4446 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, … · nonlinear optimization, power efﬁciency, quantization, resource allocation, wireless sensor networks (WSNs). I. INTRODUCTION

4446 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008

Minimizing Transmit Power for CoherentCommunications in Wireless SensorNetworks With Finite-Rate Feedback

Antonio G. Marques, Member, IEEE, Xin Wang, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE

Abstract—We minimize average transmit power with finite-ratefeedback for coherent communications in a wireless sensor net-work (WSN), where sensors communicate with a fusion centerusing adaptive modulation and coding over a wireless fadingchannel. By viewing the coherent WSN setup as a distributedspace–time multiple-input single-output (MISO) system, wepresent optimal distributed beamforming and resource allocationstrategies when the full (F-) channel state information at the trans-mitters (CSIT) is available through a feedback channel. We alsodevelop optimal adaptive transmission policies and design optimalquantizers for the finite-rate feedback case where the sensors onlyhave quantized (Q-) CSIT, or, each sensor has F-CSIT of its ownlink with the FC but only Q-CSIT of other sensors. Numericalresults confirm that our novel finite-rate feedback-based strategiesachieve near-optimal power savings based on even a small numberof feedback bits.

Index Terms—Multiple-input single-output (MISO) systems,nonlinear optimization, power efficiency, quantization, resourceallocation, wireless sensor networks (WSNs).

I. INTRODUCTION

A WIRELESS sensor network (WSN) comprises a largenumber of spatially distributed signal processing devices

(sensor nodes). In a number of application scenarios, WSNnodes are equipped with a nonrechargeable battery and thushave limited computing and communication capabilities. Whenproperly programmed and networked, nodes in a WSN can

Manuscript received October 5, 2006; revised December 23, 2007. Pub-lished August 13, 2008. The associate editor coordinating the review of thismanuscript and approving it for publication was Dr. Soren Holdt Jensen.Work in this paper was supported by the ARO Grant W911NF-05-1-0283and was prepared through collaborative participation in the Communicationsand Networks Consortium sponsored by the U.S. Army Research Laboratoryunder the Collaborative Technology Alliance Program, Cooperative AgreementDAAD19-01-2-0011. The U.S. Government is authorized to reproduce anddistribute reprints for Government purposes notwithstanding any copyrightnotation thereon. The work of A. G. Marques in this paper was partiallysupported by the C.A. Madrid Government Grant P-TIC-000223-0505. Partsof this paper were presented at the Asilomar Conference on Signals, Systems,and Computers, Pacific Grove, CA, October 2006, and the IEEE InternationalConference on Acoustics, Speech, and Signal Processing, Honolulu, HI, April2007.

A. G. Marques is with the Department of Signal Theory and Communications,Rey Juan Carlos University, Fuenlabrada, Madrid 28943, Spain (e-mail: [email protected]).

X. Wang is with the Department of Electrical Engineering, Florida AtlanticUniversity, Boca Raton, FL 33431, USA (e-mail: [email protected]).

G. B. Giannakis is with the Department of Electrical and Computer Engi-neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2008.921725

cooperate to perform advanced signal processing tasks withunprecedented robustness and versatility, thus making WSNan attractive low-cost technology for a wide range of remotesensing and environmental monitoring applications [1]. Oneof the main objectives in current WSN research is to designpower-efficient devices and algorithms to support differentaspects of network operations [6]. Various power-efficientalgorithms have been proposed for network coverage, mediumaccess control protocols, decentralized estimation and routing;see e.g., [6], [19], [24], [23], and [2]. The WSN in many ofthese works includes a fusion center (FC) with which sensorsare linked.

When these links are fading, communication performanceacross the WSN coverage area is severely degraded. A well-known approach to mitigate the adverse effects of fading relieson transmissions adapting to the full (F-) channel state infor-mation (CSI) [12], [10]. In practice, CSI at transmitters (CSIT)is typically acquired through a limited-rate feedback channelfrom the receiver, and thus, only quantized (Q-) CSIT is avail-able [9]. This finite-rate feedback model is pragmatically afford-able and is robust to channel estimation errors, feedback delayand jamming [13]. Adaptive transmissions and/or beamformingschemes based on Q-CSIT have been optimized for multiple-input multiple-output (MIMO) systems to maximize rate or re-ceive-signal-to-noise-ratio (SNR), [16], [20], or, to minimize biterror rate (BER) [25]; as well as to optimize power efficiency forsingle-input single-output (SISO) and multiuser systems [18],[19].

This paper deals with a distributed multiple-inputsingle-output (MISO) communication system in the power-lim-ited regime of a WSN where sensor transmissions arrive coher-ently at the FC [3], [4]. Timing needed to ensure coherence isassumed to have been acquired using e.g., the low-complexitysynchronization algorithm of [17]. Specifically, the sensors’average transmit power is minimized subject to average rateand BER constraints, based on three different types of CSIT:

i) F-CSIT where each channel realization is assumed avail-able at each sensor;

ii) Q-CSIT where the sensors only have quantized knowl-edge of their links with the FC; and

iii) individual (I-) CSIT where each sensor has full knowl-edge of its own channel but only quantized knowledge ofthe other sensors’ channels.

For these cases, we develop corresponding adaptive modula-tion/coding, power loading and beamforming strategies as wellas the channel quantizers needed to form the required Q-CSITas the solution of constrained optimization problems.

1053-587X/$25.00 © 2008 IEEE

MARQUES et al.: MINIMIZING TRANSMIT POWER FOR COHERENT COMMUNICATIONS IN WSNs WITH FINITE-RATE FEEDBACK 4447

Fig. 1. System model.

The rest of the paper is organized as follows. After intro-ducing modeling preliminaries in Section II, we derive inSection III optimal transmit-adaptation based on F-CSIT whichprovides fundamental limits and benchmarks the power effi-ciency based on Q-CSIT. Subsequently, we solve the optimaladaptation problem based on Q-CSIT in Section IV, whileSection V deals with the optimal design based on I-CSIT. Sim-ulated examples and comparisons are provided in Section VI,followed by concluding remarks.

Notation: We use boldface lower-case letters to denotecolumn vectors, to denote transposition, conjugate, con-jugate transposition, and the Euclidean norm. For a randomvariable will denote its probability density function(PDF), and its cumulative distribution function (CDF).Furthermore, will denote the complex-Gaussiandistribution with mean and variance the minimuminteger , and the expectation operator over .

II. MODELING PRELIMINARIES

We consider a WSN setup where sensors indexed bywish to communicate an information message (say

the value of a random variable they track or information theyrelay) to the FC; see Fig. 1. We assume that:

(as1) the information is common to all sensors and arrivescoherently at the FC.

With denoting block fading channel coefficientsbetween sensors and the FC, we further assume that:

(as2) are independent and identically dis-tributed (i.i.d.) according to a complex Gaussian distribu-tion with zero mean and unit variance, i.e., ;and each block fading channel process is ergodic;(as3) the FC feeds back to the sensors the CSI indexedby bits per channel realization, without error and withnegligible delay.

As in [3] and [4], synchronization in (as1) is assumed ac-quired using low-complexity synchronization algorithms; see,e.g., [17] and references therein. A setup where sensors have

a common message to transmit as in (as1) fits a cooperativescenario where bits from a common source encoded with suf-ficiently powerful error control codes are relayed through dis-tributed sensors to a destination. On the other hand, for WSNsdeployed to perform an estimation task, common data acrosssensors can be safely assumed identical under any one of thefollowing three operating conditions: (c1) sensors are locatedinside a small area and the information-bearing source is closeenough so that errors in recovering the source data at the sensorscan be deemed negligible; (c2) instead of multiple single-an-tenna sensors, we have a single multi-antenna sensor; or (c3)sensors exchange information to consent on the source. Besidesproviding mathematical tractability, (as2) can also hold underany one of aforementioned operating conditions (c1)–(c3). Ex-tension to correlated channels is possible but is out of the scopeof this work. Finally, (as3) can be easily guaranteed with suf-ficiently strong error control codes, since the feedback channelhas typically low rate.

Given a pool of adaptive modulation and coding (AMC)pairs, we suppose that each sensor supports a finite number

of AMC modes indexed by , with eachmode having constellation size and transmission rate

, where denotes the coding rate. To guar-antee quality of service, the rates must be deliveredwith a prescribed BER . To mitigate the effects of fading,the sensors beamform their transmitted symbol. Since a MISOsystem has multiplexing gain one (see, e.g., [9 pp. 48]), thesensors encode one information-bearing symbol per channeluse utilizing a common AMC mode. With this AMC mode,the th sensor transmits multiplied by a complex (steering)weight . Let denote the distributedbeamforming vector and the fading MISOchannel. The received symbol at the FC can be expressed as

(1)

where , and denotes the additive white Gaussiannoise (AWGN) with zero mean and variance . Notice thatboth the phase and the modulus of can be tuned to effectnot only distributed beamforming but also power allocationper fading state . Letting denote the average energy persymbol, we can write the total transmit power and receive-SNRper symbol as

(2)

(3)

where for the last equality in (2) and (3) we have assumedwithout loss of generality that . It follows from(3) that after beamforming, the MISO vector channel in (1) isfully characterized by the equivalent SISO scalar channel withnormalized power gain . The receive-SNR for theequivalent SISO system can be rewritten as .

Let denote the -bit Q-CSI codeword that theFC feeds back to the sensors per (as3). Based on , thesensors adapt their transmit-parameters to one ofprescribed modes specifying the transmission rate ,transmit power and beamforming vector .


Our goal is to optimally design the channel quantizer whichyields based on which we wish to adapt

, and , so that the total average power transmittedby all sensors is minimized subject to average rate and BERrequirements. To this end, we will first rely on F-CSIT whichcorresponds to setting in (as3).

III. SOLUTION BASED ON F-CSIT

In this section, we derive the optimal adaptive transmissionpolicy based on F-CSIT to provide insight and benchmark theQ-CSIT based design of Section IV. For , Q-CSIT be-comes F-CSIT; i.e., . Given , we wish to adapt thetransmit power , rate and beamforming vectorto minimize the average transmit power subject to prescribed re-quirements on the average rate and BER . As we shownext, the adaptation of the beamformer can be performed sepa-rately from the power and rate adaptation without loss of opti-mality (w.l.o.o.). This allows us to tackle the original problemin two separate phases: first we solve for the optimal1 adapta-tion of the beamformer ; next we introduce in theoriginal problem and solve for the optimal power and rate

adaptation.

A. Optimal Distributed Beamformer

From (2) and (3) we recognize that the selection of affectsthe scalar channel gain . Since for any AMC mode the re-quired transmit power is monotonically decreasingw.r.t. (for any given and ), to minimize the transmitpower we have to adapt per channel realization sothat is maximized. The optimal unitary beam-forming vector maximizing , hence minimizing therequired transmit power, is clearly (see, e.g., [10, Sec. 7.3.1])

(4)

and depends only on the channel phase, i.e.,.

To proceed with the optimal rate and power allocation strate-gies, we need to characterize statistically the channel in (1) whenbeamforming is adapted as in (4). When ,the channel gain is

(5)

As per (as2), adheres to a chi-squared distribution with PDF

(6)

where is the incomplete Gammafunction and . The corresponding CDF is

(7)

It is worth to recall that when optimal beamforming in(4) is implemented, the MISO channel in (1) is fully character-ized by an equivalent SISO channel with power gain dictated

1Henceforth, x will denote the optimal value of x.

by (5). This implies w.l.o.o. that solving for the optimaland is equivalent to finding the optimal and .Notice that since (and thus ) varies from one realization tothe next, rate and power will be adapted across time in orderto minimize the average transmit power under an average rateconstraint (the requirement on BER will be automaticallyaccounted for in the relationship between the power and the rateas we will see next).

B. Optimal Rate and Power Allocation

We order the AMC modes such that andlet the first mode represent the inactive mode with zero rate andpower . With denoting the instantaneousBER function, the minimum transmit power for the th AMCmode to satisfy the BER requirement can be calculated bysolving with respect to (w.r.t.) the equation

(8)

For -ary quadrature amplitude modulation (QAM), the BERcan be accurately approximated as2 [11] (e.g.,

for uncoded transmissions)

(9)

Substituting (9) into (8), the required power for the th AMCmode can be expressed as

(10)

With F-CSIT available, (10) shows that specifying the AMCmode determines not only the rate but also the power required tomeet the prescribed . Furthermore, it is easy to see that with

given the range of can be divided into consecutiveintervals with and , and the th AMCmode will be chosen if . Conversely, this meansthat once the intervals are specified for ,the rate and power allocations will be

if (11)

(12)

Letting , (11) and (12) imply that tofind the optimal rate and power allocations, we only need tosearch for the optimal which solves the following constrainedminimization problem:

where

subject to(13)

2Extensions to modulation schemes other than M-QAM are also possible.By appropriately selecting � and � , (9) can also be used to accommodate(un)coded transmissions as a BER upperbound based on the Chernoff bound[21].


where the average transmit power in the objective and theaverage rate in are calculated as the expectation of and

over all the possible realizations of . The set of constraintsensures consistency of the intervals .

Let denote the non-negative Lagrange multiplier associatedwith the rate constraint and the non-negative Lagrange multipliers associated with the constraintsin . The Lagrangian of (13) is then given by

(14)

At the optimum the necessary Karush–Kuhn–Tucker (KKT)condition is [7]

(15)

If all the constraints in are slack, i.e., , then. In this case, solving (15) w.r.t. yields for[cf. (10)]

(16)

Equation (16) expresses the optimal threshold inclosed-form as a function of the Lagrange multiplier .Upon defining , we canrewrite (16) as . As

, it iseasy to see . Therefore, the thresholds given by (16)indeed satisfy and thus they are the solutions to (15).

With specified by (16), can be calculated to satisfyusing the following algorithm.

Algorithm 1: Offline Power-Efficient Quantization (F-CSIT)

(S1.0) Let be a small tolerance level and initializewith an arbitrary positive number.

(S1.1) Calculate via (16).

(S1.2) Using (7), calculate the average rate as, and check

. If then stop; otherwise,calculate , update the multiplieras , and go to (S1.1). Parameterin the calculation of is an adaptive penaltyparameter that can be updated (per iteration)depending on convergence requirements.3

3The proposed updating scheme for � is based on the method of multipliers[5, Sec. 4.2].

Once is obtained using Algorithm 1 (that is computed of-fline), and in turn the optimal rate and power alloca-tions are determined after plugging (16) into (11) and (12).

C. On-Line Feedback and Adaptation of Transmitters

Having obtained , the following algorithm summa-rizes the online resource allocation steps the WSN has to exe-cute per channel realization:

Algorithm 2: Online Adaptation (F-CSIT)

For each channel realization :

(S2.1) The FC determines the index ofthe interval the channel gain fallsinto, and broadcasts to the sensors the F-CSITcodeword .

(S2.2) Each sensor transmits using the th AMCmode and the optimal steering weight [cf. (4) and(10)] .

Notice that even though each sensor can calculate usingonly of the feedback message, we also include infor robustness. This augmented feedback codeword reduces thecomputational burden at each sensor and also avoids the feed-back during the initialization phase needed for the sensors toacquire (recall that knowledge of is required to deter-mine the optimal power allocation and AMC mode selection).With the insights gained from the F-CSIT based benchmark, wenext derive the optimal adaptation schemes when only Q-CSITis available.

IV. SOLUTION BASED ON Q-CSIT

In this section, power-efficient sensor transmission and quan-tization schemes are derived based on Q-CSIT fed back fromthe FC to the sensors. Per fading realization, the FC quantizesto find separately the optimal according to a quantizer ,and the optimal AMC mode and according to a different quan-tizer . Concatenating the beamforming vector index

with the transmission mode index , theFC feeds back the -bit Q-CSI codeword , where

, with and .Based on this codeword , sensors adapt their transmissions andbeamforming weights to minimize their total average transmitpower.4

A. Optimal Distributed Beamformer

With only bits available, the beamforming vector ischosen from a finite set , where . As

4Although Q ( � ); Q ( � ), and pertinent adaptation schemes will be foundas solution of optimization problems, in principle we cannot claim global opti-mality among all the possible Q-CSIT designs that utilize B feedback bits. InSection VI, we simulate the performance of our design and compare it with alower bound on all Q-CSIT designs. Simulations indicate that the gap from thebound is small even for small values of B and thus demonstrate that our designexhibits near-optimal performance globally among all Q-CSIT designs.


with F-CSIT to minimize the transmit power, the optimalmaximizes the equivalent scalar channel gain in (3), i.e.,

(17)

Codebooks have been optimized for collocated MISO sys-tems [20], [16], [22]. Under various criteria, optimal codebooksminimize the maximum correlation between codewords. Basedon the Grassmanian line packing criterion, [16] further showedthat minimization of the maximum correlation is equivalent tomaximization of the minimum chordal distance which for twounitary complex vectors and is defined as [8]

(18)

In view of (18), the optimization in (17) can be expressed as

(19)

and the optimal codebook as

(20)

For arbitrary beamformer sizes and codebook sizes , nu-merical solutions of (20) are available; see, e.g., [15]. With theoptimal codebook available to both the FC and the sensors,the Q-CSI and optimal beamforming vector are then determinedas

(21)

(22)

Remark 1: Beamforming specified by (21) and (22) has beenproved to be optimal if . For detailed analysis onQ-CSIT based beamforming when , we refer the readerto [22].

Let us now turn our attention to the statistical characterizationof the equivalent channel when Q-CSIT is available. Recall thatthe gain of this equivalent channel with F-CSIT available is

for the optimal beamformer in (4). But with the Q-CSITbased optimal beamformer in (21)–(22), the equivalent channelgain becomes , where

(23)

can be interpreted as the channel gain loss due to quantization.This channel gain loss degrades the instantaneous receive-SNRto .

Based on the union bound, the CDF of can beupper-bounded tightly as [26]

(24)

where . Because and are independent[cf. (as2)], using the approximation , we canobtain the CDF of as

(25)

The PDF of can be in turn obtained as , yielding

(26)

The scalar channel gain fully characterizes the MISO channelwhen the optimal beamformer in (21)–(22) is adopted. Basedon the closed-form expressions (25) and (26), we next follow anapproach similar to what we used when F-CSIT is available toanalytically derive the optimal rate and power allocation basedon Q-CSIT.

B. Optimal Rate and Power and Allocation

When only finite-rate feedback is available, the FC needsto quantize using a finite number of regions. Towards thisobjective, identifying each quantization region with an AMCmode selection emerges as a natural first step. We considerdifferent quantization regions , and aswith F-CSIT, we associate with them the vector of thresholds

.The th transmission mode is characterized by the rate–power

pair in the quantization region (see also [14] whichdeals with throughput maximization). While is fixed for agiven AMC mode, we will select to satisfy the BER require-ment. Clearly, the average BER for the region can be ob-tained as the expected number of erroneous bits divided by theexpected number of transmitted bits; i.e.,

(27)

To satisfy the overall BER requirement , we set

(28)


It is easy to see that (28) reduces to (8) as (F-CSITcase). Furthermore, substituting (27) into (28) yields

(29)

Using an analytical expression for we derive inAppendix A, we can solve (29) for . The same expres-sion for can be used to obtain in closed-form, and thusquantify the average BER for any given quantization. Solving(29) for can be easily carried out with a one-dimensionalsearch. Letting denote this solution, the rateand power allocation can be expressed as [cf. (11) and (12)]

if (30)

(31)

Now as in the F-CSIT case, finding the optimal rate and powerallocations reduces to searching for the optimal thresholds .The corresponding optimization problem based on Q-CSIT is:

where

subject to(32)

where both transmit power in the objective as well as transmitrate in are averaged over all channel regions (quantizationstates). Notice that different from the F-CSIT based problem in(13), the loaded power here does not vary with the channel gain,but only with the region index (i.e., the power loading is fixedper AMC mode) and therefore it does not appear in the integrals.In fact, can be interpreted either as the probabilityof falling into the th quantization region or as the probabilityof selecting the th AMC mode.

Next we use the KKT conditions to find . Let denote theLagrange multiplier associated with the rate constraint and

the Lagrange multipliers corresponding to. As in (16), with , the KKT condition at the optimaldictates

(33)

Notice that using the CDF in (25), we have.

Since is an implicit function [cf. (29)],to calculate we rely on the implicit differentiation the-orem: , which

yields . Therefore,and we have

otherwise

(34)

The denominators in (34) can be evaluated analytically (seeAppendix B).

Different from the F-CSIT setup, we cannot guarantee thatthe optimal thresholds calculated from (33) always satisfy

. If by solving (33) we obtain is not slack,, and (33) does not hold. In this case, since the KKT

conditions also impose that , the optimalsolution should yield , and thus the th mode should beremoved from consideration. This implies that we need to checkthe feasibility of the solution obtained from (33). If solving (33)yields for a given , we then need to remove this thmode from the AMC pool and re-solve (33) for the remainingmodes. Notice that calculating the optimal here depends notonly on but also on the previous and the next .This prevents one from obtaining a closed-form expression for

. However, since closed-form expressions for all the termsin (33) are available, can be obtained numerically using atwo-dimensional search which is computationally affordable.

Algorithm 3: Offline Power-Efficient Quantization (Q-CSIT)(S3.0) Let denote a small tolerance, a small step size,

and the maximum value for the highestquantization threshold (e.g., a value bringing theprobability of the highest region close to 0).

(S3.1) Initialize with a small positive number and set; then calculate by solving

(33). If is not satisfied for some , set. If the obtained solution is feasible, go to

(S3.2); otherwise decrease and repeat(S3.1).

(S3.2) Based on the closed-form in (25), calculate theaverage rate .Check and if then stop;otherwise, calculate using a smallpositive adaptive constant , update the multiplierto , and go back to (S3.1).

Notice that the main computational burden in Algorithm3 pertains to the calculation of the optimal thresholds in step(S3.1), which requires a two-dimensional search. Once theoptimal as well as are calculated, the optimalquantizer can be readily determined as

(35)

With the AMC mode index given by (35), the optimal rateand power allocation are obtained via (30) and (31).



Based on the optimal beamforming and resource allocationpolicy, we outline next the online algorithm executed by the FCand the sensors per channel realization.

Algorithm 4: Online Adaptation (Q-CSIT)


(S4.1) The FC obtains andusing, respectively, (21) and (35), and broadcaststhe aggregate codeword to thesensors.

(S4.2) Each sensor transmits using the th entry ofthe optimal beamforming vector indexed by ,and loads the optimal power and rate allocationindexed by .

Notice that the optimal beamforming and resource alloca-tion configurations must be available to both the FC and sen-sors during the initialization phase. In step (S4.2) the optimaltransmit power corresponding to is calculated at the sen-sors by solving (29). This means that the sensors must know theoptimal thresholds (i.e., the FC must broadcast themduring the WSN deployment). To reduce the computational loadat the sensors, an alternative is to let the FC calculate and feedback to the sensors during the initialization phase.

V. SOLUTION BASED ON I-CSIT

So far, we derived optimal adaptive transmission strategiesbased on F-CSIT where sensors perfectly know the vectorchannel , and based on Q-CSIT where sensors have availablea quantized version of . In both cases, CSIT was obtainedthrough feedback from the FC. However, for time-divisionduplex (TDD) systems, each sensor can acquire viapilot-based channel estimation during the symmetric reversetransmission. This motivates analysis of what we call individual(I-) CSIT scenario, where each sensor has full knowledge of

, but only finite-rate is available for CSI feedback.

A. Optimal Distributed Beamforming

The adaptation in this case is similar to the one based onF-CSIT. When F-CSIT is available, (4) states that the optimalbeamforming weight at th sensor is . Onthe other hand when only I-CSIT is available, sensor has ac-cess to but the value of is unknown. To bypass thisdifficulty we define scaled beamformer and power variables as

and , respectively, so that .Now the th entry of the optimal is

, which requires only I-CSIT. The receive-SNR after optimalbeamforming based on I-CSIT is , where corre-sponds to the equivalent SISO channel gain in (5) and has PDFand CDF given by (6) and (7), respectively.

B. Optimal Rate and Power Quantization and Allocation

Given I-CSIT, the optimal (scaled) beamformer is . Toconstruct the entire steering vector , the sensors need alsothe (scaled) transmit power (or equiva-lently ), which requires knowledge of andtherefore depends on all the individual channels . With fi-nite-rate feedback, again the FC quantizes the channel gainusing a finite number of regions. Similar to Q-CSIT, the do-main of is partitioned into different quantization regions

so that the th AMC mode is employedby the sensors when . Different from Q-CSIT, now theth AMC mode is characterized by the rate–power pair ,

where is fixed per region and must be selected to satisfy theBER requirement . Upon defining

(36)

and arguing as in (27)–(35), it follows that a satisfying theprescribed BER (denoted by ) should solve

. Unlike what we had in the Q-CSITcase for , no closed-form expression is available for . How-ever, relying on the fact that is monotonically de-creasing w.r.t. to , the root can still be effi-ciently obtained via one-dimensional search.

We can now proceed to optimize resource allocationbased on I-CSIT. Given and a realiza-tion , the transmit power when the th AMC mode isselected can be found as

. To find the optimal quantizationthresholds minimizing the averagetransmit power, we need to solve

where

subject to:(37)

Letting denote the Lagrange multiplier associated withand assuming all the constraints in are satisfied with strictinequality, the KKT condition for the optimal yields

(38)


where ;and , can be obtained throughimplicit differentiation as

otherwise

(39)

Noticing the similarity between (38) and (33), we can readilydevise the counterpart of Algorithm 3 to compute andoffline.


Once the optimal thresholds are obtained, the corre-sponding can be computed at both the FCand the sensors, which then implement the following onlinealgorithm to adapt their transmissions per channel realization:

Algorithm 5: Online Channel Adaptation (I-CSIT)


(S5.1) The FC finds, and broadcasts to all

sensors.(S5.2) Each sensor transmits the common

symbol using the th AMCmode and steering weigh

.

Notice that since the beamformer based on I-CSIT does not re-quire feedback from the FC, we only need bitsfor CSI feedback, which may be significantly less than

bits required to feed back the Q-CSIT, especiallywhen is large.

VI. SIMULATIONS

In this section, we present numerical examples to assess thetransmit power consumed by the sensors when F-CSIT, Q-CSITor I-CSIT is available. The energy per symbol, system band-width and AWGN power spectral density are selected to sat-isfy . The simple cases tested include four sen-sors5 with fading links adhering to (as1). Unless otherwise spec-ified, we suppose that each sensor supports three active -aryQAM uncoded modes: 2-QAM, 8-QAM and 32-QAM plus theinactive state; i.e., the transmission rates of AMC modes are:

bits per symbol. In all simulations, we set theBER requirement to and wherever applicable, thecodebook of beamforming vector has size .

5This setup is reasonable for, e.g., a WSN organized in clusters where therole of the FC is played by a cluster-head and only a few sensors are awake percluster to minimize power consumption.

Fig. 2. Total transmit power versus total transmit rate for different CSIT sce-narios (M = 4; L = 4;N = 16).

Test Case 1 (Comparison of Transmit-Power Consumption):For variable rate requirements, Fig. 2 shows the average nor-malized transmit power per symbol (in dB) achieved by the op-timal adaptation policies based on: (i) F-CSIT, (ii) Q-CSIT, (iii)I-CSIT, and (iv) spatial (S-) CSIT. In the fourth S-CSIT case, weconsider for comparison and illustration purposes that the sen-sors implement optimal spatial beamforming based on F-CSITbut do not implement temporal power allocation across time.

From Fig. 2, we have the following interesting observations:(i) both Q-CSIT and I-CSIT based strategies can achievepower efficiency close to the optimal F-CSIT based one; (ii)the Q-CSIT based and the I-CSIT based policies yield almostidentical performance; (iii) the gap (in dB) between limited-ratefeedback based policies and the optimal F-CSIT based one re-mains almost constant for different rate requirements; and, (iv)both Q-CSIT and I-CSIT based strategies clearly outperformthe optimal S-CSIT scheme although the latter requires F-CSITwhile the former only require of few bits of feedback.

To test the importance of feedback on power efficiency, wecompare the power consumption by the optimal F-CSIT basedpolicy and that of an open-loop system without feedback. Asshown in Table I, the power consumed by the open-loop designis 20–25 dB higher than that of the closed-loop design basedon F-CSIT. As expected, CSI can largely reduce power require-ments, and thus considerably increase the lifespan of the WSN.

Test Case 2 (Power Consumption for Different Scenarios):Numerical results assessing the performance of F-CSIT,Q-CSIT, I-CSIT and S-CSIT based schemes over a wide rangeof parameter values are summarized in Table II. The simulationresults confirm: i) the near optimality of Q-CSIT and I-CSITbased policies, and ii) the significant loss in power performancethat the S-CSIT based system suffers from because it does notexploit the temporal diversity of the fading channel.

Test Case 3 (Characterizing the Optimum Solution): To gainmore insights, Table III lists the optimal quantization and re-source allocation for the three forms of CSIT. Recall that in the


TABLE IAVERAGE TRANSMIT-POWER (IN DB) FOR OPEN-LOOP AND CLOSED-LOOP SYSTEMS AS r VARIES

TABLE IIAVERAGE TRANSMIT POWER (IN DB) FOR (F, I, Q, AND S)-CSIT SCHEMES.

(REFERENCE CASE: M = 4; r = 2:5; � = 10 ; L = 4; r =[0; 1; 3; 5]; N = 16; E =N = 1; IN OTHER CASES, ONLY ONE

INDICATED PARAMETER IS CHANGED W.R.T. THE REFERENCE CASE)

F-CSIT and I-CSIT based solutions the thresholds pertain to thechannel gain whereas in the Q-CSIT solution they pertain to

.It can be seen that in all closed-loop systems, the AMC mode

with lower transmission-rate consumes the smallest averagetransmit power. We also observe that the optimal solutions tryto equalize the power price per bit in all quantization regions.It turns out that for all cases the most likely AMC mode is thethird one whose transmission rate is the closest tothe required rate . This fact is more pronounced in theF-CSIT case where the optimal allocation allows adaptationof the transmit power per AMC mode and thus, reduces theneed for adapting the rate. Interestingly, the thresholds arenoticeably different for the three cases (recall that to fairlygauge the Q-CSIT case, we should use the equivalent thresh-olds pertaining to , which are slightly larger than those for

shown in Table III). These discrepancies are due to thedifference in the SNR at the FC. Specifically, the receive-SNRis i) constant in the F-CSIT solution where with

proportional to ; ii) proportional to in the Q-CSITsolution where with constant; and iii) proportionalto in the I-CSIT solution where with constant.This also implies that the I-CSIT solution is very sensitive tosmall channel gains and, thus, it sets the threshold of the firstactive region to a relatively high value.

Test Case 4 (Effect of the Number of Feedback Bits): We haveseen that with (three active AMC modes) and ,the Q-CSIT and I-CSIT based solutions yield power efficiencyclose to the optimal F-CSIT benchmark. This is achieved using

and bits per channelrealization for Q-CSIT and I-CSIT, respectively. Next, we ana-lyze how the number of feedback bits affects the performance.

We first study the impact of varying the number of supportedAMC modes. Table IV lists the total power cost in the F-CSIT,

I-CSIT and Q-CSIT cases for different values. (When ,the sensors only support one AMC mode which does not requirefeedback; whereas for all the remaining cases, sensors support

active AMC modes plus an inactive mode indexed byfeeding back bits.) Recall that in the Q-CSIT baseddesign, we assume that beamforming modes can beemployed. As increases, we observe that i) the power con-sumption decreases for all solutions, ii) the power-gap of theQ-CSIT and I-CSIT based systems from the F-CSIT benchmarkalso decreases, and iii) the first and second increments of bringthe largest power savings. Notice also that the power consumedby the Q-CSIT solution exceeds that of the F-CSIT based systemeven as when the former only relies on a beamformingcodebook of finite size.

We next gauge the effect of varying in the Q-CSIT case.Fig. 3 plots the average transmit power versus the number offeedback bits . For comparison, we also depict thetransmit power for the corresponding I-CSIT and F-CSIT basedsolutions. Again the power consumption decreases as in-creases, while the reduction for each additional bit decreases.Note that as , and thus only power quantizationis implemented. Interestingly, we observe that when is large,the Q-CSIT based system outperforms the I-CSIT based one. Aswe already mentioned, divergence of the solutions in these twocases is due to the fact that quantizing the different variables( and ) results in different optimal quantizationdesigns. Intuitively, the advantage of Q-CSIT can be explainedbecause variations in receive-SNR are less pronounced in theQ-CSIT solution ( versus ) and this results in an optimalpolicy closer to the F-CSIT benchmark. However, it is worthto reiterate that the adaptation based on Q-CSIT requires morefeedback bits than the one based on I-CSIT, especially when theQ-CSIT solution outperforms the I-CSIT one; i.e., when islarge.

Finally, we should to point out that although theoretically thepower gap of Q-CSIT and I-CSIT relative to F-CSIT tends tozero as , our numerical results suggest that even afew feedback bits (e.g., and ) suffice to closethe gap.

Test Case 5 (Sensitivity to Synchronization, i.e., Mistiming Ef-fects): Since the synchronization among sensors assumed under(as1) is challenging to obtain, we will rely on simulated teststo gauge how sensitive is our design to synchronization errors.Although systems with instantaneous CSI at both receiver andtransmitter can cope with synchronization errors, here we testa worst case scenario where these errors are so fast that neitherthe receiver nor the nodes can account for them. The simulatedsetup consists of four sensors with random mistiming boundedby . On the one hand the FC is unable to estimate and com-pensate for these mistiming errors and thus quantizes the erro-


TABLE IIIOPTIMUM AVERAGE POWER (IN DB) AND RATE LOADING PER AMC MODE (QUANTIZATION STATE) AND

QUANTIZATION THRESHOLDS (M = 4; r = 2:5; � = 10 ; L = 4; r = [0; 1; 3; 5]; N = 16)

TABLE IVAVERAGE TRANSMIT POWER (IN DB) AS L VARIES

Fig. 3. Effect of number of feedback bits for Q-CSIT scheme (M = 4; r =2:5; L = 4).

neous channel gain. On the other hand, the sensors adapt theirtransmit configurations using codebooks designed consideringperfect synchronization (thus, suboptimum in the presence ofmistiming) and based on the erroneous information fed back bythe FC.

Based on this setup, Fig. 4 depicts BER measured at the FCfor different mistiming errors when the target rate and BER are2.5 and , respectively, for the cases of F-CSIT, Q-CSITand I-CSIT based operation. As expected, BER performancedegrades when synchronization errors occur, increasing mod-erately for small values of and exponentially for systems withvery poor synchronization (average transmit power and rate arenot presented since they remain unchanged with the FC not miti-gating for mistiming). Focusing on the Q-CSIT case, we observethat this system exhibits the best BER performance. Specificallyfor timing errors not exceeding 5%, the system performs belowthe target BER and even for errors up to 20%, the BER in less

Fig. 4. BER degradation due to synchronization mistiming for F-CSIT,Q-CSIT and I-CSIT schemes (M = 4; r = 2:5; L = 4).

than twice the required value. The reason for this behavior istwofold: i) our design is based on an upper bound [cf. (24)]which, although tight, yields a conservative design that slightlyoversatisfies the BER requirement; and ii) the quantized CSITnaturally accounts for uncertainty in channel estimation makingthe design more robust synchronization errors. More important,Fig. 4 suggests that our novel adaptive schemes are robust tolow-moderate timing synchronization errors.

VII. CONCLUSION

In a WSN entailing coherent sensor communications with afusion center, we minimized the average transmit power sub-ject to average rate and BER requirements when full (F-) CSIT,quantized (Q-) CSIT or individual (I-) CSIT is available. Withfinite-rate feedback, we optimally separated the main designin two subproblems: i) MISO channel quantization and beam-forming, and ii) rate/power quantization and allocation. By ex-ploiting the parallelism between the coherent WSN setup anda distributed MISO system, we relied on nonlinear program-ming tools to solve the programs at hand and derived the cor-responding power-efficient channel quantization and adaptivetransmission policies. Numerical results confirmed that our lim-ited-rate feedback (Q-CSIT and I-CSIT)-based solutions attainpower efficiency surprisingly close to the optimal F-CSIT based


benchmark. They outperform a S-CSIT scheme which only ex-ploits spatial diversity with F-CSIT, and offer significant powersavings relative to open loop systems that do not exploit CSIT.

APPENDIX ACLOSED-FORM BER EXPRESSION FOR Q-CSIT

Finding a closed-form expression forrequires analytical evaluation of the two integrals in-volved in (29). Using the CDF of , we have for thesecond integral that .The first integral in (29) requires solving analytically

. Utilizing (9)with , the latter can be rewritten as

, which after tediousmanipulations yields [cf. (26)]

(40)

Based on (40), we can write (29) in closed-form as

(41)

Finally, based on (28) and (41) we arrive at

(42)

which quantifies the average BER analytically when the pro-posed Q-CSIT design is in force.

APPENDIX BCLOSED-FORM OF THE POWER DERIVATIVE BASED ON Q-CSIT

For the PDF of in (26), we can write

(43)

Differentiating (9) w.r.t. , we can rewrite (43) as

(44)

where is found in closed-form as

(45)

ACKNOWLEDGMENT

The views and conclusions contained in this document arethose of the authors and should not be interpreted as repre-senting the official policies, either expressed or implied, of theArmy Research Laboratory or the U.S. Government.

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Antonio G. Marques (M’07) received the Telecom-munication Engineering degree and the Doctoratedegree (together equivalent to the B.Sc., M.Sc.,and Ph.D. degrees in electrical engineering), bothwith highest honors, from the Universidad CarlosIII de Madrid, Madrid, Spain, in 2002 and 2007,respectively.

In 2003, he joined the Department of SignalTheory and Communications, Universidad Rey JuanCarlos, Madrid, Spain, where he currently developshis research and teaching activities as an Assistant

Professor. Since 2005, he has also been a Visiting Researcher at the Departmentof Electrical Engineering, University of Minnesota, Minneapolis. His researchinterests lie in the areas of communication theory, signal processing, andnetworking. His current research focuses on channel state information designs,energy-efficient resource allocation, and wireless ad hoc and sensor networks.

Dr. Marques received several awards for his work in distinctive internationalconferences, including the International Conference on Acoustics, Speech andSignal Processing (ICASSP) 2007.

Xin Wang (M’04) received the B.Sc. degree andthe M.Sc. degree from Fudan University, Shanghai,China, in 1997 and 2000, respectively, and the Ph.D.degree from Auburn University, Auburn, IL, in 2004,all in electrical engineering.

From September 2004 to August 2006, he wasa Postdoctoral Research Associate with the De-partment of Electrical and Computer Engineering,University of Minnesota, Minneapolis. SinceSeptember 2006, he has been an Assistant Professorin the Department of Electrical Engineering, Florida

Atlantic University, Boca Raton. His research interests include medium accesscontrol, cross-layer design, resource allocation, and signal processing forcommunication networks.

Georgios B. Giannakis (F’97) received the Diplomadegree in electrical engineering from the NationalTechnical University of Athens, Greece, in 1981.From 1982 to 1986, he was with the University ofSouthern California (USC), where he received theM.Sc. degree in electrical engineering in 1983, theM.Sc. degree in mathematics in 1986, and the Ph.D.degree in electrical engineering in 1986.

Since 1999, he has been a Professor with theElectrical and Computer Engineering Department atthe University of Minnesota, where he now holds an

ADC Chair in wireless telecommunications. His general interests span the areasof communications, networking and statistical signal processing—subjects onwhich he has published more than 275 journal papers, 450 conference papers,two edited books, and two research monographs. Current research focuses oncomplex-field and network coding, multicarrier, cooperative wireless commu-nications, cognitive radios, cross-layer designs, mobile ad hoc networks, andwireless sensor networks.

Dr. Giannakis is the (co-)recipient of six paper awards from the IEEE SignalProcessing (SP) and Communications Societies, including the G. Marconi PrizePaper Award in wireless communications. He also received Technical Achieve-ment Awards from the SP Society (2000), from EURASIP (2005), a Young Fac-ulty Teaching Award, and the G. W. Taylor Award for Distinguished Researchfrom the University of Minnesota. He has served the IEEE in a number of posts,and is currently a Distinguished Lecturer for the IEEE SP Society.

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4446 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, … · nonlinear optimization, power efﬁciency, quantization, resource allocation, wireless sensor networks (WSNs). I. INTRODUCTION