Oblique Projection Based Linear Precoding for ... - IEEE Xplore

Workshop on Mobile Computing and Emerging Communication Networks

Oblique Projection Based Linear Precoding for Downlink Multi-user Multiple-input Multiple-output Communications Peng Shang, Jinsong Wu, Member, IEEE, Xudong Zhu Bell Laboratories, Alcatel-Lucent Shanghai 201206, P.R. China Email: [email protected], [email protected], [email protected]

Abstract—In this paper, we investigate linear precoding and scheduling algorithms for downlink multi-user multiple input multiple output (MU-MIMO) systems. In order to eliminate the multi-user interference and improve the bit error rate (BER) system performance, block diagonalization precoding could be combined with geometric mean decomposition (BDGMD) scheme. We propose GMD linear precoding approaches utilizing oblique projection (OP-GMD). This paper also derives system capacity in closed-form through applying water-filling power allocation algorithm into the system model. Our simulation results show that the proposed scheduling algorithms perform better with the increase of the number of transmit antennas. In terms of system capacity, the proposed OP-GMD algorithms significantly outperform the precoding scheme based on BDGMD at low SNRs. Index Terms—MU-MIMO, block-diagonalization geometric mean decomposition, linear precoding, oblique-projection geometric-mean-decomposition, orthogonal projection

I. I NTRODUCTION The increase of needs in high-volume multimedia transmission leads to heavy demands for high-data-rate wireless communications. In recent years, the multiple-input multipleoutput (MIMO) antenna techniques have become a class of dominant solutions to radically enhance the capacity and spectrum efficiency of wireless communication systems [1]. Comparing with single-user MIMO system, the spatial division multiple access (SDMA)-based multi-user MIMO (MUMIMO) scheduling techniques have been considered to be an alternative important virtual MIMO technology choices to improve the system capacity significantly, which may help multiple users transmit information at the same time, spectrum and code domain. When channel state information (CSI) is perfectly available at the transmitter, MU-MIMO systems with

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dirty paper coding (DPC) may reach the downlink broadcast channel capacity [2]. Even if approximated through using Tomlinson-Harashima precoding techniques in practice [3], DPC is still too hard to be implemented due to its exhibitive computational complexity and system cost. Thus, some near-optimal precoding algorithms with lower complexity and fairly good performance have drawn extensive attentions. Those suboptimal precoding algorithms may be categorized as codebook based and noncodebook based precoding schemes using either perfect or partial CSI. The codebook based precoding algorithms, such as unitary precoding based on Fourier transform [4], require both the transmitter and the receiver to know fixed codebooks, so that the receiver can choose the proper codeword from the codebook to precode the transmitted data from users according to CSI feedback. The other kind of suboptimal precoding algorithms is based on real time channel signal processing, which requires that the transmitter knows the CSI either perfectly or at least partly. The most popular precoding algorithms are orthogonal projection based block diagonalization (BD) and zero forcing (ZF) precoders [5]. BD precoding decomposes one multi-user MIMO channel to several independent equivalent single user channels by mapping current user’s channel matrix to other scheduled user’s null space, while ZF precoding utilizes the pseudoinverse of channel matrices to obtain complete diagonalization of the channel. Usually, ZF precoding is used for each receiver with a single antenna, while BD precoding is for each receiver with multiple antennas. The orthogonalization procedure for subchannels in ZF precoding results in the decrease of subchannel gains on projection directions, since ZF precoding for

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the case of each receiver with multiple antennas will incurred more loss in CSI than the BD precoding case. Note that BD precoding is based on singular value decomposition (SVD), and hence, the gains of different subchannels on the same user may differ in a large range. As we know, the bit error rate (BER) performance would be dominated by the weakest subchannels. Recently, geometric mean decomposition (GMD) [6] is used to resolve this problem through decomposing the MIMO channel into multiple subchannels with identical signal-to-noise (SNR). In order to both eliminate the multi-user interference and improve the system BER performance, block diagonalization precoding could be combined with geometric mean decomposition (BD-GMD) [7]–[9]. Imperfect CSI at the transmitter may result in the significant loss of the system throughput because of the residual interference among transmit streams. In addition, those methods are essentially based on orthogonal projection to construct parallel channels, so that there is notable loss in actually used channel state information. As an extension of the orthogonal projection, oblique projection has been extensively used in communication signal processing. Behrens and Scharf used oblique projection to process blind identification in SIMO system [10]. McCloud and Scharf applied oblique projection to the prediction of direction of arrival (DOA) [11]. Under the context of the combination of block diagonalization precoding and geometric mean decomposition, this paper proposes GMD linear precoding algorithms based on oblique projection (OP-GMD) to provide better capacity performance in severe channel conditions. The remainder of the paper is organized as follows. Section II introduces the mathematical preliminaries of oblique projection. In Section III, we present the system model and propose the OP-GMD precoding algorithm, and the relevant scheduling algorithms are then presented. Simulation results are presented in Section IV. Finally, we draw the conclusion in Section V. Notations: (·)T and (·)H denote transpose and transpose conjugate operations, respectively, rank(·) denotes matrix rank, tr(·) denotes the trace of matrix, ceil(x) denotes the smallest integer value greater than or equal to x, Rang(·) denotes the column space or the range of a matrix, and (x)+ denotes the max (x, 0).

perspective. In 1989, Kayalar and Weinert proposed an application of oblique projection for array signal processing, then deduced some new calculation methods and relevant iterative algorithms for oblique projection. In 1994, Behrent and Scharf worked out much more realistic calculation methods [10] aiming at the column space of a matrix. In 2000, Vandaele and Moonen derived oblique projection of the row space of a matrix by using matrix LQ decomposition [16]. The two kinds of projections, oblique projection and orthogonal projection, play a key role in adaptive signal processing. Oblique projection is the extension of orthogonal projection while orthogonal is a special case of oblique projection. We assume matrices A ∈ CN ×M , B ∈ CN ×K , M, K  N , where the column vectors of A and B are linear uncorrelated, then the projection matrix of matrix A is PA = AA† while A† = (AH A)−1 AH is the pseudo-inverse matrix of matrix A; P⊥ A = I−PA is the orthogonal projection matrix of matrix A. Oblique projection EAB is the projection along the parallel direction of Rang(B) to Rang(A): −1 H ⊥ EAB = A(AH P⊥ A PB B A)

(1)

EAB is square matrix without complex-conjugate symmetry, and has the following characteristics EAB A = A, EAB B = 0.

(2)

Because of the exchangeability between matrix A and B, −1 H ⊥ we could obtain EBA = B(BH P⊥ B PA using (1). A B) Denote R = [A, B], then the orthogonal projection of R, PR , could be decomposed into PR = EAB + EBA

(3)

III. A LGORITHM D ESIGN In this section, we describe the GMD linear precoding algorithm based on oblique projection in details. A. Precoding Design As shown in Figure 1, the considered downlink MU-MIMO system is with K users. The base station has M antennas, and the k-th user has Nr,k receiving antennas. Through multiuser scheduling and precoding techniques to cancel multiuser interference (MUI), the receiving signal vector of the k-th user is given by,

II. M ATHEMATICAL P RELIMINARIES Oblique projection was first proposed in [12] and [13], and then further investigated in [14] and [15] in mathematical

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yk = Hk Wk Dk xk + Hk + nk k = 1 · · · K,

K  j=1,j=k

W j Dj x j

(4)

where Hk ∈ CNr,k ×Nt is the channel gain matrix of the kth user, the elements of Hk are independent and identicallydistributed complex cyclic symmetric Gaussian random variable with zero mean and unit variance, xk ∈ CNk ×1 is the source signal vector for the k-th user (Nk  Nr,k ), Wk ∈ CNt ×Nk is the precoding matrix for the k-th user and the base station antenna arrays, Dk is the diagonal power allocation for the k-th user, nk is additive complex Gaussian noise (AWGN) 2 Nr,k ×Nr,k with the covariance matrix E[nk nH k ]=σ I ( I∈C is identity matrix). Denote the channel matrix of the residual   ˜ k = HT1 · · · HT , HT · · · HT T . scheduled users as H K0 k−1 k+1 K0 is the number of simultaneous users at this slot.

Fig. 1: Downlink multi-user MIMO system diagram

elements are the corresponding scheduled power values of each sub-channel for the k-th user. We assume the total system power constraint is P , then (4) could be simplified to yk = Qk Rk PH k Pk Ek xk + nk = Qk R k E k xk + n k .

(8)

Both sides of (8) are multiplied by QH k , and then we could obtain y ¯ k = QH ¯k k Qk R k E k xk + n = R k Ek x k + n ¯k ,

(9)

where n ¯ k = QH k nk . Perfect CSI is assumed at the receiver. VBLAST decoding algorithm is adopted on the receiver of the k-th user. Using V-BLAST, the most powerful signal is first detected, and from this decision, the received signal is regenerated. Then, the signal is regenerated subtracted from the received signal, and it proceeds to the detection of the second most powerful signal and so forth. Hence, the k-th user could obtain Nk independent subchannels with ak 2 channel gains  k = ak Ek x k + n k y

We now consider using a precoding method to cancel the multi-user interferences to guarantee no interference from other users in the receiving signal of each user. Let A = HH k ˜ H , then we can get the matrix Wk = EAB from and B = H k (1). According to the relations in (2), the oblique projection H precoding matrix Wk = Wk for the k-th user shall satisfy the following requirements,  Hk Wk = Hk , ∀j = k and 1  j, k  K. (5) Hk Wj = 0, We then denote the equivalent channel matrix of the k-th user as Hef f,k = Hk Wk = Hk . To maximize the gains of the worst sub-channels, GMD is chosen instead of the traditional SVD, that is Hef f,k = Qk Rk PH k ,

ak = rii,k = (

M 

σm,k )1/M , 1  m  M,

And the system capacity is calculated as COP (S) =

k∈S

where r11,k = ... = rNk Nk ,k , σm,k is the mth non-zero singular value of Hef f,k , and rank(Hef f,k ) = M . Denote Dk = Pk Ek . Ek is a diagonal matrix whose diagonal

 k∈S

max

H H T r(Wk Pk Ek EH k Pk Wk )≤P }

log det (I +

1 2 2 σ 2 ak Ek ).

(11)

Using Lagrange multiplier method to construct the function Z=

Nk 

log det (1 +

k∈S i=1

1 2 2 a E )+ σ 2 k k,i

Nk 

 H H Wk Pk Ek EH L P− k Pk Wk ii ,

(12)

k∈S i=1

where L is Lagrange multiplier. Calculate the derivative of 2 (12) with respect to Ek,i , and set it to be zero ln 2(1

1 2 σ 2 ak 2 ) + σ12 a2k Ek,i

Then we obtain

2 Ek,i

(7)

m=1

{ S∈I,



(6)

where Qk and Pk are unitary matrixes, Rk is real upper triangular matrix with uniform diagonal elements denoted as rii,k .

(10)

and

562

 k∈S

=



H − L Wk Pk PH k Wk ii = 0.

1 2 σ 2 ak

σ2

 − H a2k L ln 2 Wk Pk PH k Wk ii 1 2 σ 2 ak

σ2

 − H a2k L ln 2 Wk Pk PH k Wk ii

(13)

+ (14)

+  P.

(15)

2 Inserting the result of Ek,i into formula (11), we could have  1 log det (I + 2 a2k E2k ). (16) COP (S) = max 2 σ Ek,i

•

BD-GMD precoding based on orthogonal projection (BDGMD)

k∈S

5.5

To guarantee the existence of precoding matrix Wk , the number of selected receiving antennas of all users should be less than that of transmitting antennas of the base station. In this paper, all users are assumed to have the same number of receiving antennas, that is to say, Nr,k = Nr, ∀k. Then at the same time, the maximal number of users that system could select is Kmax = ceil(Nt /Nr ). We denote the set of all users within system as I = {1 · · · K}, and denote the set of all currently scheduled users as S ∈ I (|S|  Kmax ). B. Scheduling algorithm For oblique projection based precoding, we propose to use two different scheduling methods, exhaustive search scheduling and greedy search scheduling [17]. The proposed greedy search scheduling algorithm is described as follows:

5

System Capacity (bits/s/Hz)

4.5

OP−GMD−MAX OP−GMD−GA BD−GMD

1.5 1

0

5

10

15

20 25 30 The number of users (K)

35

40

45

50

Fig. 2: Performance comparison of sum rate capacity versus the number of users (SNR = 0dB, nt = 4, nr = 2)

28

26 System Capacity (bits/s/Hz)

step 1) Initialization: let Ω = {1, 2, · · · , K}, S ∗ = ∅, i = 1. 2 step 2) Choose first user: s1 = arg max Nk log(1 + NPk rkk ), { k∈Ω}

C1 = Ns1 log(1 + NPs rs21 s1 ), 1 users Ω = Ω − {s1 }, S ∗ =

the current channel capacity and then update the set of S ∗ + {s1 }, i = 2. step 3) To each user k ∈ Ω, let Sk∗ = S ∗ + {k}, we could obtain the current system capacity Ck∗ from formula (16), and choose the user si = arg max(Ck∗ ) in this iteration, { k∈Ω}

then update current user set Ω = Ω − {si }, S ∗ = S ∗ + {si } . step 4) When i = Kmax , the algorithm ends, and the optimal user set S ∗ can be obtained, otherwise i = i + 1 and return to step 3).

IV. S IMULATION R ESULTS The signal-to-noise ratio is defined as SN R = P/σ 2 . It is assumed that channel experiences quasi-static stationary Rayleigh fading, and base station could obtain channel state information of all users. Using system capacity criteria, three different schemes are compared in simulations:

•

3 2.5 2

Algorithm 1 Greedy scheduling algorithm

•

4 3.5

Oblique projection precoding based on exhaustive search scheduling algorithm (OP-GMD-MAX) Oblique projection precoding based on greedy search scheduling algorithm (OP-GMD-GA)

24 OP−GMD−MAX OP−GMD−GA

22

BD−GMD 20

18

16

0

5

10

15

20 25 30 The number of users (K)

35

40

45

50

Fig. 3: Performance comparison of sum rate capacity versus the number of users (SNR = 20dB, nt = 4, nr = 2) Figures 2 and 3 illustrate the sum rate capacity of the three schemes versus the number of users when system configuration is Nt = 4, Nr = 2, and at the same time, the maximal selected number of users Kmax is 2. It is notably shown in Figure 2, in a lower SNR case, the sum rate capacity for the OP-GMD precoding scheduling scheme is much higher than that for BD-GMD joint precoding scheme, and the sum rate capacity of the OP-GMD precoding scheme with suboptimal greedy search is also higher than that for BD-GMD precoding. As shown in Figure 3, in a higher SNR case, the system capacity of OP based precoding is slightly lower than that of BD-GMD system but the difference is much less than that in the lower SNR case. Note that OP is nonunitary matrix precoding, and thus the conventional water-

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conventional BD-GMD precoding scheme when SNR is low. On the other hand, when SNR is high, conventional BD-GMD system may perform a little better in the system capacity than the proposed OP-GMD linear precoder. The proposed algorithm may be used to improve the performance of system capacity for weaker channel conditions.

12

System Capacity (bits/s/Hz)

10

8

6

4 OP−GMD−MAX

0

R EFERENCES

OP−GMD−GA

2

BD−GMD

0

5

10

15

20 25 30 The number of users (K)

35

40

45

50

Fig. 4: Performance comparison of sum rate capacity versus the number of users (SNR = 0dB, nt = 6, nr = 2)

38 36

System Capacity (bits/s/Hz)

34 32 OP−GMD−MAX

30

OP−GMD−GA

28

BD−GMD

26 24 22 20

0

5

10

15

20 25 30 The number of users (K)

35

40

45

50

Fig. 5: Performance comparison of sum rate capacity versus the number of users (SNR = 20dB, nt = 6, nr = 2)

filling power allocation algorithm for BD-GMD can not be directly applied to the OP case. The settings in Figures 4 and 5 are Nt = 6, Nr = 2, and Kmax = 3, and the results of relevant performance comparisons of three schemes are similar to those in Figures 2 and 3 except using different antenna configurations. The number of synchronous transmission users increases with the increased number of transmission antennas, and the system capacity for all three scheduling schemes increases accordingly. V. C ONCLUSIONS In this paper, we have proposed oblique projection based GMD precoding algorithm. We have derived the closed-form expression of system capacity through implementing a waterfilling power allocation algorithm. The simulation results have shown that our proposed OP-GMD based precoding scheme obtains significant system capacity performance gains over the

[1] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanash, “Capacity limits of MIMO channels,” IEEE J. Select Areas in Commun, vol. 21, no. 5, pp. 684–702, Jun. 2003. [2] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 3936–3964, Sept. 2006. [3] J. Liu and W. A. Kizymien, “Improved tomlinson-harashima precoding for the downlink of multi-user MIMO systems,” Canadian Journal of Electrical and Computer Engineering, vol. 32, no. 3, pp. 133–144, 2007. [4] Samsung, “Downlink MIMO for EUTRA,” 3GPP TSG-RAN WG1 #43, R1-051353, Nov. 2005. [5] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Processing, vol. 52, no. 2, pp. 461–471, 2004. [6] Y. Jiang, W. Hager, and J. Li, “The geometric mean decomposition,” Linear Algebra and Its Application, vol. 396, pp. 528–541, 2005. [7] H. FU, T. Yao, X. Jiang, and S. Chen, “Performance enhancement for multiuser MIMO systems with block diagonalization using geometric mean decomposition,” Advances in Systems Science and Applications, vol. 6, no. 4, pp. 576–583, 2006. [8] S. Lin, W. Ho, and Y.-C. Liang, “Block diagonal geometric mean decomposition (BD-GMD) for MIMO broadcast channels,” IEEE Trans. Wireless Commun., vol. 7, no. 7, pp. 2778–2789, July 2008. [9] F. Liu, L. Jiang, and C. He, “Advanced joint transceiver design for block-diagonal geometric-mean-decomposition-based multiuser MIMO systems,” IEEE Trans. Inform. Theory, vol. 59, no. 2, pp. 692–703, Feb. 2010. [10] R. T. Behrens and L. L. Scharf, “Signal processing applications of oblique projection operators,” IEEE Trans. Signal Processing, vol. 42, no. 6, pp. 1413–1424, 1994. [11] M. L. McCloud and L. L. Scharf, “A new subspace identification algorithm for high-resolution DOA estimation,” IEEE Trans. Antennas and Propagation, vol. 50, no. 10, pp. 1382–1390, 2002. [12] F. J. Murray, “On complementary manifolds and projections in lp and lp,” Trans Amer Math Soc, vol. 43, no. 10, pp. 138–152, 1937. [13] E. R. Lorch, “On a calculus of operators in reflexive vector spaces,” Trans Amer Math Soc, vol. 45, pp. 217–234, 1939. [14] S. N. Afriat, “Orthogonal and oblique projectors and the characteristics of pairs of vector spaces,” Math PROC Cambridge Philos Soc, vol. 53, pp. 800–816, 1957. [15] K. Takeuchi, H. Yanai, and B. N. Mukherjee, The Foundations of Multivariante Analysis. Wiley, New York, 1982. [16] P. Vandaele and M. Moonen, “Two deterministic blind channel estimation algorithms based on oblique projections,” Signal Processing, vol. 80, pp. 481–495, 2000. [17] T. Cormen, C. Leiserson, R. Rivest, and C. Stein, “Introduction to algorithms,” 2003.

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