748 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 4, APRIL 2006
Noncooperative Iterative MMSE BeamformingAlgorithms for Ad Hoc Networks
Ronald A. Iltis, Senior Member, IEEE, Seung-Jun Kim, and Duong A. Hoang
Abstract—An asynchronous unicast ad hoc network is consid-ered, where each node is equipped with a receive/transmit beam-former pair (w g ) designed under a quality-of-service (QoS)SNR constraint. It is first shown that the minimum sum-powerbeamformers for the network satisfy a weak duality condition, inwhich the pairs ((gopt) (wopt) ) achieve the same sum poweras the primal network. However, the optimum receive beam-former wopt is not in general equal to (gopt) , in contrast to thecase of cellular and time-division duplexing networks. Iterativeminimum mean-square error (IMMSE) beamforming algorithmsare then proposed in which w = g is enforced. These algo-rithms are shown to be instances of the Power Algorithm inwhich g is the maximizing eigenvector of an SNR-related objec-tive matrix. The IMMSE algorithm can also be viewed as a non-cooperative beamforming game, in which the payoff includes nor-malized SNR, and the tax is related to interference caused at othernodes. The existence of fixed points (Nash equilibria) is provedfor IMMSE. Furthermore, fixed points of IMMSE are shown tosatisfy the first-order necessary conditions for optimization usinga network Lagrangian. The IMMSE game is modified to yield asequential distortionless-response beamforming algorithm, whichis shown to be convergent using a Total Interference Function.Extensive simulation results illustrate that IMMSE yields betterpower efficiency than a greedy noncooperative SNR-maximizinggame.
Index Terms—Array signal processing, game theory, least-mean-square (LMS) methods, networks.
I. INTRODUCTION
THE problem of beamforming to minimize sum powerunder quality-of-service (QoS) constraints in an ad hoc
wireless network is considered. The emphasis is on mul-tiple-input/multiple-output (MIMO) enhancement of existingwireless networks (e.g., 802.11b/g/a), in which a single symbolstream is spread on multiple antennas. Thus, the effective linearprecoder is restricted to a vector beamformer to maintain com-patibility with physical (PHY) layers originally designed for
Paper approved by A. Anastasopoulos, the Editor for Wireless Communi-cations for the IEEE Communications Society. Manuscript received May 18,2004; revised April 6, 2005, and October 14, 2005. This work was supported inpart by the National Science Foundation under Grant CCR-0073214 and GrantCCF-0429596 and by a grant from the International Foundation for Teleme-tering. This paper was presented in part at the Adaptive Sensor Array ProcessingWorkshop, Boston, MA, March 2004.
R. A. Iltis and D. A. Hoang are with the Department of Electrical and Com-puter Engineering, University of California, Santa Barbara, CA 93106-9560USA (e-mail: [email protected]; [email protected]).
S.-J. Kim was with the Department of Electrical and Computer Engineering,University of California, Santa Barbara, CA 93106-9560 USA. He is now withNEC Laboratories, Princeton, NJ 08540 USA.
Digital Object Identifier 10.1109/TCOMM.2006.873095
single antenna radios. The motivation for enforcing a QoS con-straint in the network while minimizing sum power, rather thanmaximizing decoupled [1] or centralized Shannon capacity [2],stems from cross-layer design issues. Specifically, a commonlyaccepted model for ad hoc network simulations is that packetsare captured only when the SNR exceeds a threshold [3]. Hence,a network that maximizes overall physical layer capacity mayhave less throughput than one designed under QoS constraints,due to critical links that fall below the packet capture threshold.Using power minimization as the optimization criterion is alsoclearly desirable in ad hoc networks with battery-powerednodes. Our optimization criterion is fundamentally equivalentto [4], which minimizes network sum power subject to decou-pled capacity QoS constraints. However, our formulation ofthe problem leads to a more compact Lagrangian and givesadditional motivation for the iterative minimum mean-squareerror (IMMSE) algorithm, without requiring a time-divisionduplexing (TDD) constraint. In particular, we provide poweralgorithm, Lagrangian fixed point, and game-theoretic inter-pretations of IMMSE that motivate its choice over greedySNR-maximization algorithms.
The MIMO ad hoc network considered here assumes uni-casting, such that node forward packets only to ,where . It is assumed that all nodeshave arrays with the same number of elements , althoughthe algorithms developed here are not restricted to this case.Each node maintains a unit-norm receive/transmit beamformerpair with . Reciprocal channel matrices
are assumed to represent the responseat array due to the transmit array . An unnormalized beam-former is also defined where is the transmittedpower. The beamformers are designed to meet the QoS con-straint under which all links attempt to achieve a commonSNR .
The capacity-attaining linear precoders (eigencoders) andbeamformers are well known for point-to-point MIMO links[5], [6] and are readily derived through the Lagrangian incorpo-rating power constraints. Cellular topologies were consideredin [7]–[10], with array-equipped bases and single-antennamobiles. In these cases, the optimum downlink beamformeris the conjugate of the uplink receive beamformer (duality).However, in cellular systems with array-equipped mobiles,the optimum solution for the transmit beamformers has notbeen obtained in closed form, although consideration of theLagrangian motivates iterative MMSE algorithms for beam-former updating [11], [12]. IMMSE-type algorithms basedon the power algorithm have also been developed in [13] forpoint-to-point MIMO systems.
ILTIS et al.: NONCOOPERATIVE ITERATIVE MMSE BEAMFORMING ALGORITHMS FOR AD HOC NETWORKS 749
The problem of distributed beamforming in ad hoc MIMOnetworks has received less attention than point-to-point and cel-lular systems. Iterative beamforming in ad hoc networks wasimplicitly implemented via the LMS algorithm in [14] and [15],where increased throughput using modified 802.11 MAC proto-cols was demonstrated. It should be emphasized that IMMSE isa desirable strategy for ad hoc networks, as it eliminates the needfor explicit channel estimation, and exploits the availability oftraining sequences in MAC protocols. For example, RTS, CTS,and ACK packets used in MACA/MACAW [16] and modified802.11 protocols provide an extensive set of training sequencesfor adaptive beamforming [14]. IMMSE beamforming was pro-posed as part of the LEGO algorithm in [4], [17] for ad hoc net-works under a TDD constraint. It was shown in [4] that a strongnetwork duality holds for TDD networks, in which the optimumtransmit beamformers are the conjugates of the optimum re-ceive vectors, that is . However, convergence of LEGOto the global optimum is not guaranteed. Centralized greedySNR maximization in an ad hoc MIMO network was proposedin [2] and shown in practice to yield maximum Shannon ca-pacity. However, [2] requires transmission of all array outputsto a central processor, unlike the noncooperative algorithms pro-posed here. Distributed beamforming has also been studied asa noncooperative game [1], [18]. Game theory has proven tobe successful in analyzing convergence and efficiency of powercontrol algorithms [19]–[21] and code-division multiple-access(CDMA) waveform design [22], [23], although its application tobeamforming is problematic, as will be discussed in the sequel.
The key results in this paper are summarized as follows.1) Weak duality (Theorem 1), in which beamformer pairs
and both achieve the min-imum sum power holds for the ad hoc network. Strong net-work duality proven in the TDD networkof [4] does not hold due to the absence of a TDD or cellularstructure.
2) The network Lagrangian suggests an MMSE structure forthe optimum transmit beamformer solutions (Theorem 2).Analysis of the first-order necessary conditions (FONCs)shows that a greedy SNR maximization algorithm cannotachieve the optimum solution.
3) IMMSE beamforming algorithms, which enforcesatisfy the FONCs (Theorem 3) for power minimization, asshown by the Lagrangian stationary points.
4) A game-theory interpretation suggests that IMMSE mayoutperform the greedy algorithm as IMMSE includes aninterference tax.
5) An IMMSE beamforming game using Jacobi-type iter-ations is shown to have at least one Nash equilibrium(Proposition 2).
6) IMMSE always outperforms the greedy algorithm in termsof power efficiency in extensive simulation results.
7) Convergence to fixed points cannot be guaranteed forIMMSE. A modification to IMMSE, sequential distor-tionless-response beamforming (SDRB), is shown to beconvergent (although not necessarily to the optimum) byanalysis of a total interference function (TIF) (Proposition3). TIF is shown to be a beamforming analog to totalsquared correlation for CDMA networks in [24] and [25].
The weak network duality and FONC results are given in Sec-tion II. The IMMSE algorithms and game-theory interpretationsfollow in Section III. SDRB algorithms and convergence resultsare given in Section IV. Results and conclusions are given inSections V and VI.
II. WEAK NETWORK DUALITY AND OPTIMAL
BEAMFORMER STRUCTURE
The ad hoc network is defined by nodes ,where each unicasts to a single , using symbol stream
, with . Flat-fading low-mobilitychannels are assumed, so that the channel matrices arequasi-static. Hence, the received vector corre-sponding to symbol at node is given by
(1)
Note that the array output in (1) corresponds to a network whereall nodes can transmit simultaneously, in contrast to the TDDcase of [4] and [17]. The network is synchronous in the model(1), so that intersymbol interference (ISI) from nodesis not present. When half-duplex radios are used, the model (1)still applies with powers reduced by the duty cycle. How-ever, even in the half-duplex case, the network is not TDD,since transmissions for node are not restricted to odd or evenTDD frames. It is assumed that all receivers have the same noisefigure, so that without loss of generality (w.l.o.g.), the whiteGaussian noise terms have identical covariancematrices . The resulting received SNR at node due to de-sired user is
(2)
where . The QoS constraint is forall . Define the power vector . The goalis to maintain QoS while minimizing , where isthe all-ones vector.
The following theorem describes duality for this network anduses the approach of [17] and [26] along with the earlier sum-power minimization analysis of [27] for single-antenna nodes.
Theorem 1: Weak network duality: consider a network withoptimal beamformer pairs . If a feasible solutionfor the powers exists satisfying QoS, then is defined forthis optimal network by
Define a dual network by at all nodes. Thisnetwork has the same minimum sum poweras in the primal network. A set of optimal beamformer pairs
exists for feasible that minimizes , in
750 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 4, APRIL 2006
which is always a minimum variance distortionless re-sponse (MVDR) beamformer. If the set of optimal transmitbeamformers achieving is unique, then strongduality holds with . However, ingeneral for the ad hoc network.
Proof: See Appendix I. It should be emphasized that strongnetwork duality, with , always holds for TDDnetworks [4]. The lack of strong duality in ad hoc networksstems from the inability to separate the nodes into separate up-link and downlink sets, i.e., as a bipartite graph.
The general structure of the optimum beamformers isobtained via a network Lagrangian , where
. In order to simplify the Lagrangian, de-fine the unnormalized MVDR receive beamformer at nodeby
(3)
where (game theoretic notation) de-notes all beamformers for . The interferer plus noisecovariance at node is
(4)
The optimization problem is posed as
(5)
The corresponding Lagrangian is obtained by substituting theMVDR beamformer (3) into the SNR (2) and imposing the con-straints, yielding
(6)The next theorem is proven in Appendix II.Theorem 2: FONCs for optimality (5) [28], [29] correspond
to stationary points of the Lagrangian (6) which are given by thefollowing eigenvectors:
(7)
The FONCs can be satisfied by minimizing in (7). For thisnonunique solution for the FONCs, is the maximum eigen-vector given by
(8)
Equation (8) follows by multiplying both sides of (7) by .The pseudocovariance matrix is given by
(9)
where . The powers satisfying the FONCs alsosatisfy the constraints with equality and are given by
(10)
Equation (10) follows using the definition and thedefinition of SNR in the constraints in (6).
The interpretation of Theorem 2 is that one way to sat-isfy the FONCs (although the solution is not necessarilythe optimum) is to maximize link SNR [numerator on theright-hand side (r.h.s.) of (8)], while minimizing a measureof interference to other nodes, i.e., is a sum of terms
, which characterizes interferencefrom nodes to weighted by virtual powers. These virtualpowers are given by . This observation suggeststhat a well-constructed beamforming algorithm/game shouldpay an interference tax while trying to maximize the normal-ized SNR. Note that a similar MMSE/MVDR structure for theoptimal beamformers was obtained for a cellular topology viaa Lagrangian in [12].
The following corollary identifies the key weakness of agreedy beamforming algorithm/game.
Corollary 1: Consider a greedy noncooperative SNR maxi-mizing beamforming game, which is defined at node by
(11)
i.e., a greedy game attempts to minimize transmit power ateach node without considering interference incurred by otherlinks. The fixed points of the game (11) cannot satisfy theFONCs and hence cannot yield the optimal solution .
Proof: The FONCs are satisfied only when is of theform (7). The fixed point of the greedy solution in (11) is themaximum eigenvector of , which only corre-sponds to (7) when . However, can only equal when
for all achieves a feasible solution, which isa contradiction for finite SNR constraint . Since the greedysolution does not satisfy the FONCs, it cannot be optimal.
III. IMMSE ALGORITHMS
Two versions of the IMMSE beamforming algorithm aregiven in Table I, defined by either Jacobi or Gauss–Seidel-typeiterations. These algorithms are similar to LEGO [4], [17],except that TDD is not required. Consider iteration of Ja-cobi IMMSE, where the current transmit beamformers are
. Each node updates its unit-norm MMSE/MVDR receivebeamformer at subiteration , with all other
ILTIS et al.: NONCOOPERATIVE ITERATIVE MMSE BEAMFORMING ALGORITHMS FOR AD HOC NETWORKS 751
TABLE IIMMSE DISTRIBUTED BEAMFORMING ALGORITHMS
transmit beamformers held fixed. Note that iscomputed using training sequences from node transmittedusing beamformer . Node then sets its trial transmitbeamformer to for subsequenttransmission to . Node computes its receive beam-former using training sequences from and sets
. This procedure is repeated untilconvergence at iteration . In the case of the Jacobi-IMMSE,all nodes replace by only after all IMMSEupdates have been completed. For Gauss–Seidel-IMMSE, node
immediately replaces its transmit beamformer by .The power is set to meet the SNR target at the end of it-
eration , i.e., subiteration (Gauss–Seidel), using an SNR es-timate at node . For exact SNR estimates,is given by (2) with replaced by its normalized MMSE so-lution. Then, the SNR is rewritten as
(12)
For Jacobi updates, is set to meet only after all nodeshave completed their IMMSE updates.
The power algorithm interpretation of the above algorithm isgiven in the following Proposition.
Proposition 1: Consider the th overall update in the Jacobi-IMMSE algorithm. The transmit vector converges to themaximum eigenvector of the following equation:
(13)
where is the interference-plus-noise covariance matrixat node in (4). The maximum eigenvector is also the maximizerof the ratio
(14)
which follows directly by premultiplication of (13) by.
Proof: The MVDR beamformer (e.g., see [17])is given by
(15)
752 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 4, APRIL 2006
Recall that was previously set to . Thus,is equivalently
(16)
where is chosen for unit-norm . Combining (15) and(16), invoking channel reciprocity, , and setting
yields
(17)
which is a power algorithm iteration with setting. Since is always invertible from its definition (4),
maximization of the ratio (14) is equivalent to finding the max-imizing eigenvector of the matrix in (13).
The relationship of the IMMSE algorithm to the FONCs isgiven in the following theorem.
Theorem 3: Consider the fixed points of the IMMSE algo-rithm defined by
(18)
where is the maximum eigenvalue of (13). These fixed pointsare stationary points of the Lagrangian (6) and thus satisfy theFONCs for power minimization subject to QoS constraints.
Proof: See Appendix II.Although IMMSE satisfies the FONCs for optimization, it
cannot be proven that IMMSE yields the optimal minimumpower solution. Hence, even though greedy SNR maximization(11) cannot satisfy the FONC from Theorem 2, there is noguarantee that IMMSE gives a lower power solution thanthe greedy algorithm does. However, IMMSE always outper-formed the greedy algorithm in the simulations in Section V.A game-theoretic interpretation of the IMMSE algorithm isnext presented, which suggests reasons for the superiority ofIMMSE to greedy SNR maximization.
Game-theoretic interpretations of noncooperative powercontrol algorithms are well established [19]–[21], and themethodology here follows these latter references. IMMSE isthus equivalently represented as a noncooperative game withutility function as follows:
(19)
where is any continuous, concave function with a globalmaximum at zero. The game (19) results since maximizationof the r.h.s. of (14) is equivalent to maximizing the logarithm.The function is maximized when the power is selected to
satisfy the QoS SNR constraint with equality. The normalizedSNR is given using (12) by
(20)
for unit norm . The second term in is the log-normalizedSNR, and hence the payoff increases with normalized SNR or,equivalently, with lower power . However, the last termcorresponds to an interference tax, which is defined by
(21)
Since , the tax can be interpreted as the interfer-ence incurred by users from transmitter , weightedby the power of each user .
Greedy SNR maximization (11) corresponds to thesame utility function in (19), but with the tax replaced by
. Hence, the greedy beamforming gamedoes not include a tax component representing interferenceincurred by other users. It is well known that an appropriatepricing or tax strategy [19] can lead to better social behavior(e.g., lower overall sum power) in noncooperative games.Although we cannot prove that the IMMSE tax is optimum, theinterpretation of the interference tax (21) suggests that IMMSEmay lead to a better solution from a social standpoint than thegreedy game.
Fixed-point existence for IMMSE is addressed in the fol-lowing proposition.
Proposition 2: The Jacobi-IMMSE algorithm with powerconstraints has at least one fixed point (Nashequilibrium) when a feasible solution for the constrainedpowers meeting the QoS constraints exists.
Note that Proposition 2 does not imply that Jacobi-IMMSEwill converge to the fixed point, only that at least one such pointexists. Proof: the finite power algorithm iterations in Table I arewritten as follows using the unnormalized beamformer defini-tion with :
(22)where , and
. The quantity is any vector not orthogonalto the maximum eigenvalue solution. It is readily verified that(22) represents a vector transformation , whichis differentiable and therefore continuous in . Furthermore,since a feasible solution satisfying is assumedto exist, is confined to a compact, convex set defined by
. Thus, the Brouwer fixed-point theorem [30]guarantees that at least one equilibrium of (22) exists.
The structure of the IMMSE algorithm makes it difficult todetermine if Pareto-efficient solutions [19], [31] or solutionswith minimal sum power are achieved. For example, if a Nashequilibrium was Pareto-efficient, then no other equilibriumwould exist satisfying for at least one , and
ILTIS et al.: NONCOOPERATIVE ITERATIVE MMSE BEAMFORMING ALGORITHMS FOR AD HOC NETWORKS 753
for all remaining , where is definedin (19). Alternatively, it would be desirable to show thatachieved minimum over all equilibrium.
In [19], a power control game with pricing was developed thatwas proven to yield the smallest , and iterative pricing al-gorithms were developed that yielded Pareto efficiency. Unfor-tunately, the questions of Pareto efficiency and power optimiza-tion are difficult to address for the beamforming games, due tothe lack of a natural ordering of the actions. Furthermore, beam-forming lacks the quality of “strategic complementarities” [32]that are found in power control-only games, i.e., for properlydesigned utility functions/prices in power control, user alwaysbenefits by increasing its power when users increase theirs.In beamforming, if user “steers” its beam to increaseits own SNR, it can either decrease or increase the SNR of link
, depending on the relative positions of the nodes. It is thusunclear how to design an ordered action set and obtain a super-modular game for noncooperative beamforming.
IV. SEQUENTIAL DISTORTIONLESS RESPONSE
BEAMFORMING ALGORITHM
Simulations of the Jacobi-IMMSE algorithm in Table I havedemonstrated cases where the solutions for oscillatebetween multiple points depending on initial conditions. TheGauss–Seidel IMMSE algorithm has always converged insimulations, but a Lyapunov-type function has not yet beenfound in this case to guarantee convergence. However, a Lya-punov-type function (TIF) does exist for the SDRB algorithmin Table II, which plays the same role as the total squared cor-relation (TSC) in adaptive sequence design [24], [25]. SDRBuses Gauss–Seidel-type iterations in which each vector isupdated using the most recent values of . Hence, following[24], when is updated on the th pass of the algorithm, theremaining transmit beamformers correspond to
The interpretation of SDRB is as follows. The subiterationsin Table II show that is updated according to
(23)
where normalizes to unity. Thus, SDRB againcorresponds to the power algorithm, and, as ,becomes the solution to
(24)
Incorporating the power update for and recalling thatyields the equivalent asymptotic form of the transmit
The interpretation of (25) is that minimizes an interferencemeasure while maintaining a constant SNR. However, the SNRcriterion only maintains the true SNR atin the absence of multiuser interference, i.e., when .In contrast to IMMSE, the SDRB algorithm requires two typesof channel estimation: 1) an estimate at node of the inter-ference-plus-noise covariance and 2) an estimate of thechannel matrix . However, note that is only re-quired to update the power , and hence its accuracy is not ascritical.
The SDRB algorithm has a particularly simple form for thecase of rank-1 channels and, hence, may be useful for scenarioswhere strong line-of-sight paths predominate. For the rank-1case and far-field approximation, , where
is the plane-wave response at array to a point source em-anating from the reference element (zero phase) of array . Inthis case, Table II can be simplified to the following steps.
For
For
Compute normalized MMSE beamformer at node
Set transmit beamformer to
Next node
Next
754 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 4, APRIL 2006
For rank-1 channels, the power algorithm implicit in Table IIconverges in one iteration, and hence the MMSE beamformerimmediately yields the transmit . However, the power update
still requires an estimate of the channels .To prove convergence of the SDRB algorithm to a fixed point,
first define a TIF as
TIF (26)
The TIF is similar to total squared correlation in [24], exceptthat it is defined in terms of fixed steering vectors insteadof adaptive signature sequences. The interference interpretationfollows from the summands . When , eachsuch term represents the interference seen at node due to trans-mitter , multiplied by the product of the powers . Further-more, represents the additive noise power incurred by re-ceive beamformer multiplied by . Hence, TIF can indeed beviewed as the total mutual interference in the network plus thescaled sum power. The TIF after updating can be decom-posed as
TIF (27)
for the symmetric channel case , whereis a function independent of . Convergence is claimed in thefollowing.
Proposition 3: The TIF is a strictly decreasing function of theSDRB algorithm, except at a fixed point. Furthermore, the TIFis bounded from below and hence SDRB converges.
Proof: This proof follows directly from the statement ofthe SDRB. Let be the most recently updated transmit vector.Recall that minimizes under the constraint
. Now the vector satisfies the same con-straint as , but minimizes instead. Thus
(28)
which in turn implies that or that TIF is anonincreasing sequence. Finally, TIF is trivially bounded frombelow by zero from (26).
Note that Proposition 3 does not imply that SDRB convergesto the optimum solution minimizing TIF subject to the con-straints . Results in [33] for IMMSE CDMAsequence adaptation [24] suggest that the fixed points of SDRBmay include both stable optimum and unstable suboptimum so-lutions. It was further proven in [33] that a “noisy” version of theCDMA IMMSE algorithm is globally convergent to the optimalsolution in [33]. A similar noisy version of SDRB may also beglobally convergent, but the analysis of such an approach andthe nature of SDRB fixed points are beyond the scope of thispaper.
V. RESULTS
Simulations were carried out with beampatterns, power effi-ciency, TIF, and node SNR as outputs. Power efficiency is
Fig. 1. Beampatterns of Gauss–Seidel IMMSE with eight-element antennas.
Fig. 2. Transmit powers of Gauss–Seidel IMMSE with eight-element antennas.
similar to asymptotic efficiency in multiuser detection [34] andis defined as follows. In the absence of multiuser interference,the power required to attain an SNR of at node is found interms of , where is themaximum eigenvalue of matrix . The efficiencythen quantifies the excess power required to maintain a constanterror rate (assuming Gaussian multiuser interference) and is de-fined by
(29)
where is the power on the th iteration of IMMSE. In thesimulations in Figs. 1–7, a rank-3 channel was assumed witha direct path plus two additional paths each with 3-dB gainand –rad angular spread with respect to the direct path. Apathloss exponent of 2 was assumed. The target SINR is setto 10 dB.
ILTIS et al.: NONCOOPERATIVE ITERATIVE MMSE BEAMFORMING ALGORITHMS FOR AD HOC NETWORKS 755
Fig. 3. Power efficiency of Gauss–Seidel IMMSE with eight-element antennas.
Fig. 4. Power efficiency of the greedy algorithm.
Fig. 1 shows the node configuration and the beampatternswith nodes. The IMMSE algorithm in Table I uses thetrue channels and covariances to compute (i.e., zero misad-justment of the LMS/RLS beamformer is assumed.) Each nodeis modeled to have a uniform linear array (ULA) withantenna elements. The beampatterns are obtained after 300 iter-ations of the Gauss–Seidel IMMSE algorithm. It is seen that thenulls are formed toward the interferers and the beams are steeredin the directions of the multipaths of the intended receivers.
Figs. 2 and 3 depict the evolution of the transmit powerand the power efficiency at each node, again assumingzero LMS/RLS misadjustment. After about 50 iterations, theIMMSE algorithm is seen to converge to a Nash equilibrium.Fig. 3 shows that, even in the presence of interference, morethan half of the nodes achieve a power efficiency of greaterthan 70%. It is interesting to compare Fig. 3 with Fig. 4, whichcorresponds to power efficiency of the greedy SNR-maximizinggame [see (11)]. It is clearly seen that paying an interferencetax in IMMSE results in an operating point that is socially
Fig. 5. Beampatterns of Gauss–Seidel IMMSE with four-element antennas.
Fig. 6. Power efficiency of Gauss–Seidel IMMSE with four-element antennas.
more desirable. This is also confirmed by comparing the sumof the transmit powers of the IMMSE algorithm after conver-gence with that of the greedy algorithm
.Fig. 5 shows the beampatterns of a severely overloaded sce-
nario, where nodes are communicating, and the numberof antenna elements at each node is . Gauss–SeidelIMMSE was employed using the true channel matrices and co-variances to compute . The corresponding power efficiencyplot in Fig. 6 demonstrates that is generally reduced com-pared to the previous scenario due to increased multiuser inter-ference. Nevertheless, the algorithm converges, although moreslowly than in Fig. 3.
The effect of LMS/RLS misadjustment error on IMMSE per-formance is investigated next. In Fig. 7, the power efficiencyaveraged over all nodes is plotted for the exact IMMSE, LMS,RLS, and greedy algorithm cases for the node configuration in
756 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 4, APRIL 2006
TABLE IIIPOWER EFFICIENCY OF IMMSE AND GREEDY ALGORITHMS FOR VARYING NODE DENSITIES AND MULTIPATH CONDITIONS
Fig. 7. Average power efficiency with eight-element antennas.
Fig. 1. A random binary training sequence of 128 bits was em-ployed for adaptive beamforming. A standard LMS algorithmwith a step size of 0.001 was employed, and the results from100 independent runs were averaged. For the RLS case, a for-getting factor of 0.99 was used, and 100 independent runs wereaveraged. As a comparison, the average power efficiency of thegreedy SNR-maximizing algorithm is depicted. The results inFig. 7 show that the proposed IMMSE algorithms can be imple-mented via simple adaptive beamforming techniques at the costof little performance degradation.
The above results suggest that IMMSE often yields higherpower efficiencies than the greedy algorithm. Extensive sim-ulations were conducted to further compare the IMMSE andgreedy approaches as summarized in Table III. The networkwas randomly generated in a 1000 m 1000 m rectangle.Distances between the transmit and receive nodes are uni-formly distributed over 0 to 300 m. It is assumed that thereis a direct path between the transmitter and the receiver. Allother multipaths have power 3 dB below that of the direct path.The angular spreads of the multipath are uniformly distributedover , and the pathloss exponent is 2. The SINRtarget is dB. For each network configuration, 100network instances were generated. It is seen that the IMMSEalgorithm always yielded higher power efficiency than the
Fig. 8. SINR achieved (top) and TIF of SDRB (bottom).
greedy method for varying node densities and numbers ofpaths. Further, there appears to be an intermediate node densitywhere IMMSE gives the greatest improvement in efficiency.IMMSE also generally yields better results for richer multipathenvironments (four paths.)
The performance of SDRB is examined by simulations forthe node configuration used in Fig. 1. Since the SDRB cannotguarantee that the target SINR is met, the SINRs achieved areshown in Fig. 8 (top). The TIF plotted in Fig. 8 (bottom) is seento be monotonically decreasing, as claimed.
VI. CONCLUSION
To conclude, iterative beamforming algorithms were pre-sented for ad hoc networks. It was shown that the LEGO-typeIMMSE algorithms [17] correspond to power algorithm itera-tions and maximization of the ratio of SNR to an interferencefunction. Furthermore, IMMSE fixed points satisfy the FONCfor optimal power minimization. In terms of noncooperativegame theory, the IMMSE corresponds to a payoff function foreach node that is maximized when the target SNR is met subjectto power minimization. However, an interference tax is incurredby each node as a byproduct of the IMMSE which suggests thatIMMSE should yield lower sum power solutions than greedy
ILTIS et al.: NONCOOPERATIVE ITERATIVE MMSE BEAMFORMING ALGORITHMS FOR AD HOC NETWORKS 757
SNR maximization. A sequential distortionless response beam-forming algorithm was developed, and shown to converge byminimization of a Total Interference Function. The SDRB thusprovides a connection between iterative beamforming and theiterative sequence construction algorithm of [24]. Extensivesimulation results showed that IMMSE yielded always yieldedgreater power efficiency than greedy noncooperative SNR max-imization. The combination of Lagrangian and FONC analysis,game theory interpretation, and simulation results suggest thatthe IMMSE algorithm is a good noncooperative solution to adhoc network power minimization meeting QoS.
APPENDIX IWEAK NETWORK DUALITY
An optimum network is defined by receive/transmit beam-former pairs which minimize while meetingthe QoS target SNR . Here, Theorem 1 is proven to showthat the dual network has the same minimum sumpower. The proof uses the normalized gain definition
, in terms of the optimum beamformers.If a feasible solution to the power vector exists, then the
following equation must be satisfied [26] using the SNR defini-tion (2):
(30)
where is a diagonal matrix. Note that the thelement of the matrix satisfies forand otherwise. The solution for the minimum sumpower in (2) is
(31)
Now, define the permuted vector and. Similarly, . The minimum sum power
can be written in terms of these rearranged matrices/vectorsas To complete the proof,the dual network has transfer functionsunder the channel reciprocity condition . Then, thetransfer function matrix in the dual network has entries
. However, note that . Hence,
, and is just the transpose of . Furthermore,. The sum power for the dual network
is then
(32)
Again, the results of [17] and [26] applied to (32) show that.
In contrast to TDD networks, the result onlyguarantees that the optimum when the solution forthe is unique. Proof: Assume is the unique solutionfor the transmit beamformers achieving minimum . Then,the must be MVDR/MMSE and are uniquely determinedby the . Thus, if the optimum set of receive/transmitbeamformers is , the above results show that
is also optimal. The uniqueness of thenimplies . However, there is no guarantee thatthe set of beamformers achieving is unique in the ad hocnetwork, and hence strong network duality need not hold.
APPENDIX IIIMMSE AND FONC VIA THE LAGRANGIAN
The stationary points of the Lagrangian (6)with respect to (w.r.t.) each beamformer are found by direct dif-ferentiation w.r.t. the scalar componentsyielding
(33)
where the identity
has been employed. Now, substitute
(34)
into (33), where is the th row of the matrix. Re-order terms in the sum on the r.h.s. of (33) and form the vectorof derivatives to yield
(35)
Substituting the unnormalized MVDR beamformer definitioninto (35) yields
(36)
which corresponds to the eigenvector (7) with defined in (9).Next, it is proven that a fixed point of the IMMSE algorithm
satisfies the FONC (36). Recall that the IMMSE solution foris the maximum eigenvector of (13). Hence, IMMSE solves
(36) provided that: 1) and 2) can be set to ,the inverse of the maximum eigenvalue of (13). To show that 1)is satisfied, note that if we are at a fixedpoint of IMMSE. Then, the matrix in (36) becomes
(37)
758 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 4, APRIL 2006
Then, in (4) if . Thus, itmust be shown that the IMMSE fixed points of yield
or the inverse of the maximum eigenvector of (13).Condition 2) is then satisfied, since setting is thenconsistent with .
Assume that indeed satisfies .Substitute for and using (10), which yields
(38)
Using the definition of the MVDR beamformer and therelationship , the numerator in (38) be-comes
(39)
Similar manipulations in the denominator of (38) and cancella-tion of terms show that
(40)
However, recall that a fixed point of IMMSE corresponds toequaling the inverse of the maximum eigenvalue of (13),which is just the r.h.s. of (40). Hence, ,
, and IMMSE solves the eigenvector equation (36)and hence the FONC.
REFERENCES
[1] S. Ye and R. S. Blum, “Optimized signaling for MIMO interferencesystems with feedback,” IEEE Trans. Signal Process., vol. 51, no. 11,pp. 2839–2848, Nov. 2003.
[2] O. Popescu and C. Rose, “Greedy SINR maximization in collaborativemultibase wireless systems,” EURASIP J. Wireless Commun. Netw.,vol. 2, pp. 201–209, 2004.
[4] M. C. Bromberg and B. G. Agee, “Optimization of spatially adap-tive reciprocal multipoint communication networks,” IEEE Trans.Commun., vol. 51, no. 8, pp. 1254–1257, Aug. 2003.
[5] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beam-forming design for multicarrier MIMO channels: a unified frameworkfor convex optimization,” IEEE Trans. Signal Process., vol. 51, no. 9,pp. 2381–2401, Sep. 2003.
[6] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterion,” IEEE Trans. Commun., vol. 49, no. 12, pp. 2198–2206, Dec.2001.
[7] F. Rashid-Farrokhi, K. J. R. Liu, and L. Tassiulas, “Transmit beam-forming and power control for cellular wireless systems,” IEEE J. Sel.Areas Commun., vol. 16, no. 10, pp. 1437–1450, Oct. 1998.
[8] K. Wong, R. D. Murch, and K. B. Letaief, “Performance enhancementof a multiuser MIMO wireless communication system,” IEEE Trans.Commun., vol. 50, no. 12, pp. 1960–1970, Dec. 2002.
[9] E. Visotsky and U. Madhow, “Optimum beamforming using transmitantenna arrays,” in Proc. IEEE 49th Veh. Technol. Conf., 1999, pp.851–856.
[10] F. Rashid-Farrokhi, L. Tassiulas, and K. J. R. Liu, “Joint optimal powercontrol and beamforming in wireless networks using antenna arrays,”IEEE Trans. Commun., vol. 46, no. 10, pp. 1313–1324, Oct. 1998.
[11] J. Chang, L. Tassiulas, and F. Rashid-Farrokhi, “Joint transmitter re-ceiver diversity for efficient space division multiaccess,” IEEE Trans.Wireless Commun., pp. 16–17, Jan. 2002.
[12] S. Serbetli and A. Yener, “Transceiver optimization for multiuserMIMO systems,” IEEE Trans. Signal Process., vol. 52, no. 1, pp.214–226, Jan. 2004.
[13] T. Dahl, N. Christophersen, and D. Gesbert, “Blind MIMO eigenmodetransmission based on the algebraic power method,” IEEE Trans.Commun., vol. 52, no. 9, pp. 2424–2431, Sep. 2004.
[14] S. Bellofiore, J. Foutz, R. Govindarajula, I. Bahceci, C. A. Balanis, A.S. Spanias, J. M. Capone, and T. M. Duman, “Smart antenna systemanalysis, integration and performance for mobile ad hoc networks(MANET’s),” IEEE Trans. Antennas Propag., vol. 50, no. 5, pp.571–581, May 2002.
[15] S. Bellofiore, J. Foutz, C. A. Balanis, and A. S. Spanias, “Smart-an-tenna system for mobile communication networks: Part 2: beam-forming and network throughput,” IEEE Antennas Propag. Mag., vol.44, no. 8, pp. 106–114, Aug. 2002.
[16] P. Karn, “MACA—A new channel acess method for packet radio,” inProc. ARRL/CRRL 9th Computer Netw. Conf., 1990, pp. 134–140.
[17] M. C. Bromberg, “Optimizing MIMO multipoint wireless networks as-suming Gaussian other-use interference,” IEEE Trans. Inf. Theory, vol.49, no. 10, pp. 2352–2362, Oct. 2003.
[18] H. V. Poor, “Multiuser MIMO systems,” in Proc. Adaptive SensorArray Process. Workshop, Lexington, MA, Mar. 2004, [CD-ROM].
[19] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, “Efficient powercontrol via pricing in wireless data networks,” IEEE Trans. Commun.,vol. 50, no. 2, pp. 291–303, Feb. 2002.
[20] A. B. MacKenzie and S. B. Wicker, “Game theory in communications:Motivation, explanation and application to power control,” in Proc.GLOBECOM, San Antonio, TX, Nov. 2001, pp. 821–826.
[21] T. Alpcan, T. Basar, and R. Srikant, “Uplink power control as a non-cooperative game,” Wireless Netw., vol. 8, pp. 659–670, 2002.
[22] D. Reynolds and X. Wang, “Adaptive transmitter optimization for blindand group-blind multiuser detection,” IEEE Trans. Signal Process., vol.51, no. , pp. 825–838, 2003.
[23] C. W. Sung and K. K. Leung, “On the stability of distributed sequenceadaptation for cellular asynchronous DS-CDMA systems,” IEEETrans. Inf. Theory, vol. 49, no. 7, pp. 1828–1831, Jul. 2003.
[24] S. Ulukus and R. Yates, “Iterative construction of optium signature se-quence sets in synchronous CDMA systems,” IEEE Trans. Inf. Theory,vol. 47, no. 7, pp. 1989–1998, Jul. 2001.
[25] C. Rose, S. Ulukus, and R. D. Yates, “Wireless systems and interferenceavoidance,” IEEE Trans. Wireless Commun., vol. 1, pp. 415–428, Jul.2002.
[26] P. Viswanath and D. N. C. Tse, “Sum capacity of the vector Gaussianbroadcast channel and uplink-downlink duality,” IEEE Trans. Inf.Theory, vol. 49, no. 8, pp. 1912–1921, Aug. 2003.
[27] F. Rashid-Farrokhi, K. Liu, and L. Tassiulas, “Downlink power controland base station assignment,” IEEE Commun. Lett., vol. 1, no. 4, pp.102–104, Jul. 1997.
[28] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,U.K.: Cambridge Univ. Press, 2004.
[29] D. P. Bertsekas, Nonlinear Programming. Nashua, NH: Athena Sci-entific, 2004.
[30] W. Rudin, Real and Complex Analysis. New York, NY: McGraw-Hill, 1986.
[31] T. Basar and G. Olsder, Dynamic Noncooperative Game Theory, 2nded. Philadelphia, PA: SIAM, 1999.
[32] P. Milgrom and J. Roberts, “Rationalizability, learning and equilib-rium in games with strategic complementarities,” Econometrica, pp.1255–1277, Nov. 1990.
[33] P. Anigstein and V. Anantharam, “Ensuring convergence of theMMSE iteration for interference avoidance to the global optimum,”IEEE Trans. Inf. Theory, vol. 49, no. 4, pp. 873–885, Apr. 2003.
[34] S. Verdú, Multiuser Detection. New York, NY: Cambridge Univ.Press, 1998.
ILTIS et al.: NONCOOPERATIVE ITERATIVE MMSE BEAMFORMING ALGORITHMS FOR AD HOC NETWORKS 759
Ronald A. Iltis (S’83–M’84–SM’91) received theB.A. degree in biophysics from The Johns HopkinsUniversity, Laurel, MD, in 1978, the M.Sc. degreein engineering from Brown University, Providence,RI, in 1980, and the Ph.D. degree in electricalengineering from the University of California, SanDiego, in 1984.
Since 1984, he has been with the University of Cal-ifornia, Santa Barbara, where he is currently a Pro-fessor with the Department of Electrical and Com-puter Engineering. His current research interests are
in CDMA, software radio, radiolocation, and nonlinear estimation. He has alsoserved as a consultant to government and private industry in the areas of adap-tive arrays, neural networks and spread-spectrum communications.
Dr. Iltis was previously an Editor for the IEEE TRANSACTIONS ON
COMMUNICATIONS. In 1990, he was the recipient of the Fred W. EllersickAward for the Best Paper at the IEEE MILCOM Conference.
Seung-Jun Kim received the B.S. and M.S. degreesfrom Seoul National University, Seoul, Korea, in1996 and 1998, respectively, and the Ph.D. degreefrom the University of California, Santa Barbara, in2005, all in electrical engineering.
From 1998 to 2000, he served as a Korea Over-seas Volunteer with Chiang Rai Teacher’s College,Chiang Rai, Thailand. Since 2005, he has beenwith NEC Laboratories America, Princeton, NJ. Hisresearch interests lie in detection/estimation theory,spread-spectrum communications, multiple-antenna
techniques, and cross-layer design.
Duong A. Hoang received the B.S.E.E. andM.S.E.E. degrees from Hanoi University ofTechnology, Hanoi, Vietnam, in 1997 and 2001,respectively, and he is currently working toward thePh.D. degree at the Department of Electrical andComputer Engineering, University of California,Santa Barbara.
From 1997 to 2003, he was with the Research In-stitute of Posts and Telecommunications, Vietnam,where he was a Research Engineer. His areas of in-terest include communications and signal processing
