tsp2008 lp feng

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 7, JULY 2008 2821

A Semiblind Channel Estimation Approach forMIMO–OFDM Systems

Feng Wan, W.-P. Zhu, Senior Member, IEEE, and M. N. S. Swamy, Fellow, IEEE

Abstract—In this paper, a very efficient semiblind approach thatuses a training-based least square criterion along with a blind con-straint is proposed for multiple-input–multiple-output–orthogonalfrequency-division multiplexing (MIMO–OFDM) channel estima-tion. The blind constraint is derived from the linear prediction ofthe received MIMO–OFDM signal and is used in conjunction witha weighting factor in the semiblind cost function. An appealingscheme for the determination of the weighting factor is presentedas a part of the proposed approach. A perturbation analysis ofthe proposed method is conducted to justify the superiority of thesemiblind solution and to obtain a closed-form expression for themean square error (MSE) of the blind constraint, further facili-tating the calculation of the weighting factor. The proposed methodis validated through computer simulation-based experimentations,showing a very high estimation accuracy of the semiblind solutionin terms of the MSE of the channel estimate.

Index Terms—Channel estimation, multiple-input–multiple-output (MIMO) linear prediction, MIMO–orthogonal frequency-division multiplexing (MIMO–OFDM), perturbation analysis,semiblind.

I. INTRODUCTION

THE multiple-input–multiple-output–orthogonal fre-quency-division multiplexing (MIMO–OFDM) tech-

nology has been considered as a strong candidate for the nextgeneration wireless communication systems. Using multipletransmit as well as receive antennas, a MIMO–OFDM systemgives a high data rate without increasing the total transmissionpower or bandwidth compared to a single antenna system. Fur-ther, the frequency-selective problem that exists in a conventionalwireless system can be well solved by the OFDM technique inthe MIMO–OFDM system. On the other hand, the performanceof MIMO–OFDM systems depends largely upon the availabilityof the knowledge of the channel. It has been proved [1] thatwhen the channel is Rayleigh fading and perfectly known to thereceiver, the capacity of a MIMO–OFDM system grows linearlywith the number of transmit or receive antennas, whichever isless. Therefore, an accurate estimation of the wireless channel isof crucial importance to MIMO–OFDM systems.

A considerable number of channel estimation methods havealready been proposed for MIMO–OFDM systems. They canbroadly be categorized into three classes, namely, the training-

Manuscript received June 14, 2007; revised November 27, 2007. The asso-ciate editor coordinating the review of this manuscript and approving it for pub-lication was Dr. Petr Tichavsky. This work was supported by the Natural Sci-ences and Engineering Research Council (NSERC) of Canada. Part of this workwas presented at the IEEE International Symposium on Circuits and Systems,New Orleans, LA, May 2007.

The authors are with the Center for Signal Processing and Communications,Department of Electrical and Computer Engineering, Concordia Univer-sity, Montreal, QC H3G 1M8, Canada (e-mail: [email protected];[email protected]; [email protected]).

Digital Object Identifier 10.1109/TSP.2008.917354

based method, the blind method, and the semiblind one as acombination of the first two methods. First, the training-basedmethods employ known training signals to render an accuratechannel estimation [2]–[5]. One of the most efficient training-based methods is the least squares (LS) algorithm, for which anoptimum pilot design scheme has been given in [2]. When thefull or partial information of the channel correlation is known, abetter channel estimation performance can be achieved via someminimum mean square error (MMSE) methods [3]. By usingdecision feedback symbols, the Takagi–Sugeno–Kang (TSK)fuzzy approach proposed in [4] can achieve a performance sim-ilar to the MMSE methods while with a low complexity. In con-trast to training-based methods, blind MIMO–OFDM channelestimation algorithms, such as those proposed in [6] and [7],often exploit the second-order stationary statistics, correlativecoding, or other properties to give a better spectral efficiency.With a small number of training symbols, a semiblind methodhas been proposed in [8] to estimate the channel ambiguity ma-trix for space-time coded OFDM systems.

It is worth pointing out that most of the existing blind and semi-blind MIMO–OFDM channel estimation methods are based onthe second-order statistics of a long vector whose size is equal toor larger than the number of subcarriers. To estimate the correla-tion matrix reliably, they need a large number of OFDM symbols,which is not suitable for fast time-varying channels. In addition,becausethematrices involvedin thesealgorithmsareofhugesize,their computational complexity is extremely high. In contrast, alinear prediction-based semiblind algorithm that is based on thesecond-order statistics of a short vector with a size only slightlylarger the channel length has been foundmuch more efficient thanthe conventional LS methods for the estimation of frequency-se-lective MIMO channels [9]–[11]. In this paper, we will extend thelinear prediction-based semiblind approach to the channel esti-mation of MIMO–OFDM systems.

LinearpredictionhasbeenwidelyusedinblindMIMOchannelestimation and equalization [9], [12]–[17], where the key ideais to represent the received MIMO signal as a finite-order au-toregressive (AR) process under the assumption that the trans-mitted signals are uncorrelated in time [12]. Based on the ARprocess,alinearpredictionfiltercanbeobtainedtosolveasecond-order deconvolution problem for channel equalization. By com-bining the linear prediction with a higher order statistics (HOS)or the weighted LS method, some blind channel estimation algo-rithms have been derived [12], [17]. However, these algorithmsrequire a large number of signal samples and moreover, they arenot robust. Medles et al. have proposed a semiblind algorithmby incorporating a blind criterion derived from the linear predic-tion into a training-based LS cost function [9]–[11], leading to aclosed-form expression for the estimation of the MIMO channel

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2822 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 7, JULY 2008

response. It has been shown in their papers that the semiblindmethod provides a much better channel estimation performanceover thepure training-basedLSmethod.However, thesuperiorityof theproposedsemiblindmethodhasnotbeentheoretically justi-fied. Moreover, the weighting factor employed to trade off the LSand the blind criteria has not been appropriately determined. Assuch, theresultingchannelestimationperformance, thoughbetterthan that of the LS method, could be further improved. It shouldalso be noted that this semiblind method for MIMO channel esti-mation cannot be directly applied to MIMO–OFDM systems dueto different signal models in the two systems. In MIMO systems,both information data and pilots are generated and transmitted inthe time domain, whereas in MIMO–OFDM systems, signals arefirst generated in the frequency domain and then converted to thetime domain via the inverse discrete Fourier transform (IDFT).As a consequence, the training-based LS cost function for MIMOchannel estimation cannot be used for MIMO–OFDM systems.Also, the uncorrelation among the time-domain MIMO signals,which is desirable for linear prediction, may not be available inMIMO–OFDM systems. Therefore, a lot of work on the formu-lation of MIMO–OFDM signals has to be done first to develop anew semiblind estimation solution.

In this paper, we propose a channel estimation algorithmfor MIMO–OFDM systems by employing the aforementionedsemiblind strategy. First, a new training-based LS criterion forMIMO–OFDM channel estimation is obtained through propermatrix formulations. Then, by proving that the transmittedtime-domain MIMO–OFDM signals are uncorrelated, we val-idate the use of the linear prediction for the formulation of theblind criterion. By employing the LS and the blind criterion, asemiblind channel estimation solution is obtained. As a part ofthe semiblind approach, we also propose an appealing schemefor choosing the weighting factor, which is shown to be veryefficient according to our extensive computer simulations.

The second part of this paper deals with the analysis of theMIMO linear prediction and the corresponding blind constraint.We apply the perturbation theory, which has been successfullyused in analyzing the antenna-array-based signal processingalgorithms [18]–[23], to the analysis of the proposed MIMOlinear-prediction-based method, justifying the superiority ofour semiblind solution over the blind algorithms [12], [24],[25]. The perturbation analysis of the blind constraint also leadsto a novel closed-form expression for the mean square error(MSE) of the blind estimation that is essential to the calculationof the weighting factor for the semiblind estimation solution.

The rest of this paper is organized as follows. Section IIgives a brief review of the MIMO linear prediction and arelated semiblind algorithm for MIMO channel estimation.Section III presents a new semiblind approach for the estima-tion of MIMO–OFDM channels, including the formulationof the semiblind estimation problem, derivation of the blindconstraint from the linear prediction, and development ofan appealing scheme for the determination of the weightingfactor. Section IV conducts a perturbation analysis of thelinear-prediction-based semiblind method, justifying the su-perior estimation performance of the semiblind solution overa pure blind method. The analysis also yields a closed-formexpression for the MSE of the blind estimation part of the

proposed approach, facilitating the calculation of the weightingfactor. Section V comprises a number of experimentationsvalidating the proposed method, showing significant advantageof the semiblind solution over the LS method in terms of theMSE of the channel estimate. Finally, Section VI highlightssome of the distinct features of the proposed approach.

Throughout this paper, we adopt the following notations:

pseudoinverse;Kronecker product;transpose;complex conjugate transpose;linear convolution;circular convolution;Frobenius norm;a stacking of the columns of the involved matrixinto a vector, which has the following properties:

and

II. BRIEF OVERVIEW OF LINEAR PREDICTION FOR MIMOCHANNEL ESTIMATION

Consider a MIMO system with transmit andreceive antennas. The MIMO channel can be char-

acterized by an array of -tap finite-impulse response (FIR)filters given by a number of matrices

, whose th elementrepresents the channel response from the th transmit antennato the th receive antenna. Given the transmitted signal vector

, the received signal vector canbe written as , with its elementbeing given by

(1)

where denotes the linear convolution and is aspatio–temporally uncorrelated noise with variance .

We now briefly review the MIMO linear prediction and therelated semiblind channel estimation method [12]. Define

(2)

The MIMO linear predictor can be written as

(3)

where is an matrix repre-senting the th tap of the prediction filter. The covariance ma-trix of the prediction error can then be given by

(4)

where

WAN et al.: A SEMIBLIND CHANNEL ESTIMATION APPROACH FOR MIMO–OFDM SYSTEMS 2823

Further, define

It has been shown in [12] and [13] that if the transmitted signalsare uncorrelated, and moreover , onecan obtain

(5)

(6)

Based on (5) and (6), some blind algorithms have been proposedfor MIMO channel estimation [12], [24], [25]. The basic idea isto first acquire an estimate of from that of accordingto (6), and then use (5) to obtain an estimate of the channelmatrix .

Using the aforementioned linear prediction along withtraining data, a semiblind approach for MIMO channel es-timation has been developed to achieve a better estimationperformance [9]–[11]. This approach is briefly described asfollows.

Denote the null column space of as anmatrix . From (6), one can easily find that can beestimated from . Using into (5) gives

(7)

which imposes a blind constraint on the channelmatrix . Equation (7) can then be used to derive a blindconstraint for the channel vector defined by

where . Using this blindconstraint in conjunction with a training-based LS criterion [26],a semiblind cost function is then formulated as

(8)

where is a pilot signal matrix, is the corresponding re-ceived signal vector, and is a weighting factor. The mini-mization problem (8) can be easily solved under the assumptionthat is a fixed constant, giving a semiblind solution for thechannel estimate . It should be stressed that the estimation ac-curacy of the aforementioned semiblind algorithm heavily de-pends on the choice of the weighting factor. This issue, however,has not been investigated. In what follows, we will extend thesemiblind method for the estimation of MIMO–OFDM chan-nels. We will also propose a very efficient scheme for the de-termination of the value of according to the MSE of the LSestimation as well as that of the blind constraint.

III. PROPOSED MIMO–OFDM SEMIBLIND CHANNEL

ESTIMATION ALGORITHM

For the sake of simplicity, only one OFDM symbol withsubcarriers is considered. Let be the transmitted time-do-main signals at the output of the IDFT module, and be thereceived time-domain signals before the discrete Fourier trans-form (DFT) operation. If the length of the cyclic is not less thanthe channel length , after removing the cyclic prefix at the re-ceiver, the th received signal can be written as

(9)

Note that, unlike the MIMO signal model in (1), theMIMO–OFDM signal given by (9) involves the circularconvolution with a length . We would like to develop a semi-blind MIMO–OFDM channel estimation approach based on(9) and the semiblind criterion in (8). The idea of this approachhas been proposed for the first time in [27] and will now becompleted in this paper.

A. Training-Based LS Criterion

In MIMO–OFDM systems, the training signal is transmittedin the frequency domain and thus the LS criterion in (8) shouldbe modified. Following the approach in [2], the received fre-quency-domain signal at the th receive antenna can be ob-tained as

(10)

where is the noise vector with respect to the th re-ceive antenna, is the one-path channel response vector asgiven by

(11)

and is a matrix carrying on the pilot signal, withand being the number of the pilot carriers and the number

of OFDM symbols used, respectively. By defining

(12)

we have

(13)

Further, letting , , and, from (13), one can obtain a new LS criterion

(14)

which will be used for the training signal in the proposed semi-blind method.


B. MIMO Linear-Prediction-Based Blind Criterion

First, to validate the use of linear prediction for the receivedMIMO–OFDM signal, we show in Appendix A that the trans-mitted time-domain MIMO–OFDM signal is uncorrelated. Wenow use the MIMO linear prediction to obtain a blind con-straint for the channel matrix. By following the linear predictionprocess in Section II, we can obtain a time-domain representa-tion of (5)

(15)

where is given by (3) and is ablock Toeplitz matrix with the first block row as

Letting

...

. . ....

. . .. . .

.... . .

. . ....

. . ....

(15) can be rewritten as

...(16)

Using (6), the null column space of , can easily beobtained, which is then used to form

(17)

From (16) and (17), we have

(18)

which is equivalent to

(19)

Noting that , where is a known permuta-tion matrix, (19) can be rewritten as

(20)

implying that is a blind constraint for thechannel vector .

In the computation of the linear predictor and the covari-ance matrix , one has to estimate various correlation ma-

trices , and , as discussed in Section II. Con-sidering that the circular convolution is used in MIMO–OFDMsystems, a more accurate estimate of these correlation matricescan be obtained in comparison with that in MIMO systems. Forexample, the estimate of in MIMO systems is computedas

(21)

where only received signal vectorsare available for estimation. In MIMO–OFDM

systems, however, the estimation of can be modified as

(22)

where can be obtained usingwhen due to the circular convo-

lution. Because more signal samples are used in the estimate, abetter linear prediction result can be expected in MIMO–OFDMsystems. Note that, when multiple OFDM symbols are used,

can be easily calculated by averaging the results obtainedfrom each OFDM symbol using (22).

C. Semiblind Solution

Combining (14) and (20), a semiblind cost function for theestimation of the channel vector can be formulated as

(23)

where is an estimate of the blind constraint. The solution tothis minimization problem can be obtained by letting

which gives

(24)

Clearly, the performance of the semiblind algorithm depends onthe choice of . However, the selection of has not yet beendiscussed in the existing semiblind MIMO channel estimationmethods such as those in [9], [10], [26], and [28]. In this paper,we propose an explicit formula for the calculation of in termsof the estimated MSE of the training-based criterion and that ofthe blind part.

It has been shown in [29] and [30] that the weight for aweighted least squares (WLS) minimization problem can bedetermined according to the variance of the individual es-timation error involved provided that the error is Gaussiandistributed. A vector version of the WLS problem can bedescribed as

(25)


where denotes the estimate of the true vector and isthe weight for the th error term, which should be chosen as thereciprocal of the variance of , i.e.,

In order to develop a closed-form expression for the weightingfactor in (23), let us consider the following WLS problem:

(26)

where

with being the estimate of the channel vector resulting fromthe training-based LS criterion and that from the blind esti-mation part. Since the error vectors and can be consideredGaussian, the coefficients and can be determined fromthe variance of and that of , respectively, thereby the valueof can be estimated.

We first consider the variance of . From [2], one can easilyfind that the optimal pilots are given by

Thus, for the optimal pilots, one can obtain

(27)

MSE

(28)

Note that it is not possible to directly compute the variance of, since the blind solution is not available in the proposed

method. However, we may use the following variance to replacethe variance of :

MSE (29)

It will be shown through simulation study in Section V that thisapproximation gives an appropriate choice of in terms of thechannel estimation performance.

Using (28) and (29), (26) can be rewritten as [29], [30]

MSE MSE(30)

The previous minimization problem can be nearly reformulatedas the following LS problem:

MSE MSE(31)

Evidently, the solution of (31) is equivalent to that of (23) whenis chosen as

MSEMSE

(32)

What remains in the computation of is to determine theMSE . In the next section, a perturbation analysis of thelinear-prediction-based blind algorithm is performed, leadingto a novel closed-form expression for MSE .

IV. PERTURBATION ANALYSIS OF THE LINEAR-PREDICTION-BASED BLIND

CHANNEL ESTIMATION

It is known that the solution of linear prediction or subspace-based methods is always perturbed by various sources, suchas finite data length, measurement noise, etc. [18], [21]–[23].Perturbation theory has been successfully applied to the anal-ysis of subspace-based methods [19]–[21], [26]. In this section,the first-order perturbation theory is employed to analyze theMIMO linear prediction as well as the resulting blind constraint.

A. Linear Predictor With Perturbation for MIMO ChannelEstimation

Letting and, where

for , the circular convolution (9) canbe rewritten in the matrix form as

(33)

Similarly, can be expressed as

(34)

where and is ablock Toeplitz matrix with the first block

row given by . In this paper, weconsider only the perturbation due to the finite data length in thecomputation of the correlation matrices. Without loss of gener-ality, we let the variance of the signal equal 1. Using (34), the

correlation matrix with such a perturbation can be writtenas

(35)

where denotes the signal perturbation matrix

and is the perturbation matrix introduced by the noise

(36)

with

(37)

Similarly, is given by

(38)


where

with

Based on the previous expressions for the correlation matriceswith perturbation, we now derive the perturbation form of thechannel constraint given in (15) and that of the covariance ma-trix given in (4). To this end, we first derive the perturbation formof the linear predictor . Using the first-order approximation,

the pseudoinverse of given by (35) can be obtained as

(39)

where and . It should be noted

that the noise subspace of in (35) has been omitted in ob-taining (39), and such an omission would not affect the deriva-tion of the perturbation form of (15) and (4). Using (38) and(39), the linear predictor can be derived as

(40)

Noting that can be partitioned as

where , the perturbationform of the left-hand side (LHS) of (15) can be written as

(41)

Using (40) and , and noting thatwhen has a full column

rank and , one can deduce

where

(42)

Thus, (41) can be rewritten as

(43)

Clearly, (43) is the perturbation form of (15). By noting that

(42) can be reduced to

(44)

where

(45)

(46)

It is obvious from (44) that the perturbation term consistsof the signal perturbation and the perturbation in-troduced by the noise .

We now turn to the derivation of the perturbed version of (6).The perturbed form of (4) can be written as

(47)

where

(48)

It is easy to show that

(49)

where

with

Using (40), the second term of the right-hand side (RHS) of (47)can be expressed as

(50)

From


one can get

(51)

Using (49) and (50) along with (51), the covariance matrix givenby (6) has the following perturbation form:

(52)

where

(53)

We now show that the perturbation term can be split intothe signal perturbation and the perturbation due to the noise. Byusing

the first four terms of the RHS of (53) equalwith

(54)

Therefore, (53) reduces

(55)

where

(56)

It should be mentioned that in the noise-free case, all the pertur-bation terms in the previous equations introduced by noise canbe ignored. As such, (43) and (52) can be simplified as

(57)

and

(58)

respectively. In conventional blind algorithms [12], [24], [25],an estimate of is first acquired from and is then usedto estimate the channel matrix . This approach would give

rise to a large estimation error due to the presence of perturba-tion terms as seen from (57) and (58). In the proposed semiblindapproach, however, an ideal nulling constraint on the channelmatrix has been obtained from the ideal null space of ,which is not affected by the perturbation terms. Therefore, thesemiblind method using MIMO linear prediction is superior tothe blind algorithms [12], [24], [25]. It should also be pointedout that in the presence of noise, although both the semiblind andthe blind methods are subject to the noise perturbation terms,the semiblind method still outperforms the blind one, since theperturbation introduced by the noise is, in general, significantlysmaller than the signal perturbation.

B. MSE of Blind Channel Estimate

To derive the MSE expression, the perturbed versionof (18) needs to be determined. By partitioning as

, with each submatrixbeing an ma-

trix, we can get the perturbation form of (16)

(59)

where

(60)

Then, using (52) and (55) along with the first-order approxima-tion [20], one can get the estimate of

(61)

Substituting (61) into (17) and using (59), we obtain

(62)

where

(63)

(64)

Clearly, (62) is the perturbed version of (18). From (62), theMSE of the blind criterion can be calculated as

MSE(65)

In Appendix B, we investigate the computation of andto obtain a closed-form expression for MSE in terms

of the correlation of the transmitted signal, the correlation of thenoise as well as the channel matrix.

V. SIMULATION RESULTS

We consider a MIMO–OFDM system with two transmit andfour receive antennas. The number of subcarriers is set to 512,the length of cyclic prefix is ten, and the length of the linear


predictor is . In our simulation, the quaternary phase-shiftkeying (QPSK) modulation is used and a Rayleigh channel mod-eled by a three-tap MIMO–FIR filter is assumed, in which eachtap corresponds to a 2 4 random matrix whose elements are in-dependent identically distributed (i.i.d.) complex Gaussian vari-ables with zero mean and unit variance. Because the blind con-straint of the proposed semiblind algorithm is derived directlyfrom as seen from (5) and (6), it is necessary to consider inthe simulation study the effect of on the channel estima-tion performance. To this end, we define the following metric:

and conduct our investigation with respect to different ranges of.It is seen from (32) that one has to choose the value of MSE

to determine the parameter . Although a closed-form expres-sion for MSE has been obtained in Section IV, it requires thetrue channel matrix, which is, however, unknown in practice.Therefore, we first estimate the channel matrix using the LSmethod and then utilize the preliminary estimate to compute theMSE and thereby the value of . To evaluate the proposed al-gorithm, we also calculate the value of MSE based on the truechannel matrix, which gives a reference for the choice of . Thesemiblind solution thus obtained is called the reference algo-rithm. The estimation performance is evaluated in terms of theMSE of the estimate of the channel matrix given by

MSE

where is the number of Monte Carlo iterations andand are true and estimated channel vectors with respect to

th Monte Carlo iteration, respectively.Experiment 1: MSE Versus Weight Factor : In the first

experiment, the channel estimation performance in terms ofthe plot of the MSE versus is investigated. The simulationis based on four randomly generated channel matrices, eachhaving a value of falling within the range of

, or . For each channel matrix,three runs of the transmission of one OFDM symbol at 512subcarriers, of which 32 are used as pilot for training purpose,are performed. Fig. 1 shows the MSE plots of the semiblindalgorithm with respect to different choices of in the range of

for the four channels at a signal-to-noise ratio (SNR) of15 dB. The two points identified by “ ” and “ ” in each runindicate the results from the proposed and reference algorithms,respectively. It is observed that the proposed scheme for thedetermination of gives a competitive MSE result. It is alsoseen that the use of the preliminary LS estimate of the channelmatrix, instead of the true channel characteristic, would sufficefor the calculation of the value of .

Experiment 2: MSE Versus : In this experiment, the channelestimation performance in terms of the MSE versus is investi-gated. The simulation is undertaken based on 5000 Monte Carloruns of the transmission of one OFDM symbol on 512 subcar-riers at an SNR of 15 dB. Fig. 2 shows the MSE plots resulting

Fig. 1. MSE versus � for different ranges of � in which the two points indi-cated by “�” and “�” for each run are obtained from the proposed and referencealgorithms, respectively.

from the proposed as well as the reference algorithm along withthat from the LS estimation, indicating a high consistency of


Fig. 2. MSE versus the value of �.

Fig. 3. MSE versus SNR.

the two semiblind methods. It is noted that the proposed semi-blind algorithm significantly outperforms the LS method. It isalso clear that the MSE performance of the semiblind estima-tion depends on the value of . A better estimation performanceis reached when varies from 0.3 to 0.7, which represents a typ-ical mobile communication scenario, where the first arrived pathis comparable to or stronger than other paths [31]. From Fig. 2,one can find, when is larger than 0.55, the MSE becomes alittle larger with the increase of . We believe the possible reasonis that the full column rank assumption of for the proposedsemiblind algorithm cannot be always satisfied in the experi-ment simulations, since the Rayleigh channel is generated forthe Monte Carlo iterations. With the increase of the value of ,especially when is larger than 0.55, and becomemore and more insignificant. As a result, the probability for thecase does not have a full column rank becoming larger, in-creasing the MSE a little.

Experiment 3: MSE Versus SNR: Now, we examine thechannel estimation performance as a function of the SNR.

Fig. 4. MSE versus pilot length for different number of OFDM symbols.

Again, the simulation involves 5000 Monte Carlo runs of thetransmission of one OFDM symbol. Fig. 3 shows the channelestimation results of the three methods, when . It isseen that the performances of the two semiblind algorithms arevery close and both can achieve a gain of nearly 3.3 dB overthe LS method regardless of the level of the SNR.


Fig. 5. BER versus SNR.

Experiment 4: MSE Versus Pilot Length: Here, we investi-gate the channel estimation performance of the proposed semi-blind algorithm versus the number of OFDM symbols as wellas the number of pilot subcarriers per symbol, in comparisonwith that of the LS method. The number of OFDM symbols isset to be from 1 to 4, and in each case, the number of pilot sub-carriers per OFDM symbol varies from 8 to 48. Note that theproposed method can easily be applied to the case of multipleOFDM symbols, where the calculation of , and in(24) should be based on the multiple OFDM symbols. Fig. 4shows the MSE plots from 500 Monte Carlo iterations for anSNR of 15 dB, when . It is seen that for the same numberof OFDM symbols, the performance of all the algorithms im-proved with increasing number of pilot subcarriers. Again, theperformance of the proposed method is very close to that of thereference method, both being superior to the LS method by 6 and4 dB when the number of pilot subcarriers is 8 and 48, respec-tively. It implies that the proposed semiblind method is moreadvantageous for pilot signals of a shorter length. Furthermore,it is observed that the performance improvement of the proposedsemiblind method over the LS method remains almost the samewith the increase of the number of OFDM symbols employedfor channel estimation.

Experiment 5: BER Versus SNR: In this experiment, the biterror rate (BER) performance of the MIMO–OFDM system isinvestigated by using the estimated channel matrix and an or-dered vertical-Bell Laboratories layered space time (V-BLAST)decoder. The simulation involves 5000 Monte Carlo runs of thetransmission of one OFDM symbol with eight pilot subcarriers.Fig. 5 shows the BER performance versus the SNR for the LSmethod and the proposed semiblind method, when . It isseen that the performance of the proposed semiblind algorithmis superior to the LS method by 2–5 dB.

VI. CONCLUSION

A semiblind MIMO–OFDM channel estimation approachthat incorporates a linear-prediction-based blind criterion intothe LS method was proposed. A practical yet very efficient

scheme was presented for the determination of the weightingfactor of the semiblind cost function. The perturbation analysisof the MIMO linear prediction justified the advantage of thesemiblind method over the pure blind estimation, and led toa closed-form expression for the MSE of the blind criterion.The proposed method was simulated and compared with thetraining-based LS method, showing a significant improvementin terms of the MSE of the channel estimate.

APPENDIX APROOF OF UNCORRELATION OF TRANSMITTED TIME-DOMAIN

MIMO–OFDM SIGNAL

Generally speaking, the frequency-domain signal inMIMO–OFDM systems before the DFT module can beconsidered as an i.i.d. Gaussian process with zero mean andvariance , implying that the frequency-domain signal at the

th antenna is uncorrelated in both time and spacedomains. Given and being the frequency-do-main signals at the th and th antennas, respectively, thecorresponding time-domain signals can be written as

Consider the correlation between the two time-domain sig-nals, i.e.,

(A-1)Obviously

if (A-2)

If , (A-1) reduces to

ifif

(A-3)

It is clear from (A-2) and (A-3) that the transmitted time-domainMIMO–OFDM signal is uncorrelated.

APPENDIX BA CLOSED-FORM EXPRESSION OF MSE

We formulate first . Defining

and using (63), we can derive

(B-1)


where

(B-2)

(B-3)

with

It is clear from (64) that, to obtain , one needs to firstcompute , which can be given by (see Appendix C fordetails)

(B-4)

where and are two transformation matrices which canbe easily constructed as shown in Appendix C. Using (64) and(B-4) gives

(B-5)

where

(B-6)

(B-7)

with and . Notethat has been replaced by in (B-5), since the signalperturbation term has been eliminated.

Finally, by using (B-1) and (B-5) in (65) and noting thatand , the MSE of the blind

criterion can be derived as

MSE

(B-8)

where

with being given by

The closed-form expression of is derivedin Appendix D.

APPENDIX CDERIVATION OF (B-4)

By defining a transformation matrixsuch that its submatrices, being of

the same size, satisfy ,(60) can be rewritten as

(C-1)

from which we have

(C-2)

We now simplify by introducing another transformationmatrix . Our idea is to construct such that

which leads to

(C-3)

To obtain such a , we define a-dimensional vector, denoted as

where is an matrix whose elements areall unity. The matrix of size

can then be constructed according to thefollowing rules.

1) If the th element of vector is zero, the th row ofis set to a zero vector.

2) If the th element of vector is 1, then only one elementin the th row of is 1 and all the other elements are zero.The position of the element 1 shifts to the right by one entrywith respect to the previous row with an element of 1. Ifthe element 1 in the previous reference row happens at thelast column, the next element 1 returns to the first column.Note that the first element in the first row of is always


1. As an example, considering , , and, we have the equation shown at the bottom of the

previous page. Then, the corresponding is readily givenby the equation shown at the bottom of the page.

APPENDIX DDERIVATION OF

Here, we derive the autocorrelation and cross-correlation ma-trices of and . First, weconsider . For a medium to high SNR, the perturbationmatrix of the noise autocorrelation matrix can be neglected.Thus, from (36), one can get

Since the noise element is circularly symmetric withequal variance for the real and imaginary part, one canverify that and

. In addition, itcan be easily verified that .Therefore, can be calculated by

(D-1)

To this end, the following two correlation matrices are to bedetermined:

(D-2)

(D-3)

To compute , one can obtain from (37)

(D-4)

where . Substituting (D-4)into (D-2) gives

(D-5)

where . It canbe easily shown that is a partial identity matrix.Further, it can be proved that

(D-6)

where

which is also a partial identity matrix. Thus, substituting (D-6)into (D-5) gives

(D-7)

Similarly, one can derive

(D-8)

Thus, by using (D-7) and (D-8) into (D-1), one can obtain

(D-9)

As a result, can now be explicitly computed in terms of, and the channel matrix .

In a similar manner, one can derive the expressions for


where

with

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Feng Wan received the B.E. degree from WuhanUniversity, China, in 2000 and the M.E. degree fromInstitute of Electronics, Chinese Academy of Sci-ences, China, in 2003, both in electrical engineering.Currently, he is working towards the Ph.D. degree atConcordia University, Montreal, QC, Canada.

Since 2004, he has been a Research Assistantin Concordia University. His research interestsare in the area of digital signal processing, sta-tistical estimation theory and its applications inmultiple-input–multiple-output (MIMO), orthog-

onal frequency-division multiplexing (OFDM), and smart antenna wirelesscommunications.

Mr. Wan was awarded the Graduate Fellowship and the International TuitionFee Remission Award from Concordia University.

Wei-Ping Zhu (SM’97) received the B.E. andM.E. degrees from Nanjing University of Postsand Telecommunications, Nanjing, China, in 1982and 1985, respectively, and the Ph.D. degree fromSoutheast University, Nanjing, China, in 1991, all inelectrical engineering.

He was a Postdoctoral Fellow from 1991 to1992 and a Research Associate from 1996 to 1998at the Department of Electrical and ComputerEngineering, Concordia University, Montreal, QC,Canada. During 1993–1996, he was an Associate

Professor at the Department of Information Engineering, Nanjing University ofPosts and Telecommunications. From 1998 to 2001, he worked in hi-tech com-panies in Ottawa, ON, Canada, including Nortel Networks and SR Telecom,Inc. Since July 2001, he has been an Associate Professor at the Electricaland Computer Engineering Department, Concordia University. His researchinterests include the areas of digital filters, filter banks, speech and imageprocessing, and signal processing for wireless communication.

Dr. Zhu was an Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS

AND SYSTEMS—PART I: FUNDAMENTAL THEORY AND APPLICATIONS from Oc-tober 2001 to December 2003. Currently, he serves as an Associate Editor ofCircuits, Systems and Signal Processing.

M. N. S. Swamy (S’59–M’62–SM’74–F’80) re-ceived the B.Sc. (honors) degree in mathematicsfrom Mysore University, Mysore, India, in 1954, theDiploma in electrical communication engineeringfrom the Indian Institute of Science, Bangalore,India, in 1957, and the M.Sc. and Ph.D. degreesin electrical engineering from the University ofSaskatchewan, Saskatoon, SK, Canada, in 1960 and1963, respectively. In August 2001, he was awardeda Doctor of Science in Engineering (Honoris Causa)by Ansted University, Tortola, British Virgin Islands,

“in recognition of his exemplary contributions to the research in electrical andcomputer engineering and to engineering education, as well as his dedicationto the promotion of signal processing and communications applications.”

Currently, he is a Research Professor and the Director of the Center forSignal Processing and Communications in the Department of Electrical andComputer Engineering, Concordia University, Montreal, QC, Canada, wherehe served as the Chair of the Department of Electrical Engineering from1970 to 1977 and Dean of Engineering and Computer Science from 1977 to1993. He has also taught at the Electrical Engineering Department, TechnicalUniversity of Nova Scotia, Halifax, NS, Canada, and the University of Calgary,Calgary, AB, Canada, as well as at the Department of Mathematics, Univer-sity of Saskatchewan. He has published extensively in the areas of numbertheory, circuits, systems, and signal processing, and holds five patents. Heis the coauthor of two book chapters and four books: Graphs, Networks andAlgorithms (New York: Wiley, 1981), Graphs: Theory and Algorithms (NewYork: Wiley, 1992), Switched Capacitor Filters: Theory, Analysis and Design(London, U.K.: Prentice-Hall, 1995) and Neural Networks in a SoftcomputingFramework (New York: Springer-Verlag, 2006). A Russian Translation of thefirst book was published by Mir Publishers, Moscow, Russia, in 1984, whilea Chinese version was published by the Education Press, Beijing, China, in1987. He was a founding member of Micronet from its inception in 1999 as aCanadian Network of Centers of Excellence until its expiration in 2004, andalso its coordinator for Concordia University.

Dr. Swamy is a Fellow of the Institute of Electrical Engineers (U.K.),the Engineering Institute of Canada, the Institution of Engineers (India),and the Institution of Electronic and Telecommunication Engineers (India).He has served the IEEE in various capacities such as the President-Elect in2003, President in 2004, Past-President in 2005, Vice-President (Publications)during 2001–2002, Vice President in 1976, Editor-in-Chief of the IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS—PART I: FUNDAMENTAL THEORY

AND APPLICATIONS from June 1999 to December 2001, Associate Editor ofthe IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS during June 1985–May1987, Program Chair for the 1973 IEEE Circuits and Systems Symposium,General Chair for the 1984 IEEE Circuits and Systems Symposium, Vice-Chairfor the 1999 IEEE Circuits and Systems Symposium, and a member of theBoard of Governors of the CAS Society. He is the recipient of many IEEECircuits and Systems Society awards including the Education Award in 2000,Golden Jubilee Medal in 2000, and the 1986 Guillemin–Cauer Best PaperAward.

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