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A Random Access Scheme with Physical-layer

Network Coding and User Identification

Jasper Goseling

Stochastic Operations Research, University of Twente, The Netherlands Department of Intelligent Systems, Delft University of Technology, The Netherlands

j.goseling@utwente.nl

Abstract—A slotted random access scheme is proposed that is based on physical-layer network coding. The scheme uses signature codes that enable the receiver to detect which users are active in each round and which linear combination of packets is received. Feedback enables in each round, one of the users to drop a packet, keeping the queue sizes limited. It is proven that for a broad class of feedback mechanisms the scheme is stable in the sense that the receiver can eventually decode all packets. Numerical results demonstrate that the scheme performs well in terms of the expected queue size, the maximum delay as well as the expected number of retransmissions per packet.

I. INTRODUCTION

Starting with the introduction of the ALOHA protocol, most work on multiple access protocols for wireless com-munications has been based on the assumption that simul-taneous transmission attempts by more than one user lead to a destructive collision that does not provide any useful information to the receiver, cf. [1]. Only recently, it was observed that even though a collision of packets does not provide any useful information directly, it might still be useful to store the received signal. Indeed, if the receiver obtains one of the collided packets in a later round, it can use successive cancellation decoding on the stored signal. If only two packets collided, the receiver obtains a clean signal from which it can decode an information packet from another user. This approach was first proposed in [2] in which users transmit the same packet multiple times and is known as contention resolution diversity slotted ALOHA (CRDSA). Among the follow-up work on CRDSA are suggestions on how to improve performance by optimizing how packets are repeated over time [3], [4], methods to improve performance by interleaving packets [5], sliding window approaches [6] and an unslotted version of CRDSA [7]. Finally, in [8] it was proposed to, opposed to repeating packets over time, employ a forward error correcting code over the packets and instead transmit coded packets.

Starting from CRDSA and its generalizations, it is a natural follow-up observation that instead of using successive interfer-ence cancellation, the receiver might try and decode a linear combination of the colliding packets by means of physical-layer network coding. The concept of physical-physical-layer network coding is studied in, for instance [9]–[15]. See [16] for an overview of known results and a survey of literature. Note that there are various flavours of physical-layer network coding. In [9]–[12] the aim is to obtain linear combinations reliably.

In contrast, in [13]–[15] one is satisfied with a noisy version of these linear combinations. Our interest in the current paper is in reliable physical-layer network coding.

It was shown in [17], [18] that at high SNR physical-layer network coding for random access is feasible. For finite SNR this approach was studied in [19], [20]. The type of physical-layer coding employed in [19], [20] is based on [13]–[15]. Therefore, there is a positive error probability that the receiver is not correctly decoding the desired linear combination of packets. An approach giving reliable communication at finite SNR has been proposed in [21], [22]. The goal of [21], [22] was to achieve reliable communication at high throughput in the absence of feedback from the receiver to the users.

One of the assumptions in [21], [22] as well as in [17], [18] is that the receiver knows which users are transmitting. In the current work we leverage this assumption by making use of signature codes that allow the receiver the identify the active users from the received signal. Signature codes are a particular kind of multiuser code in which each of the users has only one non-zero codeword. From the received signal the individual codewords of all active users can be recovered, hence identifying these users. The observation leading to the approach presented in the current paper is that physical-layer network coding reduces the channel to a noiseless adder channel, which is a channel for which multiuser codes have been extensively studied, see for instance, [23]–[27] or the references therein. The use of physical-layer network coding and signature codes was considered in [28] for broadcast in networks.

Another difference with previous work is that we take into account feedback that can be provided from the receiver to the users. In particular, we consider the case that the receiver can provide acknowledgements targeted at specific users. The scheme that is presented in the current paper uses feedback to instruct, in each round, one of the users that its packet can succesfully be decoded and that the user should not retransmit this packet anymore. We will demonstrate, similarly to [29], that even though the receiver might not be able to decode the packet immediately, it will be able to successfully decode all packets eventually as long as the sum of the arrival rates is less than one.

One aspect of our scheme is that users retransmit their packets in each time slot until they receive an acknowl-edgement from the receiver. Retransmissions of packets are

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undesirable, since these transmissions require energy at the users. A drawback of the schemes presented in [2]–[8] is that the number of retransmissions is large. We will show that the number of retransmissions incurred by the scheme presented in the current paper is favorable compared to these other schemes.

Another aspect of our scheme is that the receiver has a choice in which user it acknowledges. In [18] a closely related scheme is presented in which stability of the system is proven for two specific acknowledgement policies. We provide a stability proof for a large class of acknowledgement policies. The outline of the remainder of this paper is as follows. In Section II we define the model. Section III provides the required definitions and results on reliable physical-layer net-work coding as well as signature codes. A detailed description of our approach is given in Section IV. Stability of the scheme is proven in Section V and a numerical evaluation of performance is given in Section VI. Finally, in Section VII we conclude with a discussion of the results that are presented in the current paper and an outlook on future work.

II. MODEL

There are N users, a subset of which is communicating to a receiver over an AWGN channel with unit gains and unit variance noise. Time is slotted. Communication takes places over rounds of B channel uses. Users are synchronized in the sense that they know when a round starts. Let Xk[t] ∈ R and Y [t] ∈ R denote the signal transmitted by user k and the signal obtained by the receiver, respectively, in time slot τ , i.e.,

Y [τ ] = N X

k=1

Xk[τ ] + Z[τ ], (1) where {Z[τ ]} is white Gaussian noise with unit variance. Within a block, all users need to satisfy an average power constraint P .

Packets arrive at the users according to an arrival process {Ai(t)}, where Ai(t), i ∈ {1, . . . , N }, denotes the number of packets that arrive at user i in round t. The random variables Ai(t) are independent and identically distributed over t and independent for different users. Let E[Ai(t)] = λi denote the mean number of packets that arrive at user i per round. Out of all N users at most K are active in the sense that packets arrive at these users, to be transmitted to the receiver. Let K ⊂ {1, . . . , N } denote the set of active users, i.e., λi = 0 unless i ∈ K. The set of active users K is fixed over time, but not known to either the receiver or the users.

Each user keeps a queue of packets that have arrived. We denote by Wi(t) the number of packets in the queue of user i at time t. Let W(t) = (W1(t), . . . , WN(t)). The receiver does not have knowledge of the states of the queue at the users and users have knowledge about their own queue only. All packets have the same size.

There is a feedback link from the receiver to the users. Feedback is noiseless and received by all users. We do not provide any inherent limitations on the amount of feedback

that is provided by the receiver. However, our interest is in strategies that provide a limited amount of feedback at the end of each round.

III. PRELIMINARIES

A. Physical-layer network coding

One key ingredient of the strategy proposed in this paper is to use computation codes to achieve reliable physical-layer network coding within each round. We provide a short introduction to this technique and a result from [9] that will be needed later. We refer the reader to [9], [10] or [16] for technical details as well as a survey on the significant body of related work.

To set the stage, we consider N transmitters, each having an block of data to transmit. Moreover, we think of these data blocks as being represented as strings of length L over a finite field Fq. That is, we denote the data of transmitter k as

Mk= (Mk(1), Mk(2), . . . , Mk(L)), (2) where Mk(j) ∈ Fq.

Each transmitter can encode its data into a string of B real numbers satisfying an average power constraint P. We define the rate of the resulting code, which is the same for all transmitters, by

R = L log2q

B bits per channel use. (3) The real-valued strings of length B, denoted as

Xk = (Xk[1], . . . , Xk[B]) (4) are then transmitted element-wise across the multiple-access channel.

The decoder, upon observing the real-valued string Y = PN

k=1Xk+ Z of length B, is asked to provide an estimate sequence ( ˆM (1), ˆM (2), . . . , ˆM (L)) in such a way as to min-imize the probability of the event

 ˆM (1), . . . , ˆM (L(i))6= K X k=1 Mk(1), . . . , K X k=1 Mk(L(i))  . In this sense, the receiver recovers a function (namely, the sum) of the original messages, which is why this approach is referred to as computation coding. We refer to the rate R as the computation rate and say that it is achievable if the probability of the above event can be made arbitrarily small by increasing B. The next result provides the best known achievable computation rate.

Theorem 1 ( [10], Thm. 2). For the standard AWGN multiple-access channel with N users the following computation rate is achievable: R = 1 2log + 2  1 N + P  . (5)

The scheme we will employ in the present paper uses the above computation rate. The achievable strategy for this rate

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was developed in [9], [10]. It involves using one and the same code at all encoders, namely,

Xk = F (Mk), (6)

for all k = 1, . . . , K, and we refer to F (·) as the computation code.

B. Signature code

Another ingredient of the strategy that is proposed is the use of a signature code. A signature code for K out of N users is a set of N codewords {S(1), . . . , S(N )} such that

X

i∈U

S(i) 6=X i∈V

S(i), (7)

for any U, V ⊂ {1, . . . , N }, |U | ≤ K, |V | ≤ K and U 6= V . We will denote such a code as a NK-signature code.

The following result from [24] appears also in, for in-stance, [26].

Theorem 2 ( [24]). If Ls satisfies Ls≥

4K log2N

log2K (8)

then there exists a NK-signature code of length Ls.

A trivial construction of a signature code would be to use a code of length N and assign to user n the n-th unit vector. The usefulness from the above results stems from the fact it significantly reduces the code length.

The code construction given in [26] leading to Theorem 2 is binary, i.e., S(i) ∈ {0, 1}Ls, where L

s is the length of the code. The author is not aware of constructions of non-binary MN-signature codes, for M  N . For N = M , such constructions do exist, for instance, as presented in [27]. Since the interest in this paper is in the case that M  N , we will use a binary signature code, which is then mapped to Fq by mapping 1 to the additive unit in Fq.

IV. STRATEGY

Codes: Within each round a computation code is used to achieve reliable communication. All users employ the same code, which is used over all rounds, and which operates over field Fq, where q is sufficiently large. Recall that packets are of a fixed length. We denote by Ld the number of q-ary symbols in a packet. Let S : {1, . . . , N } → Fq denote a signature code of length Ls. The messages that are transmitted by users consist of a signature codeword concatenated with a data packet, i.e., of Ls+ Ld symbols from Fq. Computation code F : FLs+Ld

q → R

B is employed over the concatenated sequence. We choose B through

(Ls+ Ld) log2q B = 1 2log + 2  1 K + P  , (9)

which by Theorem 1 ensures reliable communication in a round.

Transmissions: Suppose that in round t, users Kt⊂ K have a non-empty queue. All users from the set Kttransmit the first packet in their queue. Let D(k, `) denote the first packet in the queue of customer k. User k constructs the contatenation of its signature and D(k, `), i.e., user k ∈ Kt constructs the message Mk as Mk =  S(k) D(k, `)  . (10)

The user transmits Xk = F (Mk). Users that have an empty queue do not transmit.

Decoding: The receiver obtains X

k∈Kt

F (Mk) + Z, (11)

where Z = (Z[1], . . . , Z[B]). Since F is a physical-layer network code, the receiver can decode

X k∈Kt  S(k) D(k, `)  , (12)

which decomposes into the sums of the signatures P

k∈KtS(k) and the sums of the codewords

P

k∈KtD(k, `).

Note thatP

k∈KtS(k) is over Fq, whereas the signature codes

from [24] are designed for sums over integers. In [30] it was demonstrated that from the sum over Fq we can also retrieve the integer sum of the messages. Therefore, the receiver can decode the signatures. More precisely, by Theorem 2, the receiver is able to obtain Kt fromPk∈KtS(k).

Feedback: The receiver selects any k ∈ Kt and sends k as feedback to the users. We do not specify in detail the selection mechanism employed by the receiver to select a user from Kt. The only condition that we impose is that the selection mechanism depends only on the current state of the system as known by the receiver. We do not make any additional assumptions on the selection mechanism, i.e., the acknowledgement policy, employed by the receiver to select a user from Kt.

Retransmissions: User k, upon receiving the feedback, removes the packet that it transmitted in that round from its queue. If it has more packets in its queue it will start transmitting those in the next round. The users that transmitted a packet and did not receive feedback will retransmit that packet in the next round.

Retrieving the original packets: Now, over rounds the receiver collects linear equations (over Fq) of data packets D(k, `). Note, also, that in each round it acknowledges one degree of freedom. This implies that if the queues of all users are empty, the total number of data packets that needs to be decoded at the receiver equals the number of linear equations that is available. Hence, the receiver can decode the system of linear equations. We will prove in the next session that the expected time until the receiver can decode is finite. Moreover, we will show in Section VI by means of simulation that the decoding delay is small.

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V. STABILITY

Let W(t) = (W1(t), . . . , WN(t)) denote the number of packets in each queue at the start of round t. From our assumption on the feedback mechanism, the process {W(t)} is a Markov chain. We will prove in this section that this chain is positive recurrent. First, note that we can express the progression of W(t) as

W(t + 1) = W(t) + A(t) − U(t), (13) where

A(t) = (A1(t), . . . , AN(t)) (14) denote the arrivals, An(t) with mean λn, and

U(t) = (U1(t), . . . , UN(t)) , (15) represents the packet that is dropped by the user. More precisely, if PN

n=1Wn(t) > 0 then Un(t) = 1 for exactly one n ∈ Kt and zero otherwise. If P

N

n=1Wn(t) = 0 and, consequently Kt= ∅, Un(t) = 0 for all n = 1, . . . , N .

The following result deals with stability of the proposed scheme. Note, that the receiver is guaranteed to be able to decode all packets that it received so far if W(t) = 0. Therefore, the system is stable if the {W(t)} is positive recurrent.

Theorem 3. If PN

i=1λi < 1 then the process {W(t)} is positive recurrent.

Proof:We will prove that Foster’s condition, cf. [31], is satisfied for the Lyapunov function

f (w) = N X i=1 wi !2 . (16)

This implies that we need to show that

E[f (W(t + 1) − f (W(t))|W(t) = w] ≤ −1 (17) for all but a finite number of states w.

We have E[f (W(t + 1)) − f (W(t))|W(t) = w] (18) = E   N X i=1 (Ai(t) − Ui(t)) !2 W(t) = w   − E " 2 N X i=1 (Ui(t) − Ai(t)) N X i=1 wi W(t) = w # (19) = N X i=1 λi− 1 !2 − 2 1 − N X i=1 λi ! N X i=1 wi. (20) Therefore, (17) is satisfied if N X i=1 wi≥ 1 1 −PN i=1λi . (21) 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 PN i=1λi P (#Tx = 1) P (#Tx ≤ 2) P (#Tx ≤ 3)

Fig. 1. Fraction of packets delivered in 1, 2 and 3 transmissions. (K = 32)

0.2 0.4 0.6 0.8 1 2 3 4 PN i=1λi A v erage number of transmissions per pack et

Fig. 2. Average number of transmissions per packet. (K = 32)

VI. EVALUATION

In this section we present numerical results on the per-formance of the proposed scheme for the case of Bernoulli arrivals, i.e.,

P (At(i) = 1) = 1 − P (At(i) = 0) = λi. (22) We consider the case of equal arrival rates, i.e., λi= λ for all users i ∈ K.

A. Number of transmissions

The number of times a packet is retransmitted is an im-portant measure of energy efficiency. We have depicted with the solid line in Figure 1 the fraction of packets that is delivered without retransmissions, i.e., in a single transmission. Similiarly, the figure depicts the fraction op packets delivered in at most 2 and 3 transmissions. The average number of transmissions that is used is depicted in Figure 2. In Figures 1

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0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 PN i=1λi Maximum delay

Fig. 3. Upper bound on the maximum delay in terms of the expected length of a busy period. (K = 32) 0.2 0.4 0.6 0.8 0 5 · 10−2 0.1 PN i=1λi Expected queue size

Fig. 4. Expected queue size at one user. (K = 32)

and 2 the acknowledgement policy that is used to choose one of the active users uniformly at random and K = 32.

In CRDSA [2] the number of transmissions is two for all packets. For CSA and related strategies [3]–[6] the expected number of transmissions is even larger. Figures 1 and 2 demon-strate that for low to moderate load the demon-strategy proposed in the current paper outperforms these strategies in the sense that the expected number of transmissions is smaller than two. B. Decoding delay

The next performance measure that we consider is the max-imum decoding delay, which we define as the time between arrival of packet and the succesfull decoding of the packet at the receiver. It was observed in Section IV it is guaranteed that the receiver can decode all packets as soon as no user has a packet to transmit. Therefore, the length of a busy period, i.e., the time between the first arrival of a packet at any user after

a period in which no user had packets to transmit, and the time that the whole system is empty again, provides an upper bound on the decoding delay. In Figure 3 we have depicted the expected length of a busy period for K = 32 users.

The strategy that we have presented depends on acknowl-edging packets that can be dropped by the users. These packets might not have been decoded by the receiver yet. However, even though the decoding delay might be significant, the queue sizes at the users are small. This is demonstrated in Figure 4 in which the expected queue size for a user is depicted.

VII. DISCUSSION

We have presented a multiple access scheme that is us-ing both physical-layer network codes and signature codes. Among one of the important directions for future research is a generalization of the scheme to allow for any number of active users, i.e., to allow for N = M . While, the current scheme supports this in theory, the performance will not be favorable. The reason is that it would require signature codes of length almost N . As an alternative, a mixture of the current strategy with other schemes will have to be considered. This will provide other means of dealing with more than K users, for instance, by using a tree splitting approach.

ACKNOWLEDGEMENT

This work was supported by the Netherlands Organization for Scientific Research (NWO), grant 612.001.107.

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