Robust Communication via Decentralized Processing With Unreliable Backhaul Links

Abstract

A source communicates with a remote destination via a number of distributed relays. Communication from source to relays takes place over a (discrete or Gaussian) broadcast channel, while the relays are connected to the receiver via orthogonal finite-capacity links. Unknowns to the source and relays, link failures may occur between any subset of relays and the destination in a nonergodic fashion. Upper and lower bounds are derived on average achievable rates with respect to the prior distribution of the link failures, assuming the relays to be oblivious to the source codebook. The lower bounds are obtained by proposing strategies that combine the broadcast coding approach, previously investigated for quasi-static fading channels, and different robust distributed compression techniques. Numerical results show that lower and upper bounds are quite close over most operating regimes, and provide insight into optimal transmission design choices for the scenario at hand. Extension to the case of nonoblivious relays is also discussed.

Full text

Page 1

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 20114187
Robust Communication via Decentralized Processing
With Unreliable Backhaul Links
Osvaldo Simeone, Member, IEEE, Oren Somekh, Member, IEEE, Elza Erkip, Fellow, IEEE,
H. Vincent Poor, Fellow, IEEE, and Shlomo Shamai (Shitz), Fellow, IEEE
Abstract—A source communicates with a remote destination via
a number of distributed relays. Communication from source to re-
lays takes place over a (discrete or Gaussian) broadcast channel,
while the relays are connected to the receiver via orthogonal fi-
nite-capacitylinks.Unbeknownsttothesourceandrelays,linkfail-
ures may occur between any subset of relays and the destination in
a nonergodic fashion. Upper and lower bounds are derived on av-
erage achievable rates with respect to the prior distribution of the
linkfailures,assumingtherelays tobeoblivioustothesourcecode-
book. The lower bounds are obtained by proposing strategies that
combine the broadcast coding approach, previously investigated
for quasi-static fading channels, and different robust distributed
compression techniques. Numerical results show that lower and
upperboundsarequitecloseovermostoperatingregimes,andpro-
vide insight into optimal transmission design choices for the sce-
nario at hand. Extension to the case of nonoblivious relays is also
discussed.
Index Terms—Broadcast coding, distributed source coding, era-
sure channel, relay channel, robust channel coding.
I. INTRODUCTION
W
hand is smaller than, or of the same order of magnitude of, the
linkoutageduration.Suchfailuresareoftenunpredictabletothe
transmitter, typically because of the absence of sufficiently fast
feedbacksignaling,tosimplifytransmitterdesign,orsimplybe-
causepacketlossesmaybecausedbyremoteeventssuchasnet-
work congestion, see, e.g., [1]. In these situations, conventional
channel coding is not effective in coping with link failures.
In the context of wireless channels, where outage is caused
by poor fading conditions, a standard approach considers
IRELESS or wired link failures are of a nonergodic na-
ture whenever the delay tolerated by the application at
Manuscript receivedDecember 21, 2008; revisedOctober 25, 2010; accepted
January 12, 2011. Date of current version June 22, 2011. This work was sup-
ported in part by the U.S. National Science Foundation under Grants CCF-
0914899, CNS-09-05398, CNS-06-26611, CCF-0635177, CNS-0905446, and
IIP-1032035, in part by a Marie Curie Outgoing International Fellowship, in
part by the NEWCOM++ network of excellence, and in part by the Israel Sci-
ence Foundation. The material in this paper was presented in part at the IEEE
Information Theory Workshop, Volos, Greece, June 2009.
O. Simeone is with the CWCSPR, New Jersey Institute of Technology,
Newark, NJ 07102 USA (e-mail: osvaldo.simeone@njit.edu).
O.SomekhandH.V.PoorarewiththeDepartmentofElectricalEngineering,
Princeton University, Princeton, NJ 08544 USA (e-mail: orens@princeton.edu;
poor@princeton.edu).
E. Erkip is with the Department of ECE, Polytechnic Institute of New York
University, Brooklyn, NY 11201 USA (e-mail: elza@poly.edu).
S.Shamai(Shitz)iswiththeDepartmentofElectricalEngineering,Technion,
Haifa, 32000, Israel.
Communicated by H. Bölcskei, Associate Editor for Detection and Estima-
tion.
Digital Object Identifier 10.1109/TIT.2011.2145830
fixed-rate transmission, for a given signal-to-noise-ratio (SNR),
and evaluates the best possible tradeoff between rate and
reliability (outage) [2]. However, in a number of important
applications, one may accept variable-rate data delivery, as in
the case of video broadcasting: the receiver will simply experi-
ence variable reception quality according to the current channel
state, and benefit from potentially good fading conditions
[3]–[7]. As first proposed in [6], such variable-rate delivery can
be achieved, without channel state information, by layering a
number of transmission streams via superposition coding. This
strategy is referred to as the broadcast (BC) coding approach.
Layering can then be optimized in terms of average achievable
rate with respect to a given prior distribution over the fading
gains [6].
The issue of nonergodic link failures has also been widely
studied in the context of wired networks, especially in recent
years in the field of network coding, see, e.g., [9]. When the
wired network is used for conveying information regarding cor-
related sources, the problem of transmission in the presence
of (unpredictable) link failures is one of robust source coding,
which has been studied in [10] and [11] for the case of a single
encoder and in [12] and [13] for multiple distributed encoders.
Theseworksshowthatitisgenerallyadvantageoustomakepro-
vision for the entire range of possible link conditions in order to
fully exploit the available tradeoffs in the reconstruction quality
at the receiver. This conclusion and approach are apparently
synergic with the BC coding strategy of [6]. Such a synergy,
also exploited in [3]–[7], and references therein in a different
context, discussed later, motivates this paper.
We consider a scenario in which a single source communi-
cates with a remote destination via a number of relays, also re-
ferredtoasagentsinrelatedliterature.Communicationbetween
source and agents is over a broadcast channel, either discrete or
Gaussian, while the agents are connected to the destination via
orthogonal limited-capacity channels. The scenario can be seen
asaspecialcaseofamultirelaychannel,withoutadirectlinkbe-
tween source and destination, and with no multiaccess interfer-
enceatthedestination.Inthissense,itisrelatedtothe“diamond
network” of [14], to “Aref networks” [15], where the broadcast
channel is deterministic, and to primitive relay channels [16],
where one relay is available and there is a broadcast channel
from source to relay and destination. In [17] the multirelay net-
work described above was studied under the assumption that
the relays are oblivious to the codebook used by the source; that
is, processing at the relays cannot depend on the specific code-
book selected by the source, as in, e.g., compress-and-forward
0018-9448/$26.00 © 2011 IEEE

Page 2

4188 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011
Fig. 1. Single transmitter (encoder) communicates to a remote receiver (decoder) via ?
links. The number of functioning links ? is unknown to source and relays (uninformed source and relays) and satisfies ? ? ? ? ? . Links are on or off for
the entire duration of the codeword (nonergodic scenario). Agents may also be oblivious to the codebook used by the source (oblivious agents) as in [17].
relays connected to the destination through unreliable finite-capacity
or amplify-and-forward achievable strategies. This assumption
is of particular relevance for nomadic applications, in which no
signalling is in place to exchange information regarding mod-
ulation and coding used at the source, or in networks with in-
expensive relays whose processing cannot adapt to the specific
source operation.
In this paper, we consider the multirelay channel of Fig. 1.
Unlike the works considered above, we assume that the links
between the relays and the destination are unreliable, suffering
from nonergodic failures, and the current state of the links is
unknown to the encoders (source and relays). This assumption
complicates significantly the problem and calls for the adoption
of robust coding technique at both the source and the relays.
A related model with unreliable (nonergodic) connectivity
was studied in [12] and [13] in the context of distributed source
compression. In these works, a number of agents measure sam-
plesofasourceprocess,independentandidenticallydistributed,
i.i.d., over time, via independent white Gaussian noise chan-
nels.Adescriptionofthemeasurementsisprovidedviaseparate
encoding by each agent to the destination over finite-capacity
links, so as to enable the receiver to reconstruct an estimate of
the source (the CEO problem [18]). Unbeknownst to the agents,
the links to the destination may not be functioning, and robust
distributed compression strategies must be devised to cope with
the different possible connectivity conditions. Notice that, un-
like the model of interest in the current work, the goal of [12]
and [13] is to reconstruct a given fixed source and not to design
the source coding strategy for reliable communications. There-
fore, our model combines both issues related to robust source
coding as in [12] and [13], but also of channel coding.
The basic idea behind our approach to the analysis of the
systeminFig.1is toexploitthesynergybetweentheBC coding
approach of [6] at the source, which allows for variable-data
delivery to the destination depending on the current connec-
tivity conditions, and the robust distributed compression strate-
gies of [12] and [13]. It is noted that a related idea was put
forth in [3]–[5], in which the BC coding approach was com-
bined with successive-description compression techniques for
transmission of a Gaussian source over a slowly fading channel
without channel state information.
The organization of the paper and its main contributions are
as follows. We formalize the problem in Section II and derive
upper bounds on the average achievable rate, in the sense of [6]
and [7], for the system in Fig. 1 in Section III. Average achiev-
able rates based on BC coding and robust compression are de-
rived in Section IV, which are shown via numerical results in
Section V to perform close to the derived outer bounds and to
provide relevant gains with respect to conventional strategies.
Finally, we study the case of nonoblivious agents in Section VI.
Notation: The notation
interval
, with the convention that if
then. Similarly, the subscript notation
notes the vector
with the same convention that,
if. In general, lower-case letters represent
instances of the random variables denoted by the corresponding
upper-case letters. Moreover, using standard notation, we will
sometimesusesuperscriptstodenoteindexboundsinsequences
as in
. The use of the superscript will be made
clear by the context. Probability distributions are identified by
their arguments, e.g.,
standarddefinitionsforinformationmeasuresasdefinedin[19].
with integersrepresents the
de-
. We use
II. SYSTEM MODEL
We consider the decentralized communication scenario of
Fig. 1, in which a source communicates to a destination via
“agents” or relays, connected to the receiver via orthog-
onal finite-capacity (backhaul) links of capacity
connection from the source to the destination is available. The
channel from source to relays is memoryless and either discrete
or Gaussian. For the former case, the signal
by the agent
at time instant
of a symmetric memoryless channel defined by the conditional
distribution
, with input
length
. Symmetric here means that the observations
different
are statistically exchangeable, see, e.g., [10]–[12].
For the Gaussian case, we similarly have the input-output re-
lationship
. No direct
received
is the output
and block
for
(1)
with
being the th transmitted symbol and the noise
being i.i.d. over bothand . We assume an average
input power constraint of
:. In describing the
model below, we will use the notation for the discrete model,
but it is understood that the extension to the Gaussian model (1)
is immediate.
Toaccountforanomadicscenarioand/ortosimplifytheoper-
ations at the relays, we assume, as in [17], that the relays are not
informed about the codebooks used by the transmitter (obliv-
ious agents). As formalized in Section II-A, this condition can
be modelled by assuming that the channel codebook is gener-
ated at the source based on a random key
at the relays, but is available at the destination. For reference,
Section VI also considers the case of nonoblivious agents.
that is not available

Page 3

SIMEONE et al.: ROBUST COMMUNICATION VIA DECENTRALIZED PROCESSING4189
The model described above coincides with the one studied in
[17] . Here, however, we are interested in investigating the sce-
nario in which the backhaul links from the relays to the destina-
tionareaffectedbynonergodicfailures.Specifically,following
[12], we assume that only a number
tioning in a given coding block, while the remaining
are erased, e.g., in outage, for the entire duration of the current
transmission, i.e., this is a nonergodic scenario. As in [12], we
assume that the number of functioning links
anteed to be larger than a minimum value
with probability one. Moreover, we define
the probability that
as
ties
in a vector
to the discussion above, this probability is zero for
We remark that, by the symmetry of
model) and (1) (Gaussian model), the system configuration for
a given
depends only on the number
tive and not on which links are active. Finally, in keeping with
the models for distributed source coding of [12] and [13], we
are interested in scenarios in which no instantaneous informa-
tion regarding the current state of the unreliable links, i.e., the
value of
, is available a priori to the source and the agents
(uninformed source and agents). More precisely, the only in-
formation that is available at the source and the relays is the
probability mass function , which represents the a priori state
of knowledge of the source and agents on the state of the unre-
liable links. This assumption is appropriate in scenarios where
feedbackisnotavailableortheoutageeventsaredifficulttopre-
dict, e.g., when the channel has a short coherence time.
We are interested in average achievable rates, where the av-
erage is taken with respect to the a priori connectivity proba-
bility vector
. Specifically, we consider a degraded message
structure in which the overall source message of rate
channel use] is split into submessages
of rates
. When links are active, with
the receiver decodes messages
. Notice that the more links are active the more bits
(and messages) are decoded. An average rate
of links are func-
is always guar-
so that
and collect the probabili-
. Notice that, according
.
(discrete
of active links ac-
[bits/
, respectively, i.e.,
,
of total rate
(2)
issaidtobeachievableifallrates
able, in the sense that, when
code guarantees decoding of
probability of error for all
in [6], the average rate (2) does not have the operational sig-
nificance of an ergodic rate, since the channel is nonergodic. It
is instead a measure of the rate that could be accrued with re-
peated and independent transmission blocks, or of the expected
rate. The analysis presented below can also accommodate dif-
ferent criteria, such as the outage capacity, in which a zero rate
is tolerated with a given probability. We refer also to [7] and [8]
for further discussion on capacity definitions for “nonergodic”
scenarios. The setting is formalized in the following.
aresimultaneouslyachiev-
links are active, the given
with vanishingly small
. We remark that, as
A. Formal Setting
Denoting by
channel in Fig. 1 with oblivious1relays is defined by the fol-
lowing elements:
• The encoder performs a (stochastic) mapping
the messages
to a codeword
with
the size of each coding block, a code for the
from
, namely
(3)
The codebook
is indexed by a random key
, which runs over all possible codebooks of
. The keyis chosen randomly at the
beginningofthecommunicationsession,and isrevealedto
thedestination,butnottotherelays.Noticethatthismodel,
in which coding is stochastic due to the random key
merely a way of formalizing the fact that the relays have
no prior knowledge of the codebook, and it does not entail
any real overhead. To elaborate, as in [17], the probability
of choosing a codebook
depends on a measure
codewords as
size
, is
indexed by key
over the space of the
(4)
where the product is taken over the message sets
for
over, the measure
. More-
is assumed to factor as
for a given single-letter proba-
bility distribution
the absence of information regarding
the signal transmitted by the source
. Reference [17] shows that, in
, i.e., at the relays,
is distributed i.i.d.
according to a distribution
and,
similarly, the received signals
Lemma 1]. The source does not know the current number
of active links.
• Each th relay
(oblivious relays), maps the received sequence
into an index
as
ping does not depend on the current state of the link or the
number
of active links. We remark that in Section VI,
we consider a scenario in which the relays are aware of the
source codebook.
• When
links are active, the decoder decodes
messages
knowledge of the codebook key
dices
over theactive links. These can be assumed by
symmetry to be
. The decoding function can be
written as
appear i.i.d., see [17,
, unaware of the codebook
via a given mapping
. Notice that this map-
based on its
and the received in-
(5)
1The nonoblivious case will be treated in Section VI.

Page 4

4190IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011
The probability oferror when
is averaged over
linksare active,which
, is defined as
(6)
An average rate
codes such that all rates
are achievable, i.e.,
erage capacity
rates (2).2
(2) is achievable if there exists a sequence of
for
as. The av-
is the supremum of all average achievable
III. REFERENCE RESULTS
In this section, we start the study of the system presented
above by deriving an upper bound on the capacity
emphasized that the upper bound is valid under the given as-
sumption of oblivious relays. Moreover, it is noted that, as in
[17], for the Gaussian model, we restrict the input distribution
to be Gaussian with no claim of optimality. We refer to [17]
for further discussion on the suboptimality of the Gaussian dis-
tribution. The upper bound below also motivates the use of a
BCcodingapproachatthesourceandofcompress-and-forward
(CF) at the relays, as exploited in the transmission strategies
considered in the rest of the paper, see Remark 3.1.
. It is
Proposition 3.1: (Cooperative Relays): The following is
an upper bound on the average capacity
model:
for the discrete
(7)
where the rates
(8a)
(8b)
(8c)
are calculated with respect to a joint distribution
(9)
and the maximization is taken with respect to the marginals
and that factor as
(10a)
2The average capacity can be seen as identifying the hyperplane tangent to
the region of all achievable rates ???????
vector ?.
in the direction specified by
and satisfy
(11)
Moreover, for the Gaussian model, (7) is an upper bound, under
the constraint that the input distribution is Gaussian, with
(12)
for
specttoparameters
and
, where the maximization is taken with re-
with
.
Remark 3.1: As discussed in the sketch of the proof below,
the upper bounds of Proposition 3.1 are obtained by assuming
that all of the
relays that are connected to the corresponding
active links can fully cooperate in processing their received sig-
nals. Notice that this implies that they are also informed of
whichlinksareactive.Theupperboundscanthenbeinterpreted
as stating that, under this assumption, the best way to operate at
the source is to use a standard BC code characterized by aux-
iliary random variables
crete case or powers
case. Such variables or powers correspond to the transmission
of message
to be decoded at the receiver when
Notice that the variables
satisfy the Markov chain condi-
tion (10a), or equivalently
a regular degraded broadcast channel [24]. Moreover, the re-
sult in Proposition 3.1 also proves that fully cooperative relays
can employ CF techniques without loss of optimality to com-
municate to the receiver: The auxiliary variables
for the quantization codebook used by the
relays when the active number of links is
eter
is the corresponding compression noise power for the
Gaussian case. In fact, from standard rate-distortion consider-
ations, (11) is easily interpreted in this sense as necessary and
sufficienttoguaranteesuccessful compressionforall
proof below). Notice that the optimality of CF in this context is
a consequence of the obliviousness assumption, as detailed in
the proof in Appendix A. Finally, we remark that the optimiza-
tionproblemin(7)fortheGaussiancase(12)correspondstothe
maximization of the weighted sum-rate of a broadcast channel,
whichcanbesolvedusingstandardtechniques[21].Wewilluse
this intuition regarding the appropriateness of BC coding and
CF when designing transmission strategies in the next section.
We also notice that a similar conclusion regarding optimality of
BC coding for a different setting was presented in [7].
, for the dis-
for the Gaussian
.
as for
account
cooperating
and param-
(seethe
Proof: (Sketch): Here we provide a simple proof for the
bound (7), (12) for the Gaussian channel, as the derivation is
more direct. The discrete model is discussed later and proved
in Appendix A. Assume that the relays are perfectly cooper-
ating so that, when
links are active, they can be seen as

Page 5

SIMEONE et al.: ROBUST COMMUNICATION VIA DECENTRALIZED PROCESSING4191
a unique compound agent with
the signals are statistically equivalent, there is no loss in gener-
ality in this choice of
. It is easy to see that the compound
agent can be equivalently considered as having scalar measure-
measurements , since
ments:
, because there is no performance
lossinprojectingthereceivedsignal overthesignalspace,since
the noise in (1) is uncorrelated over the agents. As aforemen-
tioned, we limit the analysis, as in [17], to random coding with
Gaussianinputsatthesource.From[17],itisknownthattheop-
timal operation at the compound agent, which is clearly aware
of the capacity
toward the destination, is to quantize to a
rate
bits/source symbol the received signal via a Gaussian
test channel
pendent of
. From standard arguments in rate-distortion
theory, in order to have vanishing probability of error in the
quantization process, as the block size
[cf. (11)], thus obtaining
. As a result, since the source is not informed
about the current value of
, the equivalent channel can be
seen as a degraded Gaussian broadcast channel, in which the
destinations observe received signals
equivalent noise variances
ticethatsuchvariancesareclearlydecreasingwith
the capacity region for the Gaussian broadcast channel, bound
(12) then easily follows. As a final remark, it is noted that (7)
and (12) for the Gaussian channel can also be obtained from
the corresponding discrete result of Proposition 3.1, proved in
Appendix A, by setting auxiliary variables
independent for
, andas discussed earlier.
withinde-
increases, we can set
with
. No- for
.Recalling
and
IV. ACHIEVABLE RATES
In the following, motivated by the upper bound of Propo-
sition 3.1 , we propose achievable schemes based on the BC
coding strategy of [6] and CF at the relays. In [6], a BC strategy
was proposed to deal with uncertain fading conditions. The
basic idea is that of treating all the possible channel fading
states that might occur as distinct users, thus effectively con-
verting the fading channel into a degraded broadcast channel.
In this paper, the same principle is leveraged to operate over
the channel at hand, which presents unknown connectivity con-
ditions from the relays to the receiver. Specifically, the source
transmits a superposition of
for . When
. As far as the operation at the relays is concerned,
due to the fact that codebook information, i.e., the key
Section II-A, is not available at the relays, here we will as-
sume that CF relaying is implemented. Notice that this coin-
cides exactly with the strategy that was proved to be optimal
for the setting of Proposition 3.1, see Remark 3.1. The dif-
ferent techniques proposed in the following differ in the way
the CF strategy is implemented in terms of compression at the
agents and decompression/decoding at the receiver, and entail
increasing levels of complexity.
codewords of rates
, the receiver decodes
of
A. Broadcast Coding and Single-Description Compression
(BC-SD)
In this section, we consider a transmission strategy based on
BC coding and single-description (SD) compression at the re-
lays. In other words, each relay sends over the backhaul link a
single index (description), which is a function of the received
signal. Moreover, we consider first separate decompression/de-
coding at the decoder, and then a potentially more effective, but
more complex, joint decompression/decoding approach. A per-
formance comparison that shows the performance-complexity
tradeoff of these schemes is provided via numerical results in
Section V for the Gaussian model.
1) Separate Decompression/Decoding (BC-SD-S): Here we
propose a strategy based on separate (S) decompression/de-
coding. The compression/decompression scheme is inspired by
the technique used in [12], see also [10], for robust distributed
source coding in a CEO problem. The technique works by
performing random binning at the agents, as is standard in dis-
tributed compression, see, e.g., [20]. Moreover, the binning rate
is selected so that the receiver can recover with high probability
the compressed signals on the
the realized value of
as long as
assumption. In other words, design of the compression scheme
targets the worst-case scenario of
should more than
links be active
sponding compressed signals would also be recoverable at the
receiver, since, by design of the binning rate, any subset of
descriptions can be decompressed [12]. After decompression is
performed, the receiver uses all the
the relays to decode the codewords up to the
the layers with rates
with
active links irrespective of
, as guaranteed by
. Notice that,
, the corre-
signals obtained from
th layer, that is,
.
Proposition 4.1: (BC-SD-S): The average rate (2) is achiev-
able for the discrete model with
(13a)
(13b)
(13c)
where the variables at hand satisfy the joint distribution
(14)
with
condition
being the same for every, and the
(15)