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Collision resolution in multiple access networks with physical-layer network coding and distributed fountain coding

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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 particularly 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.
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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¨uzand 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 trafc 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 xed 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
xed 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 nite (i.e. nite total bandwidth or nite 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
coefcient 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, kM,
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
coefcients, 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 rst 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 xsx1x2...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)) = 2k
2πσ2
k+1
2
i=1
(k
2i1)
m=1
e|y(t)do(2i1,m)Th|2
2σ2,(3)
See [11] and [12] for an extension to higher order modulations.
where do(2i1,m)is a column vector containing one (the
m-th)ofthek
2i1possible permutations over ksymbols
(without repetitions) of an odd number (2i1) of symbols
with value “+1”. As for the case with xs=0we have:
f0(y(t)) = 2k
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 nd the following
expression for the LLR:
L(t)=ln
k+1
2
i=1 (k
2i1)
m=1 e|y(t)do(2i1,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
103
102
101
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 gure 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 rst 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 dene:
Tk=Time when the k-th innovative packet is decoded
by the receiver
Xk=TkTk1, 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
1p1
+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)k1+M×RT D. (8)
C. PHY NC Collision Recovery with Full Coordination
In this scheme all nodes access the network in the rst 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=p1k
{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 xed for
all Mand is bounded above by 2E.
Proposition:Wehave M+1
1p1E[TTT]M+2M1
2M1
1pM.
Proof : The probability that a non innovative message is
produced when Mkinnovative packets have already been
received, assuming that all transmission patterns are equiprob-
able, is where 2k[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 nd an
upper and a lower bound for E[TTT ]
M
k=1
1
(1 p1)(1 2k)E[TTT]
M
k=1
1
(1 pM)(1 2k)
1
1p1M+M
k=1 2k
12ME[TTT]M+2M
k=1 2k
1pM
M+1
1p1E[TTT]1
1pMM+2M1
2M1.(10)
So we have:
M+1
1p1
+RT D E[TTD]M+2M1
2M1
1pM
+RT D. (11)
IV. NUMERICAL RESULTS
We compare the performances of the four schemes described
in terms of the average throughput, dened 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 signicant 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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The combination of unlicensed spectrum, cheap wireless interfaces and the inherent convenience of untethered computing have made 802.11-based networks ubiquitous in the enterprise. Modern uni- versities, corporate campuses and government ofces routinely de- ploy scores of access points to blanket their sites with wireless Internet access. However, while the ne-grained behavior of the 802.11 protocol itself has been well studied, our understanding of how large 802.11 networks behave in their full empirical complex- ity is surprisingly limited. In this paper, we present a system called Jigsaw that uses multiple monitors to provide a single unied view of all physical, link, network and transport-layer activity on an 802.11 network. To drive this analysis, we have deployed an in- frastructure of over 150 radio monitors that simultaneously capture all 802.11b and 802.11g activity in a large university building (1M+ cubic feet). We describe the challenges posed by both the scale and ambiguity inherent in such an architecture, and explain the algo- rithms and inference techniques we developed to address them. Fi- nally, using a 24-hour distributed trace containing more than 1.5 billion events, we use Jigsaw's global cross-layer viewpoint to iso- late performance artifacts, both explicit, such as management inef- ciencies, and implicit, such as co-channel interference. We believe this is the rst analysis combining this scale and level of detail for a production 802.11 network.