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Seek and Decode: Random Multiple Access with Multiuser Detection and Physical-Layer Network Coding

Authors:

Abstract

We present a novel random multiple access scheme that combines joint multiuser detection (MUD) with physical-layer network coding (PLNC) over extended Galois fields (EGF). We derive an analytical bound to the throughput at the system level and present simulation results for the decoding at the physical level in both fast fading and block fading channels. We adopt a cross layer approach in which a non-binary joint multiuser decoder is used in combination with PLNC at slot level, while the use of EGF increases the system diversity at frame level. The results we present are encouraging and suggest that the combination of these two interference management techniques can significantly enhance the performance of random multiple access systems.
Seek and Decode: Random Multiple Access with
Multiuser Detection and Physical-Layer Network
Coding
Giuseppe Cocco, Stephan Pfletschinger
German Aerospace Center - DLR
Oberpfaffenhofen, D-82234, Wessling, Germany
Centre Tecnol`
ogic de Telecomunicacions de Catalunya – CTTC
Parc Mediterrani de la Tecnologia, Av. Carl Friedrich Gauss 7 08860, Castelldefels, Spain
giuseppe.cocco@dlr.de, stephan.pfletschinger@cttc.es
Abstract—We present a novel random multiple access scheme
that combines joint multiuser detection (MUD) with physical-
layer network coding (PLNC) over extended Galois fields (EGF).
We derive an analytical bound on the throughput at the system
level and present simulation results for the decoding at the
physical level in both fast fading and block fading channels.
We adopt a cross layer approach in which a non-binary joint
multiuser decoder is used in combination with PLNC at slot level,
while the use of EGF aims at increasing the system diversity at
frame level. The results we present are encouraging and suggest
that the combination of these two interference management
techniques can significantly enhance the performance of random
multiple access systems.
I. INTRODUCTION
Random access systems (RAS) are at the same time an
opportunity and a challenge. Opportunity because they do not
require (or require little) coordination among the transmitters,
which, among other advantages, makes it possible to live
together with large delays, typical, for instance, of satellite
communication networks. However, if on the one hand the
lack of coordination can be seen as an asset, on the other hand
it brings about the issue of signals from different transmitters
interfering at the receiver. The problem of collisions in RAS
has been tackled in different ways like exploiting the differ-
ence in the power of the received signals [1] or the application
of multiuser detection (MUD) methods as in the code-division
multiple access (CDMA) systems [2]. Multi-packet reception,
i.e., the capability for the receiver to decode more than one
packet from a collision, has been and still is an active research
field. In [3], an overview of the main multiuser detection
techniques is presented. The impact of multi-packets reception
capability in slotted ALOHA systems has been studied in
[4]. Another approach proposed in the literature consists in
having each transmitter sending multiple replicas of the same
packet within a frame. The receiver tries to decode the packets
that do not experience collision [5] or subtracts the decoded
packets from the slots where their replicas are [6][7]. The
scheme proposed in [6] has been enhanced in [8] by inducing
fluctuations in the received power in order to allow iterative
hard interference cancelation within single slots. Recently the
possibility of decoding functions of colliding signals has been
studied in [9], where the linearity of error correction codes
has been applied for the decoding of the bitwise XOR of
the colliding signals in the two-way relay channel (TWRC)
under the assumption of equal codes at both end nodes. This
approach is one of the possible implementations of the wider
concept known as physical-layer network coding (PLNC). The
performance limits for the decoding of the sum of colliding
signals have been studied from an information theoretical
perspective and assuming lattice codes in [10][11]. Most part
of the literature on PLNC focuses on the TWRC. In [12]
a generalized sum-product algorithm has been proposed for
PLNC in the MAC phase of the TWRC. In [13] a quaternary
decoding approach for the MAC phase of the two-way relay
channel has been proposed, showing that there is an advantage
in obtaining the bitwise sum by combining the previously
estimated individual messages rather than directly decoding
the sum from the analog signal. In [14] it has been proposed
to apply PLNC in slotted random access systems by decoding
the bitwise XOR of all colliding signals within a slot and then
trying to recover all transmitted packets within a frame using
matrix manipulations in GF (2). In [15] and [16] an enhanced
scheme based on PLNC over extended Galois fields has been
proposed, showing an increased system diversity.
In the present paper we propose a random multiple ac-
cess scheme for symbol-synchronous slotted ALOHA systems
named Seek and Decode (S&D) in which the transmitters
pre-encode their information messages multiplying them by
a random coefficient in an extended Galois field while the
receiver tries to decode any linear combination in GF (2)
from the set of colliding bursts within each slot. The receiver
employs a hybrid between a joint multiuser decoder and a
PLNC decoder. Once the whole frame has been processed at
the physical layer, the receiver uses the whole set of linear
combinations available to retrieve all messages transmitted
within the frame by using matrix manipulation techniques over
the same extended Galois field used in the pre-coding stage.
The use of an extended Galois field in the pre-coding stage
increases system diversity. We derive an upper bound on the
throughput at the system level and present numerical results
for the number of innovative messages decoded within a slot in
a block fading channel. FER curves for the fast fading channel
are also presented.
II. SYSTEM MODEL
Let us consider a random multiple access network with a
population of Mtransmitting terminals T1,...,TM, and one
receiver R. In the rest of the paper we will use interchangeably
the terms “transmitting node”, “terminal node” and “transmit-
ter”. Time is divided into slots. We define a packet uas a
block of RN information bits. Each terminal generates packets
according to a Poisson process of intensity G
Mpackets per
slot, where Gis the overall load offered to the network in
packets per slot. Each time a packet ui=[ui,1,...,u
i,RN ]
is generated at terminal Ti, it is channel encoded using an
encoder of rate Rcreating a codeword ci=[ci,1,...,c
i,N ]of
Nsymbols. The same channel code is used by all transmitting
nodes. The codeword ciis then mapped to a binary phase-shift
keying (BPSK)-modulated burst xiand transmitted over the
channel. We consider BPSK modulation for simplicity, but
other kinds of modulations can also be used. We assume that
the burst duration is approximately equal to that of a slot.
Transmissions are organized in frames of Sslots each. We
further assume that the transmitters are synchronized such
that all signals transmitted within a slot add up with symbol
synchronism at the receiver. At the receiver side, Rtries to
decode as many linearly independent messages as possible
by applying both joint multi-user detection and physical layer
network coding. In order to increase system diversity at the
frame level, we assume that a pre-coding, such as the one in
[16], is applied by each of the terminals before the channel
encoding. The main innovation in the present work is in the
type of decoder used and in the fact that the receiver tries
to obtain all possible linear combinations in GF (2) from the
signals colliding within a slot. Such linear combinations are
then used by the receiver to recover the whole frame, treating
the set of decoded linear combinations as a system of equations
in GF (2n).
III. RANDOM ACCESS WITH MUD AND PLNC
In the present section we describe the proposed random
access scheme named Seek and Decode (S&D). The trans-
mitter side is the same as in [16]. The main innovation is
in the decoding process at both slot level and frame level.
The receiver processes one slot at a time in the analog
domain trying to decode either single messages or some linear
combination of the colliding signals. We briefly recall the
operations at the transmitter side presented in [16] and then
move to the description of the receiver side.
A. Transmitter Side
Each message is transmitted more than once within a frame,
i.e., several replicas of the same message (bursts) are trans-
mitted. Assume that node ihas a message uito deliver to R
during a given frame, i.e., node Tiis an active terminal. Before
each transmission, node ipre-encodes uias depicted in Fig. 1.
The message to be transmitted is divided into sub-blocks.
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Fig. 1. Pre-coding, channel coding and modulation scheme at the transmitter
side. Pre-coding consists in dividing the message into blocks of nbits
(indicated as ur
iin the figure) and multiply each of these blocks by the same
coefficient randomly chosen in GF (2n). The sub index jindicates the slot
within a frame in which the replica of message uiis transmitted. A different
coefficient αij is used for each replica.
Each sub-block is multiplied by a coefficient αij GF (2n).
Coefficient αij,j∈{1,...,S}is chosen at random in each
time slot jwhile it is fixed for all sub-blocks within a message.
Note that the pre-coding does not have any impact on the
decoding process at the physical layer. The multiplication of
uiby αij is needed to increase diversity at the frame level and
does not modify the number of information bits transmitted.
After the multiplication, the message is channel-encoded, a
header is attached and the modulation takes place. The header
can be generated using a pseudonoise sequence generator such
as the ones used in CDMA. In practice the coefficients αij can
be generated using a pseudo-random number generator. In a
given frame the active node chooses a different seed and uses
as many outputs of the generator as the number of replicas
transmitted within the frame. Each seed is associated to a
certain header, which is assumed to be detected by the receiver
using the cross-correlation properties of the header 1. The same
header is used within a given frame by an active node. In this
way the receiver can detect in which slots a certain node is
transmitting and derive the coefficients used in the different
replicas from the header. The header is also used to perform
the channel estimation of each of the transmitters. A more
detailed analysis of the issues related to header detection and
channel estimation can be found in [16] and [17].
B. Receiver Side
The main innovation of the proposed scheme with respect to
previous works is at the receiver side. In the literature of ran-
dom access, and up to our knowledge, when a receiver receives
two or more interfering signal, it can either use some kind
of interference cancelation or, as in physical layer network
1Note that other signatures can also be used by the nodes to allow Rfor
the identification of the transmitters.
coding, try to decode a function of the colliding messages 2.
Most of the multiuser detection techniques found in literature
can be categorized as parallel (PIC) or serial (SIC). Often
such methods are iterative and alternate a detection phase
to an estimation phase. In the proposed scheme the receiver
applies a joint decoder which tries to recover simultaneously
all messages involved in the collision. An FFT-based belief
propagation decoder over the vectorial combination of all
message bits, which is described in detail in the companion
paper [19], has been adopted. The decoder jointly estimates all
the single messages and then calculates the bitwise XOR of
any subset of the estimated messages. It is important to notice
that, as shown in [13], the sum in GF (2) of a set of estimated
messages can be correct even if the estimated messages taken
individually contain errors. A cyclic redundancy check (CRC)
can be used for error detection. Note that, due to the linearity
of the code, the XOR of the CRCs relative to a set of messages
is a valid CRC for the XOR of the messages in the set. Here
we assume ideal error detection at the receiver for ease of
exposition.
Given a slot with a collision of size k, the receiver tries to
decode kindependent linear combinations in GF (2) of the
colliding signals. One possible way to proceed, although not
necessarily the optimal one, can be the following. For each
slot with a collision of size k, the decoder tries to decode
single messages. If less than kmessages are decoded correctly,
the decoder tries to decode the sum in GF (2) of pairs of
messages (there are k
2possible combinations) stopping when
a total of klinearly independent combinations are decoded. If
still less than klinear combinations have been decoded, sums
of three messages are considered. The process goes on until
either enough linear combinations have been decoded or all
possible combinations have been exhausted. The total number
of linear combinations that the decoder can try to recover is
k
i=1 k
i=2
k1.
C. Example
In the following we illustrate the S&D scheme with a toy
example. Let us consider a frame with S=2slots and four
active nodes. Let us assume that nodes 1and 2transmit in
both slots, each time choosing at random their pre-coding
coefficients. Node 3only transmits in the first slot while node
4transmits only in the second, as illustrated in Fig. 2. Let us
assume that the S&D decoder is able to output only two linear
combinations in each of the two slots as shown in the picture.
The receiver tries, then, to recover all information messages
u1...,u4. The decoding is possible if the coefficient matrix
2In [18] a practical implementation of a system using both PNC and MUD
in the multiple access channel of a WLAN has been presented. Unlike in the
present work, in [18] only the case of two colliding signals is considered, a
relaying setup is assumed and a different multi-user detector (in which joint
detection is performed but not joint decoding) is adopted.
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Fig. 2. Example of decoding at the physical layer in S&D with a two-slots
frame and four active terminals. Nodes 1and 2transmit in both slots, each
time choosing at random their pre-coding coefficients. Node 3only transmits
in the first slot while node 4transmits only in the second.
Ain GF (2n)(shown below) has full rank.
A=
00α2,1α1,1
0α3,10α1,1
00α2,2α1,2
α4,20α2,20
.
Note that matrix Ais rank deficient if coefficients are chosen
in GF (2) (i.e., all coefficients shown in the matrix above are
equal to 1), while it can be full rank in some extended Galois
field. For instance, in GF (4) the matrix
A=
0011
0201
0023
1020
is full rank. If, by choosing the coefficients at random, the
above matrix is obtained it would be possible to decode the
whole frame. The probability of obtaining a full rank matrix
increases with the field size. Finally, we note that in the
example the average number of packets decoded per slot is
2.
IV. THROUGHPUT ANALYSIS
In the present section we derive an upper bound to the
system throughput, defined as the average number of de-
coded messages per time slot, for the proposed scheme. The
throughput depends on the repetition strategy chosen. For
mathematical tractability we assume a general scheme in
which each active node transmits in each slot with probability
p, fixed for all nodes.
A. Upper Bound to Decoding Probability
In our simulation results we observed that the probability
of correct decoding for the sum of a subset of messages with
cardinality ifrom a collision of size k,0<ik, is a function
of both iand kin both fast fading and block fading channels.
We define:
Pr{decode sum of imessages form collision of k}pk,i.
pk,i can be upper bounded as follows:
pk,i ¨pk,i ˜pk,i,
where
¨pk,i max
Sk,i
pk,i,
Sk,i being one of the k
isubsets of imessages among the k,
while
˜pkmax
i¨pk,i.
We found through simulations that pk,i is lower than or equal
to the probability to decode the sum of the istrongest signals
among the k. Thus, ˜pkis the maximum across all subset sizes
i,i∈{1,...,k}, of the probability to decode the sum of
the istrongest signals. ˜pkis used in the following for the
derivation of the upper bound on the throughput. By applying
both joint MUD and PLNC, the receiver can obtain up to
ηk2k1different linear combinations in a slot with a
collision of size k. At most kof the decoded combinations
are linearly independent. For ease of calculation we assume
in the following that the decodability of a given combination
is independent of the decoding of any other within the same
slot3. The number of combinations (linearly independent or
not) decoded in a slot is a random variable. We indicate such
variable with . Let us now indicate with kthe number of
combinations decoded in a slot when the collision size is k.
kis a Binomial random variable with parameters ˜pk,i and ηk,
i.e., kB(ηk,˜pk). The mean and variance of kare ηk˜pk
and ηk˜pk(1˜pk), respectively. The mean value of ,E[]=,
is then:
=
Ntx
k=1 Ntx
kpk(1 p)Ntxkηk˜pk,(1)
while the mean squared value of is:
E[2]=
Ntx
k=1 Ntx
kpk(1 p)Ntxk×
×ηk˜pk(1 ˜pk)+(ηk˜pk)2.(2)
Finally, the variance of σ
2
can be calculated using expres-
sions (1) and (2) as σ2
=E[2]2. The total number of
combinations decoded in the whole frame is a random variable
given by the sum of the numbers of combinations decoded
in all slots, which are i.i.d. random variables and for which
we just calculated the mean and the variance. Practical values
for Scan be on the order of 100, which is large enough to
approximate the sum of Si.i.d. random variables as a Gaussian
3In general this is only an approximation, since giving the correct decoding
of a subset of individual messages, any combination of such messages can
also be decoded. However, it can happen that the single messages can not be
decoded while the sum can (e.g., this is true for certain code rates and if two
signals have the same channel amplitude as shown in [13].)
variable having mean Sand variance . From expressions
(1) and (2) it can be seen that the mean and the variance of
depend on the number of active terminals in the frame. For
this we indicate with (Ntx)fr and σ(Ntx )2
the mean and
the variance of , respectively. As mentioned in Section II
we assume Poisson arrivals with an overall offered load of G
packets per slot. An upper bound on the normalized throughput
can be calculated by assuming p=2
n1,nbeing the size of
the Galois field of the coefficients used in the pre-coding step,
and assuming that all combinations decoded within a frame
are obtained using independently drawn coefficients for each
message in each equation4.Ifnis large, the probability pNtx
+
to decode a number of combinations larger than or equal to
the number of active nodes within a frame is given by (3)
pNtx
+=
Ntx=1
(GS)Ntx eGS
Ntx!
S(2Ntx 1)
m=Ntx
e(m(m)fr)2
2σ(m)2
2πσ(m)2
.(3)
Using (3) we can obtain an upper bound on the normalized
system throughput ΦUB when a large field size is used. The
expression for ΦUB is given by Eqn. 4 at the top of next page.
V. N UMERICAL RESULTS
A. Joint Decoding in the Fast Fading Channel
For a first evaluation of the joint decoding approach, we
considered the simultaneous decoding of k∈{2,3,...,8}
packets in a fast Rayleigh fading symmetric multiple-access
channel, given by
yn=
k
i=1
hi,n ·xi,n +wn,w
n∼N(0,1).(5)
The fading coefficients hi,n are i.i.d. Rayleigh distributed
with average signal-to-noise ratio SNR = E[h2
i,n]and the
decoder employs joint decoding of all kmessages as described
in [19]. Due to the fast fading and the symmetric channel,
the ktransmitted packets experience approximately the same
channel quality and we consider the word error rate of the
combined messages. In other words, we count a word error if
any of the kmessages is not correctly decoded. Fig. 3 shows
the simulation results for the CCSDS LDPC code [20] of rate
R=0.4and message length RN = 1024 bits. In Fig. 3
we can see that the simultaneous decoding of several packets
is possible and requires only a moderate SNR increase for a
growing collision size.
While these results verify the functioning of the joint
decoding approach, for a practical random access scheme the
assumption of fast fading is not realistic and we therefore
apply block fading for the following throughput evaluations.
4Note that in practice each message has the same coefficient for all
combinations within a given slot.
ΦUB =1
S
Ntx=1
Ntx (GS)Ntx eGS
Ntx!
S2Ntx
1
m=Ntx
e
(m(m)fr)2
2σ(m)2
2πσ(m)2
=G
Ntx=0
(GS)Ntx eGS
Ntx!
S2Ntx
1
m=Ntx
e
(m(m)fr)2
2σ(m)2
2πσ(m)2
.(4)
2 4 6 8 10 12 14 16 18 20
104
103
102
101
100
k=2 k=3 k=4 k=5 k=6 k=7 k=8
SNR [dB]
WER
Fig. 3. Word error rate for joint decoding in symmetric fast Rayleigh fading
channel. An error occurs when one of the kmessages can not be correctly
decoded.
B. Seek & Decode in the Block Fading Channel
As described in the previous sections, in the S&D scheme
the receiver uses a joint decoder to recover as many linearly
independent combinations of colliding signal as possible in
each slot and then, exploiting the pre-coding of the transmitted
messages at the receiver side, try to decode the whole frame
using standard matrix manipulations techniques. The linearly
independent combinations may be either single messages or
sums in GF (2) of any number of messages from 2to k,k
being the collision size in the slot. One could guess that it
might not be necessary to decode sums of messages since the
joint decoder is highly efficient and thus all messages could
be directly decoded from each slot. However, our simulation
results show that for a range of SNR and collision sizes of
practical interest, decoding sums of messages does increase
significantly the number of linearly independent combinations
decoded in a slot. This is shown in Fig. 4, where the average
number of innovative packets decoded in a slot is plotted
against the average SNR for the proposed seek and decode
method (S&D in the figure) and for a plain joint decoding
scheme (JD in the figure), in which only single messages are
decoded. For these simulations, we assumed block Rayleigh
fading where the fading coefficients are constant during a
packet transmission. A packet is said to be innovative if it can
not be obtained as a linear combination in GF (2) of previously
decoded packets. Different curves for different collision sizes
kare shown. It can be seen how for larger kand mid-low SNR
values the gain of S&D with respect to JD is significant (more
than 25% gain at about 11 dB for k=6). The gain derives
from the fact that, in a non negligible number of cases, it is
possible do decode correctly the sum of two or more messages
from a collision even if the single messages can not be decoded
5 0 5 10 15 20 25 30
0
1
2
3
4
5
6
k=1
k=2
k=3
k=4
k=5
k=6
SNR [dB]
Innovative packets per slot
S&D
JD
Fig. 4. Average number of innovative packets decoded in a slot plotted
against the average SNR for the proposed seek and decode, S&D, and for
joint decoding, JD. Different curves for different collision sizes kare shown.
It can be seen how for larger kand mid-low SNR values the gain of S&D
with respect to JD is significant (gain of around 20% at 10 dB for k=4).
with a joint decoder. Note also that, once the decoding at the
physical layer is finished, the receiver is left with Ssets of
equations. Each of these sets derives from the decoding at the
physical layer of each slot. Note also that, even if some (or all)
of the sets of equations have not a full rank associated matrix,
(i.e., not all of the colliding signals within a block can be
decoded), it may still be possible to recover all the messages
transmitted in a frame by applying matrix manipulations over
the equivalent associated matrix in GF (2n).
VI. CONCLUSIONS
We proposed a new random multiple access scheme for
symbol-synchronous slotted aloha random access systems.
Each node transmits several replicas of the same message
within a frame after a pre-multiplication by a (pseudo)-
randomly chosen coefficient in GF (2n). The receiver tries
to decode as many linear combination in GF (2) of signals
colliding in each slot as possible and then tries to recover all
the messages transmitted within a frame treating the linear
combinations decoded in the whole frame as a single system
of equations in GF (2n). We presented analytical results for
the throughput at system level and simulation results for the
decoding process at the physical level. As future work we plan
to optimize the multiple access scheme taking into account the
decoder performance, which is a function of the collision size
and the specific linear combination within a collision, with the
aim of maximizing the system throughput and minimizing the
packet error rate. At the physical level we plan to consider
higher order modulations and different decoding approaches.
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Synchronization and Channel Coding, August 2011, CCSDS 131.0-B-
2.
... On the PHY layer, previous investigations focused on applying Compressive Sensing (CS) in multi-user detection contexts by exploiting the capabilities of sparse signal processing [2], [3]. Furthermore, joint MAC-and PHY-layer protocol design has been identified as one important enabler to support MMC [4], [5], [9] and [4] has shown that the combination of Coded Random Access (CRA) and Compressed Sensing Multi-User Detection (CS-MUD) is a promising joint protocol to enhance system performance. Therefore, CS-MUD not only is known as an efficient PHY layer technique for sparse Multi-User Detection (MUD) problem [1]- [3] but also serves as promising PHY layer enabler in joint MAC-and PHY-layer protocol design [4], [5]. ...
... With (7) updated by (9), the probability term is bounded by 1/β due to γ > 0. In the case, the domain of γ has changed by the probability bounding term, which yields the effective noise variance σ 2 eff and optimal γ eff for the LASSO algorithm considering a discrete BPSK alphabet. ...
... The conventional brute-force approach, where the AP uses all decoders to decode different linear combinations of packets from the superimposed signals, leads to high system complexity when N is large. In addition to NCMA, the same decoding complexity issue applies to all NOMA systems with PNC, including random multiple access systems [10] and coordinated multiple access systems [11] in the literature. ...
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This paper presents a low-complexity physical-layer network coding (PNC) enabled non-orthogonal multiple access (NOMA) system with the help of deep neural networks (DNN). NOMA allows multiple users to send packets simultaneously to a common access point (AP) using the same frequency band. In PNC-enabled NOMA systems, the AP decodes not only individual packets of different users by conventional multiuser decoding (MUD) techniques, but also different linear combinations of individual packets by PNC decoding, referred to as PNC packets. Prior works showed that the decoded PNC packets could significantly improve the system throughput. However, when the number of simultaneously transmitting users increases, the number of possible PNC packets also increases exponentially, leading to high decoding complexity if the AP tries to blindly decode all possibilities. Therefore, this paper exploits DNNs to reduce the decoding complexity. Specifically, we find that the decoding results of different linear combinations are mainly affected by the relative phase offsets among the wireless signals of different users. Hence, we can use a DNN to learn the relationships between the relative phase offsets and the decoding results. Since our DNN can learn the decoding patterns, i.e., which linear combinations are more likely to be decoded given the same relative phase offsets, the AP can attempt to decode only a subset of all the linear combinations, thus reducing the decoding complexity. Experimental results show that our DNN-assisted decoding scheme reduces the decoding complexity by more than 30% compared with the traditional brute-force approach, while maintaining almost the same throughput performance.
... Beyond SIC, another promising strategy, which is inspired by physical-layer network coding (PLNC), consists in encoding and decoding linear combinations of packets [14]- [17]. By collecting enough linearly independent packet combinations within a contention period (frame), the receiver might be able to resolve all individual packets. ...
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Hinging on ideas from physical-layer network coding, some promising proposals of coded random access systems seek to improve system performance (while preserving low complexity) by means of packet repetitions and decoding of linear combinations of colliding packets, whenever the decoding of individual packets fails. The resulting linear combinations are then temporarily stored in the hope of gathering enough linearly independent combinations so as to eventually recover all individual packets through the resolution of a linear system at the end of the contention frame. However, it is unclear which among the numerous linear combinations---whose number grows exponentially with the degree of collision---will have low probability of decoding error. Since no analytical framework exists to determine which combinations are easiest to decode, this makes the case for a machine learning algorithm to assist the receiver in deciding which linear combinations to target. For this purpose, we train neural networks that approximate the error probability for every possible linear combination based on the estimated channel gains and demonstrate the effectiveness of our approach by numerical simulations.
... The solution falls under the category of coded random access [36], where features of channel coding are exploited both at the slot and frame level. In particular, the scheme partially presented in [37], [38] is extended for massive access, with emphasis on the transmission of short packets. The proposed scheme assumes a minimum coordination that ensures packet synchronization. ...
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... where R, λ h , and F Proof: The theorem follows from Lemma 3 and Lemma 4 by considering p i = F s (F u ( p i−1 )), where F u (·) and F s (·) are given by Eq. (26) and Eq. (23), respectively. ...
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