Wireless Peer-to-Peer Mutual Broadcast via Sparse Recovery

Wireless Peer-to-Peer Mutual Broadcast via Sparse Recovery

arXiv:1101.0294v1 [cs.IT] 31 Dec 2010

Lei Zhang and Dongning Guo Department of Electrical Engineering & Computer Science Northwestern University, Evanston, IL 60208, USA Abstract—Consider a problem frequently seen in wireless peerto-peer networks: Every node has messages to broadcast to its peers, namely, all nodes within one hop. Conventional schemes allow only one transmission to succeed at a time in a twohop neighborhood, because 1) simultaneous transmissions in the neighborhood collide, and 2) affordable radio is half-duplex and cannot simultaneously transmit and receive useful signals. In this paper, we propose a novel solution for the peer-to-peer mutual broadcast problem, which exploits the multiaccess nature of the wireless medium and addresses the half-duplex constraint at the fundamental level. The defining feature of the scheme is to let all nodes send their messages at the same time, where each node broadcasts a codeword (selected from its unique codebook) consisting of on-slots and off-slots, where it transmits only during its on-slots, and listens to its peers through its own off-slots. Each node decodes the messages of its peers based on the received superposed signals through its own off-slots. Decoding can be viewed as a problem of sparse support recovery based on linear measurements (with erasures). In case each message consists of a small number of bits, an iterative message-passing algorithm based on belief propagation is developed, the performance of which is characterized using a state evolution formula in the limit where each node has a large number of peers. Numerical results demonstrate that the message-passing algorithm outperforms popular compressed sensing algorithms such as CoSaMP and AMP. Furthermore, to achieve the same reliability for peer-topeer broadcast, the proposed scheme achieves three to five times the rate of ALOHA and carrier-sensing multiple-access (CSMA) in typical scenarios.

I. I NTRODUCTION Consider a frequent situation in wireless peer-to-peer networks, where every node wishes to broadcast messages to all nodes within its one-hop neighborhood, called its peers. We refer to this problem as wireless mutual broadcast. Such broadcast traffic can be dominant in many applications, such as messaging or video conferencing of multiple parties in a spontaneous social network, or on a disaster site or battlefield. Wireless mutual broadcast is also critical to efficient network resource allocation, by exchanging messages between nodes about their demands and local states, such as the queue length, the channel quality, the code and modulation format, and request for certain resources and services. There have been numerous works on the design of medium access control (MAC) protocols for wireless peer-to-peer networks (see the surveys [1], [2] and references therein). State-of-the-art designs either apply ALOHA-type randomaccess schemes or use a mixture of random access and scheduling/reservation. For example, the ad hoc mode of the 802.11 protocol is based on carrier-sensing multiple-access

with collision avoidance (CSMA/CA), where nodes contend for the channel by broadcasting a request to peers before one node is scheduled for transmission in its neighborhood. Such conventional designs stipulate that only one packet transmission can succeed at a time in a neighborhood. This is because of two considerations: One is the half-duplex constraint due to physics, i.e., affordable radio cannot receive useful signals at the same time over the same frequency it is transmitting, so that if two peers transmit simultaneously, they do not hear each other. The other is the simplification of the physical layer to a collision model, where simultaneous transmissions interfere with and destroy each other. In this paper, we exploit recent advances in communications and signal processing to propose a novel, alternative paradigm for the physical and MAC layers for peer-to-peer broadcast. The defining feature of the scheme is to allow all nodes to send their messages simultaneously over a frame interval, during which each node broadcasts a codeword consisting of on-slots and off-slots. The on-off codeword also functions as a duplex mask, because the node only emits energy during its on-slots, and receives the superposition of its peers’ signals through its off-slots. By using interleaved on- and off-slots, a node’s transmissions are scheduled at the timescale of the symbol level, rather than at the frame level as is done in existing schemes. The key observation that underlies the new design is that a node’s transmissions can be regarded as erasures of its own received signal at known positions. With proper errorcontrol coding, each node can receive and then decode the messages from all its peers, at the same time it broadcasts its message to all its peers. The focus of this paper is peer-to-peer mutual broadcast where each message consists of a small number of bits, where the proposed duplex scheme is shown to significantly outperform ALOHA and CSMA in terms of data rate. For example, consider a network with Poisson distributed nodes, where each node has nine neighbors on average, and that the signal-to-noise ratio per active symbol is 10 dB. To achieve a message error probability of 10−2 or less, the proposed scheme needs only 200 symbol transmissions, which is merely 1/5 and 1/3 of that required by ALOHA and CSMA-type randomaccess schemes, respectively. This is largely due to the fact that retransmissions due to collision consume many resources in ALOHA and CSMA networks, whereas the proposed signaling is suitable for multiaccess channels. Decoding of the multiuser signal is in fact a problem

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of support recovery based on linear measurements, since the received signal (with erased symbols eliminated) is a noisy superposition of peers’ codewords selected from their respective codebooks. There are many algorithms developed in the compressed sensing (or sparse recovery) literature to solve the problem, the complexity of which is polynomial in the size of the codebook (see, e.g., [3]–[6]). In this paper, an iterative message-passing algorithm with linear complexity is developed based on belief propagation (BP). Its performance is characterized using a state evolution formula in the largesystem limit, where the number of peers each node has tends to infinity. Numerical results show that the message-passing algorithm outperforms two popular sparse recovery algorithms, namely, compressive sampling matching pursuit (CoSaMP) [3] and approximate message passing (AMP) [4]. The remainder of the paper is organized as follows. Section II describes the proposed coding scheme. The messagepassing decoding algorithm is developed and analyzed in Section III. In Section IV, competing random-access schemes are described and performance bounds are provided. Numerical comparisons are presented in Section V. Section VI concludes the paper. II. E NCODING FOR M UTUAL B ROADCAST Suppose all transmissions use the same single carrier frequency. Let time be slotted and all nodes be perfectly synchronized. Each node k is assigned a unique codebook of 2l on-off signatures (codewords) of length Ms , denoted by {S k (1), . . . , S k (2l )}. For simplicity, let each element of each signature be generated randomly and independently, which is 0 with probability 1 − q and 1 or −1 with probability q/2. Node k broadcasts its l-bit message (or information index) wk ∈ {1, . . . , 2l } by transmitting the codeword S k (wk ). For simplicity, let us assume that the physical link between every pair of neighboring nodes be an additive white Gaussian noise channel of the same signal-to-noise ratio (SNR), denoted by γ. Let the set of peers of node k be denoted by Nk , the population of which is its cardinality |Nk |. The received signal of node k, if it could listen over the entire frame, is then described by √ X S j (wj ) + W k (1) Ye k = γ j∈Nk

where W k is Gaussian noise consisting of independent identically distributed (i.i.d.) entries of zero mean and unit variance. For simplicity, transmissions from non-neighbors, if any, are accounted for as part of the additive Gaussian noise. Without loss of generality, we focus on the neighborhood of node 0 and omit the subscript k in (1). Suppose |N0 | = K and the neighbors of node 0 are indexed by 1, 2, . . . , K. The total number of signatures of all neighbors is N = 2l K. Due to the half-duplex constraint, however, node 0 can only listen during its off-slots, the number of which has binomial distribution, denoted by M ∼ B(Ms , 1 − q), whose expected value is E {M } = Ms (1 − q). Let the matrix S ∈ RM ×N consist of columns of the signatures from all neighbors of node 0, observable during the M off-slots of node 0, and then

p normalized by Ms q(1 − q) so that the expected value of the l2 norm of each column in S is equal to 1. Based on (1), the M -vector observed through all off-slots of node 0 can be expressed as √ (2) Y = γs SX + W where γs = γMs q(1 − q), and X is a binary N -vector for indicating which K signatures are selected to form the sum in (1). Precisely, X(j−1)2l +i = 1{wj =i} for 1 ≤ j ≤ K and 1 ≤ i ≤ 2l . Note that the sparsity of X is exactly 2−l , which can be very small.
The average system load is defined as β = N/ Ms (1 − q) . In general, the decoding problem each node k faces is to identify, out of a total of 2l |Nk | signatures from all its neighbors, which |Nk | signatures were selected. This requires, of course, every node to have knowledge of the codebooks of all neighbors. One possible solution is to let the codebook of each node be generated using a pseudo-random number generator using its network interface address (NIA) as the seed, so that it suffices to acquire the NIAs of all neighbors. This, in turn, is a neighbor discovery problem, which has been studied in [7]– [9]. In particular, the compressed neighbor discovery scheme proposed in [8], [9] is based on the same type of signalling described in this paper, where nodes simultaneously transmit there respective signatures which identify them, and efficient discovery algorithms have been developed. The assumption of perfect synchronization can be relaxed in practice. It suffices to have all communicating peers be approximately symbol synchronized, so that the timing difference (including the propagation delay) is much smaller than the symbol interval. This can be achieved by using distributed algorithms based on average consensus [10]. We also note that if sophisticated joint detection and decoding techniques are used, there is no need for synchronized transmission, as long as the timings can be acquired. It should also be noted that switching the carrier on and off at high rate is feasible in practice, because the response time of RF circuit is in nanoseconds. For instance, on-off signalling over sub-millisecond slots is a defining feature of the second generation cellular systems such as GSM. In time-hopping impulse radio, the carriers are switched on and off at submicrosecond interval [11], which is at least tens of times faster than needed by the proposed scheme (in microseconds). III. S PARSE R ECOVERY (D ECODING ) VIA M ESSAGE PASSING The problem of recovering the support of the sparse input X based on the observation Y has been intensively studied in the compressed sensing literature (see, e.g., [12]–[15]). In this section, we develop an iterative message-passing algorithm based on belief propagation, and characterize its performance in a certain limit. The computational complexity of the algorithm is in the order of O(M N q), i.e., it is proportional to the number of nonzero elements in the signature matrix S, which is of the same order as the complexity of the most popular algorithms in the compressed sensing literature, including CoSaMP and AMP.

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Y1 

Y2 

Y3

Y5 

Y4

 m ˆ k

 mk

X1 

Fig. 1.

X2 

X3 

X4 

X5

X6

X7

X8

X9

X10  X11

X12

The Forney-style factor graph of coded mutual broadcast.

A. The Factor Graph Belief propagation belongs to a general class of messagepassing algorithms for statistical inference on graphical models, which has demonstrated empirical success in many applications including error-control codes, neural networks, and multiuser detection in code-division multiple access (CDMA) systems. In order to apply BP to the coded mutual broadcast problem, we construct a Forney-style bipartite factor graph to represent the model (2), which is rewritten as yµ =

√

γs

N X

sµk xk + wµ

Algorithm 1 Message-Passing Decoding Algorithm 1: Input: S, Y , γs , β, q. 2: Initialization: P P 2 3: Λ ← − log(2l − 1), L ← µ |∂µ|, L2 ← µ |∂µ| −1/2 4: R ← 2γs Y −S×1 5: m ˆ 0µk ← 0 for all µ, k 6: Main iterations: 7: for t = 1 to T − 1 do 8: for all µ, k with sµk 6=
P0 do t−1 ˆ νk 9: mtkµ ← tanh Λ2 + ν∈∂k\µ tanh−1 m 10: end for P P 11: Qt ← L12 µ |∂µ| j∈∂µ (mtjµ )2   βL2 t −1 12: At ← γ4s + N qL (1 − Q ) 13: for all µ, k with sµk 6= 0 do P
14: m ˆ tµk ← tanh At sµk (rµ − j∈∂µ\k sµj mtjµ ) 15: end for 16: end for
P −1 17: mk ← tanh Λ m ˆ Tνk−1 for all k ν∈∂k tanh 2 + 18: Output: w ˆj = arg maxi=1,...,2l m(j−1)2l +i , j = 1, . . . , K

(3)

k=1

where µ ∈ 1, 2, . . . , M and k ∈ 1, 2, . . . , N index the measurements and the “input symbols,” respectively. For simplicity, we ignore the dependence of the symbols {Xk } for now, which shall be addressed toward the end of this section. Each Xk then corresponds to a symbol node and each Yµ corresponds to a measurement node, where the joint distribution of all {Xk } and {Yµ } are decomposed into a product of M + N factors, one corresponding to each node. For every (µ, k), symbol node k and measurement node µ are connected by an edge if sµk 6= 0. A simple example is shown in Fig. 1 for 5 measurements and 3 neighbors each with 4 messages, i.e., M = 5, K = 3, and N = 3 × 4 = 12. (The actual message chosen by each neighbor for broadcast is marked by a dark node.) B. The Message-Passing Algorithm In general, an iterative message-passing algorithm involves two rounds in each iteration, where a message (or belief, which shall be distinguished from an information message) is first sent from each symbol node to every measurement nodes it is connected to, and then a new set of messages are computed and sent in the reverse direction, and so forth. The algorithm performs exact inference within finite number of iterations if there are no loops in the graph (the graph becomes a tree), while it provides in general a good approximation for loopy graphs as the one in the current problem. For convenience, let ∂µ (resp. ∂k) denote the subset of symbol nodes (resp. measurement nodes) connected directly to measurement node µ (resp. symbol node k), called its neighborhood. Also, let ∂µ\k denote the neighborhood of measurement node µ excluding symbol node k and let ∂k\µ be similarly defined. The message-passing algorithm, which decodes the information indexes w1 , . . . , wK , is described as Algorithm 1.

In the following, we derive Algorithm 1. It is a simplification of the original iterative BP algorithm, which iteratively computes the marginal a posteriori distribution of all symbols given the measurements, assuming that the graph is free of t cycles. For each k ∈ ∂µ (hence µ ∈ ∂k), let Vkµ (x) x∈{0,1} represent the message from symbol node k to measurement t node µ and Uµk (x) x∈{0,1} represent the message in the reverse link at the t-th iteration. Each message is basically the belief (in terms of a probability mass function) the algorithm has accumulated about the corresponding symbol based on the measurements on the subgraph traversed so far, assuming it is a tree. Let pX (x) denote the a priori probability: pX (1) = 2−l 0 0 and pX (0) = 1 − 2−l . Let Uµk (0) = Uµk (1) = 1/2 for all (k, µ). One iteration of BP computes: Y t−1 t Vkµ (x) ∝ pX (x) Uνk (x) (4a) ν∈∂k\µ

for x = 0, 1 and all (k, µ) with sµk 6= 0, and then  X 1 √ t Uµk (x) ∝ exp − yµ − γs sµk x 2 (xj )∂µ\k  X
2 Y √ t − γs sµj xj Vjµ (xj ) j∈∂µ\k

(4b)

j∈∂µ\k

P where (xj )∂µ\k denotes sum over xj = 0, 1 for all j ∈ ∂µ\k, and V (x) ∝ u(x) means that V (x) is proportional to u(x) with proper normalization such that V (0) + V (1) = 1. The complexity of computing the sums in (4b) is exponential in |∂µ| = O(qN ), which is in general very high for the problem at hand. However, as qN  1, the computation carried out at each measurement node admits good approximation by using the central limit theorem. A similar technique has been used in the CDMA detection problem, for fully-

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connected bipartite graph in [16]–[19], and for a graph with large node degrees in [20]. To streamline (4a) and (4b), we use a transformation √ bk = 2xk − 1 P so that bk ∈ {−1, 1}. Also, let σ = 2/ γs , rµ = σyµ − j∈∂µ sµj (which corresponds to line 4 of Algorithm 1), and define a set of equivalent messages:   t Vkµ (1) 1 t t mtkµ = Vkµ (1) − Vkµ (0) = tanh log t (5a) 2 Vkµ (0)   t Uµk (1) 1 t t t m ˆ µk = Uµk (1) − Uµk (0) = tanh log t . (5b) 2 Uµk (0) Using Gaussian approximations, one can reduce the messagepassing algorithm to iteratively computing the following messages:   X Λ + tanh−1 m ˆ t−1 (6a) mtkµ = tanh νk 2 ν∈∂k\µ  ! X m ˆ tµk = tanh At sµk rµ − (6b) sµj mtjµ  j∈∂µ\k

as seen in Algorithm 1. Here, Λ = log

pX (1) = − log(2l − 1) pX (0)

is the log-ratio of the a priori probability,  −1 At = σ 2 + βδ(1 − Qt )

(7)

(8)

and Qt =

1 X X X t 2 (mjµ ) L2

(9)

k µ∈∂k j∈∂µ

where L2 = µ |∂µ|2 . After T −1 iterations of computing (6), the approximated posterior mean of bk can be expressed as   X Λ −1 T −1 + tanh m ˆ νk . (10) mk = tanh 2 P

ν∈∂k

The detailed derivation is relegated to Appendix A. We now revisit the assumption that X has independent elements. In fact, X consists of K sub-vectors of length 2l , where the entries of each sub-vector are all zero except for one position corresponding to the transmitted message. Algorithm 1 outputs the position of the largest element of each of the K sub-vectors of [m1 , . . . , mN ]. In fact the factor graph Fig. 1 can be modified by including K additional nodes, each of which puts a constraint on one sub-vector. Slight improvement over Algorithm 1 can be obtained by carrying out message passing on the modified graph. C. Performance Characterization of Algorithm 1 It turns out that the iterative decoding dynamics of Algorithm 1 can be quantified using a state evolution in the (large-system) limit, where K, M → ∞ with the average system load β and all other parameters kept constant. As demonstrated in Section V using numerical results, the state

evolution provides a good approximation for the performance of systems of moderate size, at least at relatively low SNR. For simplicity, we declare decoding success if node 0 can correctly recover all the messages its peers transmitted; otherwise it is regarded as a decoding error. The probability of decoding error averaged over all realizations of all possible messages, signatures and noise is denoted by Pes . The main result of the large-system analysis is summarized in the following proposition: Proposition 1: In the large-system limit, with T − 1 iterations of message-passing decoding described in Algorithm 1, the decoding error probability Pes is given by  Z ∞ √ 2h √ i2l −1 dx K 1 √ e− 2 (x− E) 1 − Q(x + E) Pes =1 − 2π −∞ (11) √ R∞ where Q(x) = x exp(−z 2 /2) dz/ 2π and E = η T γs /4 is obtained using the following recursive equation: η t+1 =

1 1+

1 4 γs βE

1 t 4 γs η




where initially η 0 = 0, and  n o2  √ X − E X | EX + W E(E) = E

(12)

(13)

denotes the minimum mean-square error of estimating the input X ∈ {1, −1} with pX (1) = 2−l in Gaussian noise W ∼ N (0, 1). Precisely,   X Z 1 + xr √ xΛ −z2 √ tanh z E + E + e 2 dz E(E) = 1 − 2 2 2π x=±1 where r = tanh(Λ/2). The proof of Proposition 1, which is relegated to Appendix B, makes use of the idea of density evolution originally developed for analyzing graphical error-control codes. A similar technique has also been used in [16], [18]–[20] to analyze CDMA systems. Proposition 1 is basically a singleletter characterization of the performance in the large-system limit (cf. [21]). That is, from the view point of an individual neighbor, the mutual broadcast system with message-passing decoding is asymptotically equivalent to sending a simplex code through a scalar Gaussian channel with some degradation in the SNR, where the degradation factor is determined from the iterative formula depending on the number of iterations. IV. R ANDOM -ACCESS S CHEMES In this section we describe two random-access schemes, namely slotted ALOHA and CSMA, and derive lower bounds on the minimum number of symbol transmissions required for achieving a given error probability. The result will be used in Section V to compare with the performance of mutual broadcast via sparse recovery. Let each coded frame carry a single message at the rate of the single-user channel capacity. A message is assumed to be always decoded correctly if no collision occurs. Each broadcast period consists of a number of frames to allow for

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retransmissions. Without loss of generality, consider node 0 and let K denote the number of its neighbors. Node 0 shall receive K messages. An error event is defined as the situation where one or more peers of node 0 have had no chance to transmit their frames to node 0 without collision until the end of the broadcast period. We denote the total number of symbol transmissions of ALOHA and CSMA by Ma and Mc , respectively. Let L denote the total number of bits encoded into a frame, which includes an l-bit message and a few additional bits which identify the sender. This is in contrast to broadcast via sparse recovery, where the signature itself identifies the sender (and the message). Over the additive white noise channel with SNR γ, in order to send L bits reliably through the channel, the number of symbols in a frame must exceed 2L/log2 (1 + γ).

(14)

A. Slotted ALOHA In slotted ALOHA, each node chooses independently with the same probability p to transmit in every frame interval. The error probability is lower bounded by the probability that one particular peer cannot successfully transmitting its frame without collision by the end of n frame intervals: n    KK e K n (15) P ≥ 1 − p(1 − p) ≥ 1− (K + 1)K+1 where the second inequality becomes equality if and only if p = 1/(K + 1). Therefore, by (14) and (15), in order to achieve a given error probability Pe , the total number of symbol transmissions Ma must satisfy Ma ≥

2L log Pe · .  KK log2 (1 + γ) log 1 − (K+1) K+1

(16)

B. CSMA As an improvement over ALOHA, CSMA lets nodes use a brief contention period to negotiate a schedule in such a way that nodes in a small neighborhood do not transmit data simultaneously. Consider the following generic scheme: Each node senses the channel continuously. If the channel is busy, the node remains silent and disables its timer; as soon as the channel becomes available, the node starts its timer with a random offset, and waits till the timer expires to transmit its frame. Clearly, the node whose timer expires first in its neighborhood captures the channel and transmits its frame. To analyze the performance of CSMA, we use the Mat´ern hard core model [22]. Let Φ = {Xi } be a Poisson point process with intensity λ on R2 , which describes the locations of nodes in the network. Let {τi } be i.i.d. random variables with probability density function f (τ ), which represent the e = {(Xi , τi )} is an independently marked timer offsets. Then Φ Poisson point process. For any node Xi ∈ Φ, let its neighborhood be denoted by e : |Xj − Xi | ≤ R, j 6= i} for some NXi = {(Xj , τj ) ∈ Φ R > 0, whose cardinality is KXi = |NXi |. Node Xi will

transmit if and only if τi < τj for all (Xj , τj ) ∈ NXi . By symmetry, its transmission probability is pXi = 1/(KXi + 1). Recall that node 0 has K neighbors. Suppose the locations of these neighbors are , zK . Define Kmax = z1 , z2 , . . .P K max Kz1 , Kz2 , . . . , KzK and K = i=1 Kzi /K. After n contention periods, the error probability is lower bounded by the probability that the neighbor of node 0 who has the most number of neighbors has not had a chance to transmit:  n  n 1 1 e P ≥ 1− ≥ 1− (17) Kmax + 1 K +1 where the second inequality is because Kmax ≥ K. Therefore, in order to achieve a given error probability Pe , the total number of symbol transmissions Mc must satisfy Mc ≥

2L log P1e log Pe 2L
· ≥ K (18) log2 (1 + γ) log 1 − 1 log2 (1 + γ) K+1

where the first inequality is by (14) and
(17), and the second 1 inequality is because of log 1 − x+1 ≥ − x1 for x > 0. The lower bound (18) dependents on the realization of K. The expected value of K, conditioned on that node 0 has K neighbors, is presented in the following result, which is proved in Appendix C. Proposition 2: If node 0 has K neighbors, then √ √
3 3
3 3 2 E K = 1− (K − 1) + λR + 1. (19) 4π 4 Therefore, by (18) and (19), the expected number of symbol transmissions required to achieve the error probability Pe must satisfy √  
2L log P1e 3 3 E(Mc ) ≥ K− K − 1 − λπR2 . (20) log2 (1 + γ) 4π Note that this lower bound is quite loose since the hidden terminal problem is neglected and the contention overhead is also ignored. V. N UMERICAL R ESULTS In order for a fair comparison, we assume the same power constraint for both the sparse-recovery-based mutual broadcast scheme and random-access schemes, i.e., the average transmitted power in each active slot (in which the node transmits energy) is the same. Recall that, in the proposed scheme, where the frame length is Ms , the SNR in the model (2) is γs = γMs q(1 − q). We first compare the number of symbol transmissions required by the three schemes, denoted by Ms , Ma and Mc , respectively, for node 0 to achieve the same error probability Pe . We consider the special case where the number of neighbors of node 0 is K = λπR2 , which is the average over all nodes. We always choose q = 1/(K + 1) in the sparserecovery-based scheme. First consider the case K = 4 and that each node has a message of l = 10 bits, so that Kl = 40 bits are to be collected by node 0. In random-access schemes, at least 2 additional bits are needed to identify a sender, so we let L = 12.

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0

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ALOHA CSMA Message Passing CoSaMP AMP MP Approximation

SNR=10dB

−1

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Probability of error

Probability of error

Sparse Recovery via Message Passing ALOHA CSMA

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SNR=0dB

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−3

10

2

3

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−3

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2

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Fig. 2.

Performance comparison with 4 neighbors and 10 bits per node.

Fig. 4. System-level performance comparison with 10 bits and on average 4 neighbors per node.

0

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ALOHA CSMA Message Passing CoSaMP AMP MP Approximation

−1

Sparse Recovery via Message Passing ALOHA CSMA

SNR=10 dB −1

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Fig. 3.

Performance comparison with 9 neighbors and 5 bits per node.

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Number of symbol transmissions

Fig. 5. System-level performance comparison with 10 bits and on average 9 neighbors per node.

In Fig. 2, the error performance of random-access schemes computed from (16) and (20) are compared with that of the proposed mutual broadcast scheme with three different sparse recovery algorithms (message passing, CoSaMP, and AMP). The simulation results show that the message-passing decoding algorithm noticeably outperforms CoSaMP and AMP at both SNRs, γ = 0 dB and γ = 10 dB. Moreover, the sparse-recovery-based scheme with message-passing decoding requires much fewer number of symbol transmissions than ALOHA and CSMA to achieve the same error probability. At γ = 0 dB, we also make a comparison between the simulation result and the large system approximation from (11) for the message-passing algorithm. The approximation is seen to provide a good characterization of the performance of the message-passing algorithm. This is observed for several systems of moderate size at low SNRs (some additional supporting numerical evidence is, however, omitted due to space limitations). Note that the performance of ALOHA is much worse than that of CSMA at γ = 10 dB and is omitted

for better illustration. Fig. 3 repeats the simulation with the number of neighbors changed to K = 9 and the message length per node changed to l = 5. In this case, random-access schemes require each node to transmit at least 4 additional bits so that node 0 can identify its sender. The relative performance of the different schemes remains similar as in the previous case depicted in Fig. 2. In particular, the proposed mutual broadcast scheme with message-passing decoding has significant gain compared with random-access schemes. The large-system analysis also approximates the performance of the message-passing algorithm very well at 0 dB SNR. In the remainder of this section, we consider some systemlevel experiments. Suppose there are 1000 nodes in a wireless network, and they are uniformly distributed in a 100 × 100 square. Assume that each node has a sensing radius of R, then each node has an average number of Kav = πR2 /10 neighbors

7

by considering periodic boundary condition. We always choose q = Kav1+1 in our sparse-recovery-based scheme. For better performance in ALOHA-based random-access scheme, we let each node Xi transmit in each frame interval with probability of 1/(KXi + 1). The metric for performance comparison between different schemes is the probability for one node to miss a given neighbor, averaged over all pairs of neighbors in the network. First we consider one realization of the network where each node has Kav = 4 neighbors and each node has a message of l = 10 bits. Suppose the identification overhead in randomaccess schemes is ignored and L is set to 10. We still consider γ = 0 dB and 10 dB. As shown in Fig. 4, in the same system setting, the proposed sparse-recovery-based scheme needs much less symbol transmissions than both ALOHAand CSMA-based random-access schemes. Similar results can be observed in Fig. 5 when simulating one realization of the network where Kav = 9 and l = 10 at 10 dB SNR. To achieve the error rate of 0.01, the proposed scheme requires 200 symbol transmissions, whereas CSMA and ALOHA require about 700 and 1, 000 symbol transmissions, respectively. VI. C ONCLUDING R EMARKS The idea of using on-off signalling to achieve full-duplex communication using half-duplex radios applies to general peer-to-peer networks, and is not limited to mutual broadcast focused on in this paper. The sparse recovery algorithms are of course suitable for the situation where the broadcast messages consist of a relatively small number of bits. The proposed scheme can serve as a highly desirable sublayer of any network protocol stack to provide the important function of simultaneous peer-to-peer mutual broadcast. This sub-layer provides the missing link in many advanced resource allocation schemes, where it is often assumed that nodes are provided the state and/or demand of their peers. It is also interesting to note that the proposed scheme departs from the usual solution where a highly-reliable, capacityachieving, point-to-point physical-layer code is paired with a rather unreliable MAC layer. By treating the physical and MAC layers as a whole, the proposed scheme achieves better overall reliability at much higher efficiency. A PPENDIX A D ERIVATION OF A LGORITHM 1 We derive (6) from (4) and (5). Using (5a) and (7), (4a) can be rewritten as   t−1 X U (1)  Λ 1 log νk (21) mtkµ = tanh  + t−1 2 2 U (0) νk ν∈∂k\µ t which becomes (6a). By noting that Vjµ (xj ) = (1+mtjµ bj )/2, it is easy to check using (5b) that (4b) can be rewritten as   P Q 1+mtjµ bj bk pµ (rµ |b) j∈∂µ\k b 2  1+mt b  (22) m ˆ tµk = P Q jµ j p (r |b) µ µ b j∈∂µ\k 2

where pµ (r|b) denotes a conditional Gaussian density:   1 1 2 exp − 2 (r − ∆µk − sµk bk ) (23) pµ (r|b) , √ 2σ 2πσ 2 P where ∆µk = j∈∂µ\k sµj bj depends on all elements of b but bk . Therefore, BP can be implemented as iteratively computing the messages mt and m ˆ t using (6a) and (22), respectively. The key to the simplification is to recognize that ∆µk is approximately Gaussian, due to the central limit theorem, whose mean is X t vµk = sµj mtjµ (24) j∈∂µ\k

and variance is X

t 2 (σµk ) =

s2µj 1 − (mtjµ )2




(25)

j∈∂µ\k

=

β β X (|∂µ| − 1) − (mtjµ )2 . Nq Nq

(26)

j∈∂µ\k

By the law of large numbers, we can further approximate t 2 ) by its average over all (µ, k) pairs with sµk 6= 0, i.e., (σµk
t 2 (σµk ) ≈ βδ 1 − Qt (27) P where δ = L2 /(N qL), L = µ |∂µ|, and Qt is given by (9). t Note also that Vjµ (xj ) = (1 + mtjµ bj )/2 is a probability mass function, the right hand side of (22) is exactly the posterior mean of bk over a certain probability measure. Using the Gaussian approximation and change of probability measure, (22) can be reduced to (6b) where At is given by (8). A PPENDIX B T HE L ARGE -S YSTEM A NALYSIS OF M ESSAGE PASSING We derive the recursive expressions for η t in Proposition 1 in the limit where K, M → ∞ with fixed ratio. The analysis is similar to that in [16], [18], [19] developed for CDMA. The key difference here is that the underlying factor graph is not fully connected and we need to take into account the variation of the parameters L and L2 in different realizations of the graph. In fact, the asymptotic theory developed here can be made fully rigorous as in [20] if the graph is also made sufficiently sparse by letting q vanish at proper rate. Due to space limitations we only provide an outlinePof the analysis. ˆ t = tanh−1 m ˆt Let h ˆ tνk and htkµ = νk ν∈∂k\µ hνk . In the large-system limit, the node degrees become increasingly large, so that htkµ converges to a Gaussian random variable by P ˆt . the central limit theorem. Moreover, htkµ ≈ htk , ν∈∂k h νk t t Assume bk hkµ ≈ bk hk is independently sampled from a Gaussian distribution with mean E t and variance F t . Define 1 X X X bj mtjµ . (28) Mt = L2 k µ∈∂k j∈∂µ

t+1

t+1

We relate M and Q (defined by (9)) approximately to E t and F t . Starting with M t+1 , by (28) and (6a), we have   1 X X X bj Λ t t+1 M = tanh + bj hjµ L2 2 k µ∈∂k j∈∂µ

8

 X 1 X bj Λ t ≈ tanh + bj hj |∂µ| L2 j 2 µ∈∂j    bj Λ t ≈ E tanh bj hj + 2  √  X Z 1 + br bΛ −z2 √ tanh z F t + E t + = e 2 dz 2 2 2π b=±1 (29) where r = tanh( Λ2 ). Similarly, Qt+1 can be approximated as  √  X Z 1 + br bΛ −z2 √ tanh2 z F t + E t + Qt+1 ≈ e 2 dz. 2 2 2π b=±1 (30) On the other hand, the mean and variance at the t-th update can be approximated as follows: 1 X 1 X X ˆt bk htk = bk hµk Et ≈ (31) N N k k µ∈∂k  
1 X X ˆ t
2 1 t t 2 F ≈ E hµk − N |∂k| k µ∈∂k 1 X X ˆ t
2 ≈ hµk . (32) N k µ∈∂k

By (23), one can rewrite rµ as: rµ =

N X

k

1 L 2 = + σ . Ms q(1 − q) L2

(37)

By the same procedure, the second and third terms within the brackets of (36) can be approximated as X 1 X X rµ sµj mtjµ L2 k µ∈∂k

j∈∂µ\k

1 X X X 1 Mt ≈ bj mtjµ = (38) L2 Ms q(1 − q) Ms q(1 − q) k µ∈∂k j∈∂µ

and sµk bk + nµ

(33)

k=1

where nµ is a Gaussian white noise with zero mean and variance σ 2 . In the following, we express E t and F t using M t and Qt by (6b) and (33): X
1 X X t Et ≈ A sµk bk rµ − sµj mtjµ N k µ∈∂k j∈∂µ\k  X At X X 1 = + sµk bk sµj bj N Ms q(1 − q) k µ∈∂k j∈∂µ\k  X + sµk bk nµ + sµk bk sµj mtjµ . (34) j∈∂µ\k

Due to independence and symmetry, it is not difficult to see that the contribution of the last three terms within the brackets in (34) vanishes when summed over k and µ ∈ ∂k. Moreover, M ∼ B(Ms , 1 − q) and |∂µ| ∼ B(N, q) for all µ, thus L/(Ms N ) → q(1 − q) and L2 /(Ms N 2 ) → q 2 (1 − q) almost surely in the large-system limit. Therefore, we have δ → 1 almost surely and (34) can be approximated as  −1 At L Et ≈ ≈ σ 2 + β(1 − Qt ) (35) N Ms q(1 − q) by (8). By (6b), F t defined in (32) can be calculated: X X  X (At )2 Ft ≈ rµ2 − 2rµ sµj mtjµ N Ms q(1 − q) k µ∈∂k j∈∂µ\k  X 2  + sµj mtjµ . (36) j∈∂µ\k

To treat the first term rµ2 within the brackets, we use (33), 1 X X 2 rµ L2 k µ∈∂k  X 1 1 X X X + sµj bj = sµi bi L2 Ms q(1 − q) k µ∈∂k j∈∂µ i∈∂µ\j  1 X X 2 + 2sµj bj nµ + nµ L2 k µ∈∂k X X |∂µ|  X 1 + E sµj bj sµi bi ≈ Ms q(1 − q) L2 k µ∈∂k i∈∂µ\j 
L X |∂k| + 2sµj bj nµ + E n2µ L2 L

 X 2 1 X X sµj mtjµ L2 k µ∈∂k

j∈∂µ\k

Qt 1 X X X (mtjµ )2 = . (39) ≈ L2 Ms q(1 − q) Ms q(1 − q) k µ∈∂k j∈∂µ

Substitute (37)–(39) into (36), F t is approximated as:  L L2 t t 2 F ≈ (A ) σ2 + N Ms q(1 − q) N Ms2 q 2 (1 − q)2  × (1 − 2M t + Qt ) ≈

σ 2 + β(1 − 2M t + Qt )  2 . σ 2 + β(1 − Qt )

(40)

Hence we have the iterative update equations (29), (30), (35) and (40). Starting with E 0 = F 0 = 0, we can compute all values of M t , Qt , E t and F t at the t-th iteration. It can be shown that E t = F t implies M t = Qt so that E t+1 = F t+1 . At the final iteration of the message-passing algorithm, we decode the message from the k-th neighbor as w ˆk = arg maxi=1,...,2l hT(k−1)2l +i . Since bk hTk are asymptotically i.i.d. random variables with Gaussian distribution of mean E T and variance F T = E T , the probability of successfully decoding one node is expressed as
2   Z ∞ x − ET 1 √ Psucc = exp − 2E T 2πE T −∞ l    2 −1 x + ET × 1−Q √ dx . (41) ET

9

The decoding error probability can thus be approximated as:

R EFERENCES

Pe = 1 − (Psucc )K .

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(42)

A more compact recursive equation in (12) can be √ obtained for an alternative parameter η t = σ 2 E t . Denote Y = E t X + W where W ∼ N (0, 1) and the prior log-likelihood ratio of X is Λ. It is easy to compute that √
(43) E {X|Y } = tanh E t Y + Λ/2 so that the minimum mean-square error E(E t ) is obtained as 2 X Z 1 + xr  √ z2 xΛ
t t √ 1 − tanh E z + E + e− 2 dz 2 2 2π x=±1 which expands to E(E t ) = 1 − 2M t+1 + Qt+1 = 1 − Qt+1 by (29), (30) and the fact that M t = Qt . Note that E t = η t γs /4 because σ 2 = 4/γs , (12) follows from (35). The error performance (formula (11)) follows from (41) and (42). A PPENDIX C P ROOF OF P ROPOSITION 2 By the stationarity of the Poisson point process, it can be assumed without loss of generality that node 0 is at the origin. The reduced Palm distribution of Φ\{0} given that 0 ∈ Φ is the same as that of Φ [22]. Further conditioned on that node 0 has exact K neighbors, the point process can be considered as the combination of two independent processes: 1) A binomial point process of K points in B0 = {x ∈ R2 : |x| ≤ R}: The process is formed by K independent points z1 , z2 , . . . , zK uniformly distributed in B0 ; 2) A restricted Poisson point process in B0c , i.e., Φ0 = Φ|B0c , where B0c is the complementary of B0 : The intensity measure of the process is Λ|B0c (·) = Λ(· ∩ B0c ), where Λ(dx) = λdx is the intensity measure of Φ. denote the number of neighbors of zi in B0 Let Kzini and Kzout i and B0c , respectively. Then we have    E K = E Kzin1 + E Kzout + 1. (44) 1 Let r = |z1 | and θ = arccos(r/(2R)), then the overlapped area of B0 and the disc centered at z1 with radius R is 2R2 θ− R2 sin 2θ. Thus, Z 2π Z R  1 2R2 θ − R2 sin 2θ E Kzin1 = (K − 1) rdrdθ 2 πR 0 πR2 0   √ = 1 − 3 3/(4π) (K − 1). (45) For fixed z1 , the expected number of z1 ’s neighbors in B0c is Z λ f (x)dx = λ(πR2 − 2R2 θ + R2 sin 2θ). (46) B0c

Therefore, averaged over all realizations of z1 , we have Z 2π Z R  1 E Kzout = λ(πR2 − 2R2 θ + R2 sin 2θ)rdrdθ 1 πR2 0 0 √ = (3 3/4)λR2 . (47) Proposition 2 follows by substituting (45) and (47) into (44).