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Cyber Journals: Multidisciplinary Journals in Science and Technology, Journal of Selected Areas in Telecommunications (JSAT), July Edition, 2012 Adaptive Tree Search Detection with Variable Path Expansion Based on Gram-Schmidt Orthogonalization in MIMO Systems Wei Hou, Student Member, Tadashi Fujino, and Toshiharu Kojima, Member, IEEE forcing (ZF) or minimum mean squared error (MMSE) with the Abstract—This paper proposes new adaptive tree search lattice-reduction (LR) technology can offer a remarkable detection with variable path expansion based on Gram-Schmidt complexity reduction with performance loss [4]-[7]. Numerous (GS) orthogonalization (GSO) in MIMO systems. We adopt the suboptimal detection techniques have been investigated to GSO procedure to reduce the channel matrix instead of the QR-decomposition in the conventional QRM-MLD. This detection approximately approach the ML performance with relatively scheme combined the GSO reduction with the M-algorithm, what lower complexity, such as the sphere detection (SD) and the we call GSM-MLD, can achieve near-ML performance as the MLD with QR Decomposition and M-algorithm (QRM-MLD) conventional QRM-MLD. The proposed detection method is a [8]-[16]. To looking for the suboptimal detection algorithm with breadth-first algorithm and performs the adaptive tree search with the near optimal performance and the affordable complexity variable path expansion in the GSM-MLD. In this paper, we costs for MIMO gains faces a major challenge. introduce a path metric ratio function to evaluate the reliability for all the survived branches. The survived but lower reliable The conventional QRM-MLD is one solution to relatively branches adopt parts of the constellation points as the candidates reduce the complexity while retaining the ML performance. The into the next detection layer. The proposed detection algorithm number of M in the QRM-MLD is defined as the number of the reduces the complexity by adaptively decreasing the computation survived branches in each detection layer of the tree search, of the path metric for the low reliable candidates. The numerical which is a tradeoff between the complexity and the performance. results exhibit that the proposed scheme achieves near-ML Furthermore, the value of M should be large enough to ensure performance with relatively lower complexity compared to the conventional QRM-MLD. that the correct symbols exist in the survived branches under the ill-conditioned channel, in particular for the large size MIMO Index Terms—Adaptive signal processing; Gram-Schmidt (GS) and the high modulation order. Hence, the conventional orthogonalization (GSO); QRM-MLD; MIMO; tree search. QRM-MLD still requires high complexity in the high Eb/N0 region [10]. To overcome this drawback, numerous methods I. INTRODUCTION with adaptively controlling the survived branch M have been M ultiple-input multiple-output (MIMO) technology has attracted attention in wireless communications, since it provides significant increases in data throughput and the high proposed in [11]-[13]. These schemes still have the problem that needs to accurately and dynamically measure SNR for optimal setting of the number of survived branch in each layer. spectral efficiency [1]-[3]. MIMO systems employs multiply In this paper, we first present a detection scheme combined antennas at both ends of the wireless link, and hence can the Gram-Schmidt (GS) orthogonalization (GSO) reduction increase the data rate by transmitting multiple data streams. To with the M-algorithm, which we call the GSM-MLD. This exploit the potential gains offered by MIMO, signal processing scheme has such features that it achieves near-ML BER involved in a MIMO receiver requires a large computational performance like the QRM-MLD with lower computational complexity in order to achieve the optimal performance. The complexity. The channel matrix is reduced using the GSO maximum likelihood (ML) detection (MLD) is known as the procedure, and meanwhile a transform matrix is created. In optimal receiver in terms of minimizing bit error rate (BER). contrast to the QR decomposition of the channel matrix in the However, the complexity of MLD obstructs its practical QRM-MLD, which R retains the property of the channel matrix, implementation. The common linear detection such as zero the column vectors of the GS-reduced channel matrix are purely orthogonal for the GSM-MLD. The GS-reduced channel matrix spans the same subspace as the columns of the original channel Manuscript received July 10, 2012. Wei Hou is with the University of Electro-Communications, Tokyo, Japan matrix. The transform matrix is an upper triangular matrix with (phone: +81 42-443-5235; fax: +81 42-443-5291; e-mail: houwei@ uec.ac.jp). unity diagonal entries. Tadashi Fujino, emeritus professor, is with the University of Based on the GSM-MLD, we propose novel adaptive tree Electro-Communications, Tokyo, Japan (e-mail: tad-fujino@mwb.biglobe.ne. jp). search detection with variable path expansion based on GSO in Toshiharu Kojima is with the University of Electro-Communications, Tokyo, the MIMO systems. The proposed algorithm retains the same Japan (e-mail: kojima. toshiharu@ uec.ac.jp). breadth of the tree search as the GSM-MLD to achieve the 1 near-ML performance, and however the number of the possible Re(sc ) Re(y c ) Re(z c ) branches is adaptively controlled. The adaptive scheme avoids a s , y ,z (4) large amount of the path metric evaluations and sorting to Im(sc ) Im(y c ) Im(z c ) reduce the computational complexity. We also analyze the complexity of the proposed detection. The proposed detection Letting n2Nr and m2Nt, we define the dimension of the can considerably decrease the complexity in the high Eb/N0 real-valued channel matrix H to be nm. The dimensions of the region. vectors in (4) are given as yn, zn and sm, where The remainder of this paper is organized as follows. Section II denotes the finite set of the real-valued transmitted signals. This presents the system model and the conventional QRM-MLD set is given by {1, 3,..., ( K 1)} for K-QAM algorithm. Section III explains the GSM-MLD algorithm. In (Quadrature Amplitude Modulation). Given y and the channel Section IV, we propose an adaptive tree search scheme to the matrix H, the ZF soft estimate of the transmitted signals is GSM-MLD in MIMO systems. Section V gives numerical expressed as results and discussions. Finally, we summarize and conclude the paper in Section VI. Notations: Matrices and vectors are denoted by bold-face s(ZF) H† y (HT H)1 HT y (5) letters. AT, A1 and A† are used to denote the transpose, inverse, and pseudo-inverse of a matrix A, respectively. The The concept of the QRM-MLD is to apply a tree search to real and imaginary parts are denoted as Re[·] and Im[·]. The detect the symbols in a sequential manner [10]. The channel operator [·] is the quantization. ||·|| represents the Frobenius matrix H applies the QR decomposition as HQR, where Q is a norm. ai,j denotes the entry at the i-th row and the j-th column of unitary matrix: i.e., QTQIm, and R is an mm upper triangular A. matrix. The QR decomposition is executed by the modified GS algorithm (MGS) in [17]. The R retains the property of the II. SYSTEM MODEL AND CONVENTIONAL QRM-MLD channel matrix H. Then, we pre-multiply both the hand sides of Consider a multiple antenna system with Nt transmit and Nr (2) by QT as (NrNt) receive antennas. The signals are transmitted over a y QT y QT (QRs z) Rs z (6) rich scattering flat fading channel. Assume that the receiver has perfect knowledge of the channel state information (CSI). The with expressing R as received signal vector yc [ y1 , , yNr ]T Nr1 is expressed as c c y c Hc sc z c (1) r11 r12 r1, m r22 r2, m R (7) where yic is the received signal at the i-th receive antenna. The transmitted signal vector is denoted as sc [s1 , , sNt ]T ΩNt1, c c O rm,m where each symbol s c at the j-th transmit antenna is chosen j from a finite subset of the complex-valued integer set Ω. Let where z QT z . The ML detector searches over the whole set Hc [h1 , , hc t ] denote the NrNt channel matrix. We assume c N of transmitted signals sm, and decides the transmitted signal that the entries of H c are of the i.i.d. complex Gaussian process ˆ s(ML) in terms of the minimum Euclidean distance (ED) to the with zero mean and unity variance. The noise vector received vector y. The ML detection can be formulated as zc [ z1 , , zNr ]T Nr1 is the additive white Gaussian noise c c (AWGN) vector, of which each entry is assumed to be zero s(ML) arg min y Hs ˆ 2 arg min y Rs 2 mean and variance of N0, the one-sided noise power spectral s m s m (8) density. arg min im 1| yi mi ri , j s j |2 As the system model in (1) is complex-valued, treating the j s m real and imaginary parts separately, the system model can be rewritten as where i | yi mi ri, j s j |2 denotes the branch metric in the j i-th layer. The accumulated branch metric i mi j is j y Hs z (2) defined as the path metric from the m-th layer down to the i-th layer. For each detection layer of the tree search in the with the real-valued channel matrix and the real-valued vectors QRM-MLD, there are three major operations: Re(Hc ) Im(Hc ) nm Candidate Expansion: Expand the children nodes from H (3) Im(Hc ) Re(Hc ) each survived branch. The candidates for the children nodes consist of all the constellation points. 2 Path metric evaluations: There are M K possible ˆ original matrix H. Using the GS-reduced channel matrix H branches for K-QAM in each layer. Calculate the path and Tˆ , we have metric for all the possible branches. Sorting and retaining: Sort the path metric and retain M ˆ ˆ ˆ y Hs z (HT)(T1s) z Hv z (9) branches with the smallest path metric from M K possible branches. The rest of branches discard. ˆ where H ˆ HT and v ˆ ˆ T1s with expressing T1 as Let Λi(l) denote the l-th smallest path metric of the survived path i(l) after the operations of sorting and retaining, where 1 12 13 1, m l[1,M] and Λi(1)Λi(2) ... Λi(M). Correspondingly, the partial 1 23 2,m transmitted signal ŝi(l) based Λi(l) is expressed as ŝi(l)[ŝi (l),…, ˆ T1 (10) ŝm(l)]T . The same operations are executed until the first layer. 1 m 1,m The output of the QRM-MLD is ŝ1(1)[ŝ1(1),…, ŝm(1)]T as the O 1 final estimate of transmitted signal. Although the exhaustive tree search of the QRM-MLD ˆ With the orthogonal column vectors of H , the soft estimate should visit M K nodes in each detection layer instead of of v is derived as ( K )mq 1 nodes in the i-th layer for the full MLD. The conventional QRM-MLD reduces the exponentially growing ˆ ˆ ˆ ˆ ˆ v T1s (ZF) H† y (H T H)1 H T y complexity to a linear growing complexity while retaining the T h ˆ ˆ h2 ˆ hm (11) ML performance. However, the conventional QRM-MLD still 1 2, , , y ˆ || h1 || ˆ 2 ˆ || h m || 2 requires high complexity in the high Eb/N0 region. || h 2 || III. GSM-MLD ˆ ˆ or vi hiT || hi ||2 y, i [1, m] (12) Based on Fujino et al.’s previous work of the GSO based Then, the soft estimate of ŝ is obtained by performing the lattice-reduction aided detection in MIMO systems [4,5], we following recursion as introduce the GSM-MLD algorithm. The column vectors of channel matrix H are first sorted in ascending order in length. [vi ] : i m Then, they are weakly reduced using the GSO procedure shown si ˆ (13) [vi j i 1 i , j s j ] : i m 1,...,1 ˆ m in Table I. This algorithm transforms the channel matrix H to ˆ create the GS-reduced channel matrix H and the transform ˆ ˆ matrix T . The column vectors of H are mutually orthogonal, A. Definition of Metric in GSM-MLD and the transform matrix T ˆ is an upper triangular matrix with The GSM-MLD applies a fixed number of M in each ˆ unity diagonal entries and det{ T }. Note that this algorithm detection layer as the QRM-MLD, starting from the last entry in Table I is computationally-simple since it weakly reduces the ˆ of s. Since T1 is an upper triangular matrix, the entry si column vectors of H without the size reduction in the LLL depends on the decided estimates ŝj’s where j[i1,m]. We algorithm [4]. define the branch metric i : i[1,m] in GSM-MLD as TABLE I. GRAM-SCHMIDT ORTHOGONALIZATION ˆ ˆ || hi ||2 | vi si |2 || hi ||2 | si si |2 , i m ˆ ˆ (1) Begin Input H [h1,..., hm ], T : I m [t1,..., t m ] . i (14) ˆ 2 ˆ 2 ˆ 2 || hi || | vi si mi 1 i , j s j | || hi || | si si | , i m ˆ ˆ 2 ˆ j Set h p h p , p [1, m]. (2) for p:2 to m where sm vm and si vi mi 1 i, j s j for i=m1,…,1. The j ˆ (3) for q:p1 down to 1 ˆ ˆ path metric Λi: i[1,m] is the accumulated branch metric, which hTh p (4) p,q q is defined as ˆ || h ||2 q (5) ˆ ˆ ˆ ˆ ˆ ˆ h p : h p p, qhq , t p : t p p ,q t q ˆ i mi j mi || h j ||2 | si si |2 ˆ j j (6) end i , i m (15) (7) end (8) End i i 1 , i m ˆ The upper triangular matrix T with unity diagonal entries The Λi is the partial Euclidean distance (PED). In the ˆ ˆ is invertible. The column vectors of the matrix H HT are GSM-MLD, Λi(l) still denotes the l-th smallest path metric. orthogonal and span the same subspace as the columns of the Correspondingly, the partial transmitted signal ŝi(l) based on 3 Λi(l) should be expressed as [ŝi(l),…, ŝm(l)]T. Three major According to MLD, the final estimate of transmitted signal is operations are the same as the conventional QRM-MLD. The determined by the path with the smallest path metric. To a output of the GSM-MLD is ŝ1(1)[ŝ1(1),…, ŝm(1)]T as the final certain degree, we can apply the PED to evaluate the reliability estimate of transmitted signal. of all the survived paths in a detection layer of the tree search. B. Computational Complexity In that sense, we introduce a ratio function among the path metrics in the i-th layer, where i[1,m], defined as We here use the floating point operations (flops) for the measure of the complexity, which defines one addition, one i(l ) subtraction, one multiplication, and one division for real-valued i (l ) , l 1, M (17) i(1) number to take one flop. For the m-th layer, expanding K branches, m in (14) requires two multiplications and one where Λi(1) denotes the smallest path metric after sorting the subtraction, and it consumes 3 flops expressed by ( m ) 3 . survived branch in the i-th layer. Note that the layer number i is For the (m1)-th layer down to the first layer, M branches are decreased such that i:m down to 1 successively. In general, the retained from M K possible branches in the i-th layer, where survived path i(1) with the high probability should be the i[1,m1]. For a survived branch, si in (14) requires (mi) correct path if the channel is better-conditioned. Hence, we multiplications and (mi) subtractions. Hence, the complexity assume that the survived path i(1) has the most possible to be for the computing of si is expressed as (si ) 2(m i) . For a correct path. In terms of the path metric ratio i(l) in (17), possible branch, Λi in (15) requires one addition, which we indirectly evaluate the reliability of the l-th branch in the i-th express the complexity as ( i ) 1 . ( i , i ) ( i ) layer. That is, if Λi(l) is much larger than Λi(1) and thus i(l) is ( i ) 4 denotes the total complexity for the computations larger, it illustrates that the correct path with lower reliability is of the branch metric i in (14) and the path metric Λi in (15). the l-th path. The complexity of the GSM-MLD GSM-MLD which The ratio function i(l) can be the measure of evaluating the excludes the complexity of the GSO reduction and the reliability for the l-th branch. In order to adaptively control the computation of v in (11) can be derived as candidates expansion according to i(l), we assume that the GSM-MLD K ( m ) im1 [ M 1 ( si ) M K ( i , i )] number of the candidates should be a integer between 1 and m -th layer Survived Branches Path Expansion K in the (i1)-th layer. That means the number of candidates 3 K im 1 M 2(m i) M K 4 1 (16) from a parent node is determined by the path metric ratio in the previous layer. We define the number of the candidates as 3 K M (m2 m) 4M K (m 1) i1(l) for the l-th survived branch in the (i1)-th layer. In order to find a proper rule to adaptively assign the candidates for a IV. PROPOSED ADAPTIVE TREE SEARCH SCHEME IN survived branch, we consider a decision function of the i-th GSM-MLD layer as In this section, we propose an adaptive tree search scheme in i the GSM-MLD. The proposed algorithm retains the same (i) C , i 1, m (18) breadth of the tree search as the GSM-MLD to achieve the m near-ML performance. On the other hand, we perform adaptive tree search scheme to reduce the complexity, and to overcome where C is a constant to be predetermined, which is the tradeoff the drawback which the fixed number of tree search algorithm between the BER performance and the computational requires high complexity in the high Eb/N0 region. In the complexity. The parameter (i) is depended on the detection adaptive tree search scheme, we introduce a path metric ratio layer i. Since the tree search starts with the last entry of s, the without the necessary to accurately and dynamically measure path metric at first in the larger numbered layer is insufficient to SNR. According to the reliability of each survived branch, reflect the whole channel condition. To retain the correct path, assign a suitable candidates expansion from a parent node. To the parameter (i) is defined to be proportional to the value of decrease the number of lower reliable possible branch, thereby the detection layer i. Using the variable decision function, the avoid a large amount of the path metric evaluations and sorting. value (m) is maximum as C. Correspondingly, the value (1) A. Reliability Evaluation is minimum as C/m. The decision value becomes strict as the In this subsection, we derive the reliability evaluation (RE) detected layers increase. The various decision based on the for all the survived branches in each layer. As above mentioned, layer number significantly reduces the number of candidates in the estimate of entry si depends on the decided estimates ŝj’s the smaller numbered layer seen in the Section V. where j[i1,m]. Hence, the wrong estimate existing in the For K-QAM, the number of the finite set for the real-valued decided estimates may cause more wrong estimates of the transmitted signals is K .We compare i(l) with {(i), 2 transmitted signal in the following recursion detection. (i), ... , ( K 1) · (i)}. Then we have 4 GSM-MLD with the proposed detection with C{2, 4, 8} for 16QAM and 64QAM, respectively. The results illustrate that the CDF curve of the proposed detection closely approaches the that of GSM-MLD as the value of constant C increases. The constant C4 is almost optimal value between the BER performance and the complexity. GSM-MLD (M16) B. Proposed Detection Scheme Proposed Detection (M16, C2) Proposed Detection (M16, C4) As an example, Fig. 3 illustrates an adaptive tree search Proposed Detection (M16, C8) scheme from the i-th layer to the (i1)-th layer. In the (i1)-th layer, first perform the path expansion from M survived branches in the i-th layer. Since the adaptive tree search scheme is executed, the branch metric and the path metric can be Fig. 1 The CDF of the minimum path metric at Eb/N010dB for 16QAM. expressed as i(1,1) , , i(1, i 1(1)) , i(2,1) , , i(2, i 1(2)) , i(1 ,1) , , i(1 , i 1( M )) 1 1 1 1 M M and i(1,1) , , i(1,1 i 1(1)) , i(2,1) , , i(2, i 1(2)) , i(1 ,1) , , i(1 , i 1( M )) , 1 1 1 M M respectively. Note that i 1 : k [1, i 1 (l )] represents the (l , k ) branch metric expanded from the l-th branch in the (i1)-th GSM-MLD (M64) layer. We calculate the path metric for the possible branches as Proposed Detection (M64, C2) i(,1 ) i(1k ) i(l ) . Hence, lM1 i 1 (l ) denotes the total lk l, Proposed Detection (M64, C4) number of all the children nodes in the (i1)-th layer, which Proposed Detection (M64, C8) should be equal to or less than M K . Next, sort lM1 i 1 (l ) path metrics and select M with the smallest path metric. Based on the sorted i)1 , calculate the number of candidates (l Fig. 2 The CDF of the minimum path metric at Eb/N015dB for 64QAM. expansion i2(l), l[1,M], for the next layer. The proposed adaptive tree search scheme is summarized as follows: K for i (l ) [0, (i)] Step 1: Set a fixed value of M. For K-QAM, if K <M, define i 1 (l ) K x for i (l ) ( x (i), ( x 1) (i)] (19) a layer number q such that ( K )mq 1 should be equal to or more than M in order to select M branches with the 1 for i (l ) ( K 1) (i) smallest path metric among all of the possible branches. Then, the candidates from the m-th layer down to the where i[2,m] and x[1, K 2 ]. Let (i) denote the basic q-th layer are all the constellation points. unit to divide i(l) into K regions. Then, according to i(l) in Step 2: Start the adaptive candidate selection scheme from the which region resolves the number of candidates i1(l). q-th layer. According to q(l) and (q), the number of Ranking the constellation points with the nearest distance to the candidates q1(l) for the l-th survived branch in the l) si(1 obtained in (14), the candidates in the (i1)-th layer (q1)-th layer is obtained in (19). Hence, the number of consist of the nearest constellation point up to the i1(l)-th the possible branches in the (q1)-th layer is from M to nearest constellation point. In the case of 16QAM, if si(1 2.5, l) M K. the order of candidates is {3,1,1,3}. If i1(l)2, the Step 3: Proceed to the next stage of the (q1)-th layer. Rank the candidate selection from the constellation points is {3, 1}. constellation points for the l-th survived branches with Due to the definitions of the branch metric and the path metric in the GSM-MLD, the ED can be expressed as the nearest distance to sql)1 in (14). According to q1(l), ( we select the candidates from the constellation points ˆ and calculate the path metric for the possible branches. || y Hs ||2 im 1 || hi ||2 | si si |2 ˆ (20) M branches are retained with the smallest path metric to the next layer. The same operations are executed until The maximum likelihood detection is very simple to implement the first layer. since the decision criterion depends on the ED. This detection Step 4: Obtain the detection result of the estimate scheme minimizes the probability of bit error when the transmitted messages are equally likely. Since the proposed ŝ1(1) =[ŝ1(1),…, ŝm(1)]T. detection expects to achieve the near-ML performance as GSM-MLD, we first investigate the cumulative distribution C. Complexity Analysis function (CDF) of the minimum path metric. In Figs. 1 and 2, The proposed detection reduces the complexity of the path we plot the CDF of the minimum path metric compared the metric evaluations with less possible branches. The additional 5 i-th layer i(1) i(2) i( M ) i(1,1) 1 i(1, i 1 (1)) 1 i(2,1) 1 i(2, i 1(2)) 1 i(1 ,1) M i(1 , i 1( M )) M i1 (1,1) i(1,1 i 1 (1)) i1 (2,1) i(2, i 1(2)) 1 i1 ,1) (M i(1 , i 1( M )) M · Sorting and retaining: i(1)1 i(1 ) . M (i1)-th layer · Calculate i 1 (l ), l [1, M ] and compare with { (i 1), 2 (i 1), ,( K 1) (i 1)} . · obtain i 2 (l ), l [1, M ] . i1 (1) i1 (2) i1 ) (M Fig. 3 Example of the adaptive tree search scheme from the i-th layer to the (i1)-th layer. complexity A is the computations for the path metric ratio in of the different detection algorithms are measured by the BER (17), which require a complexity of (M1)(q1) flops. If we fix characteristics and the complexity. The complexity of the tree the value of the constant C, (i) in (18) is predetermined. Hence, search detection is determined by the amount of the path metric the computational complexity of (i) is neglect. The complexity evaluations. of the proposed detection consists of three parts: the fixed complexity from the m-th layer down to the q-th layer, the A. BER with Perfect CSI various complexity from the (q1)-th layer down to the first Figs. 4 and 5 show the BER characteristics versus Eb/N0 using layer, and the above additional complexity. The fixed the full MLD, the conventional QRM-MLD, the GSM-MLD complexity of the proposed detection F can be derived as and the proposed detection, respectively. The value of M in the proposed detection is the same as that in the QRM-MLD and ( K )m i ( si ) the GSM-MLD, i.e. M16 for 16QAM and M64 for 64QAM, F K ( m ) im 1 q respectively. The constant C in the decision function is ( K )m i 1 ( i , i ) (21) assigned as C{2, 4, 8}. m 1 m i m i 1 3 K i q 2(m i ) ( K ) 4 ( K ) As seen in Fig. 4, we chose M16, which is large enough for the 16QAM in the 44 MIMO system, and hence the BER The various complexity of the proposed detection is curves of the GSM-MLD and the QRM-MLD totally achieve V varied with the number of the children nodes, derived as the ML performance. For the proposed detection, the BER curve with C8 is almost equivalent to the BER characteristics of the GSM-MLD or the QRM-MLD. The proposed detection V iq11 M ( si ) lM1 i (l ) ( i , i ) with C2 has less possible branches in each layer, and hence (22) iq11 2M (m i) 4 lM1 i (l ) the BER curve is about 1dB worse than the BER of the QRM-MLD at a BER of 10-5. The BER curves of the QRM-MLD and the GSM-MLD with where lM1 i (l ) denotes the total number of the children nodes M64 for 64QAM are equivalent to the BER characteristics of in the i-th layer. the full MLD in Fig. 5. For the proposed detection, the BER As a result, the complexity of the proposed detection Prop. curves with C achieve a near-ML performance. The which excludes the complexity of the GSO reduction and the proposed detection with C2 remarkably reduces the possible computation of v in (11) can be derived as branches in each layer, and hence the BER curve is about 0.5dB worse than that of the QRM-MLD at a BER of 10-5. Prop. A V F B. Computational Complexity ( M 1)(q 1) iq11 2M (m i) 4 lM1 i (l ) (23) We evaluated the average number of possible branches in 3 K im 1 2(m i) ( K ) m i 4 ( K ) m i 1 q each layer for the proposed detection with C{2, 4, 8}, seen in Figs. 6 and 7. For the QRM-MLD or GSM-MLD, the number of the possible branches in each layer is fixed to 64 if M16 for V. NUMERICAL RESULTS 16QAM and 512 if M64 for 64QAM, respectively. In Fig. 6, The computer simulations were carried out for 16QAM and the average number of the possible branches in the adaptive 64QAM in the 44 MIMO system, respectively. We assume the stage is varied within a certain range from 16 to 64 for 16QAM. channel is the typical flat Rayleigh fading. The performances In particular, for the curve with C2 in Fig. 6(a), the average 6 Full MLD Full MLD QRM-MLD (M16) QRM-MLD (M64) GSM-MLD (M16) GSM-MLD (M64) Proposed Detection (M16, C2) Proposed Detection (M64, C2) Proposed Detection (M16, C4) Proposed Detection (M64, C4) Proposed Detection (M16, C8) Proposed Detection (M64, C8) Fig. 4 The Eb/N0 vs. BER characteristics: 16QAM and mn8. Fig. 5 The Eb/N0 vs. BER characteristics: 64QAM and mn8. (a) Proposed Detection (M16 and C2) (a) Proposed Detection (M64 and C2) (b) Proposed Detection (M16 and C4) (b) Proposed Detection (M64 and C4) (c) Proposed Detection (M16 and C8) (c) Proposed Detection (M64 and C8) Proposed Detection at Eb/N07dB with BER101 Proposed Detection at Eb/N011dB with BER101 Proposed Detection at Eb/N012dB with BER102 Proposed Detection at Eb/N017dB with BER102 Proposed Detection at Eb/N016dB with BER103 Proposed Detection at Eb/N021dB with BER103 Proposed Detection at Eb/N018dB with BER104 Proposed Detection at Eb/N023dB with BER104 Proposed Detection at Eb/N021dB with BER105 Proposed Detection at Eb/N026dB with BER105 Fig. 6 The average number of possible branches in each layer in tree search at Fig. 7 The average number of possible branches in each layer in tree search at various Eb/N0: 16QAM. various Eb/N0: 64QAM. 7 QRM-MLD (M64) QRM-MLD (M16) GSM-MLD (M64) GSM-MLD (M16) C8 C8 C4 C4 C2 C2 Proposed Detection (M16) Proposed Detection (M64) Fig. 8 The average complexity comparison for three detection schemes: Fig. 9 The average complexity comparison for three detection schemes: 16QAM and mn8 64QAM and mn8. number of the possible branches is close to 16 if the BER diagonal entries, the soft estimate of s is directly obtained in characteristics are less than 102. Furthermore, the BER curve (14) with no division operation. It is convenient to rank the of the proposed detection with C2 is about 1dB worse than constellation points according to s in the adaptive stage. The that of the full MLD at a BER of 105. It should be noticed that adaptive tree search scheme is performed using the path metric the number of the low reliable possible branches in the ratio function, and thus the number of the possible branches in proposed detection with C4 in Fig. 6 (b) is halved or more each layer of adaptive stage is remarkably reduced. Hence, the reduced, compared to the fixed number of 64. The BER curve computational complexity of the proposed detection is much of the proposed detection with C4 is about 0.2dB worse than lower than the conventional QRM-MLD, especially in the high that of the full MLD at a BER of 105. In addition, the BER of Eb/N0 region. From Figs. 8 and 9, the complexity of the the proposed detection with C8 shown in Fig. 4 can retain the proposed detection at a BER of 10-5 is about 40% and 64% near-ML performance. The number of the possible branches smaller than that of the QRM-MLD for 16QAM and 64QAM, can remarkably reduce in the high Eb/N0 region. respectively. Fig. 7 shows the average number of possible branches in each layer for the proposed detection for 64QAM. The average VI. CONCLUSIONS number of the possible branches in the adaptive stage is varied within a certain range from 64 to 512. Similar to 16QAM, the In this paper, introducing the Gram-Schmidt Orthogonaliza- tion procedure to reduce the channel matrix, we proposed a curves with C2 in Fig. 7(a) are close to 64 if the BER MIMO detection scheme using the adaptive tree search with characteristics are less than 102, and correspondingly the BER variable path expansion in the GSM-MLD algorithm. The curve with C2 in Fig. 5 has about 0.5dB performance loss adaptive tree search scheme is to adaptively control the compared to the full MLD at a BER of 105. If the channel is candidates for each survived branch in the tree search. We better-conditioned, the average numbers of the possible adopted a path metric ratio function to evaluate the reliability for branches with C in the adaptive stage are in the range all the survived branches. To decrease the number of the low from 64 to 128, which is much smaller than the fixed number of reliable candidates in each layer, a large amount of the computation for the path metric is avoided. Hence, the 512. Meanwhile, the BER curves with C achieve a complexity of the proposed detection should be reduced. In near-ML performance. From Figs. 6 and 7, the adaptive particular in the high Eb/N0 region, the complexity of the decision threshold in each detection layer is determined by the proposed detection is about 60% and 36% of that of the constant C. QRM-MLD for 16QAM and 64QAM, respectively. The According to the average number of possible branches, we proposed detection can provide the near-ML performance with present the computational complexity of the proposed detection relatively lower complexity. As a result, it is worthy for in Figs. 8 and 9 for 16QAM and 64QAM, respectively. Due to applying even to the high modulation order. M16 for 16QAM and M64 for 64QAM, the layer number q REFERENCES in (21)-(23) is set as qm1. We calculate the complexity of [1] G. J. Foshini, “Layered space-time architecture for wireless QRM-MLD excluding the complexity of QR-decomposition communication in the fading environment when using multiple antennas,” and the computation of QTy in (5) [10]. From the numerical Bell Labs Tech. J., vol.1, no.2, pp. 41–59, Autumn 1996. results, the GSM-MLD has the same complexity with the [2] A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of conventional QRM-MLD. Since the GSO reduction is MIMO communications-A key to gigabit wireless,” Proc. IEEE, vol. 92, no. 2, pp. 198–218, Feb. 2004. computationally-simple and the transform matrix is with unity 8 [3] G. D. Golden, G. J. Foshini, R. A. Valenzuela, and P. W. Wolniansky, Toshiharu Kojima received B.E. and M.E. degrees from The University of “Detction algorithm and initial laboratory results using V-BLAST Electro-Communications, Tokyo, Japan in 1983, 1987, respectively. He space-time comunication architecture,” Electron. lett., vol. 35, no. 1, pp. received Ph.D. degree in communication engineering from Osaka University, 14-16, Jan. 1999. Osaka, Japan in 1998. He was with the 26th Japanese Antarctic Research [4] T. Fujino, “Gram-Schmidt Combined LLL lattice-reduction aided Expedition as a research assistant of The University of Electro- detection in MIMO systems,” REV J. on Electron. and Commun., vol. 1, Communications from 1984 through 1986. He joined Mitsubishi Electric no. 2, pp. 106114, AprilJune, 2011. Corporation in 1987. He worked on the research and development of digital [5] T. Fujino, S. Wakazono, and Y. Sasaki, “A Gram-Schmidt based satellite communication systems and digital mobile communication systems in the Information Technology R & D Center of Mitsubishi Electric Corporation. lattice-reduction aided MMSE detection in MIMO systems,” IEEE Global He is now with Graduate School of Informatics and Engineering, The Commun. Conf. 2009 (Globecom’09), Honolulu, USA, Dec. 2009. University of Electro-Communications, Tokyo, Japan. His research interests [6] X. Wang, Z. He, K. Niu, W. Wu and X. Zhang, “An improved detection are in the areas of the signal processing for wireless communications, based on lattice reduction in MIMO systems”, Proc. IEEE Symposium n modulation and demodulation and forward error correction. Dr. Kojima is a Personal, Indoor, Mobile and Radio Commun. (PIMRC’06), Helsinki, member of the IEEE. Finland, Sep. 2006. [7] X. Ma and W. Zhang, “Performance analysis for MIMO system with lattice-reduction aided linear equalization,” IEEE Trans. Commun., vol.56, pp. 309318, Feb. 2008. [8] C. Windpassinger, L. Lampe, R. F. H. Fischer, and T. Hehn, “A performance study of MIMO detectors,” IEEE Trans. Commun., vol.5, pp. 20042008, Aug. 2006. [9] L.G. Barbero, J.S. Thompson, “ Fixing the complexity of the sphere decoder for MIMO detection,” IEEE Trans. Commun., vol.7, pp. 2131 -2142, June 2008. [10] W. H. Chin, “QRD based tree search data detection for MIMO communication systems,” Proc. IEEE VTC’05 Spring, vol. 3, pp: 1624-1627, May 2005. [11] H. Matsuda, K.Honjo, T.Ohtsuki, “Signal detection scheme combining MMSE V-BLAST and variable K-best algorithms based on minimum branch metric,” Proc. IEEE VTC’05 fall, pp. 19-23, Sep. 2005. [12] B. Kim, K. Choi, “SNR measurement free adaptive K-Best algorithm for MIMO systems,” Proc. IEEE WCNC 2008, pp: 628-633, Apr. 2008. [13] J. Pons, and P. Duvaut, “New approaches for lowering path expansion complexity of K-best MIMO detection algorithms,” Proc. IEEE ICC’09, Dresden, Germany, June 2009. [14] H. Kawai, K. Higuchi, N. Maeda, and M. Sawahashi, “Adaptive control of surviving symbol replica candidates in QRM-MLD for OFDM MIMO multiplexing,” IEEE J. on Sel. Areas in Commun., vol. 24, no. 6, June 2006. [15] B. Kim, and K. Choi, “A very low complexity QRD-M algorithm based on limited tree search for MIMO systems,” Proc. IEEE VTC’08 Spring, pp. 1246 - 1250 , May 2008. [16] S. Lei, C. Xiong, X. Zhang, and D. C. Yang, “Adaptive control of surviving branches for fixed-complexity sphere decoder,” Proc. IEEE VTC’10 Spring, May 2010. [17] G. H. Golub, and C. F. V. Loan, “Matrix Computations,” 3rd ed. Baltimore, MD: John Hopkins University Press, 1996. Wei Hou received the B.S. degree in electrical engineering from Dalian Maritime University, Dalian, China, in 2004 and M.S. degree in Information and Communication Engineering from Beijing University of Posts and Telecommunications, Beijing, China, in 2007. She is now working towards Ph. D. degree in The University of Electro-Communications, Tokyo, Japan. Her current research lies in the area of signal processing for wireless communications including signal estimation algorithm. Tadashi Fujino received B.E. and M.E. degrees in electrical engineering and Dr. Eng. degree in communication engineering from Osaka University, Osaka, Japan, in 1968, 1970, and 1985, respectively. Since April 2011, he has been a professor emeritus of The University of Electro-Communications (UEC), Tokyo, Japan. In 20002011, he was an ordinary professor in wireless communications at the UEC. Before then, he had been with Mitsubishi Electric Corporation, Tokyo, Japan since 1970, where he devoted in R&D in the wireless communications area such as digital satellite communications and digital land mobile communications. His major works include the feasibility study and the hardware development of the 120 Mbps trellis coded modem for TDMA system. This is the first development in the world. His current interests include the signal detection in MIMO systems such as lattice-reduction aided detection. He wrote a single authored book “digital mobile communication,” and three co-authored books. He received Meritorious Award from The ARIB (The Associate of Radio Industries and Businesses of Japan) of MPT of Japan, in 1997. Prof. Fujino is a fellow of IEEE. 9

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