i is obtained by zeroing e li+1 column of H i+1. (H i ) j is e j row of H i. γ i is a reference vector indicating e post-processing noise amplification of H i. The searching order is totally determined by e vectors num and index. Vectornum divides e searching into FE stage and SE stage, and vectorindex indicates e order of FE and SE. Remars 1: One should note at e definition of( ) is exactly e same as at in [1], [6]. It is necessary to explain e left pseudo inverse of a matrix in our case, we assume a n m (n m) matrix Λ wi a1, a 2 Λ, e left pseudo inverse of Λ should meet e following conditions: Λ Λ = diag(λ 1, λ 2,, λ m ), where λ i = 1, if i / {a 1, a 2,,a }, λ i = 0 if i {a 1, a 2,,a }.,,a columns zeroed < m, It is possible at Λ is not unique, e one wi e smallest Frobenius norm is selected for V-BLAST. Obviously, for e optimal Λ, e a1, a 2,,a rows of Λ are zero. We assume Λ 1 is e matrix obtained by removing a1, a 2,,a zero columns from Λ and we also assume at Λ 2 is e matrix obtained by removing a1, a 2,,a zero rows from Λ, en we have Λ 1 = Λ RFSD RFSD [7] generalizes e FSD problem as follows: choosing a suitableh NT p wi e minimal post-processing noise amplification in following SE stages by zeroing suitable p column(s) of H. There are C N T p N T = C p N T possible channel matrices H NT p at level N T p, we denote H r, r = 1, 2,, C p N T to distinguish em, different value r corresponds to different detection order O r, O r (s) denotes e index of e detected signal at level s. The corresponding channel matrix in SE stage at level s, represented by H r s,, 2,, N T p 1, can be obtained by performing OSIC [1] on H r. We denote αr s as a reference vector to indicate e post-processing noise amplification as follows: α r s =[ (H r s) 1 2, (H r s) 2 2,, (H r s) N T 2 ] T (2) where(h r s) j is e j row of (H r s). α r s(o r (s+1)),, α r s(o r (N T )) are ignored and set to 0 since signal components x Or (s+1),, x Or (N T ) are detected and canceled in previous levels s+1,, N T. Denoting c H rs =(H r s), =arg(min jα r s), j O r (s+1),...,o r (N T ) (3) e order which minimizes e post-processing impact on SE stage being chosen, e process can be written as: arg min r c rs 2 (4) The optimal order can be recorded by index, RFSD is performed according to p, index and c rs. The ordering process of RFSD wi p=2 can be expressed as follows: Preordering o f RFSD index=[ ],H NT =H, r=0, sum=+ f or m=1:1:n T loop 1 f or l = m+1 : 1 : N T loop 2 r=r+ 1, S r =[m, l], sum=0 O r = S r,h=h NT,H(:,l)=0,H(:,m)=0,H r =H f or i=n T p : 1 : 1 (H r i ) =((H r i )H H r i ) 1 (H r i )H de f ine α r i =[ (Hr i ) 1 2, (H r i ) 2 2,..., (H r i ) N T 2 ] T c H ri =(Hr i ),=arg(min jα r i ), j {1, 2,...,N T} O r H r i (:,)=0,Hr i 1 =Hr i, O r =[, O r ] sum=sum+ c ri 2 i f sum<sum sum=sum,index=o r 4. SRFSD RFSD does have bo excellent performance and robustness. However, one notes at we need calculate (N T p)c p N T left pseudo inverses. To reduce e complexity, we consider e reference vector at level N T p only. Reference vector α r 2 can be expressed as: α r 2 = tr[(h r ) ((H r ) ) H ] (5) 117

3 Antenna array (4,4), QPSK RFSD(1,1,1,4) SRFSD(1,1,1,4) FSD(1,1,1,4) Antenna array (5,4) QAM 16QAM 10 1 BER QPSK BER Figure 2: Performance of SRFSD (QPSK, N T = 4,N R = 4) 10 4 RFSD(1,1,1,1,M) SRFSD(1,1,1,1,M) FSD(1,1,1,1,M) Lemma: namely α r 2 is e upper bound of N T p c rs 2, c rs 2 tr[(h r ) ((H r ) ) H ] (6) Proof : Assuming O r is an arbitrary detection order, O r (s) represents e index of e detected signal component at level s, s=n T, N T 1,, 1. It is natural at H r s(:,o r (i))=0 for i = N T, N T 1,, s+1. We consider e following two optimization problems. Problem I: Find a suitable N R 1 vector v s 1, (2 s N T p) which minimizing e following criterion for a given q, 1 q s 1. minj 1 =vs 1 H v s 1 s.t. 1. f or all q,1 s 1 vs 1 H Hr s 1 (:,O (7) r())=0 2. vs 1 H Hr s 1 (:,O r(q))=1 Problem II: Find a suitable N R 1 vector v s, (2 s N T p) which minimizing e following criterion for a given q, 1 q s 1. minj 2 =vs H v s s.t. 1. f or all q,1 s vs H H r s(:,o r ( (8) ))=0 2. vs H H r s(:,o r (q))=1 Problem II is equivalent to minj 2 =vs H v s s.t. 1. f or all q,1 s 1 vs H H r s 1 (:,O r())=0 2. vs H H r s 1 (:,O r(q))=1 3. vs H H r s(:,o r (s))=0 The two problems are exactly e same except at prob- (9) Figure 3: Performance of SRFSD (N T = 5,N R = 4) lem II has one more constraint (Constraint 3) an problem I, which means at e optimal solution of problem II is just a suboptimal solution to problem I, namely min (vs 1 H v s 1) min(vs H v s ). Actually, e optimalvs 1 H andvh s are given by (H r s 1 ) O r (q) and (Hr s) O r respectively. Thus, we have e (q) following inequality: (H r s 1) O r (q) 2 (H r s) O r (q) 2... (H r ) O r (q) 2 (10) The signal component wi index O r (s 1) is detected at level s 1 if detection order O r is adopted, which means at q=s 1, we have (H r s 1) O r (s 1) 2 (H r ) O r (s 1) 2 (11) which in turn leads to (H r s 1) O r (s 1) 2 (H r ) O r (s 1) 2 (12) Add (H r ) O r () 2 to bo sides of inequality (12), e right side becomes tr[(h r ) ((H r ) ) H and e left side becomes: (H r s 1) O r (s 1) 2 + (H r ) O r () 2 (13) O r is just an arbitrary detection order, for e order proposed by RFSD, formula (13) is N T p c rs 2, finally, N T p c rs 2 tr[(h r ) ((H r ) ) H ] is derived. The purpose of SRFSD is to minimize e upper bound of N T p c rs 2 by choosing a proper detection order O r at 118

4 determines which layers need a FE. It can be described as follows: Preordering o f SRFSD index=[],h NT =H, r=0 f or m=1:1:n T loop 1 f or l = m+1 : 1 : N T loop 2 r=r+ 1, S r =[m, l] H=H NT, H(:,l)=0,H(:,m)=0,H r =H (H r ) =((H r )H H r ) 1 (H r )H =argmin r tr[(h r ) ((H r ) ) H ] Remars 2: 1. SRFSD listed above can find an optimal set S which contains e indexes of layers needing to be fully expanded. The detection order of oer layers can be obtained exactly lie SE in traditional FSD. 2. We can learn at only C p N T + N T p left pseudo inverses need to be calculated. The number is 2N T 1 when p=1, a little more an traditional FSD which needs to calculate N T left pseudo inverses Antenna array (5,4), QPSK RFSD(1,1,1,1,4) SRFSD(1,1,1,1,4) FSD(1,1,1,1,4) Antenna array (5,4), QPSK RFSD(1,1,1,1,4) SRFSD(1,1,1,1,4) FSD(1,1,1,1,4) SIMULATION RESULTS The BER performance of SRFSD has been examined by means of Monte-Carlo simulation. The simulated MIMO systems are V-Blast systems wi different configurations of antennas. In addition, E b /N 0 is defined as E b /N 0 = P σ 2 log 2 M, E b/n 0 is represented in db in e figures of is paper. Groups of simulation have been done to compare SRFSD wi RFSD, FSD, and in terms of BER and flops/symbol, e results are shown in Fig.2-5. Fig.3-5 simulate a 5 4 system wi difference modulation schemes (where e left pseudo inverse H is still calculated by e MATLAB function pinv( ), we just use it as a demonstration alough e left pseudo inverse of H dose not exit.), it is shown by Fig.3-5 at, compared wi RFSD, SRFSD can still achieve excellent BER performance wi greatly reduced complexity, even rough N T > N R. In addition, Fig.4-5 shows at almost bo RFSD and SRFSD have lower complexity an. Fig. 2 is obtained based on a 4 4 QPSK system. The BER curves of FSD, RFSD, SRFSD and ML, in Fig. 2, are overlapped by one anoer. In conclusion, SRFSD, not only retains e robustness of RFSD, but also sharply reduces e complexity of RFSD wi little performance degradation. 6. CONCLUSION The selection of e signal components to be detected at FE stage is a ey factor for FSD, RFSD and SRFSD. RFSD is based on e selection in FE stage wi e smallest post processing impact on SE stage. It is better an original FSD in terms of BER and robustness, but it introduces substantial computational complexity. To alleviate e complexity of RFSD, a simplified RFSD is proposed in is paper by selecting e signal components in FE stage to minimize e upper bound of e power of e interference in SE stage. Simulation shows at SRFSD not only retains e robustness of RFSD, but also sharply reduces e complexity of RFSD Figure 4: Complexity comparison (QPSK, N T = 5,N R = 4) 7 x 104 Antenna array (5,4), 16QAM RFSD ZF(1,1,1,1,16) SRFSD ZF(1,1,1,1,16) FSD ZF(1,1,1,1,16) Antenna array (5,4), 16QAM RFSD ZF(1,1,1,1,16) SRFSD ZF(1,1,1,1,16) FSD ZF(1,1,1,1,16) Figure 5: Complexity comparison (16QAM, N T = 5,N R = 4) 119

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