IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 7, JULY

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 7, JULY 1999 2007

Redundant Filterbank Precoders and EqualizersPart II: Blind Channel Estimation,

Synchronization, and Direct EqualizationAnna Scaglione,Student Member, IEEE,Georgios B. Giannakis,Fellow, IEEE,and Sergio Barbarossa,Member, IEEE

Abstract—Transmitter redundancy introduced using finite im-pulse response (FIR) filterbank precoders offers a unifying frame-work for single- and multiuser transmissions. With minimalrate reduction, FIR filterbank transmitters with trailing zerosallow for perfect (in the absence of noise) equalization of FIRchannels with FIR zero-forcing equalizer filterbanks, irrespectiveof the input color and the channel zero locations. Exploitingthis simple form of redundancy, blind channel estimators, blocksynchronizers, and direct self-recovering equalizing filterbanksare derived in this paper. The resulting algorithms are compu-tationally simple, require small data sizes, can be implementedonline, and remain consistent (after appropriate modifications),even at low SNR colored noise. Simulations illustrate applica-tions to blind equalization of downlink CDMA transmissions,multicarrier modulations through channels with deep fades, andsuperior performance relative to CMA and existing output di-versity techniques relying on multiple antennas and fractionalsampling.

Index Terms—Blind channel estimation, block transmissions,filterbanks, intersymbol interference, minimum mean-square er-ror communication receivers, nondata aided synchronization,precoding, self-recovering equalization, zero forcing.

I. INTRODUCTION

REDUNDANCY at the transmitter buildsinput diversityindigital communication systems and is well motivated for

designing error correcting codes (e.g., [2]). Recently, however,input diversity has been exploited also for ISI suppressionusing precoders operating in the complex (as opposed toGalois) field, [3], [14], [19], [24], [26], [27], [32]. Differentprecoding schemes are possible. Multiplying the input by aknown periodic sequence offers a precoding scheme that doesnot require any increase of the transmission rate, although theconstellation’s modulus is affected and equalization of FIRchannels with FIR equalizers is impossible [3], [26]. On theother hand, repeating input symbols as in [27] leads to FIRequalizers but reduces information rate by half. Combining

Manuscript received January 6, 1998; revised January 18, 1999. Thiswork was supported by the National Science Foundation under GrantCCR-9805350. It was presented in part at Allerton’97 and ICASSP’98. Theassociate editor coordinating the review of this paper and approving it forpublication was Dr. Truong Q. Nguyen.

A. Scaglione and S. Barbarossa are with the Infocom Depart-ment, University of Rome “La Sapienza,” Roma, Italy (e-mail:[email protected]; [email protected]).

G. B. Giannakis is with the Department of Electrical and ComputerEngineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:[email protected]).

Publisher Item Identifier S 1053-587X(99)04656-5.

desired features, filterbank precoders and blind equalizerswere proposed in [14] to minimize the rate reduction andobviate channel zero restrictions imposed by spatio-temporaloutput diversity methods that rely on fractional sampling(FS) and/or multiple-antenna reception [13], [15], [22], [28],[33]. However, identifiability in [14] was established only forwhite inputs and linear equation or subspace algorithms weredeveloped for simple precoders (see also [19]). In a deter-ministic multirate framework, filterbanks fornonblindchannelequalization were also proposed in [32] under restrictions onthe channel zeros.

In the companion paper [24], we have shown that re-dundant filterbank precoders offer a unifying discrete-timemodel that encompasses a wide range of digital modulationand coding schemes [24]. Those include periodic and linecodes, orthogonal frequency-division multiplexing (OFDM)and discrete multitone (DMT) [5], [30], fractional sampling[28], (de-)interleaving, as well as multiuser transmissions suchas TDMA, FDMA, CDMA, and the most recent discretewavelet multiple access (DWMA) schemes [23] (see also [1]and [31]). However, self-recovering (orblind) approaches tochannel estimation, synchronization, and equalization were notaddressed in [24]. Redundant precoding brings input diversitysimilar to that available with training sequences that havebeen exploited recently in a semi-blind channel estimationframework [18].

Motivated by the generality and importance of filterbankprecoders, this paper builds on [14] and [24] and proposesnovel deterministic methods for blind channel estimation,block synchronization, and direct equalization without impos-ing restrictions on channel zeros (Section III). The determin-istic solution, as opposed to statistical methods, is particularlyappealing for transmissions over slowly varying channels.The methods are also applicable to random inputs, white orcolored (important for coded transmissions). General precodersequipped with trailing zeros (TZ) offer computationally simplealgorithms and lead to zero-forcing (ZF) or minimum mean-square error (MMSE) equalizing filterbanks that accept alsoadaptive implementations. Exciting options become availablefor blind equalization in OFDM, which is a transmissionscheme known to suffer from deep channel fades (see also [6]).The importance of increasing the robustness of OFDM againstfrequency selective fading is testified by its applications:OFDM is currently used in digital audio broadcasting (DAB)[8], and it has been selected for European digital terrestrial

1053–587X/99$10.00 1999 IEEE

2008 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 7, JULY 1999

television broadcasting (DTTB) [9], and, in its DMT version[4], for the asymmetric digital subscriber loop (ASDL) inthe United States. Our methods are also directly applicablein TDMA and CDMA systems for the blind equalization ofthe downlink channel. Consistency analysis and modificationsneeded at low SNR are given in Section IV. In Section V, theproposed methods are tested via simulations and comparedwith alternative methods that employ space diversity [13],[15], [22], [33], fractional sampling (FS) [28], or they exploitapriori knowledge of the symbol constellation, e.g., the constantmodulus algorithm (CMA) [16].

II. PRELIMINARIES

We will consider the same discrete-time multirate transmit-ter model presented in [24] for the baseband communicationsystem reported in Fig. 1. We will also assume Nyquist signal-ing pulses. Downsamplers and upsamplers perform blocking(i.e., multiplexing) and unblocking (demultiplexing) opera-tions. With , the ratio represents theamount of redundancy introduced. The input to the upsamplerof the th branch is . It represents the

th symbol in the th block of symbols, whereas in themultiuser case, it stands for theth user’s bits. We will use thesame notation as in [24], denoting the transmit data sequence[summation of all the sequences at the output of eachbranch] as , the noise-free channel output sequence as

, and the noisy received signal as ,where is additive noise (independent of the transmittedsymbols) characterized by the covariance matrix and,finally, by the equalizing filterbank output. The blockdata model in [24] expresses the input–output relationship inmatrix form. Let us define the vectors andthat are defined as in

(1)

The vectors , , , and , are all definedas

(2)

where the channel vector andprecoder and equalizing matrices and ,

whose elements are, respectively,, for and

, i.e., the columns of theth ( th) matrix( ), contain the th ( th) segment of length ( )

of the filters’ impulse responses ( ).Introducing the channel matrices aswell, for , we can express (see [24]for details) the transmitted block-data sequence as

, the noise-free blocked channel output as, and the equalized block sequence

as

(3)

We assume, as in [24], the following:a0) Channel is th-order FIR with .a1) For a given , the pair is chosen to satisfy

and .In [24, Th. 1], extra conditions on the precoding strategy

that guarantee the existence of FIR linear block equalizers thatare independentof the channel zero locations are established.The condition , according to the definition of ,implies that or, in words, thatthe interblock interference (IBI) is limited only to subsequentblocks. Existence of FIR linear equalizers requires, in any case,redundant precoding that is obtained setting , while thecomplexity of the precoding strategy and equalization dependson the ratio . In this paper, we consider the specificclass of precoders that allows exact FIR-ZF equalization,fulfilling the requirements of [24, Th, 2]. Specifically, weassume the following.

a2) Precoder filters have trailing zeros (TZ), i.e.,, , and are linearly

independent, i.e., rank , which guarantees one-to-onemapping and, thus, recovery of from the coded symbols

.In force of a2), the precoder is of order zero,

implying that , and assumes the form

(4)

where is a full-rank matrix with elementsfor , whereas indicates1 an

matrix of zeros that creates the guard interval oftrailing zeros. The insertion of a null guard time interval is notnew in digital communications. DVB systems, for example,send a null guard time interval at the beginning of each frame,for synchronization purposes, whose duration varies from 168to 1297 s [9]. Exact knowledge of is not required as long as

is chosen to satisfy , where is a known upperbound on the channel order. Taking into account the FIR natureof both and , it is not difficult to verifythat TZ prevent IBI, i.e., (see also [24]); thus,assumptions a0)–a2) imply that , where,according to the definition of , matrix is

......

......

. . .. . .

(5)

and is Toeplitz lower triangular matrix with first columnand first row .

1In general, we will use the symbol0 to indicate null vectors and matriceswithout specifying their dimensions unless they are not obvious.

SCAGLIONE et al.: REDUNDANT FILTERBANK PRECODERS AND EQUALIZERS PART II 2009

Fig. 1. Multirate discrete-time transmitter/channel/receiver model.

Denoting by the Toeplitz matrix formed by thefirst columns of

......

.. ....

. . ... .

. . ... .

.... . .

.. ....

(6)

we have that ; therefore

(7)

and the received block data model is [24]

(8)

Henceforth, we will focus on the block-by-block transmissionand reception framework described by the model in (8).Equation (8) shows that in the absence of noise, there is also aone-to-one mapping between input and output blocks. Thanksto a1) and a2), consecutive data blocks anddo not interfere with each other. Hence, is a sufficientstatistic with respect to , and then, the linear equalizercan be without loss of generality (w.l.o.g.) of order zero, i.e.,

. Indicating the equalizing matrix as, (3) becomes

(9)

We report the statement of [24, Th. 2].Theorem—TZ Precoders:Under a0)–a2), there exists a zero

order equalizing filterbank such that .Moreover, the minimum-norm ZF filterbank is unique and isgiven by

(10)

where denotes pseudo inverse.Later on, we will make use of the following persistence-

of-excitation (p.e.) assumption on the transmitted symbols:

a3) There exists an , such that the matrixhas full rank . Note that

as , tends to the input correlation

matrix . In general, a3) is satisfied even for colored(e.g., coded) inputs provided that their spectra arenonzero for at least frequencies (modes). Recall thatp.e. assumptions like a3) are needed even with nonblindidentification problems.

Given blocks of data only, and based ona0)–a3), theobjectiveof this paper is threefold:

i) identify the channel ;ii) estimate directly the equalizer matrix in (9);iii) derive blind synchronizers to make block-coherent re-

ceiver processing possible.

Our goal for blind channel estimation, besides the evidentpurpose of avoiding the periodic transmission of bandwidthconsuming training sequences, is also instrumental for i) send-ing channel status information (CSI) back to the transmitter,whenever a feedback channel is available, to optimize thetransmission strategy, as in [24], or ii) for deriving zero-forcing(ZF), decision-feedback (DF), or minimum-mean square error(MMSE) equalizers that also rely on CSI. Even with moderatenumber of filters in the precoder, the maximum likelihood(ML) receiver implemented with Viterbi’s algorithm has pro-hibitively large complexity, which motivates the search forlinear (and preferably low order FIR) equalizing filterbanks.ZF solutions offer (almost) perfect symbol recovery in (highSNR) noise-free environments, and their performance in termsof error probability is easily computable. At low SNR, a vectorMMSE (or Wiener) equalizer is better motivated than the ZFsolution (10) and can be derived by minimizing

tr . The MMSE solutionis obtained by equating to zero the gradient of withrespect to (see [24]) and is

(11)

In the absence of noise,, which is satisfied if and only if and therefore,

[c.f. (10)] . Observe, however, thatin (11) does not tend to in (10), as ,

and matrix tends to become singularbecause the matrix has rank .

Note that [24, Th. 2] poses no constraints on the channelzeros. In contrast, FIR-ZF equalizers in [13], [15], [22], [27],[28], [32], [33] do not exist for certain configurations ofchannel zeros on the unit circle, and more important, their


performance degrades even when channels have zeros close tothose noninvertible configurations. If an upper boundis only available on the channel order, Theorem 2 holds truewith replacing in a1). Because, for a fixed bandwidth,the information rate depends only on the ratio , weunderscore that under a1), the rate reduction can be madearbitrarily small by selecting (and thus ) sufficientlylarge. Of course, this comes at the cost of increased decodingdelay and greater complexity so that bandwidth efficiency isgenerally traded with computational complexity and maximumallowable delay.

We wish to emphasize that the computation of the equalizingmatrices and from (10) and (11) requires knowl-edge, or estimates, of the channel. In the ensuing sections,we will derive self-recovering channel estimation and directequalization methods that take advantage of the redundancyintroduced at the transmitter by the insertion of trailing zeros.The latter can be interpreted as a form ofdistributed training.Since both data and trailing zeros are exploited for channelidentification and direct equalization, the proposed approachescan be classified as semi-blind (see also [18]). On the otherhand, our methods rely on the received data only, and fromthis point of view, our receivers are self-recovering (or blind).

III. B LIND SYMBOL RECOVERY

Blind channel estimation is well motivated for wirelessenvironments where the multipath channel changes rapidlyas the mobile communicators move. Self-recovering equaliza-tion schemes are important to avoid frequent retraining andthus increase bandwidth efficiency. In our block-transmissionschemes, both channel estimation (Section III-A) and directequalization (Section III-B) assume knowledge of the begin-ning of each block, which is a subject we also address lateron (Section III-C).

A. Blind Channel Estimation

Under a2), and collecting data vectorsin a matrix, we arrive at

(12)

where is defined as in a3). From (6), it is sufficient thatone channel coefficient is different from zero to ensure thatrank , and this, along with a2) and a3), implies thatrank . Therefore, the nullity of the matrixis , and the eigendecomposition

(13)

yields the matrix , whose columns span the nullspace . Because in (12) is full rank,

, where stands for range space. However, sinceis orthogonal to , it follows that

(14)

where denotes the th column of , andis the Toeplitz matrix in (6) denoting convolution. Because

convolution is commutative, (14) can be written as

(15)

where each is an Hankel matrix formed byas

......

......

(16)

Our result and the corresponding algorithm rely on (15) andare summarized in the following.

Theorem 1: Let a0)–a3) hold true. Starting from the datamatrix , we form the matrix as in(12)–(15). Channel vector can then be obtained as the unique(within a scale factor) null eigenvector of in (15).

Proof: We will prove channel identifiability by contra-diction. Let us assume that (15) admits two different solutions

and , where represents the true channel. Hence

(17)

It is easy to verify that the left null space of the convolutionmatrix , which is a Toeplitz matrix [see (6)], is spanned by

Vandermonde vectors of the form ,where , are the distinct2 roots of the channeltransfer function . Therefore, there exists an

full-rank matrix such that ,where is a matrix formed with the Vandermondevectors . If the vectors and , with ,must satisfy (17), then

(18)

Using , we can rewrite (18) as, or since is full rank

(19)

Equation (19) implies that (where is a complex con-stant) as a consequence of the fact that only impulse responseshaving the same roots (and with the same multiplicity) sharethe same Vandermonde (or generalized Vandermonde) basisfor the null space of . In fact, if two polynomials havingthe same order share all the roots, their coefficients have to beproportional by a constant factor. This proves that the channelcan be identified up to a scalar factor as in all blind channelidentification methods.

The result and proof of Theorem 1 is similar in spirit with[22]. However, as we mentioned earlier, our method entailsno assumptions on the FIR channel zeros, and generalizesthe special TDMA-like result of [19] to arbitrary precoderssatisfying a2) and with respect to [19] it is deterministic.Another distinct feature is our method’s behavior under chan-nel order mismatch. If the channel order is underestimated,similar to existing algorithms, our algorithm is not expected towork. However, in contrast to most output diversity methods,

2If H(z) has multiple roots, it is possible to extend appropriately thedefinition using the so-called generalized Vandermonde vectors (see e.g., [21]).


we also verify in the simulations section that our methodis interestinglynot affected by channel order overestimation.Since in most applications the maximum delay is known,an upper bound on the maximum channel order is readilyavailable. Furthermore, the method is robust with respect toDoppler shifts modeled as a multiplication of the transmitsequence by in contrast with the standard channelestimation techniques of OFDM systems, based on pilot tones.

With the channel matrix obtained from (15), we canproceed to determine either the ZF equalizer filterbank from(10) or the MMSE equalizer from (11). In fact, it is possibleto derive MMSE equalizers involving a delayed decision (withor without feedback) or pursue the computationally complexbut optimal ML receiver. Our channel estimation algorithm issummarized as follows.

1) With blocks of data, form the matrixas in (12).

2) Determine the eigenvectors , , corre-sponding to the null eigenvalues of the matrix(to be replaced by the smallest eigenvalues for the caseof noisy observations).

3) Estimate the channel vectoras the nontrivial solutionof the system of linear equations in (15).

4) Use the channel estimateto form the matrix in(6).

5) Equalize the data with .

We focus next on direct blind equalizers that do not evenrequire channel estimation as a first step, and being linear, theylend themselves naturally to online self-recovering algorithms.Recall, however, that channel estimation is indispensable fordesigning the optimal precoders [24].

B. Direct Blind Equalization

From (7), according to [24, Th. 2], the ZF-block equalizer iswell defined because the system of linear equations isalwaysinvertible, and the solution that minimizes the error norm isgiven by , where denotes matrix pseudoinverse.Intuitively, even deep fades can be equalized because thepresence of guard bits allows us to equalize the channel bysolving anoverdeterminedsystem of linear equations. In fact,each block of input symbols is mapped to a block of

data. Even if thebest ZF equalizer, in termsof numerical stability of the result and noise enhancement,is given by , infinite many matrices solve the sameoverdetermined linear system of equations in (7), and theyare all potential equalizers. In the following, we will showthat some of these equalizers are self recovering, in the sensethat they can be built directly form the received data withoutrequiring channel estimation.

Collect blocks to form the data matrix asin (12), and define to arrive at

(20)

Hence, one possible equalizer is .Because the matrix is lower triangular Toeplitz,it follows easily (by forming ) that the inverse

is also lower triangular Toeplitz. Thus, all the rowsof can be obtained from the last row . Relying on (20),we will show how can be determined directly from thereceived data matrix in (12).

Let denote a square shift matrix

......

.. ....

.... . .

. . .(21)

Since is not invertible, to simplify the derivations, we willuse the symbol to denote

(22)

where superscript is transpose. Matrix , in analogy withthe delay factor of the -transforms, is a delay matrix,whereas is the advance matrix similar to.

Due to its lower triangular Toeplitz structure, using adelay matrix , the rows of can be successively relatedas , whichimplies that

(23)

However, focusing on the last rows of (20), it follows thatfor , , we have ,which, after employing (23), leads to

(24)

We will show that the nullity ; thus, (and hence) can be determined from (24). In summary, we establish the

following result.Theorem 2: Let a0)–a3) hold true. Then, , and

can be identified from (24) as the unique (within a scalarambiguity) null eigenvector of . With as the throw, the lower triangular Toeplitz matrix can be built andused in (10) to obtain directly.

Proof: To show that , observe first that thematrix has dimensionality and that .Thus, our goal is to prove that there are exactlylinearly independent rows or columns that would imply that

rank . Note that by construction, (24)guarantees that but not its uniqueness. Because

, we know that . Since ,matrix can be decomposed as

(25)

where stands for the Kronecker product. In view of assump-tion a3), matrix is of rank ; therefore, itdoes not affect the dimensionality of the range space of,which depends only on the matrix , , , .The effect of multiplying successively from the left-handside by is to shift down the elements of each column,


adding every time one zero on top of it; thus, we have (26),shown at the bottom of the page. Observe that the last, andonly the last, column of is linearly independent of allthe columns of for , and eachblock adds to the matrix only one independent column.The matrix in (26) has thus exactly linearlyindependent columns. Since all other columns are replicas ofthese , rank or, equivalently,

.Our direct blind equalization algorithm consists of the

following steps.

1) Form with blocks of data asas in (12).

2) Form the matrixwith transpose of the shift matrix

defined in (21).3) Determine the eigenvector corresponding to the

null (smallest for noisy observations) eigenvalue of thematrix .

4) Equalize the data with , where the throw of is given by for .

Neither the channel estimator in Theorem 1 nor the equalizerimplied by Theorem 2 invoke any restrictions on the channelzeros or rely on any statistical input assumption (e.g., white-ness as in [14]). Thus, the input can be recovered exactly in theabsence of noise. Contrary to CMA, all symbol constellationsare allowed, and as long as the information rate is preserved,the symbol rate can be reduced in order to decrease theequivalent discrete-time channel order. The latter simplifiesthe channel estimation and equalization tasks considerably.Because only data blocks are required in , andeach is , the minimum number of symbols requiredis .

Because the blind channel-equalizerhas a lower triangularToeplitz form, it is easy to observe that the correspondingequalizer can be written as

(27)

where is lower triangular Toeplitz, and it is builtwith first rows and columns of . More specifically, isthe inverse of the matrix built with the first rows of

. On the other hand, if we change the time reference so thatthe null guard interval is considered at the beginning of each

block instead of at its end, each block of data can be written as

......

.. .. . .

......

. . .. . .

.. .. . .

.... . .

. . ... .

. . ....

. . .. . .

.. .. . .

...

(28)

where the matrix is now Toeplitz upper triangular,and its inverse is, consequently, upper triangular and Toeplitz.In addition, in this case, it is possible to derive the equalizingmatrix blindly, directly from the data, by forcing to zerothe first elements of and arriving at the dual versionof the algorithm described before for obtaining, where thelast two steps as modified as follows.

3) Determine the eigenvector associated with the nulleigenvalue of , where

4) Equalize the data with , where the firstrow of is , and the genericth row is simplya right shifted replica of , i.e., , with

.

Although the two solutions derived under the two differenttime references are in principle identical, they yield differentperformance in the presence of noise, as a function of thechannel zero location. In fact, it is well known that in theequalization of SISO FIR channels, having transfer function

, it is important to optimize the choice of the delayleading to a stable equalizing filter with minimum norm.

Specifically, denoting by the equalizing filter impulseresponse and by its transfer function, the optimal delayresults from the solution of the optimization problem

argmin with

(29)

In the presence of white noise, this choice minimizes the noisevariance at the equalizer output. In our transmission block

......

.... . .

......

. . ....

. . ....

. . ... .

.... . .

......

. . .. . .

..... .

. . ....

(26)


context, we face the same problem concerning the optimaldelay, except that is now confined between the values 0 and

. Introducing the notation

(30)

the counterpart of (29) in our case is

argmin subject to

and (31)

Since is a square matrix, the ZF constraint implies, and therefore, .

Without loss of generality, we assume that thechannelroots are ordered in the sense of decreasing amplitudes,i.e., . We are now able to solve theoptimization problem stated in (31) via the following theorem.

Theorem 3: For , the optimal delay thatguarantees and ,

is equal to the number of zeros outside the unitarycircle, i.e., . In particular

.(32)

Proof: The matrix , due to its Toeplitz structure, canalways be decomposed as

(33)

It is straightforward to verify that ; hence, wecan write the factorization of as

(34)

Similar to , consider-ing that approximately for , (becausediag ), we can write

(35)

The approximation in (35) is valid in the sense that the ratioof the error norm over the matrix norm tends to zeroas increases. From (35), we infer that canbe factorized as3

(36)

3In deriving this equation, we exploited the propertyJJJm = 0 form > M � 1 and the expansion(III � rJJJ)�1 = M�1

m=0rmJJJm, which

is valid for any r.

From this decomposition, recalling that all matrix normssatisfy the submultiplicative property , wehave

(37)

Therefore, to prove , it is sufficient tofocus on the behavior of the factors or

, proving thator , respectively, onlywhen coincides with the number of zeros of outsidethe unit circle. In fact, if we consider the Frobenious norm

, which is known to be always greater than or equal tothe 2-norm (see [17, p. 57]), we can write the upper limitfor the 2-norm of each factor in (37) as

(38)

Therefore, recalling that the roots are ordered in the sense ofdecreasing radii, we have

(39)

if because in such a case, all the roots outsidethe unit circle are inverted, whereas the other ones are not.Considering that the left-hand side of (37) is certainly non-negative, this proves that when

. Conversely, if , there exists at least onefactor in (37) that is not invertible when goes to . Infact, since, in general, (e.g., [17, p. 57]),

and , if ,there certainly is a term with an infinite 2-norm. An unbounded2-norm for one factor implies that this factor is not invertible,leading to , and this completes theproof of the theorem.

Interestingly, this theorem provides, as a by-product, a blinddeterministic method for identifying the number of channelzeros outside the unit circle: a task that is impossible in thestandard SISO setup with output second-order statistics. In fact

(40)


Theorem 3 implies that enhances the noise whenever thereare channel zerosinside the unit circle. Hence, even if both

and estimates are consistent, as will be proved in thefollowing, the average (over several channels) performance interms of bit error rate of both equalizers ,

will not achieve performance similar tothe MMSE ZF equalizer given by - .Therefore, whenever the estimate of or , we canestimate indirectly, by using the approach proposed in[25]. Alternatively, we can estimate the channel, up to a scale,using the equation as the null vectorsin Theorem 1 and then build, with this channel estimate, theoptimal equalizer. Channel identification is guaranteed in thiscase, and even if we carry out the estimate using a single nullvector instead of , and are unique.

Remark 1: We have shown in [24] that with, the filterbank precoder of Fig. 1 reduces

to the digital OFDM transmitter [5]. Under a2), the trailingzeros TZ-OFDM (which are detailed in [24]) can be equalizedblindly even when has unit circle zeros located at

; this is a case where deep fades deteriorate perfor-mance of conventional OFDM (see also [6] for a blind LMSadaptive solution).

C. Direct Blind Synchronization

Equation (8) assumes that block synchronization has beenaccomplished. Although techniques relying on training dataare possible, it is of interest to achieve block synchronizationblindly, which is a task complementing our blind equalizernicely (see also [27] for a statistical method). A deterministicblind approach is proposed herein after observing that matrix

in (24) becomes full rank when the receiver is notblock synchronous. Specifically, we propose to retrieve apossible time-offset of symbols between transmitter andreceiver by checking the rank properties of the matrix

as a function of the delay. Each matrix is obtained similar to (12) by collecting

vectors , , ,for .

The following theorem summarizes our blind block syn-chronization result.

Theorem 4: If blocks at the receiver are off bysamples and we form matrices for each possible shift

, then can be found as

(41)

where denotes the minimum eigenvalue. Inthe noise-free case, we guarantee identifiability by showingthat , whereasfor .

Proof: We will assume, without loss of generality, thatand that the offset is positive since the extension of

the proof for is straightforward.It is instrumental for the proof to represent the convolution

in the transmit-filterbank through the matrix-vector multipli-

cation between the matrix

......

.. .. . .

......

. . .. . .

. . .. . .

(42)

and the input blocks . For a shift , the precoded blockis given by the vector

(43)

where is a positive shift matrix similarto (21). There are two cases that have to be treated separately:i) and ii) .

i) : We will prove that in this case, the matrix isfull rank. Therefore, the matrix is necessarily fullrank, and thus, its minimum eigenvalue is strictly positive. For

, the vector can be partitioned as

(44)

where stands for a nonzero element. Collectingconsecutive vectors, we form the matrix as

(45)

where and have dimensionalitiesand , respectively. The matrix can be writtenequivalently as

(46)

where is a matrix obtained by , eliminating thecentral columns [i.e., from the th to the

th column] that multiply the null elements ofthe input matrix, i.e., as shown in (47) at the bottom of thenext page. The matrix in (47) is full rank since bothand are nonzero. Relying on the persistence-of-excitationassumption a3), we infer that there exists a value ofsuch that the matrix is full row rank, implying that

[and thus ] is full rank.ii) : In this case, we will prove that the matrix

is not full rank for , but is still full rank, as inthe previous case. In fact, the input vector now hasmore than zeros

(48)


and correspondingly, the matrix is

(49)

In addition, in this case, we can write in terms of productsbetween the matrices and with their zero entriesremoved. However, since there are more zeros in the sequence,the resulting matrix will not be full row rank. In fact, in thiscase, is a matrix, whose structure is similar tothat obtained by eliminating the first columns from (47).Following the same steps used in Section III-B to show that

, we can factorize the matrix as

(50)

By virtue of a3), is a square full rank matrix;hence, , , , . Keeping inmind the structure of , it follows by inspection (it is just moretedious to write it) that again, as observed about (26), everyblock adds one, and only one, independent column tothe previous one. Thus, since rank , because thereare independent columns for each block, and since

, the matrix , , , is full row rank.Hence, for .

Our blind block synchronization algorithm follows thesesteps.

1) With blocks of data, form the matrix ,, , as a function of the delay

for . Each matrix is obtained bycollecting vectors ,

, , , for .2) For each such that , determine the smallest

eigenvalue of the matrix .3) Determine the time offset in the block synchroniza-

tion as in (41).

To augment the accuracy of the block time synchronizationalgorithm, we could exploit the dual solution and also collect

matrices and the corre-sponding . Similar steps prove that in theabsence of noise, .

IV. NOISY CASE—CONSISTENCY

In Section III-A, we established that, in the absence of noise,the channel can be identified exactly. In this section, we provethat also in the presence of noise, we can identify the channeluniquely, using the received data covariance matrix. Whenstationary additive noise is present, the data correlationmatrix is given by [cf. (8)]

(51)

In practice, the ensemble correlation matrices are replaced bysample averages, which converge in the mean square senseto true correlation matrices since in (8) is mixing [input

has finite moments and has finite memory]; thus

m.s.s.as

(52)where m.s.s. stands for mean-square sense convergence. As-suming that the symbols and noise have zero mean, thecorrelation matrix coincides with the covariance matrix. Wenow establish consistency for both channel identification anddirect equalization methods.

A. Channel Identification

Assuming that the noise covariance matrix is known,we prove that it is still possible to identify the channel, up toa constant factor, as the number of samplesgoes to infinity.The proof is developed in two steps.

Step S1)

S1) In the absence of noise, we reformulate the blindidentification algorithm presented in Section III-A interms of the covariance matrix of the receiveddata instead of the matrix used in Theorem 2.

S2) In the presence of noise, we use the covariance matrixinstead of if the noise is white; otherwise,

we prewhiten the noise and then use the eigendecom-position of , where is any matrix thatfactorizes as .

......

......

. . ....

. . ....

. . .. . .

......

. . .

. . ....

. . ....

......

. . .. . .

......

. . ....

. . .. . .

.... . .

......

. . .

(47)


Step S1) Since the covariance matrix is fullrank, assumptions a2) and a3) imply that rank

. From (12), it follows that; hence, we also have rank . There-

fore, the noise subspace of has dimension .The noise subspace is thus spanned by the lastcolumns

of the matrix , which decomposes thecovariance matrix as , assuming that theeigenvalues are written in a decreasing order. Sinceis fullcolumn rank, is full row rank, and hence, the matrix

is full row rank. Therefore, the left null spaceof coincides with the left null spaceof . Once has been obtained, we can proceed exactly asin (14).

Step S2) Here, we show how to determine from thecovariance matrix of the noisy data , assuming that thenoise covariance is known.

If the noise is correlated, it is necessary to whiten it bymultiplying the received data vector by any matrix suchthat . The covariance matrix of the prewhiteneddata is

(53)

Defining the eigendecompositions

(54)

and

(55)

we obtain

(56)

implying that coincides with . Let us partition

as , , where contains the eigenvectorsassociated with the smallest eigenvalues of .Since , the eigenvectors

have to lie in the null space of .Hence

(57)

which implies that is a basis for or,equivalently, that is related to through theproduct by an invertible matrix. Notice however, that, as faras channel estimation is concerned, nothing prevents us fromusing the matrix directly in (14) instead ofbecause the proof of identifiability holds for any basis of

.Finally, because is a continuous function of

and the estimate (52) of is consistent, it follows that theproposed channel estimation algorithm is also consistent.

B. Direct Equalization

Similar to the previous subsection, we will also present ourconsistency results in two steps (noise-free and noisy cases).

Step S1) Using (12) and (20) and adopting the sameapproach as in Section III-B, we find that

(58)

Hence, arguing as in (23), we infer that ,which implies that for

. Therefore,

(59)

where

(60)

Matrix will be shown to have nullity . Infact, matrix is nothing but

(61)

It follows from (61) that , , ,because is full rank, and it

has been already shown in Section III-A [see (26)] that ,, , , which leads to our assertion that

. Indeed, if the noise were not present, (andhence ) could have been obtained (within a scale factor) asthe null eigenvector of .

Step S2) In the noisy case, we define ( ) asin (60) with ( ) replacing . The procedure is nowexactly equivalent to the one shown before for the channelestimation method, with the only difference that the null spaceof has dimensionality one.

Factorizing , we can prewhiten the noise bymultiplying the data matrix by . Hence, using (51), wecan write

(62)

where ( ) denotes the eigenvector (eigenvalue) ma-trix of . As in the previous case, we observethat , and thus, the minimumeigenvector of spans . In fact, if wedefine and rewrite (60) as ,we obtain . The latter, along with (62),yields

(63)

which shows that is the eigenvector ofcorresponding to its unitary eigenvalue, which, in force of (62),is also its minimum eigenvalue. Based on, we can find that

, and thus, the direct equalizing matrixreliesonly on the noisy covariance matrix and knowledge ofthe noise covariance .


(a) (b) (c)

Fig. 2. (a) Minimum-phase channel. (b) Channel with two zeros on the unit circle. (c) Nonminimum-phase channel (theoretical value: solid line, averagevalue obtained by simulation:+, SNR = 1).

Notice that even if the noise is white, a prewhiteningof the data is required becausediag , and its spectral factor in (63) is

diag (64)

With sample averages only available, consistency of ourdirect equalizer estimator follows easily from the definition

and the fact that

(65)

As before, the sample average tends to the ensembleaverage as goes to infinity.

At this point, a natural question arises: Should we use inpractice the “deterministic” approaches of Section III or their“statistical” counterparts in Section IV? Our recommendationdepends on SNR versus complexity tradeoff’s. When the SNRis high, the deterministic solution in Section III-C should bepreferred, and the minimal number of blocks shouldbe used for computational simplicity. However, at low SNR,

should be chosen large enough to obtain reliable estimatesin (65).

Remark 2: Our blind channel estimation and direct equal-ization algorithms in Sections III and IV end up solvingeigenvalue problems of the form , which can be turnedinto solving systems of linear equations like afterfixing the first entry of to unity. Although it is beyondthe size and scope of this paper, it is evident that adaptivevariants of our batch algorithms are possible. In the contextof OFDM, an adaptive LMS solution to direct equalizationwas reported recently in [6]. Notice, however, that with ourmultirate framework, both LMS and RLS channel estimatorsand equalizers are possible for general transmission schemesfalling under the umbrella of redundant filterbank precoding.

V. SIMULATIONS—COMPARISONS

In this section, we test our proposed algorithms and comparetheir performance with existing input- and output-diversity

methods. To make a fair comparison with methods not usingany precoding, we have normalized the precoder matrixto be unitary. In all examples, the filters of the precoderare ; see also [24] fordifferent precoder choices.

Example 1—Blind Synchronization:Using the transmissionscheme shown in Fig. 1, with and

, we have estimated average value and standarddeviation of as a function of ,using 100 independent realizations. In general, the behaviorof depends on the channel. To analyze the impact of thechannel transfer function on , we have considered threetypical classes corresponding to the following:

1) a minimum-phase channel transfer function with zeros(0.8, 0.8, 0.5 , 0.5 );

2) a channel with unit circle zeros (1,1, 0.7 , 0.7 );3) a nonminimum phase channel with zeros (1.2,1.2,

0.7 , 0.7 ).

The results are shown in Fig. 2, where the theoreticaldelay value, obtained by substituting the matricesby their expected values (solid line), is reported togetherwith the simulation results (). We observe a fairly goodagreement between theory and simulation. Interestingly, peaksand valleys of for the minimum-phase channel appearreversed for the nonminimum-phase channel. It is importantto observe that even if exhibits an almost flat behavior inthe neighborhood of for certain channels [see Fig. 2(a)],the absolute value of the gradient of around is highin at least one direction. This consideration could be exploitedto improve the performance of the method by incorporating thegradient into the estimation procedure, but this goes beyondthe scope of this paper.

The noise effect on the synchronization procedure is illus-trated in Fig. 3(a) and (b), where the average (solid line)is plotted together with its true value plus/minus the standarddeviation (dashed-lines), as a function of, for two differentchannels at SNR 6 dB. It is important to notice that not onlythe minimum average value of is achieved for , asexpected, but that the standard deviation assumes its minimumvalue at the same position as well.


(a) (b) (c)

Fig. 3. (a) Channel with zeros at (1.2,�0.9, 0.7j, �0.7j). (b) Minimum-phase channel with zeros at (0.8,�0.8, 0.5j, �0.5j) (average value: solidline, average value plus/minus the standard deviation: dashed line, SNR= 6 dB, M = 16, N = P = 19). (c) Synchronization error rate versusSNR (dB), M = 8, L = 4, and N = P = 12.

TABLE ISET OF CHANNEL RESPONSES

h1(n) 0 �1:280� j0:301 1:617 + j2:385 0:178+ j0:263

h2(n) �1:023� j0:501 0:106+ j1:164 1:477+ j1:850 �0:482� j0:523

h3(n) 0 �0:282+ j0:562 0:371� j1:001 0:041� j0:110

h4(n) �0:227+ j0:487 0:031� j0:211 0:336� j0:866 �0:110+ j0:271

Performance of the proposed synchronization algorithm hasbeen tested by simulating 400 independent trials. Fig. 3(c)shows the synchronization error rate as a function of the SNRfor the two channels of Fig. 3(a) and (b). The system pa-rameters were . From Fig. 3(c),we observe that the synchronization procedure has betterperformance for channels characterized by more accentuatedselectivity [case (a)]. This observation is also justified bylooking at the behavior of shown in Fig. 3(a) and (b).

It is important to point out that the proposed deterministicblind synchronization method is able to provide satisfactoryerror rates also at low SNR with relatively short data records.Such a performance makes it an attractive choice for synchro-nizing block (e.g., multicarrier) transmissions through rapidlyfading channels.

Example 2—Blind Channel Estimation versus [33]:Wecompare now our blind channel estimation method with thedeterministic method proposed in [33], which is denotedhereafter as the XLTK method. The XLTK method has beenimplemented for the two basic configurations of single-inputmultiple-output (SIMO) systems obtained using either spacediversity (antenna array) or fractional sampling at the receiver.

In the first case, we used four antennas, with the same fourchannels of [33, Table II], reported in Table I for convenience.For our algorithm, we used only one antenna, with channelimpulse response given by the second column of Table I. Wechose and . Fig. 4(a) shows the meansquare error (MSE) obtained in the blind channel estimationusing our approach from (12) (solid line) and the XLTKmethod (dashed line) at various SNR levels. In both cases,100 data samples were used for estimation. The advantageof the proposed approach is evident, although we used onlyone antenna instead of the four-sensor array used in [33].This brings out that introducing redundancy at the transmitter

improves the estimation performance, with respect to methodsrelying upon output redundancy, whereas at the same time,the receiver complexity is reduced. Moreover, although thealgorithms use the same amount of data, the XLTK algorithmis heavier from a computational point of view. The maincost associated with our blind channel estimation procedureis, in fact, that related to the SVD of the matrix

, which is necessary to obtain thevectors . Then,we have to solve an homogeneous linear system ofequations. It is important to remark that our method is robustagainst channel order mismatching, provided that we use anoverestimate. Conversely, the XLTK method is particularlysensitive to channel order mismatching, and then, it needsa channel order estimate at the beginning (e.g., see [33, p.298]). This requires the SVD of matrices, whose dimensionis , where is thenumber of sensors in its array-based implementation, or theoversampling factor is the channel order overestimate, and

is the number of information symbols. Therefore, assuming, i.e., the same number of symbols for both systems,

we see that for , the XLTK method needs to compute theSVD of larger matrices, and this is indeed the case consideredin our Example 2, where . Furthermore, the numberof unknowns with the XLTK method increases proportionallyto .

Next, we compare our method with the XLTK algorithmusing a single antenna at the receiver side and two channelsobtained by fractional sampling. In Fig. 4(b), the two differenttechniques are analyzed by introducing the same amount ofredundancy at the transmitter in the form of excess-bandwidthfor the XLTK method, controlled by the roll-off factor of theraised cosine impulse response, and in the form of trailingzeros for the proposed strategy. With rolloff equal to 0.5,the information rate is , where indicates


(a) (b) (c)

Fig. 4. (a) Mean square error of the blind estimation (XLTK-4 antennas: dashed line; SGB: solid line). (b) Proposed method (solid line); XLTK-fractionalsampling with roll-off factor equal to 0.5 (dashed line) and 1 (dotted line). (c) MSE of proposed method (solid line) and XLTK-fractional sampling(dashed line), with channel order overestimated by two samples.

the available bandwidth as well as the maximum theoreticalinformation rate. Clearly, it is possible to have the sameinformation rate in our scheme by settingand assuming an ideal Nyquist characteristic for the pulseshaping filter (i.e., roll-off factor equal to 0). The channelimpulse response at sampling rate has been computedas the convolution of the raised-cosine impulse responsewith roll-off 0.5 with the channel impulse response obtainedby interleaving the four channels reported in Table I. Theresulting impulse response has been truncated by settingto zero the values below 10% of the maximum. Denotingby the continuous time impulse response, the resultingsamples at rate are ,

, , ,, , ,, , ,, , ,

, and the two-channel impulse responsesof the equivalent SIMO model are thus of order .Therefore, the order of the channel at the symbol-rate is also

. In the implementation of our method, we have usedthe impulse response corresponding to the even samples of

, i.e., , ,, , ,

, . Imposing the constraint, we have , , and thus,

our method requires the minimum amount of data samplesequal to . We have used the same amount ofdata to test the performance of both methods, again, given interms of mean square error of the channel impulse responseestimates. Moreover, we have simulated the XLTK methodusing rolloff equal to 1, which simply means double bandwidthwith respect to the information rate . The channel samplesare , ,

, , ,, , ,

, , ,, and the subchannel is of order .

From Fig. 4(b) we observe that the XLTK algorithm suffersfrom a threshold effect at low SNR and that the proposedmethod outperforms the XLTK algorithm as well, when arolloff factor equal to 1 is used in the XLTK implementation,indicating that the structured redundancy introduced at thetransmitter by the filterbank leads to better performance.

One more important remark about the comparison be-tween our method and XLTK algorithm concerns the effectof channel order overestimation on the performance. Morespecifically, we plot, in Fig. 4(c), the MSE on the channelestimation obtained with our method (solid line) and XLTKmethod (dashed line) when the channel order has been over-estimated by two samples. We observe that our method isrobust to order overestimation, whereas the performance ofXLTK depends strongly on knowing the exact channel order.The channel used for the XLTK method is the same as inFig. 4(b), with rolloff factor equal to 1. In our case, the systemparameters are , and the same channel of orderas in Fig. 4(b) was used, but in this case, , and

. This feature is particularly important for applications,where only an upper bound on the channel order is available.

Example 3—Direct Blind Equalization:Here, we com-pared our direct blind equalizer with the constant modulusalgorithm (CMA) [16] and with the subspace method proposedin [15], which will be referred to as the GT method (see also[13] for an independent derivation of the same method).

First, we simulated blocks of 8-PSK symbols, eachblock with ( information symbols andTZ’s). The third-order channel had zeros at 0.9, , and theSNR at the receiver was fixed at 25 dB. With this relativelylow SNR, we implemented the deterministic direct equalizerapproach summarized in Theorem 3 using the minimal numberof samples . To compare with CMA, Fig. 5 showsscattering diagrams obtained in the following cases:

a) without equalization;b) with the ZF equalizer of Theorem 1;c) with the MMSE equalizer of (2);d) with the CMA.


(a) (b)

(c) (d)

Fig. 5. Scattering diagrams. (a) Before equalization. (b) After direct ZFequalization. (c) After MMSE equalization (SNR= 25 dB). (d) Afterequalization based on the CMA (SNR= 1, equalizer length 30 and stepsize 0.56� 10�4).

For the CMA, we used a 30-tap equalizer, and the adaptationrule was run over 20 000 samples at SNR . In thissituation, CMA suffers from the zeros on the unit circle so that,even in the absence of noise, there is no convergence towardthe correct equalizer. Conversely, in our case, the convergenceis guaranteed, and the performance is satisfactory, even witha relatively small amount of data. One more important remarkis that CMA requires a prohibitively increasing number ofsamples for high order constellations, whereas the number ofdata required for our method to converge isindependentofthe constellation size.

Next, we compare our method with the method of [15] and[13] using space diversity built with a two-sensor array. Thetwo impulse responses are , ,

, and, , ,. In our case, we have used only the second channel.

The SNR was fixed at 20 dB in both cases. The systemparameters were . The scatteringdiagrams for a QPSK transmission obtained with both methodsare reported in Fig. 6. Indeed, the performance of [13] and[15] depends strongly on the closeness of the zeros of the twochannels. In the present case, the zeros corresponding to thetwo channels are located in the positions depicted in Fig. 6(c)(“ ” refers to and “ ” to ). From Fig. 6, we observe thatour method provides better performance than [13] and [15],despite the use of only one sensor instead of two, even if thezeros of the two channels are not excessively close.

Example 4—Direct Blind Equalization versus Blind ChannelEstimation: In Section III, we have proposed a blind channelestimation method and a direct equalization method. Clearly,in the first case, the channel estimate can also be used to

derive the equalizer coefficients, according to (10), using theestimate instead of . In this section, we compare thesetwo approaches. Since the performance of the blind directequalizer strongly depends on the channel zero locations, weprovide results averaged over several independent channels.More specifically, we considered the so-called “Vehicular A”model, which was adopted for the wideband CDMA in UMTSto test its performance [10]. The channel model is

(66)

where the rays amplitudes are independent complex Gauss-ian random variables with zero mean and variances(ex-pressed as 10 log ): [0, 2.5, 6.5, 9.5, 12.5, 13,

15.5, 25.5, 50, 21.5, 25.5]; is a Nyquist pulse.The delays are [0, 0.25, 0.5, 0.75, 1, 1.2, 1.7, 1.9, 2.4, 2.7]s,and the sampling rate is 4.096 Mcps. In Fig. 7(a), we reportthe BER versus the ratio . The symbol constellation isQPSK, and the BER has been averaged over 100 independentchannels. Specifically, in Fig. 7(a), the solid line refers to theZF equalizer, where the data are equalized by, whereis estimated by using our blind method; dashed and dottedlines refer to an OFDM scheme, where the FFT outputs aredivided by the corresponding channel transfer function valuesestimated using our method; the dashed line refers again toOFDM, but in the ideal situation where the channel is supposedperfectly known at the receiver; finally, a dotted line refers tothe blind direct equalizer. Both OFDM schemes insert cyclicprefixes to simplify the equalization. From Fig. 7(a) we noticethat at high SNR, the blind method based on the channelidentification outperforms all other methods, including theideal OFDM scheme that assumes perfect knowledge of thechannel. In fact, if some channels have zeros close to or on theunit circle, some of the OFDM symbols cannot be recoveredreliably. This explains the smaller slope of the BER curve,at high SNR, with respect to our blind ZF method based onchannel estimation, which, thanks to the trailing zeros, doesnot experience such a problem. Conversely, the blind methodbased on direct equalization presents the worst performance,and the reason for this is that some of the independent channelrealizations are nonminimum phase, and thus, the delayshould be properly chosen, according to the theory developedin Section III-B, whereas we assumed for all channelrealizations. In Fig. 7(b), we tested the previous equalizationalgorithms against Rice fading channels, i.e., channels witha line-of-sight path. In particular, with reference to (66), weassumed , as in [9, p. 43] for the fixedchannel, for each realization, whereas the statistics of thepath, with , are the same as before. As we can see,the performance of the direct equalizer improves considerablywith respect to the Rayleigh fading channel case. Indeed, athigh SNR, the direct equalizer is the method exhibiting the bestperformance. This behavior is due to the fact that with the Ricemodel and the same power profile as before, the probabilityof incurring into a nonminimum phase channel decreases, andthus, the choice is optimal in most cases.


(a) (b) (c)

Fig. 6. Scattering diagrams obtained with (a) this paper’s method and (b) the GT method.

(a) (b)

Fig. 7. Average BER versusEs=N0 using blind channel estimation (solid line), direct equalization (dotted line), OFDM with null guard time intervaland channel perfectly known (dashed line) or estimated via our blind method (dashed and dotted line). (a) BER averaged over 100 independent Rayleighfading channels. (b) BER averaged over 100 independent Rice fading channels.

Example 5—Downlink CDMA:We have simulated thedownlink channel for a CDMA system, where the usercodes are the Walsh–Hadamard codes, and we comparedthe performance of our blind methods with a RAKE receiverin Fig. 8. The curves report the BER averaged over 100independent Rayleigh fading channels, assuming the samechannel model as in Fig. 7. Specifically, the solid line refersto our blind estimation method, the dotted line refers tothe direct equalizer, and the thick solid line refers to aRAKE receiver that assumes perfect knowledge of the channelimpulse response. We can observe the superior performanceof the method based on the channel estimation over the RAKEreceiver at high SNR, i.e., when the multiuser interference isthe dominating disturbance, whereas the poor performance ofthe direct equalizer is only due to the lack of optimizationwith respect to the delay.

VI. CONCLUSIONS

Building on the general precoding framework establishedin [24], we have proposed blind deterministic methods forchannel identification, direct equalization, and synchroniza-

tion. Comparing the performance with alternative proceduresthat introduce redundancy at the receiver in the form of spacediversity or fractional sampling, we have shown that introduc-ing redundancy at the transmitter offers distinct advantages.

Although the proposed methods have been derived for time-invariant channels, their deterministic structure also makesthem suitable for application over time-varying channels,provided that the channel’s coherence time is greater than thetime interval necessary to transmit blocks of data, whereis uniquely determined from the length of the precoding filtersand the channel memory.

As far as a priori knowledge of the channel memory isconcerned, it is important to remark that the proposed methodis robust even when the channel order is overestimated. Sincethe maximum delay due to multipath propagation is generallyknown, depending on the carrier frequency, bandwidth, andapplication (e.g., indoor versus outdoor), the assumption aboutthe upper bound on the order of the discrete-time equivalentchannel is perfectly reasonable in practice.

Other important features of the proposed procedures are thelack of constraints on channel zero locations and the simplicityof the receiver, relative to existing blind approaches which


Fig. 8. Average BER versusEs=N0 for the downlink channel of a CDMAsystem employing Walsh–Hadamard codes using blind channel estimation(solid line), direct equalization (dotted line), and a RAKE receiver assumingperfect knowledge of the channel (thick solid line). The BER is averaged over100 independent Rayleigh fading channels.

require multiple sensors or fractional sampling that assumesexcess bandwidth.

REFERENCES

[1] A. Akansu, M. Tazebay, and R. Haddad, “A new look at digitalorthogonal transmultiplexers for CDMA communications,”IEEE Trans.Signal Processing,vol. 45, pp. 263–267, Jan. 1997.

[2] S. Benedetto, E. Biglieri, and V. Castellani,Digital TransmissionTheory. Englewood Cliffs, NJ: Prentice-Hall, 1987.

[3] A. Chevreuil and P. Loubaton, “Blind second-order identification of FIRchannels: Forced cyclo-stationarity and structured subspace method,”IEEE Signal Processing Lett.,vol. 4, pp. 204–206, July 1997.

[4] J. S. Chow, J. C. Tu, and J. M. Cioffi, “A discrete multitone transceiversystem for ADSL applications,”IEEE J. Select. Areas Commun.,vol. 9,pp. 895–908, 1991.

[5] L. J. Cimini, “Analysis and simulation of a digital mobile channel usingorthogonal frequency division multiple access,”IEEE Trans. Commun.,vol. COMM-30, pp. 665–675, July 1985.

[6] M. de Courville, P. Duhamel, P. Madec, and J. Palicot, “A leastmean-squares blind equalization technique for OFDM systems,”Annal.Telecom.,pp. 4–11, 1997.

[7] B. W. Dickinson, “Efficient solution of linear equations with bandedToeplitz matrices,”IEEE Trans. Acoust., Speech, Signal Processing,vol.ASSP–27, pp. 421–423, Apr. 1979.

[8] Euro. Telecommun. Stand., “Radio broadcast systems: Digital audiobroadcasting (DAB) to mobile, portable and fixed receivers,” ETS 300401, Mar. 1994.

[9] , “Radio broadcast systems for television, sound and data services:Framing structure, channel coding and modulation for digital terrestrialtelevision,” ETS 300 744, Mar. 1997.

[10] , “Wideband direct-sequence CDMA (WCDMA)—Evaluationdocument (3.0),” ETSI SMG, Meeting 24, Madrid, Spain, Dec. 15–19,1997.

[11] K. Fazel and G. P. Fettweis, Eds.,Multi-Carrier Spread Spectrum.Boston, MA: Kluwer, 1997.

[12] D. T. Gavel, “Solution to the problem of instability in banded Toeplitzsolvers,” IEEE Trans. Signal Processing,vol. 40, pp. 464–466, Feb.1992.

[13] D. Gesbert, P. Duhamel, and S. Mayrargue, “On-line blind multichannelequalization based on mutually referenced filters,”IEEE Trans. SignalProcessing,vol. 45, pp. 2307–2317, Sept. 1997.

[14] G. B. Giannakis, “Filterbanks for blind channel identification andequalization,”IEEE Signal Processing Lett.,vol. 3, pp. 184–187, June1997.

[15] G. B. Giannakis and C. Tepedelenlioglu, “Direct blind equalizers ofmultiple FIR channels: A deterministic approach,”IEEE Trans. SignalProcessing,vol. 47, pp. 62–74, Jan. 1999.

[16] D. N. Godard, “Self-recovering equalization and carrier-tracking in two-dimensional data communication system,”IEEE Trans. Commun.,vol.COMM-28, pp. 1867–1875, 1980.

[17] G. H. Golub and C. F. Van Loan,Matrix Computation. Baltimore,MD: Johns Hopkins Univ. Press, 1989, pp. 79–81.

[18] A. Gorokhov and P. Loubaton, “Semi-blind second-order identificationof convolutive channels,” inProc. Int. Conf. Acoust., Speech, SignalProcess.,Munich, Germany, 1997, vol. V.

[19] S. D. Halford and G. B. Giannakis, “Blind equalization of TDMAwireless channels exploiting guard time induced cyclostationarity,” inProc. First IEEE Signal Process. Workshop. Wireless Commun.,Paris,France, Apr. 16–18, 1997, pp. 117–120.

[20] A. Erdogan, B. Hassibi, and T. Kailath “H-infinity equalization ofcommunication channels,” inProc. ICASSP,Seattle, WA, May 1998,vol. VI, pp. 3433–3437.

[21] L. Lupas, “On the computation of generalized Vandermonde matrixinverse,” IEEE Trans. Automat. Contr., vol. 20, pp. 559–561, 1975.

[22] E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspacemethods for the blind identification of multichannel FIR filters,”IEEETrans. Signal Processing,vol. 43, pp. 516–525, Feb. 1995.

[23] S. D. Sandberg and M. A. Tzannes, “Overlapped discrete multitonemodulation for high speed copper wire communications”IEEE J. Select.Areas Commun.,vol. 13, pp. 1571–1585, 1995.

[24] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbankprecoders and equalizers Part I: Unification and optimal designs,”IEEETrans. Signal Processing,this issue, pp. 1988–2008; see alsoProc. 35thAnnu. Allerton Conf.,Monticello, IL, Sept. 1997, vol. II, pp. 1134–1137.

[25] A. Scaglione, S. Barbarossa, and G. B. Giannakis, “Inverting overde-termined Toeptlitz matrices with application to channel equalization inblock transmission systems,” inProc. 32nd Asilomar Conf. Signals, Syst.,Comput.,Pacific Grove, CA, Nov. 1998.

[26] E. Serpedin and G. B. Giannakis, “Blind channel identification andequalization using modulation induced cyclostationarity,”IEEE Trans.Signal Processing,vol. 46, pp. 1930–1944, July 1998.

[27] M. K. Tsatsanis and G. B. Giannakis, “Transmitter induced cyclosta-tionarity for blind channel equalization,”IEEE Trans. Signal Processing,vol. 45, pp. 1785–1794, July 1997.

[28] L. Tong, G. Xu, and T. Kailath, “Blind identification and equalizationbased on second order statistics: A time domain approach,”IEEE Trans.Inform. Theory,vol. 40, pp. 340–349, 1994.

[29] J. J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimationof time and frequency offset in OFDM systems,”IEEE Trans. SignalProcessing,vol. 45, pp. 1800–1805, July 1997.

[30] S. B. Weinstein and P. M. Ebert, “Data transmission by frequencydivision multiplexing using the discrete Fourier transform,”IEEE Trans.Commun. Technol.,vol. COMM-19, pp. 628–634, Oct. 1971.

[31] G. Wornell, “Emerging applications of multirate signal processing andwavelets in digital communications,”Proc. IEEE,vol. 84, pp. 586–603,Apr. 1996.

[32] X.-G. Xia, “New precoding for ISI cancellation using nonmaximallydecimated multirate filterbanks with ideal FIR equalizers,”IEEE Trans.Signal Processing,vol. 45, pp. 2431–2441, Oct. 1997.

[33] G. Xu, H. Liu, L. Tong, and T. Kailath, “A least-squares approach toblind channel identification,”IEEE Trans. Signal Processing,vol. 43,pp. 2982–2993, Dec. 1995.

Anna Scaglione(S’97), for photograph and biography, see this issue, p. 2005.

Georgios B. Giannakis(F’96), for photograph and biography, see this issue,p. 2006.

Sergio Barbarossa(M’88), for photograph and biography, see this issue, p.2006.

Date post:	18-Feb-2022
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 7, JULY

Documents