BLIND JOINT IDENTIFICATION AND EQUALIZATION OF WIENER-HAMMERSTEIN COMMUNICATION CHANNELS USING PARATUCK-2 TENSOR DECOMPOSITION

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1 5th European Signal rocessing Conference (EUSICO 2007), oznan, oland, September 3-7, 2007, copyright by EURASI BLIND JOINT IDENTIFICATION AND EQUALIZATION OF WIENER-HAMMERSTEIN COMMUNICATION CHANNELS USING ARATUCK-2 TENSOR DECOMOSITION Alain Kibangou and Gérard Favier LAAS-CNRS, University of Toulouse, 7 avenue Colonel Roche, 3077 Toulouse, France I3S, University of Nice Sophia Antipolis, CNRS, 2000 route des Lucioles, Les Algorithmes, Euclide B, B 2, Sophia Antipolis Cedex phone: +(33) , +(33) , akibango@laasfr, favier@i3sunicefr ABSTRACT In this paper, we consider the blind joint identification and equalization of Wiener-Hammerstein nonlinear communication channels By considering a special design of the input signal, we show that output data can be organized into a third-order tensor We show that the obtained tensor has a ARATUCK-2 representation We derive new results on uniqueness of the ARATUCK-2 model by considering structural constraints such as Toeplitz and Vandermonde forms for some of its matrix factors We also constrain the input signal to belong to a finite alphabet Then an Alternating Least Squares (ALS) algorithm is proposed for estimating the factors of the ARATUCK-2 model and therefore the parameters of the Wiener-Hammerstein channel and the unknown input signal The performances of the proposed joint identification and equalization method are illustrated by means of simulation results INTRODUCTION Linear approximation is often inappropriate or inadequate for modelling practical systems Therefore, it is necessary to consider nonlinear models Among them the Volterra model has been the subject of several works [] However, Volterra model suffers generally of a huge parametric complexity By taking the structural property of the considered nonlinear plant into account, it is sometimes possible to use block-structured nonlinear models such as Wiener- Hammerstein models These models that combine a memoryless nonlinearity with linearly dispersive elements have been successfully employed in various areas including digital communications systems [2, 3, 4] There are relatively few works in the area of blind identification of Wiener-Hammerstein or block-structured nonlinear systems Existing identification methods mainly concern the stochastic case, the input signal being generally assumed to be Gaussian (see [5] for example), and make use of higher order statistics, or exploit the cyclostationarity of the input signal [6] In this paper, we consider a deterministic approach based on a third-order tensor decomposition Tensor analysis is used in many areas of science and engineering Since few years, the use of tensor concepts in signal processing has gained more attention, motivated by the field of higher order statistics When dealing with tensors, two families of models are generally encountered: ARAFAC [7, 8] and TUCKER models [9] Different models sharing some properties of both ARAFAC and TUCKER models have been proposed in the dedicated literature, as for instance the ARATUCK-2 model [0] In this paper, we show that the output data of a Wiener-Hammerstein communication channel can be organized into a third-order ARATUCK-2 tensor with structural constraints for its matrix factors in Toeplitz and Vandermonde forms After deriving uniqueness properties of this constrained ARATUCK-2 decomposition, a blind joint identification and equalization algorithm is proposed for such communication channels The paper is organized as follows In Section 2, we briefly recall some tensor models The link between Wiener-Hammerstein model and the ARATUCK-2 decomposition is stated in Section 3 Then in Section 4, the ARATUCK-2 uniqueness properties are considered before describing the estimation procedure in Section 5 Some simulation results are provided to illustrate the performances of the proposed algorithm in Section 6 Conclusions and future work are given in Section 7 Notations: Matrix pseudo inverse T Matrix transpose Khatri Rao product F Frobenius norm I N (N N) Identity matrix A p (resp A p ): pth column (resp row) of the matrix A D p (A): Operator that forms a diagonal matrix from the pth row of the matrix A D(a): Operator that forms a diagonal matrix from the elements of the vector a So, we have D p (A) = D(A p ) x p denotes a vector with the entries of x = (x x 2 x M ) T raised to power p, ie x p = (x p xp 2 xp M )T 2 TENSOR DECOMOSITIONS Let X be a three-way array, also called a I I 2 I 3 third-order tensor, the elements of which are x i,i 2,i 3, i j =,2,,I j, j =,2,3 In analysis of X the difference between independent and interacting factors is illustrated by the two following decompositions: N x i,i 2,i 3 = a i,n b i2,n c i3,n, n = N 2 N 3 N x i,i 2,i 3 = a i,n b i2,n 2 c i3,n 3 h n,n 2,n 3 n = n 2 = n 3 = The first representation is called ARAFAC(ARallel FACtors analysis)[7] or CANDECOM (CANonical DECOMosition)[8] It exhibits independence between factor contributions to each data point On the contrary, in the second representation, called TUCKER-3 [9], the factors interact in their contributions to the data The TUCKER-3 representation is then more general than ARAFAC In general, TUCKER models are not unique whereas uniqueness conditions exist for ARAFAC [] A family of models sharing features of the two above mentioned models has been suggested in the literature: the so-called ARATUCK-2 model [0] given by N N 2 x i,i 2,i 3 = a i,n c A i 3,n h n,n 2 c B i 3,n 2 b i2,n 2 () n = n 2 = By slicing the I I 2 I 3 tensor X along its third mode, we get the following matrix representation of the ARATUCK-2 model: X i3 = AD i3 (C A )HD i3 (C B )B T R I I 2, i 3 =,2,,I 3, (2) 2007 EURASI 56

2 5th European Signal rocessing Conference (EUSICO 2007), oznan, oland, September 3-7, 2007, copyright by EURASI where A, B, C A, and C B are matrices with the respective dimensions I N, I 2 N 2, I 3 N, and I 3 N 2 H is a N N 2 matrix called kernel-matrix Uniqueness of the ARATUCK-2 decomposition has been proved when N = N 2 provided A, B, and H are full column rank matrices and H has no zero elements [0] To the best of the authors knowledge, few applications and works are devoted to the ARATUCK-2 decomposition In the sequel, we show how this decomposition can be used for jointly identifying and equalizing Wiener-Hammerstein channels, and we give uniqueness conditions when N N 2 3 ARATUCK-2 MODELLING OF WIENER-HAMMERSTEIN COMMUNICATION CHANNELS The Wiener-Hammerstein model, depicted in Fig, is one of commonly used block-oriented nonlinear structures We denote by u(n), y(n), v (n), v 2 (n), and e(n), the input signal, the output signal, intermediate variables, and the additive noise respectively Assuming that the nonlinearity is continuous within the considered dynamic range, then, from the Weierstrass theorem, it can be approximated to an arbitrary degree of accuracy by a polynomial C() of finite degree, the coefficients of which are c p So, the Wiener-Hammerstein model is constituted by u (n) v ( n) v 2 ( n) y(n) l () C() g() Wiener model Hammerstein model Figure : Wiener-Hammerstein model a polynomial C(), sandwiched between two linear filters with impulse response l() and g() and memory M l and M g respectively, ie, v (n) = M l l(i)u(n i), v 2 (n) = c p v p (n), and i=0 p= M g y(n) = g(i)v 2 (n i) + e(n) For characterizing the parametric representation of each block of the Wiener-Hammerstein sys- i=0 tem, we define the following vectors: l = (l(0) l() l(m l )) T, g = ( g(0) g() g(m g ) ) T, and c = (c c 2 c ) T One can note that if the linear subsystems l() and g() are respectively scaled by α and α 2 then the polynomial defined by the coefficients c p = c p α p α leads to the same input-output equation In other words, 2 the linear subsystems can be defined up to a scaling factor In order to remove such an indeterminacy we enforce the linear subsystems to be monic, ie l(0) = g(0) = This constraint is not restrictive Indeed, if the actual linear subsystems are not monic then they can be determined up to the first coefficient of their respective impulse responses One can also note that this constraint ensures the unicity of the polynomial subsystem With these specifications, the equivalent Volterra representation is given by: M p y(n) = h p ( j,, j p ) u(n j k ) + e(n), (3) p= j,, j p =0 k= where M = M l + M g, and M g p h p ( j,, j p ) = c p g( j ) l( j k j), (4) j= k= with j k =,,M, and k =,, p To remove the effect of the non-diagonal Volterra kernel coefficients on the output signal, assuming that the memory M and the nonlinearity degree are well-known, the input signal is designed such that: { φr s(n) i f m = u((nr + r )M + m ) = 0 i f m = 2,,M where φ r, r =,2,,R, are distinct non-zero real valued known coefficients whereas s(n) are the informative unknown symbols to be transmitted, belonging to a finite alphabet set Λ = { } Q/2 ±λ q q= This choice of the input signal corresponds to a data transmission by block Each block r associated with a symbol s(n), is of dimension M and contains this symbol coded by a constant φ r and (M ) zeroes that allow to remove the non-diagonal Volterra kernel coefficients effect on the output signal Then, the output signal is given by: y((nr + r )M + m ) = h p (m,m,,m )φr p s p (n) + e((nr + r )M + m ) p= By using (4) and denoting y m,n,r = y((nr + r )M + m ) and e m,n,r = e((nr + r )M + m ), we get: M g y m,n,r = c p g( j )l p (m j)φr p s p (n) + e m,n,r, (5) p= j= or equivalently M l y m,n,r = c p g(m j)l p ( j )φr p s p (n) + e m,n,r (6) j= p= The indices m, r, and n represent respectively the position m in the block r associated with the symbol s(n) By defining the following correspondences: (i,i 2,i 3,n,n 2,I,I 2,I 3,N,N 2 ) (m,n,r, j, p,m,n,r,m l,), and a i,n = g(m j), c A i 3,n = φ r l( j ), h n,n 2 = l p ( j ), c B i 3,n 2 = c p φr p, and b i2,n 2 = s p (n), we can conclude that (6) is a ARATUCK-2 model of the form () We first reduce our discussion to the noiseless case We define the M N matrices Y r as: y,,r y,2,r y,n,r y 2,,r y 2,2,r y 2,N,r Y r = y M,,r y M,2,r y M,N,r Using the above stated correspondences and their equivalent matrix formulations, ie(a,c A,H,C B,B) (G,F,L,C,S), equation (2) becomes : Y r = GD r (F)LD r (C)S T, (7) where G is a M M l Toeplitz matrix the first row and column of which are respectively given by G = ( g(0) 0 0 ) and G = ( g T 0 0 ) T,we also note G = T (g), S is a N column-wise Vandermonde matrix: S = ( s s 2 s ), s = ( s() s(2) s(n) ) T, L is a M l column-wise Vandermonde matrix: L = ( l 0 l l ), F is a R M l rank one matrix: F = φl T, φ = ( φ φ 2 φ R ) T C is a R matrix: C = ΦD(c), Φ = ( φ 0 φ φ ) 2007 EURASI 57

3 5th European Signal rocessing Conference (EUSICO 2007), oznan, oland, September 3-7, 2007, copyright by EURASI The M N R tensor Y, defined by the slices Y r, r =,2,,R, is a ARATUCK-2 model characterized by structural constraints concerning the matrix factors G, L, and S that are in Toeplitz and Vandermonde forms respectively One can note that uniqueness results given in [0] cannot be applied for such a ARATUCK-2 model since M l in general In the sequel we investigate uniqueness properties of the ARATUCK-2 model (7) by exploiting structural constraints on the involved matrices 4 UNIQUENESS ISSUE OF THE ARATUCK-2 MODEL WITH STRUCTURAL CONSTRAINTS Before considering the uniqueness issue, we recall the useful property of admissibility in the Vandermonde sense of a given alphabet and a lemma on a related identifiability result concerning bilinear decompositions [2] Definition: Let V be the th-order Vandermonde matrix associated with the finite alphabet Λ = { } Q/2 ±λ q q= defined by λ λ λ Q/2 λ Q/2 λ 2 λ 2 λq/2 2 λ 2 Q/2 V = λ ( ) λ λq/2 ( ) λq/2 The alphabet Λ is said to be admissible of order in the Vandermonde sense if Q and if the only admissible permutation matrices Π of order Q satisfying V Π(V V I Q ) = 0 are Π = I Q and Π = J Q, J Q being a Q Q block diagonal matrix constituted with Q/2 blocks defined as J 2 = ( 0 0 Lemma Let Y = HU T, where H is an arbitrary full rank matrix, U is a column-wise Vandermonde matrix, constructed from a finite alphabet set Λ = { } Q/2 ±λ q q= If the alphabet Λ is admissible of order in the Vandermonde sense and if each symbol of Λ appears at least once in the first column of U then U and H can be uniquely identified up to a diagonal matrix T = diag(β, β 2,,β ), β = ± In the sequel, we consider the following assumptions: A: g(0) = and g(m g ) is nonzero This implies that any bilinear decomposition as GK, G being in Toeplitz form and K with no particular structure, is unique [3] A2: Λ is an admissible alphabet of order in the Vandermonde sense and each symbol of Λ appears at least once in the sequence {s(n)} n=,2,,n A3: The weights φ r, r =,2,,R, and the coefficients of the polynomial c p, p =,2,, are nonzero A4: At least coefficients of the impulse response l() are distinct and non zero, and (M l ) The uniqueness property of the ARATUCK-2 model is stated in the theorem below Theorem Consider a M N R third-order tensor Y with the ARATUCK-2 structure given in (7) Suppose there is an alternative representation of Y with matrices of the same size and structural form Y r = ḠD r ( F) LD r ( C) S T,r =,2,,R Then, according to assumptions A-A4, the two representations are necessarily linked by the following relations: ) Ḡ = G, F = F, L = L, S = S, C = C, (8) with = diag(β, β 2,,β ), β = ± roof: see the Appendix Thanks to the uniqueness property of the ARATUCK-2 decomposition stated by the Theorem, we can estimate the vectors g, l, and c describing the different blocks of the Wiener-Hammerstein system Obviously, both g and l are uniquely determined from G and L respectively From C = ΦD(c) we can deduce c Φ being known, the coefficients of the polynomial subsystems are then blindly estimated up to a sign 5 ESTIMATION OF THE ARATUCK-2 FACTORS As for ARAFAC and TUCKER decompositions, the estimation of the ARATUCK-2 decomposition factors can be carried out using the Alternating Least Squares (ALS) algorithm, as it was first proposed in [4] where the different factors were obtained by minimizing the cost function f (G,F,L,C,S) = R Ỹr GD r (F)LD r (C)S T 2 F, Ỹ r being the r= noisy version of Y r In this section, we propose a new method for estimating the matrix factors of the ARATUCK-2 model We define the matrices L 2 = D(l)L and C 2 = D(φ)C Then the slices Y r and the unfolded matrix Y R defined in (4) can be rewritten as: Y r = GL 2 D r (C 2 )S T, Y R = (C 2 GL 2 )S T We also consider the unfolded matrices associated with the first and second modes For the first mode, the slices are given by Y m = SD m (GL 2 )C T 2, m =,,M and we have Y M = ( Y T T M) YT = (GL2 S)C T 2 For the second mode, we have Y n = C 2 D n (S)L T 2 GT, n =,,N, and Y N = ( Y Ṭ T N) YṬ = (S C2 )L T 2 GT For estimating the factors S, G, L 2, and C 2, we alternatively minimize the cost functions J R = Ỹ R (C 2 GL 2 )S T 2 F, J N = Ỹ N (S C 2 )L T 2 GT 2 F, and J M = Ỹ M (GL 2 S)C T 2 2 F, where Ỹ R, Ỹ N, and Ỹ M are respective noisy versions of Y R, Y N, and Y M Then, we can deduce C and F The proposed parameter estimation algorithm is now described 5 Estimation of S By minimizing J R with respect to S, we get: ( ) T Ŝ LS = (C 2 GL 2 ) Ỹ R In order to enforce the Vandermonde structure, Ŝ is constructed from the first column of Ŝ LS after its projection onto the alphabet Λ: Ŝ = (Ŝ LS, Ŝ 2 LS, Ŝ LS, ) (9) 52 Estimation of G One can note that Y T N = GL 2(S C 2 ) T = GX, with X = L 2 (S C 2 ) T We define Θ = ( T (X ) T T (X RN ) T ) T, where T (X k ) denotes the M M g Toeplitz matrix constructed from X k G being a Toeplitz matrix, we get: vec(y T N ) = Θg Then J N can equivalently be written as: J N = vec(ỹ T N ) Θg 2 2 Its minimization with respect to g yields: ĝ = Θ vec(ỹ T N ) (0) The entries of ĝ are first divided by the first entry and then Ĝ = T (ĝ) 53 Estimation of C Since C 2 = D(φ)C = D(φ)ΦD(c), for estimating C, it is necessary to get c One can note that Y M = (GL 2 S)C T 2 = (GL 2 S)D(c) C T, with C = D(φ)Φ Then 2007 EURASI 58

4 5th European Signal rocessing Conference (EUSICO 2007), oznan, oland, September 3-7, 2007, copyright by EURASI J M can be rewritten as J M = vec(ỹm ) ( C (GL 2 S) ) c 2 2 As a consequence: and Ĉ = ΦD(ĉ) 54 Estimation of L and F ĉ = ( C (GL 2 S) ) vec(ỹm ), () By minimizing J N with respect to L 2, we get: In Fig 2, corresponding to a noiseless case, the output NMSE is plotted with respect to the number of iterations These results are averaged over 00 random initializations One can note the convergence behavior of the algorithm The convergence speed of the proposed algorithm increases with the convergence threshold value σ In the sequel we set σ = 0 In presence of noise, the choice of the estimated parameters initial values becomes more crucial We consider 0 random initializations and for each one 00 noise sequences are generated The results given in the following plots are averaged values over all of the initializations and noise sequences L 2,LS = G Ỹ T N ((S C 2 ) T ) Noticing that the estimated vector ˆl is given by the first column of L 2,LS : ( ( ˆl ) ) = G Ỹ T N (S C 2 ) T, (2) all the entries of this vector are divided by its first element before constructing the following estimated factors : ˆF = φˆl T and ˆL = ( ˆl 0 ˆl ˆl ) Output NMSE SNR=2 db SNR=27 db SNR=33 db 55 Wiener-Hammerstein identification algorithm Step 0: Initialize g,c, l, with random values and deduce the initial values for G, C 2, and L 2 Step : Estimate S using (9) If a threshold value σ is reached by the cost function J R, project the elements of the first column of S into the finite alphabet Λ Step 2: Estimate g using (0) and then construct Ĝ = T (ĝ) Step 3: Estimate c using () and then deduce C 2 Step 4: Estimate l using (2) and then construct L 2 Step 5: If a convergence criterion is reached, the algorithm is stopped Else, return to Step The algorithm is stopped when the estimates of both g and l do not significantly vary during consecutive iterations 6 SIMULATIONS We consider a Wiener-Hammerstein system such that the linear filters and the polynomial coefficients are, respectively, given by l = (, 03 0), g = (, 05 02), c = (2, 08 05) The input signal is a 6-AM one, which is admissible of order = 3 in the Vandermonde sense We evaluate the performance of the proposed algorithm by means of two measures: the output normalized mean square error, ie Ỹ R (Ĉ 2 Ĝ ˆL 2 )Ŝ T 2 F / Y R 2 F, and the parameters normalized mean square error We consider the transmission of N = 60 unknown symbols with a repetition factor R = 3 A white Gaussian noise was added to the channel output Output NMSE σ = 0 σ = 0 2 σ = 0 3 σ = Figure 2: Output NMSE in the noiseless case Figure 3: Output NMSE in noisy cases In Fig 3, one can note the effect of the projection on the alphabet, which leads to get an NMSE corresponding to the additive noise level However, this projection can deteriorate the algorithm performances especially for lower SNRs arameters NMSE Figure 4: arameters NMSE in a noisy case (SNR=30 db) In Figure 4, we can see convergence on the estimation of the subsystems parameters The estimation of the polynomial subsystem is less precise than that of the linear ones In Figure 5, we evaluate the equalization performance in terms of symbol error rate We compare the proposed scheme with a non-blind one suggested in the literature [5] In such approach, the channel parameters are assumed known One can note that the performance significantly lowers when the SNR decreases For high SNR the proposed algorithm gives performances close to those obtained with the non-blind scheme 7 CONCLUSION In this paper, we have proposed a blind joint identification and equalization algorithm for a Wiener-Hammerstein type channelthis algorithm uses a particular input design or precoding that yields a tensorial representation of output data We have shown l g c 2007 EURASI 59

5 5th European Signal rocessing Conference (EUSICO 2007), oznan, oland, September 3-7, 2007, copyright by EURASI Symbol Error Rate ARATUCK 2 based blind method olynomial roots based non blind method where t i,i;r, and x i,i;r denote respectively the diagonal entries of T r and X r As a consequence, t,;r = t 2,2;r = = t Ml,M l ;r = x,;r and x,;r = x 2,2;r = = x,;r = t,;r, ie t,;r x,;r = Then equations (6)-(8) can be written as: D r ( F) = t,;r D r (F), L = x,;r t,;r L = L, and D r ( C) = x,;r D r (C) By considering all values of r, we get: F = 2 F and C = 2 C We also have F = 2 φl T = φ l T Since from L, l can be uniquely determined, necessarily φ = φ and then 2 = I R As a consequence: C = C REFERENCES SNR (db) Figure 5: Error probability on symbols detection that the obtained tensor is a constrained ARATUCK-2 one Its factors exhibit the Toeplitz and Vandermonde structures We have established uniqueness conditions for such a ARATUCK-2 tensor model Then an iterative LS type algorithm has been proposed for estimating the factors of the tensor model From these factors we deduce the channel parameters and the input signal The initialization of the proposed algorithm remains a crucial question We intend to study this issue and that of the robustness to noise of the proposed scheme 8 AENDIX For any slice Y r, we can write: Y r = GZ r, with Z r = D r (F)LD r (C)S T By concatenating all the R slices from left to right, we get: Y = (Y Y R ) = GZ, Z = (Z Z R ) (3) Then according to assumption A and using the result in [3], we know that this bilinear decomposition is unique, ie Ḡ = G In the same way, by concatenating the slices from up to down, we get where Y R = ( Y Ṭ Y Ṭ R) T = WS T, (4) GD (F)LD (C) W = (5) GD R (F)LD R (C) Thanks to assumptions A3 and A4, by construction W is a full column rank matrix Therefore, using assumption A2 and Lemma, the above bilinear decomposition is unique up to a diagonal matrix = diag(β,β 2,,β ), β = ± Thus we can write: Y r = ḠD r (F)LD r (C) S T,r =,2,,R Let T r and X r be non-singular matrices An alternative representation of the tensor Y can be obtained by defining D r ( F) = D r (F)T r (6) L = T r LX r (7) D r ( C) = X r D r (C) (8) In order to keep the matrices in the left side of (6) and (8) diagonal, it is obvious that T r and X r must be diagonal Recall that L is a column-wise Vandermonde matrix with s in its first row and column Since L must have the same Vandermonde structure, then t,;r x,;r L = t Ml,M l ;r x,;r t,;r x,;r =, and L T = =, t,;r x,;r [] M Schetzen, The Volterra and Wiener theories of nonlinear systems John Wiley and Sons, Inc, New-York, 980 [2] N Wiener, Nonlinear problems in random theory New-York: Technology press of MIT and John Wiley and Sons Inc, 958 [3] E Biglieri, A Gersho, R Gitlin, and T Lim, Adaptive cancellation of nonlinear intersymbol interference for voiceband data transmission, IEEE J Select Areas Commun, vol 2, pp , 984 [4] X N Fernando and A Sesay, Adaptive asymmetric linearization of radio over fiber links for wireless access, IEEE Trans on Vehicular Technology, vol 5, no 6, pp , November 2002 [5] S rakriya and D Hatzinakos, Blind identification of LTI- ZMNL-LTI nonlinear channel models, IEEE Trans on Signal rocessing, vol 43, no 2, pp , December 995 [6], Blind identification of linear subsystems of LTI- ZMNL-LTI models with cyclostationary inputs, IEEE Trans on Signal rocessing, vol 45, no 8, pp , August 997 [7] R Harshman, Foundation of the ARAFAC procedure: models and conditions for an "explanatory" multimodal factor analysis, UCLA working papers in phonetics, vol 6, pp 84, 970 [8] J Caroll and J Chang, Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decomposition, sychometrika, vol 35, pp , 970 [9] L Tucker, Some mathematical notes of three-mode factor analysis, sychometrika, vol 3, pp 279 3, 966 [0] R Harshman and M Lundy, Uniqueness proof for a family of models sharing features of Tucker s three-mode factor analysis and ARAFAC/CANDECOM, sychometrika, vol 6, no, pp 33 54, 996 [] J Kruskal, Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics, Linear Algebra Applicat, vol 8, pp 95 38, 977 [2] A Kibangou, G Favier, and A de Almeida, Blind identification of series-cascade nonlinear channels, in ASILO- MAR Conference on Signals, Systems, and Computers, acific Grove, CA, USA, October 30-November 2, 2005, pp [3] A Kibangou and G Favier, Wiener-Hammerstein systems modelling using diagonal Volterra kernels coefficients, IEEE Signal roc Letters, vol 3, no 6, pp , June 2006 [4] R Bro, Multi-way analysis in the food industry models, algorithms and applications hd dissertation, University of Amsterdam, Netherlands, 998 [5] A Redfern and G Tong Zhou, A root method for Volterra system equalization, IEEE Signal rocessing Letters, vol 5, no 5, pp , EURASI 520