IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 9, SEPTEMBER 2014 4923

Reachback WSN Connectivity: Non-CoherentZero-Feedback Distributed Beamforming

or TDMA Energy Harvesting?Konstantinos Alexandris, George Sklivanitis, Student Member, IEEE, and Aggelos Bletsas, Senior Member, IEEE

AbstractThis work is motivated by the reachback connec-tivity scenario in resource-constrained wireless sensor networks(WSNs): a single terminal at maximum power cannot estab-lish a reliable communication link with the intended destina-tion. Thus, neighboring distributed transmitters should contributetheir radios and transmission power, in order to achieve reliabletransmission of a common message. This work is particularlyinterested in low-SNR scenarios with unreliable feedback chan-nels, no channel state information (CSI), and commodity radios,where carrier phase/frequency synchronization is not possible.Concrete non-coherent maximum likelihood and energy detectionreceivers are developed for zero-feedback distributed beamform-ing. The proposed receivers are compared with non-coherent en-ergy harvesting reception, based on simple time-division multipleaccess. It is shown that the proposed zero-feedback distributedbeamforming receivers overcome connectivity adversities at thelow-SNR regime. This is achieved by exploiting signals align-ment of M distributed transmitters (i.e., beamforming), evenwith commodity radios, at the expense of network (total) powerconsumption. Application scenarios include resource-constrainedWSNs or emergency radio situations.

Index TermsNon-coherent receivers, reachback connectivity,wireless sensor networks, zero-feedback beamforming.

I. INTRODUCTION

W IRELESS Sensor Networks (WSNs) are typicallyequipped with low-complexity, battery-operated radiosand low-cost isotropic antennas that generate undirected and

Manuscript received July 25, 2013; revised February 2, 2014; acceptedMay 28, 2014. Date of publication June 12, 2014; date of current versionSeptember 8, 2014. This work was supported in part by the European Union(European Social Fund-ESF), and in part by the Greek national funds throughthe Operational Program Education and Lifelong Learning of the NationalStrategic Reference Framework (NSRF)Research Funding Program: Thales.Investing in knowledge society through the European Social Fund. The as-sociate editor coordinating the review of this paper and approving it forpublication was T. Hou.

K. Alexandris was with the Telecom Laboratory, School of Electronicand Computer Engineering, Technical University of Crete, Chania 73100,Greece. He is now with EURECOM, 06410 Biot, France (e-mail: kalexan-dris@isc.tuc.gr).

G. Sklivanitis was with the Telecom Laboratory, School of Electronicand Computer Engineering, Technical University of Crete, Chania 73100,Greece. He is now with the Signals, Communications, and Networking Re-search Group, Department of Electrical Engineering, University at Buffalo,The State University of New York, Buffalo, NY 14260-1920 USA (e-mail:gsklivan@buffalo.edu).

A. Bletsas is with the Telecom Laboratory, School of Electronic andComputer Engineering, Technical University of Crete, Chania 73100, Greece(e-mail: aggelos@telecom.tuc.gr).

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

Digital Object Identifier 10.1109/TWC.2014.2330295

relatively weak signals. Distributed transmit beamforming (orsimply distributed beamforming), i.e., cooperative transmissionfrom two or more distributed terminals, such that the phasesof the transmitted signals align and offer a constructive gaintowards the intended destination receiver, has been proposed asa means to boost the power of the transmitted signal and im-prove connectivity in resource-constrained WSNs. Distributedbeamforming could in principle offer high directivity, when thenetwork of terminals is designed to operate as a virtual antennaarray.

However, several key challenges need to be addressed.Beamforming setups utilize powerful optimization tools [1], [2]that require some type of prior knowledge, either in the form ofchannel state information (CSI) or its second order statistics, inorder to minimize the total transmission power and maximizethe received signal-to-noise ratio (SNR). Phase alignment at thereceiver depends on carrier and packet synchronization, whichplay crucial role in the realization of power beamforming gains[3]. However, in distributed (i.e., network) setups, synchro-nization is quite challenging, since each terminal has its ownlocal oscillator and the network topology is usually unknown.Furthermore, in the case of low SNR scenarios or fast-fadingenvironments where channel estimation often fails but packet-level synchronization is still feasible, non-coherent receptionseems an ideal solution.

Several techniques for distributed beamforming have beenproposed, including multi-bit (or even single-bit) closed-loopfeedback between receiver and distributed transmitters, as de-scribed in [4][6]. Another approach includes an interference-limited spread-spectrum scheme across the distributed nodesthat maintains the beamforming properties of the network [7].Work in [8] discusses a new timing and phase synchroniza-tion method and evaluates its precision in distributed multi-user multiple-input multiple-output (MU-MIMO) setups usingwireless open-access research platform (WARP) radios. Phaseand time synchronization between the distributed transmittersis achieved with a master-slave setup. Synchronization and sig-nal generation are implemented in a field-programmable-gate-array (FPGA). Moreover, a master-slave architecture for carriersynchronization was investigated in [9]; it was shown that evenwith phase errors on the order of 60, SNR gains of 70% arepossible. Finally, work in [10] revisits 1-bit feedback distributedbeamforming [4] and discusses a scalable synchronization ar-chitecture which is based on receivers wireless feedback andan extended Kalman filter at the transmitters for frequencylocking. A proof-of-concept implementation on commercial

1536-1276 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

4924 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 9, SEPTEMBER 2014

software-defined radios was also provided. A comprehensivereview of distributed beamforming can be found in [11] andreferences therein. It can be safely said that most prior art ondistributed beamforming requires either CSI at the distributedtransmitters (e.g., [12]) or feedback (from the receiver) avail-ability or ability to access the transmitters radio module forcarrier phase adjustments.

Furthermore, blind eigenvalue-based detectors exploiting re-cent random matrix theory [13][15] or subspace trackingmethods [16], are not always an option, since a significantamount of data (e.g., a large number of transmitted sym-bols) and increased computational effort are required; suchrequirements may not be practically feasible in low-complexity,resource-constrained WSN terminals.

Finally, capacity-related results for centralized multiple-input multiple-output (MIMO) non-coherent reception in [17],suggest a signal structure through unitary space-time modula-tion (USTM) [18], [19]. However, such designs are created forcentralized multi-antenna transmitters where there are no dif-ferent carrier frequency offsets (CFOs) among the transmittingelements. Therefore, those structures are not directly applicableto the distributed setup, considered in this work.

In sharp contrast to prior art, this work studies distributedbeamforming in a non-traditional fashion, assuming:

no CSI availability, no reliable receiver-based feedback, no access to the physical layer for carrier phase adjust-

ments (commodity WSN radio transmitters).This work is motivated by network partitioning problems,

where a network subset is disconnected from the rest of thenetwork, i.e., each terminal alone cannot communicate witha distant receiver, outside its immediate neighborhood (this isalso known as the reachback communication scenario). Thatmay occur in resource-constrained WSNs or emergency radiosituations, e.g., firefighters radios that collaborate in order totransmit a common emergency information message outside aburning building.

In such cases, feedback from outside the subset may not bereceived reliably, while commodity radios, typically utilized inWSNs, may not offer access to the transmitted carrier phase.Work in [20] and [21] showed that zero-feedback beamformingwith unsynchronized carriers is possible and provided analysisresults in terms of signal alignment probability, signal align-ment delay and respective beamforming gains. However, nospecific receivers were proposed. Zero-feedback beamforminggains will be offered if the distributed terminals can trans-mit packets at the same time. Such packet-level simultaneoustransmission is possible with a simple protocol, where trans-missions are dictated by a master (maestro) terminal, at thevicinity of the distributed transmitters, as experimentally shownin [22].

Inability to acquire CSI and establish a reliable feedbackchannel, both impose significant constraints and offer a chal-lenging problem, that may be initially considered unsolv-able: the terminals can either employ zero-feedback distributedbeamforming, where each node transmits at maximum power- in which case a concrete receiver is required - or the nodes

(a)

(b)

Fig. 1. Transmission schemes. (a) Zero-feedback distributed beamforming.(b) TDMA.

transmit in a round-robin fashion, i.e., with time-division mul-tiple access (TDMA) (Fig. 1); in the latter case the receivergathers signal energy from multiple, distributed transmitters (asopposed to single terminal transmission) in order to achievereliable reception. This work particularly focuses on the lowsignal-to-noise ratio (SNR) regime and poses the followingquestion: can zero-feedback distributed beamforming outper-form TDMA at the low SNR regime, via constructive signaladdition with commodity radios, at the expense of total powerconsumption?

As shown in this work, the answer is positive. Specific non-coherent maximum likelihood and energy detection receiversfor the zero-feedback distributed beamforming are presented,and compared with non-coherent energy harvesting (TDMA-based) reception (Fig. 1). Analytical bit error rate (BER) re-sults are also presented. For completeness, USTM is brieflydiscussed.

Section II introduces the definitions, the basic idea andbriefly discusses USTM in the context of distributed transmit-ters. Section III presents the proposed zero-feedback distributednon-coherent receivers and their BER performance, Section IVprovides the TDMA receiver and its BER performance andSection V offers the numerical results. Finally, Section VIconcludes this work.

Notation: Upper and lower case bold symbols denote matri-ces and column vectors, respectively; IN denotes the N Nidentity matrix; 0NN denotes the N N zero matrix; ()Tdenotes transpose; () denotes complex conjugate; () denotestranspose complex conjugate; rank(A) denotes the rank ofmatrix A; x CN (,) denotes that random vector x iscomplex Gaussian with mean vector and covariance matrix1; x N (,) denotes that random vector x is Gaussianwith mean vector and covariance matrix 2; G(k, ) denotes

1The probability density function (p.d.f.) of a N -dimensional x is given by:fX(x) =

1N det()

exp{(x )1(x )}.2The p.d.f. of a N -dimensional x is given by: fX(x) =

1(2)N det()

exp{(1/2)(x )T1(x )}.

ALEXANDRIS et al.: REACHBACK WSN CONNECTIVITY 4925

Fig. 2. System setup with M distributed transmitters.

the Gamma distribution with parameters k, 3; erfc() stands forthe complementary error function4; [a/b] stands for the integerdivision operator; a mod b stands for the modulo operator; a|bstands for a divides b i.e., if a|b then b mod a = 0; a b standsfor a does not divide b (b mod a = 0).

II. SYSTEM MODEL AND BASIC IDEA

This work considers M distributed terminals (Fig. 2) that si-multaneously transmit a common symbol towards a destinationterminal at a given frequency band. All M terminals:

use on-off keying (OOK) modulation, with signal set X ={x0, x1}, where x0 = 0 and x1 =

E1;

operate over Rayleigh, flat-fading channels hm=

Amejm CN (0, 1), independent across different

mT = {1, . . . ,M} (with Am real and m [0, 2); are equipped with non-ideal local oscillators, (i.e., man-

ufacturing inaccuracies result to offsets from the nomi-nal oscillation frequency) thus carrier frequency offsets{fm}mT are introduced per transmitter-receiver link.

CFO parameters {fm}mT are assumed to be independentand identically distributed (i.i.d.) random variables accord-ing to N (0, 2f ). The standard deviation f is set to f =E[f2m] = fc ppm, where fc denotes the nominal carrier

frequency and ppm denotes the frequency skew of the clockcrystals, with typical values of 120 parts per million (ppm).Finally, reception of the kth information symbol at the destina-tion occurs in the presence of additive complex white Gaussiannoise (CWGN), wk CN (0, 2):

yk= xk

Mm=1

hme+j2fmkTs + wk = xk + wk, (1)

where xk X and 1/Ts is the symbol-transmission (baud) rate.In classic beamforming setups, the transmitted signal per

antenna element is multiplied by a complex shaping param-eter, such that the aggregate received signal power is strongat a given direction (e.g., towards the destination) and weak

3The p.d.f. is given by: fX(x; k, )= 1k 1

(k) xk1 e x

u(x), where u() denotes the unit step function and (k) = (k 1)!for any positive integer.

4The error complementary function is given by: erfc(x) = 2

+x

et2dt.

Fig. 3. Zero-feedback distributed beamforming views transmitted signals asrotating phasors with non-zero alignment probability, i.e., there are time instantswhere signals from distributed transmitters can constructively add.

towards other directions (hence the term beamforming). Inlinewith the basic assumption of this work that commodity radiomodules are assumed, where access to the physical-layer signalis not readily available, the model above does not include theshaping parameters at each transmit antenna. However, thebeamforming effect can be achieved with commodity radiodue to the constructive addition of multiple signals transmit-ted by distributed terminals. Specifically, this work exploitsthe distributed nature of the system setup and particularlythe existence of different CFO parameters {fm}mT pertransmitter-receiver link. None of the above holds in the caseof a centralized multiple-input single-output system (MISO),where all transmitting antennas share a common oscillator andfm = f, m T .

More specifically, the idea behind zero-feedback distributedbeamforming is based on signal alignment at the receiver andrespective power maximization. The received signal poweraccording to Eq. (1) is given by:

|xk|2=xk

(M

m=1

hme+j2fmkTs

)2

=x2k

{M

m=1

A2m

+ 2m =i

AmAi cos(2(fmfi)kTs+mi).(2)

The cosine term inside the braces is not necessarily positive,since its value depends on the pairwise CFO and channel phasedifferences among the different links.

Each transmitted signal Ame+j(2fmkTs+m) (see Eq. (1))can be viewed as a phasor, with angular rotating speed propor-tional to the respective CFO fm. Thus, there is a non-zeroprobability that all phasors (signals) align, since they rotatewith different angular speeds. For example, consider M = 2distributed transmitters with carrier frequency offsets f2 =2f1 = f0 and channel phase difference 1 2 = at timeinstant t = t0, i.e., the two signals add destructively at thereceiver (Fig. 3). It can be easily seen that at time t = t0 +1/f0, the two transmitted signals will be aligned, i.e., they willadd constructively, offering beamforming gain, provided thatthe same information symbol is repetitively transmitted by bothtransmitters and the wireless channel fading parameters remainconstant; in other words, a zero-feedback distributed setupcan create an alignment event that offers beamforming gain,

4926 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 9, SEPTEMBER 2014

even with commodity radios (hence the term zero-feedbackdistributed beamforming).

In [20] the authors analytically calculated the alignmentprobability as a function of time, for M signals/phasors withina sector of angle 0, discussed the expected number of sym-bols where alignment occurs, the required average length ofrepetition and studied the feasibility of such schemes. It wasshown that such steady-state alignment probability depends onthe repetition length and not on the clock frequency skew (inppm) or the wireless channels phase offsets {m}mT , spark-ing interest on research for non-coherent reception. Frequencyskew (in ppm) only affects how fast steady-state alignmentprobability will be achieved [20]. Non-coherent reception isideal for low SNR scenarios or fast-fading environments wherechannel estimation often fails but packet-level synchronizationis still feasible. This work extends zero-feedback distributedbeamforming proposed in [20], by offering concrete, non-coherent receivers.

Parameter L denotes the number of transmitted symbolsper block (block-length). The term phase used in this workdescribes the duration of L symbols after which a new phasebegins and the fading coefficients are changed independentlyfrom the previous ones (quasi-static fading). CFO parametersare assumed random but constant, during one phase.5 Finally,it is assumed that L f Ts 1, since L must be kept low(so that 1/L is large, as will be further explained below), whilef Ts is significantly smaller than unity for typical values. Forinstance, for L = 3, crystals of 2 ppm (2 106) and binarymodulation rate of 1 Mbps at 2.4 GHz, Lf Ts=0.0144 1.This assumption will be relaxed in the analysis and numericalresults sections.

Moreover, the average SNR per mth transmitter antenna perkth time slot is defined as:

SNR =E[x2k]

E [|wk|2] =E122

. (3)

It is noted that when the transmitters are allowed to si-multaneously transmit different symbols, the resulting schemecorresponds to distributed space-time coding, fundamentallydifferent than the beamforming setup of this work. Work in[19] studied the problem of non-coherent reception in classicMIMO systems with unitary space-time modulation; due to theco-located setup, different CFOs among different links werenaturally not incorporated in their model. Given that the MIMOdesign in [18] is non-coherent, we study for completenessits MISO special case, in the context of distributed termi-nals, where CFO parameters {fm}mT are prominent. TheRayleigh fading coefficients are assumed to be constant forT symbols and CWGN is added at the receiver. For a singlereceiver and M transmitting antennas, the model in [18] simpli-fies to a y vector of length-T , where its tth element is given by:

yt =

M

Mm=1

hme+j2fmtTsstm + wt, (4)

5CFO typically changes with temperature; the latter can be assumed constantfor a number of transmitted bits.

Fig. 4. Simulation BER performance using USTM for M = 2, T = 8 andR = 1 bit/symbol, for the conventional, centralized (CFO-free) and distributed(CFO-limited) case (as in this work).

for t {1, . . . , T}. Coefficient represents the expected SNRat the receiver antenna and stm stands for the (t,m)th elementof the T M space-time matrix S. The systematic design of Sis presented in [19].

The existence of CFOs and the distributed counterpart vastlychanges the design requirements. Fig. 4 depicts BER perfor-mance of USTM, for the cases with and without CFOs; con-stellation of 2RT signals was assumed, with R = 1 bit/symboland T = 8. The unitary space-time signals were constructed forM = 2 transmitting antennas, K = 1 (dimension of the blockcode), q = 257 (arithmetic base [19, Table I]). The SNR at thesingle receiving antenna per time slot is [19, Eq. (1)]. WithoutCFOs, USTM achieves reduced BER, while for the distributedcase (i.e., presence of {fm}mT ), performance is degraded,as expected, since USTM has been designed for the centralized,CFO-free MIMO case.

Therefore, different non-coherent transmission schemes (in-cluding the USTM methodology) need to be devised for thedistributed setup. From that perspective, the distributed zero-feedback beamforming receivers of this work target a newlyformulated problem, which could be of potential academic andindustry interest.

III. DISTRIBUTED TRANSMIT BEAMFORMING RECEIVERS

Repetitive transmission exploits signal alignment event, asexplained above. The M distributed transmitters simultane-ously transmit the same information symbol for L slots, whilethe channel values remain unchanged (Fig. 5). The achievedrate is 1/L and according to the system assumptions, the binaryhypothesis test is given by:

H0 : y =w,

H1 : y =gx1 +w, (5)

where

g= [g1 gl gL]T , (6)

ALEXANDRIS et al.: REACHBACK WSN CONNECTIVITY 4927

Fig. 5. Repetitive transmission scheme. The M distributed transmitters simul-taneously transmit the same information symbol for L slots, while the channelparameters remain unchanged.

and

w= [w1 wl wL]T . (7)

The random variable gl=M

m=1 hme+j2fmlTs , l{1, 2,

. . . , L}, is proved to be distributed according to CN (0,M) (seeAppendix A, Lemma 1). The noise vector elements are i.i.d.according to wl CN (0, 2) for l {1, . . . , L}.

This scheme is used both in Section III-A and B for thederived detectors.

A. Heuristic Detector

The slots, where signal alignment occurs, are not a prioriknown. Thus, a subset of slots cannot be pre-selected fordetection but instead all L symbols are taken into account, usinga square-law technique:

yy =Ll=1

|yl|2. (8)

Under H0, the squared L2 norm of y is a Gamma-distributedrandom variable, as a sum of i.i.d. exponentials:

H0 : yy =

Ll=1

|wl|2 = w G(L, 2). (9)

Under H1 and given {fm}mT , the squared L2 norm of y,is a sum of correlated, identically Gamma-distributed randomvariables, i.e.,

H1|{fm}mT :

yy =Ll=1

|yl|2 =Ll=1

l, l G(1,Mx21 +

2), (10)

and ij is the correlation coefficient between i and j

ij =cov[i, j ]var[i]var[j ]

, i = j, i, j {1, 2, . . . , L}

=x41

{M + 2

k =n cos [2Ts(fk fn)(i j)]

}(Mx21 +

2)2 .

(11)

The sum in the ij calculation above is performed over all(M2

)possible CFO pairs (fk,fn), for k, n T .

A closed form for the p.d.f. of the sum of correlated Gammais provided in [23, Eq. (5)] while in [24], is offered as a functionof the L L matrix K,

K =

1

12 . . .

1L

21 1 . . .2L

.

.

. . . .L1

L2 . . . 1

, (12)for the special case where K is positive definite and ij > 0. Inour problem, K is not necessarily positive definite and ij maybe negative. Thus, relevant analytical results in [23] and [24]are not applicable in this work.

Instead, the detection threshold of the binary test is calculatedwith a heuristic method, taking advantage of the known statis-tics under H0. The non-coherent heuristic detector is given by:

yy =Ll=1

|yl|2H1 1(k). (13)

In order to estimate an appropriate value for threshold 1, theprobability of error under H0, P (e|H0), is considered, i.e., theerror of deciding that x1 =

E1 was transmitted instead of

the correct x0 = 0. The considered threshold is given by:

1(k) = E[w] + k

var[w] = 2[L+ kL], k > 0, (14)

where k is a positive parameter selected through simulations, inorder to minimize the probability of error and random variablew was defined in Eq. (9). An upper bound of parameter k isacquired by calculating the probability of error under H0 asfollows:

P (e|H0) 1(L 1)!

(L,

1(k)

2

) , (15)

where for example = 106 and (a, z) = (a) (a, z) = +z t

a1etdt;(a) > 0, (a, z) is the incomplete Gammafunction [25, p. 260, Eq. (6.5.2)] and (a) is the Gammafunction [25, p. 255, Eq. (6.1.1)]. Such k from Eq. (15) is onlyan upper bound and does not optimize the overall BER, sinceP (e|H1) is not taken into account. Near-optimal k will be foundthrough simulations, such that both P (e|H1) and P (e|H0) areconsidered.

B. Maximum-Likelihood Non-Coherent Detector forFully-Correlated Equivalent Channel Taps

The maximum-likelihood detector derived in this para-graph is based on fully-correlated6 equivalent channel taps7

{gl}Ll=1 =

1M {gl}Ll=1.

6The elements of a vector x = [x1 x2 . . . xN ]T are fully-correlated, if thecorrelation coefficient xixj = 1, i, j {1, 2, . . . , N}.

7At this point and throughout this paper, the term equivalent channel tapswill stand for {gl}Ll=1.

4928 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 9, SEPTEMBER 2014

Theorem 1: The random vector g is distributed according toCN (0,M), where =[1 1]T , if Lf Ts0 (in thesense of e2[(kl)fTs]2 1 for k = l, k, l {1, 2, . . . , L}).

Proof: The random vector g is defined as g =

1M g,

where

g = [g1 gl gL]T , l {1, 2, . . . , L}, (16)

and random variable gl =

1M

Mm=1 hme

+j2fmlTs CN (0, 1). For notational convenience, random vectors

h= [h1 hM ]T , (17)

and

e= [f1 fM ]T , (18)

are defined. The random variables {gl}Ll=1 are correlated andtheir L L covariance matrix is expressed as C = E[gg]. The(k, l)th element of covariance matrix C, for k, l {1, , L},is analytically computed as follows:

Eh,e [gkgl ]

= Eh,e

[(1

M

Mm=1

hme+j2fmkTs

)

(

1

M

Mn=1

hne+j2fnlTs

)]

=1

M

Mm=1

Eh,e

[|hm|2e+j2fm(kl)Ts

]hm,fm

=indep.

1

M

Mm=1

Ehm

[|hm|2]Efm [e+j2fm(kl)Ts]

=1

M

22f

Mm=1

+

e

+j42ffm(kl)Tsf2m

22f dfm. (19)

The integral above in Eq. (19) is computed according to[26, p. 163. Eq. (7.7.6)]:

I = limx

+x

e

+j42ffm(kl)Tsf2m

22f dfm

=

1

2

22fe

2[(kl)fTs]2

limx erfc

(1

22fx j

2(k l)fTs

)

=

22fe2[(kl)fTs]2 . (20)

From Eqs. (19), (20), the (k, l)th element of covariance mat-rix C becomes:

Eh,e [gkgl ] = e

2[(kl)fTs]2 , (21)

and the matrix C is analytically described as:

C =E[gg]

=

1 e2[(1L)fTs]2...

.

.

.

.

.

.

e2[(L1)fTs]2 1

. (22)Note that the (k, l)th element of matrix C, e2[(kl)fTs]2

1, for k = l, if the exponent 2[(k l)fTs]2 0. A suf-ficient condition for the above approximation is Lf Ts0.The square included in the exponent accelerates convergenceof the exponential term to unity, when the sufficient con-dition L f Ts 0 is satisfied. In that case, all the ele-ments of random vector g are fully-correlated (in the senseof e2[(kl)fTs]2 1 for k = l, k, l {1, 2, . . . , L}), sincetheir correlation coefficient

gkgl 1, for k = l. Considering

this case, the random vector g can be replaced by the randomvector g0 CN (0,), where g0 CN (0, 1) and =[1 1]T . Exploiting the above, it can be directly concludedthat g is distributed according to CN (0,M).

Corollary 1: For the case of fully-correlated equivalentchannel taps {gl}Ll=1 (in the sense of e2[(kl)fTs]

2 1for k = l, k, l {1, 2, . . . , L}), g is distributed according toCN (0,M).

In many real-world WSNs scenarios, the condition L f Ts 0 is satisfied. For instance, if f = 2.4 GHz 2 ppm =4.8 kHz, Ts = 1 s (i.e., rate 1 Mbps for binary modulation)and L = 4, then e2[(kl)fTs]2 1, for k = l. This is afrequent case, assuming high transmission rate in RF bands anda typical value of 2 ppm (2 106) for clock crystals and smallL for repetitive transmission in order to avoid rate degradation.

Using Corollary 1 and under hypothesis H1 : y CN (0,Mx21 +

2IL), as an affine transformation of independentcircularly-symmetric complex Gaussian random vectors andunder H0 : y CN (0, 2IL). The non-coherent ML receiver,assuming equiprobable symbols, is described by the followingexpression:

fy|H1H1 fy|H0 , (23)

which is simplified to the following expression:

yDyH1 2 = 2 ln

[det

(IL +

Mx21

2

)], (24)

where D = IL (IL +(Mx21/2))1.It is noted that for not fully-correlated equivalent channel

taps {gl}Ll=1, the p.d.f. of g is not known. Given {fm}mT ,the random vector g can be written as:

g = Ah, (25)

where h is given from Eq. (17) and the LM matrix A isgiven by:

A =

e+j2f1Ts e+j2fMTs...

.

.

.

.

.

.

e+j2f1LTs e+j2fMLTs

. (26)

ALEXANDRIS et al.: REACHBACK WSN CONNECTIVITY 4929

Consequently, given the CFOs, g is distributed according to theconditional p.d.f. fg|A(g|A)=fg|{fm}mT (g|{fm}mT )CN (0,AA), as a linear combination of a circularly-symmetric complex Gaussian vector h CN (0, IM ). How-ever, the p.d.f. of A is not known, and thus, a closed form forthe unconditioned p.d.f. of g cannot be derived.

Therefore, for partially correlated and uncorrelated equiva-lent channel taps, a heuristic receiver is proposed by replacingthe term of Eq. (24) with C:

yGyH1 3 = 2 ln

[det

(IL +C

Mx212

)], (27)

where C is given by Eq. (22) and G=IL(IL+C(Mx21/

2))1.1) BER Performance Analysis:Theorem 2: Assuming fully-correlated equivalent channel

taps and equiprobable hypotheses, the average BER for the MLnon-coherent detector is given by:

P (e) =1

2[1 Fr (H0 , 2) + Fr (H1 , 2)] , (28)

where under hypothesis Hi, i {0, 1}, Fr(Hi , 2) is the CDFof yDy. Furthermore, analytical form of CDF Fr(Hi , 2)is given in Appendix B. Vector Hi contains the eigenvaluesof a 2L 2L matrix (Hi)1/2E(Hi)1/2, r= rank(E), E=[

D 0LL0LL D

], H0 =

[(1/2)2IL 0LL

0LL (1/2)2IL

]and H1 =[

(1/2)(Mx21 + 2IL) 0LL

0LL (1/2)(Mx21 + 2IL)

].

Proof: Assuming equipropable hypotheses, BER iswritten as:

P (e) =

1i=0

P (e|Hi)P (Hi)

=1

2

[P (yDy 2|H0) + P (yDy < 2|H1)

], (29)

where P (e|Hi) for i = 0, 1 are calculated by the CDF of yDydescribed in Appendix B-Eq. (37).

IV. NON-COHERENT ENERGYHARVESTING (TDMA) RECEIVER

A time-slotted protocol among M distributed terminals isused to schedule transmission to the intended destination. Mdistributed terminals transmit the same symbol using time-division multiplexing for L slots (one phase). Each distributedterminal transmits separately from the others the same symbolfor [L/M ] slots. In that way, the receiver augments the receivedenergy, in order to reliably detect each information symbolat the expense of transmission rate. If M does not divide L(M

L), the remaining slots are allocated to the mth terminal,

that is selected randomly (uniformly) (Fig. 6). Assuming CFOcorrection at the receiver, the signal model is expressed as:

y = hx+w, (30)

Fig. 6. Non-coherent energy harvesting (TDMA) scheme.

where h = [h1 h1 [L/M ]

hM hM [L/M ]

]T , if M |L and h =

[h1 h1 [L/M ]

hM hM [L/M ]

hm hm L mod M

]T , if M L. Finally, ran-

dom variable hm CN (0, 1),m {1, . . . ,M} and randomvector w CN (0, IL).

A. Maximum-Likelihood Non-Coherent Detector

Given the hypotheses, Eq. (30) can be written as:

H0 : y =w,

H1 : y =Bhx1 +w, (31)

where

B=

1 0 0 0.

.

.

1 0 0 00 1 0 0

.

.

.

0 1 0 0.

.

.

0 0 0 1.

.

.

0 0 0 1

extra rows

[ LM ] rows (1st block)[ LM ] rows (2nd block)[ LM ] rows (M th block)}L mod M rows

and h CN (0, IM ) according to Eq. (17).Each block of [L/M ] rows of matrix B corresponds to the

mth user transmission. If M L, then the extra rows of matrixB are selected to be the same with one of the [L/M ] rows of themth user block. Thus, the extra rows correspond to a differentmth user which is selected uniformly.

Under H1 : y CN (0,BBx21 + 2IL) as an affinetransformation of independent circularly-symmetric complexGaussian random vectors and under H0 : y CN (0, 2IL).Similarly to the zero-feedback distributed beamformingscheme, by assuming equiprobable symbols, the non-coherent

4930 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 9, SEPTEMBER 2014

receiver is based on the maximum-likelihood method (seeEq. (23)) and is given by:

yRyH1 = 2 ln

[det

(IL +BB

x21

2

)], (32)

where R = IL (IL +BB(x21/2))1

.

1) BER Performance: Using the methodology in Sec-tion III-B1, the CDF of complex quadratic form yRy isneeded to describe the probability of error under each hypoth-esis. The theorem below provides BER analysis of the TDMAreceiver both for the case of M |L and M L.

Theorem 3: Assuming equiprobable symbols, BER closedform both for the cases of M |L and M L is given by:

P (e)=

12 [1Fr (H0 ,)+Fr (H1 ,)] , if M |L,1

2M

Mj=1

[1Fr

(jH0 ,

)+Fr

(jH1 ,

)], if M

L,

(33)where Fr(, ) is, the CDF of yRy (given at the Appendix B).E =

[R 0LL

0LL R

], H0 =

[(1/2)2IL 0LL

0LL (1/2)2IL

]and H1 =

[(1/2)(BBx21+

2IL) 0LL0LL (1/2)(BBx21+

2IL)

].

Under hypothesis Hi, i {0, 1}, vectors Hi (case for M |L)and jHi (case for M L) contain the eigenvalues of the2L 2L matrix (Hi)1/2E(Hi)1/2, where for the caseof M |L, matrix E is based on R constructed by B withoutincluding any extra rows and for the case of M L, matrixE is based on R constructed by B with extra rows (i.e., theL mod M rows of the jth user block). Finally, r = rank(E).

Proof: Considering the cases of M |L, M L and assumingequiprobable symbols, the analysis follows as:If M |L, BER is computed as:

P (e) =1

i=0

P (e|Hi)P (Hi) = 12

1i=0

P (e|Hi)

=1

2

[P (yRy |H0) + P (yRy < |H1)

]. (34)

If M L, BER is computed as:

P (e) =1

2

1i=0

P (e|Hi) = 12

Mj=1

1i=0

P (e Txj |Hi)

=1

2

Mj=1

1i=0

P (e|Txj , Hi)P (Txj |Hi) P (Txj)

, (35)

where Txj denotes the event of the jth user transmission atthe extra allocated slots. Since the extra slots are allocated

Fig. 7. BER performance for ZF-DBF and TDMA transmission schemes(L = 4).

uniformly, the probability P (Txj) is set to P (Txj) = 1M . Con-sequently, Eq. (35) becomes:

P (e) =1

2M

Mj=1

1i=0

P (e|Txj , Hi)

=1

2M

Mj=1

[P (e|Txj , H0) + P (e|Txj , H1)]

=1

2M

Mj=1

[P (yRy |Txj , H0)

+P (yRy < |Txj , H1)]. (36)

Using the derived closed form CDF of yRy, as described inAppendix B-Eq. (37), under each hypothesis and given the jthuser transmission (implying R construction with extra rows inB, the L mod M rows of the jth user block, if M L or no extrarows if M |L), Eq. (34) and Eq. (36) result in Eq. (33).

Parameter r is the same for both the cases of M |L and M L,since for the case of M L, the addition of extra rows in matrixB leaves the rank of matrix B unchanged and thus the rank ofmatrix R is also the same.

V. NUMERICAL RESULTS

Both simulation and analytical BER results are presentedwith SNR per transmitter antenna per time slot, as defined inEq. (3), fc = 2.4 GHz, Ts = 1 s (i.e., 1 Mbps) and 2 ppm(2 106) clock crystals. For these values, the received sam-ples at the destination receiver are fully-correlated and ex-ploited in the appropriate detector (Figs. 710). Block-lengthparameter L was kept relatively small (on the order of 34),so that rate degradation 1/L was also kept small. Therefore,blind eigenvalue-based detectors are not comparable, since theyrequire large block-length.

ALEXANDRIS et al.: REACHBACK WSN CONNECTIVITY 4931

Fig. 8. BER performance for ZF-DBF and TDMA transmission schemes(L = 3).

Fig. 9. BER performance for ZF-DBF transmission schemes with differentnumber of M distributed terminals.

Fig. 10. BER performance for ZF-DBF transmission schemes in different Ltime intervals.

Fig. 7 shows BER as a function of SNR per transmitterantenna per time slot for the zero-feedback distributed beam-forming (ZF-DBF) and the energy harvesting (TDMA) scheme,M = 2 distributed transmitters and L = 4 symbols. It is shownthat analysis and simulation results agree. The ZF-DBF ML re-ceiver based on fully-correlated equivalent channel taps resultsin better performance than the heuristic receiver, as expected.Furthermore, the ZF-DBF ML receiver for fully-correlatedequivalent channel taps outperforms the TDMA receiver forSNR values up to 5 dB. Better performance at lower SNR ofZF-DBF is due to its beamforming gain, at the expense of totaladditional transmission power (by a factor of M for each slot),compared to TDMA. The latter performs better at higher SNRdue to the diversity offered by the M independent transmitter-receiver channels. For comparison reference purposes, BERperformance for single symbol non-coherent detector (ZF-DBFML detector of Eq. (24) for L = 1) is also depicted.

Fig. 8 demonstrates BER simulation and analytical resultsfor the ZF-DBF and TDMA schemes, M = 2 distributed trans-mitters and smaller L value (L = 3 symbols). For the case ofZF-DBF receivers, the expected number of symbols (out ofL = 4) with M = 2 aligned signals within at most 0 = /4is 1, assuming that out of this sector 0, the signals are not con-sidered aligned. This implies that there is one slot on averagewith beamforming gain in L = 4 time slots. In other words, theminimum repetitive transmission length L should be selectedin order to guarantee signal alignment during at least one slotout of L. For L = 3, the expected number of symbol slots withsignal alignment can be easily obtained using [20, Eq. (12)]and is strictly smaller than 1. Thus, by reducing the number ofslots to L = 3, the achieved rate (1/L) is increased, howeveralignment is not guaranteed and BER performance is degraded,as Fig. 8 depicts. Furthermore, for L = 3 the ZF-DBF receiveroutperforms TDMA for SNR values smaller than 6 dB; TDMAperformance is degraded by 1 dB compared to L = 4, since lessslots reduce the effects of diversity. On the other hand, smallerL improves rate. Thus, for all schemes, there is a trade-offbetween better rate and reliable communication, with ZF-DBFoffering smaller BER (and thus better reachback connectivity)at lower SNR, at the expense of total transmission power.However, in reachback connectivity scenarios, using the batteryof the neighboring terminal for distributed transmission may bethe only valid option.

Fig. 9 provides simulation and analytical BER results for theZF-DBF scheme for L = 4 symbols and different number of Mdistributed terminals. For larger values of M , signal alignmentoccurs with smaller probability, which decreases exponentiallywith M [20]; BER is reduced with increasing number oftransmitters, at the expense of total transmission power; again,trading total (network) transmission power with connectiv-ity (and respective communication reliability) is preferablein reachback connectivity scenarios; in those cases one nodetransmitting alone at maximum power does not suffice; instead,zero-feedback beamforming could be employed, where the un-connected distributed transmitters could contribute their radiosand transmission power.

Fig. 10 depicts BER performance for the ZF-DBF schemefor a different number of symbols L and M = 2 distributed

4932 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 9, SEPTEMBER 2014

Fig. 11. BER performance for ZF-DBF and TDMA transmission schemesincluding different cases of equivalent channel taps correlation.

terminals. It can be easily seen that as L increases, BERperformance is also improved, since more transmissions of thesame information symbol offers reliability, at the expense oftotal power consumption and rate degradation.

Fig. 11 presents BER performance for different cases ofequivalent channel taps correlation. Both partially correlatedequivalent channel taps with Ts = 1 s, 20 ppm (20 106)clock crystals and uncorrelated equivalent channel taps withTs = 0.4 ms, 2 ppm (2 106) clock crystals are considered.The selection of these parameters results in a different co-variance matrix C (see Eq. (22)). Fully-correlated equivalentchannel taps offer a matrix C of ones, uncorrelated equivalentchannel taps create a matrix C equal to the identity matrix andpartially correlated equivalent channel taps provide a matrixC with elements valued between 0 and 1. Fig. 11 depicts theZF-DBF detector (described in Eq. (24) and Eq. (27) respec-tively) for all the equivalent channel taps correlation types.Furthermore, TDMA receiver BER performance provided, isthe same for all correlation cases, since it is independent of{fm}mT due to coarse and fine CFO correction conductedat the receiver. Both for the heuristic and ZF-DBF receiver,partially correlated and uncorrelated equivalent channel tapsoffer better BER performance compared to the fully-correlatedcase, since instantaneous deep fading or signals destructiveaddition does not affect all the received samples. ZF-DBF re-ceiver is optimal only for the case of fully-correlated equivalentchannel taps, thus heuristic receiver performs better for theuncorrelated equivalent channel taps. On the other hand, it isnoted that for partially correlated equivalent channel taps, ZF-DBF still dominates the latter. Finally, the heuristic receiverfor the uncorrelated equivalent channel taps outperforms allthe other schemes, at the low SNR regime, alleviating thereachback communication problem.

VI. CONCLUSION

This work has presented concrete non-coherent receiversfor zero-feedback distributed beamforming and compared them

with non-coherent detection of a TDMA-based scheme. It wasmotivated by resource-constrained WSNs, where one nodetransmitting at maximum power cannot reliably communicatewith the intended far-reaching destination, as in reachbackconnectivity problems. Moreover, it was shown that the pro-posed zero-feedback distributed beamforming receivers over-come connectivity adversities, at the low-SNR regime. Thisis achieved by exploiting signals alignment of M distributedtransmitters (i.e., beamforming), at the expense of network(total) power consumption. No (transmitter or receiver) CSI,no receiver feedback for carrier/phase synchronization and onlycommodity radio hardware were assumed, in sharp contrast toprior art. On the other hand in high SNR cases, where connec-tivity is not an issue and one node is used per time slot, TDMAoutperforms the other schemes and ensures reliability due tomulti-user diversity. Finally, a discussion of USTM schemeswith and without CFO was also offered, pointing towards newresearch directions.

APPENDIX APDF OF THE COMPLEX RANDOM VARIABLE gl

Lemma 1: The random variable gl=M

m=1hme+j2fmlTs ,

l {1, 2, . . . , L}, is distributed according to CN (0,M).Proof: Given {fm}mT , gl CN (0,M) as a linear

combination of circularly-symmetric complex Gaussian ran-dom variables {hm}mT CN (0, 1). Thus, fgl|{fm}mT(gl|{fm}mT ) CN (0,M), which is independent of CFOs{fm}mT . By taking the expectation over {fm}mT , thePDF of gl is given by:

fgl (gl) =Ee[fgl|e(gl|e)

]= fgl|e (gl|e)

+

fe (e) de

= fgl|e (gl|e) ,

where e = [f1 . . . fM ]T .

APPENDIX BCDF OF A COMPLEX QUADRATIC FORM yAy

Lemma 2: Let yAy the complex quadratic form of L1 y, where y CN (0,C), C is real, A is real and A = AT .Then, the CDF of yAy is given by:

Fr (, z) =

+i=0

(1)ici zr2+i

(r2 + i+ 1

) , (37)where (z) =

+0 t

z1etdt denotes the Gamma func-tion, vector = [1 r]T contains the eigenvaluesof 2L 2L matrix 1/2E1/2, =

[12C 0LL

0LL 12C]

, E =[ A 0LL0LL A

]and r = rank(E).

ALEXANDRIS et al.: REACHBACK WSN CONNECTIVITY 4933

The coefficients ci(i 0) can be calculated recursivelythrough the relation:

ci=

r

j=1

(2j) 12 , i = 0,

1i

i1j=0

dijcj , i > 0,(38)

where di(i 1) is expressed as follows:

di=

1

2

rj=1

(2j)i, i 1. (39)

Proof: Let a complex random vector y CN (0,C). Ifmatrix C is real, then the real-valued equivalent random vectory can be expressed as [27]:

y=[{y}T {y}T

]T N (0,), (40)

where the real covariance matrix =[

12C 0LL

0LL 12C]

. Define

yR= {y}, yI = {y} and E =

[ A 0LL0LL A

], then:

yAy = (yTR jyTI )A(yR + jyI)=yTRAyR + jyTRAyI jyTI AyR + yTI AyI ,

yTEy =yTRAyR + yTI AyI .

Thus, iff A = AT , then yAy = yTEy. Consequently,yAy yTEy, and using [28, Theorem 4.2b.1], we concludein Eq. (37).

REFERENCES[1] N. D. Sidiropoulos, T. N. Davidson, and Z. Q. Luo, Transmit beamform-

ing for physical layer multicasting, IEEE Trans. Signal Process., vol. 54,no. 6, pp. 22392251, Jun. 2006.

[2] V. H. Nassab, S. Shahbazpanahi, A. Grami, and Z. Q. Luo, Distributedbeamforming for relay networks based on second-order statistics of thechannel state information, IEEE Trans. Signal Process., vol. 56, no. 9,pp. 43064316, Sep. 2008.

[3] S. Song, J. Thompson, P.-J. Chung, and P. Grant, BER analysis fordistributed beamforming with phase errors, IEEE Trans. Veh. Technol.,vol. 59, no. 8, pp. 41694174, Oct. 2010.

[4] R. Mudumbai, B. Wild, U. Madhow, and K. Ramchandran, Distributedbeamforming using 1 bit feedback: From concept to realization, in Proc.Allerton Conf. Commun., Control Comput., Sep. 2006, pp. 10201027.

[5] M. Johnson, K. Ramchandran, and M. Mitzenmacher, Distributed beam-forming with binary signaling, in Proc. IEEE Int. Symp. Inf. Theory,Jul. 2008, pp. 890894.

[6] R. Mudumbai, D. R. Brown, U. Madhow, and H. V. Poor, Distributedtransmit beamforming: Challenges and recent progress, IEEE Commun.Mag., vol. 47, no. 2, pp. 102110, Feb. 2009.

[7] A. Kalis and A. Kanatas, Cooperative beamforming in smart dust: Get-ting rid of multihop communications, IEEE Pervasive Comput., vol. 9,no. 3, pp. 4753, Jul.Sep. 2010.

[8] H. Balan, R. Rogalin, A. Michaloliakos, K. Psounis, and G. Caire,AirSync: Enabling distributed multiuser MIMO with full spatial mul-tiplexing, IEEE/ACM Trans. on Netw., vol. 21, no. 6, pp. 16811695,Dec. 2013.

[9] R. Mudumbai, G. Barriac, and U. Madhow, On the feasibility ofdistributed beamforming in wireless networks, IEEE Trans. WirelessCommun., vol. 6, no. 5, pp. 17541763, May 2007.

[10] F. Quitin, M. Rahman, R. Mudumbai, and U. Madhow, A scalable ar-chitecture for distributed transmit beamforming with commodity radios:Design and proof of concept, IEEE Trans. Wireless Commun., vol. 12,no. 3, pp. 14181428, Mar. 2013.

[11] J. Uher, T. Wysocki, and B. Wysocki, Review of distributed beamform-ing, J. Telecommun. Inf. Technol., vol. 2011, no. 1, p. 78, Mar. 2011.

[12] Y. Lebrun et al., Performance analysis of distributed ZF beamforming inthe presence of CFO, EURASIP J. Wireless Commun. Netw., vol. 2011,p. 177, Nov. 2011.

[13] S. Chatzinotas, S. K. Sharma, and B. Ottersten, Asymptotic analysis ofeigenvalue-based blind spectrum sensing techniques, in Proc. IEEE Int.ICASSP, May 2013, pp. 44644468.

[14] Y. Zeng and Y.-C. Liang, Maximumminimum eigenvalue detection forcognitive radio, in Proc. IEEE 18th Int. Symp. PIMRC, 2007, pp. 15.

[15] Y. Zeng and Y.-C. Liang, Eigenvalue-based spectrum sensing algorithmsfor cognitive radio, IEEE Trans. Commun., vol. 57, no. 6, pp. 17841793,Jun. 2009.

[16] C. G. Tsinos and K. Berberidis, Adaptive eigenvalue-based spectrumsensing for multi-antenna cognitive radio systems, in Proc. IEEEICASSP, May 2013, pp. 44544458.

[17] T. L. Marzetta and B. M. Hochwald, Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading, IEEE Trans. Inf.Theory, vol. 45, pp. 139157, Jan. 1993.

[18] B. M. Hochwald and T. L. Marzetta, Unitary space-time modulation formultiple-antenna communication in Rayleigh flat-fading, IEEE Trans.Inf. Theory, vol. 46, pp. 543564, Mar. 2000.

[19] B. Hochwald, T. Marzetta, T. Richardson, W. Sweldens, and R. Urbanke,Systematic design of unitary space-time constellations, IEEE Trans. Inf.Theory, vol. 46, no. 6, pp. 19621973, Sep. 2000.

[20] A. Bletsas, A. Lippman, and J. Sahalos, Simple, zero-feedback, dis-tributed beamforming with unsynchronized carriers, IEEE J. Sel. AreasCommun., vol. 28, no. 7, pp. 10461054, Sep. 2010.

[21] A. Bletsas, A. Lippman, and J. Sahalos, Zero-feedback, collaborativebeamforming for emergency radio: Asymptotic analysis, Mobile Netw.Appl. (MONET), vol. 16, no. 5, pp. 589599, Oct. 2011.

[22] G. Sklivanitis, K. Alexandris, and A. Bletsas, Testbed for non-coherentzero-feedback distributed beamforming, in Proc. IEEE ICASSP,May 2013, pp. 25632567.

[23] M. S. Alouini, A. Abdi, and M. Kaveh, Sum of gamma variates andperformance of wireless communication systems over Nakagami-fadingchannels, IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 14711480,Nov. 2001.

[24] J. F. Paris, A Note on the Sum of Correlated Gamma Random Variables,Mar. 2011. [Online]. Available: http://arXiv.org/abs/1103.0505

[25] M. Abramowitz and I. A. Stegun, Handbook of Mathematical FunctionsWith Formulas, Graphs, Mathematical Tables. New York, NY, USA:Dover, 1964.

[26] F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbookof Mathematical Functions, 1st ed. New York, NY, USA: CambridgeUniv. Press, 2010.

[27] R. Gallager, Circularly-Symmetric Gaussian Random Vectors, Jan. 2008.[Online]. Available: http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf

[28] A. Mathai and S. Provost, Quadratic Forms in Random Variables: The-ory and Applications. New York, NY, USA: Marcel Dekker, 1992,ser. Statistics: A Series of Textbooks and Monographs.

Konstantinos Alexandris received the Diploma andM.Sc. degrees (with distinction) in electronic andcomputer engineering from the Technical Universityof Crete, Chania, Greece, in 2012 and 2014, respec-tively. He was with the Telecommunications CircuitsLaboratory (TCL), cole Polytechnique Fdrale deLausanne, Switzerland, as a Research Assistant foreight months. He is currently working toward thePh.D. degree with the Mobile Communications De-partment, EURECOM, Biot, France. His research in-terests are in the areas of wireless cellular networks,

wireless communications, wireless sensor networks, software-defined radioimplementations, and software-defined networking. He has received fellowshipawards for his undergraduate studies. He was the recipient of the 20112012Best Diploma Thesis Award on Advanced Wireless Systems, presented bythe IEEE Vehicular Technology Society and Aerospace and Electronic SystemsSociety joint Greece Chapter.

4934 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 9, SEPTEMBER 2014

George Sklivanitis (S11) received his Diploma inelectronic and computer engineering from the Tech-nical University of Crete, Chania, Greece, in 2010.He is currently working toward the Ph.D. degree inelectrical engineering at the University at Buffalo,The State University of New York, and is a ResearchAssistant with the Signals, Communications andNetworking Research Group. His research interestsare in the area of wireless communications, signalprocessing, and wireless networking with an empha-sis on real-time adaptive signal processing, massive

MIMO communications, cognitive radio, software-defined radio networks, andunderwater acoustic communications. He is a Student Member of the IEEECommunications and Signal Processing Societies. He was the winner of the2014 Nutaq Software-Defined Radio Academic US National Contest.

Aggelos Bletsas (S03M05SM14) received theDiploma degree (with excellence) in electrical andcomputer engineering from Aristotle University ofThessaloniki, Thessaloniki, Greece, in 1998 and theS.M. and Ph.D. degrees from the MassachusettsInstitute of Technology, Cambridge, MA, USA, in2001 and 2005, respectively. He was at MitsubishiElectric Research Laboratories, Cambridge, as aPostdoctoral Fellow and at RadiocommunicationsLaboratory, Department of Physics, Aristotle Univer-sity of Thessaloniki, as a Visiting Scientist. He joined

the School of Electronic and Computer Engineering, Technical University ofCrete, in the summer of 2009, as an Assistant Professor and was promoted toAssociate Professor at the beginning of 2014. His research interests span thebroad area of scalable wireless communication and networking, with emphasison relay techniques, backscatter communications and RFID, energy harvesting,radio hardware/software implementations for wireless transceivers, and lowcost sensor networks. His current vision and focus is on single-transistor front-ends and backscatter sensor networks, for large-scale environmental sensing.He is the Principal Investigator of project BLASE: Backscatter Sensor Net-works for Large-Scale Environmental Sensing, funded by the General Secre-tariat of Research and Technology Action Proposals evaluated positively fromthe 3rd European Research Council (ERC) Call. He is also a ManagementCommittee Member and a National Representative in the European UnionCOST Action IC1301 Wireless Power Transmission for Sustainable Electron-ics (WiPE). He is an Associate Editor of the IEEE WIRELESS COMMUNI-CATIONS LETTERS since its foundation and a Technical Program CommitteeMember of flagship IEEE conferences. He holds two patents from USPTO andhe was recently included in http://www.highlycitedgreekscientists.org/. He wasthe corecipient of the IEEE Communications Society 2008 Marconi Prize PaperAward in Wireless Communications, Best Paper Distinction at ISWCS 2009,Siena, Italy, Second Best Student Paper Award at the IEEE RFID-TA 2011,Sitges, Barcelona, Spain and Best Paper distinction at IEEE Sensors Conf.2013, Baltimore, MD, USA. Two of his undergraduate advisees were winnersof the 20092011 and 20112012 Best Diploma Thesis Contest, respectively,among all Greek Universities on Advanced Wireless Systems, awarded byIEEE VTS/AES joint Greek Chapter. At the end of 2013, he was awarded theTechnical University of Crete 2013 Research Excellence Award.

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