Computationally Efficient Approximated Matrix Inversion with Application to Crosstalk Precoding in Downstream ∗ VDSL Li Youming
Institute of Communication Technology, Ningbo University Ningbo 315211, P. R. China
[email protected]
Amir Leshem
School of Engineering, Bar-Ilan University Ramat Gan 52900, Israel
[email protected]
ABSTRACT
Fang Liming
Huawei Technologies Co. Ltd., P. R. China
[email protected]
Several crosstalk cancelers have been proposed recently. Ginis and Cioffi [1] proposed a Tomlinson Harashima type FEXT precoding with high complexity due to nonlinear processing. Cendrillon et. al [2] have noted that it is sufficient to use a linear precoding due to the diagonal dominance property of the FEXT channel matrix, still their zero-forcing (ZF) solution requires matrix inversion at each tone of the multichannel DSL system. Leshem and Li [3], [4] proposed a simplified linear precoding scheme, by approximating the matrix inversion using power series expansion. This approach has a much lower implementation complexity. This paper presents an alternative precoder which has the same complexity as the simplified linear precoder [4] and a better performance. The new precoder, splits the FEXT channel matrix as a sum of a tridiagonal matrix plus the matrix formed with the remaining elements. The inverse matrix is then approximated using this decomposition. A fast implementation for computing this approximation is also proposed. The structure of the paper is as follows. The system model and some matrix notations are given in Section 2. Section 3 reviews the proposed FEXT precoding methods. In section 4 we propose an efficient implemnetation of the new precoder. Computer simulation results based on measured DSL channels are presented in Section 5. We end up with some conclusions.
An algorithm for approximating crosstalk channel matrix inversion is proposed in this paper. The algorithm decomposes the crosstalk channel matrix into a tridiagonal matrix plus a residual matrix composed of the remaining elements. By using diagonal dominance property, the inverse of the crosstalk channel matrix is approximated. Based on these results we propose a novel precoding method with lower computational complexity for canceling the crosstalk in downstream VDSL. Computer simulation results based on measured channel data are provided to verify the efficiency of the proposed algorithm.
Categories and Subject Descriptors F.2 [Analysis of algorithms and problem complexity]: Numerical Algorithms and Problems
General Terms Algorithms,performance
Keywords Crosstalk cancellation, digital subscriber line, matrix splitting, vectoring.
2.
1.
Assuming a synchronized Discrete Muli-Tone modulation (DMT) the channel model at each tone can be describe by:
INTRODUCTION
Crosstalk is the major limiting noise in very high bit rate digital subscriber line (VDSL) systems. For downstream transmission, as the receiving modems of different users are located at different place, the crosstalk cannot be cancelled at the receiver side. However, the transmitting modems are co-located at the central office (CO) or optical network unit (ONU). This allows crosstalk precoding to be applied at the transmitter.
CHANNEL MODEL AND NOTATIONS yk = Hk xk + nk
(1)
T where xk = [x1k , ..., xN k ] is the vector of symbols transmitted on n tone k, and xk is the symbol transmitted by user n on tone k. Similarly yk = [yk1 , ..., ykN ]T is the vector of symbols received on tone T k. The vector nk = [n1k , ..., nN k ] is the additive noise experienced
by the receivers on tone k. The crosstalk channel matrix is denoted as Hk = [hn,m ]. The diagonal element hn,n is the direct chank k nel from transmitter n to receiver n, while the off-diagonal element hn,m is the crosstalk channel from transmitter m to receiver n. k For notational convenience, the crosstalk channel matrix is denoted as Hk = [hij ]. The following two matrix splitting notations are important for developing our algorithms. The first splitting is called diagonal line splitting which is defined as
∗This research is supported by ZJNSF project (Y105619), NBNSF(2006A610002), and Huawei company project
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Hk = Dk + Ek
(2)
where Dk is a matrix from the diagonal elements of crosstalk channel matrix defined by (3) and Ek is the matrix containing off diagonal elements of Hk :
429
Dk =
h11 ..
.
(3)
hN N
0
h21 Ek = . .. hN 1
h12 .. . .. . ···
··· .. .
h1N .. .
0
h(p−1)N 0
hN (N −1)
,
(4)
The second splitting is called tridiagonal line splitting which is formed as follows: Hk = DIII + EIII
(5) Figure 1: Crosstalk cancellation for up and downstream transmission
where DIII is a band structured matrix defined by (6) and EIII is the matrix containing remaining elements of Hk defined in (7): h11 h12 0 ··· 0 h21 h22 h23 · · · 0 0 h32 h33 · · · 0 (6) DIII = . . . . . .. .. .. .. .. 0 0 0 · · · h(N −1)N 0 0 0 ··· hN N
EIII
3.
0 0 h31 .. .
= h (N −1)1 hN 1
0 0 0 .. .
h13 0 0 .. .
h(N −1)2 hN 2
h(N −1)3 hN 3
··· ··· ··· .. . .. . ···
h1N h2N h3N .. . 0 0
3.2
For first order precoder based on diagonal line splitting, we have
(7)
−1 −1 Wk = (D−1 k − Dk Ek Dk )Dk
PROPOSED CROSSTALK PRECODER
2 sˆk = sk − (Ek D−1 k ) sk + nk
As is no zero, this precoder does not cancel the crosstalk completely, causing some loss as compare to ZF precoder. For second order precoder based on diagonal line splitting, we have ¡ −1 ¢2 −1 −1 −1 Wk = (D−1 Dk )Dk (13) k − Dk Ek Dk + Dk Ek Detailed analysis about second order precoder is given in [4].
The zero-forcing precoder estimates the transmitted symbols by multiplying the transmitted symbol vector in the transmitter side with the following filter matrix Wk .
4.
TRI-DIAGONAL LINE PRECODER
In this section, we propose a novel precoder. The basic idea is still to use simple matrix splitting to approximate the inverse of crosstalk channel matrix.
(8)
4.1
ZF precoder cancels the crosstalk completely and causes negligible PSD changing of the transmitted signal. In this way, we can estimate the symbols in the receiver side as sˆk = sk + nk
(12)
2 (Ek D−1 k ) sk
Zero-forcing precoder
Wk = H−1 k Dk
(11)
As Dk is a diagonal matrix, its inversion requires only N divisions. Compare to ZF precoder, diagonal line precoder has much lower computational complexity. With diagonal line precoder, we have the estimated symbols in the receiver side as
A typical scenario of a VDSL system with crosstalk canceler for upstream and downstream transmission are shown respectively in Figure 1 (a) and (b). For upstream transmission, we design the receiver filter matrix Wk in the receiver side to cancel the crosstalk, and for downstream transmission, our purpose is to design a crosstalk precoding matrix Wk in the transmitter side that completely eliminates the crosstalk in the receiver side. We only consider downstream transmission in this paper. We now review two crosstalk precoding techniques.
3.1
Diagonal line precoder
The basic idea of the diagonal line precoder is to split the crosstalk channel matrix as the diagonal matrix plus the matrix with zero elements in the main diagonal line and approximate the inverse of crosstalk channel matrix in a simplified form. Using the diagonal dominance property of the matrix Hk and the matrix decomposition (2), the inversion of Hk has the following power series expansion: ¡ −1 ¢2 −1 −1 −1 −1 H−1 Dk + . . . (10) k = Dk − Dk E k Dk + Dk E k
The new precoder
Based on matrix splitting (5) and power series expansion (10), we have the following first order inversion approximation
(9)
−1 −1 −1 D−1 III − DIII EIII DIII ≈ Hk
The main disadvantage of the zero-forcing precoder is from the high complexity of matrix inversion in each tone.
(14)
From this approximation, we form the following precoder matrix filter
430
Wk =
(D−1 III
−
−1 D−1 III EIII DIII )DIII
Let λ = max |lij | , the following approximation result: 1 1 λ λ 1 λ2 −1 . . . . LII ∼ .. .. .. .. .. .. λN −2 . . 1 ··· N −1 M −2 2 λ λ ··· λ λ 1
(15)
With this approximation, the estimated symbols in the receiver side are given by 2 sˆk = sk − (EIII D−1 III ) sk + nk
(16)
Comparing(15) with (11), we find that the inversion of the tridiagonal matrix is the most complex. In the following subsection we will discuss the fast implementation of the inversion process.
4.2
Fast implementation of the tri-diagonal precoder
The inversion of tri-diagonal linear precoder matrix structured, can be approximated by the following technique. First, we perform an LU decomposition of the matrix DIII : DIII = LII UII where
1 l1 =
LII
UII
v1
=
w1 v2
1 .. .
..
.
lN −2
1 lN −1
(18)
1
w2 v3
..
.
..
.
wN −2 vN −1
wN −1 vN
(19)
This decomposition can be implemented in the following steps: First from comparing the elements of first row in both side of(17), we have v1 = h11 , w1 = h12 , and the first column we l1 = hv21 . 1 Generally, from comparing the i-th row and i-th column, we respech . tively have vi = hii − li−1 vi−1 , wi = hi(i+1) , and li = (i+1)i v1 In this way, we can obtain the decomposition in (N-1) multiplications, divisions, and additions. Then the inversion of LII has the following apparent expression:
L−1 II
=
1 u21 .. . ui1 .. . uN 1
PERFORMANCE COMPARISON
In this section we compare the performance of the proposed tridiagonal linear precoder with the ZF and the first order diagonal linear precoder method. Comparison is based on France Telecom experimental data of 8 VDSL lines over 4 different distances 75m, 150m, 300m, and 590m. No precoder case is also considered for comparison purpose. Each modem has a coding gain of 3.8 dB, noise margin of 6 dB which results in a gap of 12 dB for BER of 10−7 . We have performed 200 independent runs, and in each run, the transfer function and FEXT are taken randomly from the 28 pairs of the given length. Figure 2, Figure 3, and Figure 4 depict the estimated Cumulative Distribution Function (CDFs) of the achievable rates of ZF precoder with respectively to no canceler, diagonal line precoder, and tridiagonal line precoder methods when the stand VDSL downstream frequency band 0.1-3.75MHz, 5.2-8.5MHz are used. For each method, we just estimated CDFs of the achievable rates in one group of four transmitter distances of 75m, 150m, 300m, and 590m. As we use 8 VDSL lines in our simulation, the results for other four lines are omitted here due to similar results.
1 l2
As the crosstalk channel matrix has diagonal dominance property, λ
