Expectation Propagation Based Iterative Groupwise Detection for ...

IEEE WCNC'14 Track 1 (PHY and Fundamentals)

Expectation Propagation Based Iterative Groupwise Detection for Large-Scale Multiuser MIMO-OFDM Systems Sheng Wu∗§ , Linling Kuang†§ , Zuyao Ni‡§ , Jianhua Lu∗, Defeng (David) Huang¶ , and Qinghua Guo¶ ∗ Department

of Electronic Engineering, Tsinghua University, Beijing, China Space Center, Tsinghua University, Beijing, China ‡ Research Institute of Information Technology, Tsinghua University, Beijing, China § Shenzhen Key Laboratory of Wireless Broadband Communication and Signal Processing, Shenzhen, China ¶ School of Electrical, Electronic and Computer Engineering, the University of Western Australia, Australia  School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Australia Email: [email protected], {kll, nzy, lhh-dee}@mail.tsinghua.edu.cn, [email protected], qinghua [email protected] † Tsinghua

Abstract—For the spatially correlated multiuser MIMOOFDM channels, the conventional iterative MMSE-SIC detection suffers from a considerable performance loss. In this paper, we use the factor graph framework to design robust detection algorithms by clustering a group of symbols to combat the spatial correlation and using the principle of expectation propagation to improve message passing. Furthermore, as the complexity of detection becomes one of the issues in the design of large-scale multiuser MIMO-OFDM systems, we propose a low-complexity approximate message-passing algorithm by opening the channel transition node, which eliminates the expensive matrix inversions involved in the MMSE-SIC based algorithms. Finally, numerical results are presented to verify the proposed algorithms.

I. I NTRODUCTION Multiple-input multiple-output orthogonal frequencydivision multiplexing (MIMO-OFDM) is a key technology for many wireless communication systems, due to its high spectral efficiency [1]. Recently, large-scale multiuser MIMO systems with tens to hundreds of antennas at the base-station have attracted much attention [2]–[4]. The motivation to consider the large-scale MIMO systems comes from the growing demands for higher throughput and better quality-of-service in the next-generation multiuser wireless communication systems [2]. Turbo MIMO detection can achieve near-optimal performance by iteratively exchanging probabilistic information about the coded bits between a soft-input soft-output (SISO) detector and a SISO channel decoder [5]. The SISO detector employing maximum a posteriori probability (MAP) algorithm is optimal in terms of performance but has an exponential complexity. To reduce the complexity, there has been considerable interest in the suboptimal SISO detectors [6]–[9]. To further reduce the complexity, the minimum meansquare error based soft interference cancellation (MMSE-SIC) can be applied to MIMO detection [10], [11]. Nevertheless, the conventional iterative MMSE-SIC receiver suffers from a c 978-1-4799-3083-8/14/$31.00 2014 IEEE

978-1-4799-3083-8/14/$31.00 ©2014IEEE

236

considerable performance loss when applied to the spatially correlated fading channels. To overcome this performance degradation, groupwise detection combining MMSE filtering and optimal MAP detection can be employed [12]. In this paper, we use a unified factor-graph framework to develop detection algorithms that are robust against spatial channel correlation. First, the users’ symbols are divided into several disjoint groups, and each group of symbols is considered as a cluster node [13] in the corresponding factor graph. Then the principle of expectation propagation [14] is used to find the approximate Gaussian messages, where the symbol belief, rather than the message itself, is approximated by a Gaussian probability density function (PDF) using minimum Kullback-Leibler (KL) divergence criterion. Finally, to achieve low-complexity detection for the large-scale multiuser MIMO-OFDM systems, we open the channel transition node in the underlying factor graph, and then formulate an efficient message-passing algorithm that can avoid the matrix inversions in the MMSE-SIC based algorithms in [10] and [12]. We note that the scope of this work is different from that of [12], [15] and [16]. Particularly, [12] employed the groupwise detection to improve the MMSE-SIC algorithm, and [15] and [16] derived the MMSE-SIC algorithm based on the principle of expectation propagation, while our work employs a unified framework to design detection algorithms combining the two above mentioned processing techniques and focuses on low-complexity message-passing algorithm for the large-scale multiuser MIMO-OFDM systems. Throughout the paper, we use the following notations. The superscripts T and H denote the transpose operation and the conjugate transpose operation, respectively. The symbol I and 0 denote an identity matrix and a zero matrix, respectively. Also, ln (·) denotes the natural logarithm; NC (x, m, V ) denotes the Gaussian PDF of x with mean m and covariance matrix V ; and x \ xi denotes the symbols in x with xi excluded. Furthermore, Ep(x) [·]

IEEE WCNC'14 Track 1 (PHY and Fundamentals)

denotes the statistical expectation operation with respect to the distribution p (x), and  [x] takes the real part of x. Finally, er ∈ RNR ×1 denotes the standard basis vector with the unique one in the rth entry.

code constraints

c11;1 Ă

cQ 1;1

c11;2 Ă

Á1;1

Consider the uplink of a multiuser system with N independent users, where each user is equipped with NT antennas and the receiver is equipped with an array of NR ≥ N NT antennas. OFDM is employed for each user to combat multipath interference. With a 2Q -ary constellation set χ, each Q interleaved coded bits are mapped into one symbol. In the frequency domain, the symbols transmitted from the ith transmit antenna within the nth user are denoted T by xin = xin (1) , xin (2) , . . . , xin (K) , i = 1, 2, . . . , NT , where xin (k) ∈ χ denotes the symbol transmitted at the kth subcarrier, and K is the number of subcarriers in the OFDM system. Each transmitter applies a K-point IFFT to each data block xin and adds a cyclic prefix (CP) before transmission. These NT × N blocks from all the N users are transmitted simultaneously over the MIMO multipath channel. At the receiver, the CP is removed and each data block is converted into the frequency domain via a K-point FFT. It is assumed that these N transmitters and the receiver are synchronized and all channel coefficients are known at the receiver. Then the received signal at the kth subcarrier y (k) ∈ CNR ×1 can be written as

c1g;1 Ă

x1;2

Ã1

cQ g;1

c1g;2 Ă

c1G;1 Ă

Ãg

xg

x1

c1 cQ G;1 G;2

xG;1 Ă

Q

Ă cG;2

ÁG;2

ÁG;1

xg;2

xg;1 Ă

code constraints cQ g;2

Ág;2

Ág;1

Á1;2

x1;1

II. S YSTEM D ESCRIPTION

code constraints cQ 1;2

xG;2

ÃG xG

f

Fig. 1. Factor graph of the multiuser MIMO-OFDM system for one subcarrier, NG = NT = 2, G = N . Note that the code constraints are across all the subcarriers.

 y and the a priori log likelihood ratios (LLRs) λa cqg,i fed back from the decoders, the task of detector is to generate extrinsic LLR for each coded bit cqg,i [10]



q 

 p cqg,i = 1 | y  − λa cqg,i . λe cg,i = ln q (3) p cg,i = 0 | y Therefore, we need the a posteriori marginal prob to calculate

ability p cqg,i | y = c\cqg,i ,x p (c, x | y). Although exact

 evaluation of p cqg,i | y is computationally prohibitive for the problem sizes of interest, it can be approximately evaluated by message passing on the underlying factor graph. For the presentation of factor graph and message-passing algorithm, we will use the same convention as in [18] and [19], to which we refer the reader for an in-depth review. According to the facts that c → x → y is a Markov chain and the user’s codewords are generated independently, the joint probability p (c, x | y) can be factorized as  φg,i (cg,i , xg,i ) p (cg,i ) , (4) p (c, x | y) ∝ p (y | x)

y (k) = H (k) x (k) + n (k) , (1) T  NT 1 T (k), . . . , x (k), . . . , x (k) where x (k) = x11 (k), . . . , xN 1 N N denotes all the symbols associated with the kth subcarrier, and n (k) ∈ CNR ×1 denotes the additive Gaussian noise with zero mean and covariance matrix σn2 I. The Kronecker spatial g i fading correlation model is employed for the frequencydomain MIMO channel, which is given by [17] where ∝ denotes equality up to a scale, c denotes coded bit   L−1 vector corresponding to symbol vector x, and φg,i (cg,i , xg,i )  1 1 −j2πlk Rl2 H l S l2 exp H (k) = , (2) represents the mapping constraint δ (ϕ (cg,i ) , xg,i ), where K l=0 ϕ (cg,i ) maps cg,i to xg,i , and δ (·) is the Kronecker delta function. The a priori probability NR ×NR where L denotes the number of channel taps, Rl ∈ C

q  p (cg,i ) = Q q q   N NT ×N NT c 1 + exp λ cg,i is calculated exp c λ a a g,i g,i q=1 and S l ∈ C denote the receive and transmit

q  by the a priori LLRs λ c fed back from the dea correlation matrices of the lth channel tap, respectively, and g,i NR ×N NT coders. The probabilistic structure characterized by the facdenotes the time-domain channel matrix Hl ∈ C torization (4) is illustrated by the factor graph in Fig. 1. comprised of independent complex Gaussian entries, which denotes the grouping constraint In this factor graph, ψ g is assumed to be independent for different l. For notational   T simplicity, subcarrier index k is henceforth omitted. ψg (xg,1 , . . . , xg,NG , xg ) = δ xg , [xg,1 , . . . , xg,NG ] , and f denotes the channel transition function III. I TERATIVE G ROUPWISE D ETECTION U SING  

−N NT 1 2 E XPECTATION P ROPAGATION p (y | x) = πσn2 exp − 2 |y − Hx| . (5) σn A. Factor Graph Representation The symbol vector x is divided into G disjoint groups T {xg , g = 1, 2, . . . , G}, such that xg = [xg,1 , . . . , xg,NG ] contains NG symbols that will be jointly detected. We assume identical sized groups, such that GNG = N NT . Let cg,i denote the coded bits corresponding to the symbol xg,i , and cqg,i denote the qth coded bit in cg,i . Based on the received signal

237

B. Groupwise MMSE-EP Detection Let us investigate the message-passing algorithm on the factor graph in Fig. 1. To reduce complexity, we choose to pass messages from the top to the bottom of Fig. 1 and back immediately, thereby avoiding inner iteration. Once the decoders update extrinsic information and pass downward, a

IEEE WCNC'14 Track 1 (PHY and Fundamentals)

new iteration starts. Using the sum-product updating rule [18], the message from the channel transition node f to the grouping node ψg in the tth iteration takes the form   μtf →ψg (xg ) = p (y | x) μtψg →f (xg ) . (6) x\xg

g =g

However, (6) needs exponential time to marginalize out the random vector x except xg , as the symbols take on values in the discrete set χ. If we consider xg in (6) as a vector of continuous random variables and approximate the message μtψg →f (xg ) with a complex Gaussian PDF μ ˆtψg →f (xg ) =   NC xg ; mtψg →f , V tψg →f , where the parameters mtψg →f

and V tψg →f will be defined later, μtf →ψg (xg ) can be calculated analytically by integration as follows   μtf →ψg (xg ) = p (y | x) μ ˆtψg →f (xg ) x\xg  g =g (7)   = NC

H ig xg ; mtξg , V tξg

,

where ig includes the symbol indices belonging to the gth symbol group and H ig ∈ CNR ×NG is composed of the corresponding columns of H associated with  the indices set ig , ξg is defined by ξg = y − g =g H ig xg + n , and the parameters in (7) are given

by mtξg = y − g =g H ig mtψg →f and V tξg = σn2 I +

H t μtf →ψg (xg ) can be g =g H ig V ψg →f H ig . Furthermore,   viewed as another Gaussian PDF NC xg ; mtf →ψg , V tf →ψg  −1  −1 H with covariance matrix V tf →ψg = H ig V tξg H ig  −1 H and mean mtf →ψg = V tf →ψg H ig V tξg mtξg . Using the Sherman–Morrison–Woodbury formula [20], it is shown in the Appendix that mtf →ψg and V tf →ψg become −1 H −1

 H −1  y − Hmtx H ig H ig V tf mtf →ψg = H ig V tf + mtψg →f , V tf →ψg

 H −1 −1 = H ig V tf H ig − V tψg →f ,

V tf

μtψg →f

(9)

=

238



Q NG   exp cqg,i λta cqg,i

 . (xg ) = 1 + exp λta cqg,i i=1 q=1

(12)

To find the approximate message μ ˆ tψg →f (xg ) in (7),  one approach is to minimize the KL divergence  t t ˆψg →f (xg ) , then the parameters of KL μψg →f (xg )  μ μ ˆtψg →f (xg ) are given by [14]  mtψg →f = αi μtψg →f (xg = αi ) , (13) αi ∈χNG



V tψg →fg =

t αi αH i μψg →f (xg = αi )

NG

αi ∈χ

−

mtψg →f



mtψg →f

H

(14) .

For the algorithm formed by (8), (9), (13) and (14), we will refer to it as MMSE-EX. When NG ≥ 2, MMSE-EX is identical to the groupwise MMSE detection proposed in [12], and when NG = 1, it degenerates to the conventional MMSESIC [10]. Besides, we can utilize the principle of expectation propagation to find the approximate message μ ˆtψg →f (xg ). First, an intermediate belief is defined by β t (xg ) 

μtψg →f (xg ) μt−1 f →ψg (xg ) xg ∈χNG

μtψg →f (xg ) μt−1 f →ψg (xg )

,

(15)

and then it will beapproximated by  a Gaussian PDF denoted by βˆt (xg ) = NC xg ; mtxg , V txg , where the parameters of βˆt (xg ) can be obtained by for the mini  the moment matching t t ˆ mum KL divergence KL β (xg )  β (xg ) , namely mtxg =  H   t t Eβ t (xg ) [xg ] and V txg = Eβ t (xg ) xg xH . g − mxg mxg Finally, according to the EP principle [14], μ ˆtψg →f (xg ) is updated by μ ˆtψg →f (xg ) =

(8)

H t σn2 I + and g H ig V ψg →f H ig T  t t t mx = mψ1 →f , . . . , mψG →f . Finally, with the messages μtf →ψg (xg ) and μtψg →f (xg ), the LLRs of coded bits corresponding to symbol vector xg are calculated by

tg (xg ) t μψg →f (xg )

q 

 xg :cqg,i =1 e t λe cg,i = ln − λta cqg,i , t (x ) g t g μψg →f (xg ) xg :cqg,i =0 e (10) for i = 1, . . . , NG , q = 1, . . . , Q, where tg (xg ) and μtψg →f (xg ) are given by  −1 t V xg tg (xg ) = − xH f →ψg g    (11) −1 t H t + 2 xg V f →ψg mf →ψg , where

and

βˆt (xg ) , (xg )

μt−1 f →ψg

(16)

and the parameters of μ ˆtψg →f (xg ) can be derived by using the canonical form of Gaussian PDF and the Sherman–Morrison–Woodbury formula V tψg →f = (a)

 

V txg

= I +V

−1 t xg

−1 −1  − V t−1 f →ψg

 −1  t−1 t V f →ψg − V xg V txg ,

(17)

  −1 −1  t−1 mtxg − V t−1 m mtψg →f =V tψg →f V txg f →ψg f →ψg     −1 (b) t = I + V txg V t−1 f →ψg − V xg    −1 t t−1 t−1 t × mxg − V xg V f →ψg mf →ψg . (18)

IEEE WCNC'14 Track 1 (PHY and Fundamentals)

xg

x1 Ă



xG

r

μtψg →xg (xg ) μt−1 xg →ψg (xg )

Ă

Ă

Ă

Ă

˚t (xg )  β

Ă

f Ă

f1

μt−1 fr →xg (xg ) as follows

Ă

Ă

fr

Ă

Ă Ă

fNR

Fig. 2. The channel transition node f is opened to reveal its structure.

Using (a) in (17) and (b) in (18) can avoid the numerical problems incurred by V txg , which may become singular along with iterations. We will refer to the proposed algorithm formed by (8), (9), (17) and (18) as MMSE-EP. C. Low-Complexity Groupwise AMP-EP Detection Both the MMSE-EX algorithm and the MMSE-EP algorithm need to calculate the inversion of the square matrix V tf ∈ CNR ×NR as shown in (8) and (9), which makes the computational complexity prohibitively high for a large-scale MIMO system. Therefore, it is desirable to design a lowcomplexity solution. The channel transition node f in the factor graph in Fig. 1 can be further factorized as ⎛  2 ⎞     1   T exp ⎝− 2 eT H ig xg  ⎠ , (19) p (y | x) ∝ r y − er   σ n r g and its structure is revealed in Fig. 2, where the node with “=” represents the cloningnode as in [18], and node  f  2 r

 1  T T denotes fr (yr | x) ∝ exp − σ2 er y − er g H ig xg  . n Similarly, the message μtfr →xg (xg ) from the node fr to the cloning node of xg can be calculated by   μtfr →xg (xg ) = fr (y | x) μ ˆtxg →fr (xg ) x\xg  g =g (20)   = NC

t t eT r H ig xg ; zfr →xg , τfr →xg

,

  where μ ˆtxg →fr (xg ) = NC xg ; mtxg →fr , V txg →fr is the approximate message replacing the original message μtxg →fr (xg ), and the parameters zft r →xg and τftr →xg are given by ⎛ ⎞  ⎝y − zft r →xg = eT (21) H ig mtxg →fr ⎠ , r

xg ∈χNG

μtψg →xg (xg ) μt−1 xg →ψg (xg )

t−1 and the parameters mt−1 xg →ψg and V xg →ψg hold following relationships −1   1 H = H ig er eT V t−1 r H ig xg →ψg t−1 τfr →xg (25) r H

= H ig Λt−1 H ig ,  −1  zft−1 H H r →xg t−1 V t−1 m = H ig er = H ig Γt−1 . xg →ψg xg →ψg t−1 τfr →xg r (26) t−1 where Λ is a diagonal matrix with   T

1

, . . . , τ t−11 on its diagonal and fN →xg   t−1 R T zft−1 →xg zf →xg NR 1 Γt−1 = τ t−1 , . . . , τ t−1 . Then a Gaussian PDF f1 →xg fN →xg R   βˆt (xg ) = NC xg ; mtxg , V txg is found by minimizing the   ˚t (xg )  βˆt (xg ) , and the parameters KL divergence KL β of βˆt (xg ) are obtained by matching the moments of ˚t (xg ) and βˆt (xg ), namely mt β [xg ] and ˚t xg = Eβ  H (xg )   t − mtxg mtxg . Finally, the V xg = Eβ˚t (xg ) xg xH g parameters of approximate message μ ˆtxg →fr (xg ) are obtained by using μ ˆtxg →fr (xg ) = βˆt (xg ) /μt−1 fr →xg (xg )  −1 −1  −1 H t t t−1 T V xg →fr = V xg − τfr →xg H ig er er H ig τft−1 1 →xg

H

=V

t xg

+

t V txg H ig er eT r H ig V xg

, H t τft−1 − eT r H ig V xg H ig er r →xg  (27) t−1 z H fr →xg H ig er . mtxg →fr = V txg →fr (V txg )−1 mtxg − t−1 τfr →xg (28) By using (21), (22), (27), and (28), it is not hard to show that  τftr →xg = σn2 + κtr,g λtr,g − κtr,g λtr,g , (29) g

⎝ τftr →xg = σn2 + eT r

 g =g

zft r →xg = yr −

⎞ H ig V txg →fr H ig ⎠ er .

(22)



t t t t ηr,g  λr,g  + ηr,g λr,g .

H

To find the approximate message (xg ), we also use the principle of expectation propagation. First, the belief of xg is defined by μtψg →xg (xg ) and μt−1 xg →ψg (xg ) =

239

(30)

g t t where κtr,g = eT r H ig V xg H ig er , λr,g =

μ ˆtxg →fr

(23)

As μt−1 fr →xg (xg ) , r = 1, . . . NR are Gaussian PDFs, we can have   t−1 t−1 μt−1 (24) xg →ψg (xg ) ∝ NC xg ; mxg →ψg , V xg →ψg ,

g =g

⎛

.

zft−1 r →xg τft−1 r →xg

τft−1 r →xg t−1 τfr →xg −κtr,g

, yr =

t T t eT κtr,g . With (25) and r y, and ηr,g = er H ig mxg − t q  (26), LLRs λe cg,i are calculated by (10).

IEEE WCNC'14 Track 1 (PHY and Fundamentals)

Based on the above discussions, a low-complexity approximate message-passing (AMP) algorithm for the tth iteration between the decoders and the detector is summarized briefly as follows:     −1

IV. S IMULATION R ESULTS

N =1 G

−2

10

ρ=0.9

BER

1) With V t−1 and V t−1 mt−1 xg →ψg xg →ψg xg →ψg calculated in the last iteration and the downward message μtψg →xg (xg ), update mtxg and V txg via (23),   mtxg = Eβ˚t (xg ) [xg ] and V txg = Eβ˚t (xg ) xg xH g −  H −1  = 0 and mtxg mtxg . Note that V t−1 xg →ψg −1  V t−1 mt−1 xg →ψg xg →ψg = 0, when t = 1. t 2) Update τfr →xg via (29) and zft r →xg via (30).  −1 3) Update V txg →ψg via (25) and  −1 Vt mtxg →ψg via (26), calculate LLRs g  t x g →ψ λe cqg,i via (10), and pass LLRs to the decoders. We will refer to the above algorithm   as AMP-EP. Roughly, the complexity of AMP-EP is O N NT NR + NNNGT 2NG Q per subcarrier per iteration, of MMSE  and the complexity EX and MMSE-EP is O NR3 + NNNGT 2NG Q per subcarrier per iteration. Compared with the MMSE-EX and MMSE-EP, the complexity of AMP-EP is decreased by about NR times as N NT ≤ NR .

−3

10

ρ=0.7

−4

10

6

240

8

9

10 11 Eb/N0[dB]

12

13

14

Fig. 3. BER performance of 64 × 64 16QAM system, rate-1/2 (113, 171)oct RCC, ρ = 0.7 and ρ = 0.9.

−1

10

−2

10

MMSE−EX MMSE−EP AMP−EP NG=2

−3

We consider the uplink of a multiuser system with N = 32 independent users, and each user is equipped with NT = 2 transmit antennas. For each user, the transmission is based on OFDM with K = 128 subcarriers and a Gray-mapped 16QAM modulation, and 8192 information bits are encoded by a rate-1/2 recursive convolutional code (RCC) with generator polynomial [G1 , G2 ] = [133, 171]oct , and then interleaved by an S-random interleaver with S = 32 [21]. The channel model in all simulations is a 16-tap Rayleigh fading MIMO channel with equal tap power. The receive antennas are spaced sufficiently apart so that they can be regarded as being spatially uncorrelated, and the two transmit antennas within the same user are spatially correlated with correlation coefficient 0 ≤ ρ ≤ 1, while the transmit antennas from different users are spatially uncorrelated. Mathematically, R = I, S i,j = ρ for |i − j| = 1, S i,j = 0 for |i − j| > 1, and S i,i = 1 (due to limited space, we only consider this channel condition). For the groupwise detection, heuristically, we will set NG = 2 and assign the symbols within one user to a group for joint detection. In the following figures, “NG = 2” means that the corresponding algorithm (one of the MMSE-EX, MMSE-EP and AMP-EP) employs groupwise detection while “NG = 1” means that the corresponding algorithm employs symbol-wise detection, and the number of iterations is set to 10. Fig. 3 shows the bit error rate (BER) performance of considered algorithms in the strong spatially correlated channels with ρ = 0.7 and ρ = 0.9. When groupwise detection (NG = 2) is employed, both the proposed AMP-EP and MMSE-EP outperform the MMSE-EX, and the AMP-EP achieves the best

7

BER

−1

MMSE−EX MMSE−EP AMP−EP NG=2

−1

10

10

NG=1 MFB −4

10

4

4.5

5

5.5 Eb/N0[dB]

6

6.5

7

Fig. 4. BER performance of 64 × 64 16QAM system, rate-1/2 (113, 171)oct RCC, ρ = 0.

performance. It means that the AMP-EP can achieve better performance in the 64 × 64 16QAM system while the complexity is decreased by many times. As the complexity of AMP-EP with groupwise detection is still less than that of MMSE-EP (MMSE-EX) with symbol-wise detection in the this system, the AMP-EP is always set to employ the groupwise detection to increase the robustness against the spatial correlation in this strong spatially correlated channels. On the other hand, compared to the cases using the symbol-wise detection, the MMSE-EX and MMSE-EP using groupwise detection can obtain about 0.5 dB to 2.5 dB gain at BER = 10−4 . Fig. 4 illustrates the performance of the considered algorithms in the spatially uncorrelated channel with ρ = 0. The matched filter bound (MFB) obtained by exactly removing all the interferences is used as a lower bound on the BER. Similar to the observations in the spatially correlated channels, the AMP-EP outperforms the MMSE-EX and the MMSEEP whether the groupwise detection is employed or not. We can also find that both the proposed AMP-EP and MMSE-EP achieve the MFB at BER = 10−4 and outperform the MMSE-

IEEE WCNC'14 Track 1 (PHY and Fundamentals)

EX by about 0.8 dB at BER = 10−4 . V. C ONCLUSION We have used the unified factor-graph framework to design detection algorithms for the large-scale multiuser MIMOOFDM systems by combining the groupwise detection and the principle of expectation propagation. The resultant groupwise MMSE-EP detection offers robustness against the spatial channel correlation. Furthermore, we have developed a lowcomplexity message-passing algorithm by opening the channel transition node, which eliminates the matrix inversions in the MMSE-SIC based algorithms and achieves appealing performance. In future work, we will extend our proposed algorithms to the joint channel estimation and decoding by following the divergence minimization method in [22]. A PPENDIX Note that hereafter the iteration index t is omitted as it does not incur any ambiguity. By using the Sherman–Morrison–Woodbury formula [20], V −1 ξg can be rewritten as follows   H −1 V −1 ξg = V f − H ig V ψg →f H ig  −1 H H −1 −1 −1 =V −1 + V H − H V H H ig V −1 V ig ig ig f f ψg →f f f . (31)   H

By substituting (31) into V f →ψg = H ig V −1 ξg H i g can have  H H −1 V f →ψg = H ig V −1 f H ig + H ig V f H ig

−1

, we

−1 −1 H  H −1 −1 × V −1 − H V H H V H ig ig ig f ig f ψg →f −1  H = H ig V −1 − V ψg →f . f H ig (32)   H

By substituting V f →ψg = H ig V −1 ξg H ig

−1

and mξg =y− H

Hmx + H ig mψg →f into mf →ψg = V f →ψg H ig V −1 ξg m ξg , we can immediately have H

mf →ψg =V f →ψg H ig V −1 ξg (y − Hmx ) + mψg →f .

(33) By substituting (31) and (32) into (33), mf →ψg can be rewritten as   −1 H H H + Ξ H ig V −1 mf →ψg = H ig V −1 ig y y (y − Hmx ) + mψg →f ,

(34)  H where Ξ = −V ψg →f + I − V ψg →f H ig V −1 × H ig y   −1 H −1 V −1 = 0. ψg →f − H ig V y H ig ACKNOWLEDGMENT This work was partially supported by the National Nature Science Foundation of China (Grant Nos. 91338101,

241

61021001, and 61132002), the National Basic Research Program of China (Grant No. 2013CB329001), the Co-innovation Laboratory of Aerospace Broadband Network Technology, and also by the Australian Research Council’s Discovery Projects DP1093000, DECRA Grant DE120101266. R EFERENCES [1] G. L. St¨uber, J. R. Barry, S. W. McLaughlin, Y. G. Li, M. A. Ingram, and T. Pratt, “Broadband MIMO-OFDM wireless communications,” Proc. IEEE, vol. 92, no. 2, pp. 271–294, Feb. 2004. [2] F. Rusek, D. Persson, B. K. Lau, E. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and challenges with very large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60, Jan. 2013. [3] J. Hoydis, S. ten Brink, and M. Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?” IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 160–171, Feb. 2013. [4] W. Feng, Y. Wang, N. Ge, J. Lu, and J. Zhang, “Virtual MIMO in multi-cell distributed antenna systems: Coordinated transmissions with large-scale CSIT,” IEEE J. Sel. Areas Commun., vol. 31, no. 10, pp. 2067–2081, Oct. 2013. [5] B. M. Hochwald and S. ten Brink, “Achieving near-capacity on a multiple-antenna channel,” IEEE Trans. Commun., vol. 51, no. 3, pp. 389–399, Mar. 2003. [6] S. B¨aro, J. Hagenauer, and M. Witzke, “Iterative detection of MIMO transmission using a list-sequential (LISS) detector,” in Proc IEEE Int. Conf. on Commum. (ICC), 2003, pp. 2653–2657. [7] E. Larsson and J. Jald´en, “Fixed-complexity soft MIMO detection via partial marginalization,” IEEE Trans. Signal Process., vol. 56, no. 8, pp. 3397–3407, Aug. 2008. [8] C. Studer and H. B¨olcskei, “Soft-input soft-output single tree-search sphere decoding,” IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 4827– 4842, Oct. 2010. [9] Q. Guo, L. Fang, D. D. Huang, and S. Nordholm, “A soft-in soft-out detection approach using partial Gaussian approximation,” in Proc. Int. Conf. on Wireless Commum. and Signal Process. (WCSP), 2012. [10] X. Wang and H. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul. 1999. [11] C. Studer, S. Fateh, and D. Seethaler, “ASIC implementation of softinput soft-output MIMO detection using MMSE parallel interference cancellation,” IEEE J. Solid-State Circuits, vol. 46, no. 7, pp. 1754– 1765, Jul. 2011. [12] M. Grossmann and C. Schneider, “Groupwise frequency domain multiuser MMSE turbo equalization for single carrier block transmission over spatially correlated channels,” IEEE J. Sel. Topics in Signal Process., vol. 5, no. 8, pp. 1548–1562, Dec. 2011. [13] F. Kschischang, B. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. [14] T. P. Minka, “A family of algorithms for approximate Bayesian inference,” Ph.D. dissertation, Massachusetts Institute of Technology, 2001. [15] J. Hu, H.-A. Loeliger, J. Dauwels, and F. Kschischang, “A general computation rule for lossy summaries/messages with examples from equalization,” in Proc. 44th Allerton Conf. Communication, Control, and Computing, 2006, pp. 27–29. [16] M. Senst and G. Ascheid, “How the framework of expectation propagation yields an iterative IC-LMMSE MIMO receiver,” in Proc. Global Telecomm. Conf. (GLOBECOM), 2011, pp. 1–6. [17] L. Schumacher, K. Pedersen, and P. Mogensen, “From antenna spacings to theoretical capacities-guidelines for simulating MIMO systems,” in Proc. IEEE Int. Symp. PIMRC, 2002, pp. 587–592. [18] H.-A. Loeliger, “An introduction to factor graphs,” IEEE Signal Process. Mag., vol. 21, no. 1, pp. 28–41, Jan. 2004. [19] H. Wymeersch, Iterative Receiver Design. Cambridge, U.K.: Cambridge Univ. Press, 2007. [20] G. H. Golub and C. F. Van Loan, Matrix computations. Johns Hopkins University Press, 1996. [21] C. Heegard and S. B. Wicker, Turbo coding. Boston, MA: The Kluwer, 1999. [22] T. Minka, “Divergence measures and message passing,” Microsoft Research Cambridge, Tech. Rep., 2005.