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Practical issues in multi-user physical layer network coding

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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.
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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 efciency 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 ef-
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
xed 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(tlTs),(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
signicant 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 ltering and sampling
at R, in case of a collision of size k(assuming, without loss
of generality, the rst 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 ltering 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 identication 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 xsx1x2...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
2i1)
m=1 e|r(tl)do(2i1,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 coef-
cients of the ktransmitters at time tl(which change at each
sample due to frequency offsets), while do(2i1,m)and
de(2i, m)are column vectors containing one (the m-th) of the
k
2i1or 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 eld which is appended to the
message before channel coding, called CRC eld.Asthe
CRC operations are done in GF (2) and by the linearity of
the channel encoder, the CRC eld 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)
iTstϕ(n)
i)
+βir(t)
k
l=1
bl(t)ˆ
A(n)
lej(2π
Δν(n)
lTstϕ(n)
l),(6)
Maximization step -fori=1,...,k calculate
min
A,Δν
Npre
t=1 bi(tp(n)
i(t)Aej(2πΔνTst+ϕ)
2,(7)
where pi(t)is the preamble of burst iafter the matched lter,
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
109
108
107
106
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[|
ΔνΔν|2].Esis the average energy per transmitted symbol for each
node. The modied 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
104
103
102
101
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]2.Esis the average energy per transmitted symbol for each node.
510 15
104
103
102
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
AA|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 gure 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 rst symbol of the bursts that rst arrives at R.We
dene the relative delay (RD) ΔTiof node ias the temporal
distance between the peak value of the rst pulse of burst i
and T. In other words, the burst which arrives rst 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
0ΔTmax 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 lter in case of three colliding
bursts with no timing offsets, i.e., ΔT1T2T3=0. The transmitted
signals after the matched lter 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
gure, 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 lter is:
y(t)=
k
i=1
si(t)+w(t),(8)
where,
si(t)=Ai
N
l=1
bi(l)g(tlTsΔTi)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 lter 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(tllTsΔTi)ej(2πΔνitl+ϕi),(11)
p(t)being the raised cosine pulse, is the convolution
operator and n(t)is the noise process after ltering 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 rst 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 rst symbol of each
of the users, weighted by the relative channel coefcient, 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 lter 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 lter 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 lter in case of three colliding
bursts with timing offsets ΔT1=0,ΔT2=Ts/6and ΔT3=Ts/4.
The transmitted signals after the matched lter 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 gure, 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 gure, 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 kLLRsispassedtothe
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
rTm).(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 simplied 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 tTi, then the
channel coefcient of burst qthat is used in the LLR is:
heq
q(t)=Aqej(2πΔνqTsΔTi+ϕq)pTiΔTq),(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 ltering the received signal using a lter 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
104
103
102
101
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 signicantly 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
103
102
101
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
signicantly 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.
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