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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.

QELWV QELWV

&KDQQHO

&RGLQJ

0RGXODWLRQ

QELWV

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

k−1.

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.

/LQHDUHTXDWLRQVLQ

6'

GHFRGHU

6'

GHFRGHU

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<i≤k, 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

ηk2k−1different 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., k∼B(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)Ntx−kηk˜pk,(1)

while the mean squared value of is:

E[2]=

Ntx

k=1 Ntx

kpk(1 −p)Ntx−k×

×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 Sσ. 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

n−1,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 e−GS

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 e−GS

Ntx!

S2Ntx

−1

m=Ntx

e

−(m−(m)fr)2

2σ(m)2

2πσ(m)2

=G

∞

Ntx=0

(GS)Ntx e−GS

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

10−4

10−3

10−2

10−1

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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