# Stationary Interference Cancellation in Upstream Coordinated DSL Using a Turbo-MMSE Receiver

Authors: Issam Wahibi; Meryem Ouzzif; Jérôme Le Masson; Samir Saoudi
Journal: International Journal of Digital Multimedia Broadcasting
 (2008)

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## Abstract

We investigate the scenario of an upstream coordinated DSL transmission in
presence of spatial-correlated noise. Joint signal processing helps mitigate this noise and reduce
internal interference effects between coordinated users. We propose to couple noise whitening with
a mean-squared error iterative receiver in order to approach the matched filter bound of the DSL
coordinated system. The convergence of the iterative scheme in this scenario is predicted using
EXIT charts under realistic transmission conditions.


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## Body

## 1. Introduction

Faced with the mounting need of residential and
professional customers to benefit from new high data-rate multimedia services,
access  network operators are improving their offers thanks to diverse
high-speed technologies. Very high speed digital subscriber line (VDSL2) allows
tens of megabits per second to be sent to customers over copper lines for short
distances. This technique, associated with an optical termination situated
close to the customer premises, can supply this need while saving on
infrastructure costs. In this configuration, far-end crosstalk (FEXT) coupling
between copper lines can represent a major performance limitation.Crosstalk,
in the case of a coordinated DSL transmission, can be separated into two
categories. The first, calledin-domain crosstalk, comes from the
vectored system whereas the second category of crosstalk, called out-of-domain
crosstalk, originates from outside the coordinated system when two or more
access network operators share the same binder [1]. When the DSL systems
transmit continuously, the crosstalk noise can be considered stationary.
Several techniques have been proposed for in-domain crosstalk cancellation in
the upstream link. In the literature, Ginis and Cioffi use a zero
forcing-successive interference cancellation (ZF-SIC) structure for FEXT
cancellation [2],
while Cendrillon et al. show that a linear zero-forcing (ZF) canceller
achieves near optimal performance [3].Out-of-domain crosstalk or
equivalently external interference presents a spatial correlation on
the receiver side. A whitening filter based on the Cholesky decomposition is
applied to mitigate its impact and is followed by a successive interference
cancellation (SIC) structure to help reduce the inherent interference of the
equivalent channel [1]. A power allocation method that mitigates the external
crosstalk effect in vectored DSL systems has been proposed in [4].In the presence of strong external interference, the
previously presented interference vectored cancellation schemes perform far
from the optimum. In this paper, we propose to
carry out after the whitening filter an iterative processing based on the
minimum mean-squared error (MMSE) criterion
using a priori information [5–7]. The iterative detection of
multiple-input multiple-output (MIMO) signals using the MMSE detector is
presented in [8].The remainder of the paper is organized as follows.
The system model is described in Section2. Maximum throughput upper bound for
coordinated DSL systems in presence of external interference is derived in
Section 3. The ZF-SIC and the linear iterative MMSE detectors are examined in
Section 4. Numerical results are analyzed in Section 5 and conclusions are
drawn in Section 6.


## 2. System Model

A coordinated
DSL system is depicted in Figure1. N coordinated lines are colocated in the same
Optical Network Unit (ONU). Because of crosstalk between pairs of the same
binder, the coordinated signals interfere with each other and undergo
interference from L different external sources. The N coordinated lines can benefit from a joint
signal processing at the receiver side in the upstream direction.Figure 1
External crosstalk environment.Such as the
recently adopted standard VDSL2, the transmission is based on a discrete
multitone (DMT) modulation applied to data that were first coded and
interleaved before being mapped to complex QAM symbols. According to the known
DMT modulation principle, the constellation size is determined on each tone as
a function of its signal-to-noise ratio. It is assumed that the cyclic prefix
length exceeds the maximum delay of the channel and that the coordinated
transmissions are perfectly synchronized. Thus, the transmission can be modelled in the frequency domain. For each tonek, the received signal   yk is therefore written as(1)yk=Hkxk+Akzk+wk,k=1,…,K,where K is the number of tones. The vector xk=[x1k,…,xNk]T contains the complex symbols transmitted on
tone k for the N coordinated users. The vector zk=[z1k,…,zLk]T contains the frequency components of the
external transmitted signals on tone k. wk is the vector of additive white Gaussian noise
(AWGN) elements on each line. If the transmit power spectral density (PSD) on
each coordinated line i on tone k is denoted by σxik2,
the autocorrelation of the transmitted spatially uncorrelated signal is a
diagonal matrix Dxk=ℰ{xkxkH}=diag(σx1k2,…,σxNk2). In a similar way, the external sources are spatially uncorrelated and
their autocorrelation matrix is Dzk=ℰ{zkzkH}=diag(σz1k2,…,σzLk2). The white noise PSD is assumed equal for the different coordinated
lines. Therefore, the noise autocorrelation matrix is ℰ{wkwkH}=σwk2IN. The matrix Hk is the frequency response of the coordinated
MIMO channel on tone k.
The diagonal element hi,ik of matrix Hk is the attenuation coefficient of the line i and the off-diagonal element hi,jk is the FEXT coupling coefficient between
transmitter j and receiver i on tone k.
Thanks to the physical properties of the cable, Hk is diagonally dominant. This property is
inherent to the fact that the direct channel is stronger than the crosstalk
coupling channels which implies diagonal elements greater than the off-diagonal
elements in the channel matrix. The N×L matrix Ak contains the coupling coefficients between the
external sources and the coordinated lines in the frequency domain. For reasons
of clarity, the tone index is dropped in the following sections.


## 3. Noise Mitigation and Maximum Throughput Upper Bound

The resulting
noise termn=Az+w in (1) is spatially correlated. Its different
components cannot be directly known by the receiver but its covariance matrix Rnn can be measured. It is written as
follows:(2)Rnn=E{nnH}=ADzAH+σw2IN.In this section, an upper bound
of the multiuser DSL system maximum throughput is derived when noise whitening
is carried out.


### 3.1. Noise Whitening

Noise
mitigation is obtained through linear filtering of the received signal in the
frequency domain. The noise part of the resulting whitened signal has a
diagonal correlation matrix. The computation of the whitening filter can be
carried out using different methods among which two are described. The first
method is based on the Cholesky decomposition of the noise correlation matrixRnn.
The whitening filter can be expressed by(3)W=G−1, where Rnn=GGH and G is a lower triangular matrix. The upper
triangular structure of the whitening filter allows successive noise cancelling
to be carried out when the channel matrix H is diagonal as proposed in [1].The second method is based on the inverse square root
of the noise covariance matrixRnn given by(4)W=(Rnn)−1/2.Eldar and Oppenheim show in
[9] that this solution
minimizes the mean-squared error between the original and whitened data.
Therefore, it is used as a whitening solution with the iterative receiver
described in the next section. In order to avoid matrix inversion, the
computation of the noise whitening filter could be implemented iteratively as
suggested in [10]. The
received signal after whitening is expressed by(5)yˇ=Wy=Hˇx+nˇ.Noise nˇ has a covariance matrix Rnˇnˇ=IN which overcomes the noise correlation due to
external noise in DSL coordinated systems. Unfortunately, the equivalent
channel matrix Hˇ looses the diagonal dominance property useful
for linear crosstalk cancellation.


### 3.2. Matched Filter Bound (MFB)

In order to determine the matched filter bound of the
multiuser coordinated DSL receiver, a single user is assumed to be
transmitting. At the receiver side, the signal is received on the direct line
and on the other lines belonging to the coordinated system through crosstalk
between lines. Thanks to whitening at the reception, the noise is uncorrelated
and has the same power level on the different lines. This system is therefore
equivalent to a channel withN diversity branches. It is shown in [11] that the maximum ratio
combiner (MRC) is the optimum receiver for such a transmission scheme. The MRC
linearly combines the individually received branch signals so as to maximize
the signal-to-noise ratio for the considered user. The MRC output signal for
user i can be written as(6)yiMFB=∑j=1N|hˇj,i|2xi+∑j=1Nhˇj,i*nˇj.With the assumption that the
different users of the coordinated system do not interfere with each other, the
maximum throughput of user i on a given tone is expressed
as(7)CiMFB=log2(1+1Γ∑j=1N|hˇj,i|2σxi2),where Γ is the is the signal-to-noise ratio gap to
capacity.In practice, all the coordinated users transmit simultaneously
and interfere with each other. These interferences have to be dealt with using
signal processing techniques that can improve the system performance upper
bounded by the maximum throughput (7).


## 3.1. Noise Whitening

Noise
mitigation is obtained through linear filtering of the received signal in the
frequency domain. The noise part of the resulting whitened signal has a
diagonal correlation matrix. The computation of the whitening filter can be
carried out using different methods among which two are described. The first
method is based on the Cholesky decomposition of the noise correlation matrixRnn.
The whitening filter can be expressed by(3)W=G−1, where Rnn=GGH and G is a lower triangular matrix. The upper
triangular structure of the whitening filter allows successive noise cancelling
to be carried out when the channel matrix H is diagonal as proposed in [1].The second method is based on the inverse square root
of the noise covariance matrixRnn given by(4)W=(Rnn)−1/2.Eldar and Oppenheim show in
[9] that this solution
minimizes the mean-squared error between the original and whitened data.
Therefore, it is used as a whitening solution with the iterative receiver
described in the next section. In order to avoid matrix inversion, the
computation of the noise whitening filter could be implemented iteratively as
suggested in [10]. The
received signal after whitening is expressed by(5)yˇ=Wy=Hˇx+nˇ.Noise nˇ has a covariance matrix Rnˇnˇ=IN which overcomes the noise correlation due to
external noise in DSL coordinated systems. Unfortunately, the equivalent
channel matrix Hˇ looses the diagonal dominance property useful
for linear crosstalk cancellation.


## 3.2. Matched Filter Bound (MFB)

In order to determine the matched filter bound of the
multiuser coordinated DSL receiver, a single user is assumed to be
transmitting. At the receiver side, the signal is received on the direct line
and on the other lines belonging to the coordinated system through crosstalk
between lines. Thanks to whitening at the reception, the noise is uncorrelated
and has the same power level on the different lines. This system is therefore
equivalent to a channel withN diversity branches. It is shown in [11] that the maximum ratio
combiner (MRC) is the optimum receiver for such a transmission scheme. The MRC
linearly combines the individually received branch signals so as to maximize
the signal-to-noise ratio for the considered user. The MRC output signal for
user i can be written as(6)yiMFB=∑j=1N|hˇj,i|2xi+∑j=1Nhˇj,i*nˇj.With the assumption that the
different users of the coordinated system do not interfere with each other, the
maximum throughput of user i on a given tone is expressed
as(7)CiMFB=log2(1+1Γ∑j=1N|hˇj,i|2σxi2),where Γ is the is the signal-to-noise ratio gap to
capacity.In practice, all the coordinated users transmit simultaneously
and interfere with each other. These interferences have to be dealt with using
signal processing techniques that can improve the system performance upper
bounded by the maximum throughput (7).


## 4. Multiuser Detection

After
whitening as described in Section3, the different transmitted signals have
to be estimated from the whitened signal yˇ expressed by (5). In such a scenario, the
linear processing for the coordinated upstream DSL system proposed in [3] is no longer efficient since
the equivalent channel matrix Hˇ is not diagonally dominant. In this section,
we present two multiuser detection methods that will be investigated in this
transmission configuration. The first method uses successive interference
cancellation based on ZF criterion as suggested in [2] for vectored DSL systems.
The second method consists of an iterative MMSE receiver as proposed for
wireless MIMO systems in [8].


### 4.1. ZF-Based SIC

The successive
interference canceller investigated for the multiuser detection is based on the
QR decomposition of the equivalent channel matrix after whitening as
follows:(8)Hˇ=QR,where Q is a unitary matrix and R is upper triangular. The whitened vector yˇ is multiplied by matrix QH which results in the following
output:(9)y¯=QHyˇ=Rx+n¯,where n¯=QHnˇ and Rn¯n¯=IN.
Since matrix R is upper triangular, successive interference
cancellation can be carried out as follows:(10)x^i=1ri,i(y¯i−∑j=i+1Nri,jdec(x^j)),i=N,…,1,where dec(⋅) denotes symbol decision that might include
channel decoding. When symbol xi is being estimated, it is assumed that the
decisions about symbols x^j with j=i+1,…,N were error-free. Therefore, the
signal-to-noise ratio for user i is expressed as(11)SNRi=|ri,i|2σxi2.Since matrix Q is unitary, the norm of each column vector i of matrix Hˇ is equal to its dual column vector i norm from matrix R.
Therefore, the previous expression can be rewritten as(12)SNRi=(∑j=1N|hˇj,i|2−∑k=1i−1|rk,i|2)σxi2.The maximum throughput attained
by the ZF-based SIC for user i on one given tone is(13)CiZF-SIC=log2(1+1Γ(∑j=1N|hˇj,i|2−∑k=1i−1|rk,i|2)σxi2).


### 4.2. Iterative MMSE Receiver

The iterative
receiver presented here in the context of multiuser DSL transmissions is based
on MMSE detection with a priori information. Such a receiver was introduced
for turbo-equalization purposes in [6, 12] and then adapted for MIMO systems [8]. Information exchanged
between the MMSE receiver and the soft-input soft-output (SISO) channel decoder
is represented by log-likelihood ratios (LLRs) [13] denoted by ℒ(⋅).
The LLR on a binary value b is defined by(14)ℒ(b)=ln(P{b=1}P{b=0}).As depicted in Figure 2, the
MMSE detection provides soft symbols x˜ using the channel output yˇ and a priori information in the form of soft symbols x^.
During each iteration and for each user, enhanced LLR values ℒad are calculated from the previous ones ℒed by the channel-decoding
stage.Figure 2
Iterative MIMO receiver.The decoder requires soft inputs represented by the
LLR values of each binary element. It is therefore necessary to convert complex
symbols coming from the MMSE detector to LLR values on their binary elements.
The output of the MMSE detector can be expressed by(15)x˜=g⋅x+η,where g is a bias given by (23) and η is noise with a variance of ση2.
The transmitted symbols x are chosen in QAM-constellations 𝒳.
The conditional probability P{x˜|x} is given by(16)ℒ(bi|x˜)=ln(P{bi=1|x˜}P{bi=0|x˜}),ℒ(bi|x˜)=ln(∑x∈𝒳1iP{x|x˜}∑x∈𝒳0iP{x|x˜}),ℒ(bi|x˜)=ln(∑x∈𝒳1iP{x˜|x}P{x}∑x∈𝒳0iP{x˜|x}P{x}),  where 𝒳bi are subsets of constellation 𝒳 whose bit of index i is equal to b.In the case of demodulation without a priori information, all symbols are equiprobables. Therefore(17)ℒ(bi|x˜)=ln(∑x∈𝒳1iP{x˜|x}∑x∈𝒳0iP{x˜|x}),ℒ(bi|x˜)=ln(∑x∈𝒳1iexp(−|x˜−g⋅x|2/ση2)∑x∈𝒳0iexp(−|x˜−g⋅x|2/ση2)).The decoder delivers LLR values for the decoded binary
elements. This information has to be translated to its equivalent complex
symbols in order to be fed back to the MMSE detector. The LLR-to-symbol mapping
is performed according to(18)x^=E{x|ℒad},x^=∑s∈𝒳sP{x=s|ℒad},where ℒad is the a priori LLR value for each binary element contained in
the transmitted symbol x.The sets of interleavers{Π1⋯ΠN} and {Π1−1⋯ΠN−1} are, respectively, used to arrange the LLR
values in the correct order before channel decoding and soft mapping.The ouput of the MMSE receiver with a priori information can be expressed for each useri of the system by [14](19)x˜i=λipiH(yˇ−Hˇx^+x^ihˇi),where hˇi is the ith column of the equivalent channel matrix Hˇ.
Equation (19) corresponds to interference cancellation followed by MMSE
equalization where(20)pi=(HˇVHˇH+IN)−1hˇiis a vector containing the
equalization coefficients. It takes into account the reliability of the a
priori information by means of the
matrix V=diag(ν^12,…,ν^N2) with(21)ν^i2=∑s∈𝒳i|s|2P{x=s|ℒa}−|x^i|2.The coefficients λi are computed as follows:(22)λi=σxi21+(σxi2−ν^i2)βi,βi=piHhˇi.The bias gi and the variance ση2 defined in (15) can be expressed
by(23)gi=λiβi,ση2=var(x˜i−gix^i).At the first iteration in the MMSE receiver, there is
no a priori information andν^i2=σxi2.
This results in the classical MMSE receiver. In case of perfect a priori information, the interference in the received
signal is completely removed and as matrix V becomes null, the receiver acts as the MRC
described in Section  3.2. Then, from (7) and considering a
convergence of the iterative process, the maximum throughput of the system with
the iterative MMSE receiver on a given tone is expressed as(24)CiMMSE=log2(1+1Γ∑j=1N|hˇj,i|2σxi2).


## 4.1. ZF-Based SIC

The successive
interference canceller investigated for the multiuser detection is based on the
QR decomposition of the equivalent channel matrix after whitening as
follows:(8)Hˇ=QR,where Q is a unitary matrix and R is upper triangular. The whitened vector yˇ is multiplied by matrix QH which results in the following
output:(9)y¯=QHyˇ=Rx+n¯,where n¯=QHnˇ and Rn¯n¯=IN.
Since matrix R is upper triangular, successive interference
cancellation can be carried out as follows:(10)x^i=1ri,i(y¯i−∑j=i+1Nri,jdec(x^j)),i=N,…,1,where dec(⋅) denotes symbol decision that might include
channel decoding. When symbol xi is being estimated, it is assumed that the
decisions about symbols x^j with j=i+1,…,N were error-free. Therefore, the
signal-to-noise ratio for user i is expressed as(11)SNRi=|ri,i|2σxi2.Since matrix Q is unitary, the norm of each column vector i of matrix Hˇ is equal to its dual column vector i norm from matrix R.
Therefore, the previous expression can be rewritten as(12)SNRi=(∑j=1N|hˇj,i|2−∑k=1i−1|rk,i|2)σxi2.The maximum throughput attained
by the ZF-based SIC for user i on one given tone is(13)CiZF-SIC=log2(1+1Γ(∑j=1N|hˇj,i|2−∑k=1i−1|rk,i|2)σxi2).


## 4.2. Iterative MMSE Receiver

The iterative
receiver presented here in the context of multiuser DSL transmissions is based
on MMSE detection with a priori information. Such a receiver was introduced
for turbo-equalization purposes in [6, 12] and then adapted for MIMO systems [8]. Information exchanged
between the MMSE receiver and the soft-input soft-output (SISO) channel decoder
is represented by log-likelihood ratios (LLRs) [13] denoted by ℒ(⋅).
The LLR on a binary value b is defined by(14)ℒ(b)=ln(P{b=1}P{b=0}).As depicted in Figure 2, the
MMSE detection provides soft symbols x˜ using the channel output yˇ and a priori information in the form of soft symbols x^.
During each iteration and for each user, enhanced LLR values ℒad are calculated from the previous ones ℒed by the channel-decoding
stage.Figure 2
Iterative MIMO receiver.The decoder requires soft inputs represented by the
LLR values of each binary element. It is therefore necessary to convert complex
symbols coming from the MMSE detector to LLR values on their binary elements.
The output of the MMSE detector can be expressed by(15)x˜=g⋅x+η,where g is a bias given by (23) and η is noise with a variance of ση2.
The transmitted symbols x are chosen in QAM-constellations 𝒳.
The conditional probability P{x˜|x} is given by(16)ℒ(bi|x˜)=ln(P{bi=1|x˜}P{bi=0|x˜}),ℒ(bi|x˜)=ln(∑x∈𝒳1iP{x|x˜}∑x∈𝒳0iP{x|x˜}),ℒ(bi|x˜)=ln(∑x∈𝒳1iP{x˜|x}P{x}∑x∈𝒳0iP{x˜|x}P{x}),  where 𝒳bi are subsets of constellation 𝒳 whose bit of index i is equal to b.In the case of demodulation without a priori information, all symbols are equiprobables. Therefore(17)ℒ(bi|x˜)=ln(∑x∈𝒳1iP{x˜|x}∑x∈𝒳0iP{x˜|x}),ℒ(bi|x˜)=ln(∑x∈𝒳1iexp(−|x˜−g⋅x|2/ση2)∑x∈𝒳0iexp(−|x˜−g⋅x|2/ση2)).The decoder delivers LLR values for the decoded binary
elements. This information has to be translated to its equivalent complex
symbols in order to be fed back to the MMSE detector. The LLR-to-symbol mapping
is performed according to(18)x^=E{x|ℒad},x^=∑s∈𝒳sP{x=s|ℒad},where ℒad is the a priori LLR value for each binary element contained in
the transmitted symbol x.The sets of interleavers{Π1⋯ΠN} and {Π1−1⋯ΠN−1} are, respectively, used to arrange the LLR
values in the correct order before channel decoding and soft mapping.The ouput of the MMSE receiver with a priori information can be expressed for each useri of the system by [14](19)x˜i=λipiH(yˇ−Hˇx^+x^ihˇi),where hˇi is the ith column of the equivalent channel matrix Hˇ.
Equation (19) corresponds to interference cancellation followed by MMSE
equalization where(20)pi=(HˇVHˇH+IN)−1hˇiis a vector containing the
equalization coefficients. It takes into account the reliability of the a
priori information by means of the
matrix V=diag(ν^12,…,ν^N2) with(21)ν^i2=∑s∈𝒳i|s|2P{x=s|ℒa}−|x^i|2.The coefficients λi are computed as follows:(22)λi=σxi21+(σxi2−ν^i2)βi,βi=piHhˇi.The bias gi and the variance ση2 defined in (15) can be expressed
by(23)gi=λiβi,ση2=var(x˜i−gix^i).At the first iteration in the MMSE receiver, there is
no a priori information andν^i2=σxi2.
This results in the classical MMSE receiver. In case of perfect a priori information, the interference in the received
signal is completely removed and as matrix V becomes null, the receiver acts as the MRC
described in Section  3.2. Then, from (7) and considering a
convergence of the iterative process, the maximum throughput of the system with
the iterative MMSE receiver on a given tone is expressed as(24)CiMMSE=log2(1+1Γ∑j=1N|hˇj,i|2σxi2).


## 5. Numerical Results

This section
is dedicated first to the analysis of the maximum throughput of the different
detection schemes presented in Section4. In a second step, the iterative MMSE
receiver behavior is examined in realistic DSL transmission conditions.


### 5.1. Performance Evaluation

Derived
maximum throughput expressions for the ZF-based SIC and iterative MMSE receiver
in (13) and (24), respectively, help compare the expected performance of these
structures. For this purpose, a scenario of four-coordinated users and two
external crosstalk interferers as depicted in Figure 1 is considered.
Transmission parameters mimic the VDSL2 setup and are reported in Table 1.Table 1
System setup.
Coordinated line length600 mExternal line length75 mFrequency band[7.09–10.18] MHzTransmitted signal PSDℰx=−60 dBm/HzExternal interferer PSDℰz=−60 dBm/HzWhite Gaussian noise PSDℰw=−140 dBm/HzTransmission gap9.8 dBPerformance results in terms of allocated bits per
tone for each user are drawn in Figure3. Maximum throughput of the
uncoordinated upstream transmission are represented by the diamond curves. The
attained maximum throughput by the ZF-SIC scheme is represented by the starred
curves and the iterative MMSE performance after convergence is depicted by the
crossed curves. The difference in throughput between the coordinated system users
is due to the channel structure and the position of the copper pairs in the
binder. Moreover, the throughputs decrease with frequency because of the
channel attenuation. ZF-SIC performance of user 1 attains the iterative MMSE
maximum throughput (24) which has been shown to be equal to the MFB, C1MMSE=C1ZF-SIC.
Conversely, user 4 does not benefit from the multiuser processing in the ZF-SIC
scheme, and its performance is equivalent to single-user
transmission performance. On the opposite, all users in the
iterative MMSE receiver benefit equally from the multiuser processing to
approach the MFB.Iterative MMSE
and ZF-SIC bit allocation in the coordinated system.
(a)(b)(c)(d)


### 5.2. Iterative MMSE Convergence Analysis

The previous
results in terms of maximum throughputs assume the convergence of the MMSE
iterative processing that reaches the matched filter bound. A practical
approach for the iterative MMSE convergence analysis is to use extrinsic
information transfer (EXIT) charts introduced by ten Brink in [15]. This allows the exchange
of information to be displayed and the transfer characteristics of an iterative
process are represented on a chart.The mutual information between a binary elementx∈{−1,+1} and its weighted information ℒ is given by(25)I(ℒ,x)=12∑x∈{−1,+1}∫−∞+∞fℒ(l|x)log2(2fℒ(l|x)fℒ(l|+1)+fℒ(l|−1))dl.It is characterized by the
probability density function fℒ(ℒ|x).
This quantity can be approached by a histogram of LLR values (14) which are
assumed to follow a Gaussian distribution.Each block in the iterative MMSE receiver is
characterized by a transfer function linking its input and output mutual
information [15]. The
mutual information transfer function for the detector is plotted with its a
priori input IA,det on the abscissa axis and its extrinsic output IE,det on the ordinate axis. The decoder component
transfer function is plotted with its a priori input IA,dec on the ordinate axis and its extrinsic output IE,dec on the abscissa axis. The decoder is common to
all coordinated users, whereas each user has its own detector transfer
function. In the sequel, the convolutional used code is G(D)=((1+D+D2),(1+D2)) for which a maximum a posteriori MAP decoding
algorithm is implemented. Data coding and decoding are carried out in the
frequency domain; and each codeword corresponds to a DMT block. Interleaving
helps decorrelate data between the detector and the decoder.Figure4 shows an EXIT chart for two-coordinated
receiving users with one external crosstalk interferer. The mutual information
transfer functions are computed using blocks of one hundred DMT symbols. The
same parameters setup reported in Table 1 except for the frequency band that is
chosen within [7.3–8.9] MHz is considered. This frequency band results
in a bit allocation ranging between 1 and 6 bits per tone.Figure 4
EXIT chart for two-coordinated users and one
external interferer.In an EXIT
chart representation, a decoding trajectory will take the form of a
stair-shaped curve. The trajectory goes between one of the detector mutual
information transfer functions and the decoder transfer function along the
different iterations. In the case of two-coordinated lines and one external
line, we can prove that the output of the MMSE detector of user 1 depends only
on the input of the MMSE detector of user 2 and vice versa, as shown in the
appendix. Consequently, the first user trajectory intersects, respectively, the
first detector transfer function for the odd iterations and the second detector
transfer function for the even iterations. The second user trajectory behaves the
opposite way. This progress of the trajectories'
iterations is illustrated in Figure4. The difference between the
MMSE detector transfer functions for both users is caused by the structure of
the equivalent channel matrix Hˇ.
The departing point is low for user 2 because the equivalent channel matrix
off-diagonal element hˇ2,1 magnitude is of the same order as the direct
path hˇ2,2 magnitude, whereas the off-diagonal element hˇ1,2 magnitude is small compared to the direct path hˇ1,1 magnitude.The same scenario with four-coordinated users and two
external crosstalk interferers considered in Section5.1
is examined regarding convergence of the iterative MMSE process for the setup
given in Table 1. The associated EXIT chart is depicted in Figure 5. The
resulting EXIT chart and the position of the detector mutual information
transfer functions indicate that the system converges within few iterations and
therefore the matched filter bound performance will be attained.Figure 5
Convergence
of the iterative system for four-coordinated users with two external crosstalk
interferers.


## 5.1. Performance Evaluation

Derived
maximum throughput expressions for the ZF-based SIC and iterative MMSE receiver
in (13) and (24), respectively, help compare the expected performance of these
structures. For this purpose, a scenario of four-coordinated users and two
external crosstalk interferers as depicted in Figure 1 is considered.
Transmission parameters mimic the VDSL2 setup and are reported in Table 1.Table 1
System setup.
Coordinated line length600 mExternal line length75 mFrequency band[7.09–10.18] MHzTransmitted signal PSDℰx=−60 dBm/HzExternal interferer PSDℰz=−60 dBm/HzWhite Gaussian noise PSDℰw=−140 dBm/HzTransmission gap9.8 dBPerformance results in terms of allocated bits per
tone for each user are drawn in Figure3. Maximum throughput of the
uncoordinated upstream transmission are represented by the diamond curves. The
attained maximum throughput by the ZF-SIC scheme is represented by the starred
curves and the iterative MMSE performance after convergence is depicted by the
crossed curves. The difference in throughput between the coordinated system users
is due to the channel structure and the position of the copper pairs in the
binder. Moreover, the throughputs decrease with frequency because of the
channel attenuation. ZF-SIC performance of user 1 attains the iterative MMSE
maximum throughput (24) which has been shown to be equal to the MFB, C1MMSE=C1ZF-SIC.
Conversely, user 4 does not benefit from the multiuser processing in the ZF-SIC
scheme, and its performance is equivalent to single-user
transmission performance. On the opposite, all users in the
iterative MMSE receiver benefit equally from the multiuser processing to
approach the MFB.Iterative MMSE
and ZF-SIC bit allocation in the coordinated system.
(a)(b)(c)(d)


## 5.2. Iterative MMSE Convergence Analysis

The previous
results in terms of maximum throughputs assume the convergence of the MMSE
iterative processing that reaches the matched filter bound. A practical
approach for the iterative MMSE convergence analysis is to use extrinsic
information transfer (EXIT) charts introduced by ten Brink in [15]. This allows the exchange
of information to be displayed and the transfer characteristics of an iterative
process are represented on a chart.The mutual information between a binary elementx∈{−1,+1} and its weighted information ℒ is given by(25)I(ℒ,x)=12∑x∈{−1,+1}∫−∞+∞fℒ(l|x)log2(2fℒ(l|x)fℒ(l|+1)+fℒ(l|−1))dl.It is characterized by the
probability density function fℒ(ℒ|x).
This quantity can be approached by a histogram of LLR values (14) which are
assumed to follow a Gaussian distribution.Each block in the iterative MMSE receiver is
characterized by a transfer function linking its input and output mutual
information [15]. The
mutual information transfer function for the detector is plotted with its a
priori input IA,det on the abscissa axis and its extrinsic output IE,det on the ordinate axis. The decoder component
transfer function is plotted with its a priori input IA,dec on the ordinate axis and its extrinsic output IE,dec on the abscissa axis. The decoder is common to
all coordinated users, whereas each user has its own detector transfer
function. In the sequel, the convolutional used code is G(D)=((1+D+D2),(1+D2)) for which a maximum a posteriori MAP decoding
algorithm is implemented. Data coding and decoding are carried out in the
frequency domain; and each codeword corresponds to a DMT block. Interleaving
helps decorrelate data between the detector and the decoder.Figure4 shows an EXIT chart for two-coordinated
receiving users with one external crosstalk interferer. The mutual information
transfer functions are computed using blocks of one hundred DMT symbols. The
same parameters setup reported in Table 1 except for the frequency band that is
chosen within [7.3–8.9] MHz is considered. This frequency band results
in a bit allocation ranging between 1 and 6 bits per tone.Figure 4
EXIT chart for two-coordinated users and one
external interferer.In an EXIT
chart representation, a decoding trajectory will take the form of a
stair-shaped curve. The trajectory goes between one of the detector mutual
information transfer functions and the decoder transfer function along the
different iterations. In the case of two-coordinated lines and one external
line, we can prove that the output of the MMSE detector of user 1 depends only
on the input of the MMSE detector of user 2 and vice versa, as shown in the
appendix. Consequently, the first user trajectory intersects, respectively, the
first detector transfer function for the odd iterations and the second detector
transfer function for the even iterations. The second user trajectory behaves the
opposite way. This progress of the trajectories'
iterations is illustrated in Figure4. The difference between the
MMSE detector transfer functions for both users is caused by the structure of
the equivalent channel matrix Hˇ.
The departing point is low for user 2 because the equivalent channel matrix
off-diagonal element hˇ2,1 magnitude is of the same order as the direct
path hˇ2,2 magnitude, whereas the off-diagonal element hˇ1,2 magnitude is small compared to the direct path hˇ1,1 magnitude.The same scenario with four-coordinated users and two
external crosstalk interferers considered in Section5.1
is examined regarding convergence of the iterative MMSE process for the setup
given in Table 1. The associated EXIT chart is depicted in Figure 5. The
resulting EXIT chart and the position of the detector mutual information
transfer functions indicate that the system converges within few iterations and
therefore the matched filter bound performance will be attained.Figure 5
Convergence
of the iterative system for four-coordinated users with two external crosstalk
interferers.


## 6. Conclusion

In this paper,
the combination of external crosstalk whitening and iterative MMSE processing
is examined for upstream coordinated DSL systems. The iterative MMSE receiver
achieves the MFB performance under the assumption of perfect a priori information. Unlike the ZF-SIC, all users in
the iterative MMSE receiver benefit equally from the multiuser processing to
approach the MFB. EXIT charts are used to analyze the behavior of this scheme
for upstream coordinated DSL systems in the presence of correlated external
noise. Convergence of the iterative MMSE receiver for coordinated DSL systems
is obtained in realistic conditions.



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*Source: 428037-2008-09-23.xml*