Collision Resolution in Multiple Access Networks with Physical-Layer ...

Collision Resolution in Multiple Access Networks with Physical-Layer Network Coding and Distributed Fountain Coding G. Cocco∗† , C. Ibars∗ , D. G¨und¨uz∗ and O. del Rio Herrero¶ ∗ Centre Tecnol` ogic de Telecomunicacions de Catalunya – CTTC Parc Mediterrani de la Tecnologia, Av. Carl Friedrich Gauss 7 08860, Castelldefels – Spain ¶ European Space Agency / ESTEC Noordwijk – The Netherlands {giuseppe.cocco, christian.ibars, deniz.gunduz}@cttc.es, [email protected]

Abstract—We propose two new protocols based on physical layer network coding for collision resolution in multiple access networks. When a collision occurs the receiver decodes the sum of the collided packets and after a number of transmissions, equal to or slightly higher than the number of original packets, it can recover all of them. One of the proposed protocols based on fountain codes can resolve collisions by sending out just one acknowledgement (ACK), thus being particulary suited to networks with large round trip delays such as satellite networks. We carry out a comparison of the average delay achieved by the proposed schemes with other access techniques, and show how the performance can be improved with little coordination at the receiver.

sum) of the collided packets [5]. An information theoretical analysis of such a scheme has been carried out in [6], [7] and [8]. We propose two new schemes for collision recovery in multiple access networks based on PHY NC. We perform a mathematical analysis of the average delay for the proposed schemes and compare their throughput with other schemes in terrestrial and satellite scenarios, taking into account the effects of both noise and interference.

Index Terms—physical-layer network coding, fountain codes, multiple access, collision resolution, satellite communications.

Let us consider a system with M transmitters T1 , ....., TM and one receiver R. Each transmitter Ti has an independent message ui = [ui (1), ...., ui (K)], consisting of K binary symbols of information ui (j) ∈ {0, 1} for j = 1, . . . , K, to deliver to the receiver. We assume that each terminal Ti uses the same linear channel code of fixed rate r = K N to protect its message ui obtaining the codeword xi = [xi (1), ..., xi (N )], where xi (t) ∈ {0, 1} for t = 1, . . . , N and N is the number of symbols in a codeword. For ease of exposition a BPSK modulation is considered. Each codeword xi is BPSK modulated (using the mapping 0 → −1, 1 → +1) thus obtaining the transmitted signal vector si = [si (1), ..., si (N )] with si (t) ∈ {−1, +1} for t = 1, . . . , N . We assume a slotted network model in which time is divided into time slots of fixed duration equal to N channel symbols. The same model can be adopted for systems that use channel access techniques such as FDMA or CDMA if we assume that the number of channels is finite (i.e. finite total bandwidth or finite number of orthogonal codes). In this case the model considered describes the access in each of the frequency/code channels. We consider a block fading channel model in which the channel from each transmitter to the receiver has a Rayleigh distribution, independent from other channels. The value of each channel coefficient remains constant for a block of N channel uses, and changes independently from one channel block to the next one. The received signal at R when k transmitters, k ≤ M , access the channel simultaneously, is:

I. I NTRODUCTION Multiple access systems are an essential component of wireless communications. Popular examples are the multiple access to an access point in wireless local area networks (WLANs), access to a base station in a cellular system and multiple access to a satellite. The sharing of a common wireless medium makes these systems prone to collisions of the transmitted signals at the receiver, thus introducing delays and limiting systems capacity. Measurements show how in WLANs about 10% of sender-receiver traffic is involved in collisions [1]. The situation is even worse in the satellite context, where collision avoidance is not possible. The possibility of recovering packets involved in a collision has been addressed in works such as [2], where interference cancellation at the receiver is performed by exploiting information about phase shift and channel attenuation for each of the colliding signals. Performance of such technique has been studied in several works, such as in [3] and [4], in which delay and throughput were evaluated in the high SNR regime. Another technique used for collision recovery is physical layer network coding (PHY NC), that has been largely applied to the two-way relay channel (TWRC). In the TWRC two nodes communicate through a relay. In case of a collision the relay can decode a function (e.g. the modulo † G. Cocco is partially supported by the European Space Agency under the Networking/Partnering Initiative.

978-1-4577-0539-7/11/$26.00 ©2011 IEEE

II. S YSTEM M ODEL

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y = h1 s1 + h2 s2 + ... + hk sk + w = hT S + w,

(1)

ICASSP 2011

where hT = [h1 , . . . , hk ] is a vector containing the channel coefficients, assumed to be circularly symmetric complex Gaussian random variables accounting for fading and path loss, and S is a matrix obtained stacking up signal vectors s1 . . . sk , while w is an addictive white Gaussian noise (AWGN) process with variance σ 2 .

where do (2i − 1, m)  is a column vector containing one (the k possible permutations over k symbols m-th) of the 2i−1 (without repetitions) of an odd number (2i − 1) of symbols with value “+1”. As for the case with xs = 0 we have:

A. Collision Recovery with Physical Layer Network Coding

where de (2i,  k m) is a column vector containing one (the mpossible permutations over k symbols (without th) of the 2i repetitions) of an even number (2i) of symbols with value “+1”. Finally using (3) and (4) in (2) we find the following expression for the LLR: ⎧ ⎫ o Th 2 k | ⎪ ⎪ (2i−1 ) − |y(t)−d (2i−1,m) ⎨  k+1 ⎬ 2  2 2σ m=1 e i=1 . (5) L⊕ (t) = ln 2 e T k h| (2i) − |y(t)−d (2i,m) ⎪ ⎪ ⎩  k+1 ⎭ 2  2σ2 m=1 e i=1

The last equality follows from the symmetry of the XOR operator provided that xk (t)’s are independent and identically distributes (i.i.d.) with P r[xk (t) = 1] = P r[xk (t) = 0] = 12 . Equation (2) reduces to the calculation of the ratio of the likelihood functions of y(t) for the cases xs (t) = 1 and xs (t) = 0. We indicate these functions as f1 (y(t)) and f0 (y(t)) respectively. Functions f0 (y(t)) and f1 (y(t)) are Gaussian mixtures:  2  (2i−1)  − |y(t)−do (2i−1,m)T h|2 2−k  2σ2 f1 (y(t)) = √ e , 2πσ 2 i=1 m=1 k+1

∗ See

k

[11] and [12] for an extension to higher order modulations.

(3)

(4)

We evaluate the frame error rate (FER) values for different numbers of transmitters using these LLR values in case of symmetric channels. A non-systematic LDPC code with rate 1/2 and codeword length equal to 480 symbols has been used in the simulations. The results are depicted in Fig. 1. 0

10

1 transmitter 2 transmitters 5 transmitters 20 transmitters −1

10

FER

We assume that when a collision occurs the signals from the transmitters add up with symbol synchronism. Symbol synchronization can be achieved considering orthogonal frequency division multiplexing (OFDM) modulation, that can help to counteract the delay spread in signal propagation. Each node randomly accesses the channel with probability q which leads to a certain collision probability. We further assume full channel state information at the receiver (CSIR) in each time slot. This can be achieved using a CDMA-encoded preamble, assuming that the probability that two nodes use the same code is negligible [9]. The receiver also needs to know which of the nodes are transmitting in each time slot. This can be accomplished by adding in the preamble of the first time slot the seed for the random number generator (RNG) used by each node to determine its transmission sequence [10]. When a collision occurs at the receiver, it tries to decode the bit-wise XOR of the transmitted messages. This can be done by feeding the decoder with the log-likelihood ratios (LLR) for the received signal. Such LLRs can be calculated as follows in case LDPC codes and BPSK modulation are used ∗ . When signals from k transmitters collide, the received signal at R is given by (1). Without loss of generality we order the transmitters involved in the collision from T1 to Tk . Each codeword xi is calculated from ui as xTi = uTi G, where G is the K by N generator matrix of the common code. All nodes use the same matrix G. Starting from y the receiver R wants to decode the codeword xs  x1 ⊕ x2 ⊕ . . . ⊕ xk ,, where ⊕ denotes the bit-wise XOR. In order to do this we must feed the LDPC decoder of R with the vector L⊕ = {L⊕ (1), ..., L⊕ (N )} of LLRs for xs . We have:   P r [xs (t) = 1|y(t)] L⊕ (t)  ln P r [xs (t) = 0|y(t)]   P r [y(t)|xs (t) = 1] = ln . (2) P r [y(t)|xs (t) = 0]

(2ik )  k+1 2   |y(t)−de (2i,m)T h|2 2−k  2σ2 f0 (y(t)) = √ e− , 2πσ 2 i=1 m=1

−2

10

−3

10

0

2

4

6 SNR (dB)

8

10

12

Fig. 1. FER for decoding the XOR using LLRs for different numbers of transmitters when all channel gains are equal. The SNR indicated in the picture is that of a single user transmission. Note that for a given SNR the FER slightly with the number of transmitters.

III. M ULTIPLE ACCESS We propose two schemes based on PHY NC to address collisions over the channel and compare them with an ideal TDMA scheme (centralized scheduling) and a random access scheme. Our figure of merit is the total delivery time TT D (M ) which denotes the average delay needed for the receiver to recover all M messages. The last two schemes are studied in [4] in the case of an erasure channel, while we consider channel model (1). We explicitly consider the effect of the round-trip delay in communications. We can divide TT D (M ) into two parts, the first part (TACK ) due to the round-trip-delay (RTD) needed for the acknowledgements (ACKs) sent by the receiver to be received by the transmitters, and the second part (TT T ) needed for packet transmission. TACK accounts for all the propagation delays in the transmissions while TT T accounts for the duration of the packet transmission time and the time needed for retransmissions. Furthermore, we assume

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that ACK’s are always correctly received by all the terminals. We can further divide TT T into M portions where the k-th portion represents the number of transmission slots needed to deliver the k-th innovative packet to the destination. An innovative packet is a packet that can not be obtained as a linear combination of previously received ones. For instance, in a system without collision recovery all decoded packets are innovative, while in a system where the receiver decodes linear combinations of the collided packets (which is the case for the proposed protocols) this is not always the case. Let us define: • Tk = Time when the k-th innovative packet is decoded by the receiver • Xk = Tk − Tk−1 , with T0 = 0. Hence TDT can be calculated as: TT D (M ) = TT T + TACK =

M 

Xk + TACK .

(6)

k=1

In the following we adapt the delay expressions for the centralized and the random access system studied in [4] for our setup and obtain expressions for the delays in the proposed schemes.

Nodes randomly access the channel with probability q and an ARQ protocol is adopted for collision recovery. In this case a packet is correctly received only if just one node accesses the channel and the packet is correctly decoded: 1 + M × RT D. kq(1 − p1 )(1 − q)k−1

In this scheme all nodes access the network in the first slot. If R can decode the XOR of the collided packets it sends out an ACK to a randomly chosen node, which stops transmitting. The process goes on in a similar way, so every time that a packet is decoded by R one of the transmitters stops transmitting. This guarantees that all decoded packets are innovative. When the receiver has decoded M innovative packets, it can reconstruct all of the original messages. As shown in Fig. 1, keeping constant the power of each transmitter, pk slightly increases with the number of colliding nodes. Thus TT D is: M  k=1

1 + M × RT D. (1 − pk )

k=1

(9)

M

 1 1 ≤ E[TT T ] ≤ −k (1 − p1 )(1 − 2 ) (1 − pM )(1 − 2−k ) k=1

  M  M −k  M + 2 k=1 2−k 2 1 M + k=1−M ≤ E[TT T ] ≤ 1 − p1 1−2 1 − pM

(8)

C. PHY NC Collision Recovery with Full Coordination

E[TT D ] =

2M −1 Proposition: We have M+1 1−p1 ≤ E[TT T ] ≤ 1−pM . Proof : The probability that a non innovative message is produced when M − k innovative packets have already been received, assuming that all transmission patterns are equiprobable, is where 2−k [14]. In case an innovative message is produced and transmitted there is a probability that the destination can not decode the message due to noise. This probability depends on the number of collided signals and can assume values in {p1 , . . . , pM }. So we can easily find an upper and a lower bound for E[TT T ]

M 

B. Random Access

k=1

Again we assume that the nodes randomly access the channel with probability q. The receiver tries to decode the XOR of the transmitted messages and, in case it is an innovative packet, it stores it. When the receiver gets M innovative packets it is able to decode all original messages and sends out a block ACK acknowledging all the transmitters. This technique is inspired by the same principle of fountain codes, a class of erasure correction codes in which a sender with a block of M messages to deliver to one or more destinations randomly selects and transmits linear combinations of its messages [13]. In general M +E, E > 0, transmissions are needed to produce M linearly independent packets, with E becoming negligible as M grows. In fact, the probability δ(E) of not to decode the M original packets after M + E transmissions is fixed for all M and is bounded above by 2−E . M −1

Let us consider a system with centralized scheduler in which only one node transmits its packet during a given time slot and repeats it in successive slots until an ACK is received. In a channel with a probability of not correctly decoding the packet equal to p1 , the average number of time slots needed for each of the M transmitters to receive the ACK for its packet is:   1 + RT D . (7) E[TT D ] = M 1 − p1

M 

D. PHY NC Collision Recovery with Partial Coordination

M+ 2

A. Centralized Scheduling

E[TT D ] =

As in centralized scheduling, every transmission can provide an innovative packet with a given probability. A lower bound can be easily found from (9) by setting pk = p1 ∀k ∈ {1, . . . , M }, i.e. E[TT D ] in this scheme is lower bounded by the E[TT D ] in of centralized scheme.

M +1 1 ≤ E[TT T ] ≤ 1 − p1 1 − pM

  2M − 1 M + M−1 . 2

(10)

So we have: M

M + 22M −1 M +1 −1 + RT D ≤ E[TT D ] ≤ + RT D. 1 − p1 1 − pM

(11)

IV. N UMERICAL R ESULTS We compare the performances of the four schemes described in terms of the average throughput, defined as M/E[TT D ], as a function of the number of transmitters M . Two scenarios are considered: a land scenario, characterized by an RTD which can be neglected with respect to the duration of the packet, and a satellite scenario, with an RTD that is significant with respect to the packet duration. Note that, due to slotted access, one time slot is lost for the reception of each ACK even if the

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Throughput (packets/s)

2000

1500

the number of users, while all other schemes have an almost constant throughput. This is because the other schemes have delays that rapidly increase with M , due to the higher number of ACK needed.

Centralized scheduler Random access PHY NC full coordination PHY NC partial coord. (l.b.) PHY NC partial coord. (u.b.) PHY NC partial coord. (montecarlo)

V. C ONCLUSIONS

1000

500

0

2

4

6 8 10 Number of transmitters (M)

12

14

Fig. 2. Average throughput as a function of the number of nodes in the network in terrestrial communications. The lower and upper bounds shown are obtained from the upper and lower bound for the delays calculated in the previous section, respectively.

Throughput (packets/s)

20

15

Centralized scheduler Random access PHY NC full coordination PHY NC partial coord. (l.b.) PHY NC partial coord. (u.b.) PHY NC partial coord. (montecarlo)

In this paper we have considered two new schemes for collision recovery in multiple access networks based on PHY NC. We have carried out an analysis of the delays for the proposed methods taking into account the effect of both noise and interference and have showed how the scheme with partial coordination based on fountain codes achieves a higher throughput with respect to the other schemes for a number of nodes greater than or equal to three in the land scenario and for the whole set of considered node numbers in the satellite scenario. The scheme with full coordination closely approaches the performance of ideal TDMA scheduling in networks where the propagation delay is negligible with respect to the frame duration. We have also reported the calculation of the log-likelihood ratios for PHY NC for a generic number k of colliding signals and obtained FER curves for different values of k.

10

R EFERENCES 5

0 1

2

3 Number of transmitters (M)

4

5

Fig. 3. Average throughput as a function of nodes in the network in GEO satellite communications. The lower and upper bounds shown are obtained from the upper and lower bound for the delays calculated in the previous section, respectively.

duration of the RTD can be neglected. We assume Gaussian channels with SNRs between each of the transmitters and the receiver equal to 5 dB and use packet loss rate curves reported in Section II. The access probability q is chosen as to minimize the number of transmissions for a given number of users. We refer to the ETSI standards universal mobile telecommunication system (UMTS) and satellite-UMTS (SUMTS) for the time slot durations and the RTDs in the terrestrial and the satellite cases [15]. In Fig. 2 the average throughput for a terrestrial network is shown. For a number of nodes greater than or equal to three the scheme with partial coordination achieves the highest throughput among the considered schemes. This is because the lower E[TACK ] of the scheme compensates for the need of more than M decoded packets to obtain all the messages. The superiority of the centralized scheme compared to the one with full decoding is due to the lower FER of single user reception with respect to multiuser reception. As in the collision recovery methods addressed in [4], the partial and full coordination schemes do not require centralized scheduling, thus saving complexity due to control signalling. In Fig. 3 the average throughput is shown for a GEO satellite communication system with respect to the number of transmitters in the network. The scheme with partial coordination shows a linear increase of throughput with

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