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Practical Issues in Multi-User Physical Layer

Network Coding

G. Cocco∗,N.Alagha

¶,C.Ibars

∗and S. Cioni¶

∗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@cttc.es, nader.alagha@esa.int, christian.ibars@cttc.es, stefano.cioni@esa.int

Abstract—We address several implementation issues related to

multi-user physical layer network coding, in which the symbol-

synchronous collision of an arbitrary number of signals is

decoded. In particular we study the effect of frequency and phase

offsets, the imperfect symbol synchronization of the colliding

signals and the estimation of frequency and phase offsets and

amplitudes in the presence of more than two colliding signals.

I. INTRODUCTION

The throughput of multiple access systems is limited by the

collisions that take place when more than one node accesses

the channel in the same time slot. This limitation is particularly

problematic in satellite networks with random access, where

the long round-trip time (RTT) greatly limits feedback from

the receiver, for example to perform load control or to request

retransmissions. The spectral efﬁciency of multiple access

systems can be increased by exploiting the collided signals.

In Contention Resolution Diversity Slotted ALOHA (CRDSA)

[1], for instance, the collided signals are exploited using an

iterative interference cancelation (IC) process. In CRDSA each

packet is transmitted more than once and uncollided packets

are subtracted from slots in which their replicas are present.

Another technique that allows to extract information from

colliding signals is physical layer network coding (PHY NC).

PHY NC was originally proposed to increase spectral efﬁ-

ciency in two-way relay communication [2] by having the re-

lay decoding the collision of two signals under the hypothesis

of symbol, frequency and phase synchronism. Several studies

have been reported in the literature about synchronization

issues, gain analysis and ad-hoc modulation techniques for

PHY NC in the case of two colliding signals [3][4][5]. In [6]

PHY NC has been applied in the satellite context for pairwise

node communication. In [7] and [8] it has been proposed

to apply PHY NC to determine the identity of transmitting

nodes in case of ACK collision in multicast networks by

using energy detection and ad-hoc coding schemes, under

the hypothesis of phase synchronous signals superposition at

the receiver. In [9] the decoding of multiple colliding signals

over generally complex channels has been studied from an

information theoretical point of view. In [10] PHY NC has

been applied for collision resolution in ALOHA systems with

feedback from the receiver, under the assumption of frequency

synchronous transmitters. Although considerable attention has

been dedicated to PHY NC in the case of two colliding

signals, several important implementation issues are still to

be addressed for the case of higher order collisions. Our main

contributions with respect to previous work are the following:

•We take into account frequency and phase offsets at the

transmitters when applying PHY NC for an arbitrary

number of colliding signals. Up to our knowledge, the

issue of frequency offsets in PHY NC has been previously

addressed only for the case of two colliding signals. See,

e.g., [11], [12] and references therein.

•We show the feasibility of channel estimation for PHY

NC in the presence of more than two colliding signals,

unlike previous works where only two colliding signals

were considered (see, e.g., [13]), and evaluate the perfor-

mance of the decoder in case of estimation errors..

•We study the effect of non perfect symbol synchronism

on the decoder FER for an arbitrary number of colliding

signals and propose four different methods to compensate

for such effect.

The rest of the paper is organized as follows. In Section

II we present the system model. Section III describes how

the channel decoding works in case of an arbitrary number

of colliding signals with independent frequency and phase

offsets. Section IV deals with channel estimation and error

detection. Section V is dedicated to the effect of imperfect

symbol synchronization on the decoder performance in case of

multiple colliding signals, and different schemes to overcome

such effects are presented. In Section VI we present the

numerical results, while Section VII contains the conclusions.

II. SYSTEM MODEL

Let us consider the return link (i.e. the link from a user

terminal to the satellite/base station) of a multiple access

system with Mtransmitting terminals, T1, ....., TM, and one

receiver R. Packet arrivals at each transmitter are modeled as

a Poisson process with rate G

M, which is independent from one

transmitter to the other. Each packet ui=[ui(1), ...., ui(K)]

consists of Kbinary symbols of information ui(ξ)∈{0,1},

for ξ=1,...,K. We assume that, upon receiving a mes-

sage, each terminal Tiuses the same linear channel code of

ﬁxed rate r=K

Nto protect its message ui, obtaining the

codeword xi=[xi(1), ..., xi(N)],wherexi(l)∈{0,1}for

l=1,...,N. 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

si(t)=

N

l=1

bi(l)g(t−lTs),(1)

where Tsis the symbol period, bi(l)is the BPSK mapping of

xi(l)and g(t)is the square root raised cosine (SRRC) pulse.

The signal si(t)is called burst.

In the following we will refer to a time division multiple

access (TDMA) scheme. However, the techniques proposed in

the following can be also applied to other access schemes, such

as multi-frequency-TDMA (MF-TDMA), in which a frame

may include several carriers, or code division multiple access

(CDMA), where PHY NC can be used to recover collisions

in each of the code sub-channels. It should be noted that the

proposed technique still relies on single carrier transmission

of each user terminal. From the user terminal perspective no

signiﬁcant change is required. Transmissions are organized in

slots. When more than one burst is transmitted in the same

slot a collision occurs at the receiver. A collision involving

ktransmitters is said to have size k. We assume symbol-

synchronous transmissions, i.e., in case of a collision, the sig-

nals from the transmitters add up with symbol synchronism at

R. The received signal before matched ﬁltering and sampling

at R, in case of a collision of size k(assuming, without loss

of generality, the ﬁrst kterminals collide), is:

y(t)=h1(t)s1(t)+... +hk(t)sk(t)+w(t),(2)

where si(t)is the burst transmitted by user i,w(t)is a complex

additive white Gaussian noise (AWGN) process while hi(t)

takes into account the channel from terminal ito the receiver.

hi(t)can be expressed as:

hi(t)=Aiej(2πΔνit+ϕi),(3)

where Ai=|hi|is a lognormally distributed random variable

modeling the channel amplitude of transmitter i, while Δνi

and ϕiare the frequency and phase offsets with respect to

the local oscillator in R, respectively. We assume that the

amplitude Aiand the frequency offset Δνiremain constant

within one frame [1] while ϕiis a random variable uniformly

distributed in [−π,+π]that changes independently from one

slot to the other. The fact that ϕichanges from one slot to the

other is due to the phase noise at the transmitting terminals

[1]. Assuming that the frequency offset is small compared to

the symbol rate 1/Ts(ΔνTs1), the sample taken at time

tlafter matched ﬁltering of signal y(t)is:

r(tl)=h1(tl)q1(tl)+... +hk(tl)qk(tl)+n(tl),(4)

where q(t)=s(t)⊕g(−t), while n(tl)’s are i.i.d. zero mean

complex Gaussian random variables with variance N0in each

component. Note that even in case a BPSK modulation is used,

as we are assuming in this paper, both the I and Q components

of the received signal are considered by the receiver. This is

because the phases of the users have random relative offsets

and thus both components carry information relative to the

useful signal. The random relative offsets must be taken into

account by the decoder, as they cannot be eliminated by the

demodulator. We consider this more in detail in Section III.

In order to exploit the information extracted from a col-

lision, the receiver needs knowledge of the identity of the

transmitters, as well as the full channel state information

at each time slot. As we are considering a random access

scheme, the knowledge about nodes identity cannot be avail-

able aprioriat the receiver. Instead, nodes identity must be

determined by Rstarting from the received signal, even in

case a collision occurs. This can be achieved by having the

transmitting nodes adding an orthogonal preamble in each

transmitted burst, assuming that the probability that two nodes

use the same preamble is negligible [1]. We discuss the issue

of node identiﬁcation and channel estimation more in detail

in Section IV.

III. MULTI-USER PHYSICAL LAYE R NETWORK CODING

In this section we describe the way the received signal is

processed by the receiver Rin case of a collision of signals

having independent frequency and phase offsets.

When a collision of size koccurs, i.e., kbursts collide in

the same slot, the receiver tries to decode the bit-wise XOR of

the ktransmitted messages. This can be done by feeding the

decoder with the log-likelihood ratios (LLR) for the received

signal. The calculation of the LLRs for a collision of generic

size kin case of BPSK modulation was presented in [10]. In

the following we include the effect of frequency offset in the

calculation of the LLRs, which was not taken into account in

[10].

When signals from ktransmitters collide, the received signal

at Ris given by (2). Each codeword xiis calculated from ui

as xi=C(ui),whereC(.)is the channel encoder operator. All

nodes use the same linear code C(.). Starting from r(t),the

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

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

decoder of Ris fed with vector L⊕=[L⊕(1), ..., L⊕(N)] of

LLRs for xs,where:

L⊕(l)=ln⎧

⎪

⎨

⎪

⎩k+1

2

i=1 (k

2i−1)

m=1 e−|r(tl)−do(2i−1,m)Th(tl)|2

2N0

k+1

2

i=1 (k

2i)

m=1 e−|r(tl)−de(2i,m)Th(tl)|2

2N0

⎫

⎪

⎬

⎪

⎭

,(5)

h(tl)being a column vector containing the channel coefﬁ-

cients of the ktransmitters at time tl(which change at each

sample due to frequency offsets), while do(2i−1,m)and

de(2i, m)are column vectors containing one (the m-th) of the

k

2i−1or k

2ipossible permutations over ksymbols (without

repetitions) of an odd or even number of symbols with value

“+1”, respectively. Equation (5) is derived considering that

an even or an odd number of symbols with value +1 adding

up at Rmust be interpreted by the decoder as a 0 or a 1,

respectively. If the decoding process is successful, Robtains

the message usu1⊕...⊕uk.

An important issue in multiple access systems is the capa-

bility of the receiver to determine whether the received bursts

are correctly decoded or not. A common practice in packet

networks is the use of a cyclic redundancy check (CRC), which

allows to detect a wrong decoding with a certain probability.

Some CRC’s are based on a ﬁeld which is appended to the

message before channel coding, called CRC ﬁeld.Asthe

CRC operations are done in GF (2) and by the linearity of

the channel encoder, the CRC ﬁeld in the message obtained

by decoding a collision of size kis a good CRC for us,

which is the bitwise XOR of the messages encoded in the k

collided signals. This allows to detect decoding errors, within

the limits of the CRC capabilities, also in collided bursts. The

implementation aspect of what type of CRC should be used

is out of scope of this paragraph.

IV. CHANNEL ESTIMATION AND NODE IDENTIFICATION

For each frame the receiver Rneeds to know which of

the active terminals is transmitting in each slot and must

have channel state information for each of the users. Both

needs are addressed including an orthogonal preamble, such

as the spreading codes used in CDMA, at the beginning of

the burst. The use of an orthogonal preamble was proposed

in [1] for the estimation of the phase in collided bursts. In

[1] frequency offset and channel amplitude are derived from

the clean bursts (i.e. bursts that did not experience collisions)

and assumed to remain constant over the whole frame. Unlike

in [1], the method we propose does not rely only on clean

bursts. Thus the frequency offset and the amplitude of each

transmitter must be estimated using the collided bursts for

each frame. Due to the strong interference, the typically used

channel estimators (e.g. Rife-Boorstyn and Mengali-Morelli

for the frequency [14]) may not be the optimal choice. Instead

of estimating the channels one by one considering the other

signals as interference, we opt for a joint channel estimation,

in which channels are estimated jointly.

In order to prove the feasibility of channel estimation in

such conditions we show the results we obtained using the

Estimate Maximize (EM) algorithm. We adopted the approach

described in [15], where the EM algorithm is used to estimate

parameters from superimposed signals. In [15] two examples

were proposed related to multipath delay estimation and

direction of arrival estimation. We apply the same approach

to estimate amplitudes, phases and frequency offsets from

the baseband samples of the received signal in case of a

collision of size k. The algorithm is divided into an Estep,

in which each signal is estimated, and an Mstep, in which

the mean square error between the estimation made at the

Estep of current iteration and the signal reconstructed using

parameters calculated in previous iteration is minimized with

respect to the parameters to estimate. Formally, once initialized

the parameters with randomly chosen values, at each iteration

we have the following two steps:

Estimation step -fori=1,...,k calculate

ˆp(n)

i(t)=bi(t)ˆ

A(n)

iej(2π

Δν(n)

iTst+ˆϕ(n)

i)

+βir(t)−

k

l=1

bl(t)ˆ

A(n)

lej(2π

Δν(n)

lTst+ˆϕ(n)

l),(6)

Maximization step -fori=1,...,k calculate

min

A,Δν,ϕ

Npre

t=1 bi(t)ˆp(n)

i(t)−Aej(2πΔνTst+ϕ)

2,(7)

where pi(t)is the preamble of burst iafter the matched ﬁlter,

A,Δνand ϕare tentative values for the parameters to be

estimated, Npre is the preamble length, bi(t)∈ {±1}is the

t-th symbol in the preamble of the i-th node and Tsis the

sampling period, taken equal to the symbol rate. βiare free

parameters that we arbitrarily set to βi=0.8,fori=1,...,k.

510 15

10−9

10−8

10−7

10−6

Es/N0(dB)

Frequency estimation MSE

5 transmitters

1 transmitter

MCRLB 1 transmitter

Fig. 1. Mean squared error (MSE) of the frequency offset estimation, i.e.

E[|

Δν−Δν|2].Esis the average energy per transmitted symbol for each

node. The modiﬁed Cramer-Rao lower bound (MCRLB) for the case of one

transmitter is also shown for comparison.

We evaluated numerically the performance of the EM esti-

mator assuming that phase offsets are uniformly distributed

in [−π, +π], frequency offsets are uniformly distributed in

[0,Δνmax]with Δνmax equal to 1% of the symbol rate on the

channel (1/Ts), and amplitudes are log-normally distributed.

Figures 1, 2 and 3 show the mean squared error (MSE)

of the estimation error for frequency, phase and amplitude,

respectively. Amplitude error is normalized to the actual

amplitude value while phase error is normalized to π.Inthe

simulations we used as preambles Walsh-Hadamard words of

length 128 symbols. The EM algorithm was run twice starting

from randomly chosen initial values of the parameters and

taking as result the values of the parameters that lead to the

minimum of the sum across the signals of the error calculated

in the last E step. This was done in order to reduce the

probability to choose a “bad” local maximum, which is a

510 15

10−4

10−3

10−2

10−1

Es/N0(dB)

Relative phase estimation MSE

5 transmitters

1 transmitter

Fig. 2. MSE of the phase offset estimation normalized to π,i.e.E[|ˆϕ−

ϕ|2]/π2.Esis the average energy per transmitted symbol for each node.

510 15

10−4

10−3

10−2

Es/N0(dB)

Relative amplitude estimation MSE

5 transmitters

1 transmitter

Fig. 3. MSE of the amplitude estimation normalized to the actual amplitude

of the channel, i.e. E[|ˆ

A−A|2/A2].Esis the average energy per transmitted

symbol for each node.

problem that affects all the “hill climbing” algorithms. For

each run 6 iterations were made.

V. P ERFORMANCE WITH IMPERFECT SYMBOL

SYNCHRONIZATION

In Section II we assumed that signals from different re-

ceivers add up with symbol synchronism at the receiver in case

of a collision. In Fig. 4 an example is shown of received signal

and sampling instants in the case of three nodes transmitting

with no timing offsets. The transmitted signals, which are also

shown, modulate the sets of symbols [-1 1 -1], [-1 1 1] and [

-1 -1 -1]. The situation depicted in the ﬁgure is an illustrative

one, as in a real system both I and Q signal components

are present, signals may have different amplitudes, phase and

frequency offsets for each of the bursts and the signal is

immersed in thermal noise. However, in a real system there

will always be a certain symbol misalignment, which grows

larger as the resources dedicated to the synchronization phase

diminish (see, e.g., [16] and references therein for examples

of synchronization algorithms). Being able to cope with non

perfect symbol synchronism can bring important advantages,

such as less stringent constraints on signal alignment, with

consequent savings in terms of network resources needed

for the synchronization. In this section we study the effect

of non perfect symbol synchronization and propose possible

countermeasures. Let us consider a slotted multiple access with

knodes accessing the channel at the same time. We assume

that each transmitter has its own phase and frequency offsets.

We further assume that each burst falls completely within the

boundaries of a time slot, i.e., no burst can fall between two

consecutive time slots. Let us call Tthe time at which the

peak of the ﬁrst symbol of the bursts that ﬁrst arrives at R.We

deﬁne the relative delay (RD) ΔTiof node ias the temporal

distance between the peak value of the ﬁrst pulse of burst i

and T. In other words, the burst which arrives ﬁrst at the

receiver is used as reference, i.e., has RD equal to 0.We

assume SRRC pulses with roll off factor αare used. We further

assume that all RD’s belong to the interval [0,ΔTmax], with

0≤ΔTmax ≤Ts/2. In case of a collision of kbursts, the

200 300 400 500 600 700

−4

−3

−2

−1

0

1

2

time

Amplitude

signal transmitter 1

signal transmitter 2

signal transmitter 3

received signal

sampling times

Fig. 4. Received signal after the matched ﬁlter in case of three colliding

bursts with no timing offsets, i.e., ΔT1=ΔT2=ΔT3=0. The transmitted

signals after the matched ﬁlter in case of collision-free reception are also

shown. The transmitted symbols are: [-1 1 -1], [-1 1 1] and [-1 -1 -1] for

transmitter 1, 2 and 3, respectively. For sake of clarity, frequency and phase

offsets as well as channel amplitudes were not included in the plot and the

signals were considered as real. The samples, shown with grey circles in the

ﬁgure, are taken at instants corresponding to the optimal sampling instants for

each of the signals as if they were received without experiencing collision.

received signal before the matched ﬁlter is:

y(t)=

k

i=1

si(t)+w(t),(8)

where,

si(t)=Ai

N

l=1

bi(l)g(t−lTs−ΔTi)ej(2πΔνit+ϕi),(9)

Nbeing the number of symbols in the burst, g(t)is the

square root raised cosine pulse and w(t)represents an AWGN

process. The samples taken after the matched ﬁlter at times tl

are:

r(tl)=y(t)⊗g(−t)|t=tl=

k

i=1

qi(tl)+n(tl),(10)

where,

qi(tl)=Ai

N

l=1

bi(l)p(tl−lTs−ΔTi)ej(2πΔνitl+ϕi),(11)

p(t)being the raised cosine pulse, ⊗is the convolution

operator and n(t)is the noise process after ﬁltering and

sampling. Note that in (11) the exponential term is treated as

a constant. This approximation is done under the assumption

that ΔνTs1, i.e., the exponential term is almost constant

over many symbol cycles.

The sampled signal is then sent to the channel decoder.

It is not clear at this point which is the optimal sampling

time, as the optimal sampling time for each of the bursts taken

singularly may be different. Moreover, sampling the signal just

once may not be the optimal choice. Actually, as we will show

in next section, the performance of the decoder is quite poor

in case a single sample per symbol is taken.

In the following we propose several techniques to mitigate

the impairment due to imperfect symbol synchronization. We

assume that Rhas knowledge of the relative delays of all

the transmitters, which can be derived through the orthogonal

preambles. We further assume that Rhas perfect CSI for each

of the transmitters. Without loss of generality and for ease of

exposition, from now on we will refer to the sampling time

for the symbol number 1.

A. Single sample

a) Mean Delay: The ﬁrst method we present is Mean

Delay (MD). In MD the received signal is sampled just once

per symbol. The sampling time is chosen to be the mean of

the relative delay, i.e.:

TMD =1

k

k

m=1

ΔTm.(12)

The sample r(TMD)is then used to calculate the LLR’s as in

Eqn. (5). ISI is not taken into account.

B. Multiple samples

In the following we describe four different methods that use

ksamples per symbol, kbeing the collision size.

We start by describing two methods in which the symbol is

sampled ktimes in correspondence of the RD’s. Due to the

non perfect synchronization, when the signal is sampled in

ΔTithe sample obtained is the sum of the ﬁrst symbol of each

of the users, weighted by the relative channel coefﬁcient, plus

a term of ISI due to signals sj,j ∈{1,...,k},j =i,which

are sampled at non ISI-free instants. As the LLR’s need the

channels of each of the users, the ISI should be taken into

account. However, the ISI is a function of many (theoretically

all) symbols, and can not be taken into account exactly. In

Fig. 5 the received signal after the matched ﬁlter is shown in

the case of three colliding bursts with timing offsets ΔT1=

0,ΔT2=Ts/6and ΔT3=Ts/4. The transmitted signals

after the matched ﬁlter in the case of collision-free reception

are also shown. The symbols transmitted by each terminal are

250 300 350 400 450 500 550 600 650

−4

−3

−2

−1

0

1

time

Amplitude

signal transmitter 1

signal transmitter 2

signal transmitter 3

received signal (after MF)

sampling times

Fig. 5. Received signal after the matched ﬁlter in case of three colliding

bursts with timing offsets ΔT1=0,ΔT2=Ts/6and ΔT3=Ts/4.

The transmitted signals after the matched ﬁlter in the case of collision-free

reception are also shown. The transmitted symbols are: [-1 1 -1], [-1 1 1] and

[-1 -1 -1] for transmitter 1, 2 and 3, respectively. The samples, shown with

grey circles in the ﬁgure, are taken at instants corresponding to the optimal

sampling instants for each of the signals as if they were received without

experiencing collision. Unlike in the case of perfect symbol alignment, here

more than one sample per symbol is taken.

the same as in Fig. 4. The samples, shown with grey circles

in the ﬁgure, are taken in correspondence of the RD’s, which

coincide with the optimal sampling instants for each of the

signals as if they were received without experiencing collision.

b) Mean LLR: In Mean LLR (ML) the received signal

is sampled ktimes in the instants correspondent to ΔTi,i=

1,...,k. For each of the samples the LLR’s are calculated

as in (5). Then the average of the kLLR’sispassedtothe

decoder.

c) Mean Sample: As in ML, also in Mean Sample (MS)

r(t)is sampled ktimes in correspondence of the relative

delays. The difference between the two methods is that in

MS the samples are averaged out to obtain the mean sample:

r(t)= 1

k

k

m=1

r(ΔTm).(13)

Finally, r(t)is used in the (5) instead of r(t).

d) Uniform Sampling: In Uniform Sampling (US) the

signal is sampled ktimes as in previous methods, but the

sampling times do not correspond to the RD’s. The sampling

times are chosen uniformly in [0,ΔTmax], i.e. in case of k

transmitters the samples are taken at intervals of ΔTmax /(k−

1). Then, as in MS, the samples are averaged out and used in

the calculation of the LLRs. This method has the advantage

that receiver does not need the knowledge of the RD’s in order

to decode and the sampling itself is simpliﬁed as it is done

uniformly in each symbol.

e) Equivalent Channel: The received signal is sampled k

times in the instants correspondent to ΔTi,i=1,...,k.Inthe

method Equivalent Channel (EC) the amplitude variation of

the channel of each user due to imperfect timing is taken into

account for the current symbol. Note that the ISI is not taken

into account, but only the variation in amplitude of present

symbol due to imperfect timing is accounted for. Assuming

that the received signal is sampled at time t=ΔTi, then the

channel coefﬁcient of burst qthat is used in the LLR is:

heq

q(t)=Aqej(2πΔνqTsΔTi+ϕq)p(ΔTi−ΔTq),(14)

p(t)being the raised cosine pulse. After the sampling, the k

samples per symbol are averaged together and used in the LLR

instead of r(t). This sampling procedure is equivalent (apart

from the ISI) to ﬁltering the received signal using a ﬁlter which

is matched not to the single pulse, but to the pulse resulting

from the delayed sum of Mpulses.

VI. NUMERICAL RESULTS

In Fig. 6 the FER curves for different collision sizes

obtained using the LLRs derived in Section III are shown. The

plots are obtained using the duo-binary turbo code adopted in

ETSI digital video broadcasting (DVB) - return channel via

satellite (RCS) standard with rate 1/2and codeword length

equal to 1504 symbols. The phase offsets ϕiare random

variables uniformly distributed in [−π, +π]while frequency

offsets are uniformly distributed in [0,Δνmax]with Δνmax

equal to 1% of the symbol rate 1/Ts. The FER for the case

of estimated channels using the EM algorithm is also shown.

0 5 10 15

10−4

10−3

10−2

10−1

100

Eb/No in dB

FER

k=1

k=1 (estimated channel)

k=2

k=2 (estimated channel)

k=3

k=3 (estimated channel)

k=5

k=5 (estimated channel)

Fig. 6. FER for the XOR of transmitted messages for different numbers of

transmitters. Ebis the energy per information bit for each node. The DVB-

RCS duo-binary turbo code with rate 1/2and codeword length 1504 symbols

is used by all nodes. Phase offsets are uniformly distributed in [−π, +π],

frequency offsets are uniformly distributed in [0,Δνma x ]with Δνmax equal

to 1% of the symbol rate on the channel. Amplitudes are constant and equal

to 1.

In Fig. 7 the frame error rate is shown for the case of

5 transmitters with delays uniformly distributed in [0,T

s/4].

Constant channel amplitudes were considered, while phases

and frequency offsets are i.i.d. random variables in [0,2π]and

[0,Δνmax]respectively, where Δνmax is equal to 1/(100Ts).

The results for the 5 different methods described in Section V

are shown together with the FER for the case of ideal symbol

synchronism. The methods that use more than one sample

per symbol perform signiﬁcantly better than MD, which uses

only one sample per symbol. Among the methods based on

oversampling, MS and EC perform slightly better than the

other two. The FER of all methods present a lower slope w.r.t.

the ideal case. The loss is about 1 dB at FER =10

−2for

the methods that use oversampling.

5 5.5 6 6.5 7 7.5

10−3

10−2

10−1

100

Eb/No (dB)

FER

Mean Delay

Mean LLR

Mean Sample

Uniform Sampling

Equivalent Channel

Ideal synchronism

Fig. 7. Frame error rate for decoding a collision of size 5 with independent

frequency and phase offsets across the transmitters and delays uniformly

distributed in [0,T

s/4]. The results for the 5 different methods are shown

together with the FER for the case of ideal symbol synchronism. Oversampling

signiﬁcantly improves the FER with respect to the case of single sample.

The two methods that exploit knowledge of relative delays, i.e. MS and EC,

perform slightly better than the others. The FER of all methods present a

lower slope w.r.t. the ideal case, losing about 1 dB at FER =10

−2for the

methods that use more than one sample.

VII. CONCLUSIONS

We have carried out an analysis of several physical layer

issues related to multi-user PHY NC. We extended the anal-

ysis on and proposed countermeasures against the effects

of physical layer impairments on the FER when applying

PHY NC to decode the collision of a generic number of

signals. In particular, we took into account frequency and

phase offsets at the transmitters which, up to our knowledge,

have been previously addressed only for the case of two

colliding signals. Finally, we showed the feasibility of channel

estimation for PHY NC in the presence of more than two

colliding signals and studied the effect of non perfect symbol

synchronism on the decoder FER, proposing four different

methods based on oversampling to compensate for such effect,

which dramatically reduce the FER at the receiver with respect

to the case in which a single sample per symbol is used.

REFERENCES

[1] E. Casini, R. De Gaudenzi, and O. d. R. Herrero, “Contention

resolution diversity slotted ALOHA (CRDSA): An enhanced random

access scheme for satellite access packet networks,” IEEE Trans. on

Wireless Comm., vol. 6, no. 4, pp. 1408–1419, Apr. 2007.

[2] S. Zhang, S. Liew, and P. Lam, “Physical layer network coding,” in

ACM MOBICOM, Los Angeles (CA), U.S.A., Sep. 2006.

[3] F. Rossetto and M. Zorzi, “On the design of practical asynchronous

physical layer network coding,” in IEEE Workshop on Signal Proc.

Advances in Wireless Comm., Perugia, Italy, June 2009.

[4] R. H. Y. Louie, Y. Li, and B. Vucetic, “Practical physical layer

network coding for two-way relay channels: Performance analysis and

comparison,” IEEE Trans. on Wireless Comm., vol. 9, no. 2, pp. 764–

777, Feb. 2010.

[5] J. H. Sorensen, R. Krigslund, P. Popovski, T. Akino, and T. Larsen,

“Physical layer network coding for FSK systems,” IEEE Comm. Letters,

vol. 13, no. 8, Aug. 2009.

[6] F. Rossetto, “A comparison of different physical layer network coding

techniques for the satellite environment,” in Advanced Satellite Multi-

media Systems Conf. (ASMS), Cagliari, Italy, Sep. 2010.

[7] M. Durvy, C. Fragouli, and P. Thiran, “Towards reliable broadcasting

using ACKs,” in IEEE Int’l Symp. on Info. Theo. (ISIT), Nice, France,

June 2007.

[8] C. H. Foh, J. Cai, and J. Qureshi, “Collision codes: Decoding super-

imposed BPSK modulated wireless transmissions,” in IEEE Consumer

Comm. and Networking Conf., Las Vegas (NV), U.S.A., Jan. 2010.

[9] B. Nazer and M. Gastpar, “Reliable physical layer network coding,”

Proceedings of the IEEE, vol. 99, no. 3, pp. 438–460, Mar. 2011.

[10] G. Cocco, C. Ibars, D. G ¨und ¨uz, and O. d. R. Herrero, “Collision

resolution in slotted ALOHA with multi-user physical-layer network

coding,” in IEEE Vehicular Technology Conf. (VTC Spring), Budapest,

Hungary, May 2011.

[11] D. Maduike, H. D. Pﬁster, and A. Sprintson, “Design and imple-

mentation of physical-layer network-coding protocols,” in Asilomar

Conference on Signals, Systems and Computers, Paciﬁc Grove (CA),

U.S.A., Nov. 2009.

[12] L. Lu and S. C. Liew, “Asynchronous physical-layer network coding,”

IEEE Trans. on Wireless Comm., vol. 11, no. 2, pp. 819–831, Feb. 2012.

[13] M. Jain, S. L. Miller, and A. Sprintson, “Parameter estimation and

tracking in physical layer network coding,” in IEEE Global Telecomm.

Conf., Houston (TX), U.S.A., Dec. 2011.

[14] M. Morelli and U. Mengali, “Feedforward frequency estimation for psk:

A tutorial review,” Eur. Trans. Telecomm., p. 103116, 1998.

[15] M. Feder and E. Weinstein, “Parameter estimation of superimposed

signals using the EM algorithm,” IEEE Trans. on Acoustics, Speech

and Signal Processing, vol. 36, no. 4, pp. 477–489, Apr. 1988.

[16] R. D. J. Van Nee, “Timing aspects of synchronous CDMA,” in IEEE

Int’l Symp. on Personal, Indoor and Mobile Radio Comm., The Hague,

The Netherlands, Sep. 1994.