Blind identification of multipath channels: a … › files › UGllcnJlIERVSEFNRUw=_00928700...linear system with outputs Under assumptions M2 and M4, the channel response is a causal

1468 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001

Blind Identification of Multipath Channels: AParametric Subspace Approach

Lisa Perros-Meilhac, Éric Moulines, Member, IEEE, Karim Abed-Meraim, Pascal Chevalier, andPierre Duhamel, Fellow, IEEE

Abstract—In this paper, blind identification of single-input mul-tiple-output (SIMO) systems using second-order statistics (SOS)only is considered. Using the assumption of a specular multipathchannel, we investigate a parametric variant of the so-called sub-space method. Nonparametric subspace-based methods require aprecise estimation of the model order; overestimation of the modelorder leads to inconsistent channel estimates. We show that theparametric subspace method gives consistent channel estimateswhen only an upper bound of the channel order is known. A newalgorithm, which exploits parametric information on the channelstructure, is presented. A statistical performance analysis of theproposed parametric subspace criterion is presented; limitedMonte Carlo experiments show that the proposed algorithm issecond-order optimal for a large class of channels.

I. INTRODUCTION

I N MOBILE or high-frequency (HF) radiocommunicationcontexts, signals currently propagate from an emitter to a re-

ceiver through different paths, due to reflection, diffraction, andscattering on physical objects in the environment (mobile com-munications) or to reflections on several ionospherical layers(HF transmissions).

We focus on “noncooperative blind” techniques, meaning thatthe channel estimation is achieved based only on the channeloutputs without resorting to training data, which is necessary forpassive listening applications but also of great interest in manyother situations.

For ten years, many methods have been developed to blindlyidentify single-input multiple-output (SIMO) systems from thesecond-order statistics (SOS) of the data (see [29 ] and the ref-erence therein). An important class of blind SOS-SIMO algo-rithms are based on the so-called subspace technique. A dis-tinctive advantage of the subspace algorithms is to be “deter-ministic” as the channel coefficients can be estimated withouterrors at the infinite signal-to-noise ratio (SNR) limit.

Manuscript received August 5, 1999; revised November 1, 2000. L.Perros-Meilhac was supported in part by the CNRS under Grant 550027 andThomson-CSF Communications. The associate editor coordinating the reviewof this paper and approving it for publication was Prof. Michail K. Tsatsanis.

L. Perros-Meilhac, É. Moulines, and K. Abed-Meraim are with Image andSignal Processing Department, Ecole Nationale Supérieure des Télécommu-nications, Paris, France (e-mail: [email protected]; [email protected];[email protected]).

P. Chevalier is with Thomson-CSF Communications, TCC/BSR/TBR/TSI,Gennevilliers, France (e-mail: [email protected]).

P. Duhamel is with the Laboratoire de Signaux et Systmes, Gif sur Yvette,France (e-mail: [email protected]).

Publisher Item Identifier S 1053-587X(01)01427-1.

However, as shown by many authors, subspace methods havesome severe drawbacks. In particular, they are highly sensi-tive to modeling errors: an overdetermination of the channellength leads to inconsistent estimates. In practical situations, theexact channel order should be estimated from the data, which isproved to be a rather involved task (moreover, most of the time,the concept of channel order is ill defined [20]; see also [10] and[34] for some possible adaptations of subspace ideas leading torobust algorithms).

In addition, subspace methods in their original formulationdo not take into account thea priori information that is, most ofthe time, available in digital transmission scenarios, such as theknowledge of the pulse shape or prior knowledge on the struc-ture of the propagation channel. It is a well-known fact in statis-tical estimation that improved performance can be expected ifone can put enough constraint on the model structure, leading toa reduction in the number of “free” parameters. Modificationsaiming at incorporating in the blind subspace method the knowl-edge of the pulse shape generate the reduction of the number ofunknown parameters, thus reducing the overall estimation vari-ance [5]–[7], [18], [22], [27].

The introduction of the propagation channel structure in thesubspace method is the main purpose of this paper. In many ap-plications, the propagation channel can be modeled as a specularchannel with a finite number of rays, each one being character-ized by its delay, its complex attenuation, and its direction of ar-rival. The delay spread may vary from few symbols (GSM link)to more than a decade of symbols (HF link) or several decadesof chips (UMTS link). Specular channel is a typical situationfor which the “direct” parameterization in terms of the impulseresponse coefficients is inefficient. With a parametric channelmodel, the blind identification problem reduces to the blind esti-mation of the channel parameters (attenuations, relative delays,spatial signatures).

This problem has received considerable attention over thepast few years. Most existing methods rely on preliminaryestimates of the impulse response (see, e.g., [31]; joint angleand delay estimation is considered, e.g., in [6], [26], and [32]).In the (SOS-SIMO) blind context, however, these two-stepmethods are “nonrobust” to channel overestimation, beingbased on inconsistent estimates. Recently, direct parametricSOS-SIMO methods have been proposed in [9], [12], [28], and[33]. On the other hand, blind estimation of time delay has alsobeen considered in [2], [13], and [16]; the methods proposedin these contributions are different from those developed inthis contribution, being based on cyclo correlation and cyclicspectrum.

1053–587X/01$10.00 © 2001 IEEE

PERROS-MEILHACet al.: BLIND IDENTIFICATION OF MULTIPATH CHANNELS 1469

We propose a parametric version of the subspace method[21], exploiting a specular model of the propagation channeland the prior knowledge of the pulse shape filter. If the pulseshape filter is not precisely known (e.g., in a passive listeningcontext), solutions proposed in [22] can be adapted. The pro-posed algorithm proceeds in “one-pass,” exploiting directly theparameterization (and not in a post-processing stage), leading tothe so-called “parametric” subspace approach (in contrast withthe “unstructured” plain subspace algorithm). The key result ofthis contribution is that parametric subspace methods lead toconsistent estimates of the channel parameters as soon as anupper bound of the channel order is known (which is, of course,weaker and, in practice, much more relevant than the assump-tion of a known order). The price to be paid (as in most “para-metric subspace methods”) is that the minimization is no longera convex problem. A multidimensional search algorithm is re-quired, the typical dimension of the problem being the numberof path delays. Some results developed here have been intro-duced in [24] and [25].

The paper is organized as follows. The assumptions made onthe data model are given in Section II. Section III reviews theso-called “plain” subspace method and presents (Theorem 1)a new characterization of the set of solutions (extension of [1,Theorem 6]) when the channel order is overestimated. In Sec-tion IV, a new parametric subspace algorithm is presented, andthe consistency of this algorithm is addressed (Theorem 2), witha special emphasis on the practical situation where the channelorder is overestimated. In Section V, the statistical performanceanalysis of the parametric subspace method is assessed. It isshown, under mild conditions on the probability distribution ofthe input signal, that the parameter estimates are consistent andasymptotically normal. The asymptotic covariance matrix is ex-pressed in closed form. Finally, in Section VI, some numericalcomparisons of the asymptotic covariance with the Cramér–Raolower bound (for Gaussian input) are provided. Possible exten-sions to improve the performance are suggested.

II. M ODEL AND ASSUMPTIONS

Under standard assumptions (linear modulation over a lineartime-invariant channel), the baseband representation of the con-tinuous signal received onsensors may be expressed as

(1)

where are the transmitted symbols, andis the symbolperiod; the observation vector and the composite channelimpulse response are defined as column vectors

is defined similarly to . The following is assumed inthe sequel.

H1) The source signal is a sequence of indepen-dent and identically distributed (i.i.d.) random vari-ables with zero-mean, unit variance.

H2) is a stationary temporally and spatially whiteGaussian noise, with zero-mean and second-order mo-ments and .Moreover, is independent from .

The vector includes the effects of the pulse-shaping filterdenoted and of the propagation channel associated witheach sensor. The channel is assumed to be specular, i.e., the sumof a small number of rays. Each ray is parameterized by a delay

and a spatial attenuation factor. The major assumptions wemake on the multipath scenario are listed as follows.

M1) The propagation channel is a finite superposition ofrays.

M2) The delay spread is finite, , and an upper-bound of the delay spread is known.

M3) The signal is narrowband with respect to the array aper-ture.

M4) The pulse-shaping filter is known and has finitesupport for .

M5) The channel is stationary over the length of the obser-vation interval. Doppler shift and residual carriers areneglected.

Let denote the number of delays. Under the above statedassumptions, the response is expressed as

(2)

where and are the unknown delay and spatial signatureassociated with theth path. If the multipath is directional,

, where is the array response to a point source fordirection , and is the fading factor. In this paper, we processthe array response as an arbitrary vector of dimension. Theobservation vector is sampled with period .For , we define

By combining spatial and temporal diversity, we form a SIMOlinear system with outputs

Under assumptions M2 and M4, the channel response is acausal finite impulse response system of duration with

. Then, we can define the SIMOresponse , and we may express as

(3)

In the sequel, it is also convenient to consider the vectorof dimension obtained

by stacking the coefficients of the polynomial vector .Similarly, stacking successive samples of the arrayresponse, we may rewrite (3) as

(4)


where ,, and

For the developments that follow, it is convenient todefine the -transform of the oversampled channel

. It is easily seen that

(5)

where . From (2), maybe written as

(6)

where is the -transformof the pulse shape filter sampled with rate and phase .

III. SUBSPACEMETHOD

In this section, we first give a description of the subspace al-gorithm, assuming that the channel order is known. Since thiscondition is rarely met, we then consider the case where thechannel order is overestimated, and we propose an extended ver-sion of the subspace identifiability conditions presented in [1,th. 6]. This extended version forms the basis of the structuredsubspace algorithm presented in the following section.

For nowtrue value of the parameter of interest;estimated value;current value.

For any matrix , we denote by Span the linear spacespanned by the columns of and by Null the kernel of .Under assumptions H1 and H2, the covariance matrix of thereceived signal may be written as

(7)

It is well known (see, e.g., [19]) that when is irreducible,i.e., for all , then for all , rank

. Hence, the eigendecomposition of the covariancematrix reads

diag

where are the singular values of, and contains the associated left singular vectors.

The columns of form an (arbitrary) orthogonal basis span-ning the orthogonal complement of Span, which is referredto as the noise subspace. Denote as the orthogonalprojector onto the noise subspace of . By (7), the noisesubspace corresponds to the left null space of ; thus

Span Null (8)

In addition, it has been shown in [1] and [21] that is up to anirrelevant scalar factor the unique solution of the linear equation

under the “order constraint” .Moreover, since the matrix depends linearly on , wemay write

Vec (9)

where is ablock-Toeplitz ma-

trix with , i.e., is the th column of. Equation (8) implies that Span Null .Thus, if , the response is identified up to a mul-

tiplicative constant. The case where the channel order is un-derestimated has been studied in [20]. This point isparticularly relevant when applying the unstructured subspacemethod to channels with small leading and trailing terms. Aswe will see below, the situation is rather different in the para-metric case because the channel structure restores identifiabilitywhen the channel order is overestimated. In this case, it seems(as shown in the simulations) to be a safe practice to overesti-mate the channel length. We will now study the impact of suchoverestimation on the plain subspace identifiability.

Assume that , which is the estimated channel order, is largerthan , i.e., . It has been shown in [1] that the solutionsof the linear equations

(10)

are given by , where is an arbitrary scalarpolynomial of degree less than . This result, however,gives only a partial answer to the overestimation problem be-cause (10) implicitly assumes that is known. In practice, foran estimated value, we form by taking an (arbitrary) or-thogonal subset of vectors in the nullsubspace. If , the subspace spanned by is “strictly”included in the noise subspace, i.e.,

, where . Theorem 1 below shows that the

solutions of the equations are also given by. This new result is better suited to the prac-

tical situation of interest.Theorem 1: Let be a irreducible polynomial

vector of degree , and let be apolynomial vector of degree . Let and let

and be two matrices verifying

Span Null (11)

Span Span (12)

Rank (13)

Then

(14)

where is a scalar polynomial of degree .Proof is given in Appendix A. Thus, if the system is overmod-

eled, then the subspace method leads to an inconsistent estimateas is arbitrary.


TABLE IOUTLINE OF THE PARAMETRIC SUBSPACEALGORITHM

In practice, ( ) samples are ob-served, and an estimate of may be constructed from thefollowing estimate of the covariance matrix:

(15)

by taking the eigenvectors associated wiht thesmallest eigenvalues of. We solve (9) in a least-square

sense, minimizing

(16)

where , under the nontriviality con-straint .

IV. PARAMETRIC SUBSPACEMETHOD

A. A Parametric Subspace Algorithm

Under assumptions M1 to M5, a -ray specularchannel of order , may be expressed as [see (6)],

, and the associated vectorreads , where and

. is the Kronecker product, andis a matrix defined as

......

Let be the estimated number of delays and bethe estimated channel order. The parametric subspace estima-tion consists of minimizing in the following criterion:

(17)

under the constraints and for . Sev-eral constraints are possible; for example,

is imposed in [25].Under the constraint , the minimum in of (17) for

fixed is the normalized eigenvector corresponding to the min-imum eigenvalue of matrix . Thus,the criterion (17) reduces in to

(18)

where denotes the minimum eigenvalue of matrix. We obtain an estimate of minimizing the reduced

criterion (18) of parameters under the constraint .Then, an estimate of can be obtained directly as theeigenvector corresponding to the minimum eigenvalue of

.Note that when two or more components of the vectorare

identical, then the rank of the matrix degenerates, andthe function is zero. To avoid these ill-formed solutions,we force the constraint in the criterion by using, insteadof (18), the following normalized criterion:

(19)

which is nonzero when matrix is singular. Takingguarantees that the criterion remains bounded in the neighbor-

hood of the spurious singular solutions. Taking forcesthe criterion to in that neighborhood. In practice, it appearsthat is a good choice. Perhaps surprisingly, it is shownin Section V that this normalization does not influence the lim-iting variance of the estimator, i.e., the normalization does not“asymptotically” affect the position of the minima of the func-tion nor the “shape” (the curvature) of the criterion in the neigh-borhood of these minima.

The steps of the proposed algorithm are summarized inTable I. Theorem 2 shows that under stated assumptions, theestimators are consistent, even when the channel order

and the number of delaysare overestimated.


B. Identifiability of the Parametric Model with Channel OrderOverestimation

A multipath propagation channel is characterized by anumber of rays , a set of distinct delays( for ), and a set of spatial attenuation factors

. We use the followingnotation:

Moreover, we define for every the set

i.e., the set of delays that differ from by aninteger multiple of the symbol period. Note that by construction,

. Finally, we assume that the following property(commented further) is verified. [For a set, card denotesthe cardinal of .]

Proposition 1 (P1): For any -uplets suchthat for and such that for all

card (20)

the matrix is full rank.Theorem 2: Let and be two multipath chan-

nels, and let and

. Assume that

(21)

where is a scalar polynomial of degree. Assume, in addition, that

I1) card ;I2) Rank ;I3)

;I4) where

card ;then , and

for

for

(22)

The proof is presented in Appendix B. We have seen in Sec-tion III that an overestimation of the channel order in the sub-space method introduces a polynomial indetermination, whichis denoted , i.e., . From (5), this is equiv-alent to . This theorem means thatby imposing the specular structure, the intrinsic subspace in-determination is raised to an irrelevant (and unavoidable in theblind context) translation factor . Thus,

, meaning that up to a multiplicative constant andan unidentifiable offset (equal to an integer number periods), the

parametric subspace estimate is consistent. We also have a con-sistent estimation of the channel parameters.

Comments:In most scenarios, card ,i.e., the differential delays are not integer multiple of the symbolperiod. In such case, assumption I1 always holds. AssumptionI1 indicates that when the differential delay between two raysof the true response is a multiple of the period symbol, weneed spatial diversity to identify these delays. Indeed, note that

and, where . Thus,

by multiplying the true response by a polynomial scalar , wecreate a new parametric response having “spurious” rays.The differential delay between spurious rays and the associated“true” ray is an integer multiple of the symbol period. Hence,in this case, a true and a “spurious” ray may correspond to thesame time and then cannot be separated using only temporal di-versity. Spatial diversity allows us to separate them.

Assumption I4 shows that we can obtain a consistent estimateof the channel parameters, even when the number of rays is over-estimated. In most scenarios , (see the first remark), andthe number of delays can be overestimated by a factor less thantwo.

Assumption I2 is classical and means that we have distinctangles and unambiguous space manifold. (See [32] on the ef-fective rank of ).

As assumption I3 is very technical, we propose to derive itfor some particular cases. First, if (which is the mostoften considered situation in the literature), then I3 may be re-placed by . Note that in this case, the system re-mains identifiable, regardless of the channel order overestima-tion factor . Now, let us take and, more precisely,

; then, assumption I3 may be replaced by. Thus, perhaps surpris-

ingly, if the channel order is overestimated by a factor compat-ible with this condition, then it is also possible to identify morerays than sensors. In particular, if (i.e., the channel orderis known), then I3 becomes .

Assumption I3 assumes that the property P1 is verified, i.e.,that when the number of delays falling in atime interval is “compatible” with the de-gree of , then is full rank.P1 is gen-erally theoricaly verified, but in practice, performance dependson the condition number of the matrix . We propose tostudy this point by simulation, as in [31]. We considerrays eq-uispaced in time between 0 and such that is the maximumnumber of delays verifying (20). The modulation waveform is araised-cosine pulse with excess bandwidth, truncated to zerooutside the interval ( ). We observe thesingular value of the matrix as a function of (seeFig. 1), (see Fig. 2) and (see Fig. 3).

In Figs. 1 and 2, one can see that for and , thelarge singular values are all the same. Thus, in a “bandlimited”context, we cannot tolerate more rays by increasing the over-sampling factor or the order of the pulse shape filter (as-sumption I3 should thus be interpreted carefully). Fig. 3 showsthat the resolution power depends on the excess bandwidth. Fora given , one can defined the effective value of (in func-tion of the noise power).


Fig. 1. Singular values ofG (��) for various oversampling factorp,B = 3,� = 0:25.

Fig. 2. Singular values ofG (��) for various half-pulse shape length factorB, p = 2, � = 0:25.

V. ASYMPTOTIC PERFORMANCEANALYSIS

In this section, we establish the asymptotic distribution of theparametric subspace estimator and derive an explicit expressionfor the covariance matrix of its estimation errors.

Theorem 3: The sequence of estimatesis asymptoticallynormal with mean and covariance matrix

(23)

Re Tr

(24)

where andare two shift matrices defined as

Fig. 3. Singular values ofG (��) for various excess bandwidth factor�,p = 2, B = 3.

and and

(25)

(26)

(27)

Proof: See Appendix C.The expression of the covariance matrix involves the Hessian

matrices of the criterion at the true values of the delaysand of the noise projector. Their calculations are made in [23].We get

Re(28)

(29)

where

Vec

with

and

1) Remarks:Note that the covariance of the estimator goesto zero as the noise variance tends to zero. This is related to the


so-called “deterministic” estimation properties of the subspacealgorithm; in absence of noise, the noise subspace can be esti-mated without errors from a finite number of samples, and thechannel can be identified up to an arbitrary scalar factor (pro-vided the identifiability conditions are satisfied).

We can see that the normalization factorused to avoid spurious solutions does not influence the

limiting variance of the estimates.The asymptotic variance of the estimate otherwise depends in

a rather intricate way on the emitting filter (and on the deriva-tives of the emitting filter), on the delays, and on the attenua-tions. To get a better understanding of the actual performanceof the algorithm, we perform simulation in different scenarios.

VI. PERFORMANCEEVALUATION

In this section, the performance of the proposed parametricsubspace method is assessed. An 8-PSK signal is modulated bya raised-cosine waveform with roll-off factor , trun-cated to a length of symbol periods. The signal is re-ceived on identical omnidirectional antennas, spaced byhalf a wavelength; standard far-field propagation conditions areassumed. The multipath propagation channel is characterizedby a set of delays and spatial signatures

. For the simulations, ,where and are, respectively, theth path attenuationfactor and angle of incidence, and is the steering vector ofa uniform linear array. We stress that this information is not ex-ploited by the identification algorithm, which assumes arbitraryarray geometry and propagation conditions. The oversamplingfactor is (compatible with the signal bandwidth). Thesignal is corrupted by an additive Gaussian noise. The SNR isdefined as

SNR (30)

i.e., the average signal in the useful bandwidth divided by thenumber of sensors. The number of rays is always assumed to beknown. The minimization of the cost function 19 with respectto is made using the iterative process described in Table I.To assess the robustness of the algorithm with respect to thechannel order estimation error (which is known to be criticalfor the “plain” subspace method), the channel order is overesti-mated in all simulations. We set (regardless of the “true”channel length, which is ). The size of theanalysis window is . The number of samples is setto .

A. Experimental Validation of the Asymptotic PerformanceAnalysis

We compare theoretical expressions obtained in Section Vwith empirical estimates obtained by Monte Carlo simulations.For each experiment, Monte Carlo simulations areperformed. We denote as the Monte Carlo mean-square es-timation error ( MSE)

(31)

Fig. 4. Asymptotic mean-square error versus SNR forM .


This quantity is compared with the square-root of the trace of theasymptotic covariance matrix Tr and to theGaussian Cramér–Rao lower bound (CRB), which provides atheoretical bound for all estimation procedures based on second-order moments (see [1, App. C] and references therein). In theseexperiments, we use the following parameters:

The SNR is varied between 0 and 30 dB. Fig. 4 (respectively,Figs. 5 and 6) gives for (respectively, and )

, , and the CRB as a function of the SNR. The error barsrepresent the standard deviation of.

These figures demonstrate a close agreement between theo-retical and experimental values, even for these reasonably small



Fig. 7. Performance versus the estimated channel order^L.

sample sizes. It is worthwhile to note the close fit between theproposed estimator and the Gaussian CRB, showing that, atleast for the first two scenarios considered (a single ray and twowell-separated rays), our algorithm is almost second-order effi-cient.

B. Numerical Study

In this section, the SNR is set to 15 dB. Fig. 7 gives the per-formance of proposed parametric subspace method and of the“plain” subspace method as a function of the assumed channelorder for the propagation channel . Comparison is given

in terms of mean square error between the estimated channeland the true channel . The true value of the channel order is

. First, this simulation confirms theresult of [20], which is recalled in Section III, i.e., performanceof the “plain” subspace method is better for an underestimatedvalue of the channel order. On the other hand, as ,from Theorem 3, we know that the channel remains identifiable,

Fig. 8. Asymptotic mean-square error versus the differential delay.

Fig. 9. Asymptotic mean-square error versus the gradient power.

whatever the order estimation. This point is asserted by this sim-ulation. Note also that the performance is not significantly af-fected by the order overestimation, showing that the algorithmis robust to order overestimation.

Now, we investigate performance of the algorithm as a func-tion of the propagation channel characteristics. The number ofrays , and . In Fig. 8, the two rays havethe same power, and the differential delay is varied. In Fig. 9,the two delays are , and the power gradient be-tween the two rays is varied: . Fig. 8 shows thatperformance of the proposed criterion are affected when the dif-ferential delay between the two rays becomes small.

VII. I MPROVEMENTS OF THEMETHOD FOR BLIND

EQUALIZATION

A. Generalized Algorithm

The previous section showed that performance is affected bysmall differential delay. We suggest here a new algorithm that


avoids this problem when the parameter of interest is the globalchannel response (and not the value of the differentialdelay). Using the principle proposed in [3] for direction-of-ar-rival estimation, we consider a generalized model of the propa-gation consisting for closed delays in a linear combination of

and of its gradient w.r.t. to . Denote as the numberof delays that would be too close to be estimated separately

. Then, there exists a permutation functionsuchthat

(32)

Define the gradient . Then, eventually, with somemodifications on the index of the rays, but without loss of gen-erality, the first Taylor-order expansion of yields

(33)

where and are vectors. Hence, proceeding as in Sec-tion IV and using the same notations, we get

(34)Letting , , and be the estimated values, we define thefollowing generalized criterion:

, where

B. Joint Number of Delays and Channel Response Estimation

The previous algorithm requires estimates of the number ofdelays and estimates of the number of couples of delays. Wesuggest here the selection of these parameters using a criterionbased on the equalized signals. More precisely, for a given es-timate of (associated with a given number of delaysandof couples ), we compute an equalized signal ,using e.g., an MSE equalizer. Then, we defined a criterion basedon these equalized signals. Because the modulation here is ofconstant modulus, it is appropriate to use a criterion assessingthe distance between the modulus equalized signal and the nom-inal value of this module

This criterion is used in the CMA algorithm for blind identifica-tion. Then, the joint algorithm may be summarized as follows:

• For all values of between 1 and and for all valuesof between 1 and , do the following.

— Estimate and

.— Obtain an estimate of the channel responseusing

(34).— Calculate the value of .

• .

At each step, the delays are estimated by a monodimensionalsearch using the previously obtained delays as described in Step4 of Table I.

C. Random Test Evaluation

The performance of the algorithm is assessed by the followingrandom test evaluation. bursts of eight-PSKsymbols are simulated. For each burst, the number of rays andthe delays are chosen randomly within limits ,

. The angle of incidence are fixed, andfor . To make the experiment more

realistic, we assume that the power of the raysdecreaseslinearly with the delay in such a way that ,and . The phase of the complex attenu-ation factor is also random. These values are typical of theSTANAG 4285 modems used for long-distance communicationover the ionospheric channel at 2400 b/s. The signal is receivedon antennas. The pulse shape is truncated at a lengthof symbol periods. The channel order is set at a valuecompatible with the maximum delay spread .The proposed blind equalization algorithm is compared withthe SIMO block-constant modulus algorithm (CMA) [14]. Asimulation result is given in symbol error rate as a function ofthe SNR. The proposed algorithm significantly outperforms theCMA (see Fig. 10).

VIII. C ONCLUSION

In this paper, assuming the knowledge of the pulse-shapefilter, a parametric subspace method has been presented for theblind identification of specular propagation channels. Contraryto the classical subspace method, the parametric method ex-ploits the specular structure of the propagation channel, whichmakes it very robust to channel order overestimation. By the re-duction and the normalization of the parametric subspace crite-rion, we have obtained a simpler algorithm that does not sufferfrom “spurious” global minima. Identifiability conditions havebeen given and show that the estimates are consistent in mostsituations of practical interest. The analytical performance anal-ysis, which was confirmed by limited Monte-Carlo simulations,shows the asymptotic behavior of the method compare with theGaussian CRB.

APPENDIX APROOF OFTHEOREM 1

The proof given here uses the theory of the rational subspace.This theory is not recalled here, but we will use vocabulary andnotations as presented in [1].


Fig. 10. Performance of proposed equalization algorithm and CMA.

Denote as the rational subspace spanned by andas the dual space of . Let

be a minimal polynomial basis of with indices, where deg . This implies (see

Forney [8]) that every polynomial vector in writes, where are scalar polynomials,

and deg deg . Using [8,th. 3] (see also [1] and [5]), we have ; thussince, under the stated assumptions , we have

. Denote as the polynomial ma-trix of size defined by the “shifted”versions of , .

forms a basis over of, where denotes the linear subspace

over of polynomial vectors of degrees less than or equal tobelonging to .It has been shown in [1] (Theorem 4) that

is isomorphic to Null , i.e., every polynomialvector is associated with a single vector

in Null . This last result means that theset of vectorsassociated with and, thus, defined as

forms a basis

of Span .Now, we show by contradiction that for all ,

there exists a scalar polynomial , degsuch that the vector , associated with ,belongs to Span . Assume that this is false forsome ; then, Span Span andRank Rank Span .As Span and fromRank , we get

Rank

Now, by assumption Rankas . This results in a contradiction.

Thus, for all , there exists a polynomial ,deg such that the vector associated with

verifies

Thus, belongs to the dual space of , i.e., .Hence, by definition of , .

APPENDIX BPROOF OFTHEOREM 2

First, let us introduce some notations. If, then we denote

. Let denoteand use .

The polynomial not being identically equal to zero,there exists such that . Let

and be,respectively, the set of the true and of the estimated delays.In a first step, we will show that for any

, i.e., .Under I1, it holds that card , and

thus, card , and we can extract a subsetsuch that . Then, under

I2, we may define an oblique projector verifying

We introduce following notations: for alland for all .

Noting that and, the equation

may be rewritten as

(35)

Applying to (35) yields

(36)

Define . The set containsall the delays appearing in (36). We will show that under I3, theset of delays verifies the condition given by (20). Fromthe structure of [using card ], wededuce that

card

card


Hence, for all , it holds that

card

card

Defining the function of,, one can see that

if

and

if

and

Thus, for all ,. Hence, under I3, for all , , and thus,

we obtain the required result, i.e., for all

(37)

The set of distinct delays of satisfies the conditions ofthe property P1. Thus, if a delayappears only one time in theset , then from P1, the coefficient affecting in(36) is equal to zero. Thus, implies that

. However, as has been constructed insuch a way that , then . Repeatingthe same argument for shows that . Wecan assume that for all .

Now, assume that exists such that . Pro-ceeding as above, it may be shown that , and thus,

. However

card

card

(38)

card

(39)

Thus, under I4, card card , whichresults in a contradiction, and .

Finally, plugging the previous result into (35) yields

(40)As , from property P1, are independent.Hence, for , and for

.

APPENDIX C

The estimator of the delays are obtained by solving the equa-tion . By expanding this relation around

, the true value of the delays, and of the noise projector,we may write

(41)

where denote the gradient ofand the Hessian matrix. Notethat , and thus, the Jacobian

may be expressed as

(42)

On the other hand, standard results on the perturbation of theeigenprojectors (see [17]) show that

(43)

where is the pseudo-inverse of . By collecting theprevious relations, we obtain

where

On the other hand, from (4) and (15), we have

where

and

As and , bykeeping only the first order in the noise level, we get

Tr(44)

where . On the other hand, we needthe following lemma.


Lemma 1: Under assumptions H1 and H2, for any arbitrarydeterministic matrices in :

Tr Tr

Tr (45)

Tr Tr (46)

where , andare two shift matrices defined as

and . Theproof is easy by inspection. Relation (46) follows from thecircular property of the noise.

From Lemma 1 and (44), we obtain

Re Tr

Hence, we deduce the expression of the covariance matrix.Moreover, under assumptions H1 and H2, the estimatedcovariance matrix is asymptotically normal with rate ofconvergence by an application of the Hoeffding andRobbins theorem for -dependent sequences (see, e.g., Brock-well and Davis [4, Th. 6.4.2, p. 213]). Because the parametricsubspace estimator is related to through an infinitelydifferentiable mapping [see (44)], is also asymptoticallynormal with rate .

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[35] M. Wax and T. Kailath, “Detection of signals by information theoriccriteria,” IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP-33,pp. 387–392, Apr. 1985.

Lisa Perros-Meilhac was born in Cagnes sur mer,France, in 1975. She received the M.Sc. degree inelectrical engineering and the Postgraduate degreein digital communication systems from Ecole Na-tionale Superieure des Telecommunications (ENST),Paris, France in 1997. Since October 1997, she hasbeen pursuing the Ph.D. degree with the Image andSignal Processing Department, ENST, with supportfrom CNRS and Thomson-CSF Communications.‘

Her research concerns blind array signal pro-cessing for digital communications.


Éric Moulines (M’91) was born in Bordeaux, France, in 1963. He received theM.S. degree from Ecole Polytechnique, Paris, France, in 1984 and the Ph.D. de-gree from Ecole Nationale Superieure des Telecommunications (ENST), Paris,in signal processing in 1990.

From 1986 until 1990, he was Member of the Technical Staff at CNET,working on signal processing applied to low-bit rate speech coding andtext-to-speech synthesis. Since 1990, he has been with ENST, where he ispresently a Professor. His teaching and research interests include statisticalsignal processing and speech processing. Currently, he is engaged in researchin various aspects of statistical signal processing including, among others,single and multichannel ARMA filtering and modeling, blind signal processingfor digital communications, characterization and estimation of point processeswith application to high bit-rate data traffic modeling, low bit-rate speechcoding, and speech transformation. He is an Associate Editor ofSpeechCommunication.

Dr. Moulines is an Associate Editor the IEEE TRANSACTIONS ON SIGNAL

PROCESSING. He is a Member of the IEEE Committees on Speech and StatisticalSignal and Array Processing.

Karim Abed-Meraim was born in Algiers, Algeria,in 1967. He received the M.S. degree from EcolePolytechnique, Paris, France, in 1990 and thePh.D. degree from Ecole Nationale Superieure desTelecommunications (ENST), Paris, in 1995 in thefield of signal processing and communications.

He currently is Associate Professor at the Signaland Image Processing Department, ENST. Hisresearch interests include statistical signal pro-cessing, system identification, multiuser detection,space-time coding, (blind) array processing, and

performance analysis.

Pascal Chevalierwas born in 1962 in Valenciennes,France. He received the M.Sc. degree from Ecole Na-tionale Suprieure des Techniques Avances (ENSTA),Paris, France, and the Ph.D. degree from Universityof Paris-Sud in 1985 and 1991, respectively.

Since 1991, he has been in industry (studies,experimentations, expertise, management), teachingactivities, both in French engineering schools(Supelec, ENSTA) and French Universities(Cergy-Pontoise), and research activities. Hispresent research interests are in array processing

techniques, either blind or informed, second- or higher order, spatial-orspatio-temporal, time-invariant or time-varying especially for cyclostationarysignals, linear or nonlinear and, particularly, widely linear for noncircularsignals, for applications such as TDMA and CDMA radiocommunicationsnetworks, satellite telecommunications, spectrum monitoring, and HF/VUHFpassive listening. He is author or co-author of more than 50 papers in journalsand conferences, eight patents, and several book chapters.

Dr Chevalier is a member of the THOMSON-CSF Technical and ScientificCouncil and of EURASIP and is a senior member of the Societ des Electricienset des Electroniciens (SEE).

Pierre Duhamel (S’87–F’98) was born in Francein 1953. He received the Ing. degree in electricalengineering from the National Institute for AppliedSciences (INSA), Rennes, France, in 1975 and theDr.Ing. degree in 1978 and the Dr.Sci. degree in1986, both from Orsay University, Orsay, France.

From 1975 to 1980, he was with Thomson-CSF,Paris, France, where his research interests werein circuit theory and signal processing, includingdigital filtering and analog fault diagnosis. In1980, he joined the National Research Center in

Telecommunications (CNET), Issy les Moulineaux, France, where his researchactivities were first concerned with the design of recursive CCD filters.Later, he worked on fast Fourier transforms and convolution algorithms andapplied similar techniques to adaptive filtering, spectral analysis, and wavelettransforms. From 1993 to September 2000, he was a Professor at EcoleNationale Superieure des Telecommunications (ENST), Paris, with researchactivities focused on signal processing for communications. He was head ofthe Signal and Image Processing Department from 1997 to 2000. He is nowwith the Laboratoire de Signaux et Systemes, Gif sur Yvette, France, wherehe is developing studies in signal processing for communications (includingequalization, iterative decoding, and multicarrier systems) and signal/imageprocessing for multimedia applications, including source coding-image coding,multichannel sound coding, joint source/channel coding, watermarking, andaudio processing.

Dr. Duhamel was chairman of the DSP committee of the IEE SignalProcessing Society from 1996 to 1998, was an Associate Editor of the IEEETRANSACTIONS ONSIGNAL PROCESSINGfrom 1989 to 1991, and was AssociateEditor for the IEEE SIGNAL PROCESSINGLETTERS. He was a Guest Editorfor the special issue of the IEEE TRANSACTIONS ONSIGNAL PROCESSINGonwavelets. He is now a member of the SP for Communications Committee.He was an IEEE Distinguished Lecturer for 1999 and is co-general chair ofthe 2001 International Workshop on Multimedia Signal Processing, Cannes,France. The paper on subspace-based methods for blind equalization, which heco-authored, received the “Best paper award” from the IEEE TRANSACTIONS

ON SIGNAL PROCESSINGin 1998. He was awarded the “Grand Prix FranceTelecom” by the French Science Academy in 2000.

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