Achievable Rates and Scaling Laws of Power-Constrained Wireless Sensory Relay Networks

Abstract

A wireless sensory relay network consists of one source node, one destination node and multiple intermediate relay nodes. In this paper, we study the achievable rates and the scaling laws of power-constrained wireless relay networks in the wideband regime, assuming that relay nodes have no a priori knowledge of channel-state information (CSI) for both the backward channels and the forward channels. We examine the achievable rates in the joint asymptotic regime of the number of relay nodes n, the channel coherence interval L, and the bandwidth W (or the SNR per link rho). We first study narrowband relay networks in the low SNR regime. We investigate a relaying scheme, namely amplify-and-forward (AF) with network training, in which the source node and the destination node broadcast training symbols and each relay node carries out channel estimation and then applies AF relaying to relay information. We provide an equivalent source-to-destination channel model, and characterize the corresponding achievable rate. Our findings show that when rhoL, proportional to the transmission energy in each fading block, is bounded below, the achievable rate has the same scaling order as in coherent relaying, thus enabling us to characterize the scaling law of the relay networks in the low SNR regime. We then generalize the study to power-constrained wideband relay networks, where frequency-selective fading is taken into account. Again, the focus is on the achievable rates by using AF with network training for information relaying. In particular, we examine the scaling behavior of the achievable rates corresponding to two power allocation policies across the frequency subbands at relay nodes, namely, a simple equal power allocation policy and the optimal power allocation policy. We identify the conditions under which the scaling law of the wideband relay networks can be achieved by both power allocation policies. Somewhat surprising, our findings indicate that these two power allocati-
-
on policies result in achievable rates of the same scaling order, and the scaling law can be characterized under the condition that L/W, proportional to the energy per fading block per subband, is bounded below, and that W is sublinear in n

Full text

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4084 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 9, SEPTEMBER 2006
Achievable Rates and Scaling Laws of
Power-Constrained Wireless Sensory Relay
Networks
Bo Wang, Student Member, IEEE, Junshan Zhang, Senior Member, IEEE, and Lizhong Zheng, Member, IEEE
Abstract—A wireless sensory relay network consists of one
source node, one destination node and multiple intermediate relay
nodes. In this paper, we study the achievable rates and the scaling
lawsofpower-constrainedwirelessrelaynetworksinthewideband
regime, assuming that relay nodes have no a priori knowledge of
channel-state information (CSI) for both the backward channels
and the forward channels. We examine the achievable rates in
the joint asymptotic regime of the number of relay nodes ?, the
channel coherence interval, and the bandwidth ? (or the
SNR per link ?). We first study narrowband relay networks in
the low SNR regime. We investigate a relaying scheme, namely
amplify-and-forward (AF) with network training, in which the
source node and the destination node broadcast training symbols
and each relay node carries out channel estimation and then
applies AF relaying to relay information. We provide an equiv-
alent source-to-destination channel model, and characterize the
corresponding achievable rate. Our findings show that when ??,
proportional to the transmission energy in each fading block, is
bounded below, the achievable rate has the same scaling order as
in coherent relaying, thus enabling us to characterize the scaling
law of the relay networks in the low SNR regime. We then gen-
eralize the study to power-constrained wideband relay networks,
where frequency-selective fading is taken into account. Again, the
focus is on the achievable rates by using AF with network training
for information relaying. In particular, we examine the scaling
behavior of the achievable rates corresponding to two power
allocation policies across the frequency subbands at relay nodes,
namely, a simple equal power allocation policy and the optimal
power allocation policy. We identify the conditions under which
the scaling law of the wideband relay networks can be achieved
by both power allocation policies. Somewhat surprising, our
findings indicate that these two power allocation policies result in
achievable rates of the same scaling order, and the scaling law can
be characterized under the condition that ???, proportional to
the energy per fading block per subband, is bounded below, and
that ? is sublinear in ?.
Index Terms—Cooperative relaying, network training, scaling
law, wideband regime, wireless sensory relay network.
Manuscript received September 21, 2005; revised February 22, 2006.
This research is supported in part by Office of Naval Research under Grant
N00014-05-1-0636 and National Science Foundation through the CAREER
award ANI-0208135. The material in this paper was presented in part at the
43rd Annual Allerton Conference on Communication, Control and Computing,
Monticello, IL, September 2005.
B. Wang and J. Zhang are with the Department of Electrical Engineering,
Arizona State University, Tempe, AZ 85287 USA (e-mail: bo.wang@asu.edu;
junshan.zhang@asu.edu).
L. Zheng is with the Department of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
(e-mail: lizhong@mit.edu).
Communicated by R. R. Müller, Associate Editor of Communications.
Digital Object Identifier 10.1109/TIT.2006.880029
I. INTRODUCTION
T
technologicaladvances insensordesign, coupledwith advances
in data processing and wireless communications, are creating a
new realm of possibilities for the development of sensor net-
works in many application domains. A sensor network is often
designed to carry out a set of high-level data acquisition and in-
formation processing tasks. The raw sensor measurements are
obtained and then need to be transported to a data collecting/fu-
sion center to be analyzed, which requires reliable networking
capabilities in disadvantaged wireless environments. There are
many sensor network models developed for different applica-
tions: multihop sensor networks, many-to-one sensor networks,
sensory relay networks—to name a few.
In this paper, we consider power constrained wireless sen-
sory relay networks in the wideband regime, because of their
ability of overlay with other legacy networks and the advantage
that the use of a larger bandwidth can offer significant power
savings. In sensor networks, most sensor devices have limited
power supply (e.g., battery supply), and lower power consump-
tion of sensor devices is of critical importance. Wideband com-
munications has recently garnered much attention (e.g., ultra-
wideband (UWB) systems) [24]. Indeed, UWB radios are ex-
pectedtobeinexpensiveandlow-power,andareidealforsensor
networkapplications.Itisofgreatinteresttoinvestigatethefun-
damental limits in such systems, such as the capacity and en-
ergy efficiency. Thus motivated, we focus on the sensory relay
networks in which each node is power constrained and the fre-
quency bandwidth for communication may be large. Following
[21], “the wideband regime” here encompasses all scenarios
where the information bits transmitted per receive dimension
are small. Two key challenges in power constrained wideband
relay networks are as follows: 1) coherent relaying may not be
feasible due to the low SNR and the corresponding cooperative
relaying strategies should be pursued; and 2) when frequency
bandwidth is large, it remains open what power allocation poli-
cies across the subbands work well at relay nodes.
As depicted in Fig. 1, a wireless sensory relay network con-
sists of one source-destination pair and
measurement data at the source node are sent to the destination
node, many intermediate sensors can serve as relay nodes to co-
operativelyrelaytheinformation. (This modelis wellmotivated
by event-driven sensor networks, in which the data gathered by
the source node can be relayed by other sensors in a coopera-
tive manner.) The cooperation among relay nodes can yield di-
versity gain to enhance the data transport capacity between the
HE last decade has witnessed a tremendous growth of in-
terest in wireless sensor networks. Current and anticipated
relay nodes. When the
0018-9448/$20.00 © 2006 IEEE

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WANG et al.: ACHIEVABLE RATES AND SCALING LAWS OF POWER-CONSTRAINED WIRELESS SENSORY RELAY NETWORKS4085
Fig. 1. Wireless sensory relay network model.
source node and the destination node. As is standard (e.g., [1],
[4], [7], [8], [14]), we consider a two-hop relaying strategy, in
which each relay node first receives the signal from the source
node in the first hop and then forward a processed signal to
the destination node in the second hop. We refer the links be-
tween the source node and the relay nodes as the backward
channels, composing the first hop; and those between the relay
nodes and the destination node as the forward channels, corre-
sponding to the second hop. We assume that every link experi-
ences Rayleigh fading. We note that multiple source–relay–des-
tinationlinksoffermultiplespatialroutingpathsforinformation
relaying, pointing to potential cooperative diversity gain.
Clearly, The key to reap the inherent cooperative diversity
gain in such networks is to devise good relaying strategies. Re-
cently, “decode-and-forward (DF)” and “amplify-and-forward
(AF)” have emerged as two popular candidates for two-hop
information relaying. In the decode-and-forward strategy, the
relay nodes first extract the information bits from the received
signals, re-encode it, and then transmit the new encoded bits in
the second hop. For instance, one DF scheme can be to select
the most reliable source-relay-destination path to transmit the
signal. In general, the optimal DF strategy is highly nontrivial
and the complexity increases significantly as the number of
relay nodes grows. Alternatively, simple amplify-and-forward
relaying can be used to relay information and has low com-
plexity. Intuitively speaking, to fully reap the cooperative
diversity gain, a good AF relaying scheme should reduce the
phase-offset across different source-relay-destination links as
much as possible, and this phase-aligning operation should
enable coherent combining of transmitted signals from relay
nodes at the destination node. The focus of this study is to
investigate the “right” relaying strategies for achieving high
throughput in wireless relay networks in the wideband regime.
In particular, we attempt to address the following critical issues:
• When would “amplify-and-forward” work well in relay
networks in the wideband regime? What is the “right” am-
plify-and-forward scheme to close the gap between co-
herent relaying and noncoherent relaying?
• What is the “right” power allocation strategy at the relay
nodes thatwould maximizethescalingorder oftheachiev-
able rates of relay networks in the wideband regime?
• Whenwoulditbepossibletoachievethesamescalinglaws
as in coherent relay networks?
Recall that the wideband regime encompasses all scenarios
where the information bits transmitted per receive dimension
are small. One key challenge in relay networks in the wideband
regimeisthatcoherentcommunicationsmaynotbefeasibledue
to the low SNR. To obtain a clear understanding of the impact
of low SNR on cooperative relaying, we make a more realistic
assumption that at relay nodes there is no a priori knowledge of
channel state information (CSI) for both the backward channels
and the forward channels, and this is one of key features dis-
tinguishing our model from the existing ones. As a result, each
relay node needs to estimate the channel conditions prior to the
data transmission, in order to carry out the phase-alignment as
in coherent relaying. In particular, the channel estimation of the
forward links is more challenging because it is a multiaccess
channel. A naive approach is to carry out channel estimation of
the forward links separately, which would lead to a significant
reduction of the throughput and is clearly not energy-efficient.
To overcome this difficulty, we study the following strategy for
AF relaying. As depicted in Fig. 2, at the beginning of the first
hoptransmission,thesourcenodefirstbroadcastscommonpilot
symbols, followed by its data transmission. Based on the re-
ceived signals corresponding to the pilot symbols, each relay
nodes estimates its own backward channel condition. Then at
the beginning of the second hop, the destination node (not the
relay nodes) sends out pilot symbols (We assume that the for-
ward links are reciprocal in both directions in each subband.)
Then, each relay node estimates its forward channel condition.
After thechannel estimationis done, each relaynode carries out
phasealignmentbyusingitschannelestimatesofboththeback-
ward channel and the forward channel, amplifies the received
data signal under the given power constraints, and forward the
processed signal to the destination node. For convenience, we
refer this strategy as amplify-and-forward (AF) with network
training. We emphasize that the above channel estimation ap-
proach makes use of the broadcast nature of wireless transmis-
sions, and is an energy-efficient candidate for network training.
For instance, this network training approach can be applied by
all source–destination pairs in a collocated network [6].
The main thrust of this paper is devoted to characterizing the
achievable rates and scaling laws by using AF with network
training in power-constrained relay networks in the wideband
regime. Simply put, the scaling law here is concerned with the
order of the achievable rates as the number of the relay nodes
grows,i.e.,thescalingbehaviorofthethroughput(andcapacity)
as
. Letdenote the SNR per link. It is understood that
in the wideband regime. Furthermore, it is clear that the
coherence interval of the fading channel, denoted as
key role in channel estimation. To obtain a clear understanding
of the continuum between coherent relaying and noncoherent
, plays a

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4086IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 9, SEPTEMBER 2006
Fig. 2. Amplify-and-forward with network training.
relaying, we examine the achievable rates and scaling laws in
the joint asymptotic regime of the number of relay nodes
the channel coherence interval
bandwidth
). We note that although there is no physical con-
nection amongthese parameters,the achievablerates depend on
how they scale together, rather than on each of them in isolation
[26][18].
Since a wideband channel can be decomposed into multiple
orthogonal narrowband channels, each with flat fading, we first
investigate achievable rates and scaling laws by using AF re-
laying with network training in narrowband relay networks in
the low SNR regime [25]. Our findings show that when
(which is proportional to the transmission energy per fading
block) is bounded below, AF with network training achieves the
same scaling law as in the coherent relay networks where the
CSI is known a priori. We then generalize the study to power-
constrained wideband relay networks in which frequency-se-
lective fading is inevitable. We model the wideband relay net-
works as a set of multiple narrowband relay networks under
an overall power constraint. We examine the achievable rates
and scaling laws by using AF with network training, under two
power allocation policies, respectively. Specifically, we first as-
sume that an equal power allocation policy across the frequency
subbands1is applied at each relay node. Then we examine the
optimal power allocation policy for wideband relay networks,
whereeachrelaynodeisallowedtoallocateitspoweracrossthe
subchannelsandthefadingblocks.Ingeneral,onewouldexpect
a higher achievable rate by using bursty flashing signals (more
generally, water-filling techniques). Somewhat surprising, our
findings show that the achievable rates by using AF with net-
work training have the same scaling order as those given by the
equal power allocation policy, and both policies can achieve the
scaling law under the condition that
and that
is sublinear in . That is to say, the equal power
allocation policy at relay nodes is order optimal in the sense of
achievingthescalinglawasymptotically.Roughlyspeaking,the
condition that
is sub-linear in
gree of node multiplexing per unit bandwidth
increases.
In related work, the capacity of wireless networks and the ca-
pacity of relay channels have been studied extensively in the
,
, the SNR per link(or the
is bounded below
is equivalent to that the de-
grows as
1Throughout, we use the notation of subchannel and subband interchange-
ably.
past few years (see, [1], [4], [5], [7]–[13], [16], [17], [20], [22],
[23] and the references therein). In their seminal work [10],
Gupta and Kumar investigate the throughput scaling for dense
many-to-manymultihopnetworkmodels,andtheyprovethatan
aggregated throughput of
be arbitrarily located. Grossglauser and Tse study the impact of
mobility on the throughput in mobile ad hoc networks [9]. They
show an
throughput can be achieved by using two-hop re-
laying and allowing possibly unbounded delay. Recently, wire-
less sensory relay networks has received much attention. In [7]
and [8], Gastpar and Vetterli show that the asymptotic capacity
of wireless relay networks behaves like
traffic pattern, assuming that power allocation can be carried
outacrosstherelaynodes.Bölcskei, Nabar,Oyman,andPaulraj
find in [1] that the throughput of wireless relay networks, where
each node is equipped with
Recent work [4] by Dana and Hassibi reveals that the power
efficiency of the AF scheme scales at least by a factor of
Very recent works [16], [17] by Oyman and Paulraj find that
a higher power efficiency of order
burstytransmission.Dousse,Franceschetti,andThirandiscover
in [5] that the per-node throughput remains constant as the size
of the wireless network increases, if a “small” fraction of nodes
are allowed to be disconnected.
Most relevant to our work is perhaps [1], [7], [8] which
study the scaling laws assuming coherent AF relaying. It is also
pointed out in[1] that theachievable rates by usingnoncoherent
AF relaying does not scale well at all. The study in this paper
is intended to examine the continuum between noncoherent
AF relaying and coherent AF relaying and to close the gap in
between.
The rest of this paper is organized as follows. In Section II,
wepresent themodels fornarrowbandrelaynetworks inthelow
SNR regime and power-constrained wideband relay networks,
respectively. We summarize in Section III the main results of
thispaper.InSectionIV,weobtainanequivalentsource-to-des-
tination channel model for narrowband relay networks using
AF with network training, and examine the scaling order of the
achievable rates in different scenarios. We characterize the con-
ditionsfor achievingthe scalinglawsaccordingly. InSection V,
we generalize the study to power-constrained wideband relay
networks. We analyze the achievable rates and the scaling order
corresponding to the equal power allocation policy at the relay
can be achieved if nodes can
under the relay
antennas, scales as.
.
can be achieved by using

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WANG et al.: ACHIEVABLE RATES AND SCALING LAWS OF POWER-CONSTRAINED WIRELESS SENSORY RELAY NETWORKS 4087
Fig. 3. Asymptotic regime of ?????????.
nodes, and compare the scaling behavior with that by using the
optimal power allocation policy at the relay nodes. Section VI
contains numerical examples which are used to illustrate the
achievableratesandcapacitybounds.SectionVIIconcludesthis
paper.
II. SYSTEM MODEL
In this section, we present the model for narrowband relay
networks in the low SNR regime and that for power constrained
wideband relay networks.
A. Narrowband Relay Networks in the Low SNR Regime
We impose the following assumptions for the narrowband
relay network model.
• There are
relay nodes between the source node and the
destinationnode.Thereisnodirectlinkbetweenthesource
node and the destination node. In the first hop, each relay
node listens to the transmissions from the source node, and
then forward the processed signal to the destination node
in the second hop.
• The communication bandwidth is
• The ambient noise at the relay nodes and the destination
node has power spectral density
• All nodes have an average transmit power constraint
(watts) in one fading block.
• All channels experience independent and identically dis-
tributed (i.i.d.) frequency-flat block fading. Denote the
backward channel coefficients as
channel coefficients as
are complex Gaussian random variables, with zero
mean and unit variance.
• The backward channels have a coherence interval of
symbol periods, so do the forward channels. Throughout,
we assume that
herence time is
(seconds).
(Hz).
.
, and the forward
. We assume that and
. Accordingly, the channel co-
• Within one fading block, the forward channel conditions
are symmetric in both directions, that is, the channel con-
dition from the relay node to the destination node is the
same as that from the destination node to the relay node.
Without loss of generality, we assume that the average re-
ceive SNR for each backward (forward) link is
. It is understood that
regime. On the other hand, the coherence interval
how fast the channel changes. Clearly, a larger
for more accurate channel estimation, and we would expect that
when
exceeds a certain threshold, the achievable rates of the
relay networks can scale the same as in coherent relaying case
where a priori knowledge of CSI is available at relay nodes.
In contrast, a low SNR makes channel estimation more chal-
lenging.
Toobtainaclearunderstandingofthecontinuumbetweenco-
herent relaying and noncoherent relaying, we let
with . We investigate the achievable rates in the joint asymp-
totic regime of
and . Accordingly, we can obtain a set of
achievable rates corresponding to different values of
Define
, i.e,
in the wideband
points to
is “favorable”
andscale
and.
and(1)
Remarks: Because the focus of this study is on the scaling
law, i.e., the scaling order of the capacity, it is natural to define
andonthelogarithmicscale.Althoughthereisnophysical
connection among these three parameters, the achievable rates
and their scaling behavior depend on the relation among these
three key parameters, rather than on each of them alone. Simply
put, the asymptotic regime is along a curve determined by
in the three-dimensional (3-D) space of
as illustrated in Fig. 3. Indeed, we show in Section IV that
the simultaneous scaling of the three parameters enables the
characterization of the achievable rates and their scaling in
terms of how they are related through the exponents.
,

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4088IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 9, SEPTEMBER 2006
B. Amplify-and-Forward (AF) With Network Training
As aforementioned, the channel estimation at relay nodes is
carried out in the network training phase (see Fig. 2). More
specifically,thesourcenode firstbroadcastscommonpilotsym-
bols to relay nodes at the beginning of the first hop transmis-
sion, followed by its data transmission. Then the destination
node broadcasts common pilot symbols at the beginning of the
second hop. Based on the received signals corresponding to the
pilot symbols, each relay nodes estimates its backward channel
condition and its forward channel condition, respectively. After
thechannelestimationisdone,eachrelaynodecarriesoutphase
alignment by using its channel estimates, amplifies the received
data signal under the given power constraints, and forward the
processed signal to the destination node.
1) Channel Estimation Via Network Training: We assume
that the minimum mean-square error (MMSE) estimation
method is applied to estimate
the total energy for one hop transmission over L symbols, and
be the energy for the training with
the estimate error of the MMSE estimator depends only on
([19, p. 599]), we further assume that the training can be done
using one pilot sample (symbol). We assume that
Considerthechannelestimationforthe thchannelinthefirst
hop
and. Let denote
. Since
.
(2)
where
the noise at the i-th relay node. Let
bethechannelestimatoratthei-threlaynode.,i.e.,
. By the orthogonality principle of the MMSE estimation, the
estimate error and the channel estimator are uncorrelated. Since
allchannelcoefficientsarecomplexGaussian,theestimateerror
is in fact independent from the channel estimator. Recall that
, and thus
variance of
is
is the received pilot symbol, andis
be the estimate error and
. We conclude that the
(3)
Similarly, we have that the variance of
is
(4)
2) Amplify-and-Forward at Relay Nodes: In what follows,
weestablishthechannelandsignalmodelscorrespondingtoAF
with network training. Recall that the network training is done
via usingone pilot symbol in the first hop and another one in the
second hop. During thedata transmission period in the first hop,
the information signal received by the th relay node is given by
(5)
where
is the transmitted signal from the source node, and
is complex Gaussian noise at the th relay node. In the
second hop, the th relay node applies AF with network training
to relay the received signal
. In particular, it carries out
phase alignment via multiplying
sponding transmitted signal at the th relay node is given by
with. The corre-
for(6)
where
for each relay node, and is given by
istheamplificationfactortomeetthepowerconstraint
(7)
The destination node collects signals from
the received signal is given as below
relay nodes, and
for(8)
where
Moreover, as
bility , and hence in the low SNR regime
is complex Gaussian noise at the destination node.
with proba-
(9)
C. Power-Constrained Wideband Relay Networks
In wideband relay networks, frequency-selective fading be-
comes inevitable when the frequency bandwidth increases. To
study the scaling laws of wideband relay networks, we investi-
gate an equivalent model where each wideband channel is de-
composedintoasetoffrequency-flatsubchannels.Lettheband-
width of each subchannel in wideband relay networks be
the total number of subchannels is
quency bandwidth
. Define
and
. Then, the overall fre-
(10)
Since
is fixed, we have that the total frequency bandwidth
. Again, we are interested in how the
three parametersandjointly determine the achievable
rates; and the above setting is used to obtain a clear under-
standing how the simultaneous scaling of the three key param-
eters impacts the scaling laws.
We have a few words on the exponents
and. Recall that
It is clear that
SNR” in the narrowband relay network can be viewed as the
is intimately related to. Indeed, the “low