Matrix Inversion-Less Signal Detection Using SOR Method for Uplink Large-Scale MIMO Systems

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1 Matrix Inversion-Less Signal Detection Using SOR Method for Uplin Large-Scale MIMO Systes Xinyu Gao, Linglong Dai, Yuting Hu, Zhongxu Wang, and Zhaocheng Wang Tsinghua National Laboratory for Inforation Science and Technology (TNList), Departent of Electronic Engineering, Tsinghua University, Beijing , China E-ail: Abstract For uplin large-scale MIMO systes, linear iniu ean square error (MMSE) signal detection algorith is near-optial but involves atrix inversion ith high coplexity. In this paper, e propose a lo-coplexity signal detection algorith based on the successive overrelaxation (SOR) ethod to avoid the coplicated atrix inversion. We first prove a special property that the MMSE filtering atrix is syetric positive definite for uplin large-scale MIMO systes, hich is the preise for the SOR ethod. Then a lo-coplexity iterative signal detection algorith based on the SOR ethod as ell as the convergence proof is proposed. The analysis shos that the proposed schee can reduce the coputational coplexity fro O(K 3 ) to O(K 2 ), here K is the nuber of users. Finally, e verify through siulation results that the proposed algorith outperfors the recently proposed Neuann series approxiation algorith, and achieves the near-optial perforance of the classical MMSE algorith ith a sall nuber of iterations. I. INTRODUCTION Large-scale ultiple-input ultiple-output (MIMO) is considered as a ey technology for future ireless counications [1]. Unlie the traditional sall-scale MIMO technology (e.g., at ost 8 antennas in LTE-A), large-scale MIMO exploits a very large nuber of antennas (e.g., 128 antennas or even ore) at the base station (BS) to siultaneously serve ultiple user equipents (UEs) [2]. It has been theoretically proved that large-scale MIMO can provide potential opportunity to increase the spectru and energy efficiency by orders of agnitude [3]. Hoever, soe challenging probles have to be solved to realize such attractive erits of large-scale MIMO in practice. One of the is the practical signal detection algorith in the uplin [4]. The optial detector is the axiu lielihood (ML) detector hose coplexity exponentially increases ith the nuber of transit antennas, hich aes it ipractical for large-scale MIMO systes. To achieve the (close) optial ML detection perforance ith reduced coplexity, several non-linear signal detection algoriths have been proposed. One typical category is based on the sphere decoding (SD) algorith [5], such as the fixed-coplexity sphere decoding (FSD) algorith [6]. This ind of algoriths uses the underlying lattice structure of the received signal and considers the ost proising approach to achieve the ML detection perforance ith reduced coplexity. It perfors ell for the conventional sall-scale MIMO systes, but hen the diension of the MIMO systes is large or the odulation order is high [7] (e.g., 128 antennas at the BS ith 64 QAM odulation), the coplexity is still unaffordable. Another category is based on the tabu search (TS) algorith derived fro artificial intelligence [8], such as the layered tabu search (LTS) algorith [9]. This ind of algoriths utilizes the idea of local neighborhood search to estiate the transitted signal and liits the selection of neighborhood by a tabu list. When the neighborhood range is appropriately sall and the tabu list is carefully designed, the coplexity is acceptable for largescale MIMO systes, but it suffers fro a non-negligible perforance loss copared to the optial ML detector. To ae a trade-off beteen the perforance and coplexity, one can resort to linear signal detection algoriths, such as the zero-forcing (ZF) and iniu ean square error (MMSE) algoriths, hich are near-optial for uplin ultiuser large-scale MIMO systes [4]. Hoever, these algoriths involve unfavorable inversion of a atrix of large size, hose coplexity is still high for large-scale MIMO systes. Very recently, to reduce the coplexity of atrix inversion, [10] proposed the Neuann series approxiation algorith to convert the atrix inversion into a series of atrix-vector ultiplications. Hoever, only arginal reduction in coplexity can be achieved. In this paper, e propose a atrix inversion-less signal detection algorith ith lo coplexity based on the SOR ethod [11] for large-scale MIMO systes. We first prove that the MMSE filtering atrix is syetric positive definite for uplin large-scale MIMO systes, according to hich e propose to exploit the SOR ethod to avoid the coplicated atrix inversion. We also prove the convergence of the proposed signal detection algorith to guarantee its feasibility in practice. We verify through siulation results that the proposed algorith can efficiently solve the atrix inversion proble in an iterative procedure until the desired detection accuracy is attained. To the best of our noledge, this or is the first one to utilize the SOR ethod for the signal detection in uplin large-scale MIMO systes. The rest of the paper is organized as follos. Section II briefly describes the syste odel. Section III specifies the proposed lo-coplexity signal detection algorith, together ith the convergence proof and the coplexity analysis. The siulation results of the bit error rate () perforance are provided in Section IV. Finally, conclusions are dran in Section V /14/$ IEEE 3291

2 Notation: We use loer-case and upper-case boldface letters to denote vectors and atrices, respectively; ( ) T, ( ) H, ( ) 1, and denote the transpose, conjugate transpose, atrix inversion, and absolute operators, respectively; Re{ } and I{ } denote the real part and iaginary part of a coplex nuber, respectively; Finally, I N represents the N N identity atrix. II. SYSTEM MODEL We consider a uplin large-scale MIMO syste eploying N antennas at the BS to siultaneously serve K singleantenna UEs [2], [4]. Usually e have N>>K, e.g., N = 128 and K =16have been considered in [4]. The transitted bit streas fro different users are first encoded by the channel encoder and then apped to sybols by taing values fro a odulation alphabet. Let s c =[s c,1,,s c,k ] T denote the transitted signal vector fro all K users, and H c C N K denote the flat Rayleigh fading channel atrix, hose entries are assued to be independently and identically distributed (i.i.d.) ith zero ean and unit variance [2]. Then the received signal vector y c =[y c,1,,y c,n ] T at the BS can be represented as y c =H c s c +n c, (1) here n c =[n c,1,,n c,n ] T is the noise vector hose entries are i.i.d and follo the distribution CN(0,σ 2 ). For signal detection, the coplex-valued syste odel (1) can be converted into a corresponding real-valued one as y = Hs + n, (2) here y =[Re{y c } I{y c }] T is of size 2N 1, accordingly s =[Re{s c } I{s c }] T, n =[Re{n c } I{n c }] T, and H = [ Re{Hc } I{H c } I{H c } Re{H c } ]. (3) 2N 2K At the BS, after the channel atrix H has been obtained through tie-doain and/or frequency-doain training pilots [12] [13], the tas of signal detection is to recover the transitted signal vector s fro the received signal vector y. It has been proved that the linear MMSE signal detection algorith is near-optial for uplin ulti-user large-scale MIMO systes [4], and the estiate of the transitted signal vector ŝ can be obtained by ŝ =(H H H + σ 2 I 2K ) 1 H H y = W 1 ŷ, (4) here ŷ = H H y, and the MMSE filtering atrix W is denoted as W = G + σ 2 I 2K, (5) here G = H H H is the Gra atrix. The coputational coplexity of the direct atrix inversion W 1 is O(K 3 ), hich is high for large-scale MIMO systes. III. LOW-COMPLEXITY SIGNAL DETECTION FOR UPLINK LARGE-SCALE MIMO In this section, e first prove a special property of largescale MIMO systes that the MMSE filtering atrix is syetric positive definite. Based on this property, e then propose a lo-coplexity signal detection algorith utilizing the SOR ethod to iteratively achieve the MMSE estiate ithout atrix inversion. The convergence proof is also addressed. Finally, e provide the coplexity analysis of the proposed algorith to sho its advantage over conventional schees. A. Matrix inversion-less signal detection utilizing SOR ethod Unlie the conventional (sall-scale) MIMO systes ith sall nuber of antennas, large-scale MIMO systes enjoy a special property that the colun vectors of the channel atrix are asyptotically orthogonal [4]. Based on that, e prove that the MMSE filtering atrix is syetric positive definite in the folloing Lea 1. Lea 1. For uplin large-scale MIMO systes, the MMSE filtering atrix W is syetric positive definite. Proof: Since the coplex-valued MIMO syste odel has been converted into the real-valued one, the transpose of atrix and the conjugate transpose of atrix ill be the sae, e.g., G = H H H = H T H. Thus, e have G T =(H T H) T = H T H = G, (6) hich indicates that the Gra atrix G is syetric. Meanhile, for uplin large-scale MIMO systes, the colun vectors of the real-valued channel atrix H are asyptotically orthogonal [4], i.e., the equation Hq =0 has an unique solution, hich is the 2K 1 zero vector. Thus, for any 2K 1 non-zero real-valued vector r, ehave (Hr) T Hr = r T Gr > 0, (7) hich iplies that G is positive definite. Considering (6) and (7), e can conclude that the Gra atrix G =H T H is syetric positive definite. Finally, as the noise variance σ 2 is positive, the MMSE filtering atrix W = G + σ 2 I 2K in (5) is syetric positive definite, too. The special property that the MMSE filtering atrix W in uplin large-scale MIMO systes is syetric positive definite inspires us to exploit the SOR ethod to efficiently solve (4) ith lo coplexity. The SOR ethod is used to solve N-diension linear equation Ax = b, here A is the N N syetric positive definite atrix, x is the N 1 solution vector, and b is the N 1 easureent vector. Unlie the traditional ethod that directly coputes A 1 b to obtain x, the SOR ethod can efficiently solve the linear equation in an iterative anner ithout the coplicated atrix inversion. Since atrix A is syetric positive definite, e can decopose it into a diagonal coponent D A, a strictly loer triangular coponent L A, and a strictly upper triangular 3292

3 coponent L T A. Then the SOR iteration can be described as [11] x (i+1) =(L A + 1 D A) 1 [(( 1 1)D A L T A ) ] x (i) +b, here the superscript i =0, 1, 2, denotes the nuber of iterations, and represents the relaxation paraeter, hich plays an iportant role in the convergence and the convergence rate. Note that hen =1, the SOR ethod is the sae as the ell non Gauss-Seidel ethod [11], hich eans that the Gauss-Seidel ethod is a special case of the SOR ethod. We ill discuss the selection of the relaxation paraeter in detail later in Section IV. Due to the MMSE filtering atrix W is syetric positive definite for uplin large-scale MIMO systes as proved in Lea 1, e can also decopose W as (8) W = D + L + L T, (9) here D, L, and L T denote the diagonal coponent, the strictly loer triangular coponent, and the strictly upper triangular coponent of W, respectively. Then e can utilize the SOR ethod to estiate the transitted signal vector s as belo s (i+1) =(L+ 1 [( D) 1 ( 1 ) ] 1)D LT s (i) +ŷ, (10) here s (0) denotes the initial solution, hich is usually set as a 2K 1 zero vector ithout loss of generality [11]. Then the solution to the signal detection proble (4) can be solved by the SOR ethod according to (L+ 1 D)s(i+1) = ŷ + (( 1 ) 1)D LT s (i). (11) As (L+ 1 D) is a loer triangular atrix, one can solve the equation (11) to obtain s (i+1) ith lo coplexity as ill be addressed in Section III-C. Next e ill prove the convergence of the proposed signal detection based on the SOR ethod. B. Convergence proof Lea 2. For uplin large-scale MIMO systes, the signal detection algorith using the SOR ethod is convergent hen the relaxation paraeter satisfies 0 < < 2. Proof: We define C =(L + 1 D) 1 ( 1 D D LT ) and d =(L + 1 D) 1 ŷ, here C is called as the iteration atrix. Then the SOR iteration (11) can be reritten as s (i+1) = Cs (i) + d. (12) The spectral radius of the iteration atrix C is defined as the non-negative nuber ρ(c) = ax λ n, here λ n 1 n 2K denotes the nth eigenvalue of C. The necessary and sufficient conditions for the convergence of (12) is that the spectral radius should satisfy [11, Theore 7.2.2] ρ(c) = ax 1 n 2K λ n < 1. (13) According to the definition of eigenvalue, e have Cr =(L + 1 D) 1 ( 1 D D LT )r = λ n r, (14) here r is an arbitrary 2K 1 non-zero real-valued vector. Note that (14) can be also presented as ( 1 D D LT )r =(L + 1 D)λ nr. (15) Multiply both sides of (15) by r T ill yield r T ( 1 D D LT )r = λ n r T (L + 1 D)r. (16) Then e tae transpose on both sides of (16), and another equation can be obtained as r T ( 1 D D L)r = λ nr T (L T + 1 D)r. (17) Note that D = D T as D is a diagonal atrix. Add (16) and (17) ill lead to ( r T ( 2 ) 2)D L LT r=λ n r T (L T +L+ 2 D)r. (18) Substituting (9) into (18), e have (1 λ n )( 2 1)rT Dr =(1+λ n )r T Wr. (19) Since the MMSE filtering atrix W is positive definite as proved above, the diagonal atrix D is positive definite, too. Then e have r T Dr > 0 and r T Wr > 0. Besides, e also have ( 2 1) > 0 if 0 <<2. Thus, e can conclude that (1 λ n )(1 + λ n ) > 0, hich eans λ n < 1. (20) Substituting (20) into (13), e can assert that ρ(c) < 1, so the SOR iteration (11) is convergent. It is orth pointing out that another different proof of Lea 2 can be found in [14, Theore ], hich utilizes the orthogonal transforation ith high coplexity to obtain the convergence proof, hile our ethod directly exploits the definition of eigenvalue, hich is sipler than the existing ethod [14]. C. Coputational coplexity analysis The coputational coplexity in ters of required nuber of ultiplications is analyzed in this part. It can be found fro (11) that the coputational coplexity of the ith iteration of the proposed signal detection algorith originates fro solving the linear equation. Considering the definition of D, L, and L T, the solution can be presented as s (i+1) =(1 )s (i) + (ŷ W, s (i+1) W, s (i) W ),, < >, =1, 2, 2K, (21) here s (i), s (i+1), and ŷ denote the th eleent of s (i), s (i+1), and ŷ in (4), respectively, and W, denotes 3293

4 the th ro and th colun entry of W. The required nuber of ultiplications in the coputation of (1 )s (i) and W, (ŷ W, s (i+1) W, s (i) ) is 1 and < > 2K +1, respectively. Therefore the coputation of each eleent of s (i+1) requires 2K +2ties of ultiplications. Since there are 2K eleents in s (i+1), the overall required nuber of ultiplications is 4K 2 +4K. Proposed detection algorith, N=64, K=8 Proposed detection algorith, N=128, K=16 MMSE ith exact atrix inversion, N=64, K=8 MMSE ith exact atrix inversion, N=128, K=16 TABLE I COMPUTATIONAL COMPLEXITY Conventional Neuann series approxiation algorith [10] Proposed signal detection algorith i =2 12K 2 4K 8K 2 +8K i =3 8K 3 +4K 2 2K 12K 2 +12K i =4 16K 3 4K 2 16K 2 +16K i =5 24K 3 12K 2 +2K 20K 2 +20K Table I copares the coplexity of the conventional Neuann series approxiation algorith [10] and the proposed algorith based on the SOR ethod. Since the coplexity of the classical MMSE algorith is O(K 3 ), e can conclude fro Table I that the conventional Neuann series approxiation algorith can reduce the coplexity fro O(K 3 ) to O(K 2 ) hen the nuber of iterations is i =2, but the coplexity is still O(K 3 ) hen i 3. To ensure the approxiation perforance, usually a large value of i is required to approach the final MMSE solution ŝ as ill be verified later in Section IV. So the overall coplexity is alost the sae as the MMSE algorith, hich eans only arginal reduction in coplexity can be achieved. Hoever, e can observe that the coplexity of the proposed algorith is O(K 2 ) for arbitrary nuber of iterations. And even for i =2, the proposed algorith enjoys a loer coplexity than the conventional one [10]. Additionally, e can observe fro (21) that the coputation of s (i+1) utilizes s (i+1) for =1, 2,, 1 and s (i) l for l =, +1,, 2K, hich is siilar to the Gauss-Seidel ethod [11]. Then, to another benefits can be expected. Firstly, after s (i+1) has been obtained, e can use it to overrite s (i) hich is useless in the next coputation of s (i+1) +1. Consequently, only one storage vector of size 2K 1 is required; secondly, hen i increases, the solution to (11) becoes closer to the final MMSE solution ŝ. Thus s (i+1) can exploits the eleents of s (i+1) for =1, 2,, 1 that have already been coputed in the current iteration to produce ore reliable result than the conventional algorith [10] only utilizing all the eleents of s (i) in the previous iteration. Thus, a faster convergence rate can be expected, and the required nuber of iterations to achieve a certain estiation accuracy becoes saller. Based on these to special advantages of the SOR ethod, the overall coplexity of the proposed algorith can be reduced further. IV. SIMULATION RESULTS To verify the perforance of the proposed signal detection algorith, e provide the siulation results copared ith the recently proposed Neuann series approxiation algorith [10]. The perforance of the classical MMSE Fig. 1. perforance of the proposed SOR-based signal detection algorith against the relaxation paraeter, here SNR = 4 db and i =3. Neuann series approxiation [10], i=2 Neuann series approxiation [10], i= Neuann series approxiation [10], i=5 Proposed detection algorith, i=2 Proposed detection algorith, i=3 Proposed detection algorith, i=5 MMSE ith exact atrix inversion Optial ML detection algorith SNR (db) Fig. 2. perforance coparison hen N K =64 8. algorith ith coplicated but exact atrix inversion is included as the benchar for coparison. Besides, to verify the near-optial perforance of the MMSE algorith, the perforance of the optial ML detection algorith is also provided. We consider to large-scale MIMO systes ith N K =64 8 and N K = , respectively. The odulation schee of 64 QAM is adopted. The rate-1/2 industry standard convolutional code ith generator polynoials [133 o 171 o ] is eployed, and a rando interleaver is also used to cobat the burst error. The Rayleigh fading channel odel is considered. After ulti-user signal detection, the estiated signal vector is used to extract the soft inforation (by calculate the log-lielihood ratios (LLRs)) for soft-input Viterbi decoder for channel decoding. Fig. 1 shos the perforance of the proposed SORbased signal detection algorith against the relaxation paraeter, here the signal-to-noise ratio (SNR) is 4 db, and the nuber of iterations is i =3. As shon in Fig. 1, s of the MMSE algorith are for N K =64 8, 3294

5 Neuann series approxiation [10], i=2 Neuann series approxiation [10], i= Neuann series approxiation [10], i=5 Proposed detection algorith, i=2 Proposed detection algorith, i=3 Proposed detection algorith, i=5 MMSE ith exact atrix inversion Optial ML detection algorith SNR (db) Fig. 3. perforance coparison hen N K = and for N K = , respectively, hich are the targets to be approached by selecting the optial relaxation paraeters. We can observe that the curve against loos lie a parabola, and fortunately the optial for both systes is Furtherore, e have conducted intensive siulations of different large-scale MIMO syste configurations and found that the systes ith fixed N/K (e.g., N/K =8in Fig. 1) ill share the sae optial selection of, hich indicates that e can easily obtain the optial after the syste diensions N and K have been fixed. The perforance coparison beteen the conventional Neuann series approxiation algorith [10] and the proposed SOR-based signal detection algorith hen N K =64 8 and N K = are shon in Fig. 2 and Fig. 3, respectively, here i denotes the nuber of iterations. It is clear that the perforance of both algoriths iproves ith the increased nuber of iterations. Hoever, hen the sae iteration nuber i is used, the proposed algorith outperfors the conventional one for both systes. Moreover, as e can observe fro Fig. 2, the perforance of the proposed algorith hen i =3is alost the sae as that of the conventional one hen i =5, hich indicates that a faster convergence rate can be achieved by the proposed SOR-based signal detection algorith. As e have addressed in Section III-C, a faster convergence rate eans saller nuber of iterations is required to achieve a certain estiation accuracy, so the coplexity of the proposed algorith can be reduced further. Meanhile, e can observe fro Fig. 2 and Fig. 3 that the MMSE algorith is near-optial copared to the optial ML detection algorith, and the proposed algorith ithout the coplicated atrix inversion can achieve the near-optial perforance of the MMSE algorith hen the nuber of iterations is large (e.g., i =3in Fig. 2 and Fig. 3). V. CONCLUSIONS In this paper, by fully exploiting a special channel property of the large-scale MIMO systes, e propose a locoplexity near-optial signal detection algorith based on the SOR ethod in the uplin. The SOR-based algorith can iteratively realize the MMSE solution ithout coplicated atrix inversion, hich can reduce the coplexity fro O(K 3 ) to O(K 2 ). We also prove the convergence of the proposed algorith, and siulation results sho that it can achieve the near-optial perforance of the classical MMSE algorith ith a sall nuber of iterations. Moreover, the idea of utilizing the SOR ethod to efficiently realize atrix inversion ith lo coplexity can be extended to other signal processing probles in ireless counications, such as the precoding in the large-scale MIMO systes. ACKNOWLEDGMENTS This or as supported by National Key Basic Research Progra of China (Grant No. 2013CB329201), National Natural Science Foundation of China (Grant Nos and ), Science and Technology Foundation for Beijing Outstanding Doctoral Dissertation (Grant No. 2012T50093), and the ZTE fund project CON REFERENCES [1] L. Dai, Z. Wang, and Z. Yang, Next-generation digital television terrestrial broadcasting systes: Key technologies and research trends, IEEE Coun. Mag., vol. 50, no. 6, pp , Jun [2] T. L. Marzetta, Noncooperative cellular ireless ith unliited nubers of base station antennas, IEEE Trans. Wireless Coun., vol. 9, no. 11, pp , Nov [3] L. Dai, Z. Wang, and Z. 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