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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, Oscar.del.Rio.Herrero@esa.int

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.

Index Terms—physical-layer network coding, fountain codes,

multiple access, collision resolution, satellite communications.

I. INTRODUCTION

Multiple access systems are an essential component of wire-

less 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 attenu-

ation 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.

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.

II. SYSTEM MODEL

Let us consider a system with Mtransmitters T1, ....., TM

and one receiver R. Each transmitter Tihas an independent

message ui=[ui(1), ...., ui(K)], consisting of Kbinary

symbols of information ui(j)∈{0,1}for j=1,...,K,to

deliver to the receiver. We assume that each terminal Tiuses

the same linear channel code of ﬁxed rate r=K

Nto protect its

message uiobtaining the codeword xi=[xi(1), ..., xi(N)],

where xi(t)∈{0,1}for t=1,...,N and Nis the

number of symbols in a codeword. For ease of exposition a

BPSK modulation is considered. Each codeword xiis 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 Nchannel 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 Nchannel uses,

and changes independently from one channel block to the next

one. The received signal at Rwhen ktransmitters, k≤M,

access the channel simultaneously, is:

y=h1s1+h2s2+... +hksk+w=hTS+w,(1)

3120978-1-4577-0539-7/11/$26.00 ©2011 IEEE ICASSP 2011

where hT=[h1,...,h

k]is a vector containing the channel

coefﬁcients, assumed to be circularly symmetric complex

Gaussian random variables accounting for fading and path

loss, and Sis a matrix obtained stacking up signal vec-

tors s1...sk, while wis an addictive white Gaussian noise

(AWGN) process with variance σ2.

A. Collision Recovery with Physical Layer Network Coding

We assume that when a collision occurs the signals from

the transmitters add up with symbol synchronism. Symbol

synchronization can be achieved considering orthogonal fre-

quency division multiplexing (OFDM) modulation, that can

help to counteract the delay spread in signal propagation. Each

node randomly accesses the channel with probability qwhich

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 ktransmitters collide, the received

signal at Ris given by (1). Without loss of generality we order

the transmitters involved in the collision from T1to Tk. Each

codeword xiis calculated from uias xT

i=uT

iG,whereG

is the Kby Ngenerator matrix of the common code. All

nodes use the same matrix G. Starting from ythe receiver

Rwants to decode the codeword xsx1⊕x2⊕...⊕xk,,

where ⊕denotes the bit-wise XOR. In order to do this we

must feed the LDPC decoder of Rwith the vector L⊕=

{L⊕(1), ..., L⊕(N)}of LLRs for xs.Wehave:

L⊕(t)ln Pr[xs(t)=1|y(t)]

Pr[xs(t)=0|y(t)]

=ln

Pr[y(t)|xs(t)=1]

Pr[y(t)|xs(t)=0]

.(2)

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 Pr[xk(t)=1]=Pr[xk(t)=0]=1

2.

Equation (2) reduces to the calculation of the ratio of the

likelihood functions of y(t)for the cases xs(t)=1and

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:

f1(y(t)) = 2−k

√2πσ2

k+1

2

i=1

(k

2i−1)

m=1

e−|y(t)−do(2i−1,m)Th|2

2σ2,(3)

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

where do(2i−1,m)is a column vector containing one (the

m-th)ofthek

2i−1possible permutations over ksymbols

(without repetitions) of an odd number (2i−1) of symbols

with value “+1”. As for the case with xs=0we have:

f0(y(t)) = 2−k

√2πσ2

k+1

2

i=1

(k

2i)

m=1

e−|y(t)−de(2i,m)Th|2

2σ2,(4)

where de(2i, m)is a column vector containing one (the m-

th)ofthek

2ipossible permutations over ksymbols (without

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:

L⊕(t)=ln⎧

⎪

⎨

⎪

⎩k+1

2

i=1 (k

2i−1)

m=1 e−|y(t)−do(2i−1,m)Th|2

2σ2

k+1

2

i=1 (k

2i)

m=1 e−|y(t)−de(2i,m)Th|2

2σ2

⎫

⎪

⎬

⎪

⎭

.(5)

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

1/2and codeword length equal to 480 symbols has been used

in the simulations. The results are depicted in Fig. 1.

0246810 12

10−3

10−2

10−1

100

FER

SNR (dB)

1 transmitter

2 transmitters

5 transmitters

20 transmitters

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. MULTIPLE 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 TTD(M)

which denotes the average delay needed for the receiver to

recover all Mmessages. 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 TTD(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 (TTT) needed for packet transmission. TAC K accounts

for all the propagation delays in the transmissions while TTT

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 TTT into Mportions 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:

TTD(M)=TTT +TACK =

M

k=1

Xk+TACK .(6)

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.

A. Centralized Scheduling

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 Mtransmitters to receive the ACK for its packet is:

E[TTD]=M1

1−p1

+RT D.(7)

B. Random Access

Nodes randomly access the channel with probability qand

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:

E[TTD]=

M

k=1

1

kq(1 −p1)(1 −q)k−1+M×RT D. (8)

C. PHY NC Collision Recovery with Full Coordination

In this scheme all nodes access the network in the ﬁrst slot.

If Rcan 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 Rone of the transmitters stops transmit-

ting. This guarantees that all decoded packets are innovative.

When the receiver has decoded Minnovative packets, it can

reconstruct all of the original messages. As shown in Fig. 1,

keeping constant the power of each transmitter, pkslightly

increases with the number of colliding nodes. Thus TTD is:

E[TTD]=

M

k=1

1

(1 −pk)+M×RT D. (9)

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[TTD]in this scheme is lower bounded by

the E[TTD]in of centralized scheme.

D. PHY NC Collision Recovery with Partial Coordination

Again we assume that the nodes randomly access the chan-

nel 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 Minnovative 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

Mmessages 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

Mlinearly independent packets, with Ebecoming negligible

as Mgrows. In fact, the probability δ(E)of not to decode

the Moriginal packets after M+Etransmissions is ﬁxed for

all Mand is bounded above by 2−E.

Proposition:Wehave M+1

1−p1≤E[TTT]≤M+2M−1

2M−1

1−pM.

Proof : The probability that a non innovative message is

produced when M−kinnovative packets have already been

received, assuming that all transmission patterns are equiprob-

able, 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,...,p

M}. So we can easily ﬁnd an

upper and a lower bound for E[TTT ]

M

k=1

1

(1 −p1)(1 −2−k)≤E[TTT]≤

M

k=1

1

(1 −pM)(1 −2−k)

1

1−p1M+M

k=1 2−k

1−2−M≤E[TTT]≤M+2M

k=1 2−k

1−pM

M+1

1−p1≤E[TTT]≤1

1−pMM+2M−1

2M−1.(10)

So we have:

M+1

1−p1

+RT D ≤E[TTD]≤M+2M−1

2M−1

1−pM

+RT D. (11)

IV. NUMERICAL RESULTS

We compare the performances of the four schemes described

in terms of the average throughput, deﬁned as M/E[TTD],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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2 4 6 8 10 12 14

0

500

1000

1500

2000

Number of transmitters (M)

Throughput (packets/s)

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)

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.

1 2 3 4 5

0

5

10

15

20

Number of transmitters (M)

Throughput (packets/s)

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)

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 qis 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 (S-

UMTS) 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

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.

V. CONCLUSIONS

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 net-

works 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

kof colliding signals and obtained FER curves for different

values of k.

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