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A Simple Proof of a Known Blind Define also the w@u CIA @v C u CIAblock Sylvester matrix @HA @IA @wIA „ „ Channel Identifiability Result r uCI Xa ‘r uCI r uCI


r uCI “ , with ‘
“ standing for @mA @mA transposition. Denote by ‚@r uCIA and x @r uCIA the range and Erchin Serpedin and Georgios B. Giannakis @mA the null space of the matrix r uCI, respectively, and by deg rm@zA the degree of polynomial rm@zA. We first establish the following proposition. @mA Abstract—An alternative simple proof for a well-known result encoun- Proposition 1: If deg rm@zAav, then a basis for x @r uCIA tered in blind identification and equalization of communication channels is given by the v generalized Vandermonde vectors corresponding is derived. It is shown that the coprimeness condition of a number of FIR to the zeros of rm@zA. If deg rm@zAalm `v, then a basis channels is equivalent to the full column rank condition of a certain block @mA Sylvester matrix, whose entries are the channel coefficients. As a corollary, for x @r uCIA is obtained by adding to the set of lm generalized it is also shown that in general, the null space of a block Sylvester matrix Vandermonde vectors corresponding to the zeros of rm@zA v lm is spanned by the set of (generalized) Vandermonde vectors associated vectors given by the last v lm columns of the @v C u CIA @v C with the common zeros present in all the channels. u CIAidentity matrix s. Index Terms—Equalization, SIMO channel, Sylvester matrix, Vander- Proof: First, we will obtain a characterization of the null space monde vector. @mA of r uCI, H m w I, when deg rm@zAav. We will @mA show that in this case, the null space of matrix r uCI is spanned by the generalized Vandermonde vectors corresponding to the zeros of I. INTRODUCTION polynomial rm@zA. Indeed, if r is a zero of multiplicity p of rm@zA, pI pI Recent results have shown that blind identification and equalization then rm@rAadrm@rAadr a


a d rm@rAadr aH, p p of single-input multiple-output (SIMO) FIR communication channels and d rm@rAadr TaH. The generalized Vandermonde vectors is possible from the second-order statistics of the output, provided corresponding to the zero r are given by the columns of the that a specific block Sylvester matrix constructed from the channel @v C u CIA p matrix, as in coefficients is full column rank [1], [4], [11]. It has also been recognized [1], [4], [11] that this full column rank condition relies † @mA@rAXa on the coprimeness condition of the channel transfer functions. In IH


H practice, the SIMO channel framework is obtained by introducing r I


H additional temporal/spatial diversity (fractionally sampling/multiple rP Pr


H antennas) at the receiver of a single-input single-output (SISO) Q P r Qr


H X channel. It is this form of diversity that allows blind identification and . . . equalization of SISO channels from the output second-order statistics. . .


. In this correspondence, an elementary and more direct proof of this @v C uA3rvCupCI rvCu @v C uArvCuI


result is presented, which avoids the notions of dual dynamic indices @v C u p C IA3@p IA3 and minimal polynomial basis (as is the case with [1] and [11]). A (1b) characterization of the null space of a block Sylvester matrix in terms of a set of (generalized) Vandermonde vectors corresponding to the @mA @mA The ith column of matrix † @rA is given by the vector vi @rAXa common zeros of the FIR channels is also presented. ‘H


I ir


@@v C uA3rvCuiCIa@v C u i C IA3@i IA3A“„ and is simply obtained by normalizing with @i IA3 and differentiating @mA II. MAIN RESULTS @i IA times the first column of † @rA. fdir @rAadri aHgpI Assume that w FIR channels of order less than or equal to It is easy to check that since m iaH , we have fr @mA v@mA@rAaHgpI r @mA † @mA@rAaH v are available, with at least one channel of order v. Consider uCI i iaH , and hence, uCI . Thus, @mA @mA also that the channel transfer functions are given, respectively, by ‚@† @rAA x@r uCIA. Due to its upper-triangular structure, v l @mA @mA rm@zAa laH hm@lAz , m aHY


YwI. To the mth channel, matrix r uCI is full row rank. Thus, dim‘‚@r uCIA“ a u CI, @u CIA @v C u CIA @mA @mA let us associate the Toeplitz matrix, as in and since dim‘‚@r uCIA“ C dim‘x @r uCIA“ a v C u CI,it @mA @mA follows that dim‘x @r uCIA“ a v. Consider that the zeros of r uCI Xa rm@zA are rIY


Yrs with corresponding multiplicities pIY


Yps hm@HA hm@IA


hm@vAH


H s ( iaI pi a v). Note also that the @v C u CIA v generalized H h @HA


h @v IA h @vA


H @mA @mA @mA @mA m m m Vandermonde matrix † Xa ‘† @rIA † @rPA


† @rsA“ ...... r Y


Yr r @mA † @mA a H . .. ..


.. .. . associated with all the zeros I s satisfies uCI . @mA @mA H


H hm@HA hm@IA


hm@vA We wish to establish that x @r uCIAa‚@† A, for which it is @mA (1a) sufficient to show that dim ‚@† Aav. The latter follows from Proposition 2, which we prove in the Appendix.1 Proposition 2: Matrix † @mA is full column rank, and Manuscript received October 13, 1997; revised August 6, 1998. The @mA p p associate editor coordinating the review of this paper and approving it for det † @I X vY XAa @ri rj A TaHX (2) publication was Dr. Sergios Theodoridis. s!i!j!I The authors are with the Department of Electrical Engineering, Uni- versity of Virginia, Charlottesville, VA 22903-2442 USA (e-mail: geor- [email protected]). 1 In writing the submatrix † @mA@IXvY XA, we adopt Matlab’s notational Publisher Item Identifier S 1053-587X(99)00740-0. convention.

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Proposition 2 shows that the columns of † @mA are linearly in- It turns out that even if the restriction on the channel zeros becomes @mA dependent; hence, they constitute a basis for x @r uCI), which more severe, then the matrix r uCI (for u`v I) is still rank proves the first claim of Proposition 1. For the second claim, when deficient. In the next example, any two channels of the w@aQA @mA r deg rm@zAalm `v, it is easy to check that a basis for x @r uCIA channels are coprime, and still, the matrix uCI is column rank I P Q R is obtained by adding to the set of lm generalized Vandermonde deficient. Indeed, if rH@zAaICPz CQz CRz CSz , r @zAaICHXSzI CHXQzP CHXPzQ CHXIzR r @zAa vectors corresponding to the zeros of rm@zA the set of vlm vectors I , P r @zAQr @zA v aR u a dva@w IAeIaRaPIaI given by the last v lm columns of the @v C u CIA @v C u CIA H I , , and , @mA then the resulting matrix r uCI is column rank deficient, and any two identity matrix s, i.e., a basis for x @r uCIA is given by the columns @mA @mA @mA of the channels rH@zA, rI@zA,orrP@zA are coprime. Further, for of the @vCu CIAv matrix † Xa‘† @rIA


† @rsA s@X u a v PaP, matrix r uCI is still column rank deficient. Thus, Ylm Cu CP X vCu CIA“. Using Proposition 2, it follows again that @mA the condition u ! v I is necessary for r uCI to be full column the columns of matrix † are linearly independent. In addition, it @mA rank. is straightforward to check that all the columns of † are in the @mA @mA @mA null space of r uCI, i.e., r uCI† a H. Thus, the columns of @mA @mA III. CONCLUSIONS † represent a basis for x @r uCIA, and hence, Proposition 1 has been established. An alternative elementary proof of the equivalence between the With these preliminaries, we are now ready to state and prove our coprimeness condition among the FIR channels and the full-column main result [1], [11]. rank condition of a Sylvester matrix has been derived. It has also r Theorem 1: If the polynomials rm@zA, m aHY


Yw I been shown that even if the channels are coprime, matrix uCI u v I are coprime, then the matrix r uCI is full column rank for any loses rank, unless the parameter is greater than or equal to . u u ! v I. Since represents the order of the linear zero-forcing equalizer [7], Proof: Suppose without loss of generality that channel rH@zA is [8], this result verifies the well-known fact that the equalizer length @HA @IA u CIshould be lower bounded by the channel order v. of maximum order v. We wish to show that x @r uCIA’x@r uCIA is spanned by the generalized Vandermonde vectors corresponding to the common zeros of rH@zA and rI@zA, considering their mini- APPENDIX @HA @HA @IA ROOF OF ROPOSITION mum multiplicity. Since x @r uCIAa‚@† A and x @r uCIAa P P 2 @IA @HA @IA @HA ‚@† A, it follows that x @r uCIA ’x@r uCIAa‚@† A ’ Since the computation of the generalized Vandermonde @mA ‚@† @IAA. Using Proposition 2, it follows that ‚@† @HAA ’‚@† @IAA determinant ‡ @rIY


YrsAXadet† @I X vY XA can be is spanned by the generalized Vandermonde vectors corresponding performed following the same lines used in the computation to the common zeros of rH@zA and rI@zA. Similarly, we can of a standard Vandermonde determinant from [6, p. 15], @HA @IA @PA show that @x @r A ’x@r AA ’x@r A is spanned by here, the proof is only sketched. It is easy to check that uCI uCI uCI @di‡ @r Y


YrAadri A j aH i aHY


Ypp I the generalized Vandermonde vectors corresponding to the common I s I r ar for I j , r @zA r @zA r @zA and j aPY


Ys. Thus, ‡ @rIY


YrsA can be factorized as zeros of H , I , and P . Proceeding in this way, we s p p wI @mA ‡ @rIY


YrsAa @rI rj A ‡ @rPY


YrsA. Similarly, conclude that x @r A is spanned by the generalized Van- jaP maH uCI @di‡ @r Y


YrAadri A j aHY dermonde vectors corresponding to the common zeros of polynomials it is easy to check that I s P r ar for wI @mA i aHY


YpPpj I, and j aQY


Ys. Thus, ‡ @rPY


YrsA can be rH@zAY


YrwI@zA. Since x @r uCIAa maH x @r uCIA and s p p factorized as ‡ @rPY


YrsAa jaQ@rP rj A ‡ @rQY


YrsA, the polynomials rH@zAY


YrwI@zA are coprime, it follows that and (2) follows by induction. We remark also that additional results matrix r uCI is necessarily full column rank. concerning the inversion of generalized Vandermonde matrices have As direct consequences of Theorem 1, we also have the following been presented in [2], [3], and [5], among others. corollary. Corollary 1: If the channels rH@zAY


YrwI@zA have p com- REFERENCES mon zeros, then dim x @r uCIAap, and x @r uCIA is spanned by the p generalized Vandermonde vectors corresponding to the p [1] K. Abed-Meraim, J.-F. Cardoso, A. Y. Gorokhov, P. Loubaton, and E. common zeros. Moulines, “On subspace methods for blind identification of single-input multiple-output FIR systems,” IEEE Trans. Signal Processing, vol. 45, Corollary 2: The converse of Theorem 1 also holds true, i.e., if pp. 42–55, Jan. 1997. r uCI is full column rank and u is chosen to satisfy u ! v I, [2] I. C. Goknar, “Obtaining the inverse of the generalized Vandermonde then the channels rm@zA, m aHY


Yw I are coprime. matrix,” IEEE Trans. Automat. Contr., vol. AC-18, pp. 530–532, Oct. The last issue we address is the following: What happens to the 1973. r u`vI u a dva@w IAeI [3] I. Kaufman, “The inversion of the Vandermonde matrix and the trans- rank of uCI if ? In particular, if , formation to the Jordan canonical form,” IEEE Trans. Automat. Contr., then r uCI is a square matrix. We will show by an example that vol. AC-14, pp. 774–777, Dec. 1969. r uCI loses rank even if the channels are coprime. Consider three [4] Y. Li and Z. Ding, “Blind channel identification based on second order cyclostationarity statistics,” in Proc. ICASSP, Minneapolis, MN, 1993, channels rH@zAYrI@zAYrP@zA such that u a dva@w IAeIa dvaPeI r @zA r @zA vol. 4, pp. 81–84. . Suppose also that H and I share at least [5] L. Lupas, “On the computation of the generalized Vandermonde matrix dvaPe CIcommon zeros, and rP@zA does not share any common inverse,” IEEE Trans. Automat. Contr., vol. AC-20, pp. 559–561, Aug. @HA @IA zero with rH@zA and rI@zA. Thus, dim‘x @r A ’x@r A“ ! 1975. dvaPe CI. Since dim‘x @r @HAA ’x@r @IAA“ C dim‘x @r @PAA“ ! [6] G. E. Shilov, Linear Algebra. New York: Dover, 1977. [7] D. T. M. Slock, “Blind fractionally-spaced equalization, perfect- dvaPeCICv, it follows that dim‘x @r @HAA’x @r @IAA’x @r @PAA“ a @HA @IA @PA @HA reconstruction filter banks and multichannel linear prediction,” in Proc. dim‘x @r A ’x@r A“ C dim‘x @r A“ dim‘@x @r A ’ ICASSP, Adelaide, Australia, 1994, vol. 4, pp. 585–588. x @r @IAAA ‘x@r @PAA“ !dvaPe CICv v u I ! I since [8] D. T. M. Slock and C. Papadias, “Further results on blind identification x @r @HAA’x@r @IAA and x @r @PAA are subspaces of ‚vCuCI. Thus, and equalization of multiple FIR channels,” in Proc. ICASSP, Detroit, r MI, 1995, vol. 5, pp. 1964–1967. matrix uCI is not full column rank, even though the channels are [9] D. T. M. Slock, “Blind joint equalization of multiple synchronous coprime. mobile users using oversampling and/or multiple antennas,” in Proc. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 2, FEBRUARY 1999 593

28th Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Oct. 1994, pp. 1154–1158. [10] L. Tong, G. Xu, and T. Kailath, “Blind identification and equalization based on second-order statistics: A time domain approach,” IEEE Trans. Inform. Theory, vol. 40, pp. 340–349, Mar. 1994. [11] L. Tong, G. Xu, B. Hassibi, and T. Kailath, “Blind channel identification based on second-order statistics: A frequency-domain approach,” IEEE Trans. Inform. Theory, vol. 41, pp. 329–334, Jan. 1995. [12] G. Xu, H. Liu, L. Tong, and T. Kailath, “A least-squares approach to blind channel identification,” IEEE Trans. Signal Processing, vol. 43, pp. 2982–2993, Dec. 1995. Fig. 1. Image enhancement using the 2DCBAE algorithm.

of order greater than two are essentially estimates of the cumulants of the signal alone. In this correspondence, a 2-D cumulant-based adaptive enhancer al- Two-Dimensional Cumulant-Based Adaptive Enhancer gorithm (2DCBAE) is suggested. Section II shows that the 2DCBAE algorithm coefficients converge to a special 2-D slice of the fourth- Hosny M. Ibrahim and Reda R. Gharieb order cumulant function of the signal or the feature of interest. In Section III, simulation results show the advantage of the 2DCBAE Abstract—This communication describes a novel two-dimensional (2- algorithm compared with the 2DACE and the 2DLMS algorithms. D) adaptive filtering algorithm to enhance 2-D signals of small spatial Finally, Section IV presents the conclusions. extent embedded in white or colored Gaussian noise. The coefficients of the adaptive filter converge to a special 2-D slice of the fourth- order cumulant function of the input signal. The proposed algorithm II. THE 2-D CUMULANT-BASED ADAPTIVE ENHANCER ALGORITHM is called the 2-D cumulant-based adaptive enhancer (2DCBAE). Results A conceptual implementation for the proposed 2DCBAE is shown are compared with images processed using the 2-D adaptive correlation enhancer (2DACE) and the 2-D least mean square (2DLMS) algorithms. in Fig. 1. It consists of a 2-D adaptive FIR filter whose matrix of coefficients is adapted as described below. The input signal to the adaptive filter x@mY nA with mean removed can be modeled as I. INTRODUCTION Increasing the signal-to-noise ratio (SNR) of two-dimensional x@mY nAad@mY nACv@mY nAH mY n w I (1) (2-D) signals corrupted by additive noise is an essential problem related to several areas of research in image processing and com- where d@mY nA is the signal of interest, and v@mY nA is an additive munications. If the statistics of the 2-D signal and background noise Gaussian noise. The output of the adaptive filter, enhanced image is are stationary or known, filtering the 2-D signal using fixed filters given by is satisfactory. However, in general, the statistics of the 2-D signal Pv Pv and background noise are nonstationary or unknown, making the use y@mY nAa wk@iY jAx@m iY n jA (2) of adaptive filtering imperative. Various update algorithms have been iaH jaH used to adjust the impulse response of the adaptive filter according to the learned statistical parameters. See, for example, [1]–[3]. where wk@iY jA, H iY j Pv are the adaptive FIR filter coeffi- Images corrupted by white Gaussian noise can be enhanced through cients, and k a mw Cn denotes the iteration index (i.e., the image is the 2-D adaptive line enhancer (2DALE), whose coefficients are scanned row-by-row left-to-right downward). At each iteration (time updated according to the 2DLMS algorithms [1], [2]. The 2-D index) k, the 2DCBAE algorithm updates w@iY jA according to the adaptive correlation enhancer (2DACE) algorithm described in [3] proposed formula written in matrix form as has shown improved results over the 2DLMS algorithm, especially @I A Q in the case of signals of small spatial extent embedded in white ‡k a ‡kI C ˆk x @mY nA Qpkx@mY nA PvPq (3) Gaussian noise. The adaptive filter impulse response using this k algorithm converges to the 2-D auto-correlation function of the signal where ‡k is the @Pv CIA @Pv CIAcoefficient matrix given by of interest. However, the performance of the 2DACE algorithm (4) and (5), shown at the bottom of the next page, in which the deteriorates in the case of colored Gaussian noise. @Pv CIA @Pv CIAobservation matrix and v is the maximum lag, Recently, the use of higher order statistics (HOS) in signal process- pk and qk are the second- and fourth-order moments of the image ing has moved to the forefront of interest among many researchers x@mY nA, which are recursively estimated by [4]–[7]. The main advantage of using HOS is based on the property P that for Gaussian noise (white or colored), all cumulants of order pk a pkI C@I Ax @mY nA R (6) greater than two are identically zero. Therefore, if a non-Gaussian qk a qkI C@I Ax @mY nA signal is corrupted by additive Gaussian noise, estimates of cumulants and  and  are smoothing factors that lie in the range Manuscript received April 17, 1995; revised February 24, 1997. The associate editor coordinating the review of this paper and approving it for H ( Y  ` I (7) publication was Dr. Yun Q. Shi. The authors are with the Department of Electrical Engineering, ‡ Faculty of Engineering, Assiut University, Assiut, Egypt (e-mail: which control the rate of adaptation of the coefficient matrix k. [email protected]). The reciprocal of @I A and @I A determine the time constant Publisher Item Identifier S 1053-587X(99)00750-3. associated to (3) and (6), respectively.

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