mimo1

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 1, JANUARY 2010 225

Low-Rate-Feedback-Assisted Beamforming andPower Control for MIMO-OFDM Systems

Filippo Zuccardi Merli, Xiaodong Wang, Fellow, IEEE, and Giorgio Matteo Vitetta, Senior Member, IEEE

Abstract—This paper proposes a novel solution to the prob-lem of beamforming and power control in the downlink ofa multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) system. This solution is developed intwo steps. First, we describe an adaptive beamforming techniquethat, using a stochastic gradient method, maximizes the powerdelivered to a mobile terminal. In the proposed solution, perturbedprecoding matrices are time multiplexed in the information signaltransmitted to a mobile terminal; then, the mobile terminal in-forms the transmitter, via a single feedback bit, about the pertur-bation delivering the larger power. This approach does not needpilot symbols and uses quasi–Monte Carlo methods to generatethe required perturbations with the relevant advantages of im-proving the downlink spectral efficiency and reducing the systemcomplexity with respect to other competing solutions. Then, wepropose a novel power-control algorithm that, selecting a propertransmission energy level from a set of possible values, aims tominimize the average bit error rate. This set of levels is generatedon the basis of the channel statistics and a long-term constrainton the average transmission power. Numerical results evidencethe robustness of the proposed algorithms in a dynamic fadingenvironment.

Index Terms—Adaptive transmissions, beamforming, low-ratefeedback, multiple-input–multiple-output (MIMO) orthogonalfrequency-division multiplexing (OFDM), power control, quasi–Monte Carlo (QMC), stochastic gradient algorithm.

I. INTRODUCTION

OVER THE PAST FEW years, multiple-input–multiple-output (MIMO) communication techniques have received

substantial attention, since they can offer high data ratesover multipath Rayleigh fading channels [1]. At the sametime, orthogonal frequency-division multiplexing (OFDM) hasattracted increasing interest due to its robustness againstfrequency-selective fading and its flexibility [2]. For thesereasons, MIMO-OFDM techniques have been adopted inthe standards of many emerging wireless systems, such as

Manuscript received January 8, 2009; revised June 22, 2009. First publishedSeptember 9, 2009; current version published January 20, 2010. This paperwas presented in part at the Asilomar Conference on Signals, Systems, andComputers, Pacific Grove, CA, November 4–7, 2007. The review of this paperwas coordinated by Dr. C. Ling.

F. Z. Merli was with the Department of Information Engineering, Universityof Modena and Reggio Emilia, 41100 Modena, Italy. He is now with KPMG,20124 Milano, Italy (e-mail: [email protected]).

X. Wang is with the Department of Electrical Engineering, Columbia Uni-versity, New York, NY 10027 USA (e-mail: [email protected]).

G. M. Vitetta is with the Department of Information Engineering, Universityof Modena and Reggio Emilia, 41100 Modena, Italy (e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2009.2031970

Third-Generation Partnership Project long-term evolution andWiMax, and are considered the natural choice for the imple-mentation of the next-generation mobile wireless systems toachieve high spectral efficiency at acceptable computationalcomplexity [3].

Different techniques have been proposed in the technicalliterature to allow an efficient use of the available energy andspectrum resources in MIMO systems. Most of the researchin this area has focused on space-time coding (STC), exploit-ing the independent modes offered by a MIMO propagationchannel to obtain multiple spatial channels for capacity en-hancement [4]. However, STC usually represents a form ofa blind technique, in the sense that it requires no knowledgeof the forward channel state at the transmitter side. Recently,other signal processing methods have been proposed to improvethe performance of MIMO systems; in particular, a substantialbody of literature has focused on adaptive transmission schemesmultiplexing the transmitted data vectors with appropriate pre-coding matrices to transmit directional signals (beams) [5]. Thisapproach, which is known as beamforming, aims to transmit theenergy in the direction of the main channel modes on the basisof the channel state information available at the transmitterside. This allows increasing the coverage area or capacity ofa wireless link without changing the air interface or evenincreasing the transmit power. Some beamforming techniquesfor OFDM systems have recently been proposed in [6] and [7].In particular, [6] describes various beamforming techniques formultiple-input single-output (MISO) OFDM; they exploit bothtime- and frequency-domain channel correlations for the designof the precoding codebook and the tracking in a time-varyingscenario. A multistage beamforming scheme employing threeweight matrices and based on an iterative algorithm has beenproposed in [7] for an MIMO-OFDM system. However, allthese strategies have been devised under the restrictive as-sumption that the downlink and the uplink channels are almostthe same, i.e., they can be used in time-division duplexingapplications. Unluckily, due to generally uncorrelated uplinkand downlink channels in frequency-division duplexing (FDD)systems, the antenna array weights used for the uplink ina communication system are, in general, not suitable for itsdownlink. To overcome this problem, an interesting solutionhas been proposed in [8] and [9] for a single-carrier wirelesssystem. It employs some feedback information from a mobileterminal (MT) to the serving base station (BS) to adjust thetransmit weights of the BS antennas so that the receiver powerat the MT is maximized. More specifically, dually perturbedtransmission weights are randomly generated at the BS and

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226 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 1, JANUARY 2010

time multiplexed in pilot symbols; then, the MT informs theBS about the perturbation delivering greater power. The use oflimited feedback information [8] for beamforming applicationshas also recently been adopted in [10] to devise a power-controlprocedure for an FDD system.

A more efficient exploitation of the available resources isobtained by combining beamforming with a power-controlstrategy. As far as we know, the joint problem of power controland beamforming in MIMO systems has been investigated in[11] only, where, however, a single-carrier system has beenconsidered, and it has been assumed that the downlink anduplink channels are the same.

The purpose of this paper is twofold. First, the approachto adaptive beamforming proposed in [8] and [9] for single-carrier communications is revisited and modified, extending itto a multicarrier (OFDM) transmission and adopting a novelscheme for subspace tracking and the random generation proce-dure of Gaussian perturbations. In fact, the proposed algorithmfor subspace tracking processes the received signal samplesassociated with information symbols, thus avoiding the use ofspecific pilot symbols; this improves both the system efficiencyand the tracking capability in a time-varying scenario withrespect to [8] and [9]. Moreover, the algorithm for generatingGaussian perturbations proposed in [8] and [9] is replacedwith a deterministic perturbation method based on quasi–MonteCarlo (QMC) methods [12], with the relevant advantages ofa substantial complexity reduction and, at the same time,improved performance. The second technical contribution of-fered by this paper is represented by a novel power-controlscheme enhancing the performance of a MIMO-OFDM systemthat can easily be combined with our beamforming technique.The devised power-control strategy selects a proper level ofthe transmitting energy from a list of possible candidates tominimize the bit error rate (BER). This list is generated onthe basis of the channel statistics and a long-term constrainton the average transmission power. Note that this approach isconceptually related to that proposed in [10], where, however,the power control procedure minimizes the outage probabilityin a single-carrier transmission over a fading relay channel inthe absence of adaptive antennas.

This paper is organized as follows. The signal and channelmodels are illustrated in Section II. The proposed low-rate-feedback-assisted beamforming and power-control schemes aredescribed in Sections III and IV, respectively. Some perfor-mance results are discussed in Section V. Finally, Section VIoffers some conclusions.

II. SYSTEM DESCRIPTION

We consider the downlink transmission in a MIMO-OFDMsystem employing NT transmit antennas at the BS and NR

receive antennas at the MT. The block diagram of the transmit-ter is illustrated in Fig. 1. Since the proposed scheme operateson a symbol-by-symbol basis, our observation interval can berestricted to the transmission of a single OFDM symbol; for thisreason, the symbol index is neglected in the following. Trans-mitter processing can be summarized as follows. The input datastream is mapped into a sequence of M -ary phase-shift keying

Fig. 1. BS transmitter block diagram in a MIMO-OFDM system.

or M -ary quadratic-amplitude modulation (M -QAM) symbols(where M denotes the constellation size); this sequence isturned into a series of nonoverlapping blocks (through serial-to-parallel conversion), each consisting of Nu symbols, whereNu represents the number of useful subcarriers. Then, Nvc =N − Nu virtual carriers are inserted in each block, resulting

in an N -dimensional vector a Δ= [a0, a1, . . . , aN−1]T , wherean represents the channel symbol associated with the nthsubcarrier. These data blocks are fed to an adaptive beamform-ing algorithm operating on a subcarrier-by-subcarrier basis.In other words, for the symbol transmitted over the nth sub-carrier, the algorithm generates an NT -dimensional complex

weight vector wnΔ= wn,1, wn,2, . . . , wn,NT

]T , where wn,i isthe weight for the ith transmit antenna. The signal transmittedfrom the NT antennas over the nth subcarrier frequency is thenwnan. This technique, which is employed for each subcarrier,produces a parallel stream of NT vectors (each consisting of Nelements), which are fed to a bank of NT OFDM modulatorswhere they undergo an N th-order inverse discrete Fouriertransform (IDFT) followed by cyclic-prefix (CP) insertion,parallel-to-serial conversion, and transmit filtering. In the fol-lowing, we assume that Ncp is the CP length, that the impulseresponse p(t) of the transmit filter is time limited to the interval(−NpTs, NpTs) (where Ts is the channel symbol interval, andNp denotes half of the duration of p(t) in symbol intervals), andthat its Fourier transform P (f) is the root of a raised cosine withroll-off α so that Nu = int[N(1 − α)] [13].

The OFDM signal is transmitted over a wide-sense stationaryuncorrelated scattering multipath fading channel. The tapped-delay-line model

hi,j(t)Δ=

L−1∑l=0

hi,j [l] δ(t − lTs) (1)

is adopted for the channel impulse response between the ithtransmit antenna (i = 1, . . . , NT ) and the jth receive antenna(j = 1, . . . , NR). Here, δ(t) denotes the Dirac delta function, Lis the number of channel distinct taps, and hi,j [l] is the complexgain of the lth tap. We also assume that 1) the channel isstatic over each OFDM symbol interval (quasi-static channel)and that 2) Ncp ≥ 2Np + L − 1 so that interblock interferenceis avoided in the detection of each OFDM symbol. At the

MERLI et al.: LOW-RATE-FEEDBACK-ASSISTED BEAMFORMING AND POWER CONTROL FOR MIMO-OFDM SYSTEMS 227

MT, after matched filtering and sampling,1 the N samplescollected at the jth receive antenna undergo an N th-orderdiscrete Fourier transform (DFT) producing the N -dimensional

vector rjΔ= [rj [0], rj [1], . . . , rj [N − 1]]T . It is not difficult to

show that this vector can be expressed as

rj =MEb

NT

NT∑i=1

WiAFLhi,j + nj (2)

where A Δ= diag{an, n = 0, 1, . . . , N − 1} is an N × N di-agonal matrix containing all the elements of a along its

main diagonal, hi,jΔ= [hi,j [0], hi,j [1], . . . , hi,j [L − 1]]T col-

lects the channel gains of hi,j(t) [see (1)], and FL is anN × L DFT matrix with [FL]p,q = exp[−j2πpq/N ], p =

0, 1, . . . , N − 1, and q = 0, 1, . . . , L − 1. Moreover, WiΔ=

diag{w0,iw1,i, . . . , wN−1,i} is an N × N diagonal matrix col-lecting the complex weights applied to the OFDM symbol sentby the ith transmit antenna, Eb is the total average transmitted

energy per information bit, and zjΔ= [zj [0], zj [1], . . . , zj [N −

1]]T ∼ Nc(0N , σ2zIN ) is an N -dimensional complex Gaussian

noise vector2 (generally speaking, the notation Nc(a,B) in-dicates a complex Gaussian vector having mean vector a andcovariance matrix B).

To simplify the derivation of our beamforming andpower-control algorithms, we convert (2) into a subcarrier-

based one. Specifically, the NR-dimensional vector r[n] Δ=[r1[n], r2[n], . . . , rNR

[n]]T containing the samples from all thereceive antennas at the nth subcarrier can be expressed as

r[n] =MEb

NTHnwnan + zn (3)

where zn ∼ Nc(0, σ2zINR

) is an NR-dimensional com-plex Gaussian noise vector, and Hn = [Hi,j [n]] (with i =1, . . . , NT and j = 1, . . . , NR) represents an NR × NT matrixcollecting the responses of the MIMO channel at the nth sub-

carrier frequency. Note that the N -dimensional vector Hi,jΔ=

[Hi,j [0],Hi,j [1], . . . , Hi,j [N − 1]]T collecting the values of thechannel frequency response between the ith transmit and the jthreceive antennas is given by [14]

Hi,j = FLhi,j . (4)

Given the received signal vectors r[n], n = 0, 1, . . . , N − 1,in (3), at an MT, we are interested in devising algorithms3

to 1) update the beamforming vector wn and 2) set a properenergy level at the BS. In Sections III and IV, we show thatthese objectives can be, respectively, achieved by employingbeamforming and power-control algorithms operating on asymbol-by-symbol basis and each using a single-feedback-bitscheme. In the derivation of the proposed algorithms, we focus

1The first Ncp samples (i.e., the samples associated with the CP) arediscarded from each OFDM time interval, and ideal timing is assumed.

2In the following, the noise variance σ2z is assumed known at the re-

ceive side.3Both algorithms operate on a symbol-by-symbol basis, and each of them

uses a feedback bit from the MT.

on an FDD scenario, where the downlink channel is known atthe MT and unknown at the BS. Moreover, we assume thatthe channel responses associated with distinct transmit/receiveantenna pairs are spatially uncorrelated. Finally, note that, sincethe proposed schemes operate on a subcarrier-by-subcarrierbasis, in what follows, our analysis is restricted to a singleOFDM subcarrier.

III. SINGLE-BIT-FEEDBACK ALGORITHM

FOR BEAMFORMING

In this section, we propose an algorithm that updates theweight vector wn in (3) at the BS, based on the stochasticgradient algorithm to track the dominant eigenvectors of theMIMO channel. Various enhancements to this algorithm arethen discussed.

A. Basic Algorithm

To begin, we define the cost function Jn for the nthsubcarrier

JnΔ= ‖Hnwn‖2

F (5)

where ‖ · ‖F is the Frobenious norm and introduce theconstraint

wHn wn = 1. (6)

The algorithm we propose for beamforming aims to ac-complish subspace tracking through the maximization of Jn

subject to (6); this produces the weight vector delivering largerpower [8]. Note that the maximization of (5) does not admita closed-form solution because of the large cardinality of wn,and an iterative technique based on the stochastic gradient andrequiring feedback from MTs has been proposed in [8], whichunluckily requires the transmission of pilot symbols.

In this section, we employ the stochastic gradient algorithm,but in a different fashion. In fact, our target is devising abeamforming technique that can directly exploit the weightedinformation symbols, therefore avoiding the use of pilot sig-naling and providing a substantial gain in terms of systemefficiency. Our adaptive beamforming algorithm, in its basicversion, evolves through the following steps: 1) probing thewireless channel via weighted information symbols; 2) gen-erating (based on probing results) and transmitting feedbackinformation from the MT; and 3) updating the weight vectors atthe BS. A detailed description of these steps is provided below.

Probing: The tracking of the vector wn is accomplishedat the BS via a feedback-based algorithm, which requires noknowledge about the downlink channel. The algorithm exploitsthe OFDM frame structure at the BS. Specifically, it perturbsconsecutive channel symbols with different probing vectorsand transmits them over the channel. Then, the MT receiver,using the channel state information, generates feedback infor-mation indicating the preferred direction (sign) to apply to theperturbation and sends it to the BS. The weight adaptation isperiodically carried out via probing and feedback generation(further details are given below). In addition, a single bit for


each subcarrier (or a group of adjacent subcarriers, as will bestated below) is transmitted on the feedback channel.

The algorithm for the weight update is described as follows.We assume that the symbols ae

n and aon are transmitted over two

consecutive (even and odd, respectively) OFDM symbols on thenth subcarrier after undergoing beamforming with the distinctweight vectors we

n and won, respectively. Then, the received

signal associated with these symbols is processed at the MTto generate binary feedback; this indicates which of the twoweights is preferable, in terms of received power. The timeslot lasting two OFDM symbols and encompassing both theperturbation measurement and the feedback interval is dubbedthe perturbation probing period (PPP) in the following. In eachPPP, the weight vectors are computed as

wen =

wbn + βpn

‖wbn + βpn‖

(7)

won =

wbn − βpn

‖wbn − βpn‖

(8)

respectively, with n = 0, 1, . . . , N − 1. Here, pn ∼Nc(0, INT

) is an NT -dimensional test perturbation, β isthe adaptation rate, and wb

n is the preferred weight matrixselected in the previous PPP.4 Note that 1) the normalizationsin (7) and (8) ensure that the transmitted power is constrainedand that 2) both the BS and the MT must be synchronized togenerate the probing vectors we

n and won because the MT needs

them for data detection.Feedback Generation: Let He

n and Hon denote the chan-

nel gain matrices over the even and odd OFDM intervals,respectively, of a given PPP. In that PPP, the MT estimatesthe composite channel of (He

n, Hon) with the corresponding

transmission weights wen (7) and wo

n (8). Then, the gradientextraction of the feedback binary digit

bn =sign(‖He

nwen‖2

F −‖Honwo

n‖2F

), n=0, 1, . . . , N−1

(9)

maximizing (5) (i.e., providing the largest received power) isaccomplished for the nth subcarrier. Note that (9) does notneed pilot symbols to generate the feedback bit like in [8]and [9], but it just needs an estimate of the channel, which,however, must be known even in the solutions in [8] and[9], when the information symbols are detected. It is alsoimportant to point out that the effects of imperfect channel stateinformation are automatically compensated for when selectingthe probing vectors; in other words, the presence of probingvectors makes the receiver less sensitive to channel-estimationerrors.

When the BS receives the binary feedback (9), at the be-ginning of the next PPP, it operates according to the followingsteps: 1) It selects the new vector wb

n in (7) and (8) accordingthe following updating rule:

wbn =

{we

n, if bn = +1wo

n, if bn = −1 (10)

4Note that wbn is either the odd perturbation or the even one selected from

the previous PPP.

2) it generates a new probing perturbation vector pn; and3) it updates we

n and won using (7) and (8), respectively. Note

that 1) the feedback bit is sent to the BS every D PPPs;2) the specific value of D depends on the application, partic-ularly on the expected maximum Doppler shift; 3) the updatein (10) involves a step of fixed magnitude in the directionof the largest achievable power at the receiver; and 4) theupdate algorithm, in its basic formulation, can be interpretedas an implementation of the stochastic gradient algorithm,performing a steepest descent adaptation to maximize the costfunction (5).

B. System Issues and Enhancements

Several system implementation issues and enhancementsabout the proposed beamforming algorithm are worth dis-cussing. They concern some key aspects of complexity, flexi-bility, and optimization, as illustrated below.

1) Subcarrier grouping: Equations (7)–(9) can be adoptedfor subspace tracking for each data subcarrier at the price,however, of substantial complexity. In particular, notethat both the processing load and the data rate of thefeedback channel linearly increase with the overall num-ber of subcarriers. A simple technique to mitigate theseproblems is based on the technique of subcarrier grouping[15]. This means that the OFDM tones are partitioned ingroups, each consisting of G adjacent subcarriers, and theproposed adaptive algorithm is applied only to a referencesubcarrier of each group, e.g., to that corresponding tothe center of the group. Then, the weight vector evaluatedfor the reference tone of each group is also used forthe beamforming on the other subcarriers of the samegroup. It is expected that this technique works well if thebandwidth of each group is close to the channel coherencebandwidth, i.e.,

G/NTs

1/LTs≈ 1 (11)

so that it is reasonable to select G = �N/L�. The pro-posed solution reduces the computational complexity by afactor N/G. Moreover, simulation results have evidencedthat its use does not appreciably degrade the system per-formance with respect to that achievable in the absence ofsubcarrier grouping. Finally, note that the weight vectorsand the probing vector that appears in (7) and (8) must begenerated only once for each group.

2) QMC methods: The performance provided by the pro-posed beamforming algorithm depends on the techniqueadopted to generate the perturbation vector pn in (7)and (8). A criticism expressed about classic Monte Carlo(MC) methods [16] is that entirely random points tend toform gaps and clusters, and therefore, they do not explorethe sample space in the most uniform way. To avoidthis drawback, here, we propose to employ the QMCtechniques [12]. These techniques are based on the idea ofusing more regularly distributed points in the generationof a perturbation, therefore exploring a more regular


Fig. 2. Cost function J (20) versus the step size β of the proposed beamform-ing algorithm. Various values of the normalized Doppler bandwidth BDNTs

are considered.

Fig. 3. BER performance versus the step size β of the proposed beamformingalgorithm. Various values of the normalized Doppler bandwidth BDNTs areconsidered.

space than a random point set associated with the MCapproach. Moreover, the QMC points can be computedoffline and stored in a lookup table, thus avoiding therandom-number-generation calculations at runtime. Ourcomputer simulations have led to the conclusion thatQMC methods can offer not only a substantial complexitysaving but a 1-dB energy gain as well. The detailedprocedure for generating the QMC points can be foundin [12].

3) Quantized beamforming: The single-bit-feedback algo-rithm described in the previous paragraph aims to adaptthe transmit antenna beamforming in a mobile scenario.Even if our simulation results evidence that the single-bit-feedback adaptive beamforming algorithm convergesin the presence of a random initialization (see Figs. 2 and3 and Section V), a proper initialization is expected tosignificantly improve the convergence speed. To that end,we can select the initial value of wb

n that appears in (7)and (8) from a predesigned codebook. In practice, the MTselects from such a codebook the entry that best matchesthe channel state, and the MT sends the entry indexto the transmitter through a low-rate-feedback channel.Generally speaking, the procedure for the selection ofsuch an entry from a codebook is more complicated andrequires a more intensive processing than our single-bit-

feedback beamforming algorithm. In the following, theformer is dubbed slow beamforming, whereas the latter isdubbed fast beamforming. It is suggested to partition theinformation transmission into frames, and the adaptivebeamforming procedure should periodically be initializedat the beginning of each frame to avoid consistent per-formance degradation if a wrong update direction hasbeen selected by the aforementioned fast procedure. Afterthe initialization phase, the weight vector will be succes-sively refined using the fast-beamforming procedure. Itis worth mentioning that various quantized beamformingstrategies can be adopted to generate a proper codebook,quantizing the possible values of the weight matrix. Someof these strategies are illustrated in [17] and are basedon maximizing the mean-squared weighted inner productbetween the optimum and the quantized beamformingvectors. A quantized codebook design based on channeldistribution has also been proposed in [18].

Finally, we summarize the single-bit-feedback adaptivebeamforming algorithm for MIMO-OFDM in Table I, whichalso specifies where each step is carried out.

IV. SINGLE-BIT-FEEDBACK ALGORITHM

FOR POWER CONTROL

In this section, we describe an adaptive power control algo-rithm that can easily be combined with the adaptive beamform-ing technique5 described in the previous section to improvesystem performance. The proposed technique is based on theidea of quantizing the transmit energy for each subcarrier; inpractice, we assume that this energy can take one into twodistinct levels E1 and E2. Level selection is accomplished byan MT on a subcarrier-by-subcarrier (or group-by-group) basis,exploiting the channel state information. The MT sends binary-feedback information to the serving BS.

Let us define the parameter

λnΔ= wH

n HHn Hnwn (12)

measuring the usable signal power at the receiver at the nthsubcarrier frequency (see [9, p. 1157, eq. (4)]); note that thisparameter refers to the beamformed channel resulting from thecombination of the physical channel Hn and the transmissionbeamforming vector wn. We propose to adopt the rule

En ={

E1, if λn < τE2, if λn > τ

, n = 0, . . . , N − 1 (13)

for the selection of the energy per bit En assigned to the nthsubcarrier. Here, τ is a threshold quantizing the transmittedenergy to E1 and E2. We propose to select the energy levelsE1 and E2 and the threshold τ to minimize the average BERon each subcarrier. It is important to note that these values areconstellation dependent, since the employed BER expressiondepends on the adopted constellation.6 In the following, we

5It is worth noting that the proposed solution can be adopted in any MIMOsystem employing an adaptive antenna array at the transmitter side.

6BER formulas for some common constellations are available in [19].


TABLE ISUMMARY OF THE PROPOSED ADAPTIVE ALGORITHM FOR BEAMFORMING

explain the basic design procedure using quaternary phase shiftkeying (QPSK) as an example, even if the proposed approachcan be extended to any other constellation in a straightforwardfashion.

In a MIMO-OFDM transmission employing a QPSK con-stellation, the SNR per bit for the nth subcarrier and thecorresponding average bit error probability are given by

SNRn(λn) =2Eb

σ2z

λn (14)

Pn =

∞∫0

Q(√

SNRn(x))

pn(x)dx (15)

respectively, where pn(x) denotes the probability density func-tion (pdf) of λn.

Substituting (14) in (15) and taking into account the strategy(13) yields the bit error probability

Pn =

τ∫0

Q(√

2E1x/σ2z

)pn(x)dx

+

∞∫τ

Q(√

2E2x/σ2z

)pn(x)dx (16)

that we aim to minimize with respect to E1, E2, and τ underthe long-term energy constraint

q1E1 + q2E2 = Eb (17)

with

q1Δ=

τ∫0

pn(x)dx q2Δ=

∞∫τ

pn(x)dx. (18)

The minimization of the cost function (16) under the con-straint (17) can be accomplished by resorting to the methodof Lagrange multipliers; the resulting technique produces es-

timates of the algorithm parameters E1, E2, and τ . Furtherdetails about this procedure are given in the Appendix.

Finally, it is worth pointing out that our algorithm for sub-carrier power control can be used for groups of subcarriers,adopting the same approach as the beamforming algorithm. Infact, applying the selection rule (13) on a group-by-group basisallows a reduction of both the computational load at the receiverand the number of feedback bits to be sent on the reverse link.

V. SIMULATION RESULTS

In this section, we provide simulation results to illustratethe performance of the proposed algorithms for both adaptivebeamforming and power control. The simulation setup is asfollows: 1) The OFDM system uses N = 256 subcarriers, andthe roll-off factor is α = 0.12 so that the number of useful sub-carriers is Nu = 225; 2) the length of the CP and the durationof p(t) are Ncp = 35 and 20Ts (i.e., Np = 10), respectively;3) groups of G = 15 adjacent subcarriers are used; 4) thesystem is equipped with NT = 4 transmit antennas at theBS and NR = 2 receive antennas at the MT; 5) the SNR isdefined as Eb/N0, where Eb is the average received energy perinformation bit and per antenna; 6) the channel model (1) ischaracterized by L = 16 zero-mean taps (Rayleigh fading) andthe exponential power delay profile [13]

σ2(l) =1 − exp(−1/5)1 − exp(−L/5)

exp(−l/5), l = 0, 1, . . . , 15

(19)

where σ2(l) = E{|hi,j [l]|2}; 7) channel realizations are staticover each OFDM symbol interval but change from symbolto symbol; 8) each tap of the set {hi,j [l], l = 0, 1, 2, . . .}is characterized by the autocorrelation function Rh[l] =J0(2πlBDNTs), where J0(x) is the zeroth-order Bessel func-tion of the first kind, and BD is the fading Doppler bandwidth;9) the feedback channel used for both beamforming and poweradaptation is modeled as a binary symmetric channel withcrossover probability δ = 0.01.


Fig. 4. BER performance of various coherent receivers operating over aRayleigh fading channel with BDNTs = 5 · 10−3. A QPSK format is used.

In our simulations, two different parameters have beenassessed in various scenarios, namely, the BER and theparameter

JΔ= E

(N−1∑n=0

Jn(k)J̃n(k)

)(20)

providing an average indication about the cost function ofour adaptive beamforming algorithm. Here, Jn(k) is the time-varying value of (5) (i.e., k is a time slot index), and J̃n(k) isthe corresponding value obtained over a static channel; note thatthe performance of the algorithm improves if Jn(k) gets closerto Jn(k), i.e., if the ratio of these two quantities approachesunity. Fig. 2 represents J versus the adaptive size β in (7) and(8) for different Doppler bandwidths when the channel symbols{ak[n]} belong to a QPSK constellation. These results showthat 1) as the Doppler bandwidth increases, the use of a largeradaptation parameter β is needed to track the faster changes ofthe communication channel, and 2) if β is properly selected, thealgorithm works well, even in a fast-varying scenario, since Jis close to unity.

Fig. 3 illustrates the BER performance versus β for differ-ent Doppler bandwidths. Note that the optimal values of βextracted from these simulations (i.e., the values minimizingthe BER curves) are the same as those provided in Fig. 2. Suchvalues have been adopted in the simulations to generate all theresults shown below.

Fig. 4 illustrates the BER performance for the follow-ing OFDM receivers (all endowed with ideal channel stateinformation):

1) a coherent receiver, which is dubbed “No BF and noPC,” working in the absence of beamforming and power-control schemes;

Fig. 5. BER performance of various coherent receivers operating overa Rayleigh fading channel with BDNTs = 10−3 and BDNTs = 10−2.A QPSK format is used.

2) a coherent receiver, which is dubbed “Slow BF only,” thatselects the weight vectors from a properly designed table[20], without tracking the time variations of the channelmodes via an adaptive beamforming algorithm;

3) a coherent receiver, which is “Fast BF only,” that up-dates the adaptive beamforming vectors using the al-gorithm derived in Section III but exploits a randominitialization;

4) a coherent receiver, which is dubbed “Complete BF,” thatcombines the proposed beamforming algorithm with theslow beamforming algorithm in [20] for initialization;

5) a coherent receiver, which is dubbed “Complete BF &PC,” that combines our beamforming scheme with thepower-control strategy proposed in Section IV.

In generating the data in Fig. 4, the following assumptionshave been made: 1) The channel symbols {ak[n]} belong toa QPSK constellation; 2) the normalized Doppler bandwidthBDNTs is equal to 10−2; and 3) when the slow beamform-ing is adopted, the transmission is partitioned into frames,and the initialization process for adaptive beamforming is runevery 80 OFDM symbols, resulting in a reset rate Rr = 1/80.These results evidence that the receiver adopting the slow-beamforming technique and using the proposed beamformingscheme provide a gain of 4 and 6 dB, respectively, over thesystem not employing any beamforming strategy. It is alsointeresting to note that if the transmission is partitioned intoframes and the proposed technique is initialized with a slowreset, a 4-dB gain can be obtained over that using only ourbeamforming algorithm; a further 4-dB gain is provided byadding our power-control algorithm.

The simulation results shown in Fig. 5 refer to an OFDMscheme employing QPSK modulation and adopting the full


Fig. 6. BER performance of various coherent receivers operating over aRayleigh fading channel with BDNTs = 5 · 10−3 and BDNTs = 5 · 10−2.A 16-QAM format is used.

beamforming scheme (with both slow and fast techniques)with and without our power-control strategy. Two multipathfading scenarios have been considered: one with normalizedDoppler bandwidth BDNTs = 10−3 and the other one withBDNTs = 10−2. All the other system parameters are exactlythe same as those listed for the previous figures. It is worthnoting that 1) an increase in the Doppler bandwidth fromBDNTs = 10−3 to BDNTs = 10−2 produces a performanceloss of about 7 dB at BER = 10−6 in the considered sys-tems; 2) even in a fast-fading scenario (BDNTs = 10−2), a6-dB performance gain can be achieved using the proposedbeamforming solution over a system not exploiting adaptiveantennas, and an additional 4-dB gain is provided by our power-control algorithm; and 3) a larger performance gain can beobtained in a slow-fading scenario.

Since the power-control algorithm is constellation depen-dent, we also assessed its performance with various modulationformats. Fig. 6 shows the error performance offered by anOFDM coherent receiver when a 16-QAM modulation in usedin the presence of BDNTs = 5 · 10−3 and BDNTs = 5 · 10−2.The receivers operate in a scenario equivalent to that describedfor Fig. 5. These results show that the proposed beamformingand power-control schemes provide substantial performancegains with both fast- and slow-fading channels.

Remark: All the simulation results analyzed earlier refer tosingle-bit-feedback adaptive schemes for both beamformingand power control. It is worth pointing out that, in principle,the proposed algorithms can be used with multiple-bit-feedbackschemes. Our simulation results have evidenced, however, thata small energy gain (i.e., a fraction of a decibel) is offered bythe use of a feedback with multiple bits in place of that usinga single bit. For this reason and due to space limitations, the

case of feedback with multiple bits has not been taken intoconsideration.

VI. CONCLUSION

In this paper, a novel adaptive beamforming technique forthe downlink of a MIMO-OFDM system has been derived.In the proposed approach, the transmitter generates probingbeamforming vectors using QMC methods. The receiver, usingan algorithm based on a stochastic gradient method, producesbinary feedback informing the transmitter which beamformingvector, delivering the largest power, must be used in the nextsignaling intervals. It has also been shown how the proposedbeamforming method can be integrated with a power-controlscheme based on a single-bit feedback scheme. A combinationof both algorithms results in a MIMO-OFDM system jointlytracking the channel time variations and optimizing the avail-able transmitted energy. Simulation results have evidenced thatthe proposed solutions offer excellent performance in scenariosaffected by slow and fast fading.

APPENDIX

Here, we outline the procedure for computing the parametersE1, E2, and τ , which are required by the adaptive power-control algorithm in Section IV.

The application of the method of Lagrange multipliers to theminimization of the cost function (16) under the constraint (17)involves the definition of the Lagrangian function

L =

τ∫0

Q(√

2E1x/σ2z

)pn(x)dx

+

∞∫τ

Q(√

2E2x/σ2z

)pn(x)dx

+ μq1E1 + μq2E2 − μEb (21)

and requires the solution of the system⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

dLdE1

= −12σz

√πE1

τ∫0

√xe

−E1x

σ2z pn(x)dx + μq1 = 0

dLdE2

= −12σz

√πE2

τ∫0

√xe

−E2x

σ2z pn(x)dx + μq2 = 0

dLdτ = Q

(√2E1τ

σ2z

)+ μE1 − Q

(√2E2τ

σ2z

)− μE2 = 0

dLdμ = q1E1 + q2E2 − Eb = 0.

(22)

From the third equation of (22), it is easily inferred that

μ =Q

(√2E2τ/σ2

z

)− Q

(√2E1τ/σ2

z

)E1 − E2

(23)

whereas from the fourth one, it is inferred that

E2 =Eb − q1E1

q2. (24)


Substituting (23) and (24) in the first and second equationsof (22) yields the following nonlinear system:{

f(E1, τ) = 0g(E1, τ) = 0 (25)

involving E1 and τ only.This system can iteratively be solved using the

Newton–Raphson method; the resulting procedure generates asequence of estimates {(E(i)

1 , τ (i)), i = 1, 2, . . .} of the couple(E1, τ) of unknown parameters. However, to simplify theapplication of this method, in our algorithm, the functionsf(E1, τ) and g(E1, τ) are replaced by their first-order Taylorapproximations

f(E1, τ) ≈ f(E

(i)1 , τ (i)

)+ fE1

(E

(i)1 , τ (i)

) (E1 − E

(i)1

)

+ fτ

(E

(i)1 , τ (i)

) (τ − τ (i)

)(26)

g(E1, τ) ≈ g(E

(i)1 , τ (i)

)+ gE1

(E

(i)1 , τ (i)

)(E1 − E

(i)1

)

+ gτ

(E

(i)1 , τ (i)

) (τ − τ (i)

)(27)

around the point (E(i)1 , τ (i)), when computing the new couple

(E(i+1)1 , τ (i+1)) from the last one (E(i)

1 , τ (i)). Here, fx(E, τ)(gx(E, τ)) denotes the first partial derivative of f(E1, τ)(g(E, τ)) computed with respect to the variable x. The ap-proximate representations (26) and (27) lead to the recursiveequations

E(i+1)1 = E

(i)1 + δE1 , τ (i+1) = τ (i) + δτ (28)

for the joint estimation of E1 and τ , respectively. Here, thevector δ = [δE1 , δτ ]T represents the solution of the system

J(E

(i)1 , τ (i)

)δ = −F

(E

(i)1 , τ (i)

)(29)

with

J(E

(i)1 , τ (i)

)=

⎛⎝ fE1

(E

(i)1 , τ (i)

), fτ

(E

(i)1 , τ (i)

)gE1

(E

(i)1 , τ (i)

), gτ

(E

(i)1 , τ (i)

)⎞⎠(30)

F(E

(i)1 , τ (i)

)=

⎛⎝ f

(E

(i)1 , τ (i)

)g

(E

(i)1 , τ (i)

)⎞⎠ . (31)

Note that 1) the pdf pn(τ) can be estimated (via standardmathematical tools, e.g., see [21]) from the values taken on byparameter λn over multiple realizations of the communicationchannel, and 2) the integrals appearing in the partial derivativesof (30) and (31) can be solved using MC techniques.

Then, if Nit represents the maximum number of the iter-ations in the Newton–Raphson method and E

(0)1 = Eb and

τ (0) = median(λn) are selected at the first iteration, the fol-

lowing procedure is employed for the iterative estimation ofE1 and τ :

for i = 1, 2, . . . , Nit

• Compute J(E(i)1 , τ (i)) from (30)

• Compute F (E(i)1 , τ (i)) from (31)

• Compute the new estimate E(i+1)1 of E1 and τ (i+1) of

τ from (28)end

Finally, E2 is computed from (24).

REFERENCES

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[2] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: A convenientframework for time-frequency processing in wireless communications,”Proc. IEEE, vol. 88, no. 5, pp. 611–640, Apr. 2000.

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Filippo Zuccardi Merli was born in Correggio,Italy, in July 1980. He received the Dr. Ing. de-gree (cum laude) in electronic engineering and thePh.D. degree from the University of Modena andReggio Emilia, Modena, Italy, in 2004 and 2008,respectively.

From 2006 to 2007, he was a Research Assis-tant with the Department of Electrical Engineering,Columbia University, New York, NY, focusing ondigital communication and statistical signal process-ing. More recently, he was with Deloitte for one year,

providing technology services for both the industrial and financial sectors.Since October 2008, he has been a Management Consultant with KPMG,advising clients and participating in several projects regarding informationsecurity. His scientific interests are in the broad areas of communication theoryand information management, with applications to information technologycompliance and governance.

Xiaodong Wang (S’98–M’98–SM’04–F’08) re-ceived the Ph.D. degree in electrical engineeringfrom Princeton University, Princeton, NJ.

He is currently with the faculty of the Depart-ment of Electrical Engineering, Columbia Univer-sity, New York, NY. His research interests are in thegeneral areas of computing, signal processing, andcommunications, and he has extensively publishedin these areas. Among his publications is a recentbook entitled Wireless Communication Systems: Ad-vanced Techniques for Signal Reception (Prentice-

Hall, 2003). His current research interests include wireless communications,statistical signal processing, and genomic signal processing.

Dr. Wang received the 1999 NSF CAREER Award and the 2001 IEEECommunications Society and Information Theory Society Joint Paper Award.He has served as an Associate Editor for the IEEE TRANSACTIONS ON

COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNI-CATIONS, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and the IEEETRANSACTIONS ON INFORMATION THEORY.

Giorgio Matteo Vitetta (S’89–M’91–SM’99) wasborn in Reggio Calabria, Italy, in April 1966. Hereceived the Dr. Ing. degree (cum laude) in electronicengineering and the Ph.D. degree from the Universityof Pisa, Pisa, Italy, in 1990 and 1994, respectively.

From 1992 to 1993, he was with the Universityof Canterbury, Christchurch, New Zealand, doingresearch for digital communications on fading chan-nels. From 1995 to 1998, he was a Research Fellowwith the Department of Information Engineering,University of Pisa. From 1998 to 2001, he was an

Associate Professor of telecommunications with the Department of InformationEngineering, University of Modena and Reggio Emilia, Modena, Italy, where heis currently a Full Professor of telecommunications. His main research interestsare in the broad area of communication theory, with particular emphasis oncoded modulation, synchronization, and channel equalization.

Dr. Vitetta is serving as an Editor of the IEEE TRANSACTIONS

ON COMMUNICATIONS and the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS.

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