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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