Article

Opportunistic Adaptive Transmission for Network Coding Using Nonbinary LDPC Codes

Abstract and Figures

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 of 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.
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Hindawi Publishing Corporation
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
eectiveness 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
dierent channel conditions due to fading and dierent
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 Eros 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
dierent 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 buer by
the transmitting node (source node). At each transmission,
a part of the buer, containing Kbits, is included in a
new packet of total length Nthat contains NKbits 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 eective
throughput on link idefined as
thi=rk1ppli(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 aect 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 coecients. As the combination is linear
and coecients are known, a node can decode all packets
if and only if it receives a sucient 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 dierent 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].
Anetworkgroupisformallydenedasfollows.
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’
buers. 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 dierence 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.Letusdenepacketx\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 eciency 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
γeγi(t) ,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 eects.
3. Constant Information Rate Opportunistic
Scheduling Solutions
Based on the propagation model in (5), the channel from
source to each sink will have a dierent gain. The dierence
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 Mv+ 1 sinks with
highest SNR. The transmission rate Rchosen by the source
node is the lowest capacity in the group of Mv+1
channels. The instantaneous capacity obtained during each
transmission is then
C(v)
inst =(Mv+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 eective throughput as
th =
M
i=11ppliri=1pplTr,(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 eective
EURASIP Journal on Wireless Communications and Networking 5
throughput for each sink (individual eective throughput).
Formally, for each sink i,wecalculate
r
i=arg max
ri1ppliγi,riri,(8)
where ppli(γi,ri) is the packet loss probability on the ith link
depending on the rate ri.Foreachrater
k,wedenemkas
the number of sinks for which
rir
k.(9)
At this point, for each kwe calculate the eective throughput,
setting r=rk1kwhere 1kis a mk-dimensional vector of
all ones. Finally, we choose kto maximize the eective
throughput. Note that with the CIR approach only sinks
whose optimal rate is greater or equal than the rate which
maximizes the total eective 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 eective throughput, defined
as
roptγ=arg max
r
M
i=11ppliγi,riri
=arg max
r1pplγ,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 (103)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=11p
pliγi,riri
=arg max
r1p
plγ,rTr,
(11)
where p
pli(γi,ri)takesvalue1ifγγthresh 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 2nRidierent 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 buer
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 dierent 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 M1 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 Mv+1
information sources aiming to transmit to the Mv+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(Mv+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(Mv+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,...,WM1}
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
Mj1
jM=1
min(2j1,Mj1jM)
j2=0
min(2j1j2,Mj1j2jM)
j3=0
···
min(M2j1−···− jM3,Mj1j2−···− jM3jM)
jM2=0
×M!
j1!··· jM!αj1
1αj2
2···αjM2
M2αMj1j2−···− jM2jM
M1αjM
M,
(16)
where
αj=αj(R)=e1 e2R/(j+1)/ γ e2R/ j , (17)
for v/
=M,and
αM=αM(R)=e1 e1e2R/M , (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
γln(2)2ce2c u(c),(20)
u(c) being a function that assumes value 0 for c<0and1for
c>0. Expression (19)isdicult 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}=1eRln(2)/γ
M1
v=0Rln(2)v
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/γe1e2R/M 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
ln 2 E11
γ,
(24)
where E1(x) is the exponential integral defined as
E1(x)=
1
etx
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, ...,q1}, where 0
is the additive identity.
u
iFKi
qdenotes the message of user i,oflengthKi
symbols, that is, 6Kibits. ciFN
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,...,Lq1), 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!
!ynhnμ(0)!
!
2!
!ynhnμ(k)!
!
2,(27)
where μ:FqXis the mapping function, which maps
a code symbol to a QAM constellation point, the noise is
CN(0, N0) distributed and hnis the channel coecient.
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 Pcnc\i,n=k|yn
Pcnc\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,nLc\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 103, 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
103
102
101
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
(<103) 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 eects of packet fragmentation
duetotheuseofdierent 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∩···Uj1Uc
jUj+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 USU
∩\ 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 |USU
∩\ j|linearly
independent (in GF(q)) packets in order to decode all the
|USU
∩\ 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 |USU
∩\ 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 suers 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=a1e1+···+aMeM.
The sets of missing packets in sink i,Uc
i,denea
|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
sucient 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 buer and channels from Sto each of the sinks have a
dierent SNR.
0R
N
R
3
R
2R
N
x(1)
x(N2)
x(N1)
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/(Mv+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
Kbuer. In order to match the optimal rate on the channel,
only a part of the buer 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 dierent nodes in
the network have nonoverlapping parts of the same native
packet, which could make the formation of network coding
groups more dicult. 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 sucient 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 eects 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 U1USU
∩\1U2USU
∩\2U3USU
∩\3Transmitted
0p1,p2,p3,p6p4,p5p4,p5,p6p1,p3p1,p3,p4,p5p6pγ1
4pγ2
1pγ3
6
1p1,p2,p3,p6,p4p5p4,p5,p6,p1p3p1,p3,p4,p5,p6p2pγ1
5pγ2
3pγ3
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 coecients are i.i.d. exponentially distributed ran-
dom variables with mean value γ. Their marginal pdf is then
fΓγ=1
γeγ/γuγ.(A.1)
Letussortchannelcoecients of the Mreceivers in
ascending order, namely,
γ(1)
(2) <···
(M1)
(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}(Mv+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}(Mv+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,(M1)x(2) <R,...,x(M)<R
%.
(A.7)
Note that the smaller the variable x(v), the higher the
multiplying coecient Mv+1.
We can rewrite the (A.7)as
FZ(R)=Px(1) <R
M,x(2) <R
M1,...,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{ABi},(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
jnn,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 e1e2x 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 dierence 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
Mj1
j2=0
Mj1j2
j3=0
···
Mj1j2−···− jM3jM2
jM1=0
×M!
j1!··· jM!αj1
1αj2
2···αjM2
M2αjM1
M1αjMj1j2−···−jM2jM1
M,
(A.12)
where the coecient 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{ABi}
=
1
j1=0
Mj1
jM=1
min(2j1,Mj1jM)
j2=0
min(2j1j2,Mj1j2jM)
j3=0
···
min(M2j1−···− jM3,Mj1j2−···− jM3jM)
jM2=0
×M!
j1!··· jM!αj1
1αj2
2···αjM2
M2αMj1j2−···− jM2jM
M1αjM
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)=cM1ec/γ
(M1)!γMu(c),(B.20)
14 EURASIP Journal on Wireless Communications and Networking
and finally:
Flow
Rair (c)=c
0
xM1ex/γ
(M1)!γMdx =1ecln(2)
M1
v=0cln(2)v
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
Mi+1,...,cM<R
.(B.25)
Now it is sucient to prove that the following two proposi-
tions are true
βδ,(B.26)
sδ|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/(M1)). If this is not verified there will be M1ci
for which ci>(R/(M1)) 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
M1,
...,R
M<c
M1<R
M1,MR
M+1 <c
M<R
.
(B.28)
It can be easily seen that sδ. 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
MFCR
M+1
+&FCR
M1FCR
M'M2
+FCMR
MFCMR
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 e1e2c 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/γe1e2R/M 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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Photographȱ©ȱTurismeȱdeȱBarcelonaȱ/ȱJ.ȱTrullàs
Preliminaryȱcallȱforȱpapers
The 2011 European Signal Processing Conference (EUSIPCOȬ2011) is the
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Universitat Politècnica de Catalunya (UPC).
EUSIPCOȬ2011 will focus on key aspects of signal processing theory and
li ti
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... Unlike in [10], we assume fixed slot duration as often occurs in practical situations. So channel encoding is done by keeping constant the length of the coded packets x R i and varying the information rates based on the instantaneous channel states [15]. Finally, the relay calculates the XOR of the two encoded packets, [7][13]: ...
... This is due to the lower FER determined by the asymmetric channel adaptation. The analysis of the gain originated by asymmetric channel adaptation has been carried out in [15] in terms of ergodic capacity and capacity probability density function and a broadcast transmission for a generic number of nodes. ACKNOWLEDGEMENT This is the draft version of the following article: G. Cocco, D. Gunduz, C ...
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Bell System Technical Journal, also pp. 623-656 (October)