Consensus-Based Distributed MIMO Decoding Using ... - CiteSeerX

Consensus-Based Distributed MIMO Decoding Using Semidefinite Relaxation∗ Hao Zhu, Alfonso Cano, and Georgios B. Giannakis (contact author) Dept. of ECE, University of Minnesota. E-mail: {zhuxx132,alfonso,georgios}@umn.edu

Abstract— A distributed algorithm is developed for decoding a message broadcasted from a wireless multiantenna access point to a network of single-antenna (sensor) nodes. Sensors exchange local messages to reach consensus on the transmitted message. Different from recent distributed detectors where the amount of information exchanges increases exponentially with the alphabet size (i.e., the number of hypotheses tested), the novel consensusbased approach introduced here relies on semi-definite relaxation techniques and can afford inter-sensor exchanges of polynomial order. The resultant near-optimum convexified problem is solved in a distributed fashion using the alternating direction method of multipliers. No constraint is imposed on the network topology so long as it remains fully connected. Preliminary simulations demonstrate the merits of the novel distributed detection algorithm.

I. I NTRODUCTION Consider a wireless access point (AP) equipped with multiple antennas broadcasting a common message to a network of sensors, each equipped with a single antenna, as depicted in Fig. 1. If each sensor decodes the message independently, different sensors will generally arrive at different message estimates; i.e., the network would lack consensus in decoding the broadcasted message. However, allowing sensors to exchange information can in principle lead to agreement on the decoded message while at the same time lower the probability of erroneous detection by collecting the available spatial diversity. Indeed, the detection error if only independent (i.e., local) processing is performed would be that of a multiple-input single-output MISO system, whereas consensus-based joint (i.e., global) detection has the potential to reach detection error performance attainable by a virtual colocated multiple-input multiple-output (MIMO) system, ∗

Work in this paper was prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD1901-2-0011. The U. S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon.

where the number of receive antennas equal the number of sensors in the network. Enforcing sensors to consent on the message sent by the AP through local exchanges amounts to a distributed multiple-hypotheses testing problem in which the number of hypotheses equals the number of different codewords in the message alphabet employed by the AP. Distributed multiple-hypotheses testing problems have recently been considered and solved using messagepassing, belief propagation [1] and consensus-averaging [5] schemes. The common objective in these schemes is to have all sensors agree on the hypothesis with maximum-a-posteriori (MAP) probability. If the underlying application entails a reduced number of hypotheses (as in e.g., binary radar tests) these algorithms are well motivated. If considered for decoding broadcasted messages as in here, the amount of information exchanges which increases with the alphabet size (number of hypotheses) would be prohibitively high. Specifically, information exchanges would exponentially increase with the number of antennas the AP is equipped with, and the constellation size each symbol in the message is drawn from. In this paper, we develop a distributed detection scheme that trades-off error performance for reduction in complexity and the necessary inter-sensor communication cost. The reduction is from exponential to polynomial. Supposing symbols are drawn from quadratureamplitude modulations (QAM), the resultant algorithm builds upon two notions: (i) MIMO decoding based on semi-definite relaxation [8], [4]; and (ii) distributed optimization based on the alternating direction method of multipliers [2], [7]. These techniques allow us to formulate the desired detector as the solution of convex minimization sub-problems that exhibit a separable structure and lend themselves naturally to a distributed implementation with controllable cost. II. P ROBLEM S TATEMENT Consider a wireless AP equipped with N transmit antennas. Information-bearing bits at the AP are mapped to a set of space-time T ×N matrices {Sk }K k=1 , where T

2

Fig. 1. Communication graph with receiver nodes. Black circles represent bridge nodes.

denotes time and N space, and drawn from an alphabet As . Entries of Sk are drawn from a generic QAM constellation. The set of matrices is broadcasted to a network comprising M sensors. Single-hop communications are allowed among sensors so that the mth sensor communicates solely with other sensors in its neighborhood Nm . The communication links are assumed to be ideal and symmetric. All receivers are modeled as nodes of an undirected graph, which is assumed fully connected. The k th broadcasted matrix, Sk , is received at the mth sensor in the form of a vector rm,k = Sk hm + wm,k , where (with matching sizes and indices) wm,k denotes additive complex Gaussian noise and hm is the respective AP-sensor fading channel vector. For convenience, we can alternately rewrite this input-output relationship as rm,k = Hm sk + wm,k , where sk = vec(STk ) and Hm = IT ⊗ hm .1 For equiprobable alphabet blocks, MAP detection is equivalent to maximum-likelihood (ML) detection. For a memoryless channel, each sensor can obtain a local ML estimate on the transmitted block Sk by solving the constrained optimization problem ˆ m,k := arg min krm,k − Hm sk k2 . S 2 S∈As

(1)

Similarly, the jointly (i.e., globally) optimal detector is ˆ k := arg min S

S∈As

M X

||rm,k − Hm sk ||22 .

(2)

m=1

If the communication graph has no loops, one could in principle solve (2) using backward-forward message passing between sensors. Links connecting to a given sensor could be partitioned arbitrarily into two sets, one that connects to what is termed preceding set of sensors, and another that connects to an anteceding sets of sensors. Notice that the intersection of these two sets 1

From this expression it is clear that this model can further accommodate any linear precoding operation over the QAM constellation ˜ m = Hm Φ with Φ any unitary precoding matrix. by defining H

is empty if the graph has no loops. For each sensor, the message propagation to the anteceding sensors involves transmission of its own sufficient statistic (for sensor m, the mth summand in (2)) added to the sum of messages arriving from the preceding sensors. Message propagation in the opposite direction is essentially the same, with each sensor now adding its own sufficient statistic to the sum of messages arriving from anteceding sensors to build the message sent to preceding sensors. The sum of preceding and anteceding messages along with each sensor’s own sufficient statistic is enough to have the sum in (2) locally available. This is, in short, the implementation in our context of the MAP detector proposed in [1]. The terms in the sum (2) are messages of the belief propagation algorithm in [1]. Known limitations of the belief propagation algorithm are: (i) the requirement to have a loop-free graph in the inter-sensor links; and (ii) message synchronization since each sensor needs to wait for the preceding/anteceding messages to propagate the message to the subsequent anteceding/preceding sensors. These limitations are relaxed in the consensus-average scheme proposed in [5], where the sum in (2) can be viewed as an average of values distributed among sensors. Such an average can be computed using suitable local exchanges [6]. The price paid by this approach is that convergence is assured after an infinite number of iterations. (Unlike belief propagation schemes which find the optimum solution in a finite number of steps.) Albeit optimal (in a finite or infinite number of steps), unless any specific structure of the blocks is exploited (e.g., code orthogonality [3, Ch. 3]), the sum (2) needs to be computed for every element in the alphabet As . The size of this alphabet can be prohibitively large for high number of transmit antennas and/or constellation points. This on the other hand is a well-known challenge in MIMO detection that is here aggravated by the fact that communication cost may be more costly than sensor processing. For MIMO detection, much simpler nearoptimal detectors are available using, e.g., the sphere decoding algorithm [3, Ch. 5]. This paper aims at reducing this inter-sensor communication cost by re-formulating (2) as a distributed optimization program. For that matter, we resort to an alternative suboptimal detection approach based on semi-definite relaxation [4], (SDR). SDR has gained considerable attention recently because it offers guaranteed polynomial worst-case complexity making it particularly attractive for low-medium SNR and/or large size alphabets.

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III. C ONSENSUS - BASED DISTRIBUTED DECODING We will first consider the per-block optimization task in (2) and for notational brevity we will omit the subindex k . Moreover, we will further suppose that As is a real-valued alphabet. Upon defining ˜s := [Re{s}T Im{s}T ]T T

(3) T T

˜rm := [Re{rm } Im{rm } ] · ¸ Re {H } − Im {H } m m ˜ m := H Im{Hm } Re{Hm }

(4) (5)

the optimization problem of interest is equivalent to the following real-valued problem of the form ˜ m˜s||2 , where A˜s is a real-valued min˜s∈A˜s ||˜rm − H 2 pulse-amplitude modulation (PAM) constellation. Defining X := xxT with x := [˜sT t]T and t ∈ {−1, 1}, the optimization problem in (2) can be rewritten as (see [8] for details) ˆ = arg min X

M X

X∈A˜s m=1

tr(Qm X)

(6)

where tr(·) denotes the trace operator and Qm is a matrix defined as " # ˜TH ˜ m −H ˜ T ˜rm H m m Qm := (7) ˜m 0 −˜rTm H that is built using local information. The optimization in (6) still requires exhaustive search. It is possible to re-formulate the problem in (6) as a near-ML semi-definite relaxation (SDR) problem [8] ˆ SDR = arg min X

M X

tr(Qm X)

(8)

can act as bridge sensors but inter-sensor communications in this robust topology maybe prohibitively costly. The next proposition gives a rigorous definition of this set of bridge sensors and decentralizes the solution for (8) based on this definition:2 Proposition 1 Let B be a subset of bridge sensors with the following property: (a) ∀m ∈ [1, M ] there exists at least one b ∈ B so that b ∈ Nm ; and (b) ∀b1 ∈ B there exists a receiver b2 ∈ B such that the shortest path between b1 and b2 has at most two edges. ¯ b denotes an estimate of the bridge sensor sent to If X sensor m, the optimal solution of ˆ m }M {X m=1 = arg min

M X

tr(Qm Xm )

(10)

m=1

¯ b , b ∈ B, m ∈ Nb , Xm = X

s.to:

Xm ∈ P, ∀m.

(11)

coincides with that of the (centralized) solution of (8); ˆ SDR = X ˆ m , ∀m, and thus the network arrives to i.e., X a consensus on the centralized solution. We will solve (10) in a distributed fashion using the alternating-direction method of multipliers (MoM) [2, Sec. 2.4]. The MoM exploits the decomposable structure of the augmented Lagrangian. The following algorithm describes the communication between sensors to implement the convex optimization algorithm resulting from Proposition 1. Proposition 2 For an iteration index i consider iterb (i), X (i) and X ¯ b (i) defined by the recursions ates Vm m b b ¯ b (i)], b ∈ Bm Vm (i) = Vm (i − 1)+cj [Xm (i)− X

Xm (i + 1) = arg min tr{Qm Xm Xm ∈P X T b ¯ b (i))+ cm ||Xm − X ¯ b (i)||2 ]} where + [Vm (Xm − X 2 b∈Bj P := {X ∈ R2N T+1×2N T+1 |X ≥ 0, L ≤ X(i, i) ≤ U, X 1 b ¯ P [Vm (i)+cm Xm (i+1)] X (i + 1) = b ∀i ∈ [1, 2N T ], X(2N T +1, 2N +1) = 1} (9) c β β∈Nb X∈P

m=1

with L, U depending on the constellation size [8]. We observe that still (8) can only be solved by a centralized detector. To efficiently decentralize (8), we will rely on a subset of sensors that we term “bridge sensors” and represent them in Fig. 1 with black circles; see also [7]. Bridge sensors impose consensus among neighboring (one-hop) ˆ m and percolate consensus-related insensor estimates X formation among neighborhoods. They are used to tradeoff robustness to sensor failures for reduced inter-sensor communications. If only one bridge sensor is available, we have a fusion center (FC) based topology which is not robust to FC failures. At the other extreme, all sensors

m∈Nb

b (−1) denote the initial for all receivers; and let Vm values of the Lagrange multiplier; {Xm (0)}M m=1 the ¯ local matrix estimates; and {Xb (0)}b∈B the arbitrary initial values of the consensus variables. With ideal communication links and constants {c > 0}M m=1 , the M ˆ SDR in iterates {Xm (i)}m=1 converge to the matrix X (8) as i → ∞. The three expressions in Proposition 2 solve iteratively the distributed problem in Proposition 1 and thus arrive to the centralized solution in (8). The mth sensor obtains 2 Due to space limitatons proofs of the propositions in this paper are not included.

4 b (i − local estimates Xm (i) and Lagrange multipliers Vm 1) of the optimization problem. To update the multiplier b (i), each sensor uses the received consensus variables Vm ¯ Xb (i) from all its neighboring bridge sensors. With b (i), the sensor updates the the updated multiplier Vm estimate Xm (i + 1). Finally, it suffices to transmit the b (i)+c X (i+1) to the bridge sensor which in sum Vm m m ¯ b (i + 1). Transmitting X ¯ b (i + 1) to the turn updates X neighboring sensors completes the ith iteration. Note that in Proposition 2 sensor connectivity is the only constraint imposed on the network topology; i.e., the algorithm allows for loops in the network and asynchronism when passing messages.

IV.

0

10

−1

Average BER

10

−2

10

Centralized ZF detector Distributed SDR iter=10 Distributed SDR iter=20 Distributed SDR iter=30 Distributed SDR iter=40 Centralized SDR detector (iter → ∞)

−3

10

−4

10

SIMULATIONS

In this section we test and compare the error performance of the iterates in Proposition 2 against the centralized SDR algorithm in (8) and with the zeroforcing (ZF) linear detector. This detector can be optimally implemented in a distributed fashion along the lines of [7]. To benchmark performance, we further plot the centralized solution (corresponds to an infinite number of consensus iterations) of the distributed ZF detector. Fig. 2 depicts bit-error-rate (BER) versus signal-tonoise ratio (SNR) when the AP is equipped with N = 4 antennas and the network consists of 8 nodes distributed as in Fig. 1. We assume T = 1 and elements in Sk drawn from a 16-QAM constellation; i.e., there are |As | = 216 matrices in the alphabet. The alphabet size indicates the number of messages to be sent by each sensor for the centralized ML solution to converge using the algorithms in [5] and [1]. We observe that for a relatively small number of iterations, error performance of the distributed decoder approaches that of the centralized one. V. C ONCLUDING R EMARKS We developed a distributed algorithm for MIMO decoding using a decentralized version of the semidefinite relaxation method. The resultant scheme entails messages percolating across the network to enforce consensus among receiving sensors on the message sent by the AP. The key steps include splitting the centralized optimization problem into separable convex sub-problems and employing the alternating-direction method of multipliers to solve the decentralized problem. The SDR approach incurs lower cost of inter-sensor communications compared to competing alternatives based on belief propagation and consensus averaging. Unlike the latter, the constraint relaxation step in SDR trades-off optimality for reduced communication cost. Preliminary simulations demonstrate that even a reduced number of

8

9

10

11

12 SNR

13

14

15

16

Fig. 2. BER vs. SNR for N = 4 antennas at the AP; T = 1 and entries of Sk drawn from a 16-QAM constellation.

iterations enables spatial diversity gains compared to other consensus-enforcing optimization algorithms such as distributed ZF.3 R EFERENCES [1] M. Alanyali, S. Venkatesh, O. Savas, and S. Aeron, “Distributed Bayesian hypothesis testing in sensor networks,” in Proc. of American Control Conf., 30 Jun.-2 Jul. 2004, vol. 6, page(s): 5369- 5374. [2] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods. Second Edition, Athena Scientific, 1997. [3] G. B. Giannakis, Z. Liu, X. Ma and S. Zhou, Space-Time Coding for Broadband Wireless Communications, John Wiley & Sons, Inc., 2007. [4] P. H. Tan and L. K. Rasmussen “The application of semidefinite programming for detection in CDMA,” IEEE J. on Sel. Areas Commun., Vol. 19, pp. 1442-1449, Aug. 2001. [5] R. Olfati-Saber, E. Franco, E. Frazzoli, and J. S. Shamma, “Belief Consensus and Distributed Hypothesis Testing in Sensor Networks,” Network Embedded Sensing and Control, 2006. [6] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. on Automatic Control, vol. 49, pp. 15201533, Sept. 2004. [7] I. D. Schizas, A. Ribeiro and G. B. Giannakis “Consensus in Ad Hoc WSNs with Noisy Links - Part I: Distributed Estimation of Deterministic Signals,” IEEE Trans. on Signal Processing, 2007. [8] N. D. Sidiropoulos and Z.Q. Luo, “A Semidefinite Relaxation Approach to MIMO Detection for High-Order QAM Constellations,” IEEE Signal Processing Letters, Vol. 13, No. 9, pp. 525–528, Sept. 2006. [9] J. F. Sturm “ Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods Software, Vol.11-12, pp. 625-653, 1999. [Online], Available: http://sedumi.mcmaster.ca. 3

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