Convex-optimization-based precoding for MIMO downlinks

The Journal of China Universities of Posts and Telecommunications December 2011, 18(6): 22–26 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Convex-optimization-based precoding for MIMO downlinks LI Xin-min1,2 ( ), BAI Bao-ming1 1. State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China 2. Xi’an University of Science and Technology, Xi’an 710054, China

Abstract In this paper, the design of linear leakage-based precoders is considered for multiple-input multiple-output (MIMO) downlinks. Our proposed scheme minimizes total transmit power under each user’s signal-to-leakage-plus-noise ratio (SLNR) constraint. When the base station knows perfect channel state information (CSI), suitable reformulation of design problem allows the successful application of semidefinite relaxation (SDR) techniques. When the base station knows imperfect CSI with limited estimation errors, the design problem can be solved using semidefinite program (SDP). At the same time, it can dynamically allocate each user’s SLNR threshold according to each user’s channel state, so it is more feasible than other similar SINR-based precoding methods. Simulation results show that using large SLNR thresholds, the proposed design has better bit error rate (BER) performance than maximal-SLNR precoding method at high signal-to-noise ratio (SNR). Moreover, when the base station knows imperfect channel state information, the proposed precoder is robust to channel estimation errors, and has better BER preformance than other similar SINR-based precoding methods. Keywords MIMO, signal-to-leakage-plus-noise ratio (SLNR), linear precoding, convex optimization

1

Introduction 

MIMO technique has attracted much attention due to its potential of high spectral efficiency. In the case of multiuser MIMO downlinks, the suppression of co-channel interference (CCI) can be pursued by using linear precoders at the base station (BS). Linear precoding schemes mainly includes signal-to-interference-plus-noise ratio (SINR) based methods, zero-forcing (ZF) methods and SLNR based methods. SINR based precoding methods maximize the output SINR for all users [1–3]. Due to the complexity and the coupled nature of this optimization problem, no closed form solutions are available, and the solutions can be obtained iteratively according to uplink-downlink duality. Recently, a robust precoder based on SINR is proposed, which uses convex-optimization methods to minimize the total transmit power under SINR constraints when channel Received date: 18-04-2011 Corresponding author: LI Xin-min, E-mail: [email protected] DOI: 10.1016/S1005-8885(10)60118-5

state information (CSI) is imperfectly known to the BS [4–6]. But this scheme is less feasible when channel estimation error is large, because it fixedly allocates SINR constraint values ignoring the difference of users’ channel state. ZF precoding methods can perfectly cancel the CCI by selecting each user’s equivalent channel orthogonal to other users’ channels [7–9]. These methods require the number of transmit antennas at the base station be larger than the total number of receive antennas of all users. This condition is necessary in order to provide enough degrees of freedom for the zero-forcing solution to force CCI to zero at each user. Recently, some other methods choose the precoders optimally in order to maximize the output SLNR [10–12]. Leakage is a measure of how much signal power leaks into other users. This leakage-based criterion leads to a decoupled optimization problem and admits an analytical closed form solution. Moreover, in contract to the ZF solution, the leakage scheme does not require any dimension condition on the transmit/receive antenna, and it outperforms ZF solutions.

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LI Xin-min, et al. / Convex-optimization-based precoding for MIMO downlinks

In this paper, we apply convex optimization theory to SLNR-based precoder design, and consider following two extensions: 1) We pursue all users’ fairness for designing precoders on the concept of signal leakage when the BS perfectly knows CSI. 2) We design a robust precoder when the BS knows CSI within specified uncertainty region. Our proposed design minimizes total transmit power under each user’s SLNR constraint when the BS knows perfect or uncertain CSI. This paper is organized as follows. The next section describes the multi-user MIMO system and design problem considered in this paper. The design of precoders when the BS knows perfect SCI is considered in Sect. 3, and the robust precoders with uncertain CSI is designed in Sect. 4. Simulation results is presented in Sect. 5. Finally, in Sect. 6 conclusions are presented.

2

System model and problem formulation

Consider the multi-user MIMO downlinks with a BS communicating with K users as shown in Fig. 1. Assume that the BS is equipped with nT transmit antennas and each user is equipped with a single antenna.

Fig. 1 System model of the MIMO downlinks

The signal received by the kth user is hk pk d k 

Yk

K

¦

i 1, i z k

hk pi di  nk

(1)

where d k denotes the transmitted data intended for user k, pk is an nT u 1 precoding vector for the kth user. hk represents the channel vector of size 1u nT from the BS to kth user. nk is AWGN with zero-mean and variance

V n2 . The SLNR of kth user is defined as: k

RSLNR (k )

hk pk K

¦

i 1, i z k

hi pk

(2) V

Then pk is normalized by its norm-2 to ensure pk

2 nk

Previous researches all seek to maximize each user’s SLNR to obtain optimal precoding vector pk [10–12].

2

1.

This precoding method is called maximal-SLNR scheme here. In this paper, the design of a precoder that minimizes total transmit power under SLNR constraints is considered, which can be stated as: K ½ 2 min ¦ pk 2 ° pk (3) ¾ k 1 s.t. RSLNR (k )ıJ 0 ; k 1, 2,..., K °¿ Note that this problem to be solved requires all user satisfy same SLNR constraints to ensure users’ fairness, and BS has minimum total transmit power.

3

Precoding with perfect CSI

The maximal-SLNR scheme has the problem that the suppression of co-channel interference and suppression of noise cannot be balanced especially at high SNR [12], so in our goal problem Eq. (3), we allocate large SLNR threshold value J 0 to improve BER performance at high SNR. If the BS knows perfect CSI, substituting Eq. (2) into Eq. (3), we get K ½ 2 min ¦ pk 2 ° pk ° k 1 ¾ (4) K 2 2 2 s.t. hk pk  J 0 ¦ hi pk ıJ 0V n ; k 1, 2,..., K ° °¿ i 1, i z k The above problem is a quadratically constrained quadratic program (QCQP). It is NP-hard in general. Using semidefinite relaxation (SDR), the suboptimal solutions can be generated with quite encouraging results. Define a vector of size KnT u 1 : w [ p1* p2* ... pK* ]* , where the superscript * denotes Hermitian transpose. Then we can rewrite Eq. (4) as ½ min tr( ww * ) w ° K ° § * · * * 2 (5) s.t. tr ¨ C k Qk C k  ¦ J 0 C k Qi C k ¸ ww ıJ 0V n ; ¾ ° i 1,i z k © ¹ ° k 1, 2,..., K ¿ where tr( ) denotes the trace of a matrix, Qk

2

2

23

Qi

K

¦

i 1, i z k

hi* hi , and C k

>0

" Ik

" 0@ ( I k is the

identity matrix of size nT u nT ). Let X Eq. (5) can be write equivalently as

hk* hk ,

ww * , Then

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The Journal of China Universities of Posts and Telecommunications

min tr( X ) w

s.t.

tr( Ak X )ıJ 0V ; k 2 n

X ı0, rank X

where Ak

1

C Qk C k  * k

K

¦

i 1, i z k

½ ° 1,..., K ¾ ° ¿

(6 )

J 0 C Qi C k . Note above rank * k

constraint on X is not convex and other constraints are all convex. So this problem is in a form suitable for SDR. That is, dropping the rank-one constraint, we obtain the relaxed problem min tr( X ) ½ w ° (7) s.t. tr( Ai X )ıJ 0V n2 ; k 1, 2,..., K ¾ ° X ı0 ¿ which is now expressed into a standard SDP problem, and can be efficiently solved using general purpose implementations of interior poiter methods, such as SeDuMi [13] or CVX (Cvx users’ guide for CVX version1.21, and available at http://c1319062.cdn. cloudfiles. rackspacecloud. com/cvx_ usrguide. pdf). Due to the relaxation, the matrix X c obtained by solving above SDP will not be rank one in general. There are many ways of generating good solutions to our original problem in convex optimization. We select following Gaussian randomization procedure to obtain satisfactory solutions [14– 15]. Step 1 Given an SDR solution X c , and a number of randomizations L, for l=1,2,…,L, generate ȟ l  ȃ (0, X c) Construct a QCQP-feasible point ȟl ; l 1, 2,..., L min  ȟ l* Ai ȟ l

Step 2 x l

i 1,2,..., K

Step 3 Determine l *

arg min x l* x l , output wopt l 1,2,..., L

x l* as the approximate QCQP solution.

4

Robust precoding with uncertain CSI

When the channel estimation technique used at the receiver is non-ideal, the BS will obtain the CSI with errors. Model the kth user’s channel as: hk hˆk  ek (8) where h is the pratical channel, hˆ is the estimation of k

k

the user’s channel, and

ek

is the corresponding

estimation error. We assume that estimation error lies in the ball ek İG k . uk (G ) {hk hk

hˆk  ek , ek İG k }

(9)

G1

2011

Assume each user has same estimation error, that is, G 2 " G K G . Then downlink channel is described

by H

Hˆ  E

and

E İ KG .

[hˆ1

hˆ2

... hˆK ]  [e1

e2

... eK ]

(10)

When the BS knows uncertain CSI as above, our goal is to design a robust precoding matrix that minimizes the transmit power required to ensure that each user’s SLNR constraint is satisfied. This design problem can be formulated as following robust second order cone program (SOCP): min t ½ pk , t ° K ° s.t. ¦ vec([ p1 p2 " pK ]) İt ° (11) ¾ k 1 ° V nk ] İE k hk pk , hk  uk (G ); ° [ Hpk ° k 1, 2,..., K ¿ Using Lemma 1 in Refs. [4–5] (detailed in Ref. [16]), above robust SOCP (RSOCP) can be formulated as following SDP: min t ½ pk , P , O , t ° ° tI vec([ p1 , p2 ,..., pK ]) º ª ı0 ° s.t. « » T t ¬ (vec([ p1 , p2 ,..., pK ])) ¼ ° ° T ˆ ª Ok  P k º 0 [ Hpk V nk ] °° « » ¾ « 0 K G [ pk 0]» ı0 ° Pk I « » ° ˆ K G [ pk 0]T V nk ] Ok I «[ Hp » k ° ¬ ¼ ° T ª E k hˆk pk  O º GE k pk ° « » ı0; k 1, 2,..., K ° ˆ ( E k hk pk  O ) I ¼» ¬« GE k pk ¿° (12) This problem can also be efficiently solved using software tool SeDuMi or CVX. Our method is similar to the RSOCP algorithm presented by [4–5], which minimizes the transmitted power under users’ SINR constraints. In their goal problems, users’ fairness should be satisfied. But this fairness requirement ignores each user’s instantaneous channel state condition, so it results in less feasibility of RSOCP algorithm. Our proposed scheme based on SLNR constraints has advantage over those based on SINR constraints. Because our scheme can use the existing closed form solutions in Refs. [10–11] to dynamicly allocate each user’s SLNR threshold, it is more feasible than the latter. In order to

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LI Xin-min, et al. / Convex-optimization-based precoding for MIMO downlinks

25

compare with RSOCP algorithm in the same condition, assume all users have same SLNR threshold values, that is, users’ fairness can be satisfied. Thus we can easily and quickly one-dimensionally search maximum SLNR constraints satisfied by all users according to maximum available SLNRs of each user that the closed form solutions indicate.

5

Simulation results

In the simulations, the downlink of a BS with 3 transmit antennas and 3 users each with a single antenna is assumed. All the simulations use BPSK signals with unit power as user data, and MIMO channels are Rayleigh fading channels. In our first simulation, assuming the BS knows perfect CSI ( G 0 ), we compare BER performance of our scheme with maximal-SLNR scheme, shown in Fig. 2. From this figure, it is clear that the larger SLNR threshold values, the better BER performance. Especially, when the threshold J 0 is larger than 30, the proposed precoding scheme has better BER performance than maximal-SLNR precoding method in Ref. [10], especially at high SNR.

Fig. 3 Numbers of solutions obtained by robust precoding methods in 2 000 channel realization

Finally it can be seen in Fig. 4 that the proposed scheme provides a reasonable degree of robustness to channel uncertainty compared to the maximal-SLNR scheme and outperforms the SINR based precoding scheme when it fixedly allocates SLNR constraints at G =0.03. If SLNR constraint values are allocated by dynamicly searching between 10 to 4, the BER performance of our method gets worse because the performance of such system is determined by the user with the smallest SLNR values.

Fig. 4 Fig. 2

BER performance with channel estimation error G =0.03

BER performance with perfect CSI

In our second simulation, we randomly generate 2 000 realization of channel estimations and examine the performance of each method in the presence of estimation errors. In Fig. 3, for each estimation error value G , we provide a histogram of the number, N feasible , of the 2 000 channel realizations for which each precoding scheme can generate a precoder with finite power (assume all at SNR is 10 dB). From this figure, it is clear that our proposed scheme by dynamicly allocating SLNR constraint values is more feasible than the SINR-based scheme or ours with fixed allocating SLNR.

6

Conclusions

A robust linear precoder for MIMO downlinks that minimizes the total transmit power under each user’ SLNR constraint is proposed. Although the goal problems is computationally intractable, we use semidefinite relaxation when the BS knows perfect CSI or a robust convex-optimization method when the BS knows CSI with limited estimation errors to obtain optimal solutions. Simultion results show that our methods are robust to channel estimation errors and have satisfactory BER performance.

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The Journal of China Universities of Posts and Telecommunications

Acknowledgements This work was supported by the National Natural Science Foudation of China (60972046), and the S&T Major Special Project (2009ZX03003-11-05, 2010ZX03003-003), the Scientific Research Program Funded by Shaanxi Provincial Education Commission (2010JK666).

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