Adaptive Distributed Beamforming for Amplify-and-Forward Relay Networks: Convergence Analysis

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 8, AUGUST 2014 4167

Adaptive Distributed Beamforming forAmplify-and-Forward Relay Networks:

Convergence AnalysisChang-Ching Chen, Student Member, IEEE, Chia-Shiang Tseng, Juwendo Denis, Student Member, IEEE, and

Che Lin, Member, IEEE

Abstract—For wireless amplify-and-forward (AF) relay net-works, this work focuses on the convergence analysis of adaptivedistributed beamforming schemes that can be reformulated aslocal random search algorithms via a random search framework.It is proved that under two sufficient conditions: 1) the objectivefunction of the random search algorithm is continuous and all itslocal maxima are global maxima in the considered feasible set,and 2) the origin is an interior point within the support of theprobability measure for the random perturbation, the correspond-ing adaptive distributed beamforming schemes converge almostsurely. While the second sufficient condition can be controlled bysystem designer and satisfied with relative ease, the first sufficientcondition initially seems strict. Surprisingly, further analysis onthe signal-to-noise ratio (SNR) functions in AF relay networkswith individual and total power constraints demonstrates that, inboth scenarios, local maxima are global maxima and hence, thefirst sufficient condition is satisfied. Finally, the proposed frame-work was extended to analyze adaptive distributed beamformingschemes in an asynchronous setting, and simulation results wereprovided to further validate our analysis.

Index Terms—Beamforming, convergence analysis, distributedalgorithms, feedback communications, relay network.

I. INTRODUCTION

R ELAY networks have become a topic of interest in re-cent years [2]–[8]. Distributed antennas from different

relay nodes can cooperate and form a virtual antenna arrayto achieve cooperative diversity. This is in contrast to theconventional form of spatial diversity through physical antennaarrays. Various relaying strategies have been proposed for awireless relay network [5], [9]–[11], where the amplify-and-forward (AF) strategy has attracted a lot of attention due toits simplicity [5]. When applying the AF relaying strategy,each relay scales the received signal and retransmits the scaled

Manuscript received July 27, 2013; revised November 25, 2013 and March11, 2014; accepted March 11, 2014. Date of publication March 26, 2014;date of current version August 8, 2014. This work was supported by NationalScience Council under Grant NSC99-2221-E-007-089-MY3. Part of this workwas presented at the IEEE ICASSP, Prague, Czech Republic, May 22–27, 2011.The associate editor coordinating the review of this paper and approving it forpublication was D. Niyato.

C.-C. Chen, C.-S. Tseng, and J. Denis are with the Institute of Communi-cations Engineering and Department of Electrical Engineering, National TsingHua University, Hsinchu 30013, Taiwan (e-mail: [email protected];[email protected]; [email protected]).

C. Lin is the with the Institute of Communications Engineering, NationalTsing Hua University, Hsinchu 30013, Taiwan (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/TWC.2014.2314111

version to its destination. To transmit a common message to anygiven destination in an energy efficient manner, a distributedbeamforming scheme has been proposed where proper scalingcan be designed for each relay so as to maximize the receivedsignal-to-noise ratio (SNR) at the destination [12]–[19].

Distributed beamforming coefficients in relays consist oftwo parts: the beamforming phases and the power allocation.Optimal beamforming phase alignment allows the signals for-warded by the relays to be added coherently at the destination,while optimal power allocation can further improve the receivedSNR. To obtain the optimal beamforming coefficients, fullknowledge of the channel state information (CSI) at all relaynodes is required [12], [13]. However, the cost of acquiring suchperfect CSI is too expensive and impractical. Hence, distributedbeamforming with only channel distribution information (CDI)was studied in [14], [20], [21].

Another line of research studied the case where only min-imal CSI knowledge, in the form of a one-bit feedback link,was available to assist the relays when obtaining the optimaldistributed beamforming coefficients [15]. The convergenceanalyses of this one-bit feedback adaptive scheme have beenextensively studied in [1], [15]–[18]. In all of these efforts,the authors assumed that the distributed transmitters iterativelyadjusted their beamforming coefficients based on the one-bitfeedback. Such adaptive algorithms are known as adaptivedistributed beamforming schemes. However, all the proposedanalyses only consider algorithms that seek to achieve phasealignment at the receiver while neglecting power allocation atthe relays.

Power allocation for AF relays is an important topic andhas been previously investigated [22]–[24]. Optimal powerallocation for a one-way AF relay network was derived in [22],[24], while optimal power allocation for a two-way AF relaynetwork was investigated in [23]. However, perfect CSI wasassumed available at all nodes in the network in [22]–[24]. Ina more practical scenario where only limited CSI is available,it is essential to study the joint optimization problem for bothpower allocation and phase alignment of AF relay networks.

In this paper, we generalize the framework in [18] to studysuch joint optimization problems and further show that anyadaptive distributed beamforming scheme that can be reformu-lated as a local random search algorithm converge almost surely(a.s.) if the following two sufficient conditions are satisfied:1) The objective function of the algorithm is continuous and allits local maxima are global maxima in the considered feasible

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4168 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 8, AUGUST 2014

set, and 2) the origin is an interior point in the support ofthe probability measure of the random perturbation. If theobjective function is non-negative, we can further prove thatthe considered algorithm converges in mean.

To demonstrate the applicability of our convergence analysis,we will analyze the SNR functions for AF relay networks withan individual and a total power constraint. We will show thatthe objective SNR functions for the above scenarios satisfy thefirst sufficient condition. Unlike the work in [12]–[14], wherethe analytical expressions for the globally optimal distributedbeamforming coefficients were obtained, we will primarily fo-cus on determining whether the objective SNR functions satisfythe condition where all local maxima are global maxima withinthe feasible set of interest, which is a property concerning thestructure of the problem under consideration. Our simulationswill further validate our analysis.

The remainder of this paper is organized as follows. InSection II, we will introduce the system model and the objectiveSNR function. We will then focus on the description of theproposed local random search framework together with itsconnection to adaptive distributed beamforming in Section III.We also provide the convergence analysis of the adaptivedistributed beamforming scheme, provided that it satisfies thetwo sufficient conditions in Section III. In Section IV, wewill investigate whether or not the SNR function for the AFscheme satisfies both conditions under individual and totalpower constraints. We will extend our analysis and study thecase where the relays can only asynchronously update theirbeamforming coefficients in Section V. In Section VI, we willprovide numerical results that further validate our analyses and,finally, we will conclude the paper in Section VII.

Notations: We use [·]T , 〈·, ·〉, and ‖ · ‖ to indicate the trans-pose operator, the inner product, and the 2-norm, respectively.The notation � is the element-wise inequality, i.e., a � bmeans ai ≤ bi for all i. A vector ai1,...,ik = [ai1 , . . . , aik ]

T isa permutation of the vector a formed by the correspondingelements of a, where ik ∈ {1, . . . , N} for all k. We use diag{a}to denote a diagonal matrix whose i-th diagonal entry is ai.

II. SYSTEM MODEL

We consider a synchronized half-duplex AF relay networkwith one transmitter, N relays, and one receiver where all nodesare equipped with single antenna. As depicted in Fig. 1, wedenote fi = |fi|ej∠fi and gi = |gi|ej∠gi as the slow-faded andfrequency flat channels from the transmitter to the i-th relay andfrom the i-th relay to the receiver, respectively. We also assumethat there is no direct link between the transmitter and thereceiver. Furthermore, a non-coherent communication model isconsidered where both the transmitter and the receiver have noknowledge of the CSI. However, it is assumed that each relayknows its own uplink channel fi. Furthermore, we assume thatthere is an error-free, zero-delay, and low-rate feedback linkfrom the receiver to each relay. This feedback link is used tohelp the relays adjust their beamforming coefficients during thetraining stage such that the received SNR can be improved inthe latter communication stage.

Fig. 1. System model.

For a half-duplex relay network, the communication from thetransmitter to the receiver is completed in two hops. In the firsthop, the transmitter broadcasts a common message to all therelays and the received signal at relay i is given by

ri[n] = fis[n] + vi[n]

where s[n] ∈ C is the common message with the average powerconstraint E[|s[n]|2] ≤ Ps for all n and vi[n] ∼ CN (0, 1) is theadditive white Gaussian noise (AWGN) at relay i. In this work,we focus on the training stage and assume that the signal s[n]is constant and set s[n] =

√Ps.

In the second hop, when AF relaying strategy is used, eachrelay directly applies a beamforming coefficient to the receivedsignal and forward it to the receiver. The signal transmittedfrom the i-th relay is

ti[n] = wi[n]ri[n] =αi[n]

√Pi√

1 + |fi|2Ps

ejψi[n]ri[n]

where wi[n] = (αi[n]√Pi/

√1 + |fi|2Ps)e

jψi[n] ∈ C is thebeamforming coefficient applied at relay i. We denote Pi as thecorresponding power constraint for relay i and 0 ≤ α2

i [n] ≤ 1determines the portion of the power used to forward the signalto the receiver. The term 1/

√1 + |fi|2Ps is used to normalize

the power received from the source and ψi[n] is the phase of thebeamforming coefficient. The received signal at receiver is thusgiven by

y[n] =

N∑i=1

giti[n] + z[n]

=

N∑i=1

αi[n]figiejψi[n]

√Pi√

1 + |fi|2Ps

s[n]

+

N∑i=1

αi[n]giejψi[n]

√Pi√

1 + |fi|2Ps

vi[n] + z[n] (1)

where z[n] ∼ CN (0, 1) is the AWGN at the receiver.The SNR function at the receiver in this AF relay network is

then given by

SNR (α[n],θ[n]) =

Ps

∣∣∣∣∑Ni=1

αi[n]√Pi|fi||gi|ejθi[n]√1+|fi|2Ps

∣∣∣∣21 +

∑Ni=1

α2i[n]|gi|2Pi

1+|fi|2Ps

(2)

where α[n] = [α1[n], . . . , αN [n]]T determines the power usedat each relay, and θ[n] = [θ1[n], . . . , θN [n]]T denotes the vec-tor of total phases where θi[n] = ψi[n] + ∠fi + ∠gi.

CHEN et al.: DISTRIBUTED BEAMFORMING FOR AMPLIFY-AND-FORWARD RELAY NETWORKS 4169

During the training stage, the goal is to maximize theSNR function (2) through distributed beamforming so thatthe receiver can recover the signal with minimum error inthe latter communication stage. However, since the receiverhas no knowledge of the CSI and hence the correspondingobjective SNR function, only samples of the SNR function canbe obtained at each iteration. The analysis of the consideredoptimization problem can be much simplified if we reformulateit as a more general problem stated as follows [18]:

Problem 1: Given an unknown objective function f : X →R where X ⊆ R

N , and only samples of f(.) are available, findthe global maxima of f .

Remark 1: To build the analogy between Problem 1 and theSNR maximization problem stated in (2), we should set X ≡P ×Θ, where P ⊆ R

N and Θ ⊆ RN . Moreover, the available

samples are given by f(α,θ) for any α ∈ P and θ ∈ Θ, whereboth power allocation and phase alignment are considered.

This problem cannot be solved by gradient search methodssince there is no knowledge of the objective function andits gradient at the relays in our setting. However, it can besolved by global random search algorithms in general [25]–[27]. Furthermore, if all local maxima of the objective functionare global maxima within the feasible set of interest, localrandom search algorithms, which are more efficient, can alsobe applied.

For a random search based beamforming algorithm, at then-th iteration, the phase and power coefficients of distributedtransmitter from last iteration x[n− 1] are first perturbed witha random vector δ[n], and used for transmission. The receivermeasures the SNR function corresponding to the perturbedbeamforming coefficient, i.e., x′[n] = x[n− 1] + δ[n], and usethe feedback bit to convey the information indicating whetherthe perturbation on the beamforming coefficients is beneficialto the SNR function or not; i.e.,

d[n] = 1{f(x[n−1]+δ[n])>f(x[n−1])}

where 1{·} is the indicator function defined as

1{condition} ={1 if the condition holds true0 if the condition is not true.

The distributed transmitters then update their beamformingcoefficients based on the feedback information; i.e.,

x[n] = x[n− 1] + δ[n] · 1{f(x[n−1]+δ[n])>f(x[n−1])}.

We term the entire adaptive updating process as “adaptivedistributed beamforming.”

In our previous work [1], [18], we built the connectionbetween the one-bit adaptive distributed beamforming scheme[15] and a local random search algorithm for phase alignmentonly and proved the convergence in probability. Here, wefurther generalize our previous analyses and show that anyadaptive distributed beamforming scheme that can be reformu-lated as a local random search algorithm converges a.s. if twosufficient conditions are satisfied. We also extend our analysisto the second hop of AF relay networks with only a limitedfeedback link from the receiver to all relays. Furthermore,we investigate whether or not the objective SNR functions

satisfy both sufficient conditions when joint power allocationand phase alignment is considered for AF relay networks.

Remark 2: Note that Problem 1 can also be solved byartificial-intelligence-type optimization algorithms, e.g., parti-cle swarm and simulated annealing. In fact, both particle swarmand simulated annealing can be considered as a random searchalgorithm.

Particle swarm optimization algorithm is essentially a lo-cal random search algorithm with multiple initial points toreduce the probability of the algorithm being trapped intolocal extrema. If all local maxima of the objective functionare global maxima for the considered optimization problem,our convergence analysis (to be introduced later) can also beapplied to show that any particle swarm optimization algorithmis guaranteed to converge a.s.

It is well-known that simulated annealing is a global randomsearch algorithm where the temperature determines a non-zeroprobability for the solution to jump to any point in the searchspace. Therefore, our analytical framework can also be appliedto show that any adaptive distributed beamforming scheme thatcan be reformulated into simulated annealing converges a.s.

III. CONVERGENCE OF ADAPTIVE DISTRIBUTED

BEAMFORMING SCHEMES

In this section, we provide a general proof of almost sure con-vergence for a set of adaptive distributed beamforming schemesthat can be reformulated as local random search algorithms.We consider a general function f : X → R where X ⊆ R

N iscompact. The local random search algorithm in [18] is nowbriefly described as follows:

Algorithm 1 Local random search algorithm in [18]

Initialize the algorithm by choosing x[0] ∈ χrepeat

δi[n] ∼ μn, i = 1, . . . , Niff(x[n− 1] + δ[n]) > f(x[n− 1]) then

x[n] = x[n− 1] + δ[n]else

x[n] = x[n− 1]end if

until Stopping criteria reached.

Note that δ[n] = [δ1[n], . . . , δN [n]]T , where {δi[n]} are lo-cal random perturbations generated independently from theprobability measure μn that could be time-varying and has thesupport supp(μn).

For our proof of almost sure convergence, we first state thefollowing two sufficient conditions:

(S1) The objective function f is continuous and all its localmaxima are global maxima in the considered feasible set.1

1Note that a function f along with a feasible set satisfying sufficientcondition (S1) does not imply that f is concave in the feasible set. A simpleexample is f(x) = sinx, x ∈ R.


(S2) The origin is an interior point of the support supp(μn) forall n.

Note that we do not need to specify the exact expressions ofthe objective function and the probability measure in our proofsonce both the sufficient conditions are satisfied. This makes theunderlying framework and associated proofs more general thanour previous result [18].

We now proceed to show that any adaptive distributed beam-forming scheme that can be reformulated as a local randomsearch algorithm converges a.s. if both the sufficient conditionsare satisfied.

Proposition 1: Given an objective function f(·) and a prob-ability measure μn that satisfy sufficient conditions (S1) and(S2), then for any x ∈ X \Rε, there correspond γ > 0 and0 < η ≤ 1 such that

Pr [f (D(x, δ))− f(x) ≥ γ] ≥ η

where X is the feasible set, Rε := {x ∈ X : f(x) > f(x∗)−ε} is the ε-convergence region, the mapping D : X ×supp(μn) → X is defined as D(x[n− 1], δ[n]) =x[n− 1] +δ[n] · 1{f(x[n−1]+δ[n])>f(x[n−1])}, and δ[n] is a random vectorgenerated with the probability measure μn.

Proof: We begin the proof by defining a f(x)-superlevelset Ux := {ψ ∈ X : f(ψ) ≥ f(x)}, which is the subset of Xwith all the elements in Ux mapping to function values no lessthan f(x). Note that the assumption x �∈ Rε implies that Rε ⊆Ux and since Rε is nonempty, Ux is nonempty. Furthermore,since f is continuous, the boundary points are those points thatsatisfy f(ψ) = f(x).

We claim that for any ξ > 0, there exists a set A(x) ⊂B(x, ξ) ∩ Ux with nonempty interior points, such that f(a) >f(x) for all a ∈ A(x), where B(x, ξ) is an open ball centeredat x with radius ξ. To this end, we first rule out the case wherethere are flat regions outside Rε, that is, for some x �∈ Rε, thereexists a ζ > 0 such that f(ψ) = f(x) for all ψ ∈ B(x, ζ). Thiscase cannot happen since if it does, then f(ψ) ≤ f(x) for allψ ∈ B(x, ζ) and this implies that x is a local maxima point andthus a global maxima point by the sufficient condition (S1), andthis contradicts to the assumption that x �∈ Rε.

Excluding the existence of flat regions outside Rε, we nowconsider two cases where x is either a local minimum pointor not. Note that the arguments leading to the absence of flatregions outside Rε implies that x cannot be a local maximapoint.

Case I: If x is a local minimum point, we can show that it isan interior point of Ux since the fact that x is a local minimumpoint but not a local maxima point implies that there exists aξ > 0 such that f(ψ) > f(x) for all ψ ∈ B(x, ξ) \ {x}. Thismeans that B(x, ξ) ⊆ Ux and thus x is an interior point of Ux.Note that in this case, the entire B(x, ξ) are in the interior ofUx and we choose A(x) = B(x, ξ) \ {x}, which obviously hasnonempty interior points.

Case II: If x is not a local minimum point, it can be shownthat it is a boundary point of Ux but not an isolated point.First, we show that x is not an isolated point. If x is anisolated point, it means that there exists a ζ > 0 such that forall ψ ∈ B(x, ζ) \ {x}, ψ �∈ Ux, or equivalently, f(ψ) < f(x).

This means that x is a local maximum point and thus a globalmaximum point by the sufficient condition (S1). Again, weobtain a contradiction to the assumption that x �∈ Rε. Now,since x is neither a local maximum point nor a local minimumpoint, then for any ξ > 0, we can always find a point ψ1 ∈B(x, ξ) such that f(ψ1) > f(x); i.e., ψ1 ∈ Ux and a pointψ2 ∈ B(x, ξ) such that f(ψ2) < f(x); i.e., ψ2 �∈ Ux. Thus,for any ξ > 0, B(x, ξ) contains both points inside and outsideUx. This means that x is a boundary point. Note that ψ1 is aninterior point in Ux since if it is not, then it is a boundary pointof Ux and it means f(ψ1) = f(x), which contradicts to thestatement that f(ψ1) > f(x). In this case, we choose A(x) =B(x, ξ) ∩ Ux \ C, where C := {ψ ∈ Ux : f(ψ) = f(x)} andthere are nonempty interior points in A(x). Therefore, ourclaim is indeed true.

Now we define Ω := {ω ∈ X : ω = x+ δ, δ ∈ supp(μn)},as the set by shifting supp(μn) with respect to x. Then therange of D(x, ·) for any x �∈ Rε is given by Range{D(x, ·)} =Ω ∩ Ux. Note that the sufficient condition (S2) implies that x isan interior point of Ω. Since x is either a boundary or interiorpoint of Ux, there always exist ξ > 0 such that B(x, ξ) ∩ Ux ⊆Range{D(x, ·)}. Therefore, we can conclude that A(x) ⊂Range{D(x, ·)}. This further implies that for any x ∈ X \Rε,there exists an interior point x ∈ A(x) ⊂ Range{D(x, ·)} andξ > 0 such that ∀ψ ∈ T := B(x, ξ), f(ψ)− f(x) ≥ γ(x).Then, Pr[f(D(x, δ))− f(x) ≥ γ(x)] ≥ μn(T ) =: η(x) > 0.Note that T is a function of x since x depends on x. Wecomplete the proof by letting γ = infx∈X\Rε

γ(x) and η =infx∈X\Rε

η(x) �Remark 3: In general, Proposition 1 is not true unless the

perturbation is global. In the case where a local perturbationis used, it is possible that the algorithm gets trapped in localmaxima and therefore resulting in a zero probability of im-proving the objective function. In our case, however, sufficientcondition (S1) guarantees that all local maxima are global.Consequently, the dilemna of trapping within a local maximais avoided as convergence to any local maxima is equivalent toglobal convergence.

Since there is always a non-zero probability to improve ffor each time step before the sequence reaches Rε as statedin Proposition 1, convergence of the sequence is expected. Wedescribe this more precisely in the following theorem.

Theorem 1: Let f be an objective function along with afeasible set that satisfies sufficient condition (S1) and δ[n] bea random perturbation with probability measure μn such thatits support supp(μn) satisfies sufficient condition (S2). Denote{x[n]}∞n=0 as a sequence generated by the local random searchalgorithm as described by Algorithm 1. The resulting sequenceconverges almost surely; i.e.,

Pr[limn→∞

x[n] ∈ Rε

]= 1.

In other words, f(x[n]) converges to f(x∗) almost surely,where x∗ is a global maximum point.

Proof: From Proposition 1, it states that at any time n, wehave

Pr [f (D (x[n− 1], δ[n]))− f (x[n− 1]) ≥ γ] ≥ η (3)


Since X is compact and f(·) is continuous, we can always finda positive integer p such that

f(x1)− f(x2) < pγ, ∀x1,x2 ∈ X . (4)

From (3), (4), and by setting x1 to be any point in Rε andx2 to be any starting point x[0] ∈ X \Rε, the probability thatthe sequence lies in Rε after p time steps is lower boundedas Pr[x[p] ∈ Rε] ≥ ηp since {δ[n]}∞n=0 are independent acrosstime. This implies that

Pr [x[p] �∈ Rε] ≤ 1− ηp. (5)

From (5) and the fact that {δ[n]}∞n=0 are independent acrosstime, we can derive that for each p time steps Pr[x[kp] �∈Rε | x[(k − 1)p] �∈ Rε] ≤ 1− ηp, ∀k = 1, 2, · · ·. Therefore,from any starting point x[0] ∈ X \Rε, after mp time steps forany m = 1, 2, . . ., the probability that x[mp] does not lie in Rε

can be upper bounded as follows:

Pr [x[mp] �∈ Rε]

= Pr [x[mp] �∈ Rε,x [(m− 1)p] �∈ Rε, . . . ,x[p] �∈ Rε]

= Pr [x[p] �∈ Rε]

m∏k=2

Pr [x[kp] �∈ Rε|x [(k − 1)p] �∈ Rε]

≤ (1− ηp)m. (6)

This upper bound (6) is still valid if we let the sequenceprogress any � = 0, . . . , p− 1 time steps further; i.e., Pr[x[±+�] �∈ Rε] ≤ (1− ηp)m, which leads to

∑∞m=0 Pr[x[±+ �] �∈

Rε] ≤ (1/(1− (1− ηp))) = (1/ηp) < ∞. By Borel-Cantellilemma [28], we have Pr[x[pm+ �] �∈ Rε i.o.] = 0 where i.o.represents “infinitely often.” This means that x[n] converges tox∗ a.s., where x∗ can be any point in Rε �

Once both sufficient conditions are satisfied, almost sureconvergence is assured. Also by making an additional assump-tion about the objective function, we can prove convergence inmean. This is stated in the following corollary.

Corollary 1: Consider the setting as described in Theorem 1.If the objective f is further non-negative, the resulting sequenceconverges in mean; i.e.,

limn→∞

E [|f (x[n])− f(x∗)|] = 0.

Proof: From proof of Theorem 1, the sequence{f(x[n])}∞n=0 is monotonically non-decreasing. If f is furthernon-negative, it can be shown that this sequence also convergesin mean by the Monotone Convergence Theorem [28]. �

Until now, we have shown that once both the sufficientconditions are satisfied, the convergence of the local randomsearch algorithm can be readily proved. However, there maybe concerns regarding the applicability of this convergenceanalysis since sufficient condition (S1) may seem strict at first.Surprisingly, in many practical scenarios such as AF relay net-works, the objective SNR functions along with some commonlyused power constraints indeed satisfy sufficient condition (S1).This is the main topic of the following sections.

IV. ANALYSIS OF OBJECTIVE SNR FUNCTIONS IN

AF RELAY NETWORKS WITH DIFFERENT

POWER CONSTRAINTS

In this section, we investigate whether the objective SNRfunction (2) satisfies sufficient condition (S1) under two differ-ent power constraints; i.e., an individual power constraint and atotal power constraint, respectively.

To simplify the analysis, note that for any fixed α[n], maxi-mizing the SNR function (2) with respect to θ[n] is equivalentto maximize the numerator since the denominator does notdepends on θ[n]. This implies that even when both α[n] andθ[n] are considered as variables, local maxima of the objectiveSNR function (2) can only occur when all phases are perfectlyaligned. Furthermore, based on our result in [18], it can be eas-ily shown that the SNR function (2) indeed satisfied sufficientcondition (S1) when only phase alignment is considered. Thus,in the sequel, we can focus on power allocation only to obtaina necessary condition for a local maximal point, and focus onoptimizing the following objective SNR function

SNR (α[n]) =

Ps

(∑Ni=1

αi[n]|fi||gi|√Pi√

1+|fi|2Ps

)2

1 +∑N

i=1αi[n]2|gi|2Pi

1+|fi|2Ps

. (7)

A. AF Relay Networks With Individual Power Constraints

We consider the cases when there is an individual powerconstraint for each relay. The optimization problem can bedescribed as

max

Ps

(∑Ni=1

αi|fi||gi|√Pi√

1+|fi|2Ps

)2

1 +∑N

i=1α2

i|gi|2Pi

1+|fi|2Ps

(8a)

subject to 0 � α � 1. (8b)

It is worth mentioning that if the first hop of the relaysystem is ignored, the problem will be reduced to the simpleconventional transmit beamforming problem, i.e., the sameoptimization problem with only the numerator of (8a). Inthis case, we know the optimal power allocation requires thateach relay uses full power. However, taking into considera-tion the first hop makes the problem more complex. Here wedemonstrate, by examining further the structure of the objectivefunction, that sufficient condition (S1) indeed holds true in thisscenario.

Theorem 2: The SNR maximization problem for AF relaynetworks with individual power constraints (8) satisfies (S1)since there exists only one local maximum for (8) and thus itis the global maximum.

Proof: For the sake of completeness, we restate the prob-lem formulation in [12, Section III-A] as follows. The SNRfunction (8a) can be rewritten as

SNR = Ps〈b,α〉2

1 + ‖Aα‖2 = Ps〈c,y〉21 + ‖y‖2


where

α = [α1, . . . , αN ]T

b =

[|f1||g1|

√P1√

1 + |f1|2Ps

, . . . ,|fN ||gN |

√PN√

1 + |fN |2Ps

]T

y =Aα

c =A−1b = [|f1|, . . . , |fN |]TA =diag {a}

a =

[|g1|

√P1√

1 + |f1|2Ps

, . . . ,|gN |

√PN√

1 + |fN |2Ps

]T

.

This optimization problem (8) can be decomposed into aninner and an outer optimization problem as

maxr

1

1 + r2

(max‖y‖=r

〈c,y〉)2

(9a)

subject to 0 � y � a and 0 ≤ r ≤ ‖a‖. (9b)

Now, we define ρj := cj/aj = (|fj |√

1+|fj |2Ps/|gj |√

Pj)>0 and without loss of generality, we assume ρ1 ≥ ρ2 ≥ · · · ≥ρN ≥ ρN+1 := 0. Then, we can partition the feasible inter-val of r ∈ [0, ‖a‖] in the optimization problem (9) into Nsubintervals

[0, ‖a‖] = [r0, r1] ∪ [r1, r2] ∪ . . . ∪ [rN−2, rN−1] ∪ [rN−1, rN ]

where rj for j = 0, . . . , N are defined as follows:

r0 =0

r1 = ρ−11 ‖c‖ =

√ρ−21 ‖c2,...,N‖2 + a21

r2 =√

ρ−22 ‖c2,...,N‖2 + a21 =

√√√√ρ−22 ‖c3,...,N‖2 +

2∑m=1

a2m

...

rN =

√√√√ρ−2N |cN |2 +

N−1∑m=1

a2m = ‖a‖.

Therefore, the optimization problem (8) is equivalent2 to

maxi=0,...,N−1

maxr∈Γi

1

1 + r2

(max

‖y‖=r, 0�y�a〈c,y〉

)2

(10)

where Γi := [ri, ri+1] for i = 0, . . . , N − 1. We now proceedto show sufficient condition (S1) indeed holds for problem (10)and the original problem (8).

To prove the stated theorem, we first focus on the inneroptimization problem

max‖y‖=r, 0�y�a

〈c,y〉. (11)

2In this paper, we use conventional definition of equivalent problems; i.e.,problem (P) is said to be equivalent to problem (Q) if a global optimal solutionto (P) can be readily found by a global optimal solution to (Q), and vice versa.Note that this definition does not involve properties of locally optimal solutions.

To carry on with the proof, we need to state two importantClaims on each sub-interval.

Claim 1: For any fixed r ≥ ri, local maxima occur onlywhen yj = aj for j = 1, . . . , i.

Proof: We prove this claim by induction. For i = 1, weshow that for any fixed r ≥ r1, the local maxima occur onlywhen y1 = a1, or, there are no local maxima when y1 �= a1.

First, we change the coordinates from Euclidean coordinatesto hyperspherical coordinates as

y1 = r cos θ1y2 = r sin θ1 cos θ2

...yN−1 = r sin θ1 sin θ2 . . . sin θN−2 cos θN−1

yN = r sin θ1 sin θ2 . . . sin θN−2 sin θN−1.

Since 0 � y � a, we have 0 ≤ θi ≤ π/2 for all i. Furthermore,since r ≥ r1, we have cos−1(a1/r) ≤ θ1 ≤ π/2.

Now, we show that for any feasible θ1, θ2, . . . , θN−1, if θ1 �=cos−1(a1/r), we can always improve 〈c,y〉 by decreasing θ1.Since 〈c,y〉 = ‖c‖r cosψ where ‖c‖ and r are fixed values andψ is the angle between c and y, we only need to show that cosψincreases when θ1 decreases. We can derive that

cosψ =〈c,y〉‖c‖r =

√c21 + d21‖c‖ cos(θ1 − φ1)

where d1 := c2 cos θ2 + · · ·+ cN∏N−1

i=2 sin θi > 0 and φ1 ∈[0, π/2] since cosφ1 = (c1/

√c21 + d21) > 0 and sinφ1 =

(d1/√

c21 + d21) > 0.Since 0 ≤ θ1, φ1 ≤ π/2, we have −(π/2) ≤ θ1 − φ1 ≤

π/2. If we can further show that θ1 − φ1 ≥ 0 for any fixed φ1

and all feasible θ1, we can conclude that cos(θ1 − φ1) increaseswhen θ1 decreases. This is equivalent to show that θ1,min −φ1 ≥ 0 or cos θ1,min ≤ cosφ1 where θ1,min = cos−1(a1/r).We have

cosφ1 − cos θ1,min

=c1√

c21 + d21− a1

r=

c1

r√

c21 + d21

(r − a1

√1 +

d21c21

)

≥ c1

r√

c21 + d21

(r1 − a1

√1 +

d21c21

)(∵ r ≥ r1)

=c1

r√

c21 + d21

(√ρ−21 ‖c2,...,N‖2 + a21 − a1

√1 +

d21c21

)

=a1c1

r√

c21 + d21

(√1 +

‖c2,...,N‖2c21

−√

1 +d21c21

). (12)

Let d1 := [cos θ2, sin θ2 cos θ3, . . . ,(∏N−2

i=2 sin θi) cos θN−1,

(∏N−1

i=2 sin θi)]T , where ‖d1‖2 = 1. Then, by Cauchy-Schwarz

inequality, we can derive that

d21 = 〈c2,...,N , d1〉2 ≤ ‖c2,...,N‖2‖d1‖2 = ‖c2,...,N‖2. (13)

From (12) and (13), we have cosφ1 − cos θ1,min ≥ 0. Thus, wecan conclude that 〈c,y〉 increases when θ1 decreases for r ≥ r1.


Now, assume that for i = k the argument holds, that is, forany fixed r ≥ rk, the local maximum occurs only when yj =aj for j = 1, . . . , k. We show that for i = k + 1, the argumentstill holds. Since yj = aj for j = 1, . . . , k, then for any fixedr ≥ rk+1, we consider only yj for j = k + 1, . . . , N . Define

r :=√

r2 −∑k

m=1 a2m and change the coordinates as

yk+1 = r cos θk+1

...

yN−1 = r sin θk+1 . . . sin θN−2 cos θN−1

yN = r sin θk+1 . . . sin θN−2 sin θN−1.

Following the same line of thoughts, we can show that forany feasible θk+1, . . . , θN−1, if θk+1 �= cos−1(ak+1/r), we canalways improve 〈ck+1,...,N ,yk+1,...,N 〉 by decreasing θk+1.This concludes the proof. �

Claim 2: For any fixed r ∈ Γi = [ri, ri+1], local maxima

occur only when yj = (√

r2 −∑i

m=1 a2m/‖ci+1,...,N‖)cj for

j = i+ 1, . . . , N . That is, yi+1,...,N is parallel to ci+1,...,N .Proof: From Claim 1, we know that for any r ∈

Γi = [ri, ri+1], y1,...,i = a1,...,i. We use Lagrange multipliertheorem [29] to further prove that for the rest N − i elementsyi+1,...,N , local maxima occur only when yi+1,...,N is parallelto ci+1,...,N . The optimization problem can be written as

maxyi

cTi yi (14a)

subject to yTi yi = r2i (14b)

yi � ai (14c)

yi � 0 (14d)

where ci := ci+1,...,N , yi := yi+1,...,N , ai := ai+1,...,N , and

ri :=√r2 −

∑im=1 a

2m with r ∈ Γi.

The Lagrangian associated with the optimization problem(14) is given by

L(λ0,λ,ν)=−cTi yi+λ0

(yTi yi−r2

)+λT (yi−ai)−νT yi

where λ0, λ = [λi+1, . . . , λN ]T and ν = [νi+1, . . . , νN ]T arethe Lagrange multipliers.

By Lagrange multiplier theorem, all local extrema satisfy theKKT conditions

�L(λ0,λ,ν) =λ0yi + (−ci + λ− ν) = 0 (15a)

λ �0 (15b)

ν �0 (15c)

λj(yj − aj) = 0, ∀j = I + 1, . . . , N (15d)

νjyj =0, ∀j = i+ 1, . . . , N. (15e)

First, we assume that λ0 �= 0. From (15a), we have

yj =1

λ0(cj − λj + νj), ∀j = i+ 1, . . . , N. (16)

If νj > 0, from (15e), we have yj = 0. Since yj − aj = −aj <0, from (15d), we further obtain λj = 0 and thus from (16), wecan derive that νj = −cj < 0, which contradicts to the assump-tion that νj > 0. Therefore, νj = 0 for all j = i+ 1, . . . , N and(16) becomes

yj =1

λ0(cj − λj), ∀j = i+ 1, . . . , N. (17)

If λj = 0, then from (17), we have

yj =1

λ0cj . (18)

Otherwise, if λj > 0, from (15d), we have yj = aj and thusfrom (17), we have

λj = cj − λ0aj (19)

where λ0 < (cj/aj) = ρj since λj > 0 by assumption. Now,we further prove that it is impossible that λj > 0 for any j,leading to λj = 0 for all j = i+ 1, . . . , N . Assume that forany arbitrary 1 ≤ K ≤ N − i, we have λ�k > 0 with �k ∈{i+ 1, . . . , N} for all k = 1, . . . ,K and λj = 0 for j �= �k.Without loss of generality, we assume that �1 < �2 < · · · < �K .Then from (15d), (18), and (19), we have y�k = a�k for k =1, . . . ,K, yj = (1/λ0)cj for j �= �k and λ0 < min

k∈{1,...,K}ρ�k =

ρ�K . With the above assumption and from the equality con-straint (14b), we have

λ0 =

√√√√ ‖ci‖2 −∑K

k=1 c2�k

r2 −∑i

m=1 a2m −

∑Kk=1 a

2�k

≥

√√√√ ‖ci‖2 −∑K

k=1 c2�k

r2i+1 −∑i

m=1 a2m −

∑Kk=1 a

2�k

(∵ ri ≤ r ≤ ri+1)

=

√√√√ ‖ci‖2 −∑K

k=1 c2�k

ρ−2i+1‖ci+2,...,N‖2 + a2i+1 −

∑Kk=1 a

2�k

= ρi+1

√√√√ ‖ci‖2 −∑K

k=1 c2�k

‖ci‖2 −∑K

k=1 a2�kρ2i+1

(∵ ρi+1 =

ci+1

ai+1

)

≥ ρi+1

√√√√ ‖ci‖2 −∑K

k=1 c2�k

‖ci‖2 −∑K

k=1 a2�kρ2�k

(∵ ρi+1 ≥ ρ�k , ∀�k)

= ρi+1.

Then, λ0 −maxk∈{1,...,K} ρ�k = λ0 − ρ�1 ≥ ρi+1 − ρ�1 ≥0. Since λ0 − ρ�1 ≥ 0 contradicts to λ0 < ρ�K , we con-clude that λj = 0 and yj = (1/λ0)cj for all j = i+ 1, . . . , N .Thus, from (14b) and (18), we can derive that yi+1,...,N =

(√r2 −

∑im=1 a

2m)/‖ci+1,...,N‖)ci+1,...,N . For the case λ0 =

0, it can be easily shown that the only solution is yi = ai,which can only happen when r = ‖a‖ ∈ ΓN−1, which is indeeda local minimum. �

Remark 4: Note that our Claims 1 and 2 are extensions of[12, Lemmas 1 and 2]. Although the analytical expression of


the global optimal solution was given in [12], here we examinethe validity of sufficient condition (S1), which involves thestructure of the considered problem. It is important to note thefundamental difference between the two results. Furthermore,the proof techniques used in [12] cannot be directly appliedhere. Instead, we developed new proof techniques to proveClaims 1 and 2.

From Claims 1 and 2, we can conclude that for any fixedr ∈ Γi, potential local maxima occur only when

yj =

{aj , 1 ≤ j ≤ iλcj , i+ 1 ≤ j ≤ N

where λ :=

√r2 −

∑im=1 a

2m

‖ci+1,...,N‖(20)

and it can be shown that the following relation holds:

r ∈ Γi ⇔ λ ∈ Ωi :=[ρ−1i , ρ−1

i+1

](21)

by the definition of λ.With the aid from (20) and (21), we can conclude that the

optimization problem (10) is equivalent to the one-dimensionalline search problem

maxi=0,...,N−1

maxλ∈Ωi

(∑im=1 bm + ‖ci+1,...,N‖2λ

)2

1 +∑i

m=1 a2m + ‖ci+1,...,N‖2λ2

. (22)

Further, any local optimal solution to (10) is also a correspond-ing local optimal solution to (22).

To examine (S1) for problem (22), we focus on the region Ωi

first, and start with the definitions

λi =1 +

∑im=1 a

2m∑i

m=1 bm

and

ξi(λ) =

(∑im=1 bm + ‖ci+1,...,N‖2λ

)2

1 +∑i

m=1 a2m + ‖ci+1,...,N‖2λ2

From [12, Lemma 3], it has been shown that (∂ξi/∂λ) > 0if λ < λi, and (∂ξi/∂λ) < 0 if λ > λi; i.e., the value of ξiincreases when λ increases given that λ < λi, and decreaseswhen λ increases given that λ > λi. Therefore, if ρ−1

i ≤ λi ≤ρ−1i+1, the maximum value in Ωi occurs at λ = λi. Otherwise,

if λi < ρ−1i or λi > ρ−1

i+1, the maximum value in Ωi occurs atλ = ρ−1

i or λ = ρ−1i+1, respectively.

Let us define i0 to be the smallest i such that λi < ρ−1i+1, we

now proceed to show that locations of potential local maximaare confined to the subinterval Ωi0 and that there exist only onelocal maximum, which is then the global maximum.

• Note that ρ−1N+1 = ∞ implies that i0 must exist. Since for

Ωi with i < i0, λi ≥ ρ−1i+1 by assumption, the maximum

value in Ωi occurs at λ = ρ−1i+1. This has two implications:

the function value is increasing with λ in Ωi for all i < i0and the optimal solution λ = ρ−1

i+1 for Ωi is also a feasiblesolution in Ωi+1. Therefore, with these two facts, it issufficient to consider regions Ωi with i ≥ i0.

• Furthermore, it is clear that if λi < ρ−1i+1, then λi+1 <

ρ−1i+1. Since λi0 < ρ−1

i0+1, we have λi < ρ−1i for all i > i0

and thus the maximum value in Ωi for i > i0 occurs atλ = ρ−1

i . Similarly, this implies that the function value isdecreasing with λ in Ωi for all i > i0, and the optimalsolution λ = ρ−1

i for Ωi is also a feasible solution forΩi−1. Therefore, it is sufficient to consider regions Ωi withi ≤ i0.

Combining the above two arguments, we conclude that thereexists only one local maximum which occurs in Ωi0 and thusit is also the global maximum. Hence, (S1) is satisfied for bothproblem (10) and (22). Since y = Aα is a one-to-one mapping,all local maxima in the y-domain are still local maxima in theα-domain. Hence, sufficient condition (S1) is satisfied for theoptimization problem (8). �

From Theorems 1 and 2, we can now conclude that for AFrelay networks with individual power constraints, any adaptivebeamforming scheme that can be reformulated to a local ran-dom algorithm as described by Algorithm 1 converges almostsurely to the globally optimal phase alignment and powerallocation once (S2) is further satisfied.

B. AF Relay Networks With a Total Power Constraint

In this subsection, we explore AF relay networks with atotal power constraint Pmax. The total power constraint canbe interpreted as setting Pi =

√αiPmax for all i, and further

impose constraints on the coefficients αi such that∑N

i=1 α2i ≤

1. As in the case for individual power constraint, we focus onthe structure of the objective function and drop the time indexn. Hence, the SNR maximization problem with total powerconstraint can be written as

max

Ps

(∑Ni=1

αi|fi||gi|√Pmax√

1+|fi|2Ps

)2

1 +∑N

i=1α2

i|gi|2Pmax

1+|fi|2Ps

(23a)

subject to ‖α‖2 ≤ 1. (23b)

With this reformulation, we derive the following theorem.Theorem 3: The SNR maximization problem for AF relay

networks with total power constraint satisfies (S1). Further,there is only one local maximum point, and thus it is the globalmaximum point.

Proof: We can further rewrite the SNR function (23) andconsider the following optimization problem

maxα

αTRα

1 +αTQα(24a)

subject to ‖α‖2 ≤ 1 (24b)

where

α = [α1, . . . , αN ]T

r =√

PmaxPs ·[

|f1||g1|√1 + |f1|2Ps

, . . . ,|fN ||gN |√1 + |fN |2Ps

]T

R = rrT

Q =Pmax · diag{

|g1|21 + |f1|2Ps

, . . . ,|gN |2

1 + |fN |2Ps

}.


Examining the KKT conditions of (24), it is clear that‖α‖2 = 1 for the optimal solutions by Lagrange multiplier the-orem [29] and the linear independence constraint qualification.As a result, we obtain

αTRα

1 +αTQα=

αTRα

αT (I+Q)α(∵ ‖α‖ = 1) (25)

which is a generalized Rayleigh quotient. The generalizedRayleigh quotient in (25) can be reduced to a Rayleigh quo-tient. Furthermore, the corresponding Rayleigh quotient is acontinuous function and the unit sphere is a compact set.Hence, the maximum of (25) exists and is unique. We can thenconclude that all local maxima are indeed global maxima forproblem (24). �

In this section, we showed examples of objective SNR func-tions that satisfy sufficient condition (S1). Therefore, once suffi-cient condition (S2) is satisfied, we can apply the correspondinglocal random search algorithms to obtain the optimal beam-forming coefficients for all these cases and their convergenceis guaranteed.

Note that our analysis can be generalized to the case wherethere are multiple antennas at the relays. In fact, the onlydifference between the case where there are multiple antennasin a relay and the case where there are multiple relays, eachwith single antenna is that a total power constraint should beapplied for the first case while an individual power constraintshould be applied for the second case. Since we showed thatsufficient condition (S1) holds true for both an individual anda total power constraint, it is straightforward to show that (S1)holds true for a mixture of the two power constraints.

V. ASYNCHRONOUS SCHEME

In this section, an adaptive distributed beamforming schemethat allows asynchronous updates with individual power con-straints among distributed transmitters is presented, where theconvergence analysis is performed by utilizing our proposedrandom search framework. We are able to show almost sureconvergence and convergence in mean even when synchroniza-tion among relays is not assumed.

For a general model on asynchronous systems that allowseach transmitter to update its phase at any random time in-stance, the reader is referred to [30], [31]. Here, we adopta simplified model. In particular, the asynchronous system ismodeled as an adaptive distributed beamforming scheme whereat any given time, there is a nonzero probability that each trans-mitter updates its phase. More precisely, at each time instance,each relay perturbs its beamforming coefficient independentlywith probability p = ρ% given that the random perturbationused at each relay i is δi[n] with the probability measure μi,n

and the support supp(μi,n) satisfies sufficient condition (S2).To show the convergence of this asynchronous scheme, we

define a new random perturbation δ′i[n] for each relay i asfollows:

δ′i[n] =

{0, with probability 1− pδi[n], with probability p

with the corresponding probability measure denoted by μ′i,n. It

can be observed that supp(μ′i,n) is the same with supp(μi,n).

As a result, the new probability measure μ′i,n also satisfies suf-

ficient condition (S2). This implies that the overall probabilitymeasure μ′

n of the random perturbation δ′[n] for all relays inthis asynchronous scheme satisfies sufficient condition (S2).Since both sufficient conditions (S1) and (S2) are satisfied, ourproof of almost sure convergence and convergence in mean canbe directly applied here. We want to emphasize again that byreformulating adaptive distributed beamforming schemes in ourrandom search framework, convergence behaviors can be read-ily shown via inspection of only the two sufficient conditions.This again demonstrates the generality and flexibility of ourframework.

VI. SIMULATION RESULTS

In this section, numerical results are provided to vali-date our analyses. We consider AF relay networks with anindividual and a total power constraint. In the simulation,we set N = 10 and assume that the channel coefficients fiand gi are i.i.d. CN (0, 1) for all i. We set Ps = 1 and thepower constraint for each relay to be P1 = P2 = · · · = P10 =5. We say that the algorithm converges once SNR(α,θ) ≥κSNR(α∗,θ∗), where κ = 0.9 for simplicity. For randomperturbation δθ[n] for the phase alignment, we use a time-invariant probability measure U [−Δθ,Δθ]

N , where U [·, ·] de-notes the uniform distribution and Δθ = 5◦. Meanwhile, wedefine γi[n] := αi[n]

√Pi/(1 + |fi|2Ps) and choose a ran-

dom perturbation δγ,i[n] for the power allocation with thetime-varying probability measure U [−Δ�

γ,i[n],Δuγ,i[n]], where

Δuγ,i[n] = min{0.1, (

√Pi,max − γi[n− 1])} with Pi,max :=

Pi/(1 + |fi|2Ps) and Δ�γ,i[n] = min{0.1, γi[n− 1]} for all i.

Furthermore, in the simulation, we first fix the transmit power ateach relay and focus on the phase alignment using local randomsearch algorithms. Afterwards, we fix the phases at all relaysand focus on achieving optimal power allocation.

Fig. 2(a) and (b) demonstrate the convergence behaviors ofthe AF relay networks with an individual and a total powerconstraint, respectively. Both figures indicate that the localrandom search algorithm converges from different initial pointsfor a fixed randomly generated channel realization. Note thatextensive simulations with different randomly generated chan-nel realizations have also been done and similar convergencebehaviors were obtained. Due to space limitation, we onlypresent representative figures and omit all others.

In literature, researchers have been interested in how thenumber of iterations required for the random search algorithmsto converge scales with the dimension of the problem (numberof relays in our case) [32]–[34]. In Fig. 3, we demonstrates thatthe number of iterations required for the algorithm to reach adesired level of the objective value, i.e., κ1 = 0.9, κ2 = 0.7 andκ3 = 0.5, scales linearly with the number of relays. This resultindicates that the adaptive distributed beamforming schemeswe analyzed achieve linear complexity in dimension, whichis the lowest complexity scaling achievable among all existingwork.


Fig. 2. Convergence behavior of the AF relay network (a) with an individualpower constraint, (b) with total a power constraint.

For the simulation of the asynchronous scheme describedin Section V, we use the same settings as those used forthe synchronized AF relay networks. Since the convergencebehavior of the asynchronous scheme for any ρ% is similar tothose in Figs. 2 and 3, we omit them here. Instead, we showedthe difference between the asynchronous scheme and the syn-chronous scheme, with different ρ%, in terms of the averageconvergence time in Fig. 4. It is clear that the performance ofthe asynchronous scheme is worse than that of the synchronousscheme. However, we emphasize that even when only as low as10% of the relays update their beamforming coefficients at anygiven time instance, the algorithm still converges as indicatedby the convergence analysis carried out in Section V and ournumerical simulations.

Fig. 5 demonstrates the comparison between the convergenceperformance of adaptive distributed beamforming schemes us-ing Gaussian distributed random perturbation and uniform dis-tributed random perturbation. The number of the relay nodes isset to 10 and the channel is assumed to be Rayleigh faded. FromFig. 5, the Gaussian perturbation case converges faster than

Fig. 3. Convergence behaviors of AF relay networks with individual powerconstraints.

Fig. 4. The average iterations necessary for the convergence of the asyn-chronous scheme at different ρ%.

the uniform perturbation case in the beginning but convergesnear the optimum at relatively the same speed as the uniformperturbation case. More importantly, Fig. 5 demonstrates thatadaptive distributed beamforming schemes indeed converge forboth two cases; this simulation result validates our theoreticalanalysis since both perturbations satisfy (S2).

Furthermore, we also provide the performance of adaptivedistributed beamforming schemes under time-varying channelsin Fig. 6. For the time variations, we assume that the receiveris moving at the speed of 36 kilometers per hour. The carrierfrequency and the sampling frequency is set to 900 MHz and600 KHz, respectively. To overcome such time variations, wemodified the adaptive distributed beamforming scheme suchthat for every 1200 time slots, we randomly search the optimalbeamforming phase for the first 600 time slots and the optimalpower for the remaining 600 time slots. The simulation result inFig. 6 shows that the SNR function increases significantly andconverges during the first complete search for phase and power.Although time variations of the channels make the achieved


Fig. 5. Comparison of convergence behaviors of an adaptive distributedbeamforming scheme using uniform and Gaussian perturbation.

Fig. 6. Convergence behavior of a modified adaptive distributed beamformingscheme under time-varying channels.

SNR decrease temporarily, the modified adaptive distributedbeamforming scheme can still track the optimal phase andpower successfully as indicated in Fig. 6.

VII. CONCLUSION

In this paper, we provided a general proof of almost sureconvergence and convergence in mean for adaptive distributedbeamforming schemes that can be reformulated as local randomsearch algorithms and satisfy the two sufficient conditions. Weconsidered joint power allocation and phase alignment for AFrelay networks and analyzed the corresponding objective SNRfunctions with an individual and a total power constraint. Wedemonstrated that the first sufficient condition is satisfied forall these cases. We further extended our analyses to an asyn-chronous scheme and demonstrated all desirable convergencebehaviors through our framework. Finally, through extensivesimulation experiments, our theoretical analyses were furthervalidated.

In our future work, we are interested in extending the analysisand design of adaptive distributed beamforing schemes to two-way relay channels. Furthermore, it is also of interest to studythe convergence of these schemes under time-varying channelsor even network topologies.

REFERENCES

[1] C. C. Chen, C. S. Tseng, and C. Lin, “A general proof of convergencefor adaptive distributed beamforming schemes,” in Proc. IEEE ICASSP,2011, pp. 3292–3295.

[2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity.Part I: System description,” IEEE Trans. Commun., vol. 51, no. 11,pp. 1927–1938, Nov. 2003.

[3] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity.Part II: Implementation aspects and performance analysis,” IEEE Trans.Commun., vol. 51, no. 11, pp. 1939–1948, Nov. 2003.

[4] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded pro-tocols for exploiting cooperative diversity in wireless networks,” IEEETrans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003.

[5] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity inwireless networks: Efficient protocols and outage behavior,” IEEE Trans.Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.

[6] M. Abouelseoud and A. Nosratinia, “Opportunistic wireless relay net-works: Diversity-multiplexing tradeoff,” IEEE Trans. Inf. Theory, vol. 57,no. 10, pp. 6514–6538, Oct. 2011.

[7] S. Shahbazpanahi and M. Dong, “A semi-closed-form solution to opti-mal distributed beamforming for two-way relay networks,” IEEE Trans.Signal Process., vol. 60, no. 3, pp. 1511–1516, Mar. 2012.

[8] K. Truong, P. Sartori, and R. Heath, “Cooperative algorithms for MIMOamplify-and-forward relay networks,” IEEE Trans. Signal Process.,vol. 61, no. 5, pp. 1272–1287, Mar. 2013.

[9] A. Nosratinia and A. Hedayat, “Cooperative communication in wirelessnetworks,” IEEE Commun. Mag., vol. 42, no. 10, pp. 74–80, Oct. 2004.

[10] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and capacitytheorems for relay networks,” IEEE Trans. Inf. Theory, vol. 51, no. 9,pp. 3037–3063, Sep. 2005.

[11] M. Janani, A. Hedayat, T. E. Hunter, and A. Nosratinia, “Coded coopera-tion in wireless communications: Space-time transmission and iterativedecoding,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 362–371,Feb. 2004.

[12] Y. Jing and H. Jafarkhani, “Network beamforming using relays withperfect channel information,” IEEE Trans. Inf. Theory, vol. 55, no. 6,pp. 2499–2517, Jun. 2009.

[13] I. Hammerström, M. Kuhn, and A. Wittneben, “Impact of relay gainallocation on the performance of cooperative diversity networks,” in Proc.IEEE VTC Fall, 2004, pp. 1815–1819.

[14] V. Havary-Nassab, S. Shahbazpanahi, A. Grami, and Z. Q. Luo, “Dis-tributed beamforming for relay networks based on second-order statisticsof the channel state information,” IEEE Trans. Signal Process., vol. 56,no. 9, pp. 4306–4316, Sep. 2008.

[15] 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., 2006, vol. 44, pp. 1020–1027.

[16] R. Mudumbai, J. P. Hespanha, U. Madhow, and G. Barriac, “Distributedtransmit beamforming using feedback control,” IEEE Trans. Inf. Theory,vol. 56, no. 1, pp. 411–426, Jan. 2010.

[17] J. Bucklew and W. Sethares, “Convergence of a class of decentralizedbeamforming algorithms,” IEEE Trans. Signal Process., vol. 56, no. 6,pp. 2280–2288, Jun. 2006.

[18] C. Lin, V. V. Veeravalli, and S. P. Meyn, “A random search framework forconvergence analysis of distributed beamforming with feedback,” IEEETrans. Inf. Theory, vol. 56, no. 12, pp. 6133–6141, Dec. 2010.

[19] J. Denis, C.-S. Tseng, C.-W. Lee, C.-Y. Tsai, and C. Lin, “Deterministicbisection search algorithm for distributed sensor/relay networks,” in Proc.IEEE GLOBECOM, 2012, pp. 4851–4855.

[20] J. Li, A. Petropulu, and H. Poor, “Cooperative transmission for re-lay networks based on second-order statistics of channel state infor-mation,” IEEE Trans. Signal Process., vol. 59, no. 3, pp. 1280–1291,Mar. 2011.

[21] Y. Cheng and M. Pesavento, “Joint optimization of source power allo-cation and distributed relay beamforming in multiuser peer-to-peer relaynetworks,” IEEE Trans. Signal Process., vol. 60, no. 6, pp. 2962–2973,Jun. 2012.

[22] C. Jeong, D.-H. Kim, and H.-M. Kim, “Optimal power allocation forhybrid distributed beamforming in wireless relay networks,” in Proc.IEEE VTC Fall, 2009, pp. 1–5.

[23] V. Havary-Nassab, S. Shahbaz Panahi, and A. Grami, “Optimal distributedbeamforming for two-way relay networks,” IEEE Trans. Signal Process.,vol. 58, no. 3, pp. 1238–1250, Mar. 2010.

[24] T. Q. Quek, M. Z. Win, and M. Chiani, “Robust power allocation algo-rithms for wireless relay networks,” IEEE Trans. Commun., vol. 58, no. 7,pp. 1931–1938, Jul. 2010.


[25] S. Kirkpatrick, C. D. Gelatt, Jr., and M. Vecchi, “Optimization by simu-lated annealing,” Science, vol. 220, no. 4598, pp. 671–680, May 1983.

[26] Z. B. Tang, “Adaptive partitioned random search to global optimization,”IEEE Trans. Autom. Control, vol. 39, no. 11, pp. 2235–2244, Nov. 1994.

[27] L. Shi and S. Olafsson, “Nested partitions method for global optimiza-tion,” J. Math. Oper. Res., vol. 48, no. 3, pp. 390–407, May 2000.

[28] R. A. Durrett, Probability: Theory and Examples, 2nd ed. Pacific Grove,CA, USA: Duxbury Press, 1995.

[29] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.:Cambridge Univ. Press, 2004.

[30] V. S. Borkar, “Asynchronous stocahstic approximations,” SIAM J. ControlOptim., vol. 36, no. 3, pp. 840–851, May 1999.

[31] V. S. Borkar and S. P. Meyn, “The O.D.E. method for convergence ofstochastic approximation and reinforcement learning,” SIAM J. ControlOptim., vol. 38, no. 2, pp. 447–469, Jan. 2000.

[32] G. Schrack and N. Borowski, Numerical Methods for Nonlinear Optimiza-tion. New York, NY, USA: Academic, 1972.

[33] M. Schumer and K. Steiglitz, “Adaptive step size random search,” IEEETrans. Autom. Control, vol. AC-13, no. 3, pp. 270–276, Jun. 1968.

[34] F. J. Solis and R. J. B. Wets, “Minimization by random search technique,”J. Math. Oper. Res., vol. 6, no. 1, pp. 19–30, Feb. 1981.

Chang-Ching Chen (S’09) received the B.Eng. degree in electrical engi-neering, in 2009 and the M.S. degree from the Institute of CommunicationEngineering, in 2011, all from National Tsing Hua University, Hsin Chu,Taiwan.

His research interests are in wireless communications and distributed algo-rithms in networks.

Chia-Shiang Tseng received the B.Eng. degree in electrical engineering, in2010 and the M.S. degree in Institute of Communication Engineering, in 2012,all from National Tsing Hua University, Hsin Chu, Taiwan.

His research interests are in random search theories and distributed algo-rithms in wireless networks.

Juwendo Denis (S’12) received the B.Eng. degree inelectrical engineering, in 2011 and the M.S. degreeat the Institute of Communication Engineering, in2013, all from National Tsing Hua University, HsinChu, Taiwan.

His research interests are distributed algorithms inwireless sensor networks.

Che Lin (S’02–M’08) received the B.S. degree inelectrical engineering from National Taiwan Univer-sity, Taipei, Taiwan, in 1999, the M.S. degree in elec-trical and computer engineering, in 2003, the M.S.degree in math, in 2008, and the Ph.D. degree in elec-trical and computer engineering, in 2008, all from theUniversity of Illinois at Urbana-Champaign.

Since 2008, he has been with National Tsing HuaUniversity, Hsinchu, Taiwan, where he is currentlyan Assistant Professor. He holds a U.S. patent, whichhas been included in the 3GPP LTE standard. His

research interests include feedback systems, distributed algorithms in networks,optimization theory, and information theory.

Adaptive Distributed Beamforming for Amplify-and-Forward Relay Networks: Convergence Analysis

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