Adaptive Antennas for OFDM

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Adaptive Ante~nasor OFDFred W. Vook and Kevin L. BaumMotorola Corporate Communication Research Laboratory1301 E. Algonquin RoadSchaumburg,IL 60196 USA

Abstract -- This paper explores the combination of adaptive antennas and Orthogonal Frequency Division Multiplex-ing (OFDM) for operation in a faded delay-spread channel.We consider the performance of the well-known Sample Ma-trix Inversion (SMI) algorithm for controlling an adaptiveantenna in a pilot-symbol-assisted OFDM system. Severalstrategies are considered for deploying SMI with different lev-els of tracking capability over an OFDM time-frequency slot.We propose a modification to the SMI method which incor-porates the concept of pilot-interpolation into the computa-tion of the sample covariance matrix. This modified SMIalgorithm is shown to provide performance superior to theother variations of SMI under consideration.

I. INTRODUCTIONThird and fourth generation cellular systems must be de-signed to handle the ever-increasing demand for wireless

communication in a cost-effective and flexible manner. Adap-tive antennas [1,2] promise to greatly increase cellular systemcapacity by suppressing cochannel interference, improvingcoverage quality, and mitigating multipath interference. Inthe past decade, much research has been done towards mergingadaptive antenna technology with existing cellular and PCSair-interface specifications based on AMPS, TDMA, andCDMA technology. However, the literature appears to berather silent on the topic of applying adaptive antenna tech-niques to a system that employs Orthogonal Frequency Divi-sion Multiplexing (OFDM) [ 5 , 6 ] . OFDM is a widebandmultiple access technology that offers many benefits in amobile wireless communication system. Some of the inher-ent advantages of OFDM are its flexibility, scalability, andthe ability to communicate a wideband signal without usingcomplex temporal equalization schemes.When operating in a frequency-selective environment, thebest interference suppression performance is generally ob-tained when wideband algorithms are employed by the dap-tive antenna. However, many wideband adaptive antenna al-gorithms employ FIT processing, which is also commonlyused in an OFDM demodulator to produce the narrowbandsubcarriers of OFDM. Therefore, a natural way to mergeadaptive antennas with OFDM is to employ narrowband adaptive array techniques on the demodulated subcarriers in theOFDM receiver.In a pilot-symbol-assisted OFDM system, reference-signal-based adaptive array algorithms [1,2]are appropriate forinterference suppression and equalization. Algorithms of thistype operate on the pilot symbols transmitted by the desireduser to produce a weight vector that attempts to minimize themean square error between the array output and the knownpilot sequence. For best performance, these algorithms r equire the channel and signal characteristics to be relatively

constant over the interval in which the weights are compuand applied to the array data. Algorithms which cannot trachannel variations suffer significant degradation in the error rate (BER) and signal-to-interference-plus-noisera(SINR).In an OFDM system employing an adaptive antenna algrithm on the baseband subcarriers produced by the FFTmodulator, it is difficult to guarantee that the channel will constant over the intervals in which the weights are compuand applied to the array data. Although each OFDM subcrier can be assumed to be flat faded, the presence of delspread on the channel can cause significant decorrelationthe fading processes on different subcarriers within a timfrequency slot. Even in a fixed wireless access system, teporal variations in the channel can and will occur due to motion of any surrounding objects in the system. Furthmore, the Sample Matrix Inversion (SMI) algorithm [l-for example, generally requires a number of training symbequal to at least twice the number of antenna elements to otain an average SINR within 3dB of the theoretical optimuvalue [4] . A large array therefore requires long trainingquences, and the typically low symbol rate of OFDM wreduce the likelihood of a static channel.In this paper, we explore the combination of adaptive atennas and OFDM for a high data rate system operating ifaded multipath environment. We show the performanceseveral strategies for deploying the SMI algorithm withinOFDM time-frequency slot. These strategies are designed different levels of trachng capability within a time-frequenslot. We also propose a variation on the SMI concept tprovides improved tracking performance over the other stragies being considered.

II. ANTENNA ARRAY MODELING IN OFDMWe consider an M element adaptive antenna receiving

user signals over a multicarrier communication link suchOFDM. In an OFDM system, the adaptive antenna operaon the output of the FFT demodulator as shown in FigureIn a L-subcanier OFDM system, the demodulation process one OFDM symbol interval on one receive antenna will pduce a length L vector of received data symbols, where eareceived data symbol corresponds to one of the L subcarriof the OFDM system. The channel bandwidth in OFDMtypically divided into time-frequency slots each having a scific number of symbol intervals (time duration) and an artrary number of subcarriers. Let K denote the total numbersymbols contained within a time-frequency slot. Of thesesymbols, some number P are assumed to be known pisymbols. The remaining K-P symbols are informatibearing data symbols. Figure 2 shows two example timfrequency slots, where the first is 3 subcarriers by 30 symb

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(K=90,P=32), and the second is 4 ubcarriers by 30 symbols(K= 20, P=32).We consider an uplink scenario where N users simultane-

ously transmit on the same time-frequency slot to an array ata base station. Each user ]transmitswith a single antenna, andthe receive array must separate one of these signals (the desired signal) from the other signals received by the array.Furthermore, we assume the wavelength of the RF carrier issuch that the propagation-delay-induced inter-element phaseshift experienced by an incident signal is essentially identicalon different subcarriers within the time-frequency slot.Let x,(k), m=1,2, ..,M , denote the complex baseband sig-nal sample received on the mth antenna element at the kthtime-frequency symbol within the time-frequency slot. Thebaseband samples on each antenna element at the kth symbolinstant are multiplied by a complex weight (magnitude andphase) and then summed to form the array output y ( k ) , asshown in Figure 1. Let the signal vector x(k) be the M-element column vector of antenna samples at the kth symbolinstant. Let w(k) denote the vector of complex weights ap-plied to the antenna samples at the kth symbol instant. Theoutput of the array at the kth symbol can then be written asthe vector inner-product between the weight vector and thesignal vector:where the superscriptH denotes the Hermitian transpose.

The away response vecto r for the nthuser at symbol k isdefined to be an M-element column vector a#) containingthe complex channel gains between a users antenna and theM receive antennas on the base array at symbol k. The chan-nel gains a,,@) Contained in the array response vector areassumed to take into account the large-scale path loss, thelog-normal fading, and the fast multipath fading that occursbetween the nth user and the m th antenna of the base array.The first two of these propagation effects vary rather slowlyand act to specify the user signals average incident signal-to-noise-ratio (SNR) per element. The fast multipath fadingdepends on the Doppler, delay, and angular spread of the inci-dent multipath and can cause significant variation in an m yresponse vector within a lime-frequency slot. In the multi-path propagation environmLent being considered, the differencebetween the time delays of the shortest and longest multipathcomponents is assumed to be much less than the cyclic prefixof the OFDM symbol. Hence, no intersymbol interference(ISI) or interchannel interference (ICI) is present on the demodulated subcarriers. Tlne array output is then given by

y a C ) = w aC)xaC). (1)

Ny k ) = C w H k ) a i k ) s i k ) + w xC ~ U C ) 2)i=1where n(k) is a vector containing the noise signals on the

antenna elements at time:-frequency symbol k , and s ,(k),n=1,2,...,N , be the baseband signal transmitted by the nthuserat symbol instant k.A typical weight-selection criteria is to maximize the ratioof the desired signal power to the combined power of the in-terference plus noise at the: output of the array. This ratio iscalled the signal-to-interference-plus-noise ratio, or SINR.The instantaneous SINR is defined to be the expected SINRwhen conditioned on the array response vectors and the signaland noise statistics. The instantaneous SINR at symbol kcan be shown to be:

where we define the falllowing for the case where signal 1 isthe desired signal:0 @ ( ) =a, ( )a: (C ) is the desired spatial covariancedmatrix at symbol k ,0 k)= ai k ) a r k ) + o 2 r s the undesired spa-

i =2Utial covariance matrix at symbol k . , and 6 s the per-element noise variance.

The formula for SIllrlR(k) is a Rayleigh Quotient, and themaximum value of SINR(k) is achieved with an eigenvectorassociated with the largest solution h to the following gener-alized eigenvalue problem:(D Uc:1w & ) = A @ U k ) w Uc). (4)III. THE SAMPLE MATRIX INVERSION ALGORITHM

IN OFDMWe consider a class of algorithms based on the well-

known technique of sample matrix inversion (SMI) [I-41.We apply these algorithms to the data symbols received onthe demodulated subcarriersproduced by the FFT demodulatoof an OFDM receiver. The basic SMI formula is given by:

w =@;lsl ( 5 )whereax A2x a(: )x Qc) is the sample covariance ma-p k=ltrix.S, =-

0

l P x aC )I-* aC ) is the steering vecto r.X ( k ) is a M-element column vector of baseband I+Qantenna samples received at time-frequency symbol k(i.e., received after the FFTdemodulator in the M OFDMreceivers).r(k) is the known pilot or reference symbol transmittedby the desired signal at time-frequency symbol k.The covariance matrix and steering vector are computedwith P pilot symbols in the time-frequency slot.

It is important to note that the SMI formula requiresknowledge of the desired signal pilot sequence, but does notrequire any explicit knowledge of the interference. The co-variance matrix is simply an average of the outer product ofthe total array signal vectors. The steering vector is an esti-mate of the desired users array response vector and is propor-tional to the well-known maximal ratio combining weightvector. The SMI weights basically suppress signal compo-nents contained in the covariance matrix that do not match thesignal component contained in the steering vector. Unfortu-nately, if the channel varies significantly over the time-frequency slot, the performance of this method will degrade.The question for implementing SMI becomes one of selectingthe methods for compu#ting he sample covariance matrix andthe steering vector throughout the time-frequency slot.

k=l

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The channel is assumed to be a low-mobility delay-spreadchannel, and we operate under the assumption that the channelchanges rather slowly over time, but significantly over fie-quency. In this type of system, a reasonable strategy is todeploy a sequence of pilot symbols on each subcarrier andcompute a separate SMI weight vector for each subcarrier. Asingle weight vector can be used on a subcarrier if the channelis fairly constant over the time duration of the slot. How-ever, the ability to deploy some subcarriers without pilotsequences would provide a significant savings in overhead.

This paper considers several different methods of imple-menting SMI within a time-frequency slot in which some ofthe subcarriers are deployed without pilot sequences. Thesevariations on the basic SM I formula arise depending on themethods used to compute the steering vector and the samplecovariance matrix. The algorithms that are listed below resultfrom mixing several methods for computing the steering vec-tor with several methods for computing the covariance ma-trix. In item 5 listed below, we propose a method for com-puting the covariance matrix that will be shown in the nextsection to provide better performance than the other variationsunder consideration.1 . Fixed 0 Fixed S: This version computes a singlecombining weight vector for the entire time-frequency

slot by computing (5) using all the pilot symbols in theentire slot. The assumptions behind this method are thatthe channel and interference is constant over an entiretime-frequency slot and that no attempt needs to be madeto track any channel variations within a slot.Fixed Q, - Carrier S: In this method, a single SMIweight vector is computed for each subcarrier in the slot.The weight vector for each subcarrier is computed withthe same sample covariance matrix used in the Fixed 0- Fixed S method. However, the steering vector iscomputed differently on each subcarrier. On subcarrierscontaining pilots, the sample steering vector is computeddirectly from the pilot symbols on that subcarrier. Onsubcarriers without pilots, a curve-fitting interpolationalgorithm is used to compute the entries of the steeringvector from the corresponding entries of the steering vec-tors that were computed on the subcarriers with pilots.In this study, a spline curve-fitting function was used forthis purpose, although many other alternatives are possi-ble. The motivation behind this algorithm is to see ifthe performance of the traditional SMI approach (Fixed CP- Fixed S) can be further improved by letting the steeringvector track the desired signal on a subcarrier by subcar-rier basis while leaving the covariance matrix unchangedthroughout the slot.3. Fixed Q, - Symbol S : This method computes aweight vector according to (5) for each symbol within thetime-frequency slot. The steering vector is computed ateach symbol according to a method which employs aminimum mean square error estimator of the steeringvector that assumes we have knowledge of the fading sta-tistics and the pilot sequence of the desired user. For theentire slot, a single sample covariance matrix is com-puted with a summation over all pilot symbols withinthe slot. The motivation behind this algorithm is to seeif the performance of the traditional SMI approach (FixedQ, - Fixed S) can be further improved by letting the

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steering vector track the desired signal on a symbol-bsymbol basis while leaving the covariance matrix tsame throughout the slot.Carrier Q, - Carrier S: In this method, a single SMweight vector is computed for each subcarrier. On sucarriers with pilots, the pilot symbols are used in tsummations for both the sample steering vector and tsample covariance matrix in (5) . On subcarriers withopilots, all data received on that subcarrier is used in thsummation for the covariance matrix, while the curvfitting interpolation approach from method 2 above used to compute the steering vector. The motivation bhind this algorithm is to see how well SMI performwhen applied on a subcarrier-by-subcarrier basis, espcially when some subcarriers do not contain pilot symbols. As will be shown below, this method performmuch worse than the previous three methods and providthe motivation for the next method.5. Interpolated Q, - Carrier S : This method is oproposed modification to SMI that incorporates pilot iterpolation techniques into the computation of the sample covariance matrix on subcarriers that do not contapilot symbols. In this method, the weight vectors subcarriers with pilots are identical to those used method 4 above. However, on subcarriers without plots, both the covariance matrix and the steering vecare computed by interpolating between their values computed on the subcarriers with pilots. The motivation bhind this algorithm is the rather poor performance of tCarrier Q, - Carrier S method on subcarriers thatnot have pilots, as will be shown below.

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IV. PERFORMANCEThe performance of the algorithms described above wevaluated with a four-element linear antenna array receivi

three user signals of equal average power. Figure 3 plots tuncoded BER of the desired signal (signal 1) versus the aveage received signal-to-noise atio (SNR) of all signals for tfive algorithms described above. Also plotted in Figure 3the BER that results when the array uses the Optimal Tracing weights which maximize the instantaneous SINR (3 )each symbol assuming perfect knowledge of the channIndependent fading on the antennas is assumed which meathe apparent directionality of an incident signal is not locized in the long-run to any particular region in space arouthe array. In these examples, the Doppler power spectrum feach user signal is assumed to be flat with a 15 Hz maximuDoppler frequency, which corresponds to a speed of 5 mph a 2 GHz RF carrier frequency. The power-delay profile feach user signal is assumed to be flat with a total width of p e c . The array operates with the 3 subcarrier by 30 symbOFDM slot structure shown in Figure 2. The OFDM symbol rate is 10 Ksymbols per second and the subcarrier spaciis 12.5 KHz. The modulation is QPSK, and the pilot squences of the users are uncorrelated QPSK sequences thvary from slot to slot. At each SNR value on these curve5000 independently faded time-frequency slots were simulateSince all incident signals in these examples have the samSNR per element, the incident desired signal-to-interferen

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ratio per element is equal to a constant -3 dB throughout allcurves presented below.

In Figure 3, the Fixed @ - Fixed S algorithm providesbetter performance than the other two methods that use a sin-gle fixed covariance matrix in the slot (the Fixed Q, - Sym-bol S and the Fixed @ - Carrier S algorithms). This re-sult is rather interesting because the Fixed @ - Fixed Salgorithm does not track the desired signal steering vectorwithin the slot. At first glance, one would think that if thecovariance matrix was fixed, then allowing the steering vectorto track the desired signal might improve the BER perform-ance. Figure 3 shows that if a fixed covariance matrix isused, then the performance degrades if an attempt is made totrack the steering vector within the slot, either on a subcar-rier-by-subcarrier basis (Fixed Q, - Carrier S) or on a sym-bol-by-symbol basis (Fixed Q, - Symbol S). On the otherhand, the Fixed @ - CarrierS algorithm does slightly worsethan the Fixed @ - Symbol S algorithm. This particularresult seems reasonable because the Fixed @ - Carrier Salgorithm does less tracking of the desired signal than theFixed Q, - Symbol S algorithm.

These results show that allowing the steering vector totrack while fixing the covariance matrix is not always thebest strategy. What is likely happening when we fix the co-variance matrix is that the SMI weights are unable to trackthe changes in the interference that occur over the subcarriers.With temporal and frequency variations in the interferencearray response vectors, the directionality of the interferenceis changing across the symbols of the time-frequency slot. Afixed covariance matrix is tracking the average spatial char-acteristics of the desired and interference signals, which is notacceptable in an environment where the antennas are inde-pendently faded and the interference is high-powered. Whenthe apparent directionality of the interference changes acrossthe slot, the weights must be adapted so that the interferenceis tracked in addition to the (desiredsignal.One particularly surprisiing result from Figure 3 is that theBER performance of the Carrier Q, - Carrier S algorithm issignificantly worse than any of the other methods considered.To explain this result, Figure 4 shows a single simulatedsnapshot of the instantaneous expected SINR (3) for the desired signal as a function of the received symbols within the 4by 30 time-frequency slot alf Figure 2. In this slot, the sym-bols are numbered first across time and then across subcarriersas shown in Figure 2. Symbols 1 through 30 are on subcar-rier 1; symbols 31 through 60 are on subcarrier 2, and so on.Given the large number of pilot symbols on subcarriers 1 and4, the BER performance achieved on subcarriers 2 and 3 willdominate the overall BER performance of an algorithm.As shown in Figure 4, the SINR of the Carrier Q, - Car-rier S algorithm on subcarriers2 and 3 (which do not con-tain pilots) is significantly worse than the SINR on subcarri-ers 1 and 4 (which do contain pilots). The reason for thisdegradation in the SINR can be understood from the adaptiveantenna literature. For example, Monzingo and Miller [2]show how the SINR performance can be degmded when thecovariance matrix and the steering vector are computed in amanner different from (5). A perfect steering vector and anoisy covariance matnx can be shown to perform signifi-cantly worse than (5) for the same number of samples of arraydata. However, when the covariance matrix and steering vec-

tor are both noisy, but are both computed from the same data,better suppression of the interference can be achieved. Theseresults explain what is happening with the Carrier@ - Car-rier S algorithm on the different subcarriers in Figure 4. Onsubcarriers 1 and 4, the array is using only pilot symbols tocompute the SMI weights. However, on subcarriers 2 and 3,the covariance matrix and the steering vector are not computedfrom the same array dlata. Therefore, we can expect theCarrierQ, - Carrier S algorithm to perform poorly on sub-carriers that do not contain pilots. This discussion also ex-plains why tracking the steering vector while fixing the co-variance matrix leads to worse performance than fixing bothcomponents.

These results in Figures 3 and 4 provided the motivationbehind the Interp Q, - Carrier S algorithm. On subcarrierscontaining pilots, this algorithm is identical to the Carrier@- Carrier S algorithm. On subcarriers that do not containpilots, this algorithm produces steering vectors identical tothose produced by the Carrier @ - Carrier S algorithm.However, computing tlhe covariance matrix on subcarrierswithout pilots is performed differently in the Interp Q, - Car-rier s lgorithm than in the Carrier @ - Carrier S algo-rithm. Instead of computing a covariance matrix from thedata received on subcarriers without pilots, the Interp Q, -CarrierS lgorithm computes a covariance matrix that is aninterpolation between tlhe sample covariance matrices com-puted on the subcarriers with pilots. This interpolation isperformed exactly according to the interpolation carried out forthe steering vectors. As a result, on subcarriers without pi-lots, the covariance matrix and the steering vectors in thisalgorithm are computed from the same data, unlike theCarrier Q, - Carrier S algorithm. The Interp Q, - CarrierS algorithm has a slight drop in SINR on subcarriers that donot contain pilots, but the drop is much less pronounced thanwas seen with the Carrier0 Carrier S algorithm.

Another point worth mentioning about Figure 3 is thatthe BER of all algorithims seems to level out after the SNRreaches about 20 dB. This result indicates that there is someresidual interference power that is not being suppressed evenat the high SNR levels. The system remains somewhat inter-ference-limited at the hi,gh SNRs. Two strategies can be con-sidered to improve the BER performance in this scenario.The first is to shorten the duration of the time-frequency slotso that less channel variation occurs across time. The tempo-ral channel variation across 30 symbols in time is likelycausing the BER floor in Figure 3. The second strategy is toemploy an array with more antenna elements to provide&-tional degrees of freedom for further interference suppressionand tracking. For example, if the simulations of Figure 3 arere-run with six elements instead of four, the BER floorachieved with the Interp@ - Carrier S algorithm drops fromroughly to about The additional degrees of freedomallow a fixed weight vector to place multiple nulls along thetrajectory of an interferers faded array response vectorwithin a slot.

V. CONCLUSIONIn this paper, we explored the performance of adaptive an-

tenna algorithms based on the well-known Sample MatrixInversion (SMI) algorithm for operation in a faded delay-

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spread environment. An OFDM system was considered inwhich the time-frequency slot contained pilot symbols thatpermitted the SMI calculation to be carried out for interferencesuppression and equalization of the desired signal. Severalvariations on the SMI concept were considered. An interest-ing result is that when the covariance matrix is fixed for theentire slot, tracking the steering vector appeared to provide noimprovement over a fixed steering vector computed from thesame data used in the covariance matrix. In fact, a slight deg-radation in performance occurs, a result that fits in well withprevious literature on the SMI algorithm. A new variationon the sample matrix inversion algorithm was proposed topermit better tracking of both the desired and interference sig-nals within the time-frequency slot. This method extendedthe concept of pilot-symbol interpolation to the spatial co-variance matrix of adaptive array processing. It was shownthat this technique provides improved SINR and BER per-formance over the other variations of SMI considered in thisinvestigation.

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REFERENCES

InstantaneousSINR vs Symbol Number- - . I

- 1. . ---_ _L - _ _- . - - - - -----.,-._-.I. .

- 1 _ _ _ - - --

Carrier @-Carrier Slnterp @-Carrier SFixed +Fixed S

-

20 40 60 80 100

[l] R. T. Compton Jr., Adaptive Antennas: Concepts andPe$ormance, Englewood Cliffs, NJ: Prentice Hall, 1988.[Z] R. A. Monzingo and T. W. Miller, Introduction to Adap-live Array s, New York: Wiley, 1980.[3] I. S . Reed, J. D. Mallett, L. E. Brennan, Rapid Conver-

gence Rate in Adaptive Arrays, IEEE Transactions onAerospace and Electronic System s, Vol. 10, No. 6 , No-vember 1974

[4] J. Winters, Optimum Combining in Digital MobileRadio with Cochannel Interference, lEEE Transactionson Vehicular Technology, Vol. 33, No. 3, August 1984

[SI S. B. Weinstein and P. M. Ebert, Data Transmission byFrequency Division Multiplexing Using the DiscreteFourier Transform, IEEE Transactions on Communica-tions, Vol. 19, October 1971.

[6] J. A. C. Bingham, Multicarrier Modulation for DataTransmission: An Idea Whose Time Has Come, IEEECommunications Magazine, Vol. 2 8 , pages 5-14, May1990.

v v vDemodulator Demodulator Demodulator9y@ )FIG. An M-Element Adaptive Antenna Array for OFDM.

Time (symbol Intervals)I%$1FIG. 2: Two example time-frequency slots: 3x30 and 4x30

SMI BER for s i g ~ l l1oo

t

L

io- - ptimal-Tracking. _ _ _ Fixed @-Symbo l SFi xed @-Carrier SCarrier @-Carrier So o lnterp @-Carrier S+ + Fixed @-Fixed S 1

5 10 15 20 25SNR of All Signals (dB)FIG. 3: Sample Matrix Inversion in a 3 subcarrier by 3symbol time-frequency slot. Uncoded BER is plotted versthe average SNR per antenna of all incident signals. A foelement linear antenna array receives 3 signals of equal aveage power with all having independent fading on the antenna

FIG. 4: Instantaneous SINR with a 4 element antenna arrreceiving 3 signals, each with a 20 dB SNR, and each signhaving independent fading on the antenna elements. Ttime-frequency slot is 4 subcarriers by 30 symbols.

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Adaptive Antennas for OFDM

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