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1

Seek and Decode: Random Access with

Physical-Layer Network Coding and Multiuser

Detection

Giuseppe Cocco‡, Stephan Pﬂetschinger‡∗† and Monica Navarro†

‡German Aerospace Center – DLR

∗Offenburg University of Applied Sciences

†Centre Tecnol`

ogic de Telecomunicacions de Catalunya (CTTC)

mailto: giuseppe.cocco@dlr.de

Abstract

We present a novel scheme for Slotted ALOHA random access systems (RAS) that combines

physical-layer network coding (PLNC) with multiuser detection (MUD). PLNC and MUD are applied

jointly at the physical (PHY) layer in order to extract any linear combination of messages experiencing

a collision within a slot. The set of combinations extracted from a whole frame is then processed by the

receiver to recover the original packets. A simple pre-coding stage at the transmitting terminals allows

the receiver to further decrease the packet loss rate. We present results for the decoding at the PHY layer

as well as several performance measures at frame level, namely throughput, packet loss rate and energy

efﬁciency. The results we present are promising and suggest that a cross layer approach leveraging

on the joint use of PLNC and MUD can signiﬁcantly improve the performance of RA systems in the

presence of slow fading.

I. INTRODUCTION

Random access systems (RAS) can be regarded as an opportunity and a challenge at the same

time. On the one side RAS require little coordination among the transmitters. This, among other

Part of the results presented in this article have been presented at ICC 2014 [1] and NetCod 2014 [2] conferences.

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advantages, makes it possible to live together with large delays that are typical, for instance, of

satellite communication networks. As a drawback, the lack of coordination brings about the issue

of signals from different transmitters interfering at the receiver. So far different ways of tackling

the problem of collisions in RAS have been proposed. These include exploiting the difference

in the power of the received signals [3] or applying multiuser detection (MUD) methods as in

code-division multiple-access (CDMA) systems [4]. 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 [5] an overview of the main MUD techniques is presented. The impact of

multi-packets reception in Slotted ALOHA systems has been studied in [6]. 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 as proposed in [7] or, once such clean packets have been decoded, it subtracts the

decoded packets from the slots where their replicas are [8] [9]. Recently, the possibility of

decoding functions of colliding signals has been studied in [10] and [11]. In these works the

linearity of error correction codes has been applied in the two-way relay channel (TWRC) to

decode the XOR of messages experiencing a collision. Starting from the sum of the physical

signals and assuming the same channel code at both end nodes is used, the corresponding XOR

is calculated and exploited, through an adequate MAC protocol, as side information to recover

the single messages. This approach is one of the possible implementations of the wider concept

of 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 [12] [13]. Most of the literature about PLNC focuses on the TWRC. In [14]

[15] a quaternary decoding approach for the MAC phase of the two-way relay channel has

been proposed, showing that there is an advantage in computing the XOR by combining the

previously estimated individual messages, rather than directly decoding the sum from the analog

signal. In [16] PLNC has been applied to random access systems by decoding the XOR of all

colliding signals within a slot and then trying to recover all transmitted packets within a frame

using matrix manipulations in F2. In [17] and [18], an enhanced scheme based on PLNC over

extension ﬁelds has been proposed. An information theoretical analysis of the performance of

physical-layer network coding in random access systems has been presented in [19]. Recent

variants of coded random access schemes are presented in [20], focusing on MAC aspects and

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their asymptotic performance. Details on the theoretical analysis of the different proposals based

on the coded Slotted ALOHA paradigm (with the respective frame and frameless approaches)

can be found in the references therein, together with discussions on practical implementation

aspects. In [21] a theoretical analysis of coded Slotted ALOHA systems is presented. In [22]

and [23] a practical implementation of a system that makes use of both PLNC and MUD in the

multiple-access channel of a wireless local area network is presented. Speciﬁcally, in [23] the

case of two colliding signals is considered, a relaying setup is assumed and a joint detection

(but not joint decoding) is performed. Practical solutions for the detection of active users and

the estimation of channel state information parameters are also being actively investigated [24].

Some of these techniques are based on compressed-sensing or sparse multi-user detection. Since

this is a common problem for coded random access solutions in general, we are not providing

a complete overview of recent work. Some examples and related bibliography can be found in

[25] [26].

In the present paper we propose two novel schemes to enhance throughput and packet loss

rate (PLR) of Slotted ALOHA networks that leverage on a combination of PLNC and MUD. In

the proposed schemes each information message undergoes a pre-coding stage at the transmitter

before the channel encoding. The pre-coding consists in a simple multiplication by a coefﬁcient

drawn at random from an extension ﬁeld. The receiver tries to decode at the physical (PHY) layer

any linear combination in F2from the set of colliding bursts within each slot. Once the whole

frame has been processed at the PHY layer, the receiver uses the set of linear combinations

available to retrieve all messages transmitted within the frame by using matrix manipulation

techniques over the same extension ﬁeld of the pre-coding stage. The use of an extension ﬁeld

in the pre-coding stage decreases the PLR of the system. At the PHY layer the receiver employs

a hybrid between a PLNC decoder and a MUD. Two different MUD schemes are considered in

combination with PLNC. One is a joint decoder (JD), in which all signals are decoded jointly

1. The other MUD technique we combine with PLNC is successive interference cancellation

(SIC). We present numerical results for the number of innovative (i.e., linearly independent)

1This differs signiﬁcantly from a parallel interference cancellation scheme (PIC), since in this last one several decoders are

employed in parallel estimating a different message each, while in a JD just one decoder is used, which decodes jointly all of

the messages.

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messages decoded within a slot as well as for throughput, packet loss rate and energy efﬁciency

in a framed slotted ALOHA-like scenario. Our results show that, unlike the scheme presented in

[18], the joint use of PLNC and joint decoding is robust against slow fading, which characterizes

many scenarios of practical relevance.

The rest of the paper is organized as follows. In Section II we introduce the system model.

In Section III the proposed approach is described while in Section IV we focus on the different

decoding alternatives at the PHY layer. Section V contains the numerical results, while the

conclusions are presented in Section VI.

II. SYSTEM MODEL

Let us consider a random multiple-access network with an inﬁnite population of terminals and

one receiver Rx. Time is divided into slots. Transmissions are organized in frames of Sslots

each. We deﬁne a packet uas a block of RN information bits. The user population generates

an aggregated offered trafﬁc which is modelled as a Poisson process of intensity Gpackets per

slot. Each time a packet ui= [ui,1,...,ui,RN ]is generated at terminal Ti, it is channel encoded

using an encoder of rate R, thus creating a codeword ci= [ci,1,...,ci,N ]of Nbits. The same

channel code is used by all transmitting terminals. 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. It is worth

noting that the speciﬁc modulation considered can have a signiﬁcant impact on the packet loss

rate performance of PLNC. In [27] and [28] it was shown that ﬁnding the modulation which

minimizes the message error rate is not trivial even for an uncoded system and collisions of size

2. How to optimally design the modulation constellation in a coded system and for a generic

collision size in case a joint PLNC and MUD receiver is used is a challenging open problem

which is out of the scope of the present paper. We also point out that the schemes proposed in

the following rely on channel codes and modulations already in use in commercial standards and

have the advantage of requiring little modiﬁcation at the transmitter side. Most of the additional

complexity if moved at the receiver which usually has less constraints in terms of computational

capabilities with respect to the user terminals.

We assume that the burst duration is approximately equal to that of a slot. Let us now consider

one of the Sslots of the frame. In case of a collision of Kpackets (namely, collision of size

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K) the n-th sample of the received signal can then be written as

yn=

K

X

k=1

hkxk,n +wn, wn∼ N (0,1) ,(1)

where the fading coefﬁcients are real-valued and follow a certain probability distribution with

E{|hk|2}=SNR,E{x}being the mean value of x. The fading coefﬁcients are estimated at the

receiver but are not known at the transmitters and are assumed to change in an independently

and identically distributed (i.i.d.) fashion across terminals and time slots. 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, Rx ﬁrst processes the frame at the

physical level one slot at a time. The PHY processing consists into applying a combination of

MUD and PLNC in order to decode as many linearly independent messages as possible. Once

the processing at the PHY level is completed, Rx applies a second step of decoding, which takes

place at the frame level. The linear combinations recovered from the PHY layer processing are

then treated as a system of equations in Fq,qbeing an extension ﬁeld of the kind q= 2nbc ,

nbc ∈N. The decoding process at frame level is detailed in Section III while the details of how

different combinations are extracted from the same collision are given in Section IV.

III. RANDOM ACCESS WITH PLNC AND MUD

In the present section we describe the proposed random access scheme named Seek and

Decode (S&D). The transmitter side is the same as in [18]. The main innovation is in the

decoding process at both slot level and frame level. We brieﬂy recall the operations at the

transmitter side presented in [18] and then move on to the description of the receiver side.

A. Transmitter Side

Each burst is transmitted more than once within a frame, i.e., several replicas of the same

burst are transmitted. Assume that terminal ihas a message uito deliver to Rx during a given

frame, i.e., terminal Tiis an active terminal in that frame. Before each transmission, terminal i

pre-encodes uias depicted in Fig. 1. The message to be transmitted is divided into groups of

nbc bits each. Each group of bits is mapped to a symbol in Fq,q= 2nbc , and then multiplied by

a coefﬁcient αi,j ∈Fq. The coefﬁcient αi,j,j∈ {1,...,S}, is chosen at random in each time

slot jwhile it is ﬁxed for all symbols within a message. Note that the pre-coding does not have

6

()ିଵ Gʅ

ߙ,א ,

ܰ ܴܰ ܰ

Fig. 1. Pre-coding, channel coding and modulation at the transmitter side. Pre-coding consists in mapping the message to a

vector in Fq,q= 2nbc , multiply each element of the vector by the same coefﬁcient αi,j randomly chosen in Fqand apply an

inverse mapping (Fq)−1from Fqto {0,1}. The sub index jindicates the slot within a frame in which the replica of message

uiis transmitted. A different coefﬁcient αi,j is used for each replica.

any impact on the decoding process at the PHY layer and requires little increase in complexity

with respect to a traditional scheme. The multiplication of uiby αi,j helps the decoding at

frame level, which will be described later in this section. After the multiplication, the message

is channel-encoded (block Gin Fig. 1), a header is attached and the modulation takes place

(block µin Fig. 1). Pseudo-noise sequences with good cross-correlation properties can be used in

the header in order to identify the user within the frame. Such identiﬁcation allows the receiver

to deduce the pre-coding coefﬁcients used by each transmitter as described in the following. The

coefﬁcients αi,j can be generated using a pseudo-random number generator. In a given frame the

active terminal chooses a seed for the generator and takes from it as many outputs as the number

of replicas to be transmitted. Each seed is associated to a certain header, which is detected by

the receiver using the cross-correlation properties of the header2. The same header is used within

a given frame by an active terminal. In this way the receiver can detect which slots a certain

terminal is transmitting in 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 [18], [29], [30].

2Other PHY layer signatures can also be used by the terminals to allow Rx to identify the transmitters. This is a subject

which has been extensively studied in literature and further discussion on this is out of the scope of the present work.

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B. Receiver Side: Decoding at Frame Level

According to the literature related to random access systems, when two or more signals

interfere at the receiver, this can either use some kind of interference cancelation or, as in

physical-layer network coding, try to decode a function of the colliding signals. Most of the

MUD techniques found in literature can be categorized as PIC or 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 [31], is adopted. The decoder jointly estimates all

the single messages and then calculates the XOR of any subset of the estimated messages. It is

important to notice that, as shown in [15], the sum in F2of 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. Thanks to the linearity of the channel 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 F2of the colliding signals. The total number of linear combinations that the decoder can

try to recover is PK

i=1 K

i= 2K−1. Assuming the receiver is able to reliably estimate the

random coefﬁcients and the identity of the transmitters in each slot through the packet headers

[18] [29], each decoded linear combination in F2can be interpreted at the receiver, according

to arithmetics of extension ﬁelds, as an equation in Fq,q= 2nbc . Stacking together all equations

the receiver ends up with a linear system having the form

ATU=b,(2)

where Ais the coefﬁcient matrix having Ntx rows and a number of columns that depends on

the number of combinations decoded at PHY layer, U= [u1,...,uNtx ]Tis a vector containing

the information messages transmitted by the Ntx active terminals in the frame, bis a vector

containing the output of the decoding at PHY layer and Tis the transpose operator.

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IV. DECODING AT SLOT LEVEL

In Section III we described the proposed scheme assuming a joint decoder is applied at the

PHY layer. As described in the previous section the joint decoder ﬁrst estimates all the single

messages involved in a certain collision. Afterwards, the S&D variant of the joint decoding is

applied, in which the sum in F2of any subset of the estimated messages is calculated. Since

in some cases the sum of a set of estimated messages can be correct even if the estimated

messages taken individually contain errors, the S&D approach increases the average number of

useful packets decoded from a collision with respect to a normal joint decoder.

This approach is only one of the many in which MUD can be combined with PLNC. In fact

other kinds of MUD can be adapted to the S&D scheme rather than joint decoding. Although

some of them may lose in terms of performance with respect to the joint decoding approach,

they can be attractive from a practical perspective for their lower complexity.

For completeness we recall that PLNC can be applied to recover the sum of all the collided

packets for a generic collision size directly to the received signal, i.e., without ﬁrst trying to

estimate the individual messages. We do not report here the details for a matter of space and

remaind the interested reader to [32] for further details.

In the present section we describe several alternative schemes while in Section V we numer-

ically compare their performance in terms of the number of innovative packets decoded from

a collision. Here we focus only on the decoding within a slot, while the performance at frame

level is assessed in Section V.

A. Separate Decoding

The simplest approach is to decode each packet separately, considering all other packets as

interference. As for all other schemes to follow, we assume channel state information (CSI) at

the receiver for all transmitting terminals as well as knowledge at the receiver of the transmit

alphabet, i.e. BPSK constellation 3. With this, and assuming Kbursts collide in a slot, we

can write the log-likelihood value (L-value) of user i, i ∈ {1,2,...,K}and symbol position

n, n ∈ {1,2,...,N}, as:

3A further simpliﬁcation would be to consider the interference as Gaussian noise, which would result in reduced performance

and is therefore not considered here.

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Li,n ,ln P[cki,n = 1 |yn]

P[ci,n = 0 |yn]= ln P[xi,n = 1 |yn]

P[xi,n =−1|yn].(3)

According to Eqn. (1), ynis a weighted sum of the n-th symbols of all Kcolliding signals.

Since the received symbol yndepends on all symbols, we need to marginalize over all other

users’ symbols. For this, we deﬁne the sets X(b)

i,x=µ(d) : d∈FK

2, di=bfor b∈F2,

with cardinalityX(b)

i= 2K−1. We can think of the variable das the vector of the coded bits

of all users at the same position, i.e. dn= [c1,n, c2,n ,...,cK,n]T. We obtain for the L-values

Li,n = ln Px∈X (1)

i

P[x|yn]

Px∈X (0)

i

P[x|yn]= ln Px∈X (1)

i

p(yn|x)

Px∈X (0)

i

p(yn|x)

= ln Px∈X (1)

i

exp −yn−hTx2

Px∈X (0)

i

exp −(yn−hTx)2

= jacln

x∈X (1)

in−yn−hTx2o

−jacln

x∈X (0)

in−yn−hTx2o

(4)

where jacln {x1,...,xn},ln Pn

j=1 exp (xj)denotes the Jacobian logarithm, which can be

computed recursively and for which computationally efﬁcient approximations exist [33]. These

L-values are input to a soft-input decoder, which typically is a Viterbi, a turbo or an LDPC

decoder.

B. Successive Interference Cancellation (SIC)

A straightforward and well-known extension of basic single-user decoding is SIC: if a packet

uk∗is successfully decoded, its corresponding codeword ck∗and symbol sequence xk∗are known

and can be subtracted from the received signal yn, creating a multiple-access channel (as deﬁned

in [34]) within a slot with K−1terminals. This process can be repeated until decoding of all

remaining packets fails. To avoid unneccessary computations, we can exploit the knowledge of

the instantaneous SNRs and order the users accordingly: let πbe a permutation of {1,2,...,K}

such that

hπ(1) ≥hπ(2) ≥ · · · ≥ hπ(K).(5)

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Then decoding starts with user π(1). Apart from reducing computational complexity, this ordering

is also useful to reduce the probability of undetected errors. To check the correct decoding of

a packet, usually an additional error detection code, e.g. a CRC, has to be introduced into each

message uk. Since there is a non-zero probability that an erroneous decoding is not detected,

the number of decoding attempts with low probability of success should be kept to a minimum.

C. Seek & Decode with Successive Interference Cancellation (S&D+SIC)

For a coded Slotted ALOHA system, a further decoding step after SIC is possible. Assume

that after the SIC procedure described above, K−K1packets have been correctly decoded, hence

leaving K1∈ {2,...,K}packets for which decoding failed. In this situation, the receiver can try

to decode a combined packet, which is given by the sum of two or more of the packets that have

not yet been decoded. In a typical SIC the decoding process would stop here. In the proposed

S&D approach, instead, the receiver can try to decode the sum of a subset of {1,2,...,K1},

e.g. given by K={k1, k2, . . . , kℓ} ⊂ {1,2,...,K1}. If the decoding is successful, such decoded

packet can be exploited as a side information in order to help the decoding of other packets

within the same collision. The way in which such side information is exploited resembles the

SIC process, even though the cancellation is not applied directly on the sampled signal. In

the following we detail such mechanism more in depth. Let us assume that no user in the set

{1,2,...,K1}could be decoded with the normal SIC. Then the receiver can try to decode the

sum of a subset of {1,2, . . . , K1}, e.g. given by K={k1, k2,...,kℓ} ⊂ {1,2,...,K1}. For this

subset we deﬁne the sets of constellation symbols for ℓ≥2as

X(b)

ℓ,(x=µ(d) : d∈Fℓ

2with

ℓ

X

i=1

di=b), b ∈F2,(6)

µ() being the mapping function from bits to constellation symbols, and obtain the corresponding

L-values as

LK

n= ln P

x∈X(1)

ℓ

exp −(yn−[hk1hk2···hkℓ]x)2

P

x∈X(0)

ℓ

exp −(yn−[hk1hk2···hkℓ]x)2.(7)

These L-values LK

1, LK

2, . . . , LK

Nare fed to the soft-input decoder, which, if successful, ﬁnds

the corresponding codeword Pk∈K ckor message Pk∈K uk. Note that the sum of messages or

codewords is deﬁned in the ﬁnite ﬁeld F2, which is the same as the bit-wise XOR. This concept

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of packet combining is closely related to inter-ﬂow network coding and it exploits the linearity

of the code, which can be seen by the relation

X

k∈K

ck=X

k∈K

ukG.(8)

For error detection, since CRC codes are also binary linear codes, the same CRC can be used.

For K1undecoded packets, there exist

K1

X

ℓ=2 K1

ℓ= 2K1−K1−1

combinations of two or more packets, for which a decoding attempt is possible from the L-values

deﬁned by (7). With this deﬁnition, note that the subsets X(b)

ℓonly depend on band on the

number of packets ℓbut not on their indices k1,...,kℓ. After successful decoding of a packet

sum, a subsequent idea is to re-apply interference cancellation with the packet combination.

This, however, is not directly possible since the combined codeword cK=Pk∈K ckdoes not

correspond to any received symbol sequence xkin (1) and the sum of codewords and symbol

sequences are taken over different ﬁelds, namely F2and R. However, knowledge of a combined

packet cKmight still be useful for another decoding attempt: the cardinality of the sets X(b)

ℓcan

be reduced by a factor of two by introducing the additional constraint of the known combined

packet. Then, the L-values can be recomputed and new decoding attempts (including ℓ= 1 for

individual packets) can be undertaken. This approach brings about a slight additional complexity

due to the constraint on the decoded combination. In this case, the sets X(b)

ℓwill additionally

depend on nand hence have to be computed for each coded bit.

It is interesting how such approach has strong similarities with [35]. In [35] the decoder ﬁrst

tries to decode linear combinations of a subset of the colliding messages, and then uses the

knowledge of such combination to help recovering others. In [35] as well as in the approach

just presented, the knowledge of the ﬁrst combination decoded can not be exploited by just

subtracting it from the received signal, since it does not contain a waveform corresponding to

the decoded combination. However, in both cases such side information can be exploited by the

decoder. Although there are signiﬁcant differences between the channel models in the two cases,

a joint study of the two models may lead to interesting results from a practical perspective. The

need for an in-depth analysis of the subject does not allow for an adequate assessment in the

present paper and is left as a promising matter of study for future works.

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D. Seek & Decode with Joint Decoding (S&D+JD)

From (1) we can observe that, for what concerns the detection, the received samples yn

depend on all coded bits ck,n at the same bit position but are independent of bits at other

positions. The optimum decoding approach is therefore to consider the vectorial symbols dn,

[c1,n, c2,n ,...,cK,n]Tjointly. This can be done with a joint decoder which operates on the vectors

dnor on an equivalent integer representation ¯

dnsuch that dn= bin( ¯

dn). The notation bin(b)

denotes the binary representation of the non-negative integer b. For LDPC and for convolutional

codes, such joint decoders are described in [31], [36]. The decoder input is given by the

probability vector

pn,

pn(0)

pn(1)

.

.

.

pn(2K−1)

∈R2K,(9)

where

pn(b),P[d= bin (b)|yn]∝p(yn|x=µ(bin(b))) ,(10)

for b= 0,1,...,2K−1. Let ¯

xb=µ(bin(b)), then

pn=α

exp −yn−hT¯

x02

exp −yn−hT¯

x12

.

.

.

exp −yn−hT¯

x2K−12

,(11)

where αis a scaling factor which is irrelevant for the decoding algorithm. The decoder output

is an estimate of all messages (or equivalenty of all codewords),

ˆ

Uslot =

ˆ

u1

ˆ

u2

.

.

.

ˆ

uK

.(12)

Note that ˆ

Uslot in Eqn. (12) refers to the packets transmitted within a slot, while ˆ

Uin (2)

refers to the packets transmitted in a whole frame. Making use of an error detecting code,

the receiver checks all possible packet combinations, i.e. all 2K−1non-empty subsets of

13

{ˆ

u1,ˆ

u2,...,ˆ

uK}and builds the binary matrix Aslot ∈F(2K−1)×K

2. Matrix Aslot is such that

its rows a= [a1, a2,...,aK]indicate the user indices which are contained in successfully

decoded combinations. For instance, if the combined packet c1+c3+c4is correctly decoded,

the corresponding row is a= [1,0,1,1,0,0] for K= 6. From this matrix, the number of

innovative packets decoded from the collision of size Kis calculated as its rank. This joint

decoding approach reverses the order of the S&D+SIC method: while in S&D+SIC the packet

combination is determined ﬁrst and then a decoding attempt is carried out, joint decoding ﬁrst

tries to decode all packets jointly and then the receiver checks which combinations are correct.

In order to assess the performance of the different schemes considered so far, we count the

number of innovative packets per slot. Innovative packets are either individually decoded packets

or combinations of packets which cannot be obtained by combining other decoded packets.

The number of innovative packets is the same as the number of linearly independent packet

combinations, i.e., the rank of Aslot in F2arithmetic.

Another benchmark we consider is joint decoding (JD), which consists in applying the joint

decoder without PLNC. We adopt JD and SIC as benchmarks since they allow to measure the

gains of the joint use of PLNC and MUD with respect to MUD only. The main features of the

schemes presented in this section are summarized in Table I.

It is worth noting that many other MUD methods have been proposed and proved to achieve

good performance with respect to simple SIC, such as Turbo MUD [37]. For a matter of space

all such methods can not be compared in the present paper, and a full comparison is out of the

scope of this work. Besides, our choice of the MUD schemes is motivated by the following. The

JD is the optimal decoder, in that it jointly decodes the received messages, thus achieving better

performance (average rank of Aslot) than any other decoder. The SIC has been selected due to its

low implementation complexity and practical importance. As a matter of facts, SIC is nowadays

included in commercial communications standards such as the Digital Video Broadcasting -

Return Channel to Satellite (DVB-RCS) [38].

E. Example

In the following we illustrate the S&D scheme with a toy example. Let us consider a frame

with S= 2 slots and Ntx = 4 active terminals. Let us assume that terminals 1and 2transmit in

14

both slots, each time choosing at random their pre-coding coefﬁcients. Terminal 3only transmits

in the ﬁrst slot while terminal 4transmits only in the second, as illustrated in Fig. 2. The S&D

decoder is applied at the physical layer in one of the variants presented previously in the present

section. As explained, the S&D decoder consists of a combination of PLNC and MUD and,

depending on the channel state of each of the transmitters, it may be able to decode from a

single collision a number of linearly independent combinations up to the collision size. In the

following example we assume that the decoder is able to output only two linear combinations

from each of the two slots as shown in the picture. Starting from these combinations, the receiver

tries then to recover all information messages u1,...,u4by applying another decoding stage,

this time at packet level rather than at the PHY level. The decoding is possible if the coefﬁcient

S&D

decoder

S&D

decoder

Fig. 2. Example of decoding at the PHY layer in S&D with a two-slots frame and four active terminals. Terminals 1and 2

transmit in both slots, each time choosing at random their pre-coding coefﬁcients. Terminal 3only transmits in the ﬁrst slot

while terminal 4transmits only in the second. We recall that, as shown in 1, uirepresents the mapping of the information

message uiform an RN-dimensional vector in F2to an RN/nbc -dimensional vector in Fq.

15

matrix Ain Fq(shown below) has rank equal to the number of active terminals

AT=

α1,1α2,10 0

α1,10α3,10

α1,2α2,20 0

0α2,20α4,2

.

In order to further clarify how the PHY decoder is able to obtain the system in Eqn. (2) starting

from the analog superposition of the interfering signals, let us consider the decoding of slot 1 in

the example of Fig. 2. The physical signal seen by the receiver is y1, which is the superposition

of signals x1,1,x2,1and x3,1transmitted by terminal 1, 2 and 3, respectively, each weighted

by the corresponding fading coefﬁcient. By applying the S&D decoder described previously in

this section the decoder outputs the bit-wise XOR of two different pairs of messages, namely

u′

1,1⊕u′

2,1and u′

1,1⊕u′

3,1. We recall that u′

i,1is the packet that is transmitted by terminal Ti

in slot 1 after channel encoding and modulation. Due to the pre-coding (multiplication times a

random coefﬁcient αi,1) we have u′

i,1=αi,1ui, where the multiplication is done in Fq,q= 2nbc .

According to the arithmetics of extension ﬁelds, the bit-wise XOR (sum in F2) of u′

1,1and u′

2,1is

equivalent to the equation α1,1u1+α2,1u2in Fq. The pre-coding process adds little complexity

to the transmitters and allows to achieve better results in terms of packet loss rate (PLR) as

shown in the numerical results presented in Section V.

We recall from Section III that the coefﬁcients in the matrix ATabove are chosen at random

by the four transmitters (see Fig. 1). Speciﬁcally, transmitter i, i = 1,...,4, chooses coefﬁcients

αi,j ,j= 1,2. We see in the present example that coefﬁcient α1,1is present twice in the ﬁrst

column of matrix AT. This is because the ﬁrst two rows of the matrix correspond to equations

obtained from the same slot. Note also that matrix Ais rank deﬁcient if coefﬁcients are chosen

in F2(i.e., all coefﬁcients shown in the matrix above are equal to 1), while it can be full rank

if coefﬁcients are chosen in some larger extension ﬁeld, since the probability of obtaining a full

rank matrix increases with the ﬁeld size [39]. This motivates the inclusion of the pre-coding

stage. We also note that in the example the average number of packets decoded per slot, if A

is full rank, is 2.

Note that Acontains information about the packet combinations within a given frame, but it

is not the same as the matrix Aslot deﬁned in Section IV. Matrix Ais obtained by the receiver

16

combining the Aslot matrices from all the slots in the frame and using the information relative

to the pre-coding coefﬁcients.

We stress the fact that the proposed scheme does much more than simply applying a MUD,

since any linear combination of the colliding signals decoded at the PHY layer can be exploited in

the second decoding phase at the frame level. We also note that, in principle, it would be possible

to use the soft information extracted from each slot and combine it at the frame level. Although

such approach would perform better than S&D, its complexity and memory requirements would

be much larger with respect to the S&D scheme, which has the advantage of processing each

slot only once and allows a lower complexity decoding at the frame level, since all operations

are performed over a GF of size 2nbc , which is suited to a digital implementation.

To conclude this section we recall that up to now we made the assumption that the receiver

is able to estimate the fading coefﬁcients starting from the preambles of the colliding signals.

The practical feasibility of the channel estimation, as well as other practical issues, have been

discussed in [29] and so are not dealt with explicitly here. The channel estimation based on the

estimate-maximize algorithm presented in [29] has been enhanced in [30] exploiting the cross-

correlation properties of the preamble as well as considering the presence of pilot symbols, that

are foreseen by many standards, showing that the average channel estimation error in a MUD

context can be kept reasonably low in a practical setup.

F. Complexity Considerations and Possible Combined Approaches

An important aspect in the different decoding approaches at the PHY layer is their performance-

complexity tradeoff. For the basic separate decoding scheme, complexity can be reduced by

ordering users according to their instantaneous SNR and stop decoding after the decoding of

one user has failed. This will obviously cause a slight performance loss which depends mainly on

the SNR differences and on the applied coding scheme, i.e. basically on the packet length. The

same idea can be applied to both SIC techniques, while for S&D+SIC, a packet combination can

be checked for linear independency before the decoding attempt. The complexity of S&D+SIC

in the worst case is proportional to 2K−1decoding attempts. The complexity of joint decoding

using LDPC codes is proportional to K·2Kfor belief propagation with transform-based check-

node processing [40], [41]. This complexity can be reduced on the one hand by applying joint

decoding after SIC and on the other hand by applying reduced-complexity decoding algorithms

17

[42].

V. NUMERICAL RESULTS

In this section we evaluate numerically the performance of the proposed schemes. First we

compare the different PHY layer decoding approaches presented in Section IV in terms of number

of innovative packets decoded from a single slot, then we move to the comparison of throughput,

packet loss rate and energy efﬁciency at frame level for the S&D scheme and several benchmark

systems.

A. Performance at Slot Level

We recall that innovative packets are either individually decoded packets or combinations of

packets which cannot be obtained by combining other decoded packets. Figures 3 and 4 show the

achieved number of innovative packets per slot with the described decoding techniques with 4

and 8users, that correspond to the average rank of the matrix Aslot deﬁned in Section IV. We can

see that for both cases, S&D+JD performs best and its gain with respect to the others increases

with the number of users. For a high number of users, the advantage of S&D+JD to all other

techniques is dramatic. On the other hand, we point out that, unlike S&D+JD, the S&D+SIC

scheme has the advantage that is does not require any modiﬁcation at the decoder, since only

the LLR calculation is modiﬁed with respect to a standard receiver. We further note that the

advantage of S&D+SIC over pure SIC decreases with the number of users. For sufﬁciently high

SNR, all methods beneﬁt from collided packets, which can be most clearly seen in Fig. 3 for four

users. At low SNR the average number of recovered packets per slot is close to the single-user

case, while for medium to high SNR, on average more than one packet is recovered from a

single slot. For all considered cases, the number of innovative packets tends to Kas the SNR

grows, i.e. for high SNR nearly all collided packets can be decoded.

B. Performance at Frame Level

We deﬁne the normalized throughput Tas the average number of packets decoded within a

slot averaged across the realizations. We further deﬁne the PLR as the ratio of the number of

lost packets to the total number of packets transmitted (not counting repetitions). The following

18

−5 0 5 10 15 20 25 30

0

0.5

1

1.5

2

2.5

3

3.5

4

SNR [dB]

Innovative packets per slot

S&D+JD

S&D+SIC

SIC

Sep. dec.

Single user

Fig. 3. Innovative packets decoded per slot versus average SNR in Rayleigh fading channel for a collision of size K= 4.

holds:

T=G(1 −P LR).(13)

Note that G, which represents the logical load of the network [8] [9], is independent of the number

of times a message is repeated within a frame. The physical load on the network is larger than

or equal to G. In particular, if 2 copies of the same packet are sent by each active terminal,

then the physical load is twice as large as the logical load. Since the interaction between the

frame and the PHY layers are of fundamental importance in the schemes considered here, in the

simulations the whole decoding process has been implemented. The actual decoded combinations

at the physical layer have been used as input to the decoder at the frame level. As suggested

in Section III, if rank(A)< Ntx , i.e. not all messages can be decoded in a frame, the receiver

19

10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

SNR [dB]

Innovative packets per slot

S&D+JD

S&D+SIC

SIC

Sep. dec.

Single user

Fig. 4. Innovative packets decoded per slot versus average SNR in Rayleigh fading channel for a collision of size K= 8.

applies Gaussian elimination on Ain order to extract as many packets as possible. In Fig. 5,

6 and 7 T, PLR and the energy efﬁciency are plotted against the network load G, respectively.

The energy efﬁciency is deﬁned as the ratio of the number of repetitions (which is proportional

to the total amount of energy used to transmit a packet) to the number of decoded packets (not

counting repetitions). Two repetitions and a frame with S= 10 slots have been considered for all

schemes. A Rayleigh block fading channel with 15 dB average SNR has been considered. The

LDPC code of the WiMAX standard with parameters N= 576,R= 1/2, and BPSK modulation

have been adopted. A maximum collision size of K= 7 has been set, i.e., collisions of more

than 7signals are discarded. The introduction of a maximum decodable collision size is justiﬁed

by practical issues such as complexity and power saturation at the receiver. In Fig. 5 it can be

20

0.5 1 1.5 2 2.5

0

0.5

1

1.5

2

T

G

S&D+JD, nbc=8

S&D+SIC, nbc=8

S&D+JD, nbc=2

S&D+SIC, nbc=2

Joint dec.

SIC

Sep. dec.

Slotted ALOHA

Fig. 5. Throughput in Rayleigh block fading channel, SNR=15 dB. The channel code is the WiMAX LDPC with parameters

N = 576, R= 1/2, BPSK modulation. 2 replicas of the same packet are transmitted by each user. A maximum collision size

of K= 7 has been set. Collisions of higher order are discarded. The frame size Shas been set to S= 10 slots.

seen how S&D provides signiﬁcant gains in terms of throughput with respect to the schemes

that apply MUD only. The use of a larger ﬁeld size in the pre-coding stage slightly increases the

peak throughput and enhances the PLR performance at low network loads, as shown in Fig. 5

and 6, respectively. In order to quantify such enhancement, we evaluated through Monte Carlo

simulations the probability that, once the iterative decoding stops, the rest of the packets can

be decoded through matrix inversion. In correspondence to a load of G= 2.1(for which the

peak throughput with the conﬁguration of Fig. 5 is achieved), such probability is around 4% for

coefﬁcients in F28and 0.06% for coefﬁcients in F2, i.e., the probability to decode the remaining

packets is about sixty times larger when the ﬁeld with higher cardinality is used. However, since

both probabilities are relatively small, the overall improvement on the throughput is limited.

21

0.5 1 1.5 2 2.5

10−3

10−2

10−1

100

packet loss rate

G

S&D+JD, nbc=8

S&D+SIC nbc=8

S&D+JD, nbc=2

S&D+SIC, nbc=2

Joint dec.

SIC

Sep. dec.

Slotted ALOHA

Fig. 6. Packet loss rate in Rayleigh block fading channel, SNR=15 dB. The channel code is the LDPC used in WiMAX standard

with parameters N = 576, R= 1/2, BPSK modulation. 2 replicas of the same packet are transmitted by each user. A maximum

collision size of K= 7 has been set. Collisions of higher order are discarded. The frame size Shas been set to S= 10 slots.

In ﬁgures 8, 9 and 10 the throughput, packet loss rate and energy efﬁciency for an average

SNR of 10 dB are plotted, respectively. The rest of parameters are the same as in Fig. 5. By

comparing the two sets of ﬁgures it can be seen how the channel SNR impacts the decoding at

the PHY layer, which leads to a higher throughput and lower PLR when the SNR is higher, as

expected. At both SNR values the JD scheme performs better than all others non-S&D schemes

and at 10 dB closely approaches the S&D+SIC for lower network loads, outperforming it in the

region G > 1.5. Such good performance is due to the fact that the decoding of all messages

is done jointly rather than separately as in the SIC or the separate decoding schemes. The

22

0.5 1 1.5 2 2.5

0

2

4

6

8

10

12

14

16

18

20

G

Energy/decoded packet

S&D+JD, nbc=8

S&D+SIC nbc=8

S&D+JD, nbc=2

S&D+SIC, nbc=2

Joint dec.

SIC

Sep. dec.

Slotted ALOHA

Fig. 7. Energy efﬁciency plotted against load in Rayleigh block fading channel, SNR=15 dB. The channel code is the WiMAX

LDPC with parameters N = 576, R= 1/2, BPSK modulation. 2 replicas of the same packet are transmitted by each user. A

maximum collision size of K= 7 has been set. Collisions of higher order are discarded. The frame size Shas been set to

S= 10 slots.

introduction of PLNC signiﬁcantly increases the performance of the JD scheme of up to a 13

% at both SNR values, as can be seen in Fig. 5 and Fig. 8. In all ﬁgures the Slotted ALOHA

scheme is also shown as a benchmark. In Slotted ALOHA all terminals transmit only one replica

of their message, while in all other schemes two replicas are used, i.e., twice the energy is used.

In order to compare the energy efﬁciency of the different schemes, in Fig. 7 and Fig. 10 we

show the average energy consumption per decoded message plotted against the load Gfor an

SNR of 10 dB and 15 dB, respectively. Slotted ALOHA shows a more efﬁcient energy use at

low network loads up to about 0.7. This is due mainly to the fact that in Slotted ALOHA each

23

0.5 1 1.5 2 2.5

0

0.5

1

1.5

2

T

G

S&D+JD, nbc=8

S&D+SIC, nbc=8

S&D+JD, nbc=2

S&D+SIC, nbc=2

Joint dec.

SIC

Sep. dec.

Slotted ALOHA

Fig. 8. Throughput in Rayleigh block fading channel, SNR=10 dB. The channel code is the WiMAX LDPC with parameters

N = 576, R= 1/2, BPSK modulation. 2 replicas of the same packet are transmitted by each user. A maximum collision size

of K= 7 has been set. Collisions of higher order are discarded. The frame size Shas been set to S= 10 slots.

terminal transmits half of the power used in the other schemes. However, for G > 0.7these,

and most of all S&D+JD, perform signiﬁcantly better than Slotted ALOHA in terms of energy

efﬁciency, conﬁrming the effectiveness of the proposed approach in situations characterized by a

relatively high logical network load. The combined decoder is capable of extracting much more

information from collisions than each of the two techniques taken individually. Furthermore, the

solution presented here is robust against power unbalance (actually beneﬁting from it), which

constitutes an issue if PLNC is applied with no MUD as in [18]. It is worth pointing out the

fact that the joint decoding approach is optimal within each slot if this is treated as an isolated

channel. If, instead, the slot is regarded as part of a frame and multiple replicas of the same

packet are transmitted, using PLNC jointly with joint decoder brings a signiﬁcant advantage.

24

This is not in contrast with the intuition that the joint decoder is optimal, since if it were applied

over the whole frame at once, it would lead to the best possible performance. However, the

huge increase in complexity makes such approach impractical. The advantage of our proposed

approach is that it brings signiﬁcant advantages with respect to the joint decoder applied at slot

level with a limited increase in complexity, since the whole frame is processed only once at the

physical layer, while the rest of operations are done over a ﬁnite ﬁeld. Our approach is not an

0.5 1 1.5 2 2.5

10−2

10−1

100

packet loss rate

G

S&D+JD, nbc=8

S&D+SIC nbc=8

S&D+JD, nbc=2

S&D+SIC, nbc=2

Joint dec.

SIC

Sep. dec.

Slotted ALOHA

Fig. 9. Packet loss rate in Rayleigh block fading channel, SNR=10 dB. The channel code is the WiMAX LDPC with parameters

N = 576, R= 1/2, BPSK modulation. 2 replicas of the same packet are transmitted by each user. A maximum collision size

of K= 7 has been set. Collisions of higher order are discarded. The frame size Shas been set to S= 10 slots.

alternative to other diversity schemes proposed for slotted ALOHA such as Irregular Repetition

Slotted ALOHA (IRSA) [9]. As a matter of fact the S&D approach can be used on top of IRSA.

The proposed scheme would allow either to increase the throughput for a given frame size or

25

0.5 1 1.5 2 2.5

0

2

4

6

8

10

12

14

16

18

20

22

G

Energy/decoded packet

S&D+JD, nbc=8

S&D+SIC nbc=8

S&D+JD, nbc=2

S&D+SIC, nbc=2

Joint dec.

SIC

Sep. dec.

Slotted ALOHA

Fig. 10. Energy efﬁciency plotted against load in Rayleigh block fading channel, SNR=10 dB. The channel code is the WiMAX

LDPC with parameters N = 576, R= 1/2, BPSK modulation. 2 replicas of the same packet are transmitted by each user. A

maximum collision size of K= 7 has been set. Collisions of higher order are discarded. The frame size Shas been set to

S= 10 slots.

to reduce the frame size while guaranteeing the same throughput. Similar considerations have

been presented in [43], where MUD is applied to IRSA. In order to show the gain deriving from

applying S&D on top of IRSA, we compare the throughput and PLR curves of the two schemes

for the case of a frame with 200 slots and Rayleigh fading channels with average SNR 15 dB.

In the simulation the number of replicas transmitted by a given user is chosen according to the

following degree distribution [9]:

Λ(x) = 0.5465x2+ 0.1623x3+ 0.2912x6.

26

Unlike in [43], the results shown in ﬁgures 11 and 12 have been obtained applying a combination

0 0.5 1 1.5 2

0

0.5

1

1.5

2

G

T

S&D+JD, nbc=8 on IRSA

IRSA

Fig. 11. Throughput in Rayleigh block fading channel, SNR=15 dB. The channel code is the WiMAX LDPC with parameters

N = 576, R= 1/2, BPSK modulation. A maximum collision size of K= 7 has been set for S&D only. The frame size Shas

been set to S= 200 slots.

of PLNC and MUD to IRSA rather than MUD alone. Note that the results for S&D could be

further enhanced by ﬁrst running the IRSA cancellation in the analog domain and then applying

S&D on the remaining collisions. Since the S&D would work on collisions that on average have

a lower size, its performance would enhance.

We also point out that better performance can be obtained with an approach based on a frame-

level joint detection and decoding of all of the packets rather than using a slot-based approach

as we proposed in this paper. However, such approach would imply a considerable increase in

the complexity of the decoder with respect to our method.

27

0 0.5 1 1.5 2 2.5

10−6

10−4

10−2

100

G

packet loss rate

S&D+JD on IRSA

IRSA

Fig. 12. Packet loss rate in Rayleigh block fading channel, SNR=15 dB. The channel code is the WiMAX LDPC with parameters

N = 576, R= 1/2, BPSK modulation. A maximum collision size of K= 7 has been set for S&D only. The frame size Shas

been set to S= 200 slots.

C. Discussion

In order to have a complete picture of what are the performance limits of the proposed

scheme as well as how the different parameters impact the behavior of the scheme it would

be desirable to have an analytical expression for the throughput or, equivalently, for the packet

loss rate. An approximate semi-analytical expression for the throughput has been derived in [1]

under the assumption that each active user accesses the channel in each slot with probability

1−2−nbc . Such expression is based on a bound on the probability to decode the sum of a

subset of colliding messages. Deriving a formula for the general case is quite challenging, since

it requires an analytical characterization of the packet error rate of ﬁnite length channel codes

over fading channel with no channel state information at the transmitter and in the presence of

28

interference. Furthermore, such characterization should also provide information on the behavior

of the speciﬁc code when it is applied in a MUD or PLNC context. It is in general very difﬁcult

to model the performance of a speciﬁc error correcting code over a general discrete memoryless

channel (unless the code is particularly short, or embeds a very strong structure, as in the

case of convolutional and Reed Solomon codes). For LDPC and turbo codes, in general, rather

than modeling the performance of a speciﬁc code, ensemble-based arguments are used, which

nevertheless mostly assume maximum-likelihood (rather than iterative) decoding, and hence fail

to provide a realistic model [44]. From the simulations we carried out and from the available

bounds, we can say that larger nbc lead to a higher probability of having a full rank matrix,

although, as we showed previously in the present section, the gain when going from nbc = 2 to

nbc = 8 is limited. The analysis presented in [19] can be regarded as a starting point, although the

codes and the channel model are highly abstracted and thus particular care should be used when

transposing such results to practical setups. As a ﬁnal remark, we point out that the proposed

method can be applied for channel codes of any packet length, although the performance of the

S&D decoder in general depends, apart from the speciﬁc channel code that is consider, also on

the codeword length.

VI. CONCLUSIONS

We proposed a novel cross-layer approach to random access systems that uses a hybrid PLNC-

MUD decoder at the PHY layer and a frame level decoder based on matrix manipulation over

extension ﬁelds. In the proposed scheme each terminal transmits several channel-coded replicas

of the same message within a frame after a pre-multiplication by a random coefﬁcient in an

extension ﬁeld Fq. At the PHY layer the receiver decodes as many linear combination as possible

in F2of the signals colliding in each slot. In the second decoding stage, which is carried out at

frame level, the set of combinations is treated by the receiver as a single system of equations in

Fq. We presented simulation results for throughput, packet loss rate and energy efﬁciency over

a block fading channel. The whole decoding process at both PHY and frame level has been

implemented in the simulations. Our results show that a signiﬁcant enhancement in throughput

and PLR can be achieved by combining PLNC and MUD. The combined decoder is capable

of extracting much more information from collisions than each of the two techniques taken

individually. In particular, we showed that the combination of PLNC and JD together with the

29

frame-level decoding stage, considerably enhance the JD method, despite the fact that the latter

is optimal when applied to slots in isolation. Furthermore, unlike in previously proposed schemes

based on PLNC only, the approach presented in this paper is robust against block fading.

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 PLR also

taking energy efﬁciency into account.

As a ﬁnal remark, we would like to point out that evaluating the impact of the joint use of

PLNC and MUD in random access systems is a challenging task and far from being concluded.

The present work can be regarded as a further step towards a full exploitation of these two

techniques in the Slotted ALOHA scenario.

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TABLE I

DECODING STR ATEGIES AT PHY L AYE R.

Method Description Requires Requires

pre-coding joint decoding

Separate dec. joint detection, no no

separate decoding

SIC joint detection, no no

separate decoding,

then interference

cancellation

S&D+SIC as in SIC, yes no

then detect/decode

combinations

JD joint detection, no yes

joint decoding

S&D+JD as in JD, yes yes

then combine

estimated messages