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doi:10.1016/j.sig

$This work

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(Spagna), pp. 1

by the MIUR

authors are with

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Napoli, via Cla�Correspondi

0817683149.

E-mail addre

gelli@unina.it (

f.verde@unina.

Signal Processing 84 (2004) 2021–2039

www.elsevier.com/locate/sigpro

Subspace-based blind channel identification of SISO-FIRsystems with improper random inputs$

Donatella Darsena, Giacinto Gelli, Luigi Paura, Francesco Verde�

Dipartimento di Ingegneria Elettronica e delle Telecommunicazioni, Universita degli Studi Federico II di Napoli, via Claudio 21,

Napoli 80125, Italy

Received 15 December 2003; received in revised form 10 May 2004

Abstract

In this paper, we consider the problem of blindly estimating the impulse response of a nonminimum phase single-

input single-output channel, by resorting only to the second-order statistics (SOS) of the channel output. On the basis of

a general treatment of the problem, we show that a SOS-based subspace procedure can be applied to the problem at

hand if the transmitted signal is an improper process, exhibiting some additional properties. After characterizing and

discussing these properties, we show that they are exhibited by many improper digital modulation schemes of practical

interest. Moreover, based on our unifying framework, we devise subspace-based algorithms for blind channel

identification, by addressing in particular the related identifiability issues. Finally, the theoretical expression of the

mean-squared error of the channel estimate is derived, and numerical simulations are carried out for its validation and

for comparing the performance of the proposed algorithm with that of a conventional subspace-based method.

r 2004 Elsevier B.V. All rights reserved.

Keywords: Blind channel identification; Conjugate cyclostationarity; Improper random processes; Perturbation analysis; Single-input

single-output systems; Subspace methods

e front matter r 2004 Elsevier B.V. All rights reserve

pro.2004.07.014

was presented in part at the 6th Baiona

ignal Processing in Communications, Baiona

09–114, September 2003, and it was supported

2002 Italian National Research Project. The

Dipartimento di Ingegneria Elettronica e delle

oni, Universita degli Studi Federico II di

udio 21, I-80125 Napoli, Italy.

ng author. Tel.: +39-0817683154; fax: +39-

sses: darsena@unina.it (D. Darsena),

G. Gelli), paura@unina.it (L. Paura),

it (F. Verde).

1. Introduction

Blind channel identification and equalizationtechniques in digital communications avoid thetransmission of training sequences and, thus, makemore efficient use of the available bandwidth.Earlier works in blind channel identificationconsider a single-input single-output (SISO) mod-el, obtained by sampling at the baud-rate theoutput of a linear time-invariant channel; in thiscase, it is well-known [25] that the time-invariant

d.

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autocorrelation function of the discrete-time out-put does not contain enough information toidentify nonminimum phase channels. For thisreason, blind identification algorithms based onSISO models (see, e.g., [22] and references therein)explicitly or implicitly employ higher-order statis-tics (HOS) of the channel output. However, due tothe large amount of data necessary to accuratelyestimate higher-order moments or cumulants,HOS-based algorithms suffer from a significantperformance degradation when only a limitednumber of samples of the received signal isavailable (e.g., in packet-oriented transmission).It was recognized subsequently that, for a single-input multiple-output (SIMO) channel model,blind identification can be accomplished byresorting only to second-order statistics (SOS) ofthe channel outputs (see [25] for a comprehensivereview). A SIMO model arises naturally when thereceiver employs multiple sensors, or when theoutput of a SISO continuous-time channel issampled at a rate higher than the baud rate, givingrise to multiple virtual subchannels. SIMO identi-fication methods based on SOS have attracted agreat research interest, because they are less data-consuming than HOS-based ones. Among them,the subspace approach originally proposed in [15]is perhaps the most popular one, since the channelis estimated as the solution of a linear systems ofequations. Moreover, subspace-based methods arevery robust against noise and perform channelidentification under mild restrictions on subchan-nel zeros (i.e., the ‘‘no-common zeros’’ condition).

However, SIMO channel identification techni-ques based on oversampling cannot be used in allthose situations where the excess bandwidth islimited [4]. On the other hand, the use of anantenna array in wireless communications isimpractical in the downlink (from base station tomobile), while it can significantly increase thereceiver cost in the uplink (from mobile to basestation). Thus, some recent works have addressedthe blind channel identification and equalizationproblem from a different point of view, byconsidering a baud-rate SISO model but attempt-ing to exploit some additional information, so asto avoid the need to resort to oversampling ormultiple sensors. In the modulation-induced cyclos-

tationarity (MIC) approach [3] and [21], cyclosta-tionarity is introduced in the transmitted datastream and, as a result, the autocorrelationfunction of the channel output turns out to betime-varying. This latter property is the key todevelop blind channel identification and equaliza-tion algorithms, with no restrictions on thechannel zeros and regardless of spectral propertiesof the additive stationary noise. A differentapproach, which is considered also in this paper,exploit rather the improper [16] nature of thetransmitted signal. This choice is motivated by thefact that in many modulation formats of practicalinterest [17], such as ASK, differential BPSK(DBPSK), offset QPSK (OQPSK), offset QAM(OQAM), MSK and its variant Gaussian MSK(GMSK), the transmitted symbol sequence is animproper (possibly) wide-sense conjugate cyclosta-

tionary [6] process. When the information symbolsequence is an improper random process, completecharacterization of the SOS of the channel outputrequires [16] the additional knowledge of itsconjugate correlation function (see Section 2). Inthis case, both autocorrelation and conjugateautocorrelation functions of the received signalmay contain sufficient information for blindidentification of nonminimum phase SISO chan-nels. Although from a slightly different perspec-tive, the improper nature of the transmittedsymbol sequence was perhaps first exploited forsubspace-based blind channel identification in [12].Indeed, with reference to a BPSK or MSKmodulation format, it was recognized in [12] that,besides resorting to an antenna array or over-sampling, a virtual SIMO model can be obtainedfrom a SISO one also by separately processing thereal and imaginary part of the received signal.Based on this observation, it was shown in [12] bysimulation experiments that this virtual SIMOmodel allows one to improve estimation accuracyin subspace-based channel identification. A similarapproach was independently followed in [5] withreference to GMSK modulation, where severalSOS- and HOS-based channel identification andequalization techniques are proposed. Recently, an‘‘analytical’’ method has been proposed [8]for SISO channel identification with impropersequences, which, starting from a set of samples of

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the autocorrelation and conjugate autocorrelationfunctions of the received signal in the absence ofnoise, builds a nonlinear system of equations, whichcan be solved for the unknown channel coefficients.However, the solution of such a system involves aquite complicated elimination procedure, whichalso requires a critical initialization. Moreover,when the statistics are estimated by using a shortdata record, the performances of the method arerather sensitive to the signal-to-noise ratio, and aredifficult to characterize theoretically.

In this paper, we study the feasibility ofperforming subspace-based blind channel identifica-tion for a SISO channel model with an impropersymbol sequence at its input. Unlike [5,12], theblind identification algorithm is obtained in ageneral and unified framework, by linking itsderivation to the basic theory of improper complexrandom vectors and processes [19]. This allows usto characterize the statistical properties that thesymbol sequence must possess to allow for the useof a subspace-based method. In particular, we showthat the channel impulse response cannot beestimated via a subspace-based algorithm evenwhen the transmitted symbol sequence is animproper process. To allow for blind channelidentification, the symbol sequence must be im-proper and satisfy a certain additional property: werefer to this subclass of improper random processesas strongly improper ones. By restricting attentionto this family of processes, which however encom-passes many digital modulation schemes of prac-tical interest, channel identifiability conditions arederived in general, which are subsequently particu-larized and discussed in detail for the consideredmodulation schemes. Furthermore, the unifiedtreatment considered in this paper allows us, unlike[5,8,12], to provide a detailed theoretical perfor-mance analysis of the proposed identificationalgorithm in a clear and concise manner.

The paper is organized as follows. In Section 2, weintroduce notations and briefly recall some defini-tions about improper random processes that areused throughout the paper. In Section 3, the generalsystem model, together with the basic assumptions,is introduced. The subspace-based algorithm forblind channel identification is presented in Section 4,and the identification conditions are derived. In

Section 5, the application of the proposed method tosome digital modulation schemes of practical interestis discussed. The theoretical performance analysis iscarried out in Section 6, by using a first-orderperturbative approach. In Section 7, comparativesimulation results are presented to illustrate theperformance of the algorithm and the validity of thetheoretical analysis. Finally, conclusions are drawnin Section 8.

2. Notations and preliminaries

In the rest of the paper, we will use the followingterminology and notations. Upper- and lower-casebold letters denote matrices and vectors; thesuperscripts �; T, H, �1 e y will denote theconjugate, the transpose, the hermitian (conjugatetranspose), the inverse and the Moore–Penrosegeneralized inverse (pseudo-inverse) of a matrix;the subscripts R and I stand for real and imaginaryparts of any complex-valued matrix, vector orscalar; C; R and Z are the fields of complex, realand integer numbers; Cn

½Rn� denotes the vector-space of all n-column vectors with complex [real]coordinates; similarly, Cn�m

½Rn�m� denotes thevector-space of all the n � m matrices withcomplex [real] elements; 0n; On�m and In denotethe n-column zero vector, the n � m zero matrixand the n � n identity matrix; k � k; detð�Þ andrankð�Þ denote the Frobenius norm, the determi-nant and the rank of a matrix; � denotes theKronecker product; for any A 2 Cn�m; NðAÞ;RðAÞ and R?ðAÞ denote the null space of A; thecolumn space of A and its orthogonal complementin Cn; A ¼ diag½A11;A22; . . . ;Ann� is the (block)diagonal matrix wherein fAiig

ni¼1 are diagonal

matrices; vecðAÞ associates with any matrix A thevector obtained by stacking its columns; for anymatrix A ¼ ½a0; a1; . . . ; aL�1� 2 Cn�L and positiveinteger r, wedefine the block-Toeplitz matrix

TrðAÞ ¼n

a0 a1 . . . aL�1 0n . . . 0n

0n a0 a1 . . . aL�1 . . . 0n

..

. . .. . .

. ...

0n . . . 0n a0 a1 . . . aL�1

26666664

377777752 CðnrÞ�ðLþr�1Þ; ð1Þ

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dð�Þ and E½�� denote the Kronecker delta and thestatistical averaging operator; and, finally, for anarbitrary positive integer number r and m 2 Z; wedenote with CrðmÞ 2 Rr�r the matrix whoseðk þ 1; h þ 1Þth entry is given by dðm þ h � kÞ;for k; h 2 f0; 1; . . . ; r � 1g:

Second-order moments of a zero-mean discrete-time complex random process sðnÞ can bedefined in terms of two functions: the autocorrela-tion function Rssðn;mÞ ¼

nE½sðnÞ s�ðn � mÞ� and the

conjugate correlation function Rss� ðn;mÞ ¼n

E½sðnÞ sðn � mÞ�: A complex random process sðnÞ

whose conjugate correlation function vanishesidentically, i.e., Rss� ðn;mÞ � 0 for any n;m 2 Z;is called proper, otherwise it is said improper [16].Although many complex processes arising incommunications turn out to be proper (e.g., thoseobtained as complex envelopes of bandpass wide-sense stationary signals [16]), there are someinteresting examples and applications of improperrandom processes (see [19] for a review). A greatvariety of random processes with applications tocommunication theory exhibit periodically oralmost-periodically time-variant second-order mo-ments (see [6] for a general treatment). A zero-mean random process sðnÞ whose autocorrelationfunction is periodic in n with period N1X1; i.e.,Rssðn;mÞ ¼ Rssðn þ N1;mÞ; is called (wide-sense)cyclostationary [6] of period N1; similarly, if theconjugate correlation function is periodic in n

with period N2X1; i.e., Rss� ðn;mÞ ¼ Rssðn þ

N2;mÞ; the process sðnÞ is called (wide-sense)conjugate cyclostationary [6] of period N2: Observethat, when the previous properties hold for N1 ¼ 1and/or N2 ¼ 1; the functions Rssðn;mÞ and/orRss� ðn;mÞ do not depend on n. A proper processcan be cyclostationary, whereas improper pro-cesses can be cyclostationary and/or conjugatecyclostationary.

3. The system model

Let us consider a digital communication systememploying linear modulation with baud-rate 1=T s:The complex envelope of the received signal, afterfiltering, ideal carrier-frequency recovering and

baud-rate sampling, can be expressed as

rðnÞ ¼XLc�1

i¼0

cðiÞ sðn � iÞ þ wðnÞ; (2)

where sðnÞ is the sequence of the transmittedsymbols, cðnÞ denotes the composite impulse re-sponse (including transmitting filter, channel, receiv-ing filter, and timing offset) of the linear time-invariant discrete-time signal channel, which isassumed to be a causal finite-impulse response(FIR) filter of order Lc � 141; and, finally, wðnÞ

represents additive noise at the output of thereceiving filter.

The following assumptions will be consideredthroughout the paper:

(A1)

The symbol sequence sðnÞ is a white zero-mean complex random process, whose auto-correlation function RssðmÞ ¼

nE½sðnÞ s�ðn �

mÞ� ¼ s2s dðmÞ; with s2

s ¼ E½jsðnÞj2�40; is in-dependent of n; in addition, sðnÞ is improper

and conjugate cyclostationary of period N,with conjugate correlation functionRss� ðn;mÞ ¼

nE½sðnÞ sðn � mÞ� ¼ s2

s f ðnÞ dðmÞ;where f ðnÞ is a known complex-valuedperiodic function of period N;

(A2)

wðnÞ is a white zero-mean complex randomprocess, statistically independent of sðnÞ;whose autocorrelation function RwwðmÞ ¼

n

E½wðnÞw�ðn � mÞ� ¼ s2w dðmÞ; with s2

w ¼

E½jwðnÞj2�; is independent of n; in addition,wðnÞ is proper, i.e., Rww� ðn;mÞ ¼

nE½wðnÞwðn �

m� � 0 for any n;m 2 Z;

(A3) cðnÞ is a complex-valued channel, that is,

neither cRðnÞ nor cI ðnÞ vanish identically;moreover, without loss of generality, it isassumed that cð0Þa0 and cðLc � 1Þa0:

A large number of digital modulation schemes ofpractical interest [17] satisfy assumption A1,including ASK, differential BPSK (DBPSK),offset QPSK (OQPSK), offset QAM (OQAM),MSK and its variant Gaussian MSK (GMSK)(see Section 5 for a detailed exposition anddiscussion). Assumption A2 is surely fulfilled ifthe continuous-time filter used at the receiving sidehas (approximatively) a square root raised-cosineimpulse response; more generally, A2 holds if a

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whitened matched-filter [17] is employed at thereceiver.

4. Subspace-based blind channel identification

In this section we discuss the feasibility of usingsubspace-based techniques for estimating theunknown channel impulse response, and we derivethe channel identification technique in a generalframework. More specifically, in the following, wewill exploit the second-order statistical propertiesof the improper received signal rðnÞ to derive ageneral subspace method, which allows one toblindly estimate the unknown channel vectorc¼

n½cð0Þ; cð1Þ; . . . ; cðLc � 1Þ�T 2 CLc : To this aim,

we collect Le consecutive samples of rðnÞ in thevector rðnÞ ¼

n½rðnÞ; rðn � 1Þ; . . . ; rðn � Le þ 1Þ�T 2

CLe ; which, accounting for (2), can be written as

rðnÞ ¼ C sðnÞ þ wðnÞ; (3)

where C ¼nTLe

ðcTÞ 2 CLe�K represents the wide(LeoK) channel Toeplitz matrix associated withthe channel vector c; with K ¼

nLe þ Lc � 1;

sðnÞ ¼n½ sðnÞ; sðn � 1Þ; . . . ; sðn � K þ 1Þ �T 2 CK and

wðnÞ ¼n½wðnÞ;wðn � 1Þ; . . . ;wðn � Le þ 1Þ�T 2 CLe :

Observe that, except for the trivial case when thechannel impulse response cðnÞ vanishes identically(which is in contrast to assumption A3), thechannel matrix C turns out to be structurallyfull-row rank, i.e., rankðCÞ ¼ Le: To convenientlyexploit the improper nature of the transmittedsymbols, let us consider the augmented vectorerðnÞ 2 C2Le obtained by stacking rðnÞ and itscomplex conjugate r�ðnÞ: Accounting for (3), sucha vector can be expressed as

erðnÞ ¼n rðnÞ

r�ðnÞ

" #¼

C OLe�K

OLe�K C�

" #|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Ca2Cð2LeÞ�ð2KÞ

�sðnÞ

s�ðnÞ

" #|fflfflfflffl{zfflfflfflffl}esðnÞ2C2K

þwðnÞ

w�ðnÞ

" #|fflfflfflfflffl{zfflfflfflfflffl}ewðnÞ2C2Le

: ð4Þ

Note that, due to its block diagonal structure andthe fact that C is full-row rank, also the wide

matrix Ca is full-row rank, i.e., rankðCaÞ ¼

rankðCÞ þ rankðC�Þ ¼ 2 Le:

The proposed channel identification method isbased on the eigenvalue decomposition (EVD) ofthe autocorrelation matrix eRrrðnÞ ¼

nE½erðnÞerHðnÞ� 2

Cð2LeÞ�ð2LeÞ of the augmented vector erðnÞ: Account-ing for (4) and invoking assumptions A1–A2, sucha matrix can be written aseRrrðnÞ ¼ Ca

eRssðnÞCHa þ s2

w I2Le; (5)

where eRssðnÞ ¼n

E½esðnÞesHðnÞ� 2 Cð2KÞ�ð2KÞ denotesthe autocorrelation matrix of esðnÞ; whose elementscan be expressed in terms of RssðmÞ and Rss� ðn;mÞ:Since, by assumption A1, the conjugate correlationfunction Rss� ðn;mÞ is periodic in n with period N,the matrix eRssðnÞ is periodic in n with the sameperiod N; hence it is completely specified by itsvalues for n 2 M¼

nf0; 1; . . . ;N � 1g: A subspace-

based blind identification procedure can be derivedfrom (5) only if the signal-dependent part

CaeRssðnÞCH

a 2 Cð2LeÞ�ð2LeÞ of the autocorrelationmatrix eRrr is singular, i.e., it results DðnÞ ¼

n

rank½CaeRssðnÞCH

a �o2 Le; at least for one valueof n 2 M; in this case, the signal contribution isconfined into a DðnÞ–dimensional subspace ofC2Le : Thus, it is crucial to investigate whether theprevious condition is verified; as a first steptowards this end, we provide the followingLemma.

Lemma 1. The matrix eRssðnÞ is singular for any n 2

M if and only if NpK and there exists at least one

value n 2 M such that jf ðnÞj ¼ 1:

Proof. By invoking assumption A1, eRssðnÞ can bepartitioned as follows:

eRssðnÞ ¼ s2s

IK FðnÞ

F�ðnÞ IK

� �; (6)

where the matrix FðnÞ ¼n

diag½f ðnÞ; f ðn �

1Þ; . . . ; f ðn � K þ 1Þ� 2 CK�K is periodically time-varying in n with period N. By using the formulafor the determinant of a 2-by-2 partitioned matrix[9], one obtains

det½eRssðnÞ� ¼ s2s detðIK Þ � det½IK � F�ðnÞFðnÞ�

¼ s2s

YK�1

‘¼0

1 � jf ðn � ‘Þj2� �

: ð7Þ

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Hence, for a given n, det½eRssðnÞ� ¼ 0 if and only ifthere exists at least one n 2 fn; n � 1; . . . ; n � K þ

1g such that jf ðnÞj ¼ 1: Remembering that f ðnÞ is aperiodic function, if we make the simplifyingassumption that its period NpK ; the previouscondition is satisfied for any n 2 M if and only ifthere exists at least one value n 2 M such thatjf ðnÞj ¼ 1: &

Singularity of eRssðnÞ; discussed in Lemma 1,represents a necessary condition in order to derivea subspace-based channel identification procedureon the basis of (5). In fact, since the matrix Ca isfull-row rank, if eRssðnÞ is nonsingular, it results [9]that rank½Ca

eRssðnÞCHa � ¼ rankðCaÞ ¼ 2 Le; in this

case, subspace-based channel identification basedon (5) is impossible, since the columns of thematrix Ca

eRssðnÞCHa span the whole observation

space C2Le : It is interesting to note that, accordingto Lemma 1, eRssðnÞ is surely nonsingular when thesymbol sequence sðnÞ is proper, i.e., Rss� ðn;mÞ � 0for any n;m 2 Z: in fact, in this case it resultsf ðnÞ � 0 for any value of n 2 M: Therefore,subspace-based identification based on (5) is im-

possible when sðnÞ is proper, which motivates aposteriori our choice to discuss the case of animproper random process sðnÞ: However, using anarbitrary improper random sequence sðnÞ does notassure that its associated matrix eRssðnÞ is singular,since it might happen that the conditions ofLemma 1 are not verified, that is, jf ðnÞja1 forany n 2 M: On the other hand, singularity ofeRssðnÞ represents only a necessary, but notsufficient, condition in order for Ca

eRssðnÞCHa to

be singular: indeed, since Ca is wide (LeoK), thesignal-dependent matrix Ca

eRssðnÞCHa can be non-

singular even when eRss is singular. Thus, thefollowing Lemma provides a sufficient conditionon matrix eRssðnÞ; which assures the singularity ofthe signal-dependent matrix Ca

eRssðnÞCHa regard-

less of the channel coefficients.

Lemma 2. The matrix CaeRssðnÞCH

a is singular for

any n 2 M if rank½eRssðnÞ�o2 Le for any n 2 M:Moreover, let MðnÞ denote the number of diagonal

entries of FðnÞ that lie on the unit circle in the

complex plane and assume that f ðnÞa0 for any n 2

M; it results that rank½eRssðnÞ�o2 Le for any n 2 Mif and only if MðnÞ42 ðLc � 1Þ for any n 2 M:

Proof. By using straightforward rank inequalities[9], it results that

rank½CaeRssðnÞCH

a �pminf2 Le; rank½eRssðnÞCHa �g

¼ rank½eRssðnÞCHa �

pminf2 Le; rank½eRssðnÞ�g ð8Þ

and, thus, if rank½eRssðnÞ�o2 Le; it follows that

rank½CaeRssðnÞCH

a �prank½eRssðnÞ�o2 Le; for any

n 2 M; thus CaeRssðnÞCH

a is singular. Moreover,

since f ðnÞa0 for any n 2 M; the matrix FðnÞ isnonsingular for any n 2 M and, thus, accounting

for (6), the rank of eRssðnÞ is equal to the rank of thefollowing matrix:

s�2s

IK OK�K

OK�K FðnÞ

� � eRssðnÞ ¼IK FðnÞ

FðnÞF�ðnÞ FðnÞ

� �;

(9)

where FðnÞF�ðnÞ ¼ diag½jf ðnÞj2; jf ðn � 1Þj2; . . . ;

jf ðn � K þ 1Þj2�: Owing to the particular structureof the matrix at the right-hand side of Eq. (9), letMðnÞ denote the number of diagonal entries ofFðnÞ that lie on the unit circle in the complex plane,

it is apparent that rank½eRssðnÞ� ¼ 2 K �

MðnÞo2 Le if and only if MðnÞ42 Le � 2 K ¼

2 ðLc � 1Þ; for any n 2 M: &

Observe that Lemma 2 requires that P values ofthe function f ðnÞ; say f ðn1Þ; f ðn2Þ; . . . ; f ðnPÞ; lie onthe unit circle in the complex plane, whereP¼

nfn1; n2; . . . ; nPg � M must be chosen such that

MðnÞ42 ðLc � 1Þ for any n 2 M: To simplifymatters, we focus hereinafter our attention tothe case where P � M; that is, we requirethat jf ðnÞj ¼ 1; for any n 2 M; in this case, itresults that MðnÞ ¼ K and, thus, the conditionof Lemma 2 is satisfied if: (A4) Le4Lc � 1; whichin the following will be assumed to hold. Inthis case, Lemma 2 assures that the subclass ofimproper random processes for which jf ðnÞj ¼ 1for any n 2 M allows one to perform sub-space-based blind channel identification on thebasis of (5). Motivated by this observation, sincef ðnÞ is periodic of period N, we further restrictsomehow assumption A1 by giving the followingdefinition:

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Definition 1. A random process sðnÞ satisfying A1 iscalled strongly improper if jf ðnÞj ¼ 1 for any n 2 M:

Let us observe that, according to A1 and theabove definition, an arbitrary random process sðnÞ

is strongly improper if and only if jE½s2ðnÞ�j ¼

s2s jf ðnÞj ¼ E½jsðnÞj2�; for any n 2 Z; in its turn, by

invoking the Cauchy–Schwartz inequality, thisrelation holds if and only if

Efjs�ðnÞ � aðnÞ sðnÞj2g ¼ 0;

for any n 2 Z; ð10Þ

where aðnÞ is an arbitrary complex-valued deter-ministic function of n: thus, we can equivalentlysay that the random process sðnÞ is stronglyimproper if and only if there exists a mean-square

linear dependence between sðnÞ and its conjugates�ðnÞ; for any n 2 Z:

For the sake of simplicity, in the following, wewill restrict our attention to the case where theperiodic function f ðnÞ; satisfying the constraintjf ðnÞj ¼ 1; is a simple complex exponential, thatis, we assume that f ðnÞ ¼ ej 2pbn; where b ¼ p=N

and p 2 M; the value 0pbo1 is calledthe conjugate cycle frequency [6] of the processsðnÞ: It is interesting to note that, in this case,the rank of the signal-dependent matrixCaeRssðnÞCH

a is independent of n, i.e., DðnÞ ¼ D;for any n 2 M: In fact, in this case it can be seenthat the periodically time-varying matrix FðnÞin (6) can be written as FðnÞ ¼ ej 2pbn J ; withJ ¼

ndiag½1; e�j 2pb; . . . ; e�j 2pbðK�1Þ� 2 CK�K and,

thus, accounting for the definition (4) of Ca; theautocorrelation matrix eRrrðnÞ given by (5) canbe rewritten, after straightforward algebraic ma-nipulations, aseRrrðnÞ ¼ s2

s XðnÞWðcÞWHðcÞX�

ðnÞ þ s2w I2Le

; (11)

where XðnÞ ¼n

diag½ILe; e�j 2pbnILe

� 2 Cð2LeÞ�ð2LeÞ is aknown block diagonal matrix, whereas the matrixWðcÞ ¼

n½CT; J� CH

�T 2 Cð2LeÞ�K depends on the(unknown) channel vector c 2 CLc : Since thediagonal matrix XðnÞ is nonsingular for any n 2

M; it turns out that D¼n

rank½CaeRssðnÞCH

a � ¼

rank½WðcÞ�pK ¼ Le þ Lc � 1; where the last in-equality accounts for assumption A4; moreover,since the matrix C is full-row rank, it results thatDXLe: Therefore, we conclude that the value of

the rank of the signal-dependent matrixCaeRssðnÞCH

a belongs to D 2 fLe;Le þ 1; . . . ;Le þ

Lc � 1g for any n 2 M:The main difficult in using (11) to blindly

estimate the unknown channel vector c lies in theperiodically time-varying nature of eRrrðnÞ; whichcomplicates its estimation from the received data.To remove the variability with n, we multiplyeRrrðnÞ by the known matrices X�

ðnÞ and XðnÞ;obtaining thus

Rzz ¼n X�

ðnÞ eRrrðnÞXðnÞ ¼ s2s WðcÞWðcÞH þ s2

w I2Le;

(12)

where the time-invariant matrix Rzz can be inter-preted as the autocorrelation matrix of the vectorzðnÞ¼

n X�ðnÞerðnÞ ¼ ½rTðnÞ; ej 2pbn rH ðnÞ�T 2 C2Le ; i.e.,

Rzz ¼ E½zðnÞ zHðnÞ�: On the basis of (12), thechannel vector c can be estimated by resorting tothe EVD of Rzz; which in its turn can beconsistently estimated, via batch or adaptive algo-rithms, from the received data. Following thisapproach, let l1Xl2X � � �Xl2Le

denote the eigen-values of the autocorrelation matrix Rzz; it followsthat li4s2

w; for i 2 f1; 2; . . . ;Dg; and li ¼ s2w; for

i 2 fD þ 1;D þ 2; � � � ; 2 Leg: Moreover, let ei 2

C2Le represent the eigenvector associated with theith eigenvalue li; the EVD of Rzz is given by

Rzz ¼ ES KS EHS þ EN KN EH

N; (13)

where KS ¼n

diag½l1; l2; . . . ; lD� 2 RD�D and KN ¼n

s2w I2Le�D; whereas the columns of the matrices

ES ¼n½e1; e2; . . . ; eD� 2 Cð2LeÞ�D and EN ¼

n½eDþ1;

eDþ2; . . . ; e2Le� 2 Cð2LeÞ�ð2Le�DÞ span the D-dimen-

sional subspace R½WðcÞ� (referred to as the signal

subspace) and the ð2Le � DÞ-dimensional orthogo-nal complement R?½WðcÞ� (referred to as the noise

subspace) in C2Le to the signal subspace. Byexploiting the orthogonality between signal andnoise subspaces, the following relation holds:

WðcÞWHðcÞEN ¼ Oð2LeÞ�ð2Le�DÞ; (14)

which can be regarded as a matrix equation in c;under appropriate conditions detailed in Theorem1, the solution of (14) turns out to be essentiallyunique. Since discussion of the rank of WðcÞ plays akey role, we first provide the following Lemma.

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1Note that, by construction [see the orthogonality relation

(16)], all the eigenvectors corresponding to the smallest real

eigenvalue of the matrix QHQ must be real-valued. This

structural property remains valid even when we considerbQH bQ; which is constructed by using bPN:

D. Darsena et al. / Signal Processing 84 (2004) 2021–20392028

Lemma 3. Let z1; z2; . . . ; zLc�1 denote the zeros of

the Z-transform of the channel vector c: Under

assumption A4, the matrix WðcÞ turns out to be full-

column rank, i.e., D ¼ rank½WðcÞ� ¼ K ¼

Le þ Lc � 1; if and only if the following assumption

holds: (A5) z‘az�q ej 2pb; for ‘; q 2 f1; 2; . . . ;Lc � 1g:

Proof. See Appendix A. &

We defer to Section 5 a thorough discussionabout implications of Lemma 3 and assumptionA5 in practical cases, with specific reference to theconsidered modulation schemes. The followingidentification Theorem provides sufficient condi-tions for unique channel identification.

Theorem 1 (Identifiability condition). Let PN 2

CR�2Le denote any matrix verifying NðPNÞ ¼

R?ðENÞ; and consider the matrix equation

PN WðcÞ ¼ OR�ðLeþLc�1Þ: (15)

Under assumptions A4 and A5, the following two

statements are equivalent: (1) c0ac is a solution of

(15); (2) c0 ¼ x c; with x 2 R:

Proof. See Appendix B. &

The above theorem asserts that, under assump-tions A4 and A5, the channel vector c can beuniquely determined, up to a scalar factor, fromthe ðLe � Lc þ 1Þ-dimensional noise subspaceRðENÞ of the autocorrelation matrix Rzz); notethat, unlike standard subspace-based channelidentification, the scalar factor is real rather thancomplex. As to the choice of the matrix PN;several options may be pursued: for example, onecan choose PN ¼ EH

N 2 CðLe�Lcþ1Þ�2Le or PN ¼

EN EHN 2 C2Le�2Le : Although the former choice is

undoubtedly the most obvious one, the latter onesimplifies the performance analysis of the channelestimation method for a finite sample size (thisissue is explored in Section 6), and thus we adoptthe second choice hereinafter.

By exploiting the structure of WðcÞ; Eq. (15) canbe equivalently written as a system of linear

equations in the unknown channel coefficients.To this aim, observe that (15) holds if and only ifvec½PN WðcÞ� ¼ 02KLe

; with K ¼ Le þ Lc � 1;which can be written as

vec½PN WðcÞ� ¼ Q ca ¼ 02KLe; (16)

where the parameterization matrix Q 2 Cð2KLeÞ�ð2LcÞ

depends on J and EN (see Appendix C for itsdetailed expression), whereas the vectorca ¼

n½cT

R; cTI �

T 2 R2Lc is obtained by stacking thereal cR and imaginary cI parts of the channelvector c: Note that, according to Theorem 1, theunknown vector ca can be uniquely determined upto a real scalar factor and, hence, rankðQÞ ¼

2 Lc � 1 must hold.In practice, we do not know the autocorrelation

matrix Rzz and, thus, sample estimates of the eigen-vectors spanning the signal and noise subspaces mustbe obtained from the sample autocorrelation matrix

bRzz ¼n 1

Ns

XNs�1

n¼0

zðnÞzHðnÞ

¼ bESbKS

bEH

S þ bENbKN

bEH

N; ð17Þ

where Ns denotes the estimation sample size. For theproblem at hand, the matrix bRzz converges (in meansquare and with probability one) to Rzz

as Ns approaches infinity. Let bQ denote theparameterization matrix corresponding to bPN ¼bEN

bEH

N; equality (16) holds only approximately,

that is, e¼n

vec½bPN WðcÞ� ¼ bQ ca ’ 02KLe: In this

case, an estimate bca of the unknown vector ca canbe obtained by solving the constrained minimizationproblem

bca ¼ argminx 2R2Lc

xH bQH bQ x;

subject to kxk2 ¼ 1; ð18Þ

where the constraint is necessary to avoid the trivialsolution bca ¼ 02Lc

: By invoking the Rayleigh–Ritztheorem [9], the solution of the optimizationproblem (18) is given by the eigenvector1 associatedwith the smallest eigenvalue of the matrix bQH bQ:Finally, observe that, similarly to many subspace-based algorithms (e.g., [15]), the described algorithmrequires accurate channel-length determination,

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which can be obtained by performing rank estima-tion of the sample autocorrelation matrix bRzz:

2

5. Application to some digital modulation schemes

A large number of digital modulation schemesfall into the considered framework, since theysatisfy assumption A1 and agree with Definition 1;in the following, we briefly recall some of them andparticularize the general results obtained in theprevious section, discussing some specific issues.

3For instance, with reference to the GMSK case, it results [5]

5.1. Real-valued modulations

In M-ary amplitude-shift keying (M-ASK)modulation [17], the symbols sðnÞ are modeled asa sequence of independent and identically distrib-uted (i.i.d.) real-valued random variables, eachassuming equiprobable values in S ¼ f2 p � M �

1gMp¼1: When M ¼ 2; the 2-ASK modulation

coincides with the well-known binary phase-shiftkeying (BPSK) modulation. A variation of BPSKis the differential BPSK (DBPSK) modulationscheme [17], wherein the information is encoded inthe phase differences between successive signaltransmissions, as opposed to absolute phaseencoding used in the BPSK modulation. For M-ASK, BPSK, and DBPSK, it can be easily verifiedthat Rss� ðn;mÞ ¼ RssðmÞ ¼ s2

s dðmÞ; with s2s ¼

ðM2 � 1Þ=3 for M-ASK and s2s ¼ 1 for BPSK

and DBPSK. Thus, the symbol sequence sðnÞ is astrongly improper random process with conjugatecycle frequency b ¼ 0: For the above-mentionedmodulation techniques, assumption A5 (Lemma 3)simplifies to z‘az�q; for ‘; q 2 f1; 2; . . . ;Lc � 1g;which means that, according to Theorem 1, thechannel vector c can be uniquely identified if its Z-

transform has no real zeros and no conjugate pairs

of zeros. Interestingly, this identifiability conditionalso assures the existence of a widely linear FIRzero-forcing (ZF) equalizer [7], which achieves

2Recently, subspace-based algorithms have been developed

[13], which are based on rank-revealing ULV decomposition

and do not require rank estimation. The feasibility of applying

these algorithms to our framework is an interesting research

issue, which however is outside the scope of this paper.

perfect ISI suppression in the absence of noiseeven when the signal is sampled at the baud-rate;this result contrasts with simple linear equaliza-tion, where ZF baud-rate equalization of a FIRchannel can be attained only by resorting toinfinite-impulse response (IIR) filters. Finally, notethat the authors in [8] claim that their method isable to identify the channel vector c if its Z-transform has no real zeros. However, as shown inAppendix D, this claim is based on a not entirelycorrect derivation, whereas the correct identifia-bility conditions of the method proposed in [8] arethe same as those derived in this paper.

5.2. MSK-type modulations

The minimum-shift keying (MSK) modulation[17] is a binary continuous-phase modulation(CPM) with index h ¼ 1

2: An important example

of MSK modulation is the Gaussian MSK(GMSK), which is currently employed in theglobal system for mobile (GSM) communicationsstandard. All CPM modulations belong to theclass of nonlinear modulation schemes and, thus,the received MSK signal cannot be exactlyrepresented by the linear model considered in (2).However, following the development given byLaurent (see, e.g., [17]), for an arbitrary modula-tion pulse of (approximately) finite duration L T s;the continuous-time MSK signal can be repre-sented as a linear superposition of 2L�1 amplitude-modulated signals, where the first term (principal

component) contains the most significant part ofthe signal energy.3 Thus, by considering only theprincipal component, after baud-rate sampling,the received MSK signal can be satisfactorilyapproximated by (2), where the symbol sequenceis expressed recursively [17] as sðnÞ ¼ j aðnÞ sðn �

1Þ; with aðnÞ being a sequence of i.i.d. real-valuedrandom variables taking equiprobable values inS ¼ f�1; 1g: Assuming, without loss of generality,

that the principal component of the Laurent series expansion of

the transmitted signal contains the 99.5% of the overall GMSK

signal energy. In other words, the effect of approximating the

GMSK signal with a single amplitude-modulated pulse is to

virtually generate an additive interference in the received signal,

with a signal-to-interference ratio (SIR) of about 23 dB.

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that sð0Þ ¼ 1; this recursive equation can beexplicitly solved for sðnÞ; obtaining

sðnÞ ¼ j nYn

p¼1

aðpÞ; for n ¼ 1; 2; 3; . . . : (19)

Taking into account (19), it can be easily seen thatRssðmÞ ¼ dðmÞ and Rss� ðn;mÞ ¼ ð�1Þn dðmÞ: Thus,the symbol sequence sðnÞ is a strongly improperprocess with conjugate cycle frequency b ¼ 1

2: In

this case, assumption A5 simplifies to z‘a� z�q;for ‘; q 2 f1; 2; . . . ;Lc � 1g; which means that,according to Theorem 1, the channel vector c canbe uniquely identified if its Z-transform has no

purely imaginary zeros and no anti-conjugate4 pairs

of zeros. Note that a similar identifiability condi-tion appears in [5] with reference to blind channelequalization for GSM cellular systems.

5Although more complicated QAM constellation geometries

can be considered, rectangular constellations are most fre-

5.3. Offset modulations

The term offset or staggered modulation refersto any complex scheme wherein the quadraturecomponent (Q) of the modulated signal is shiftedin time with respect to the in-phase components(I). Common examples of this family are the offset

quaternary phase-shift keying (OQPSK) modula-tion and the offset quadrature amplitude modula-

tion (OQAM) [17], which are the offset versions ofQPSK and QAM, respectively. The main advan-tage of offset modulations, with respect to theirnon-offset counterparts, is the increased band-width efficiency, which motivated their use inwireless [1,10] and cable modem systems [2].

The OQPSK transmitted signal can be regardedas composed by two BPSK signals in-quadraturefor which the phase transitions are staggered intime by T s seconds. After baud-rate sampling, thereceived OQPSK signal can be represented by (2),where the symbol sequence sðnÞ is given by

sðnÞ ¼aðnÞ; n even;

j aðnÞ; n odd

�(20)

and aðnÞ is a sequence of i.i.d. random variablesassuming equiprobable values in S ¼ f�1; 1g:

4We say that a; b 2 C are anti-conjugate if a ¼ �b�:

Rectangular5 M-ary quadrature amplitude mod-ulation (QAM) with rate 1=T s; where M ¼ 2q (qeven), can be interpreted [17] as the effect ofsimultaneously impressing, on two quadraturecarriers, two half-rate mutually independent reali.i.d. sequences aI ðnÞ and aQðnÞ assuming equiprob-able values in S ¼ f2 p �

ffiffiffiffiffiffiM

p� 1g

ffiffiffiffiM

p

p¼1 : An M-OQAM signal is obtained from an M-QAM one,by introducing between the two quadraturecomponents a time offset of one symbol periodT s: In this case, it can be seen [7] that, after baud-rate sampling, the received OQAM signal can berepresented by the model (2), where

sðnÞ ¼aI ð‘Þ; n ¼ 2 ‘;

j aQð‘Þ; n ¼ 2 ‘ þ 1:

�(21)

Due to the close similarity between (20) and (21),both OQPSK and OQAM signals satisfy assump-tion A1 and Definition 1: indeed, it results thatRssðmÞ ¼ s2

s dðmÞ and Rss� ðn;mÞ ¼ s2s ð�1Þn dðmÞ;

with s2s ¼ 1 for OQPSK and s2

s ¼ ðM � 1Þ=3 forM-OQAM. Thus, similarly to MSK, the informa-tion sequence sðnÞ for OQPSK and OQAMtransmissions is a strongly improper process withconjugate cycle frequency b ¼ 1

2: Since the con-

jugate correlation function has essentially the sameform of the MSK one, the identifiability conditionsfor OQPSK and OQAM are the same alreadydiscussed for MSK-type modulations.

6. Performance analysis

In this section, we provide the theoreticalperformance analysis of the channel identificationmethod described in Section 4, by deriving theanalytical expression of the estimation meansquare error (MSE). The basic tool for ouranalysis is the powerful first-order perturbationapproach (see, e.g., [14,23]), which allows us torelate, under the large sample size assumption, theproperties of bPN ¼ bEN

bEH

N to those of the sampleautocorrelation matrix bRzz of the preprocessed

quently used in practice since they can be easily generated and

demodulated [17].

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received vector zðnÞ ¼ X�ðnÞerðnÞ: Our derivation

holds in particular for the modulation schemesdiscussed in Section 5, and the theoretical resultswill be validated by computer simulation experi-ments in Section 7.

When the autocorrelation matrix Rzz is esti-mated from the received data as in (17), for asufficiently large sample-size Ns; the estimate canbe decomposed as bRzz ¼ Rzz þ dRzz; where dRzz isa small additive perturbation (in the Frobeniusnorm sense). Consequently, the matrix bQ can bewritten [14,23] as bQ ¼ Q þ dQ; where dQ repre-sents the resulting perturbation in the estimatedparameterization matrix, whose norm is of theorder of kdRzzk: Since the channel vector ca isestimated as the eigenvector of bQH bQ [cfr. (18)]corresponding to its smallest eigenvalue,the estimated bca can be decomposed in its turn asbca ¼ ca þ dca; where dca represents the channelestimation error, whose norm is of the order ofkdQk: Since the identification method of Section 4is able to provide the channel impulse responseonly up to a real scalar factor, to carry out theperformance analysis it is necessary to suitablynormalize the estimated vector bca: We thusnormalize bca so that its dth entry bcaðdÞ is exactlyequal to the corresponding entry caðdÞa0 in thetrue channel vector ca; where d ¼ argmaxijcaðiÞj

and, hence, we define the normalized channel erroras follows:

ðdcaÞn ¼n caðdÞbcaðdÞ

bca � ca: (22)

Reasoning as in [18], we observe that,under the small perturbation assumption, onehas

caðdÞbcaðdÞ¼

caðdÞ

caðdÞ þ dcaðdÞ¼

1

1 þdcaðdÞcaðdÞ

� 1 �dcaðdÞ

caðdÞ; ð23Þ

where here and in the sequel the symbol � denotesfirst-order equality, i.e., we neglect all the sum-mands that tend to zero, as the sample size Ns

approaches infinity, faster than the perturbationnorm kdRzzk: By substituting (23) in (22), it

follows that

ðdcaÞn � dca �ca

caðdÞdcaðdÞ

¼ I2Lc�

ca f Td

caðdÞ

� �|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

U2Cð2LcÞ�ð2LcÞ

dca; ð24Þ

where f d 2 R2Lc is the dth canonical basis vectorhaving all entries equal to zero expect for the dthone. Taking into account (24), the normalizedchannel estimation MSE is given by

E½kðdcaÞnk2� � E½dcT

a UTUdca�

¼ tracefU E½dca dcTa �U

Tg; ð25Þ

where the last equality is obtained by using theproperties of the trace operator. The channelperturbation dca can be expressed [14,23] in termsof dQ; more precisely, using (16), it turns out thatbQbca � Q dca þ dQ ca � 02KLe

; with K ¼ Le þ Lc �

1; and, therefore, it follows that dca �

�Qy dQ ca ¼ �QyðbQ � QÞca ¼ �Qy bQ ca: Conse-

quently, one obtains

E½dca dcTa � � Qy E½ðbQ caÞ ðbQ caÞ

H�ðQy

ÞH

¼ Qy Ree ðQyÞH; ð26Þ

where Ree ¼n

E½e eH� denotes the autocorrelationmatrix of the vector e ¼ bQ ca ¼ vec½bPN WðcÞ� 2C2KLe : The next step consists of expressing Ree interms of the sample autocorrelation matrix bRzz: Tosimplify matters, we assume that the mean-squarelinear dependence (10) between sðnÞ and s�ðnÞ

holds everywhere, i.e., s�ðnÞ ¼ aðnÞ sðnÞ for any n 2

Z and for any realization of the random processsðnÞ: To be consistent with A1, it turns out thataðnÞ ¼ f �1

ðnÞ ¼ e�j 2pbn; hence we are assumingexplicitly that s�ðnÞ ¼ e�j 2pbn sðnÞ: Observe thatthis subclass of strongly improper processesencompasses all the modulation formats recalledin Section 5. In these cases, it turns out that s�ðnÞ ¼J� sðnÞ e�j 2pbn which, accounting for model (4) andthe partitioned structure of X�

ðnÞ and Ca; allowsus to explicitly write zðnÞ as follows:

zðnÞ ¼ WðcÞ sðnÞ þ vðnÞ; (27)

where vðnÞ ¼n X�

ðnÞ ewðnÞ 2 C2Le : This alternativeexpression of zðnÞ; together with the simplifyingassumption that the additive noise wðnÞ is

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Gaussian, allows one to obtain a coincise expres-sion of the autocorrelation matrix Ree; which isvalid up to first-order in kdRzzk:

Lemma 4. Assume, in addition to A2, that the

additive noise wðnÞ is a Gaussian process. Under the

assumption that the signal-plus-noise eigenvalues

l1; l2; . . . ; lK of Rzz are distinct,6 the first-order

approximation of the autocorrelation matrix of the

residual vector e is given by

Ree �1

Ns

XLe�1

m¼�ðLe�1Þ

½WHðcÞYRzzðmÞ

�YWðcÞ�� � ½PN RvvðmÞPN�; ð28Þ

where we have defined the Hermitian matrix

Y¼n

ES ðKS � s2w IK Þ

�1EHS 2 Cð2LeÞ�ð2LeÞ; and the

m-lag autocorrelation matrices RzzðmÞ ¼n

E½zðnÞ zHðn � mÞ� and RvvðmÞ ¼n

E½vðnÞ vHðn � mÞ�:Moreover, it results that RzzðmÞ ¼ s2

s WðcÞCK ðmÞWH

ðcÞ þ RvvðmÞ and RvvðmÞ ¼ s2w ½I2 �

CLeðmÞ�X�

ðmÞ:

Proof. The proof of (28) is lengthy but can beconducted by following the guidelines delineated in[11,24]. As to the explicit expressions of matricesRzzðmÞ and RvvðmÞ; observe that, taking intoaccount the signal model (27) and invokingassumptions A1 and A2, it is easily seenthat RzzðmÞ ¼ WðcÞRssðmÞWH

ðcÞ þ RvvðmÞ; whereRssðmÞ¼

nE½sðnÞ sHðn � mÞ� ¼ s2

s CK ðmÞ: On theother hand, observing that, under assumption A2,it results that eRwwðmÞ¼

nE½ewðnÞ ewH

ðn � mÞ� ¼ s2w

½I2 � CLeðmÞ�; one obtains that the autocorrelation

matrix of vðnÞ ¼ X�ðnÞ ewðnÞ is given by RvvðmÞ ¼

X�ðnÞ eRwwðmÞX�

ðn � mÞ ¼ s2w ½I2 � CLe

ðmÞ�X�ðmÞ:

At this point, we can state the main result of thissection.

Theorem 2 (Normalized channel MSE). Assume, in

addition to A2, that the additive noise wðnÞ is a

Gaussian process. Under the assumption that the

signal-plus-noise eigenvalues l1; l2; . . . ; lK of Rzz

are distinct, the first-order approximation of the

6This is a very mild assumption that holds generally in

practice.

normalized channel MSE is given by

E½kðdcaÞnk2� �

x1 þ x2

Ns; (29)

where the summands

x‘ ¼n

XLe�1

m¼�ðLe�1Þ

tracefU Qyf½WH

ðcÞESðKS � s2w IK Þ

�1

�H‘ðmÞ ðKS � s2w IK Þ

�1EHS WðcÞ��

� ½EN EHN ðI2 � CLe

ðmÞÞX�ðmÞEN EH

N�g

�ðQyÞHUTg; ð30Þ

for ‘ 2 f1; 2g; with

H1ðmÞ ¼n s2

s s2w EH

S WðcÞCK ðmÞWHðcÞES 2 CK�K

(31)

and

H2ðmÞ ¼n s4

w EHS ½I2 � CLe

ðmÞ�X�ðmÞES 2 CK�K ;

(32)

are independent of the sample size Ns:

Proof. The proof of (29) can be obtained, aftersome algebraic manipulations, by substituting (28)in (26) and the result in (25). &

It can be observed from (29) that, for givennonzero values of the symbol and noise variances s2

s

and s2w; the normalized channel MSE approaches

zero as the sample size Ns tends to infinity.

7. Simulation results

In this section, the MSE performance of theproposed SISO subspace-based identificationmethod, working at symbol spacing T s; is inves-tigated and compared with the reference SIMOsubspace-based method proposed in [15] (referredto as the method of Moulines in the sequel),working at half-symbol spacing T s=2: We validateour performance analysis of Section 6 by reportingboth the analytical results and those obtained byMonte Carlo computer simulations. For compar-ison, in addition to its experimental results,the theoretical performance of the method ofMoulines, derived in [18], is also plotted.

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In all the experiments, the following simulationsetting is assumed. The input stream sðnÞ is drawnfrom an OQPSK constellation7and the additivenoise wðnÞ is a complex Gaussian process. Thesignal-to-noise ratio (SNR) is defined asSNR¼

nðs2

s=s2wÞkck2 and both the symbol and noise

sequences are randomly and independently gener-ated at the start of each Monte Carlo run. Sincethe proposed method and the method of Moulinesemploy a different discrete-time channel, to allowfor a fair comparison, we considered for bothmethods the same continuous-time channel caðtÞ

which, with the exception of experiment 5, spansLc ¼ 3 symbol periods. More precisely, we startedfrom the T s=2-sampled version of caðtÞ; i.e.,ecðqÞ ¼n caðq T s=2Þ; for q 2 f0; 1; . . . ; 2Lc � 1g; whichcan be expressed in terms of the two polyphasecomponents ec ð0Þ

ðnÞ ¼n ecð2 nÞ and ec ð1Þ

ðnÞ ¼n ecð2 n þ 1Þ;

for n 2 f0; 1; . . . ;Lc � 1g; which are the two chan-nels employed by the SIMO method of Moulines.Thus, as a by-product, we obtain the uniquesymbol-spaced channel for the proposed SISOmethod as cðnÞ ¼ ec ð0Þ

ðnÞ; n 2 f0; 1; . . . ;Lc � 1g: Thetwo channels ec ð‘Þ

ðnÞ; for ‘ ¼ 0; 1; are assigned interms of their Z-transforms:eC ð‘Þ

ðzÞ ¼ ð1 � 0:5 ej y1;‘z�1Þ ð1 � 1:2 ej y2;‘z�1Þ;

for ‘ ¼ 0; 1; ð33Þ

where y1;0 ¼ 0:5pþ g; y2;0 ¼ y1;0 þ p; y1;1 ¼ y1;0 þ

g and y2;1 ¼ y2;0 þ g; and the angular separation gcan be adjusted so as to approach the situationwhere identification fails for both methods undercomparison. Indeed, for g ¼ 0; the first polyphasecomponent ec ð0Þ

ðnÞ � cðnÞ has two purely imagin-ary zeros, which is the identifiability limit for theproposed SISO method (see discussion in Section 5for OQPSK modulation). Moreover, for g ¼ 0; thetwo subchannels ec ð0Þ

ðnÞ and ec ð1ÞðnÞ have common

zeros, which is the identifiability limit for theSIMO method of Moulines. Unless otherwisespecified, the angular separation is fixed to g ¼0:2p: The channel coefficients are scaled so thatthe two vector c¼

n½ec ð0Þ

ð0Þ;ec ð0Þð1Þ; . . . ;ec ð0Þ

ðLc � 1Þ�T

2 CLc and ec¼n ½ecð0Þ;ecð1Þ; . . . ;ecð2Lc � 1Þ�T 2 C2Lc

7We have also carried out simulations for a BPSK modula-

tion, whose results are not reported here, since they are very

similar to those of the OQPSK case.

have unitary norm. As channel estimation perfor-mance measure, denoting with x the channelvector to be estimated for any of the two methodsunder comparison, we used the normalized MSEexpressed in dB:

MSEðdBÞ ¼n

10 log10

1

Nr

XNr

i¼1

kwbxi � xk2

!; (34)

where Nr ¼ 1000 is the number of independentMonte Carlo trials conducted for each experi-ments, bxi denotes the estimate of x obtained in theith run and, denoting with xð‘Þ and bxið‘Þ the ‘thentry of x and bxi; the normalization coefficient w isgiven by w ¼ xðdÞ=bxiðdÞ; with d ¼ argmax‘jxð‘Þj:This choice is consistent with the normalizationcarried out for performance analysis in Section 6.

Experiment 1—Normalized channel MSE versus

SNR. In this experiment, we evaluate the perfor-mances of the considered methods as a function ofSNR ranging from 0 to 50 dB. The length of thetemporal window is set to Le ¼ 5 and the samplesize is equal to Ns ¼ 200 symbols. Results of Fig. 1evidence that the proposed method outperformsthe method of Moulines for all values of SNR.Moreover, observe that the theoretical expression(29) of the MSE derived in the previous sectionagrees very well with the simulation results for allvalues of SNR whereas, in accordance with theexperimental results conducted in [11,18], thetheoretical analysis for the method of Moulinesis not accurate for small values of SNRðSNRp5 dBÞ: Indeed, since our method assuresbetter performances than the identification algo-rithm of Moulines, the small perturbation assump-tion is verified in our method for a wider range ofSNR values.

Experiment 2—Normalized channel MSE versus

sample size. In this experiment, the performancesof the considered methods are studied as afunction of the sample size Ns ranging from 50to 1000 symbols. The length of the temporalwindow is set to Le ¼ 5 and SNR ¼ 20 dB: It canbe seen from Fig. 2 that exploiting the impropernature of the transmitted symbol sequence allowsone to obtain a more data-efficient channelestimation algorithm in comparison with theSIMO subspace-based method of Moulines. Note

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0 10 20 30 40 50-70

-60

-50

-40

-30

-20

-10

0

10

SNR [dB]

Nor

mal

ized

MS

E [d

B]

Moulines (experimental)Moulines (theoretical)Proposed (experimental)Proposed (theoretical)

Fig. 1. Normalized channel MSE versus SNR (Le ¼ 5 and

Ns ¼ 200 symbols).

0 100 200 300 400 500 600 700 800 900 1000-45

-40

-35

-30

-25

-20

-15

Sample size

Nor

mal

ized

MS

E [d

B]

Moulines (experimental)Moulines (theoretical)Proposed (experimental)Proposed (theoretical)

Fig. 2. Normalized channel MSE versus Ns (in symbols)

(Le ¼ 5 and SNR ¼ 20dB).

3 4 5 6 7 8 9 10-34

-33

-32

-31

-30

-29

-28

-27

-26

-25

-24

Le

Nor

mal

ized

MS

E [d

B]

Moulines (experimental)Moulines (theoretical)Proposed (experimental)Proposed (theoretical)

Fig. 3. Normalized channel MSE versus Le (in symbols)

(Ns ¼ 200 and SNR ¼ 20 dB).

D. Darsena et al. / Signal Processing 84 (2004) 2021–20392034

that the proposed method assures a normalizedMSE equal to �30 dB for a sample size as short as100 symbols, whereas the method of Moulinesrequires about 500 symbols to achieve the sameaccuracy. Also in this case, observe that thetheoretical performance analysis for the proposedmethod agrees very well with Monte Carlo results.

Experiment 3—Normalized channel MSE versus

Le: In this experiment, we evaluate the perfor-mances of the methods under comparison as afunction of the length of the temporal window Le

ranging from 3 to 10. In this case, rememberingthat Lc ¼ 3; the dimension Le � Lc þ 1 of thenoise subspace RðENÞ varies from 1 (correspond-ing to Le ¼ 3Þ to 8 (corresponding to Le ¼ 10).The sample size is equal to Ns ¼ 200 symbols andSNR ¼ 20 dB: Results of Fig. 3, besides corrobor-ating again the accuracy of the theoretical analysiscarried out in Section 6, show moreover that,compared with the algorithm of Moulines, theperformance of the proposed method is lesssensitive to the choice of the length Le of thetemporal window. Moreover, it is worthwhile tonote that the proposed method is able to reliablyestimate the channel parameters even when Le ¼

3; i.e., only one noise eigenvector is considered.Finally, it is apparent that, for the proposedmethod, employing values of Le45 does not leadto a significant improvement in channel estimationaccuracy.

Experiment 4—Normalized channel MSE versus

g: In this experiment, we compare the robustnessof the considered methods to the location ofchannel zeros, by plotting the theoretical MSE as afunction of the angular separation g ranging from0:005p to 0:3p; and for different values of SNR(recall that as g ! 0 the channel is not identifiablefor both methods under comparison). Unlike allthe other experiments, we chose not to reportMonte Carlo results, due to the extreme variabilityof the curves when the channel approaches the

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2 3 4 5 6 7-35

-30

-25

-20

-15

-10

-5

Lc

Nor

mal

ized

MS

E [d

B]

Moulines (experimental)Moulines (theoretical)Proposed (experimental)Proposed (theoretical)

Fig. 5. Normalized channel MSE versus Lc (Le ¼ 10; Ns ¼ 200

and SNR ¼ 20dB).

D. Darsena et al. / Signal Processing 84 (2004) 2021–2039 2035

non-identifiability condition. The length of thetemporal window is set to Le ¼ 5 and the samplesize is equal to Ns ¼ 200 symbols. It can be notedfrom Fig. 4 that the proposed method is able toassure better performance than the algorithm ofMoulines both in poor (i.e., small g) as well asgood (i.e., high g) channel conditions. In particu-lar, as g approaches zero, the performance of themethod of Moulines degrades faster than that ofthe proposed channel estimation technique.

Experiment 5—Normalized channel MSE versus

Lc: In the last experiment, we evaluate theperformances of the considered methods as afunction of the channel length Lc ranging from 2to 7. In this case, the Z-transforms of thepolyphase components ec ð‘Þ

ðnÞ; ‘ ¼ 0; 1; are givenby

eC ð‘ÞðzÞ ¼

YLc�1

i¼1

ð1 � ri ej yi;‘z�1Þ; for ‘ ¼ 0; 1;

(35)

where y1;0 ¼ 0:5 pþ g and r1 ¼ 0:5; y2;0 ¼ y1;0 þ pand r2 ¼ 1:2; y3;0 ¼ �y1;0 and r3 ¼ 0:8; y4;0 ¼

�y2;0 and r4 ¼ 1:5; y5;0 ¼ �0:5pþ g and r5 ¼ 1:5;y6;0 ¼ �y5;0 and r6 ¼ 1:8; whereas yi;1 ¼ yi;0 þ g;for i 2 f1; 2; . . . ;Lc � 1g: The length of the tempor-al window is set to Le ¼ 10; the sample size isequal to Ns ¼ 200 symbols and SNR ¼ 20 dB:

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.3-50

-40

-30

-20

-10

0

10

20

30

40

γ/π

Nor

mal

ized

MS

E [d

B]

Moulines (theoretical)Proposed (theoretical)

SNR=10 dB

SNR=20 dB

SNR=30 dB

Fig. 4. Normalized channel MSE versus g=p for different

values of SNR (Le ¼ 5 and Ns ¼ 200).

Results of Fig. 5 show that the normalized MSEfor both methods increases as a function of thechannel length Lc: However, the proposed methodassures a significant performance gain with respectto the method of Moulines, for all the consideredvalues of Lc:

8. Conclusion

We tackled in this paper the problem of blindlyidentifying a SISO-FIR channel via a subspace-based approach. We provided analytically thatblind channel identification is possible if thetransmitted symbol sequence belongs to a parti-cular subclass of improper random processes, i.e.,it is strongly improper, which is a propertyexhibited by many digital modulation schemes ofpractical interest. By exploiting the stronglyimproper nature of the transmitted symbol stream,we devised a general framework for subspace-based channel identification, which allowed us toaddress identifiability issues and carry out thetheoretical performance analysis in a unifyingmanner. Results of computer simulations corro-borated the theoretical performance analysis andshowed that the proposed method performssignificantly better than the conventional SIMOsubspace-based algorithm originally proposed in

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[15]. Finally, it should be noted that our generalframework allows one to extend the treatment tomodulation techniques that are more sophisticatedthan those discussed in the paper, e.g., modula-tions schemes wherein conjugate cyclostationarityis deliberately introduced in transmission. Thisissue is the subject of our current research.

A. Proof of Lemma 3

Let us define the nonsingular matrix

H¼n

diag½1; . . . ; 1zfflfflfflffl}|fflfflfflffl{Leþ1

; e�j 2pb; e�j 2p2b; . . . ; e�j 2pðLe�1Þb� 2

Cð2LeÞ�ð2LeÞ: Observe8 preliminarily that rankðWÞ ¼

rankðHWÞ ¼ rankð½CT;DT�TÞ; where

D¼nTLe

ðdTÞ 2 CLe�K denotes the Toeplitz matrix

associated with c¼n½c�ð0Þ; c�ð1Þ ej 2pb; . . . ; c�ðLc �

1Þ ej 2pðLc�1Þb�T 2 CLc : Thus, under assumption A4,applying standard results in blind channel identi-fication (see, e.g., [20]) yields that W is full-columnrank, i.e., rankðWÞ ¼ K ¼ Le þ Lc � 1; if and only

if the Z-transforms of c and c are coprime, that is,they do not share common zeros. Let

CðzÞ ¼nZ½c� ¼

XLc�1

q¼0

cðqÞ z�q

¼ cð0ÞYLc�1

q¼1ð1 � zq z�1Þ ðA:1Þ

be the Z-transform of the vector c; wherez1; z2; . . . ; zLc�1 denote the zeros of CðzÞ; it followsthat

CðzÞ ¼nZ½c� ¼

XLc�1

q¼0

c�ðqÞ ej 2pqb z�q ¼ C�ðz� ej 2pbÞ

¼ c�ð0ÞYLc�1

q¼1

½1 � ðz�q ej 2pbÞ z�1�: ðA:2Þ

Thus, from (A.1) and (A.2), it results that thepolynomials CðzÞ and CðzÞ do not share commonzeros if and only if z‘az�q ej 2pb; for‘; q 2 f1; 2; . . . ;Lc � 1g: &

8To keep notation simple, throughout the Appendices, we

replace WðcÞ with W:

B. Proof of Theorem 1

To prove the theorem, we state beforehand thefollowing lemma.

Lemma 5. Under assumptions A4 and A5, an

arbitrary vector c0 ¼n½c0ð0Þ; c0ð1Þ; . . . ; c0ðLc � 1Þ�T 2

CLc is a solution of (15) if and only if RðW0Þ �

RðWÞ; where W0¼n½ðC 0

ÞT ; J� ðC 0

ÞH�T 2 Cð2LeÞ�K ;

with C 0¼nTLe

½ðc0ÞT� 2 CLe�K denoting the Toeplitz

matrix associated with vector c0 and K¼LeþLc�1:

Proof. Observe that, owing to (15) and accountingfor the properties of PN; the columns of W belongto R?ðENÞ; that is, RðWÞ � R?ðENÞ: Due to thelinear independence of the noise eigenvectors, thesubspace R?ðENÞ has dimension equal to K; onthe other hand, by assumption, the dimension ofRðWÞ is K. Thus, it turns out that RðWÞ ¼

R?ðENÞ: On the basis of this equivalence, Lemma5 is readily proven.

If: Let RðW0Þ � RðWÞ; the columns of W0 belong

to the subspace RðWÞ and, thus, to R?ðENÞ which,accounting for the relation NðPNÞ ¼ R?ðENÞ;implies that PN W0

¼ OR�K :Only If: Let PN W0

¼ OR�K ; then the columnsof W0 belong to the subspace NðPNÞ ¼ R?ðENÞ;that is, RðW0

Þ � R?ðENÞ ¼ RðWÞ: &

Given the nonsingular matrix

H¼n

diag½1; . . . ; 1zfflfflfflffl}|fflfflfflffl{Leþ1

; e�j 2pb; e�j 2p2b; . . . ; e�j 2pðLe�1Þb� 2

Cð2LeÞ�ð2LeÞ; consider the matrices Y¼n HW ¼

½CT;DT�T 2 Cð2LeÞ�K and Y0¼n HW0

¼

½ðC 0ÞT; ðD0Þ

T�T 2 Cð2LeÞ�K ; where D¼

nTLe

ðdTÞ 2

CLe�K and D0 ¼nTLe

½ðd 0ÞT� 2 CLe�K represent the

Toeplitz matrices associated with the two vectors

d ¼n½c�ð0Þ; c�ð1Þ ej 2pb; . . . ; c�ðLc � 1Þ ej 2pðLc�1Þb�T 2

CLc and d 0¼n½c0ð0Þ�; c0ð1Þ� ej 2pb; . . . ; c0ðLc �

1Þ� ej 2pðLc�1Þb�T 2 CLc ; respectively. It can be shownthat the following equivalence holds:

RðW0Þ � RðWÞ () RðY0

Þ � RðYÞ: (A.3)

Let TLeðHÞ 2 Cð2LeÞ�K and TLe

ðH 0Þ 2 Cð2LeÞ�K be

the block-Sylvester matrices associated with the

matrices H ¼n½hð0Þ; hð1Þ; . . . ; hðLc � 1Þ� 2 C2�Lc and

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9For the sake of simplicity, we assume that s2s ¼ 1:

D. Darsena et al. / Signal Processing 84 (2004) 2021–2039 2037

H 0 ¼n½h0

ð0Þ; h0ð1Þ; . . . ; h0

ðLc � 1Þ� 2 C2�Lc ; respec-

tively, where hðkÞ ¼n½cðkÞ; c�ðkÞ ej 2pbk�T 2 C2

and h0ðkÞ ¼

n½c0ðkÞ; ðc0ðkÞÞ� ej 2pbk�T 2 C2; for k ¼

0; 1; . . . ;Lc � 1: Since TLeðHÞ and TLe

ðH 0Þ differ

from Y and Y0; respectively, by interchange ofrows, it follows that RðYÞ ¼ R½TLe

ðHÞ� and

RðY0Þ ¼ R½TLe

ðH 0Þ�: Thus, by virtue of Lemma 5and (A.3), we can state that c0 is a solution of (15)if and only if

R½TLeðH 0Þ� � R½TLe

ðHÞ�: (A.4)

By exploiting the shift property of the Toeplitzmatrices and following the guidelines delineated in[15], it can be proven that, under assumptions A4

and A5, relation (A.4) holds if and only if H 0 ¼

xH ; with x 2 C; which implies that c0ðkÞ ¼ x cðkÞ

and c0ðkÞ� ¼ x cðkÞ�; for k ¼ 0; 1; . . . ;Lc � 1: thelast two equalities show that if c0 is a solution of(15), then c0 must differ from c up to a multi-plicative scalar x; which must be a real number,i.e., x� ¼ x: &

C. Derivation of the parameterization matrix Q

By performing the following partition PN ¼

½P0N;P00

N�; with P0N;P00

N 2 Cð2LeÞ�Le ; and exploit-ing the structure of W; the matrix Eq. (15) becomesP0

N C þ P00N C� J� ¼ Oð2LeÞ�K : This equation is

equivalent to vecðPN C þ P00N C� J�Þ ¼ 02KLe

;which, by using the properties of Kroneckerproduct, can be written as ðIK � P0

NÞ vecðCÞ þ

ðJ� � P00NÞ vecðC�

Þ ¼ 02KLe: Observing that

vecðCÞ ¼ vecðCRÞ þ j vecðC IÞ and vecðC�Þ ¼

½vecðCÞ��; we finally obtain

vecðPN WÞ ¼ K1 vecðCRÞ þ K2 vecðC IÞ ¼ 02KLe;

(A.5)

where K1 ¼nðIK � P0

NÞ þ ðJ� � P00NÞ 2 Cð2KLeÞ�KLe

and K2 ¼n

j ðIK � P0NÞ � ðJ� � P00

N� �

2

Cð2KLeÞ�KLe : At this point, it is convenient to writethe Toeplitz matrices CR and C I in the followingform

CR ¼XLc�1

‘¼0

cRð‘ÞF ð‘Þ and C I ¼XLc�1

‘¼0

cIð‘ÞF ð‘Þ;

(A.6)

where F ð‘Þ ¼nTLe

ðaT‘ Þ 2 RLe�K is the Toeplitz matrix

associated with a‘ ¼n½0; . . . ; 0zfflfflfflffl}|fflfflfflffl{‘

; 1; 0; . . . ; 0�T 2 RK :Accounting for (A.6) and using again the proper-ties of Kronecker product, it can be shown thatvecðCRÞ ¼ bfL3 cR and vecðC IÞ ¼ K3 cI; with

K3 ¼nðIK � FÞG ; and F ¼

n½F ð0Þ;F ð1Þ; . . . ;F ðLc�1Þ� 2

RLe�ðKLcÞ; where the matrix G 2 RðK2LcÞ�Lc (con-taining only zeros and ones) can be easily derivedand, thus, its expression is omitted for the sakeof brevity. On the basis of the above observa-tions, equation (A.5) can be concisely written

as vecðPN WÞ ¼ Q ca ¼ 02KLe; where Q¼

n½K1;K2�

ðI2 � K3Þ 2 C2KLe�ð2LcÞand ca ¼n½cT

R; cTI �

T 2 R2Lc :

D. Discussion on the identifiability condition derived

in [8]

In this appendix, we show that the identifiabilityconditions derived in [8, Theorem V.2, p. 2201] areincomplete. Let us consider the case where theinput sequence sðnÞ is a stationary strongly real

improper random process (e.g., ASK, BPSK orDBPSK), that is, f ðnÞ ¼ 1; for any n 2 Z; in thiscase, according to assumption A1, it results thatRssðmÞ ¼ Rss� ðmÞ ¼ s2

s dðmÞ: Using our notations,the proposition considered in [8, TheoremV.2] canbe summarized as follows: prove that, in theabsence of noise, the joint use of the autocorrela-tion RrrðmÞ ¼

nE½rðnÞ r�ðn � mÞ� ¼ s2

s

PLc�1q¼0 cðqÞ

c�ðq � mÞ and the conjugate autocorrelationRrr� ðmÞ ¼

nE½rðnÞ rðn � mÞ� ¼ s2

s

PLc�1q¼0 cðqÞ cðq � mÞ

functions of the received signal rðnÞ; given by (2),allows one to uniquely (up to a scalar factor)identify the channel impulse response fcðnÞgLc�1

n¼0 :To study this problem, let us consider the Z-transforms of the functions RrrðmÞ and Rrr� ðmÞ;which are given by9

SrrðzÞ ¼nZ½RrrðmÞ� ¼

Xþ1

m¼�1

RrrðmÞ z�m

¼ CðzÞC�ð1=z�Þ; ðA:7Þ

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Srr� ðzÞ ¼nZ½Rrr� ðmÞ� ¼

Xþ1

m¼�1

Rrr� ðmÞ z�m

¼ CðzÞCð1=zÞ; ðA:8Þ

where CðzÞ ¼nZ½c� ¼

PLc�1q¼0 cðqÞ z�q denotes the

Z-transform of the channel vector c: When thepolynomials SrrðzÞ and Srr� ðzÞ are known, relations(A.7) and (A.8) can be used to identify theunknown channel Z-transform CðzÞ: To thisaim, it can be seen that the following twostatements are equivalent: (1) an arbitrary poly-nomial C0ðzÞ satisfies both (A.7) and (A.8); (2)C0ðzÞ ¼ AðzÞCðzÞ; where AðzÞ is an arbitraryrational function verifying the two relations (i)AðzÞA�ð1=z�Þ ¼ 1 and (ii) AðzÞAð1=zÞ ¼ 1: Inaddition, since the channel is a FIR system, thefunction C0ðzÞ cannot have poles except for z ¼ 0:this fact imposes that (iii) the poles of AðzÞ

coincide with the zeros of CðzÞ: It is well-knownthat a polynomial AðzÞ satisfying condition (i)admits the following factored form:

AðzÞ ¼ ZYQq¼1

1 � a�q z�1

aq � z�1; (A.9)

where Z is an unit-modulus complex number. Byvirtue of (A.9), it follows that

AðzÞAð1=zÞ ¼ Z2YQq¼1

ð1 � a�q z�1Þ ða�

q � z�1Þ

ðaq � z�1Þ ð1 � aq z�1Þ:

(A.10)

The Authors in [8] claim that, if CðzÞ has no real

zeros (which, owing to condition (iii), implies thatAðzÞ has no real poles or, equivalently, zeros), theonly function AðzÞ satisfying both conditions (i)and (ii) is the zero-order polynomial AðzÞ ¼ � 1: Inthis case, the unknown channel impulse responsecan be uniquely identified, that is, C0ðzÞ is asolution of both (A.7) and (A.8) if and only ifC0ðzÞ ¼ �CðzÞ: As it is apparent from (A.10), thisclaim is incomplete: indeed, if AðzÞ has no realpoles while possessing a conjugate pair of zeros,say, a�

q1¼ aq2

; with q1; q2 2 f1; 2; . . . ;Qg; thereexists a function AðzÞ verifying both conditions(i) and (ii), given by

AðzÞ ¼ �ð1 � a�

q1z�1Þ ð1 � a�

q2z�1Þ

ðaq1� z�1Þ ðaq2

� z�1Þ; (A.11)

which clearly does not allow unique identification ofthe channel function CðzÞ: Based on this exampleand taking into account condition (iii), the condi-tion derived by the Authors in [8] must be rephrasedas follows: the channel vector c can be uniquelyidentified from (A.7) and (A.8) if its Z-transform

has not real zeros and conjugate pairs of zeros.

References

[1] S. Benedetto, E. Biglieri, Principles of Digital Commu-

nications with Wireless Applications, Kluwer Academic/

Plenum Publishers, New York, 1999.

[2] J.A.C. Bingham, The Theory and Practice of Modem

Design, Wiley, New York, 1988.

[3] A. Chevreuil, Ph. Loubaton, Blind second-order identifica-

tion of FIR channels: forced cyclostationarity and

structured subspace method, IEEE Signal Process.

Lett. 4 (July 1997) 204–206.

[4] Z. Ding, Characteristics of band-limited channels uniden-

tifiable from second-order cyclostationary statistics, IEEE

Signal Process. Lett. 3 (May 1996) 150–152.

[5] Z. Ding, G. Li, Single-channel blind equalization for GSM

cellular systems, IEEE J. Select. Areas Commun. 16

(October 1998) 1493–1505.

[6] W.A. Gardner, Introduction to Random Processes,

McGraw-Hill, New York, 1990 (Chapter 12).

[7] W.H. Gerstacker, R. Schober, A. Lampe, Receivers with

widely linear processing for frequency-selective channels,

IEEE Trans. Commun. 51 (September 2003) 1512–1523.

[8] O. Grellier, P. Comon, B. Mourrain, P. Trebuchet,

Analytical blind channel identification, IEEE Trans. Signal

Process. 50 (September 2002) 2196–2207.

[9] R.A. Horn, C.R. Johnson, Matrix analysis, Cambridge

University Press, Cambridge, 1990.

[10] International Telecommunication Union (ITU), Radio-

communication Study Groups, Study of the required

geographic coordination distance between a HAPS

TDMA system and adjacent system IMT-2000 terrestrial

mobile stations operating in the same frequency band,

Document 8-1/368-E, May 1988.

[11] M. Kristensson, B. Ottersten, A statistical approach to

subspace based blind identification, IEEE Trans. Signal

Process. 46 (June 1998) 1612–1623.

[12] M. Kristensson, B. Ottersten, D. Slock, Blind subspace

identification of a BPSK communication channel, in:

Proc. 30th Asilomar Conference on Signals, Systems

and Computers, Monterey, USA, November 1996,

pp. 828–832.

[13] X. Li, H.H. Fan, Blind channel identification: subspace

tracking method without rank estimation, IEEE Trans.

Signal Process. 49 (October 2001) 2372–2382.

[14] F. Li, H. Liu, R.J. Vaccaro, Performance analysis for

DOA estimation algorithms: unification, simplification,

ARTICLE IN PRESS

D. Darsena et al. / Signal Processing 84 (2004) 2021–2039 2039

and observations, IEEE Trans. Aerospace Electron.

Systems 29 (October 1993) 1170–1184.

[15] E. Moulines, P. Duhamel, J.-F. Cardoso, S. Mayrargue,

Subspace methods for the blind identification of multi-

channel FIR filters, IEEE Trans. Signal Process. 43

(February 1995) 516–525.

[16] F.D. Neeser, J.L. Massey, Proper complex random

processes with applications to information theory, IEEE

Trans. Inform. Theory (July 1993) 1293–1302.

[17] J.G. Proakis, Digital Communications, McGraw-Hill,

New York, 2001.

[18] W. Qiu, Y. Hua, Performance analysis of the subspace

method for blind channel identification, Signal Process. 50

(1996) 71–81.

[19] P.J. Schreier, L.L. Scharf, Second-order analysis of

improper complex random vectors and processes,

IEEE Trans. Signal Process. 51 (March 2003)

714–725.

[20] E. Serpedin, G.B. Giannakis, A simple proof of a known

blind channel identifiability result, IEEE Trans. Signal

Process. 47 (February 1999) 591–593.

[21] E. Serpedin, G.B. Giannakis, Blind channel identification

and equalization with modulation-induced cyclostationar-

ity, IEEE Trans. Signal Process. 46 (July 1998) 1930–1944.

[22] O. Shalvi, E. Weinstein, New criteria for blind deconvolu-

tion of nonminimum phase systems (channels), IEEE

Trans. Inform. Theory 36 (March 1990) 312–321.

[23] G.W. Stewart, Error and perturbation bounds for sub-

spaces associated with certain eigenvalue problems, SIAM

Rev. 15 (October 1973) 727–764.

[24] P. Stoica, T. Soderstrom, Statistical analysis of MUSIC

and subspace rotation estimates of sinusoidal frequencies,

IEEE Trans. Signal Process. 39 (August 1991) 1836–1847.

[25] L. Tong, S. Perreau, Multichannel blind identification:

from subspace to maximum likelihood methods, Proc.

IEEE 86 (October 1998) 1951–1968.