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EURASIP Journal on Wireless Communications and Networking

Volume 2010, Article ID 517921, 15 pages

doi:10.1155/2010/517921

Research Article

Opportunistic Adaptive Transmission for

Network Coding Using Nonbinary LDPC Codes

Giuseppe Cocco, Stephan Pﬂetschinger, Monica Navarro, and Christian Ibars

Centre Tecnol`

ogic de Telecomunicacions de Catalunya, 08860 Castelldefels, Spain

Correspondence should be addressed to Giuseppe Cocco, giuseppe.cocco@cttc.es

Received 31 December 2009; Revised 14 May 2010; Accepted 3 July 2010

Academic Editor: Wen Chen

Copyright © 2010 Giuseppe Cocco et al. This is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

Network coding allows to exploit spatial diversity naturally present in mobile wireless networks and can be seen as an example

of cooperative communication at the link layer and above. Such promising technique needs to rely on a suitable physical layer in

order to achieve its best performance. In this paper, we present an opportunistic packet scheduling method based on physical layer

considerations. We extend channel adaptation proposed for the broadcast phase of asymmetric two-way bidirectional relaying to

a generic number Mof sinks and apply it to a network context. The method consists of adapting the information rate for each

receiving node according to its channel status and independently of the other nodes. In this way, a higher network throughput

can be achieved at the expense of a slightly higher complexity at the transmitter. This conﬁguration allows to perform rate

adaptation while fully preserving the beneﬁts of channel and network coding. We carry out an information theoretical analysis

of such approach and of that typically used in network coding. Numerical results based on nonbinary LDPC codes conﬁrm the

eﬀectiveness of our approach with respect to previously proposed opportunistic scheduling techniques.

1. Introduction

Intensive work has been devoted the ﬁeld of network coding

(NC) since the new class of problems called “network

information ﬂow” was introduced in the paper of Ahlswede

et al. [1], in which the coding rate region of a single

source multicast communication across a multihop network

was determined and it was shown how message mixing at

intermediate nodes (routers) allows to achieve such capacity.

Linear network coding consists of linearly combining packets

at intermediate nodes and, among other advantages [2],

allows to increase the overall network throughput. In [3],

NC is seen as an extension of the channel coding approach

introduced by Shannon in [4] to the higher layers of the

open systems interconnection (OSI) model of network archi-

tecture. Important theoretical results have been produced in

the context of NC such as the min-cut max-ﬂow theorem

[5], through which an upper bound to network capacity can

be determined, or the technique of random linear network

coding [6,7] that achieves the packet-level capacity for

both single unicast and single multicast connections in both

wired and wireless networks [3]. Practical implementations

of systems where network coding is adopted have also been

proposed, such as CodeCast in [8] and COPE in [9].

The implementation proposed in [9] is based on the idea

of “opportunistic wireless network coding”. In such scheme

at each hop, the source chooses packets to be combined

together so that each of the sinks knows all but one of the

packets. Considering the problem in a wireless multihop

scenario, each of the potential receivers will experiment

diﬀerent channel conditions due to fading and diﬀerent

path losses. At this point, a scheduling problem arises:

which packets must be combined and transmitted? Several

solutions to this scheduling problem have been proposed up

to now. In [10], a solution based on information theoretical

considerations is described, that consists of combining and

transmitting, with a ﬁxed rate, packets belonging only to

nodes with highest channel capacities. The number of such

nodes is chosen so as to maximize system throughput.

In [11], the solution [10] has been adapted to a more

practical scenario with given modulations and ﬁnite packet

loss probabilities. In both cases network coding and channel

2 EURASIP Journal on Wireless Communications and Networking

coding are treated separately. However, as pointed out in the

paper by Eﬀros et al. [12], such approach is not optimal

in real scenarios. In [13,14], a joint network and channel

coding approach has been adopted to improve transmissions

in the two-way relay channel (TWRC) in which two nodes

communicate with the help of a relay. One of the main ideas

used in these works is that of applying network coding after

channel encoding. This introduces a new degree of ﬂexibility

in channel adaptation, which leads to a decrease in the packet

errorrateofbothreceivers.

Up to our knowledge, this approach has been applied

only to the two-way relay channel. In the present paper,

we extend the basic idea of inverting channel and network

coding to a network context. While in the TWRC the

relay broadcasts combinations of messages received by the

two nodes willing to communicate, in our setup the relay

can have stored packets during previous transmissions by

other nodes, which is typical in a multihop network, and

transmit them to a set of Msinks. As a matter of fact,

in a wireless multihop network more than just two nodes

(sinks) are likely to overhear a given transmission. Due to the

diﬀerent channel conditions, a per-sink channel adaptation

is done in order to enhance link reliability and decrease

frequent retransmissions which can congest parts of the

network, especially when ARQ mechanisms are used [9]. In

particular, packet uiof length Kis considered as a buﬀer by

the transmitting node (source node). At each transmission,

a part of the buﬀer, containing Kbits, is included in a

new packet of total length Nthat contains N−Kbits of

redundancy. Network Coding combination takes place on

such packets. The value of K, which determines the amount

of redundancy to be introduced in each combined packet

(i.e., the code rate), is chosen by the source node considering

the physical channel between source node and sink i.Given

a set of channel code rates {r1,...,rs}, we propose that the

code rate in channel ibe the one that maximizes the eﬀective

throughput on link ideﬁned as

thi=rk1−ppli(rk),(1)

where ppli(rk) is the current probability of packet loss on

channel iwhen using rate rk.

In present paper, we carry out an information theoretical

analysis and comparison for the proposed method and the

method in [10], which maximizes overall throughput in a

system where opportunistic network coding is used, showing

how the ﬁrst one noticeably enhances system throughput.

Moreover, we evaluate the performance of the two methods

in a real system using capacity-approaching nonbinary low-

density parity-check (LDPC) codes at various rates (in [13,

14] parallel concatenated convolutional codes (PCCC) have

been adopted for channel coding). Numerical results conﬁrm

those obtained analytically. Finally, we consider some issues

regarding how modiﬁcations at physical level aﬀect network

coding from a network perspective.

The paper is organized as follows. In Section 2, the system

model is described. In Section 3,weproposeabenchmark

system with equal rate link adaptation. Section 4contains

the description of our proposed opportunistic adaptive

transmission for network coding. In Section 5,wecarryout

the comparison between the two methods by comparing

the cumulative density functions of the throughput and the

ergodic achievable rates. Section 6contains the description of

the simulation setup and the numerical results. In Section 7,

we consider some scheduling and implementation issues at

network level that arise from applying the proposed adaptive

transmission method, and ﬁnally in Section 8, we draw the

conclusions about the results obtained in this paper, and we

suggest possible future work to be carried out.

2. System Model

2.1. Network Level. Let us consider a mobile wireless multi-

hop network such as the one depicted in Figure 1.Wedenote

by Fqthe ﬁnite ﬁeld (Galois ﬁeld) of order q=2l.Each

packet is an element in FK

q; that is, it is a K-dimensional

vector with components in Fq. We say that a node niis the

generator of a packet piif the packet pioriginated in ni.We

say that a node is the source node during a transmission slot

if it is the node which is transmitting. We call sink node the

receiving node during a given transmission slot and desti-

nation node the node to which a given packet is addressed.

We will refer to generators’ packets as native packets.Each

node stores overheard packets. Native and overheard packets

are transmitted to neighbor nodes. For ease of exposition

and without loss of generality we assume that a collision-free

time division multiple access is in place. The number of hops

needed to transmit a packet from generator to destination

node depends on the relative position of the two nodes in

the network. In Figure 1, two generator-destination pairs

are shown (G1–D1, G2–D2). Thin dashed lines in the ﬁgure

represent wireless connectivity between nodes and thick lines

represent packet transmissions. G1 has a packet to deliver to

D1 and G2 has a packet to deliver to node D2. In the ﬁrst time

slot, generator G1 and G2 broadcast their packets p1and p2,

respectively, (thick red dash-dotted line). In the second time

slot, node 6 acts as a source node broadcasting packet p2

(thick green dotted line) received in previous slot. Note that

in this case node 6 is a source node but not a generator node.

Finally, in the third time slot, node 5 broadcasts the linear

combination in a ﬁnite ﬁeld of packets p1andp2 (indicated

in Figure 1with p1+p2). Destination nodes D1 and D2 can,

respectively, obtain packets p1and p2from p1+ p2 using

their knowledge about packets p2andp1 overheard during

previous transmissions.

In general, using linear network coding we proceed

as follows. Each node stores overheard packets, linearly

combines them and transmits the combination together with

the combination coeﬃcients. As the combination is linear

and coeﬃcients are known, a node can decode all packets

if and only if it receives a suﬃcient number of linearly

independent combinations of the same packets. At this point,

a scheduling solution must be found in order to decide which

packets must be combined and transmitted each time. In the

paper by Katti et al. [9], a packet scheduling based on the

concept of network group has been described. Such solution,

called opportunistic coding, consists of choosing packets so

that each neighbor node knows all but one of the encoded

EURASIP Journal on Wireless Communications and Networking 3

Node 1

Node 5 Node 7

Node 9

Node 3

(G2)

Node 4

(D2)

Node 8

(G1)

Node 13

Node 2

(D1)

Node 14

p2

p2

p2

p2

p2

p2

p2

p1

p1

p1

p1

1st time slot

2nd time slot

3rd time slot

Node 6

Node 10

Node 11

Node 12

p1+p2

p1+p2

p1+p2

p1+p2

p1+p2

p1+p2

p1+p2

Figure 1: Mobile wireless multihop network. Two diﬀerent information ﬂows exist between two generator-destination pairs G1–D1 and

G2–D2. Thin dashed lines represent wireless connectivity among nodes while thick lines represent packet transmissions. In the ﬁrst time slot

generator G1 and G2 broadcast their packets p1andp2, respectively, (thick dash-dotted line). In the second time slot, node 6 broadcasts

packet p2 (thick dotted line) received in previous slot. In the third time slot, node 5 broadcasts the linear combination of packets p1and p2

(p1+p2). Destination nodes D1 and D2 can, respectively, obtain packets p1andp2from p1+ p2 using their knowledge about packets p2

and p1 overheard during previous transmissions.

packets. Such approach has been implemented in the COPE

protocol, and its practical feasibility has been shown in [9].

Anetworkgroupisformallydeﬁnedasfollows.

Deﬁnition 1. AsetofnodesiscalledasizeMnetwork group

(NG) if it satisﬁes the following:

(1) one of the nodes (source) has a set U={u1,...,uM}

of Mnative packets to be delivered to the other nodes

in the set (sinks);

(2) all sink nodes are within the transmission range of

the source;

(3) each of the sink nodes has all packets in Ubut

one (they may have received them during previous

transmissions).

All native packets are assumed to contain the same number K

of symbols. A native packet is considered as a K-dimensional

vector with components in Fqwith q=2l, that is, a native

packet is an element in FK

q.

Figure 2shows an example of how a network group is

formed during a transmission slot.

Network groups appear in practical situations in wireless

mesh networks and other systems. A classical example is a

bidirectional link where two nodes communicate through a

relay. More examples can be found in [9]. In the following,

we will assume that all transmissions adopt the network

group approach; that is, during each transmission slot, the

source node chooses the packets to be combined so that each

of the sinks knows all but one of the packets. As a matter of

fact, if nodes are close one to each other it is highly probable

that many of them overhear the same packets. Nevertheless

this assumption is not necessary to obtain NC gain or to

apply the technique proposed in this paper. In Section 7,we

will extend the results to a more general case, in which a node

may not know more than one of the source packets.

We assume time is divided into transmission slots. Dur-

ing each transmission slot source node combines together

the Mpackets in Uand broadcasts the resulting packet to

sink nodes of the network group. Let us indicate with uithe

packet to be delivered to node i. The packet transmitted by

the source node is

x=

M

i=1

ui,(2)

4 EURASIP Journal on Wireless Communications and Networking

N1

N2

N3

γ1

γ2

γ3

P1

P1

P1

P2P3

P3

P3

P4

P4

P4

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

γ1

γ2

γ3

γ=

N4

(S)

Figure 2: Network group formation. N4 is going to access the

channel. Node N4 knows which packets are stored in its neighbors’

buﬀers. Based on this knowledge it must choose which packets to

XOR together in order to maximize the number of packets decoded

in the transmission slot. A possible choice is, for example, P1+ P2

which allows nodes N1andN2todecode,butnotN3. A better

choice is to encode P1+P3+P4, so that 3 packets can be decoded

in a single transmission. The diﬀerence in SNR for the three sinks

(γ1,γ2,andγ3) can lead to high packet loss probability on some of

the links if a single channel rate is used for all the sinks. γis the

vector of SNRs.

where indicates the sum in FK

q.Letusdeﬁnepacketx\jas

follows:

x\j=

M

i=1,i/

=j

ui(3)

Sink ican obtain uiby adding xand x\jin FK

q,wherex\jis

known according to our assumptions.

Note that in the network in Figure 1many aspects deserve

in-depth study, such as end-to-end scheduling of packet

transmissions on multiple access schemes. These aspects are

however beyond the scope of this paper, where we focus on

maximizing the eﬃciency of transmissions within a network

group.

2.2. Physical Level. Physical links between source and sinks

are modeled as frequency-ﬂat, slowly time-variant (block

fading) channels. The SNR of sink iduring time slot tcan

be expressed as

γi(t)=Ptx|hi(t)|2

dα

siσ2,(4)

where Ptx is the power used by source node during trans-

mission, hi(t) is a Rayleigh distributed random variable that

models the fading, dsi is the distance between source and

sink i,αis the path loss exponent and σ2is the variance

of the AWGN at sink nodes. From expression (4)itcan

be seen that the SNR at a receiver with a given dsi is an

exponentially distributed random variable with probability

density function

pγi(t)=1

γe−γi(t)/γ ,forγi(t)≥0, (5)

where γis the mean value of the SNR. We assume that

the quantities γi(t)dα

si at the various sinks are i.i.d. random

variables. In the model we are not taking into account

shadowing eﬀects.

3. Constant Information Rate Opportunistic

Scheduling Solutions

Based on the propagation model in (5), the channel from

source to each sink will have a diﬀerent gain. The diﬀerence

in link states experienced by the sinks gives rise to the

problem of how to choose the broadcast transmission rate.

In [10], an interesting solution has been proposed based

on information-theoretical capacity considerations. Sink

nodes are ordered from 1 to Mwith increasing SNR. The

solution proposed consists of combining and transmitting

only packets having as destination the M−v+ 1 sinks with

highest SNR. The transmission rate Rchosen by the source

node is the lowest capacity in the group of M−v+1

channels. The instantaneous capacity obtained during each

transmission is then

C(v)

inst =(M−v+1

)log21+γ(v),(6)

where γ(v)is the SNR experienced on the vth worse channel.

vis chosen so that (6) is maximized. Note that all sinks in

the network group receive the same amount of information

per packet. In [11], another approach is proposed in which

the source node transmits to all nodes in the NG. A practical

transmission scheme with ﬁnite bit error probability and

ﬁxed modulations is described.

3.1. Constant Information Rate Benchmark. Based on [10,

11], we deﬁne a constant information rate (CIR) system that

will be used as a benchmark to our proposed adaptive system.

Let us now deﬁne the eﬀective throughput as

th =

M

i=11−ppliri=1−pplTr,(7)

where ppl and rare two M×1 vectors containing, respectively,

the packet loss probabilities and the coding rates for the

various links, Trepresents the transpose operator and 1is

an M-dimensional vector of all ones. The quantity expressed

in (7) measures the average information ﬂow (bits/sec/Hz)

from source to sinks. ppl is an M-dimensional function that

depends on the modulation scheme, coding rate vector rand

SNR vector γ. We assume channel state information (CSI) at

both transmitter and receiver (i.e., the source knows vector γ

containing the SNR of all sinks and node iknows γi).

In the CIR system, the source calculates ﬁrst the rate

of the channel encoder which maximizes the eﬀective

EURASIP Journal on Wireless Communications and Networking 5

throughput for each sink (individual eﬀective throughput).

Formally, for each sink i,wecalculate

r∗

i=arg max

ri1−ppliγi,riri,(8)

where ppli(γi,ri) is the packet loss probability on the ith link

depending on the rate ri.Foreachrater∗

k,wedeﬁnemkas

the number of sinks for which

ri≥r∗

k.(9)

At this point, for each kwe calculate the eﬀective throughput,

setting r=rk1kwhere 1kis a mk-dimensional vector of

all ones. Finally, we choose kto maximize the eﬀective

throughput. Note that with the CIR approach only sinks

whose optimal rate is greater or equal than the rate which

maximizes the total eﬀective throughput will receive data.

4. Opportunistic Adaptive Transmission for

Network Coding

We propose a scheme in which information rate is adapted to

each sink’s channel. This can be accomplished by inverting

the order of channel coding and network coding at the

source. In order to explain our method, let us consider again

Figure 2. In the ﬁgure, a network group is depicted, in which

node 4 accesses the channel as source node (S) and nodes N1,

N2andN3 are the sink nodes.

As mentioned in Section 2, the source is assumed to know

the packets in each sink (this can be accomplished with a

suitable ACK mechanism such as the one described in [9]).

We propose a transmission scheme for a size MNetwork

Group consisting in Mvariable-rate channel encoders, a FK

q

adder and a modulator as shown in Figure 3. We assume

CSI at both ends. The transmission scheme is as follows.

Based on the SNR to sink i,γi, the source chooses the code

rate ri=K

i/N that maximizes the throughput to sink i,

i=1, ...,M. Overall, the rate vector chosen by the source

is the one that maximizes the eﬀective throughput, deﬁned

as

roptγ=arg max

r⎛

⎝

M

i=11−ppliγi,riri⎞

⎠

=arg max

r1−pplγ,rTr.

(10)

As we are under the hypothesis of independent channel gains,

optimal rate can be found independently for each physical

link. In order to apply our method to a packet network, we ﬁx

the size of coded packets to Nsymbols. Channel adaptation is

performed by varying the number of information symbols in

the coded packet. So, referring to Figure 3, once the optimal

rate r∗

i=Ki/N has been chosen for link i,i=1, ...,M, the

source takes Kiinformation symbols from native packet ui

and encodes them with a rate r∗

iencoder, thus obtaining a

packet u

iof exactly Nsymbols. Finally, packets u

1,...,u

M

are added in Fq, modulated and transmitted. On the receiver

side, sink iis assumed to know a priori the rate used by the

source for packet uias it can be estimated using CSI.

As previously stated we will assume that a constant

energy per channel symbol is used. We will not consider

the case of constant energy per information bit as packet

combination at source node is done in FK

qbefore channel

symbol ampliﬁcation.

As we will see in Section 6, in this paper, we consider

nonbinary LDPC codes which have a word error rate

characteristic (WER) versus SNR with a high slope. Thus,

the packet loss probability is negligible (≤10−3)beyonda

given SNR threshold and rapidly rises below the threshold.

The threshold depends of the code rate considered. Under

this assumption, (10) can be approximated with

roptγ=arg max

r⎛

⎝

M

i=11−p

pliγi,riri⎞

⎠

=arg max

r1−p

plγ,rTr,

(11)

where p

pli(γi,ri)takesvalue1ifγ≤γthresh and 0 otherwise,

γthresh being a threshold that depends on the rate ri.Wewill

refer to our approach as adaptive information rate (AIR),

indicating that the number of information bits per packet

received by a given sink is adapted to its channel status. The

same approximation regarding ppl will be used for the CIR

system.

5. Information Theoretical Analysis

Let us consider a system where opportunistic network coding

[9] is used. As described in Section 2, opportunistic Network

Coding consists in a source node combining together and

transmitting Mnative packets to Msinks. Each of the

sinks knows a priori all but one of the native packets (see

Figure 2). Each of the receivers can, then, remove such

known packets in order to obtain the unknown one. In

the following, we provide an outline of the achievability

for the achievable rate of the system, based on the results

in [15] for the broadcast channel with side information

[16]. In order to study the proposed adaptive transmission

method we need to introduce an equivalent theoretical

model. We model each of the Mpackets stored in the

source node as an information source.Thusanequivalent

modelforoursystemisgivenbyaschemewithasetof

Minformation sources IS ={IS1,...,IS

M}all located in

the source node, and a set of Msinks D={D1,...,DM}.

Information source ISiproduces a message addressed to sink

Diwho has side information (perfect knowledge, speciﬁcally)

about messages produced by sources in the subset IS \

ISi. This models the situation in which each of the sinks

knows all but one of the messages transmitted by source

node (see Figure 2). Figure 4depicts the equivalent model.

Let us consider the system we described in Section 4.

The theoretical idea behind such system is to adapt the

information rate of each information source ISito channel

i. Each information source ISichooses a message from a set

of 2nRidiﬀerent messages. An M-dimensional channel code

book is randomly created according to a distribution p(x)

and revealed to both sender and receiver. The number of

6 EURASIP Journal on Wireless Communications and Networking

Multiple rate LDPC

encoder for sink 1

Multiple rate LDPC

encoder for sink 2

Multiple rate LDPC

encoder for sink

Channel 1

Channel 2

Sink 1

Sink 2

Source buﬀer

Source node

Modulo

2 adder Modulator

Network group

( sinks)

..

..

.

.

.

.

.

.

U1

U2

UMChannel MSink M

M

N

N

N

K1

K2

KM

M

CSI for all sinks(γvector)

Figure 3: Transmission scheme at source node for the proposed adaptive transmission scheme: the number of information symbols per

packet addressed to a given sink is adapted to the sink’s channel status using channel encoders at diﬀerent rates. In the picture, the packet

length at the output of the various blocks is indicated.

sequences in the channel code book is 2nM

i=1Ri.Sourcenode

produces a set of Mmessages, one for each information

source in it. Given a set of messages, the corresponding

channel codeword Xis selected and transmitted over the

channel. Sink Didecodes the output Yiof his channel by

ﬁxing M−1 dimensions in the channel code book using

its side information about the set of information sources

S\ISiand applying typical set decoding along dimension i.If

we impose that for each information source Ri<I(X;Y

i)=

log2(1 + γi)whereXand Y

iare, respectively, the input

and output of a channel where only transmission to sink

Ditakes place, then an achievable rate for the system is the

sum of the instantaneous achievable rates of the various

links

Rair =

M

v=1

log21+γv.(12)

Let us now consider the scheduling solution proposed

in [10]. According to this solution, sinks are ordered from

1toMwith increasing channel quality. The M−v+1

information sources aiming to transmit to the M−v+1

sinks with best channels (i.e., sinks Dv,Dv+1,...,DM)are

selected. Each information source in the source node chooses

a message from a set of 2nR elements, where Ris chosen so

that R=log2(1 + γv). This means that only sinks whose

channels have instantaneous capacity greater than or equal to

node vcan decode their message. Only information sources

that produce messages addressed to these nodes are selected

for transmission. An achievable rate for this system can be

obtained from (12) by setting to 0 the ﬁrst vterms in the

sum, setting the others equal to log2(1 + γv) and optimizing

with respect to v

Rcir =max

v(M−v+1

)log21+γ(v),(13)

where γ(v)indicates the vth worst channel SNR. In order

to compare the two approaches, we will consider the

probability, or equivalently the percentage of time, during

which each of the systems achieves a rate lower than a given

value R, that is,

P{Rinst <R}=FRinst (R),(14)

where FRinst(R) is the cumulative density function of the

variable Rinst. In the constant information rate system such

probability is

P{Rcir <R}=Pmax

v(M−v+1

)log21+γv<R

.

(15)

EURASIP Journal on Wireless Communications and Networking 7

-dimensional

encoder

Channel 1

p(y1|x)

2

Decoder 1

Decoder 2

W1

W2

WM

Y1

Y2

YM

.

.

..

.

.

.

.

.

Source node

Sink nodes

Channel

p(y2|x)

Channel

p(yM|x)

W1

W2

WM

Decoder M

M

MX

IS-1

IS-2

IS-M

{W2,...,WM}

{W1,W3,...,WM}

{W1,W2,...,WM−1}

Figure 4: Equivalent scheme for adaptive transmission. Minformation sources {IS1,...,IS

M}are located in the source node. Information

source ISiproduces a message addressed to sink iwhich has previous knowledge of messages produced by information sources in the subset

S\ISi.p(yi|x) represents the probability transition function of the channel between the source node and sink i.

We calculated this expression for a network with a generic

number Mof nodes (see Appendix A). Such expression is

given by

FRcir (R)

=

1

j1=0

M−j1

jM=1

min(2−j1,M−j1−jM)

j2=0

min(2−j1−j2,M−j1−j2−jM)

j3=0

···

min(M−2−j1−···− jM−3,M−j1−j2−···− jM−3−jM)

jM−2=0

×M!

j1!··· jM!αj1

1αj2

2···αjM−2

M−2αM−j1−j2−···− jM−2−jM

M−1αjM

M,

(16)

where

αj=αj(R)=e1/γ e−2R/(j+1)/ γ −e−2R/ j /γ , (17)

for v/

=M,and

αM=αM(R)=e1/γ e−1/γ −e−2R/M /γ , (18)

γbeing the mean value of the SNR, assumed to be exponen-

tially distributed.

Let us now consider the cumulative density function

for our proposed system (adaptive information rate). By

deﬁnition we have

P{Rair <R}=P⎧

⎨

⎩

M

v=1

log21+γv<R

⎫

⎬

⎭

=P⎧

⎨

⎩

M

v=1

ci<R

⎫

⎬

⎭

=R

−∞ fc1(c)⊗···⊗ fcM(c)dc,

(19)

where:

fci(c)=e1/γ

γln(2)2ce−2c/γ u(c),(20)

u(c) being a function that assumes value 0 for c<0and1for

c>0. Expression (19)isdiﬃcult to calculate in closed form

for the general case. For the low SNR regime we calculated

the following expression (see Appendix B):

P{Rair <R}=1−e−Rln(2)/γ

M−1

v=0Rln(2)/γv

v!.(21)

In Figure 5, expressions (16)and(21)arecomparedfor

a Network Group of 5 nodes and an average SNR of −10 dB.

The Montecarlo simulation of our system is also plotted for

comparison with (21). At higher SNR (see Figure 6), the CDF

of AIR system is upper bounded by (16)andlooselylower

bounded by the (21) (see Appendix B). A better lower bound

is given by (see Appendix B):

F−

Rdir (R)=eM/γe−1/γ −e−2R/M /γ M.(22)

8 EURASIP Journal on Wireless Communications and Networking

00

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

Capacity

CDF

Analytic approximation AIR

Montecarlo AIR

Analytic CIR

Figure 5: Comparison between cumulative density functions in the

system with constant information rate (CIR), adaptive information

rate (AIR) and Montecarlo simulation of AIR. For each value of

R, the constant rate system has a probability not to achieve a rate

equal or greater that Rwhich is higher with respect to our system.

Equivalently, our system will be transmitting at a rate higher than R

for a greater percentage of time.

0

051015

0.1

1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Achievable rate

Analytic AIR (approx. at low SNR)

Montecarlo AIR

Analytic CIR

Lower bound AI

R

CDF N=5

Figure 6: Comparison between cumulative density functions of the

two systems with M=5 nodes and SNR =5 dB. We can see how

for the 40% of time the rate of AIR system will be above 8 bits/s/Hz

while CIR system achievable rate will be above 5.2 bits/s/Hz. At high

SNR the (21)isalooseupperboundforthe(19). A tighter lower

bound is given by 22 which is also plotted.

The ergodic achievable rate of the two systems can now

be calculated. For the constant information rate system, we

have

Rcir =E{Rcir}=+∞

−∞ cdFRcir(c)

dc dc,(23)

where FRcir(c)isgivenby(16).

11.522.533.544.55

0

1

2

3

4

5

6

7

8

9

10

Average SNR (dB)

Ergodic achievable rate

Analytical AIR

Montecarlo CIR

Figure 7: Ergodic achievable rate for AIR and CIR systems for

a Network Coding group with M=5 nodes. The high values of

the rates are due to NC gain. We see how AIR system gains about

2 bits/sec/Hz in all the considered SNR range.

As for the system with adaptive information rate, we have

Rair =E{Rair}=E⎧

⎨

⎩

M

v=1

ci⎫

⎬

⎭

=ME{ci}=Me1/γ

ln 2 E11

γ,

(24)

where E1(x) is the exponential integral deﬁned as

E1(x)=∞

1

e−tx

tdt. (25)

In Figure 7, the average achievable rate of the two

systems, assuming constant transmitted power, is plotted

against the mean SNR for AIR and CIR systems with M=5

nodes.

6. Simulation Setup and Results

In this section, we describe the implementation of the

proposed scheme using nonbinary LDPC codes and soft

decoding.

6.1. Notation. During each transmission slot the source node

combines together the packets in U(see Section 4)and

broadcasts the resulting packet to sink nodes of the network

group. In this paper, we used the DaVinci codes, that is, the

nonbinary LDPC codes from the DaVinci project [17]. For

such codes the order of the Galois ﬁeld is q=64 =26,

that is, each GF symbol corresponds to 6 bits. We denote the

elements of the ﬁnite ﬁeld by Fq={0, 1, ...,q−1}, where 0

is the additive identity.

u

i∈FKi

qdenotes the message of user i,oflengthKi

symbols, that is, 6Kibits. ci∈FN

qis the codeword of user

i,oflengthN=480 symbols, that is, 6 ·480 =2880 bits,

constant for all users.

6.2. L-Vectors. Acodewordccontains Ncode symbols. At the

receiver, the demapper provides the decoder with an LLR-

vector (log-likelihood ratio) of dimension qfor each code

EURASIP Journal on Wireless Communications and Networking 9

symbol, that is, for each codeword, the demapper has to

compute q·Nreal values.

The LLR-vector corresponding to code symbol nis

deﬁned as L=(L0,L1,...,Lq−1), with

Lkln Pcn=k|y

Pcn=0|y .(26)

For 64-QAM and a channel code deﬁned over F64, this

simpliﬁes to (see e.g., [18])

Lk=1

N0!

!yn−hnμ(0)!

!

2−!

!yn−hnμ(k)!

!

2,(27)

where μ:Fq→Xis the mapping function, which maps

a code symbol to a QAM constellation point, the noise is

CN(0, N0) distributed and hnis the channel coeﬃcient.

6.3. Network Decoding for LLR-Vectors. We w ant t o c o m pute

the LLR-vector of user i, having received yn=hnμ(cn)+wn.

c=U

i=1ciis the sum (deﬁned in Fq)ofallcodewords.

We assume that user iknows the sum of all other

codewords

c\i

U

j=1

j/

=i

cj.(28)

Then the LLR-vector of user ifor code symbol nis

L(i)

kln Pci,n=k|yn,c\i,n

Pci,n=0|yn,c\i,n =ln Pcn−c\i,n=k|yn

Pcn−c\i,n=0|yn

=ln Pcn=k+c\i,n|yn

Pcn=c\i,n|yn

=ln Pcn=k+c\i,n|yn

Pcn=0|yn

Pcn=0|yn

Pcn=c\i,n|yn

=Lk+c\i,n−Lc\i,n.

(29)

The sum in the indices is deﬁned in Fq. In Figure 8the

block scheme of the ith receiver is illustrated.

Note that in our scheme, we have inverted the order of

network and channel coding, while doing soft decoding at

the receiver. This approach has the important advantage of

allowing rate adaptation while fully preserving the advan-

tages of channel and network coding.

The network coding stage is transparent to the channel

coding scheme; that is, the channel seen by the channel

decoder is equivalent to the channel without network coding.

This is the reason why no speciﬁc design of the channel code

is required for the proposed scheme.

6.4. Rate Adaptation. For 64-QAM with the DaVinci codes

of length N=480 code symbols and rates Rc∈

{1/2, 2/3, 3/4, 5/6}, we obtain the following word error rate

(WER) curves.

For a target WER of 10−3, this leads to the SNR thresholds

of Table 1.

Soft

demapper

Networ k

decoder

Channel

decoder

yCNLRq×NL(i)

c\i

ui

User i

Figure 8: Receiver scheme for node i. The demapper provides

the decoder with Lvectors relative to received symbols. Network

decoder uses knowledge of symbol c\ito calculate L(i)vector, that is,

the Lvector of ci.

8 10121416182022242628

10−3

10−2

10−1

100

SNR (dB)

WER

AWGN channel, N=480 code symbols

Rc=1

2

Rc=2

3

Rc=3

4

Rc=5

6

Uncoded

Figure 9: Word error rate (WER) for nonbinary LDPC codes

at various rates. The high slopes of the curves allow to deﬁne

thresholds for the various rates, such that a very low word error rate

(<10−3) is achieved beyond the threshold, while it rapidly increases

before such thresholds.

6.5. Simulation Results. In the following, the channel is block

Rayleigh fading with average SNR γ.ForM=5users,sum

rates for the proposed system and for the benchmark system

are depicted in Figure 10.

Next, we consider two users, where the ﬁrst one has

average SNR γ1and the second one γ2=0.1γ1, that is, 10 dB

less. The resulting rates are depicted in Figure 11.

As before, the error rate is very low in both cases (the

adaptation is designed such that Pw<.001, and this is

fulﬁlled.)

7. Implementation

In this section, we discuss some issues arising by the

application of our proposed scheme. In particular we discuss

a generalization of network groups, in order to apply our

method to a real system, the eﬀects of packet fragmentation

duetotheuseofdiﬀerent code rates and the implications our

method has on system fairness.

7.1. Generalized Network Group. In Section 2,weassumed

that, at each transmission, the source combines so that each

of the sinks knows all but one of the packets. This assumption

can be relaxed, leading to a more general case which makes

our scheme usable in most situations arising in practice.

10 EURASIP Journal on Wireless Communications and Networking

Tab le 1: In the table the information packet length Kand the coding rate Rcare indicated for each SNR threshold. Note that for each

threshold we have: K/Rc=480, that is, all encoded packets have the same length.

K0 240 320 360 400 480

Rc01/22/33/45/61

SNR (dB) −∞ 11 14.415.917.527

10 15 20 25 30 35 40

0

Average SNR ¯

γ(dB)

5

10

15

20

25

30

Block fading, 5 users

Benchmark

Sum rate

Rate-adaptive

Figure 10: Sum rate for AIR and CIR systems for a Network Coding

group with M=5 nodes. Variable rate nonbinary LDPC codes

with 64 QAM modulation have been used. The high values of the

rates are due to NC gain. We see how AIR system gains about

2 bits/channel use in the higher SNR range. It is interesting to note

that almost the same gain has been calculated in Section 5when

considering the average achievable rates for CIR and AIR systems

with the same number of nodes at lower SNRs.

Let us consider a generalized network group of size M.The

source has a set of packets USwhile sink jhas a set of packets

Ujlacking one or more packets in US. Let us now deﬁne the

set U∗

∩\ jas

U∗

∩\ j=U1∩···∩Uj−1∩Uc

j∩Uj+1 ∩···∩UM,

(30)

where Uc

jdenotes the complement of Uj. In other words,

U∗

∩\ jrepresents all packets which are common to all sinks

but sink j. The source transmits to node jone of the packets

in the set US∩U∗

∩\ j(i.e., all packets in U∗

∩\ jwhich are

known to the source node). Thus, if we indicate with |U|the

cardinality of set U, the sink jwill need |US∩U∗

∩\ j|linearly

independent (in GF(q)) packets in order to decode all the

|US∩U∗

∩\ j|original native packets [19]. Such l.i. packets can

be obtained from the same source node or from other nodes

in the network which previously stored the packets. With

such scheme a total of maxj(|Uc

i|) transmission phases are

needed for all the sinks to know all the packets. As a special

case, if |US∩U∗

∩\ j|=1forall j, we have the NG considered

in Section 2.

0

1

2

3

4

5

6

Benchmark, user 1

Benchmark, user 2

Rate-adaptive, user 1

Rate-adaptive, user 2

R1,R2

10 15 20 25 30 35 40

Average SNR ¯

γ1(dB)

Block fading, 2 users, ¯

γ2=0.1¯

γ1

Figure 11: Comparison of the rates of two nodes belonging to

a Network Coding group with M=2 nodes in both AIR and

CIR systems. One of the nodes suﬀers from a higher path loss

attenuation (10 dB) with respect to the other. Node with better

channel in AIR system achieves higher rate with respect to node

with better channel in CIR system. The gain arises from adapting

the coding rate of each node to the channel independently from the

other nodes.

In order to understand how to proceed when more than

one packet is unknown at one or more sinks, deﬁne an M-

dimensional vector space associated to the source packet

set US. A canonical basis for this space is deﬁned as e1=

[10 ···0] ···eM=[0 ···01]. The transmitted packet is a

linear combination of this basis, x=a1∗e1+···+aM∗eM.

The sets of missing packets in sink i,Uc

i,deﬁnea

|Uc

i|-dimensional space. In the concept of network group

described in Section 2, the transmitted packet is obtained as

x=e1+··· +eM, which is linearly independent from the

subspace spanned by the packets owned by sinks 1···M.As

a result, the packets contained in each sink together with x

span the whole space IS, therefore all packets can be decoded.

In a more general case, where more than one packet

is unknown by one or more sinks, we need to transmit a

number of packets that, along with the subspaces spanned

by the packets of sinks 1 ···M, span the whole US.

Transmitting maxi(|Uc

i|) linear combinations of packets is

suﬃcient to achieve this goal.

EURASIP Journal on Wireless Communications and Networking 11

S

P1

P1

P1P2

P2

P3

P3

P3

P6

P6

P6

P4

P4

P4

P5

P5

P5

N1

N2

N3

γ1

γ2

γ3

Figure 12: In the setup the three sinks have three distinct subsets of

packets in S’s buﬀer and channels from Sto each of the sinks have a

diﬀerent SNR.

0R

N

R

3

R

2R

N

x(1)

x(N−2)

x(N−1)

x(N)

.

.

.

Figure 13: Virtual representation of event A. Random variables

x1,...,xMare sorted in ascending order in sequence x(1),...,x(M).

According to the deﬁnition of event Athe vth variable must assume

a value less or equal R/(M−v+1).

In Figure 12, an example is given which clariﬁes the

concept just described. In the setup the three sinks have

three distinct subsets of packets and channels from Sto

each of the sinks have SNRs γ1,γ2and γ3.Table2gives a

possible scheduling and transmission solution for the setup

in Figure 12 by applying the method we just described

together with channel adaptation.

In particular, during the transmission the source broad-

casts a packet obtained by adding packets pγ1

4,pγ2

1,andpγ3

6,

where pγ1

4is packet p4after channel encoding adapted to

γ1. Once sink 1 receives p4, it needs packet p5.Nextpacket

transmitted by Sis pγ1

5added with pγ2

3and pγ3

2for sinks 2 and

3, respectively. Finally packet pγ2

2is transmitted to sink 2.

7.2. Packet Fragmentation and Fairness. Our proposed solu-

tion implicitly assumes that native packets can be frag-

mented. Each native packet ucan be considered as a length

Kbuﬀer. In order to match the optimal rate on the channel,

only a part of the buﬀer uis sent over the channel during a

time slot on size Ncoded packet. In the following, we discuss

how to handle native packet fragmentation at the network

level.

Scheduling in Packet Fragmentation. When a node requests a

packet that needs to be fragmented the ﬁrst part of the packet

is always sent out ﬁrst. This avoids that diﬀerent nodes in

the network have nonoverlapping parts of the same native

packet, which could make the formation of network coding

groups more diﬃcult. Let us now consider the case in which a

given node irequests a fragment fvof a given native packet ui.

In this case, nodes belonging to its NC group do not need to

know the whole native packet. It is suﬃcient that the portion

they know of native packet uiinclude fragment fv.

Capacity and NC Group Limits. The maximum rate at which

agivennodeinanetworkgroupcanreceivedataisactually

limited by two factors. One is the capacity of the physical

link between source and node (capacity-limited rate). The

other factor that limits the transmission rate is the minimum

across the nodes of the NC group of the portion of packet

ui. If such portion has length K, then the maximum

transmission rate for packet uiduring a packet slot must be

less than K/N, otherwise not all nodes in the NC group

will be able to correctly decode the packet addressed to

them (NC group-limited rate). The last factor must be taken

into account in the formation of the NC group. In order to

avoid such situation we can impose that a packet cannot be

transmitted before it has been completely received.

Fairness Improvement. Shadowed users in a network would

probablyexperienceahighpacketlossrate.TheCIR

approach penalizes those nodes, as their channels will have

a low capacity. By adapting the rate to each of the nodes’

channel conditions we can guarantee that users which expe-

rience shadowing for a long time (e.g., because of big urban

barriers) are not totally excluded from the communication.

This is likely to increase fairness and decrease delay in the

system.

These are some side eﬀects at network level of our pro-

posed method. The global behavior of a network in terms of

aggregated throughput, reliability, delay, and fairness where

such transmission scheme is used need to be quantiﬁed by

means of analytical/numerical methods, and is beyond the

scope of this paper.

8. Conclusion

In this paper we proposed a new approach for rate adaptation

in opportunistic scheduling. Such approach applies channel

adaptation techniques originally proposed for asymmetric

TWRC communication to a network context. After system

model deﬁnition at both packet level (network group) and

physical level (channel statistics), we described previously

proposed methods for transmission scheduling in NC. We

carried out a comparison between our method (adaptive

information rate) and the scheduling method typically

used in nc (constant information rate) from a information

theoretical point of view. We obtained expression for the cdf

of achievable rates for CIR system and a lower bound for

AIR system’s cdf. We also calculated an approximation to

AIR cdf at low SNRs and showed that cdf of CIR systems

12 EURASIP Journal on Wireless Communications and Networking

Tab le 2: Scheduling solution for the setup of Figure 12.txkindicates the transmission phase. Each phase corresponds to the complete

transmission of a native packet (or a sum of native packets).

Trx p h a s e U1US∩U∗

∩\1U2US∩U∗

∩\2U3US∩U∗

∩\3Transmitted

0p1,p2,p3,p6p4,p5p4,p5,p6p1,p3p1,p3,p4,p5p6pγ1

4⊕pγ2

1⊕pγ3

6

1p1,p2,p3,p6,p4p5p4,p5,p6,p1p3p1,p3,p4,p5,p6p2pγ1

5⊕pγ2

3⊕pγ3

2

2p1,p2,p3,p4,p5,p6p1,p3,p4,p5,p6p2p1,p2,p3,p4,p5,p6pγ2

2

3p1,p2,p3,p4,p5,p6p1,p2,p3,p4,p5,p6p1,p2,p3,p4,p5,p6

is an upper bound that of AIR system. We implemented a

simulator using nonbinary LDPC codes developed in the

DaVinci project [17] and showed that our method allows

a better exploitation of good channels with respect to CIR

method. This was shown to increase throughput at each

transmission. We then discussed some issues that arise from

the modiﬁcations at physical level brought from AIR method

in a network coding scenario. Such issues will be extensively

analyzed and their impact quantiﬁed in our future works, as

well as a system-level throughput analysis gain. New coding

techniques can also be investigated in order to fully exploit

achievable throughput and fairness enhancements in AIR

systems.

Appendices

In the following, we derive the calculation for the cumulative

density function of the achievable rate for the system with

constant information rate and the approximation for the cdf

of the adaptive information rate system we proposed in this

paper. We talk about achievable rates and not capacity as we

are not optimizing with respect to power.

A. Constant Information Rate

Channel coeﬃcients are i.i.d. exponentially distributed ran-

dom variables with mean value γ. Their marginal pdf is then

fΓγ=1

γe−γ/γuγ.(A.1)

Letussortchannelcoeﬃcients of the Mreceivers in

ascending order, namely,

γ(1) <γ

(2) <···<γ

(M−1) <γ

(M).(A.2)

We will use round brackets to indicate variables sorted

in ascending order, that is, γ(1) is the smallest among

variables γ(v). As stated in Section 5, the cdf for the constant

information rate system is given by:

FRcir (R)=P{Rcir <R}

=P"max

v∈{1,...,M}(M−v+1

)log21+γ(v)<R

#.

(A.3)

Let us introduce the following notation:

xv=log21+γv,

x(v)=log21+γ(v),(A.4)

and ﬁnally

z=max

v∈{1,...,M}(M−v+1

)log21+γv=Rcir.(A.5)

Using (A.5)in(A.3)wecanwrite

Fcir(R)=P{z<R}=FZ(R),(A.6)

where FZ(R) is the cumulative distribution function of the

variable zcalculated in point R.ThefunctionFZ(R)is,by

deﬁnition

FZ(R)=P$Mx(1) <R,(M−1)x(2) <R,...,x(M)<R

%.

(A.7)

Note that the smaller the variable x(v), the higher the

multiplying coeﬃcient M−v+1.

We can rewrite the (A.7)as

FZ(R)=Px(1) <R

M,x(2) <R

M−1,...,x(M)<R

.(A.8)

Let us indicate the event inside brackets as A. Figure 13 gives

a graphical representation of event A.

We can calculate the probability of event Aby using the

law of total probability

FZ(R)=

M

i=1

P{A∩Bi},(A.9)

where Biare disjoint events partitioning the area of the

sample space to which Abelongs. Let us choose as Bithe

event “jnout of Mvariables fall in the interval [R/(n+

1), R/n]” for all n∈{1, 2, ...,M}and putting R/(M+1)=0

and M

n=1jn=M. The intersection with Aimposes on Bithe

further constraint

jn≤n,∀n∈{1, 2, ...,M}.(A.10)

Let us give an example to clarify the deﬁnitions given

up to now for the case with M=2nodes.Wehavetwo

i.i.d. random variables x1and x2. We sort them and call

the smallest one x(1) and the biggest one x(2).EventAis,

by deﬁnition: A={x(1) <R/2, x(2) <R}.EventsBi,with

i∈{1, 2, 3}are the following:

(i) B1=“2 variables fall in the interval [R/2, R]and0

variables fall in the interval [0, R/2]”;

(ii) B2=“2 variables fall in the interval [0, R/2] and 0

variables fall in the interval [R/2, R]”;

EURASIP Journal on Wireless Communications and Networking 13

(iii) B3=“1 variable falls in the interval [R/2, R]and1

variable falls in the interval [0, R/2]”.

It is easy to see that these are disjoint events which partition

the sample space, that is, they take into account all the

possible ways in which the two variables can be distributed

in the two intervals. In order to calculate the (A.9), we need

to ﬁnd the intersection between event AandeachoftheBi.

It can be easily veriﬁed that such intersection can be found

by adding to each Bithe constraint (A.10), which, for M=2,

can be expressed as “the number of variables that fall in the

interval [R/2, R] must be less than or equal to 1 and the

number of variables that fall in the interval [0, R/2] must be

less than or equal to 2”. This implies that the (A.9)isgiven

by the sum of the probabilities of events B2and B3. Note that

events Bido not consider sorted variables, as the sorting is

implicitly deﬁned in the deﬁnition of such events. This allows

to consider the variables as i.i.d, which makes calculation of

events Bieasier.

A similar calculation can be done for a generic number M

of nodes. As seen in the example, the calculation reduces to

deﬁning events Bi, choose those which describe event Aand

sum their probabilities. Such probabilities can be calculated

as follows. The probability that a generic variable xv=

log2(1 + γv) (unsorted) falls in the interval [R/(n+1),R/n]is

equal to FX(R/n)−FX(R/(n+ 1)), FX(x) being the cumulative

density function of x.FX(x) can be obtained transforming

the exponential r.v. γ

FX(x)=e1/γ e−1/γ −e−2x/γ u(x), (A.11)

where u(x) is a function that assumes value 0 for x<

0, 1 for x>0and1/2 in 0. Because of independency

among the variables, we can calculate the probability that

jnvariables fall in the interval [R/(n+1),R/n], which is

[FX(R/n)−FX(R/(n+ 1))] jn. From now on, we will indicate

with αnthe diﬀerence FX(R/n)−FX(R/n + 1). We can now

express the probability of the union of events Biwith the

formula (A.12)

M

i=1

P{Bi}

=

M

j1=0

M−j1

j2=0

M−j1−j2

j3=0

···

M−j1−j2−···− jM−3−jM−2

jM−1=0

×M!

j1!··· jM!αj1

1αj2

2···αjM−2

M−2αjM−1

M−1αjM−j1−j2−···−jM−2−jM−1

M,

(A.12)

where the coeﬃcient M!/j

1!··· jM! is the number of parti-

tions of M elements in M bins putting jnelements in bin

number n. Finally, including constraint (A.10)weobtain

expression (A.13)

FZ(R)

=

M

i=1

P{A∩Bi}

=

1

j1=0

M−j1

jM=1

min(2−j1,M−j1−jM)

j2=0

min(2−j1−j2,M−j1−j2−jM)

j3=0

···

min(M−2−j1−···− jM−3,M−j1−j2−···− jM−3−jM)

jM−2=0

×M!

j1!··· jM!αj1

1αj2

2···αjM−2

M−2αM−j1−j2−···− jM−2−jM

M−1αjM

M.

(A.13)

B. Adaptive Transmission

B.1. CDF in the Low SNR Regime. Let us indicate with cithe

(unsorted) instantaneous capacity of the link between source

and receiver i. Let us recall from Section 5that an achievable

rate for such system is

Radapt =

M

i=1

ci.(B.14)

We wish to calculate an approximation for the cdf of Cair

in the low SNR regime. By deﬁnition the cdf of Rair is

FRair (c)=P⎧

⎨

⎩

M

i=1

ci<c

⎫

⎬

⎭

, (B.15)

where

ci=log21+γi,(B.16)

γibeing an exponentially distributed random variable with

mean value E{γi}=γi=γ.

When γi1 (which is the case most of the time in

the SNR regime), we can approximate the logarithm with its

Taylor expansion at the second term, that is

ci=log21+γiγi

ln(2).(B.17)

Thus, we have

Rair =

M

i=1

ci

M

i=1

γi

ln(2)=

M

i=1

γ

i.(B.18)

Using expression (B.18) we can calculate the pdf of Rair as

fRair (c)=fγ

1(c)⊗fγ

2(c)⊗···⊗ fγ

M(c).(B.19)

By substituting the expression of fγ

1(c)in(B.19)weﬁnd

fRair (c)=cM−1e−c/γ

(M−1)!γMu(c),(B.20)

14 EURASIP Journal on Wireless Communications and Networking

and ﬁnally:

Flow

Rair (c)=c

0

xM−1e−x/γ

(M−1)!γMdx =1−e−cln(2)/γ

M−1

v=0cln(2)/γv

v!.

(B.21)

At higher SNR the (B.24) is a loose lower bound for the cdf

of Cair, in fact we have the following inequalities:

γ

i=γi

ln(2)>log21+γi=ci,(B.22)

M

i=1

γ

i>

M

i=1

ci, (B.23)

Flow

Rair (c)=P⎧

⎨

⎩

M

i=1

γ

i<c

⎫

⎬

⎭

<P

⎧

⎨

⎩

M

i=1

ci<c

⎫

⎬

⎭

=FRair (c).(B.24)

B.2. Upper Bound of cdf. We now show that the (16)upper

bounds the cdf of the achievable rate for the AIR system.

Let us start by modifying the condition in brackets in the

(B.15) that we will call condition β. We relax such condition

so that it be veriﬁed with higher probability for each R.Such

condition says that the sum of capacities in all links must not

exceed R. We want to ﬁnd a condition δso that if βis true

also δis true, but there must exist a set of events with non

zero probability for which if δis veriﬁed βis not. For this

purpose, let us put δ=A,whereAis the event that deﬁnes

the cdf of cir system (see Appendix A), that is

δ=c1<R

M,...,ci<R

M−i+1,...,cM<R

.(B.25)

Now it is suﬃcient to prove that the following two proposi-

tions are true

β⊂δ,(B.26)

∃s⊂δ|s/

⊆β,P{s}>0.(B.27)

Let us start with the (B.26). For βto be veriﬁed, at least one

of the cimust be <R/Mbecause if not the sum in βwould

be greater than R. Moreover, if we impose that cj<R/M

for a given j, there must be at least another cisuch that

ci<(R/(M−1)). If this is not veriﬁed there will be M−1ci

for which ci>(R/(M−1)) plus cj, so the total sum would be

greater than R. Iterating this Mtimes we will obtain exactly

the condition δwhich, as just shown, must be veriﬁed for the

βto be true. Now let us consider the (B.27). We can take as

condition sthe following:

s=R

M+1 <c

1<R

M,R

M<c

2<R

M−1,

...,R

M<c

M−1<R

M−1,MR

M+1 <c

M<R

.

(B.28)

It can be easily seen that s⊂δ. The minimum value for the

sum of all ciunder condition sis R(2 −2/M) which is greater

than Rfor M2. This means that s/

⊆β.Wehaveleftto

show that P{s}>0. The probability of sis a ﬁnite quantity

given by

P{s}

=FCR

M−FCR

M+1

+&FCR

M−1FCR

M'M−2

+FCMR

M−FCMR

M+1,

(B.29)

the FC(c) being the cdf of the random variable c=log2(1+γ).

We recall the expression for the FC(c)

FC(c)=e1/γ e−1/γ −e−2c/γ u(c).(B.30)

B.3. Lower Bound. Inordertoﬁndalowerboundforthecdf

of AIR system, we introduce the following constraint to the

condition inside brackets in the (B.15)

ci<R

M,∀i∈{1, 2 ...,M}.(B.31)

Adding (B.31)in(B.15) we obtain the following expression:

F−

adapt(R)=P⎧

⎨

⎩

M

i=1

ci<R,ci<R

M,∀i∈{1, 2, ...,M}⎫

⎬

⎭

=Pci<R

M,∀i∈{1, 2, ...,M}=FM

ciR

M

=eM/γe−1/γ −e−2R/M /γ M.

(B.32)

Acknowledgments

The authors would like to thank Dr. Deniz Gunduz for

the helpful discussions made during the development of

present work. This work was partially supported by the

Spanish Government through Project m:VIA (TSI-020301-

2008-3), by the European Commission by INFSCO-ICT-

216203 DaVinci (Design And Versatile Implementation of

Nonbinary wireless Communications based on Innovative

LDPC Codes) and the Network of Excellence in Wireless

COMmunications NEWCOM++ (Contract ICT-216715),

and by Generalitat de Catalunya under Grant 2009-SGR-

940. G. Cocco is partially supported by the European Space

Agency under the Networking/Partnering Initiative.

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3, pp. 497–510, 2008.

[10] H. Yomo and P. Popovski, “Opportunistic scheduling for

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Photographȱ©ȱTurismeȱdeȱBarcelonaȱ/ȱJ.ȱTrullàs

Preliminaryȱcallȱforȱpapers

The 2011 European Signal Processing Conference (EUSIPCOȬ2011) is the

nineteenth in a series of conferences promoted by the European Association for

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Centre Tecnològic de Telecomunicacions de Catalunya (CTTC) and the

Universitat Politècnica de Catalunya (UPC).

EUSIPCOȬ2011 will focus on key aspects of signal processing theory and

li ti

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OrganizingȱCommittee

HonoraryȱChair

MiguelȱA.ȱLagunasȱ(CTTC)

GeneralȱChair

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GeneralȱViceȬChair

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Technical Program Co

Ȭ

Chairs

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relevance and originality. Accepted papers will be published in the EUSIPCO

proceedings and presented during the conference. Paper submissions, proposals

for tutorials and proposals for special sessions are invited in, but not limited to,

the following areas of interest.

Areas of Interest

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ȱ

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ȱ

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PlenaryȱTalks

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Submissions

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ȱ

(UPC)

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ustr

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sonȱ

&

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hibi

ts

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(UniversityȱofȱPiraeus)

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InternationalȱLiaison

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roposa

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orȱspec

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a

l

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onsȱ

15

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ecȱ

2010

Proposalsȱforȱtutorials 18ȱFeb 2011

Electronicȱsubmissionȱofȱfullȱpapers 21ȱFeb 2011

Notificationȱofȱacceptance 23ȱMay 2011

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