Blind channel identification and equalization in OFDM ... · 96 IEEE TRANSACTIONS ON SIGNAL...

96 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 1, JANUARY 2002

Blind Channel Identification and Equalization inOFDM-Based Multiantenna Systems

Helmut Bölcskei, Member, IEEE, Robert W. Heath, Jr., Student Member, IEEE, andArogyaswami J. Paulraj, Fellow, IEEE

Abstract—Wireless systems employing multiple antennas atthe transmitter and the receiver have recently been shown to havethe potential of achieving extraordinary bit rates. Orthogonalfrequency division multiplexing (OFDM) significantly reducesreceiver complexity in multiantenna broadband systems. In thispaper, we introduce an algorithm for blind channel identificationand equalization in OFDM-based multiantenna systems. Ourapproach uses second-order cyclostationary statistics, employsantenna precoding, and yields unique channel estimates (up toa phase rotation for each transmit antenna). Furthermore, itrequires only an upper bound on the channel order, it does not im-pose restrictions on channel zeros, and it exhibits low sensitivity tostationary noise. We present simulation results demonstrating thechannel estimator and the corresponding multichannel equalizerperformance.

Index Terms—Blind equalization, cyclostationarity, MIMO,multiantenna systems, OFDM.

I. INTRODUCTION

DEPLOYING multiple antennas at both the transmitter andthe receiver of a wireless system has recently been shown

to yield extraordinary bit rates [1]–[5]. The correspondingtechnology, known as spatial multiplexing [1] or BLAST [2],[6], allows an impressive increase in data rate in a wirelessradio link without additional power or bandwidth consumption.Orthogonal frequency division multiplexing (OFDM) [7]–[9]significantly reduces receiver complexity in wireless broadbandsystems and has recently been proposed for use in wirelessbroadband multiantenna systems [4], [5], [10]. In practice,in order to get the promised increase in data rate, accuratechannel state information is required in the receiver. Thisinformation can be obtained by sending training data and esti-mating the channel [10]–[12]. The training overhead required,unfortunately, is more significant in estimating multiple-inputmultiple-output (MIMO) channels. To avoid this problem,we propose an algorithm for blind channel identification andequalization in OFDM-based MIMO systems.

Manuscript received January 26, 2000; revised October 11, 2001. This workwas supported in part by FWF under Grants J1629-TEC and J1868-TEC and byfunding from Ericsson Inc. through the Networking Research Center at Stan-ford University. The associate editor coordinating the review of this paper andapproving it for publication was Prof. Lang Tong.H. Bölcskei is with the Coordinated Science Laboratory and Department of

Electrical Engineering, University of Illinois at Urbana-Champaign, Urbana, IL61801 USA (e-mail: [email protected]).R. W. Heath, Jr. and A. J. Paulraj are with the Information Systems Labora-

tory, Department of Electrical Engineering, Stanford University, Stanford, CA94305 USA.Publisher Item Identifier S 1053-587X(02)00414-2.

Blind MIMO Channel Estimation: Blind identification andequalization of MIMO channels has been a very active area ofresearch during the past few years. Due to lack of space, wewill not summarize all existing ideas and algorithms; rather,we refer the interested reader to [13], which contains an ex-cellent overview of the subject and an extensive reference listuntil 1996. More recent references can be found, for example,in [14]–[17]. To the best of our knowledge, previous work onblind MIMO channel estimation was restricted to single-car-rier systems. The use of cyclostationary statistics to accomplishblind MIMO channel estimation has first been proposed for fre-quency-flat fading channels in [18] and [19]. More recently, theuse of conjugate cyclostationary statistics in combination withconstant-modulus antenna precoding has been suggested in [20]to accomplish blindMIMO channel estimation in the single-car-rier case. We finally note that since we are dealing with OFDMsignals, which, in practice, have a high number of subcarriers(typically between 512 and 8192), the transmit signals will beGaussian. This makes the application of source separation ap-proaches relying on the assumption that the different sourceshave different distribution functions and using higher order sta-tistics difficult (see e.g., [21] and references therein).Contributions: In this paper, using a periodic nonconstant-

modulus precoding scheme, we introduce an algorithm for theblind identification and equalization of OFDM-based MIMOsystems. Our method uses second-order cyclostationary statis-tics and identifies the matrix channel on a subchannel by sub-channel basis, i.e., each scalar subchannel is identified individ-ually. Important aspects of the proposed algorithm include thefollowing.• It requires only an upper bound on the channel order.• It does not impose restrictions on channel zeros.• It exhibits low sensitivity to stationary noise.

Our equalization method recovers the transmitted symbolstreams up to a phase rotation (which will in general be dif-ferent for different symbol streams). This remaining ambiguitycan be resolved using short training sequences.Relation to Previous Work: Blind MIMO channel identifi-

cation algorithms based on second-order statistics are, in gen-eral, able to identify the MIMO channel up to a unitary mixingmatrix only and suffer from common zeros problems. Our ap-proach overcomes both of these drawbacks and is an extensionof an idea first proposed by the authors for the single-carriercase in [22]. The algorithm introduced in [18] and [19] is re-stricted to single-carrier modulation and frequency-flat fading(i.e., no delay spread). Besides applying to OFDM, our algo-

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BOLCSKEI et al. BLIND CHANNEL IDENTIFICATION AND EQUALIZATION 97

Fig. 1. Baseband OFDM transceiver. (a) Modulator. (b) Demodulator.

rithm differs from that suggested in [20] in that we use non-constant-modulus precoding instead of constant-modulus pre-coding and cyclostationarity instead of conjugate cyclostation-arity. Moreover, our approach does not impose restrictions onchannel zeros, it applies to arbitrary symbol constellations andarbitrary stationary noise, the phase ambiguity is resolved upto a diagonal matrix of phase terms, and the performance ofour estimator does not degrade significantly if the number ofsources (number of transmit antennas) increases. A determin-istic method to estimate the channel of a space-time precodedMIMO-OFDM system exploiting the constant-modulus prop-erty of the transmitted symbols has been reported in [23].Organization of the Paper: The rest of this paper is organized

as follows. In Section II, we briefly describe broadband OFDM-based spatial multiplexing systems, and we state our assump-tions and introduce some notation. In Section III, we presentthe novel algorithm by considering the case of two transmit andtwo receive antennas. In Section IV, we describe the general-ization to an arbitrary number of transmit and receive antennas,and we discuss the design of the precoding sequences. Section Vcontains simulation results demonstrating the estimator and thecorresponding equalizer performance. Finally, Section VI pro-vides our conclusions.

II. OFDM-BASED SPATIAL MULTIPLEXING SYSTEMS

In this section, we briefly review OFDM-based spatial mul-tiplexing, and we provide some preparation for the rest of thepaper. In the following, and denote the number oftransmit and receive antennas, respectively.OFDM-Based Spatial Multiplexing: An apparent disadvan-

tage of single-carrier based spatial multiplexing systems is thefact that the computational complexity of the receiver (either avector-MLSE or a multichannel equalizer) will, in general, bevery high. The use of OFDM [7]–[9] alleviates this problem byturning the frequency-selectiveMIMO channel into a set of par-allel narrowbandMIMOchannels. Thismakes equalization verysimple. In fact, only a constant matrix has to be inverted for eachOFDM tone [4], [5], [10].In an OFDM-based spatial multiplexing system, the in-

dividual data streams are first OFDM-modulated and thentransmitted simultaneously from the antennas. Fig. 1illustrates a baseband discrete-time single antenna OFDMtransceiver. The modulator applies an -point IFFT todata symbols and prepends the cyclic prefix (CP) [7] of

Fig. 2. OFDM-based spatial multiplexing system. (OMOD and ODEMODdenote an OFDM-modulator and demodulator, respectively.)

length , which is a copy of the last samples of the IFFToutput. The overall OFDM symbol length is, therefore, givenby . Throughout this paper, we assume that

, which is usually satisfied in practice. The CP actsas a guard space between consecutive OFDM symbols andavoids intersymbol-interference (ISI) if the channel impulseresponse length is less than or equal to the length of the CP.In the receiver, the CP is first removed, and then, an -pointFFT is applied. The signals received by the individual antennasare first passed through OFDM demodulators, separated, anddemultiplexed (and potentially decoded). Fig. 2 shows anOFDM-based spatial multiplexing system.Let us next introduce some notation and some basic results

needed later in the paper. In the following, the transmitted signalcorresponding to the th antenna is givenby

where rect with

rect else.

Furthermore, denotes the (complex) data symbol trans-mitted on the th tone in the th OFDM symbol from the thantenna.Now, denote the continuous-time impulse response between

the th transmitter and the th receiver as, where and are the transmit



and the receive pulse shaping filters, is the impulse re-sponse of the propagation channel, and is the convolution op-erator. Taking samples at a rate of , where isthe subcarrier bandwidth, we obtain the discrete-time impulseresponse .The signal received at the th antenna can now be written as

(1)

where is stationary additive (po-tentially colored) noise observed at the th receive antenna.Using the notation

we can rewrite the input–output relation (1) in vector-matrixform as

where. In the following, we assume that the delay

spread of the matrix channel is taps, i.e.,

Denoting the reconstructed data symbol corresponding toas and organizing the data symbols into frequency vectorsaccording to and

,assuming perfect synchronization, it can be shown that

(2)

where is stationary additive Gaussian noise. Thanks to (2),the convolutive mixtures observed at the receive antennas canbe separated and equalized by computing

(3)

The existence of a zero-forcing (ZF) equalizer assumes that thematrices have full rank.Note that usually, in OFDM systems, forward error correction(FEC) spreads out the information bits across frequency. There-fore, if one or several of the are rank-deficientor ill-conditioned, the corresponding data vectors will not bedecoded. Rather, the receiver will attempt to make use of theFEC in the system and extract this information from the datavectors corresponding to the “well-conditioned tones.” An

alternative to ZF equalization is to use a multichannel MMSEequalizer, which combats the problem of rank-deficiency of

at the cost of degraded signal separation perfor-mance. In practice, the receiver estimates theusing pilot tones inserted in the transmitted data vectors [4],[10]. In this paper, we will be concerned with the blind esti-mation of the matrices .Note, however, that a direct estimation of thewould be very inefficient since it requires the estimation of

parameters, which can be a significant number inpractical systems. Rather, we will provide an algorithm forblindly estimating the channel impulse response matrix taps

from which we can compute estimatesof the using the FFT.This approach requires the estimation of parameters.We emphasize that our algorithm requires knowledge of anupper bound on the channel order for appropriate selection ofthe CP length.

III. BLIND CHANNEL IDENTIFICATION ALGORITHM

In this section, we will introduce the novel channel identifica-tion algorithm, andwe demonstrate the basic idea using a simpleexample with two transmit and two receive antennas. The gen-eralization to an arbitrary number of antennas will be discussedin Section IV.

A. PreparationThe basic idea of our algorithm is to perform periodic non-

constant-modulus precoding in the transmitter such that the cy-clostationary statistics allow a separate identification of the indi-vidual scalar subchannels

. This is achieved by providing each transmitantenna with a different signature in the cyclostationary domainwith the signatures chosen such that for a given cycle, all butone transmit antennas are nulled out. It is therefore possible toidentify the matrix channel on a column-by-column basis up toa constant diagonal matrix of phase rotations. After equaliza-tion according to (3), the individual symbol streams will be de-coded up to a phase rotation, which will, in general, be differentfor different symbol streams. This remaining ambiguity can beresolved using short training sequences. Alternatively, if differ-ential detection is employed, the phase rotation can be ignored.We note that periodic precoding serves to separate the MIMOchannel identification problem into scalar problems, and the re-dundancy introduced by the CP is used to blindly identify thescalar subchannels.Periodic Precoding: Our approach is based on periodic

nonconstant-modulus precoding, which consists of multiplyingthe individual data streams by -periodic precoding sequencesprior to transmission. The precoding sequences have to bedifferent for different transmit antennas. More specifically,the precoded transmit signal corresponding to the th transmitantenna is given by

(4)



where is the th -periodic precoding sequence. Inorder to keep the average transmit SNR constant, the precodingsequences have to be normalized such that

. From (4), it follows that all the (com-plex) data symbols in the th OFDM symbol transmitted fromthe th antenna are multiplied by before the IFFT is ap-plied. Equivalently, this multiplication can be performed on thetime-domain samples after the IFFT and parallel-to-serial con-version. We note that this form of periodic precoding has previ-ously been suggested by Serpedin and Giannakis in [24] to in-troduce cyclostationarity in the transmit signal, thereby makingblind channel identification based on second-order statistics insymbol-rate sampled single-carrier systems possible. The moregeneral idea of transmitter-induced cyclostationarity has beensuggested previously in [25] and [26].We emphasize that nonconstant-modulus periodic precoding

will incur a loss in spectral efficiency. The reason for this is thatfor a given average transmit power, the minimum distance be-tween symbols decreases, and hence, in order to achieve a givenbit error rate, a higher average transmit power is required. There-fore, even though the transmission rate is kept constant, there isa loss in spectral efficiency. In this sense, it is desirable to keepthe amplitude variation in the precoding sequences as small aspossible. Finally, because of the time-varying transmit SNR, thepower amplifiers in the system need a higher dynamic range,which increases hardware complexity. The impact of noncon-stant-modulus precoding on bit error rate (BER) performance ofmultiantennaOFDMsystems is further investigated bymeans ofcomputer simulations in Section V.Cyclostationary Statistics: Now, defining the cor-

relation matrix of the vector random process as1

and assuming that the data sequences are statistically inde-pendent of the noise and satisfy

it is shown in the Appendix that

(5)

where , and

diag (recall thatthe different transmit signals were assumed to be uncorrelated).The scalar correlation functions are given by

rect

rect

rectelse .

1 stands for the expectation operator, and the superscript denotes conju-gate transposition.

Now, using the -periodicity of the precoding sequences ,it can be verified that for

. Consequently, we have

(6)

which shows that is a cyclostationary vector randomprocess with period . By cyclostationary vector randomprocess with period , we mean that each of the entries inthe vector is a scalar cyclostationary random process withcyclostationarity period . Using (6), it follows from (5) that

, and hence, is a cyclostationaryvector random process with period as well.Since is -periodic in , we can expand it into a

Fourier series with respect to with the Fourier series coeffi-cient matrices given by

Next, applying a -transform with respect to , we obtain thecyclic power spectral matrices

which are shown in the Appendix to be given by

(7)

where , , and

diag

In the Appendix, it is shown that the Fourier series coefficientmatrices are given by

diag

diag (8)

where

(9)



and

(10)

B. Basic Identification AlgorithmWe will next explain our algorithm using a simple example

with two transmit and two receive antennas. Considering the 22 case simplifies the presentation and conveys the basic ideas

underlying the algorithm. The more general case with an arbi-trary number of transmit and receive antennas will be discussedin Section IV.We use the normalized four-periodic precoding se-

quences and. From (9), it fol-

lows that , and.

In the following, in order to keep the discussion more general,we will stick to the notation for the period of the precodingsequences instead of specializing to . From (7), we obtain

(11)

(12)

where , and . The basic idea of ourchannel identification algorithm now is to find cyclesand such that

and (13)

Now, since , it follows thatif and only if .

From (8), it follows that the differ only in the .Consequently, (13) can be satisfied by picking and suchthat

and (14)

Clearly, setting and satisfies (14). In the fol-lowing, in order to keep the discussion more general, we will

stick to the notation and for the cycles. Specializing (11)and (12) and using , we obtain

(15)

(16)

(17)

(18)Now, we can use a modified version of an algorithm first pro-posed by Tong et al. in [27] and later extended by Serpedin andGiannakis [24] to identify the scalar subchannels and

from and , respectively,and the subchannels and fromand , respectively. By proper design of the pe-riodic precoding sequences, we have broken up the 2 2 ma-trix channel identification problem into the identification of fourscalar subchannels.The algorithm we are using in the following to identify the

individual scalar subchannels has been used previously in CPOFDM systems [28] and in pulse shaping OFDM systems [29],[30].Wewill therefore keep the presentation of the identificationalgorithm short. Furthermore, we restrict the discussion to theidentification of . The remaining subchannel filters canbe identifiedusing the sameprocedure. Starting from(17),weget

(19)Straightforward manipulations show that (19) can be rewrittenas

(20)where

(21)

and is the length of the impulse response of the subchannelfilter . When the exact value of is not known, itcan be shown from [24] that our algorithm is not sensitive tochannel-order overestimation (and likewise fails with channelorder underestimation). Since OFDM systems require an upperbound on the channel order to correctly choose the CP length,a safe estimate of is always available and known to thereceiver. The form of the solution when the channel is over-estimated is omitted. See [24] for further discussion on thispoint. From (8) and (10), it follows that for

, and else. This implies that



Furthermore, for . From (7),it follows that the influence of stationary noise (with arbitrarycorrelation function) can be minimized by considering nonzerocycles only. In order to solve (20) for thechannel, we rewrite the equation in vector-matrix form as

(22)

with the Toeplitz matriceswith first row

and first column shown at the bottom of the page, and thediagonal matrix diag Fur-

thermore

and denotes the all-zero vector. Using(17) and (19) and [24, th. 1], it can be shown that the channel

is uniquely identifiable within a phase ambiguity (irrespec-tively of channel zeros) if and only if there is nosuch that . This implies that identifiability isguaranteed for . Since OFDM symbols arevery long in practice (i.e., is typically chosen to be at least 4times the channel length), this condition will easily be met.The channel estimate can now be found by first esti-

mating the cyclostationary statistics from a finite datarecord of length according to [19] and [31]

(23)

and then solving the following optimization problem:

(24)

where is an estimate of obtained byreplacing in (21) by the estimates .The solution of (24) is found as the eigenvector of

associated with the smallest eigen-value. The remaining subchannel filters , , and

can be identified by performing the same procedureas above for (15), (16), and (18), respectively. For a discussionof further aspects of the algorithm and an extension to themulticycle case (which potentially improves the estimatorperformance), see [27] and [28].We finally note that periodic precoding increases the cyclo-

stationarity period by a factor of , and hence, longer datarecords are required to arrive at good estimates of .Therefore, in practice, one should aim at precoding sequenceswith small . Unfortunately, to guarantee at least zero cy-cles, we need . (Recall that the cycle is not

admissible.) Larger allows for more zero cycles and, hence,makes the application of multicycle approaches possible, whichpotentially improves estimator performance.

C. Resolving the Complex Scale AmbiguityFrom (20), it can be seen that each of the scalar subchan-

nels has now been identified up to a complex constant, which,in general, will be different for different subchannels. We willnext describe a two-step procedure for resolving this remainingambiguity up to a diagonal matrix of phase rotations. Assumethat the scalar subchannel filters have been identified up to acomplex constant denoted as for the filter .Denoting as the perfect estimate of up to thescaling factor , we get from (17)

This suggests to form an estimate of as

wherewith denoting the estimate

of obtained from (24), and .The estimator performance can potentially be improved byaveraging over the interval according to

The remaining subchannel gains , , and can be es-timated by performing the same procedure as above for (15),(16), and (18), respectively. Let us next show how the phaseambiguity due to the can be reduced by considering thecross terms of . Now, denoting as the perfectestimate of including the channel gain , it followsfrom (7) that



Using , this suggests an estimation of the phasedifference as

wherewith denoting the es-

timate of , including the estimated scale factor and. Again, the estimator performance

can potentially be improved by averaging over the intervalaccording to

Further averaging can be done by considering

(25)

and performing the same procedure as above. The phase differ-ence can be estimated similarly from

and . Now, since we have estimated the channelgains and the phase differencesand , we have identified the 2 2 channel transfermatrix up to a diagonalmatrix of phase rotations. The onlyambiguities left are the or, equivalently, any ma-trix

diag

with arbitrary is a valid solution of our algorithm.Now, since

diag

we can see that equalization according to (3) recovers the indi-vidual data streams up to a phase rotation.

IV. EXTENSION OF THE ALGORITHM AND DESIGN OF THEPRECODING SEQUENCES

In this section, we describe how the new algorithm can begeneralized to an arbitrary number of transmit and receive an-tennas, and we discuss the design of the precoding sequences.

A. Arbitrary Number of AntennasChannel Identification: Starting from (7), we obtain

......

. . ....

(26)

Assume that the periodic precoding sequences have beendesigned such that for each , there is at least one

that satisfies and for. This implies that and

for . In the cyclostationary domain, this causes the thtransmit antenna to be “active” at cycle , whereas all the otherantennas are “inactive.” From (26), it then follows that

(27)The filters or, equivalently,the th column of the channel transfer matrix can nowbe identified by performing a scalar subchannel by subchannelidentification on the using the subspace-based al-gorithm discussed in Section III. This process has to be repeatedfor all columns .Resolving the Complex Scale Abiguity: Once all the scalar

subchannel filters have been identified, we can resolve the re-maining channel gain and phase ambiguities by proceeding asfollows. An estimate of can be obtained as

wherewith denoting the esti-

mate of obtained from (24) and .This procedure has to be repeated for all subchannel gains.The phase differences can be estimatedaccording to

wherewith denoting the es-

timate of including the estimated gain , and

Further averaging can be done by considering andperforming the same procedure. Now, we have identified thechannel transfer matrix up to a diagonal matrix of phaserotations, i.e., any matrix

diagwith arbitrary is also a valid solutionof our algorithm. Again, it follows that equalization accordingto (3) recovers the individual symbol streams up to a phase ro-tation.

B. Design of the Precoding SequencesWe will next discuss the design of the precoding sequences

and related tradeoffs on channel estimator performance.Simple Construction: We provide a simple construction

of periodic precoding sequences for an arbitrary number oftransmit antennas. This method shows the existence of proper



TABLE IPRECODING SEQUENCES FOR TWO, THREE, AND FOUR TRANSMIT ANTENNAS AND ZERO PATTERNS OF THE CORRESPONDING -FUNCTIONS

TABLE IIABSOLUTE VALUES FOR AND NORMALIZED PRECODING SEQUENCES

precoding sequences for any . Let bethe product of prime integers , andsuppose the squared precoding sequence for the th transmitantenna has period (and, hence, period as well). Let

denote the Fourier series coef-ficients of this squared sequence with respect to period ,and let denote the Fourier seriescoefficients of the same sequence with respect to period . Itis now easy to show that

otherwise.(28)

Thus, the th sequence has zeros in its Fourier series, exceptwhere the cycle is a multiple of . Since we have chosenthe periods to be prime, the nonzero cycles are mutually ex-clusive. Now, noting that the precoding sequences are noncon-stant-modulus, it follows that every sequence has at least onenonzero cycle with index . Hence, if cycle for sequenceis nonzero, then cycle will be zero for all the other sequences.As an example, suppose with , ,

, and hence . Antenna 1 is active for cycles, antenna 2 is active for cycles , and antenna 3

is active for cycles . Thus, we can use cycleto estimate the channels induced by the first transmit

antenna, cycles to estimate the channels induced

by the second transmit antenna, and cycles toestimate the channels induced by the third transmit antenna.While such a construction guarantees the presence of zero

cycles, it leads to large periods, especially as increases.Therefore, it is of interest to design shorter precoding sequenceswith zeros in specific prescribed locations in the spectrum. Wepresent one such construction in the following.Alternative Design Procedure: In the following, let de-

note the vectors given by

and

We then have(29)

where is the DFT matrix of size. Using2 in (29), we obtain

(30)

The design problem now consists of finding vectors withstrictly positive real elements such that the correspondinghave the desired zero pattern. Let us demonstrate how this can be2 stands for the identity matrix.



TABLE IIIABSOLUTE VALUES FOR AND NORMALIZED PRECODING SEQUENCES

TABLE IVABSOLUTE VALUES FOR AND NORMALIZED PRECODING SEQUENCES

done for the case of two transmit antennas (arbitrary number ofreceive antennas) and precoding sequences of period 4. Our goalis to design and such that and

with the being nonzero. We notethat two zeros in are not required; however, their presenceenables improved estimator performance using multicycle tech-niques (see, e.g., [24]). Therefore, we have to satisfy the fol-lowing equation:

with the elements in and being strictly posi-tive real. By inspection, one possible solution follows as

with and andwith , Re ,

and Im . This choice for the -vectors resultsin and

Re Im Re Im .Finally, the magnitude of the coefficients and can beobtained from and by taking the square root of theindividual elements. The phases assigned to the andcan be chosen arbitrarily.Tradeoffs on Estimator Performance: In the above example,

it is obvious that making and as large as possible will

improve the channel estimator performance due to better sepa-ration of the individual transmit signals’ cyclostationary statis-tics [cf. (11) and (12)]. On the other hand, in terms of overallsystem performance, it is desirable to keep the degree of am-plitude variation in the precoding coefficients as small as pos-sible in order to have the transmit SNR as constant as possibleover time (see the discussion in Section III-A). In the present ex-ample, this requirement translates to having and as smallas possible. We conclude that in general, in terms of overallsystem performance, there will be an optimum choice for theprecoding sequences, which will, however, depend on the re-maining system parameters such as the symbol constellationand the specific FEC scheme. In Section V, we will provide asimulation example (see Simulation Example 3) pertaining tothis issue. Finally, we emphasize that for a higher number oftransmit antennas, better separation properties are required inorder to avoid a degradation in estimator performance. There-fore, the choice of the precoding sequences will also depend on

.Examples of Precoding Sequences: Based on the alternative

design procedure discussed above, we designed simple pre-coding sequences with periods 4, 8, and 16 for two, three, andfour transmit antennas, respectively. These precoding sequencesare simple since they consist of two different elements andonly, where and are design parameters. The sequences

along with the zero patterns of the corresponding aresummarized in Table I. In Tables II–IV, the functions



corresponding to the normalized precoding sequences specifiedin Table I have been evaluated numerically forand 8, respectively. It is clearly seen that for bigger , betterseparation is achieved. Note that in Tables II–IV, standsfor the vector obtained by taking elementwise absolute valuesin . We conclude by noting that the minimum periodrequired using the first design approach is for ,

for , and for . Althoughthe second design procedure is less systematic, it is preferablesince it yields significantly shorter precoding sequences.

V. SIMULATION RESULTS

In this section, we provide simulation results demonstratingthe performance of the proposed matrix channel estimation al-gorithm and the corresponding multichannel equalizer. In allsimulation examples, the estimator performance was measuredin terms of the average bias given by the expression at the bottomof the page and the mean square error (MSE)

where denotes the number ofMonte Carlo trials.We simulateda system with two transmit and two receive antennas,subcarriers, and CP of length (i.e.,), and we used the length-4 precoding sequences provided

in Table I. The data symbols were i.i.d. 4-PSK symbols, andthe channel code employed was a convolutional code of rate

and constraint length 5. The signal-to-noise-ratio (SNR) forthe th data stream was defined as SNR ,where , and is the variance of the whitenoise process . Since in all simulations the SNR was chosento be the same for all data streams, in the following, we dropthe index in SNR in order to simplify the notation. Unlessspecified otherwise, all results were obtained by averaging over

independent Monte Carlo trials, where each realizationconsisted of 8256 data symbols (i.e., 516 OFDM symbols). Thesampled matrix channel we simulated is given by

(31)

In all simulation examples, the precoding sequences were nor-malized.Simulation Example 1: In the first simulation example, we

computed the average bias and theMSE of the channel estimatoras a function of SNR in decibels. Fig. 3 shows the results for

Fig. 3. Estimator performance. (a) Average bias and (b) MSE as a function ofSNR in decibels for , and , respectively.

and . We can see that the performance of theestimator generally improves with increasing SNR and that thebest estimator performance is obtained for . Furthermore,we can observe that going from to improves theestimator performance significantly, whereas a further increaseto yields less improvement.Simulation Example 2: In the second simulation example,

we investigate the effect of the length of the data record usedfor estimating the cyclic statistics on the performanceof the channel estimator. For SNR dB, Fig. 4(a) and (b)shows the average bias and theMSE, respectively, of the channelestimator as a function of for and . (Notethat in Fig. 4, the length of the data record has been specified in



Fig. 4. Estimator performance as a function of data record length in OFDMsymbols. (a) Average bias and (b) MSE for , and ,respectively.

OFDM symbols. The actual length of the data record is there-fore obtained by multiplying the number of symbols by 16.) Wecan observe that the performance of the estimator generally im-proves with increasing data record length. Similar to the pre-vious example, increasing from 2 to 4 yields a significant im-provement in estimator performance, whereas a further increaseto yields only a slight improvement. We can furthermoreconclude that, especially if short data records have to be used(i.e., is small), it is crucial to have sufficientlylarge.Simulation Example 3: For the channel estimates obtained

in Simulation Example 1, we investigate the performance ofthe corresponding multichannel equalizer. For each SNR value,we took the average of the channel estimates over all MonteCarlo runs and computed the corresponding

. Fig. 5 shows the BER as a function of theSNR for and . We can see that con-sistently yields the best performance in terms of BER. In orderto isolate the impact of nonconstant-modulus precoding on BER

Fig. 5. Multichannel equalizer performance. Bit error rate for the blindalgorithm (solid curves) and for the perfect channel knowledge case (dashedcurves) as a function of SNR in decibels for and ,respectively.

performance, we also plot the BER curves for the case of perfectchannel knowledge. We find that for and , the blindalgorithm comes very close to the perfect channel knowledgecase. For , there is a significant difference between theblind approach and the perfect channel knowledge case, whichis due to the fact that for , the channel estimator perfor-mance is very poor. We can furthermore see that for increasing, the BER performance degrades consistently.

VI. CONCLUSION

We introduced an algorithm for blind channel identificationand equalization in OFDM-based spatial multiplexing systems.Our approach uses second-order cyclostationary statistics andemploys periodic precoding. The basic idea of our method is toprovide each transmit antenna with a different signature in thecyclostationary domain to null out the influence of all but onetransmit antenna at a time. This makes a scalar subchannel bysubchannel identification of the matrix channel possible. Our al-gorithm yields unique channel estimates (up to a phase rotationfor each transmit antenna), its performance does not degradesignificantly if the number of transmit antennas is increased, itrequires knowledge of an upper bound on the channel lengthonly, it does not impose restrictions on channel zeros, and it ex-hibits low sensitivity to stationary noise. We discussed the de-sign of the precoding sequences, and we provided simulationexamples demonstrating the channel estimator performance, thecorresponding multichannel equalizer performance, and the im-pact of precoding on BER.

APPENDIX

With the received signal vector given by



we obtain

where . Now, computing thecyclic spectrum of according to

where .We next compute the correlation matrix of the transmit signal

vector given by

diag

where , and

else.

The correlation matrix of is given by

diag

diag

Next, using diag ,we get

diag

where . Let us next compute theFourier series coefficients of with respect to , i.e.,

diag

Now, substituting , where and, we obtain

diag

diag

where , and.



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Helmut Bölcskei (M’98) was born in Mödling,Austria, on May 29, 1970. He received the Dipl.-Ing.and Dr. techn. degrees in electrical engineering/com-munications from Vienna University of Technology(VUT), Vienna, Austria, in 1994 and 1997, respec-tively.From 1994 to 2001, he was with the Institute of

Communications and Radio-Frequency Engineering,VUT. Since March 2001, he has been an AssistantProfessor of Electrical Engineering at the Universityof Illinois at Urbana-Champaign. In February 2002,

he will join ETH Zurich, Zurich, Switzerland, as an Assistant Professor of com-munication theory. From February to May 1996, he was a Visiting Researcher atPhilips Research Laboratories, Eindhoven, The Netherlands. From February toMarch 1998, he visited the Signal and Image Processing Department at ENSTParis, Paris, France. His research interests include communication and informa-tion theory and signal processing with special emphasis on wireless systems,multi-input multi-output (MIMO) antenna systems, space-time coding, orthog-onal frequency division multiplexing (OFDM), and wireless networks. From1999 to 2001 he was is a consultant for Iospan Wireless Inc. (formerly GigabitWireless Inc.), San Jose, CA.Dr. Bölcskei was an Erwin Schrödinger fellow of the Austrian National Sci-

ence Foundation (FWF) from February 1999 to February 2001, performing re-search in the Smart Antennas Research Group with the Information SystemsLaboratory, Department of Electrical Engineering, Stanford University, Stan-ford, CA. He serves as an Associate Editor for the IEEE TRANSACTIONS ONSIGNAL PROCESSING.

Robert W. Heath, Jr. (S’95) received the B.S. andM.S. degrees from the University of Virginia, Char-lottesville, and the Ph.D. degree from Stanford Uni-versity, Stanford, CA, in 2001, all in electrical engi-neering.From 1998 to 1999, he was with Iospan Wireless,

Inc. (formerly Gigabit Wireless Inc.), San Jose,CA. In January 2002, he will join the Universityof Texas at Austin as an Assistant Professor. Hisresearch group—the Wireless Systems EngineeringLaboratory—will focus on the theory, design, and

practical implementation of wireless systems. His current research interestsare coding, modulation, synchronization, and equalization for multiantennawireless systems.



Arogyaswami J. Paulraj (F’91) received the Ph.D.degree from Naval Engineering College and the In-dian Institute of Technology, Bangalore, in 1973.He has been a Professor at the Department

of Electrical Engineering, Stanford University,Stanford, CA, since 1993, where he supervisesthe Smart Antennas Research Group. This groupconsists of approximately 12 researchers workingon applications of space-time signal processing forwireless communications networks. His researchgroup has developed many key fundamentals of this

new field and helped shape a worldwide research and development focus ontothis technology. His nonacademic positions included Head, Sonar Division,Naval Oceanographic Laboratory, Cochin, India; Director, Center for ArtificialIntelligence and Robotics, Bangalore; Director, Center for Development ofAdvanced Computing; Chief Scientist, Bharat Electronics, Bangalore, andChief Technical Officer and Founder, Iospan Wireless Inc., San Jose, CA. Hehas also held visiting appointments at Indian Institute of Technology, Delhi,Loughborough University of Technology, Loughborough, U.K., and StanfordUniversity. He sits on several boards of directors and advisory boards forcompanies/venture partnerships in the United States and India. His researchhas spanned several disciplines, emphasizing estimation theory, sensor signalprocessing, parallel computer architectures/algorithms, and space-time wirelesscommunications. His engineering experience included development of sonarsystems, massively parallel computers, and, more recently, broadband wirelesssystems.Dr. Paulraj has won several awards for his engineering and research con-

tributions. These include two President of India Medals, the CNS Medal, theJain Medal, the Distinguished Service Medal, the Most Distinguished ServiceMedal, the VASVIK Medal, the IEEE Best Paper Award (Joint), among others.He is the author of over 250 research papers and holds eight patents. He is aMember of the Indian National Academy of Engineering.


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