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Impulse response shortening for xDSL discrete multitone modems with linear phase filters Carlos Ribeiro1,2, Vitor Silva2 and Paulo S. R. Diniz3 1 Escola Superior de Tecnologia e Gestão, Instituto Politécnico de Leiria, 2401-951 Leiria, Portugal 2 Instituto de Telecomunicações, Departamento de Engenharia Electrotécnica e de Computadores, Universidade de Coimbra, 3030-290 Coimbra, Portugal 3 COPPE, Universidade Federal do Rio de Janeiro, Brasil cribeiro@estg.iplei.pt; vitor.silva@co.it.pt; diniz@lps.ufrj.br Abstract 2. Channel shortening with linear phase filters The problem of shortening the effective channel impulse Unlike previously proposed method [7], the maximum response in xDSL is addressed. A new approach is shortening method [4] explicitly uses the length ν of the described based on linear phase FIR equaliser filters. The cyclic prefix (CP) as the desired effective length of the mathematical framework for the LPIRS method is channel impulse response, thus minimising the channel developed and simulation results are presented for several Inter Symbolic Interference (ISI). As Melsa et al. referred VDSL loops. in [4], there is no direct relation between the achievable bit For a similar performance, the novel method offers a design rate and the channel shortening, but this is certainly a good with considerably lower computational complexity and a criteria. The coefficients of the optimal shortening filter are simpler solution, on either DSP or dedicated hardware obtained by an eigenvector decomposition of the energy implementations. function SSNR, as described next. Consider the effective channel heff in its matrix form as 1. Introduction heff = Hw, (1) Discrete Multitone Modulation (DMT) is a powerful where H is the convolution matrix built from the channel modulation scheme [1], adopted by the standardisation impulse response of length M and w is the equaliser filter of bodies [2], [3] in the development of recent Digital length t. By defining the vector hwin as a window of ν+1 Subscriber Line (xDSL) standards. One of the main issues consecutive samples of heff, starting at sample d, and the in the xDSL systems is the severe attenuation and distortion vector hwall containing the remaining samples of heff, vectors caused by the channel and other services on the same hwin and hwall are given by (2) and (3), respectively, bundle. To overcome these channel effects on the transmitted signals, equalisation techniques are commonly h win = H win w used in time [4] - [8] and frequency [9] domains. The T equalisation process can be simplified by periodically = heff (d ) heff (d + 1) ! heff (d +ν ) extending each symbol with a convenient number of prefix h( d ) h(d − 1) ! h(d − t + 1) w(0) and suffix samples [1]. Hence, circular convolution can be h(d + 1) h( d ) ! h(d − t + 2) w(1) used instead of linear convolution, so that the channel = × equalisation can then be easily performed in the frequency " " # " " domain. Unfortunately, this is only possible if the effective h(d +ν ) h(d +ν + 1) ! h(d +ν − t + 1) w(t − 1) length of the channel impulse response is shorter than the (2) guard period (cyclic prefix and suffix). In order to shorten h wall = H wall w the impulse response, several methods have been proposed T [4] - [8], based on Finite Impulse Response (FIR) = heff (0) ! heff ( d − 1) heff ( d + ν + 1) ! heff ( M + t − 2) equalisers. In this paper is proposed a novel method of equalisation for h(0) 0 ! 0 xDSL DMT modems based on linear phase FIR equalisers. " " # " w(0) The next section presents a review of the original Minimum h(d − 1) h(d − 2) ! h(d − t ) w(1) Shortening Signal-to-Noise Ratio (MSSNR) method [4] and = × its improvements [5]. Also, is described the development of h( d + ν + 1) h(d + ν ) ! h(d + ν − t + 2) " " # " " w(t − 1) the proposed Linear Phase Impulse Response Shortening (LPIRS) method and the mathematical framework, which 0 ! 0 h( M − 1) leads to an optimal solution for every type of linear phase (3) filter. The energy of these vectors can be expressed as Simulation results are presented in section 3, which clearly show that the achieved shortened signal noise ratios hT h wall = w T H T H wall w = w T Aw wall wall (4) (SSNR) are similar to yielded by other methods [4], [5], at a win =w h T h win T = w Bw , H T H win w win T (5) considerably lower computational complexity. where A and B are presumably positive definite matrices. Finally, conclusions about the new method are drawn in The solution to the shortening problem can be found by section 4, where it is discussed the influence of some design either minimising (4) or maximising (5). parameters, e.g. the filter type (symmetry) and length. 1/4 Originally, the optimal solution was found by minimising Since linear phase filters can provide a prescribed the energy of hwall [4]. To avoid an unbounded solution, a magnitude frequency selectivity with lower computational unit energy constraint was imposed on hwin [6], given by complexity than a more general nonlinear phase filters, we 2 propose to constraint the equaliser filter w to have linear h win = 1 . (6) phase (LPIRS method), as following described. Define Assuming that matrix B is positive definite, we can use w L = Lx , (16) Cholesky decomposition to obtain where wL is the linear phase equaliser filter of length t, L is ( )( ) T T B = QDQT = Q D Q D = B B , (7) an extension matrix (see Table 1 and 2) and x is a generic vector. where D is the diagonal matrix formed from the eigenvalues of B, and the columns of Q are the orthonormal eigenvectors of B. Defining the following matrix Symmetric Anti-symmetric 1 0 0 ! 0 1 0 0! 0 ( ) A ( DQ ) = ( B ) −1 A B , −1 T −1 −1 T C= Q D (8) 0 1 0 " 0 10 " 0 0 # 0 0 # the optimal solution is given as 0 1 0 0 1 0 −1 " 1 " 1 w opt = B l min , T 0 0 0 0 (9) L= L= 0 0 1 0 0 − 1 0 1 0 0 −1 0 where lmin is the unit-length eigenvector corresponding to 0 0 $ 0 0 $ the minimum eigenvalue λmin of C. The optimal shortening SNR is defined as 0 1 0 " 0 −1 0 " 1 0 0 ! 0 − 1 0 0 ! 0 w T Bw opt SSNRopt = 10 log T opt = −10 log(λ ) , (10) Table 1. Extension matrices L (t×t 2 ) for even length vectors. w Aw min opt opt which depends on the minimum eigenvalue λmin of the Symmetric Anti-symmetric composite matrix C. This approach is valid as long as B is 1 0 0 ! 0 1 0 0! 0 non-singular and B exists, which is true if the length of 0 1 0 " 0 10 " the equaliser t is shorter than ν [5]. 0 0 # 0 0 # The solution to the shortening problem was found in [5], by 0 1 0 0 1 0 maximising (5) and imposing a unit energy constraint on L=" 0 0 1 L= " 0 0 0 hwall, 0 1 0 0 −1 0 2 0 0 $ 0 0 $ h wall = 1 , 0 1 0 " 0 −1 0 " (11) − 1 0 0 ! under the assumption that matrix A is non-singular. 1 0 0 ! 0 0 Therefore, matrix A can be decomposed using Cholesky decomposition as Table 2. Extension matrices L (t×(t −1) 2 ) for odd length vectors. ( )( A = QDQT = Q D Q D = A A ) T T (12) There are four possible L matrices depending on the length and the new composite matrix C may be defined as of vector wL and type of symmetry adopted. ( ) B ( DQ ) = ( A ) −1 B A . (13) −1 T −1 −1 T As in [4] and [5], we can rewrite (4) and (5) as C= Q D h T h wall = w L T H T H wall w L = x T A L x wall wall (17) The new solution is given in [5] as −1 h T h win = w L T H T H win w L = x T B L x , win win (18) w opt = A l max , T (14) where AL and BL are given by A L = LT AL , (19) where lmax is the unit-length eigenvector corresponding to the maximum eigenvalue λmax of C. B L = L BL . T (20) The new shortening SNR is then defined as The solution proposed in [5] was the one adopted in this w T Bw opt work, since it was shown to be valid regardless of the SSNRopt = 10 log T opt = 10 log(λ ) . (15) lengths of either the CP or the equaliser. w Aw max opt opt Assuming that a proper delay d is chosen so that A is This new solution is valid for any value of t and yielding a positive definite, we can show that AL is also positive similar performance [5] as the original method [4]. definite since A common and key issue is to guarantee the non-singularity w L T Aw L = (Lx)T A (Lx) = x T (LT AL)x = x T A L x > 0 , (21) and the matrices L are full rank ( t 2 or (t − 1) 2 , of both A and B matrices. By ensuring that there is at least one non-zero sample of heff outside hwin, it guarantees, by definition, that A is positive definitive. The same principle respectively, for even or odd length filters). can be applied to matrix B. As in [5], Cholesky decomposition is applied to matrix AL, Linear phase is an important property in digital signal ( AL = Q D Q D )( )T = AL AL . T (22) Using a similar development as in [5], we define processing. It ensures the absence of phase distortion on the signal (only an integer delay). 2/4 (A ) −1 Fig. 3 shows the normalized impulse response of FSAN BL AL T −1 CL = L (23) Cabinet #3 test loop, with MSSNR and LPIRS (symmetric) equalisation. and the optimum solution for the equaliser wL is given by −1 1 w L opt = L A L l L max , T (24) 0.8 where l Lmax is the unit-length eigenvector corresponding to 0.6 the maximum eigenvalue λ L max of CL. 0.4 It is important to highlight that the dimensions of matrices AL and BL are t 2 × t 2 or (t − 1) 2 × (t − 1) 2 , thus resulting 0.2 in a substantial complexity reduction on the Cholesky decomposition and eigen analysis, as compared to [4] and 0 [5]. -0.2 3. Simulation results -0.4 0 500 1000 1500 2000 2500 3000 3500 4000 To evaluate the performance of the proposed method, several experiments were carried out on different VDSL Fig. 3. FSAN Cabinet #3 MSSNR (with dashed line) and LPIRS test loops, as described in [3] and [10]. equalised impulse response. In Fig. 1 is shown the original impulse response of the FSAN Cabinet #3 test loop, which is one of the most For performance comparison purposes, several simulations difficult VDSL test loops. Fig. 2 shows the normalized results are presented in Table 3 and 4, using the same target impulse response of the same loop after MSSNR impulse response and number of carriers, as in the previous equalisation. simulations. In Table 3, an equaliser of length t = 10 is The same input parameters were used for all simulations used, whereas in Table 4, an equaliser of length t = 19 is presented, i.e. 4096 carries, a target impulse response used instead. length of 64 samples and an equaliser w with length t = 10 . Considering the short equaliser filter lengths (low -4 complexity implementation), the results in Tables 3 and 4 x 10 show, from a practical point of view, a similar performance 3 between the MSSNR and LPIRS methods, taking in 2.5 consideration the high levels of obtained SSNR (apart from the highly demanding FSAN Cabinet #3 loop). 2 Test Loops MSSNR LPIRS 1.5 Symmetric Anti-symmetric ANSI VDSL Loop 3 70.00 69.94 69.90 1 ANSI VDSL Loop 4 38.61 38.38 36.96 ANSI VDSL Loop 7 47.18 39.17 44.79 0.5 FSAN FTTEx #3 73.65 68.36 67.47 FSAN Cab #3 11.79 7.94 8.01 0 0 2000 4000 6000 8000 10000 Table 3. Results in dB for an even length equaliser. Fig. 1. FSAN Cabinet #3 impulse response. LPIRS Test Loops MSSNR Symmetric Anti-symmetric ANSI VDSL Loop 3 71.72 71.72 71.62 1.2 ANSI VDSL Loop 4 40.33 39.92 39.45 1 ANSI VDSL Loop 7 52.90 47.93 48.23 FSAN FTTEx #3 81.53 78.11 77.18 0.8 FSAN Cab #3 12.05 8.17 9.06 0.6 Table 4. Results in dB for an odd length equaliser. 0.4 4. Conclusions A new approach to the problem of shortening the effective 0.2 channel impulse response in xDSL has been described, 0 using linear phase FIR equaliser filters. The use of the full rank extension matrices is the crucial point in this novel -0.2 method. The resulting target functions are defined with half 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 size of the original matrices, leading to substantial Fig. 2. FSAN Cabinet #3 original and MSSNR equalised impulse complexity reduction in obtaining the optimal solution. response (with dashed line). Hitherto, all efforts leading to simplified solutions for the implementation of VDSL modems are crucial, as neither 3/4 DSP nor dedicated hardware solutions are able to implement the full VDSL standards. As shown in section IV, the equaliser performance attained by the LPIRS method is similar to that yielded by the reference method proposed by Melsa et al, used as a reference. References [1] John A. C. Bingham, “ADSL, VDSL and Multicarrier Modulation, Wiley Series in Telecommunications and Signal Processing, 2000. [2] ETSI TM, “VDSL – Part 2: Transceiver Specification”, TS 101 270-2 v.1.1.1, Feb 2001. [3] T1E1.4, “VDSL Metallic Interface – Part 1: Functional Requirements and Common Specification”, Draft Trial-Use Standard, Feb. 2001. [4] Peter Melsa, Richard Younce and Charles Rohrs, “Impulse response shortening for discrete multitone transceivers”, IEEE Trans. on Communications, vol. 44, nº 12, pp. 1662-1672, December 1996. [5] Changchuan Yin and Guangxin Yue, “Optimal impulse response shortening for discrete multitone transceivers”, Electronics Letters, vol. 34, pp. 35-36, January 1998. [6] D. Daly, C. Heneghan and A. Fagan, “A minimum mean-squared error interpretation of residual ISI channel shortening for discrete multitone transceivers”, Proc. IEEE International Conference on Acoustic Speech and Signal Processing (ICASSP), 2001. 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