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IEEE TRANSACTIONS ON BROADCASTING, VOL. 51, NO. 3, SEPTEMBER 2005 305

A New ICI Matrices Estimation Scheme UsingHadamard Sequence for OFDM Systems

Hsiao-Chun Wu and Yiyan Wu, Fellow, IEEE

AbstractThe intercarrier interference (ICI) matrix for theorthogonal frequency division multiplexing (OFDM) systemsusually has a fairly large dimension. The traditional least-squaresolution based on the pseudo-inverse operation, therefore, has itslimitation. In addition, the provision of a sufficiently long trainingsequence to estimate the complete ICI matrix is not feasible,since it will result in severe throughput reduction. In this paper,we derive a lower bound for the mean-square estimation erroramong the least-square ICI matrix estimators using differenttraining sequences and prove that the minimum mean-squareerror (MMSE) optimality is attained when the training sequencesin different OFDM blocks are orthogonal to each other, regard-less of the sequence length. We also prove that the asymptoticalmean-square estimation error using the maximal-length shift-reg-ister sequences ( -sequences) as in the existing communicationstandards is 3 dB larger than that using the perfectly orthogonalsequences for ICI matrix estimation. Thus, we propose to employthe training sequences based on the Hadamard matrix to achievea highly efficient and optimal ICI matrix estimator with minimummean-square estimation error among all least-square ICI matrixestimators. Meanwhile, our new scheme involves only square com-putational complexity, while other existing least-square methodsrequire the complexity proportional to the cube of the ICI matrixsize. Analytical and experimental comparisons between our newscheme using Hadamard sequences and the existing method using

-sequences (pseudo-random sequences) show the significantadvantages of our new ICI matrix estimator. The proposed methodis most suitable for OFDM systems with large amount of subcar-riers, using high order of subcarrier modulation, and designed forhigh-end of RF frequency band, where accurate ICI estimation iscrucial.

Index TermsChannel estimation, Hadamard sequence,least-square solution, minimum mean-square error, -sequence,OFDM, transceiver design.

I. INTRODUCTION

ORTHOGONAL frequency division multiplexing (OFDM)is widely adopted by wireless local-area networks, wire-less metropolitan networks and digital video broadcasting [1].However, in some wireless OFDM-based standards, such asthe DVB-T standard [2], the number of subcarriers is usu-ally fairly large. The problem in estimating the intercarrier

Manuscript received January 31, 2005; revised March 3, 2005. Thiswork was supported in part by Information Technology Research Award forNational Priorities (NSF-ECS 0426644) from the National Science Foun-dation, Faculty Research Initiation Award from the Southeastern Centerfor Electrical Engineering Education, Research Enhancement Award fromthe Louisiana-NASA Space Consortium and Faculty Research Award fromLouisiana State University.

H.-C. Wu is with the Communications and Signal Processing Laboratory, De-partment of Electrical and Computer Engineering, Louisiana State University,Baton Rouge, LA 70803 USA e-mail: [email protected]).

Y. Wu is with the Communications Research Centre, Ottawa, Ontario, K2H8S2 Canada (e-mail: [email protected]).

Digital Object Identifier 10.1109/TBC.2005.852244

interference (ICI) matrix poses the limitation on the OFDMsystem performance with large amount of subcarriers [3]. TheICI matrix results from different factors such as frequencyoffset, sampling clock offset, phase noise and time synchro-nization error [4], [5]. In the existing OFDM technology, thosefactors have been isolated and the associated problems havebeen separately tackled one at a time [4], [5]. However, thissequential procedure in [4], [5] may not be ideal, since theaforementioned factors cannot be addressed separately and in-dependently in practice. As a matter of fact, a more realisticOFDM signal model in the frequency-domain should involveICI matrix, which can be considered as the frequency-domainOFDM channel resulting from those factors simultaneously [6].The dimension of this matrix is proportional to the number ofOFDM subcarriers, often in the order of hundreds or thousandsin a practical OFDM implementation [2], [7]. As a rule ofthumb, the required training sequence must consist of dozensof thousands of data samples, if a traditional least-square so-lution is applied [8]. However, it is impossible to transmitso many training symbols that will significantly reduce thespectral efficiency. Meanwhile, the ICI matrix is time-varyingin a mobile environment and cannot be deemed as a constantmatrix during such a long period of training sequence trans-mission [9]. Therefore, only a small subset of entries in theICI matrix are usually estimated and the number of the pilotsymbols can be greatly reduced.

The difficulty of the frequency-domain channel estimation,or ICI matrix estimation, arises in the OFDM systems, be-cause the pilot symbols have to be allocated in every OFDMblock (symbol) separately [10]. Maximal-length shift-registersequences ( -sequences) (pseudo-random sequence) with wellcontrolled autocorrelation characteristics have been widelyadopted for the existing communication systems and havebeen shown to provide the minimum throughput sacrifice andthe nearly optimal channel estimation in time domain [11].However, it can not provide similar satisfactory solution whenthe ICI matrix is to be estimated in the frequency domain,because the sequence length or the period has to be restrictedthat it can fit into an OFDM block and the time structure of

-sequences cannot be fully utilized. In this paper, we derivethe mean-square error function for the ICI matrix estimationand compare the mean-square errors between the estimatorsusing the proposed Hadamard sequences and the conventional

-sequences. We will show that the Hadamard sequence ap-proach provides much improved performance for ICI matrixestimation. The proposed method is particularly suitable forhigh order modulation system, with large amount of subcarriers,and at high RF frequency band with severe frequency offsetemerging at the local oscillators, where accurate ICI estimation

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306 IEEE TRANSACTIONS ON BROADCASTING, VOL. 51, NO. 3, SEPTEMBER 2005

is crucial. Another advantage of using Hadamard sequences isthe computational simplicity.

This paper is organized as follows. The generalized formu-lation for the demodulation of OFDM signals involving an ICImatrix is provided and the ICI matrix approximation usingthe window truncation and circular representation techniquesis introduced in Section II. In Section III, the optimality forminimum mean-square ICI matrix estimation error is derived,the analytical comparison among the training sequences ispresented; and our new ICI matrix estimation scheme using theHadamard sequences is also introduced. The simulation resultswill be shown in Section IV. Concluding remarks will be drawnin Section V.

Notations: C denotes the field of complex values; denotesa vector and denotes a matrix; denotes the statisticalexpectation; is the Hermitian adjoint of ; is thediagonalized version of by setting all of the off-diagonal ele-ments in the square matrix to be zero; ifis a column vector, then the circular-shifted operation is definedas

if is a row vector, then the circular-shiftedoperation is defined as

II. OFDM SIGNALS AND ICI MATRIXA. OFDM Signal Model in the Frequency Domain

In general OFDM systems, the cyclic prefix sequence is as-sumed to be long enough to combat the longest multipath delaysuccessfully [1], [6], [10]. Therefore we need to be concernedwith the ICI only [1], [6], [10]. Provided subcarriers in eachOFDM block (symbol), according to [6], [12][14], theblock of demodulated signals at theOFDM receiver, after the discrete Fourier transform, can bewritten as

(1)where

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(2)and

ICI weighting coefficient associated with the interfer-ence from the subcarrier to the subcarrierICI matrix assumed to be stationary within a fewOFDM blocks

the block of transmitted symbols

the block of additive white noise

Assume that the subset of the block of transmitted signals, are the pilot

symbols. Thus the column vector

(3)can be called the training sequence vector in the OFDMblock.

B. Circulant Representation and Window Truncation for LargeICI Matrices

It is noted that no information symbol can be transmittedif the complete ICI matrix is to be estimated ac-

cording to (1), (2) and (3). Since the ICI matrix estimate haselements, it would involve dozens of millions of parame-

ters to be estimated in the DVB-T systems [2]. The least-squaresolution for ICI matrix estimation is usually computationally in-feasible and unreliable when an OFDM system operates in themobile environments. Hence we need to simplify the ICI matrix

to reduce the number of underlying parameters [3].For the simplification of this problem, a good approximation

of as analyzed in [6], [13], [14], can be formulated as

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(4)

where the row vector in

(5)is the circular-shifted sequence of

by samples tothe right [15]. Thus we only need to estimate the first rowvector in under this assumption. According to (1), (2),(4) and (5), the demodulated subcarrier signal in theOFDM block can be formulated as

(6)Very often, can be further truncated as a matter of fact that,for a positive integer

(7)where denotes the modulo- value of and isthe entry in .

According to (1)(7), we can predetermine the trunca-tion length for , and then assign the pilot symbolsas .Thus, the demodulated signal at the receiver can be formu-lated as

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(8)

where , ,; as depicted in Fig. 1, the training

WU AND WU: A NEW ICI MATRICES ESTIMATION SCHEME USING HADAMARD SEQUENCE FOR OFDM SYSTEMS 307

Fig. 1. The allocation of pilot symbols and training sequence in the i OFDMblock. (a) Transmitted signal vectors ; (b) circular-shifteds or shift (s ).

sequence vector in the OFDM block canbe written as

(9)

Thus, according to (5)(9), we can obtain

(10)

where the truncated ICI vectoris defined as

(11)

A typical example of the ICI vector and is depicted inFig. 2. Once we collect blocks of demodulated signals

, according to (10), (11), we can obtain

(12)

where

(13)(14)

Thus, in every block, subcarriers can constantlycarry the pilot information while the rest of sub-carriers can still carry the information symbols to maintain aconsistent spectral efficiency all the time. Therefore, with thiscirculant representation and window truncation technique as

Fig. 2. A typical example to illustrate the circulant representation and windowtruncation. Dashed dotted curve denotes the absolute values of the elements inc ; solid curve denotes the absolute values of the elements in c , which is thecircular-shiftedc by 7 samples to the right, and the truncation window lengthcan be chosen asM 1 = 15 in this example.

described in (5), (11), the spectral efficiency of our proposedICI matrix estimation scheme is always forevery OFDM block. Our proposed circulant representation andwindow truncation technique in this section may greatly reducethe number of parameters to be estimated according to the afore-mentioned ICI matrix characteristics so the required training se-quence lengths can also be significantly reduced.

III. LEAST-SQUARE ICI MATRIX ESTIMATION ANDTRAINING SEQUENCE ANALYSIS

According to (12), the least-square estimate of the truncatedvector in the ICI matrix can be given by[8]

(15)

It is noted that the singularity problem associated within (15), should be avoided. The reliability of depends onthe training data matrix . According to Appendix A, themean-square estimation error is given by

(16)where is the noise variance and the equality exists if and onlyif . According to (15) and (A.5),the least-square estimate can be achieved as

(17)

The signal-to-noise ratio can be defined as

(18)

where is denoted as theaverage transmitted pilot symbol energy and P is the sequence


length. According to (A.3), (A.5) in Appendix A and (18), themean-square estimation error can be formulated as

(19)where . The existing OFDM standards all adoptthe -sequences as the training sequences for channel estima-tion [2]. However, -sequences are not optimal in the minimummean-square estimation error sense. In this paper, we propose touse the Hadamard sequences as the new training sequences toestimate the ICI matrix in OFDM, which can result in the op-timal ICI matrix estimates with minimum mean-square estima-tion error. The estimation error analyses for both -sequencesand Hadamard sequences are provided next.

A. Mean-Square Estimation Error Using the -SequencesIn the existing OFDM standards, the -sequences are widely

adopted as the training sequences because they can be easilygenerated using a circular-shift register and they have well con-trolled autocorrelation functions [11]. For an -sequence withperiod , the training data matrix

can be given by

(20)

where is the vector resulting from one period of-sequence and , .

Thus, the associated periodic autocorrelation function,, , , can be derived as [11]

(21)

According to (21), the correlation matrix using the-sequences with period P can be derived as

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(22)

According to Appendix C, we can obtain the mean-square es-timation error as . It isnoted that given a very long period

(23)

When the -sequences are applied in the frequency domain, thecyclic-shift time structure cannot be utilized and the sequenceperiod has to be identical to the ICI vector length, therefore theasymptotical mean-square estimation error in (23) is 3 dB largerthan the optimal value give by (19), which is achieved using theperfectly orthogonal sequences. Comparing (19) and (23), wediscover that the mean-square estimation error using the -se-quences is twice as much as the optimal value for the large ICImatrices fully loaded with pilot symbols.

B. Minimum Mean-Square Error Estimation Using theHadamard Sequences

According to (19) and (A.5) in Appendix A, the optimal es-timation error is achieved, when

. Thus, according to (18), the min-imum mean-square error (MMSE) ICI vector estimate can beobtained as

(24)

To achieve the MMSE estimate as given by (24), the trainingdata matrix should consist of orthogonal row vectors. Con-sequently, in this paper, we propose to use the Hadamard codesas the training sequences for optimal ICI matrix estimation.Hadamard code has long been adopted as the spreading codesfor multi-access communications [11], [16] or space-time blockcodes among multiple antennae [17], [18] due to its perfectorthogonality, i.e., any Hadamard codeword is completely or-thogonal to others regardless of the code length [11]. Recently,Kaiser also proposed to adopt the Hadamard sequences as thespace frequency block codes [19][22] in OFDM. To the bestof our knowledge, no existing literature has ever addressed theuse of the Hadamard sequences as the training sequences in thefrequency domain for OFDM systems. There are two ways togenerate Hadamard sequences: one is through the Hadamardgenerator in [11]; the second way is to construct the Hadamardmatrix [23], wherein the corresponding rows are the completeset of orthogonal Hadamard sequences. A Hadamardmatrix, , for , can be written in a recursiveformula as [23]

(25)

where and denotes the Kronecker product[23]. According to (25), typical examples of binary Hadamardsequences for , 2, 3 are listed in Table I. How-ever, the Hadamard training sequences derived from (25)are all real-valued. We may modulate the Hadamard sequencesusing both in-phase and quadrature-phase components to max-imize the transmission power so as to improve the transmissionsignal-to-noise ratio and mitigate the fading effect on the pilotsymbols. Therefore, for QPSK and QAM constellation symbols,we design new bi-phase -bit Hadamard sequences as follows.First we denote the original binary Hadamard sequences as vec-tors, such that

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TABLE IBINARY (SINGLE-PHASE) HADAMARD TRAINING SEQUENCES

TABLE IIBI-PHASE HADAMARD TRAINING SEQUENCES

then the new bi-phase Hadamard sequence vectors can be con-structed as

(26)

Examples of bi-phase Hadamard sequences are provided inTable II.

It is noted that the lengths of Hadamard sequences are re-quired to be radix-2. Hence we need to modify (9) and (11) forHadamard training sequences by getting rid of one more pa-rameter in both and , such that the Hadamard trainingsequence vector is

(27)and the corresponding truncated ICI vectoris shown in (28) at the bottom of the page where is a

Fig. 3. Theoretical mean-square estimation error (MSE) comparison amongdifferent training sequences.

radix-2 number to satisfy the typical Hadamard sequence length.For the sequence length of ,we propose to construct a Hadamard training data matrix

as

(29)

where , for , are given by (25) and (26).

IV. SIMULATIONTo compare our proposed MMSE ICI matrix estimator using

the Hadamard sequences and the conventional estimator usingthe -sequences, first we construct the training data matrices forthese two different schemes according to (20) and (29), respec-tively. It is noted that the sequence length of an -sequence canonly be a radix-2 number minus one, while that of a Hadamardsequence can only be a radix-2 number.

A. Comparison of Theoretical Mean-Square Estimation ErrorsUsing Different Training Sequences

Assuming that the length of the approximated ICI vector es-timate given by (15) is equivalent to, or larger than, thenumber of nonzero elements in the actual ICI vector givenby (4), or the number of parameters in is sufficient towell model . According to (19) and (23), our proposed MMSEICI vector estimator using the Hadamard sequences will reachthe optimal performance and the corresponding mean-squareestimation error is independent of the sequence length, whilethe existing ICI vector estimator using the -sequences de-pends on the sequence length or the period P. The theoreticalmean-square estimation errors versus signal-to-noise ratios areplotted in Fig. 3. Our proposed MMSE ICI vector estimatorusing the Hadamard sequences greatly outperforms the existingICI vector estimator using the -sequences especially when theSNR is low.

(28)


TABLE IIIICI MATRIX ESTIMATION SCHEMES IN COMPARISON

B. Comparison of ICI Matrix Estimation Schemes UsingDifferent Training Sequences

In this simulation, all of the system parameters comply withthe IEEE 802.11a standard [7]. The number of subcarriers ischosen to be or . The COST 207 channelmodel for rural areas in [24] is tested. As discussed in Sec-tion II-B, only the first row of the ICI matrix needs to beestimated and then the circulant representation can be appliedas

(30)

According to (30), the average ICI vector-norm-to-error-ratio(ICI-VNER) can be defined as

(31)The number of training signal blocks is identical to the se-

quence length or sequence period, namely, . Onehundred Monte Carlo trials are simulated to test four differentICI matrix estimation schemes as listed in Table III. Accordingto Table III, the spectral efficiencies of the four schemes are dif-ferent. However, the spectral efficiency differences between theconventional scheme using -sequences and our proposed newscheme using Hadamard sequences are negligible. With thenormalized frequency offset , the random fadingchannels based on the COST207 model for rural areas in [24]and the frequency-offset-only model in [13] are used to gen-erate the corresponding ICI matrices [6], [24]. The ICI-VNERcurves versus signal-to-noise ratios are depicted in Fig. 4, 8(QPSK-OFDM with 64 subcarriers), Figs. 5, 9 (64QAM with64 subcarriers), Figs. 6, 10 (QPSK-OFDM with 256 subcar-riers) and Figs. 7, 11 (64QAM-OFDM with 256 subcarriers).According to Figs. 411, our proposed ICI matrix estimator

Fig. 4. ICI vector-norm-to-error-ratio (ICI-VNER) comparison using differenttraining sequences for the QPSK-OFDM (N = 64 subcarriers, fT = 0:2)system in the COST207 channel for rural areas in [24].

Fig. 5. ICI vector-norm-to-error-ratio (ICI-VNER) comparison using differenttraining sequences for the 64QAM-OFDM (N = 64 subcarriers,fT = 0:2)system in the COST207 channel for rural areas in [24].

Fig. 6. ICI vector-norm-to-error-ratio (ICI-VNER) comparison using differenttraining sequences for the QPSK-OFDM (N = 256 subcarriers,fT = 0:2)system in the COST207 channel for rural areas in [24].

outperforms the conventional method by up to 2 dB in lowsignal-to-noise ratio conditions. It is close to the asymptotical


Fig. 7. ICI vector-norm-to-error-ratio (ICI-VNER) comparison using differenttraining sequences for the 64QAM-OFDM (N = 256 subcarriers, fT =0:2) system in the COST207 channel for rural areas in [24].

Fig. 8. ICI vector-norm-to-error-ratio (ICI-VNER) comparison using differenttraining sequences for the QPSK-OFDM (N = 64 subcarriers, fT = 0:2)system in the absence of multipath fading and Doppler drift [13].

Fig. 9. ICI vector-norm-to-error-ratio (ICI-VNER) comparison using differenttraining sequences for the 64QAM-OFDM (N = 64 subcarriers,fT = 0:2)system in the absence of multipath fading and Doppler drift [13].

mean-square error analysis as given by (19) and (23). Heuristi-cally speaking, our proposed scheme has the clear advantage forlarge subcarrier OFDM systems, when frequency offset is large,

Fig. 10. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usingdifferent training sequences for the QPSK-OFDM (N = 256 subcarriers,fT = 0:2) in the absence of multipath fading and Doppler drift [13].

Fig. 11. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usingdifferent training sequences for the 64QAM-OFDM (N = 256 subcarriers,fT = 0:2) system in the absence of multipath fading and Doppler drift [13].

and noise level is high. It is also noted that the ICI-VNER curveshave the flooring effect, because the truncation errors limit theperformances of the estimators when our proposed techniques inSection II-B are used here. It is noted that our proposed circularrepresentation technique for the ICI matrices works very wellfor the frequency-offset-only ICI model in [13] because suchICI matrices are very close to circulant matrices as discussed in[6].

C. Computational Complexity ComparisonWe also compare the computational complexities here, in

terms of complex multiplications. The numbers of subcarriersare considered as 64, 128, 256, 512, 1024, 2048, and 4096.

Four different schemes are compared, namely, (i) ICI matrixestimator using the -sequences with length ,(ii) ICI matrix estimator using the -sequences with length

, (iii) ICI matrix estimator using the Hadamardsequences with length , and (iv) ICI matrix esti-mator using the Hadamard sequences with length .According to Appendix D, the computational complexitiesin terms of subcarrier number for these four schemes are


Fig. 12. Computational complexity comparison for the ICI matrix estimatorsusing different training sequences (P is the training sequence length and thespectral efficiency is N P=N ).

provided in Table III. The illustrative curves of the requiredcomplex multiplications versus the subcarrier number forthese four schemes are plotted in Fig. 12. According to Fig. 12,the computational complexity of our proposed estimator islower than the conventional estimator by one to two orders ofmagnitude in terms of complex multiplications.

V. CONCLUSIONIn this paper, we derived the mean-square error bound for

the least-square ICI matrix estimators and proposed to usethe Hadamard sequences as the training sequences for betterICI estimation in OFDM system. In addition, we provedthat the asymptotical mean-square estimation error using theHadamard sequences is only one half as much as that using the

-sequences adopted by all of the existing OFDM standards.A novel minimum mean-square error (MMSE) ICI matrixestimator was designed using the circular representation andwindow truncation technique. Our new ICI matrix estimationscheme outperformed the conventional least-square estimationtechnique not only in the mean-square estimation error, but alsoin the computational complexity. Through numerous MonteCarlo simulations, we verified that our new MMSE ICI matrixestimator is very robust over fading channels for differentmodulation schemes, subcarrier numbers and signal-to-noiseratios. For the further improvement of our MMSE ICI matrixestimator to deal with the ICI model involving severe selectivemultipath fading, sampling clock offset and time synchro-nization error, the training sequences in the individual OFDMblocks for estimating the first ICI vector may be circular-shiftedaltogether dynamically for estimating the second, the third ICIvectors and so on. Therefore, the restriction of the estimatedICI matrix with circular representation can be easily removedby transmitting those training sequences dynamically. It canbe investigated that the ICI matrix estimation schemes willhave significant advantages over other coding schemes suchas the convolutional coding [7] and the ICI self-cancellationcoding [13] because the composite phase ambiguity is a crucialproblem of degrading the effectiveness of those coding schemesand they need the further spectral efficiency reduction [14].

The comparison between the ICI matrix estimation methodsand other coding schemes will be provided in our forthcomingpaper.

APPENDIX

A. Bound of Least-Square EstimatorsThe mean-square error of the least-square ICI vector esti-

mator as given by (15) can be derived as

(A.1)According to (15) and , (A.1) can be simpli-fied as

(A.2)

Assume that since noise is white. Then(A.2) can be further simplified as

Since is a positive-definite Hermitian matrix [8], itcan be decomposed as

(A.4)where is the eigenvector matrix and is the eigenvalue ma-trix such that

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Therefore, according to Theorem 1 in Appendix B, substitutingwith into (B.1), we obtain


(A.5)

where the equality exists if and only ifor is a diagonal matrix.

B. Proof of the Inequality in (A.5)Theorem 1: For any positive-definite Hermitian matrix

, , the following inequality exists:

Proof: Since is positive-definite Hermitian, is non-singular and its inverse matrix can be written as [23]:

(B.2)

where are the eigenvalues associated with, and are the corresponding column eigenvec-

tors. Consequently,

Let be the column vector containing all zero entries except aone as its element. It is noted that

(B.4)

Since are the eigenvectors of a Hermitian matrix, accordingto [25], the matrix expressed as

satisfies the following condition:

(B.5)

Thus, according to (B.3), (B.4), (B.5) and Theorem 2 in Ap-pendix B, we can derive

Theorem 2: Given a set of positive valuesand a matrix where ,

, , , , and ,, , then

(B.6)

where the equality in (B.6) exists if and only if is an identitymatrix.

Proof: According to [23], it is noted that

(B.7)

where is the diagonal matrix containing .The equality in (B.7) exists only when

where is an identity matrix. That is

(B.9)

where the equality exists if and only if is an identity matrix.

C. Mean-Square Estimation Error Using -SequencesGiven the correlation matrix as in (22), we can

form the eigen-decomposition as

(C.1)

where is the Fourier matrix and isthe diagonal matrix containing the discrete Fourier transformvalues as its diagonal elements and

(C.2)According to (C.2)

(C.3)

According to (20), (21), (22), (A.5), (C.1), (C.2), (C.3), themean-square estimation error using -sequences can be derivedas

(C.4)


D. Computational Complexities of Least-SquareICI Matrix Estimators

1) Computational Complexity of ICI Matrix Estimator in Eq.(15) Using Non-Orthogonal Sequences: Here we focus on thecomplex-valued multiplications only, which induce the majoritycomputational time in matrix operations. For theICI vector estimate given by (15), using nonorthogonalsequences, requires complexmultiplications and requirescomplex multiplications. Since is Hermitian butnot necessarily Toeplitz, the most efficient way to compute

in (15) involves the Cholesky outer product[26]. If , then ,where is a lower-triangular ma-trix [27]. It requires complex multiplicationstotally to construct both and . Then the product of

involves additional complexmultiplications. The final product in(15) requires complex multiplications. There-fore, the total computational complexity in (15) involves

complex multiplications.2) Computational Complexity of ICI Matrix Estimator

in (23) Using Hadamard Sequences: According to (24),the MMSE estimate involves

complex multiplications.

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Hsiao-Chun Wu (M98) received a B.S.E.E. degreefrom National Cheng Kung University, Taiwan,in 1990, and the M. S. and Ph.D. degrees in elec-trical and computer engineering from University ofFlorida, Gainesville, in 1993 and 1999, respectively.

From March 1999 to January 2001, he had workedfor Motorola Personal Communications SectorResearch Labs as a Senior Electrical Engineer.Since January 2001, he has joined the faculty inDepartment of Electrical and Computer Engineering,Louisiana State University, Baton Rouge. His re-

search interests include optimization, estimation, wireless communications andsignal processing. Dr. Wu currently serves as an Associate Editor for IEEETRANSACTIONS ON BROADCASTING.

Yiyan Wu received the M.Eng. and Ph.D. degreesin electrical engineering from Carleton University,Ottawa, Canada, in 1986 and 1990, respectively.He is a Principal Research Scientist with the Com-munications Research Centre in Ottawa, Canada.Dr. Wus research interests including multimediacommunications, digital broadcasting and commu-nication systems engineering. He is a Fellow of theIEEE, an adjunct professor of Carleton University inOttawa, Canada, a member of the IEEE BroadcastTechnology Society Administrative Committee,

Editor-in-Chief for IEEE TRANSACTIONS ON BROADCASTING, and a memberof the ATSC Board of Directors, representing IEEE.

tocA New ICI Matrices Estimation Scheme Using Hadamard Sequence forHsiao-Chun Wu and Yiyan Wu, Fellow, IEEEI. I NTRODUCTIONNotations: C denotes the field of complex values; $\harp{a}$ den

II. OFDM S IGNALS AND ICI M ATRIXA. OFDM Signal Model in the Frequency DomainB. Circulant Representation and Window Truncation for Large ICI

Fig.1. The allocation of pilot symbols and training sequence inFig.2. A typical example to illustrate the circulant representaIII. L EAST -S QUARE ICI M ATRIX E STIMATION AND T RAINING S EQUA. Mean-Square Estimation Error Using the $m$ -SequencesB. Minimum Mean-Square Error Estimation Using the Hadamard Seque

TABLEI B INARY (S INGLE -P HASE ) H ADAMARD T RAINING S EQUENCTABLEII B I -P HASE H ADAMARD T RAINING S EQUENCESFig.3. Theoretical mean-square estimation error (MSE) comparisoIV. S IMULATIONA. Comparison of Theoretical Mean-Square Estimation Errors Using

TABLEIII ICI M ATRIX E STIMATION S CHEMES IN C OMPARISONB. Comparison of ICI Matrix Estimation Schemes Using Different T

Fig.4. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usiFig.5. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usiFig.6. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usiFig.7. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usiFig.8. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usiFig.9. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usiFig.10. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usFig.11. ICI vector-norm-to-error-ratio (ICI-VNER) comparison usC. Computational Complexity Comparison

Fig.12. Computational complexity comparison for the ICI matrix V. C ONCLUSIONA. Bound of Least-Square EstimatorsB. Proof of the Inequality in (A.5)Theorem 1: For any positive-definite Hermitian matrix $\widetildProof: Since $\widetilde{A}$ is positive-definite Hermitian, $\w

Theorem 2: Given a set of positive values $\{\lambda_{1},\lambdaProof: According to [ 23 ], it is noted that $$\eqalignno{\sum_{

C. Mean-Square Estimation Error Using $m$ -SequencesD. Computational Complexities of Least-Square ICI Matrix Estimat1) Computational Complexity of ICI Matrix Estimator in Eq. (15) 2) Computational Complexity of ICI Matrix Estimator in (23) Usin

R. V. Nee and R. Prasad, OFDM for Wireless Multimedia Communicat

Transmission System for Digital Terrestrial Television BroadcastY. S. Choi, P. J. Voltz, and F. A. Cassara, On channel estimatioM. Speth, S. Fechtel, G. Fock, and H. Meyr, Optimum receiver desH.-C. Wu and G. Gu, Analysis of intercarrier and interblock inte

Wireless LAN Medium Access Control (MAC) and Physical Layer (PHYS. Haykin, Adaptive Filter Theory, 4th ed, New Jersey: Prentice-Y. Li and L. J. Cimini, Bounds on the interchannel interference L. Hanzo, M. Munster, B. J. Choi, and T. Keller, OFDM and MC-CDMR. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to SS. G. Kang, Y. M. Ha, and E. K. Joo, A comparative investigationJ. Armstrong, Analysis of new and existing methods of reducing iH.-C. Wu and X. Huang, Joint phase/amplitude estimation and symbA. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time SiS. Kaiser, OFDM code-division multiplexing in fading channels, IS. Lei and S. Roy, A rate-one nonorthogonal space-time coded OFDA. Doufexi, A. P. Miguelez, S. Armour, A. Nix, and M. Beach, UseS. Kaiser, Space frequency block coding in the uplink of broadbaR. A. Horn and C. R. Johnson, Matrix Analysis . New York: CambriA. F. Molisch, Wideband Wireless Digital Communications, New JerD. K. Faddeev and V. N. Faddeeva, Computational Methods of LineaG. W. Stewart, The decompositional approach to matrix computatioG. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed. Bal

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