# A set of N lines is bonded through co-ordination at both ends of the link

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(2) 2. Y. (F). =. CS - .- * ) *) . F−1. H direct. CP

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(17) . CS . . . . . . . F. CP ,+ /,+ 0/ 0/ F+1. ('(''( ('(''( ('(''( ('"!('"! ('"! "! "! "!. (F−1). X CS CP (F). #$$## #$$## #$$## &%$$#&%&%$$#&% &%$$#&% %& %& %&. X CS CP (F+1). X. +. CS F−1. H next. CP . CS . F. lN EXT − δ. CP F+1. δ. . (F−1). U CS CP (F). . U. + Z. (B). CS CP (F+1). U Fig. 2.. NEXT and Cyclic Extensions (F - block index). So the use of cyclic extensions, and synchronized transmission prevents inter-carrier FEXT, NEXT and echo. This allows us to perform FEXT, NEXT and echo cancellation per-tone, which requires 2N K mults/user/DMT-block, in contrast to the N K mults/user/DMT-block required by FEXT cancellation alone. Model transmission on one tone as. yk = Hk Dk sk + Hnext Duk uk + zk k ª where the diagonal matrices Dk , diag d1k , . . . , dN and Duk shape the spectrum of all users at tone k. k ©. NEXT Cancellation: NEXT and echo cancellation is performed by simple subtraction. yk − Hnext Duk uk k © ª Noise Whitening: Let E zk zH = Zk . We decompose the noise spectrum using a Cholesky decomposition k chol. Zk = Gk GH k which is then used to whiten the noise at the receiver ek y © ª −H zk e zH such that E e = G−1 = IN . k k Zk G k. =. ¡ ¢ yk − Hnext Duk uk G−1 k k. =. G−1 zk k Hk Dk sk + e. e k , G−1 Hk Dk can be transformed into a set of independent, FEXT Cancellation: Using the SVD, the channel H k parallel sub-channels[2]. Define svd e Uk Λk VkH = H k © 1 ª where Λk , diag σk©, . . . , σªkN . We now pre-code the transmitted symbols sk = Vk xk , such that xk contains © H ªthe true QAM symbols and E xxH = I . Since V is unitary, it has no effect on the power of s and hence E sk sk = IN . N k k k.

(18) 3. Algorithm 1 Iterative Vector Waterfilling init dnk = 0 ∀n, k repeat for n =Ã1 . . . N ¸+ !1/2 · 1 1 ∀k dnk = m m H dm 2 −1 hn λn − hn H [IN +P ] k m6=n hk hk k k end until convergence H An estimate of the transmitted symbols is formed by equalization with Λ−1 k Uk . Hence ¡ ¢ H −1 bk = Λ−1 yk − Hnext Duk uk x k k Uk Gk H xk + Λ−1 zk k Uk e © ª H zk is E zk zH = Λ−2 where the power of the noise term zk , Λ−1 k k Uk e k .. =. Hence the SNRs seen by each of the sub-channels is SIN Rkn = σkn 2 The total capacity of the system if we employ a standard slicer after equalization is thus X X ¯ ¯ ¯ ¯ −1 2 H −H ¯ log2 ¯IN + Λ2k ¯ = log2 ¯IN + UH k Gk Hk Dk Hk Gk Uk k. =. k X. ¯ ¢¯ ¡ ¯ log2 ¯IN + Z−1 Hk D2k HH k k. k. which is the theoretical capacity of the channel using ML detection. Thus we achieve full capacity using simple single user detection (standard slicer) plus linear pre-compensation and equalization on a per-tone basis. DSM We can convert our channel with coloured, (possibly) correlated noise into an equivalent AWGN channel Hk , G−1 k Hk . Our channel is effectively a single user, vector (MIMO) channel. The theoretical capacity is thus ¯ ¯ X ¯ ¯ C= log ¯IN + Hk D2k HH k ¯ k. We desire to maximize C as a function of {Dk }k=1,...,K under a total power constraint on each line X 2 dnk ≤ P ∀n. (3). k. Simultaneous Vector Waterfilling: This problem was addressed in [3] where it was shown that the optimal power allocation is a form of simultaneous vector waterfilling. The optimal power allocation is thus + 1/2 1 dnk = − λn. H hnk. 1. h IN +. P. m m6=n hk. H hm k. 2 dm k. i−1. hnk. . ∀n, k. (4). where hnk , [Hk ]col n . The Lagrange multipliers {λ1 , . . . , λN } are chosen such that the set of constraints in (3) are all met with equality. Iterative Vector Waterfilling: No closed form solution is known for solving the set of equations in (3) and (4). Fortunately, the optimal solution can be found using a cheap iterative algorithm[4]. λn is chosen in each inner iteration of Alg. 1 to meet the power constraint on line n (see Eq. (3)) with equality..

(19) 4. Noise Only Waterfilling: While iterative vector waterfilling allows us to find the optimal power allocation in an efficient way, we can exploit certain properties of the DSL channel to reduce the complexity of power allocation even more. Define X mH m2 Qk , IN + hm dk k hk m6=n. 1 hk . h 1H IN + .. hk N hk . =. ···. NH. i. hk. ¤ £ m 1 and hi,j where hk , hm,1 · · · hm,n−1 dkn−1 hm,n+1 dn+1 · · · hm,N dN k , [Hk ]i,j . In DSL the direct k k dk k k k k channel from a given transmitter to it’s receiver is always much stronger than the channel from that transmitter to another receiver. We call this column dominance since it ensures that hn,n À hm,n k k Using this observaton we can approximate " #+ 1/2 1 1 dnk ' − n,n 2 £ −1 ¤ λn |hk | Qk n,n. ¯ ¯ £ ¤ n,n ¯ n,n ¯ −1 Now Q−1 = ¯Q ¯ |Qk | where Qk , [Qk ]col 1:n−1 n+1:N, row 1:n−1 n+1:N , ie. Qk with row n and column n k k n,n removed. ¯ ¯ 1 ¯ ¯ hk ¯ ¯ h i ¯ ¯ . h1 H · · · hN H ¯ . |Qk | = ¯¯IN + ¯ . k k ¯ ¯ N ¯ ¯ hk ¯ ¯ n ¯ ¯ hk ¯ ¯ 1 ¯ ¯ ¯ ¯ hk ¯ ¯ . ¯ ¯ . ¯ ¯ . h i ¯ ¯ n−1 nH 1H n−1 H n+1 H NH ¯ ¯ = ¯IN + hk hk hk hk · · · hk · · · hk ¯ ¯ ¯ n+1 ¯ ¯ hk ¯ ¯ . ¯ ¯ . ¯ ¯ . ¯ ¯ N ¯ ¯ hk Dividing this into sub-matrices. ¯· ¯ A |Qk | = ¯¯ C. ¸¯ ¯ B ¯ n,n ¯ Qk ° ° h n nH ° n° 1 n−1 where A , 1 + °hk °, B , hk Mk , C , MH hk · · · hk k hk . Here M , n,n Qk = MH M + IN −1 . Using the Schur decomposition ¯ ¯¯ ³ n,n ´−1 ¯¯ ¯ n,n ¯ ¯¯ |Qk | = ¯Qk ¯ ¯A − B Qk C¯¯. n+1 hk. ···. N hk. i . Notice that. ¯ ° ° ¡ ¢−1 H n H ¯¯ £ −1 ¤ n ¯ ° n° M hk ¯ Qk n,n = ¯1 + °hk ° − hk M MH M + IN −1. hence svd. i. j. H Using the SVD M = UM ΛM VM . If hk 6= αhk ∀i, j then M will have full rank and UM and VM with be unitary matrices of size N − 1 × N − 1. This is typically the case in practice. Hence ¡ ¢−1 H ¡ ¢−1 H M MH M + IN −1 M = UM Λ2M Λ2M + IN −1 UM.

(20) 5. 997 (Symmetric) Bandplan, Alien Crosstalk A 160 Normal WF FEXT FEXT+WF NEXT+FEXT NEXT+FEXT+WF. Rate per Line (per direction) (Mbps). 140 120 100 80 60 40 20 0 300. Fig. 3.. 400. 500. 600. 700 800 Line Length (m). 900. 1000. 1100. 1200. Average Rate (per line) vs. Line Length (Alien Crosstalk A). Since the SNR in DSL is high we can approximate Λ2M + IN −1 ' Λ2M and ¡ ¢−1 H M MH M + IN −1 M ' IN −1 Hence and. £ −1 ¤ Qk n,n = 1 " #+ 1/2 1 1 dnk ' − n,n 2 λn |hk |. (5). Using this power allocation strategy, each user’s power allocation can be done independently. This considerably reduces power allocation complexity. Each user waterfills against their own direct channel and the background noise as if no interference was present. The reason that interference is explicitly ignored in (5) is because the high SNR and column dominant nature of the channel allow for near-perfect crosstalk cancellation. R ESULTS Using these techniques it is possible to support more than double the rates achieved with FEXT cancellation alone. For example, a 20 pair binder of 300m can support 3 Gbps symmetric when used in a bonded fashion with FEXT, NEXT and echo cancellation. Shown in Fig.s 3 and 4 is the average rate achieved (per line) in a bonded system employing different levels of coordination. These results were found using empirical transfer functions. This should be repeated using measured transfer functions. Note that the crosstalk cancellation gains are largest on short lines whilst the waterfilling gains are largest on long lines. Also note that crosstalk cancellation gains decrease with the severity of background noise, whilst waterfilling gains increase. For example in Fig. 4 long lines experience minimal gain from crosstalk cancellation since the noise (Alien Crosstalk Type F) is so severe. On the other hand the waterfilling gains are significant..

(21) 6. 997 (Symmetric) Bandplan, Alien Crosstalk F. Rate per Line (per direction) (Mbps). 120 Normal WF FEXT FEXT+WF NEXT+FEXT NEXT+FEXT+WF. 100. 80. 60. 40. 20. 0 300. Fig. 4.. 400. 500. 600. 700 800 Line Length (m). 900. 1000. 1100. 1200. Average Rate (per line) vs. Line Length (Alien Crosstalk F). C OMMENTS Due to spectral compatibility concerns, it may be necessary that the transmit powers dnk 2 fall within a set of spectral masks. This problem is addressed in [5] for the single line power allocation problem. It’s extension to multiple (bonded) lines is an open problem. In unbundled scenarios it becomes more difficult to cancel out-of-domain NEXT. This could be achieved using blind algorithms (blind synchronization) and successive-interference cancellation. If this is not allowed by regulatory bodies it may be possible to protect out-of-domain services, which would employ echo cancellation only, whilst still allowing large performance gains to systems which choose to employ full NEXT cancellation. This would be done by allowing full-duplex transmission in the low frequency bands where NEXT coupling is small. Optimal power allocation in this case is difficult, however good sub-optimal solutions have been proposed[6]. R EFERENCES [1] F. Sjoberg, M. Isaksson, P. Borjesson, et al., “Zipper: A Duplex Method for VDSL Based on DMT,” IEEE Trans. Commun., vol. 47, no. 8, pp. 1245–1252, Aug. 1999. [2] G. Taubock and W. Henkel, “MIMO Systems in the Subscriber-Line Network,” in Proc. of the 5th Int. OFDM-Workshop, 2000, pp. 18.1–18.3. [3] P. Viswanath, D. Tse, and V. Anantharam, “Asymptotically Optimal Water-Filling in Vector Multiple-Access Channels,” IEEE Trans. Inform. Theory, vol. 47, no. 1, pp. 241–267, Jan. 2001. [4] W. Yu, W. Rhee, S. Boyd, and J. Cioffi, “Iterative Water-filling for Gaussian Vector Multiple Access Channels,” IEEE Trans. Inform. Theory, submitted April 2001. [5] E. Baccarelli, A. Fasano, and M. Biagi, “Novel Efficient Bit-Loading Algorithms for Peak-Energy-Limited ADSL-Type Multicarrier Systems,” IEEE Trans. Signal Processing, vol. 50, no. 5, pp. 1237–1247, May 2002. [6] Autonomous DSM Mixture of Symmetric and Asymmetric Service: Bi-directional Iterative Waterfilling, ANSI Std. Contrib. T1E1.4/2002-058, Rev. 1, 2002..

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