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Performance analysis of a commonmode signal based lowcomplexity crosstalk cancelation scheme in vectored VDSL
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 63 (2012)
Abstract
In this article, we propose a vectored system by using both common mode (CM) and differential mode (DM) signals in upstream VDSL. We first develop a multiinput multioutput (MIMO) CM channel by using the singlepair CM and MIMO DM channels proposed recently, and study the characteristics of the resultant CMDM channel matrix. We then propose a low complexity receiver structure in which the CM and DM signals of each twistedpair (TP) are combined before the application of a MIMO zero forcing (ZF) receiver. We study capacity of the proposed system, and show that the vectored CMDM processing provides higher datarates at longer looplengths. In the absence of alien crosstalk, application of the ZF receiver on the vectored CMDM signals yields performance close to the single user bound (SUB). In the presence of alien crosstalk, we show that the vectored CMDM processing exploits the spatial correlation of CM and DM signals and provides higher data rates than with DM processing only. Simulation results validate the analysis and demonstrate the importance of CMDM joint processing in vectored VDSL systems.
1 Introduction
Very highspeed digital subscriber line (VDSL) systems optimize the use of available bandwidth of the telephone lines for broadband transmissions [1]. This advanced technology can support the deployment of "triple play services" by providing high data rates (potentially upto 100 Mbps) through copper wires. Twisted pairs (TPs) run from the central office (CO) to the respective customer premise equipments (CPEs) in order to provide dedicated data rates for each user. The use of the higher bandwidth frequencies (upto 30 MHz) generates electromagnetic couplings that cause significant crosstalk among TPs, which degrades the performance of VDSL systems. The crosstalk that originates from the subscribers enjoying similar DSL service in a coordinated system is referred to as selfcrosstalk [2]. In addition to the selfcrosstalk, alien crosstalk that arises from other sources (for example, radio ingress) and noncoordinated users (enjoying other DSL services) severely limits the rates obtained by the VDSL systems [3]. Differentialmode (DM) processing that uses voltage differences between the two wires of the TP for the signal transmission and reception is a conventional technique to reduce the interferences, including radio ingress. In spite of this, crosstalk is the major performance limiting factor for the highbandwidth VDSL systems. Another factor limiting data rates is the significant attenuation with large loop lengths, resulting in marginal data rates even when all upstream bands are used. We demonstrate in this article that use of commonmode (CM) signals can result in significant enhancement in rates in VDSL systems as compared to systems using DMonly processing. In addition, it can help deal with alien crosstalk. The additional CM signal can be obtained by using the arithmetic mean of the voltages, measured between each wire of the TP and ground [4]. It is shown in [4] that this CM signal can be readily obtained with very little extra hardware. The CM channel can also be realized by using an alternative reference point like binder sheath [5] or one of the wires in the binder [6].
It has been shown in [7] that coordinated processing of the DM signals in all TPs (referred to as vectoring) enables cancelation of the farend crosstalk and hence results in enormous rate increase in the DSL system. As is well known, the use of discrete multitone (DMT) modulation in the VDSL system makes tonewise processing of the received signals possible. In vectored systems with N TPs, this results in an N × N multiinput multioutput (MIMO) system at each of the 4096 tones used. In [7], it was shown that a decision feedback equalizer (DFE) based receiver can result in rates close to the single user bound (SUB). Subsequently, it was established in [8] that a simple zero forcing (ZF) receiver at each tone can help achieve rates very close to the SUB. Recently, an iterative receiver based on the SAGE algorithm has been proposed in [9] to eliminate the channel matrix inversion requirement of the ZF receiver while achieving nearoptimal performance. When alien crosstalk is present, it is clear that various linear and nonlinear receivers can be readily applied after a prewhitening filter (see [9, 10], and references therein). It is noted that noise is spatially correlated across TPs due to the introduction of crosstalk from an alien source [3]. When the ZFbased receiver [8] is applied after noise whitening, it suffers from a significant performance loss due to the illconditioning of the channel after the whitening process as discussed in [11]. Chen et al. [10] have shown that a nonlinear DFEZF can achieve data rate higher than a ZF receiver, though with an increase in computational complexity. In our previous study [9], we have shown that the SAGE based iterative receiver can effectively cancel the alien crosstalk to achieve a higher rate (with a little extra complexity). The MMSE based receivers have been proposed in [11, 12] as a tradeoff between rate and complexity by eliminating the use of a whitening filter.
Computational complexity and power consumption remain the major implementation hurdles in vectored DSL systems. Linear ZF receiver requires $\mathcal{O}\left({N}^{2}\right)$ realtime computation at each of the 4096 tones. Besides this, some offline computation is required to update the inverse matrix at each tone. This is very high for realtime implementation with restrictions in computational complexity and power consumption. Several partial cancelers have been suggested [13, 14] that selectively cancel crosstalk and tradeoff rate for reduced complexity.
As noted already, the CM channel provides an additional mode for transmission and reception of signals through the copper wires. The CM signals at the CO are a result of leakage of transmitted DM signals due to imbalances in the TPs. Clearly, the CM signals are highly correlated with the DM signals [15], and consist primarily of the crosstalk (from the neighboring wires) and external noises. This suggests that the CM signals can be used as a reference for estimation of the interferences. For this reason, the CM signals have been utilized to mitigate radio frequency interference (RFI) ingress (see [16] and references therein). Yeap et al. [17] and KamkarParsi et al. [18] have proposed CMbased adaptive cancelers to mitigate a single wideband noise in downstream transmissions. Pierrugues et al. [19] have used a CM signal that is orthogonal to the transmitted signal to mitigate a single noise source at the CPE. The proposed scheme is effective since the correlation between CM and DM noise is close to unity. By considering a singleTP DSL system, an algorithm based on a weighted combination of CM and DM signals has been presented in [20] to cancel a single disturber. The authors have provided extensive simulations and analytical results to show that the proposed algorithm is effective when adapted in the absence of the transmitted signal. These applications of CM signal at the receiver side for interference cancelation can be translated into improved capacity in DSL systems. Explicitly, capacity analysis using CMaided reception has been discussed in [4, 21, 22]. It has been shown in [4] by cable measurements, and through channel capacity expressions that the CM signals can provide significant capacity improvements, especially at high signaltonoise ratio (SNR) levels, and low CM noise power. However, the capacity drops significantly with an increase in the number of disturbers [21]. This observation has been made rigorous in [22]. Specifically, Magesacher et al. [22] have investigated the capacity of the CMaided reception by considering a single DSL user. They have shown that the exploitation of CM signal can result in doubling of capacity than that possible by using DM signals alone. Note that these works have focused on cancelation of a single disturber only, by considering a nonvectored DSL system. To further enhance the performance by canceling a large number of disturbers, CMaided receive processing combined with vectored transmissions is well motivated for the considered system [22]. Recently, transmission of signals in the CM channel along with the coreception of CM signals has also been studied in [5].
In this article, we propose a CMDM system model by vectoring the coreceived CM and DM signals for upstream transmissions at the CO. We first employ a simple low complexity receiver which linearly combines the CM and DM signals on each TP (so as to maximize SNR for each user) and then use the signallevel coordination for the resultant signals. Since the size of the resultant vectored model remains equal to that of the DM model, the crosstalk cancelation algorithms developed for the DM system can be readily applied on the proposed CMDM system model. To assess performance of the existing crosstalk cancelers on the proposed system model, we study characteristics of the equivalent CMDM channel at each tone. To do so, we model (due to nonavailability of an analytical model for the CM channel) a CM vectored channel by extending the recently proposed singleTP CM channel [18], and modify it to include the statistical dispersion (using the approach for DM channels in [23]). In particular, we show that the CMDM channel matrix is more columnwise diagonally dominant (CWDD) than the DM channel at high frequency and/or longer looplength. This fact is useful for development of efficient algorithms for crosstalk cancelation. We also obtain the SUB of the proposed system, and show that the CMDM processing increases reach of the VDSL service by providing higher rates at longer looplengths. This advantage is accrued because of the fact that attenuation in the CM channel is smaller than the attenuation in the DM channel [18]. We extend the ZF receiver of the DM system for the proposed system, derive a simple lower bound on the capacity, and show that the SUB of CMDM system is achieved by the ZF receiver. Finally, we use CMDM processing to exploit high correlation of CM and DM signals for the mitigation of alien crosstalk. We derive an upper bound on the SUB after noise whitening and show that higher datarate is attained with the considered system. Computer simulations are presented to further demonstrate the datarate performance of the vectored CMDM processing in each of the upstream frequency bands.
The remainder of this article is organized as follows. In Section 2, we describe the DSL system model. In Section 3, we discuss a model for the CM vectored channel. We present the issue of CMDM combining in Section 4, and elaborate on its applications to the vectored CMDM processing in Section 5. The CM signals are exploited for the alien crosstalk cancelation in Section 6. Section 7 presents computer simulations. Finally, Section 8 provides the main conclusions.
Notation: Superscripts ^{(d)}and ^{(c)}indicate the DM and CM signals/values, respectively. Vectors and matrices are boldface small and capital letters, respectively. A(:,i) denotes the i th column of the matrix A and A_{ ij }denotes the ij th element of matrix A. Further, diag (A) denotes a diagonal matrix consisting of the diagonal elements of A. We use ≜ to denote equality by definition. The operators tr(·),  · , and det(·) represents trace, absolute value, and determinant, respectively, while (·)*, (·)^{T}, and (·)^{†} denote the conjugate, transpose, and conjugate transpose, respectively. p(x,y) denotes the joint density function of random variable X and Y.
2 System model
We consider N VDSL users in a binder whose receivermodems are colocated at the CO. The DMT modulation used for signal transmission partitions the channel into a number of parallel subchannels, called "tones". With transmission of only DM signal over each TP, both CM and DM signals exist at the CO. The CM counterpart actually arises due to the leakage from the original DM signal transmission via differential to commonmode conversion. We assume synchronized transmissions, and that the impulse response length of channel is not more than one longer than the cyclic prefix length. Since each DMT tone represents an independent subchannel, the system model presented herein is applicable to any tone. Hereafter, we omit the tone index for brevity. Denoting the symbol transmitted by the i th TP by X_{ i }, the received DM and CM signals ${Y}_{i}^{\left(d\right)}$ and ${Y}_{i}^{\left(c\right)}$ are given by:
where ${V}_{i}^{\left(d\right)}$ is a component of the DM additive noise that includes the alien and thermal noise at the DM port. Similarly, ${V}_{i}^{\left(c\right)}$ is a component of the CM additive noise that includes the alien and thermal noise at the CM port. ${H}_{ij}^{\left(d\right)}$ and ${H}_{ij}^{\left(c\right)}$ denote the crosstalk coupling coefficients from the j th to the i th TP.
Our model in (1) and (2) is a general case of that presented in [19]. By considering downstream transmissions, Pierrugues et al. [19] have assumed that the CM signal is strongly correlated with external noise but orthogonal to the transmitted signal. As such, the CM signal receives the alien noise only and rejects (X_{ i }= 0 in (2)) the useful signal. It is also noted that the system of Equations (1) and (2) is a special case of that in [5, (1), (2)] where the CM channel using the bindersheath has also been used for signal transmissions. However, in our treatment here we focus on CM reception only, and avoid consideration of transmission over the CM channel. While this concept certainly holds interesting possibilities, we believe that transmission over the CM channel is still associated with a number of difficulties, and would require further maturing of associated technologies, issues and standards before their serious induction and deployment. We also note that exploiting reception of the CM signals in the upstream requires changes only in the equipment at the CO, and not at the CPE.
As we are extracting the DM and CM signals at each TP, the vectoring concept used previously for only DM signals [7] can be applied for the CM signals as well. Therefore, the signallevel coordination of N DM and N CM received data is used to obtain individual vectored models for DM and CM system as
where ${\mathbf{y}}^{\left(d\right)}={\left[{Y}_{1}^{\left(d\right)},{Y}_{2}^{\left(d\right)},...,{Y}_{N}^{\left(d\right)}\right]}^{T},{\mathbf{y}}^{\left(c\right)}={\left[{Y}_{1}^{\left(c\right)},{Y}_{2}^{\left(c\right)},...,{Y}_{N}^{\left(c\right)}\right]}^{T},\mathbf{x}={\left[{X}_{1},{X}_{2},...,{X}_{N}\right]}^{T},\phantom{\rule{2.77695pt}{0ex}}{\mathbf{v}}^{\left(d\right)}={\left[{V}_{1}^{\left(d\right)},{V}_{2}^{\left(d\right)},...,{V}_{N}^{\left(d\right)}\right]}^{T}$, and ${\mathbf{v}}^{\left(c\right)}={\left[{V}_{1}^{\left(c\right)},{V}_{2}^{\left(c\right)},...,{V}_{N}^{\left(c\right)}\right]}^{T}$ are Ndimensional vectors of received (DM and CM), transmitted, and noise samples (DM and CM), respectively. H^{(d)}and H^{(c)}are N × N matrices with ij th element defined as ${H}_{ij}^{\left(d\right)}={\left[{\mathbf{H}}^{\left(d\right)}\right]}_{ij}$, and ${H}_{ij}^{\left(c\right)}={\left[{\mathbf{H}}^{\left(c\right)}\right]}_{ij}$. In the following section, we formulate a CM channel model H^{(c)}and study its CWDD property. We later use this to study properties of the combined CMDM channel.
3 Commonmode channel model
The DM channel model is well established (verified through measurements) and widely adopted in literature (see [23–25], and references therein). In particular, a stochastic MIMO channel model based on Gaussian distribution is proposed in [23] that extends the conventional 99% worst case model [2]. Similarly, crosstalk models based on beta distribution [24] and sumofsinusoids [25] have been suggested to characterize the crosstalk accurately. However, an analytical model for the CM channel is not well studied in the literature. In [6], a splitpair signaling (based on the multiconductor transmissionline theory) that offers 2N  1 modes of transmissions in a binder of N pairs is presented by considering one of the wires as a joint common reference for all other wires. Although this configuration can provide significant improvement in data rate, the presence of ingress or egress limits the potential use of splitpair (usually not twisted) signalling. A CM channel model has also been suggested in [5] by considering the metallicsheath of the binder as a common reference. This configuration (of 2N modes) is more reliable than the splitpair signalling and significantly reduces the effect of ingress and egress on the binder provided the metallic sheath is grounded properly. A CM channel has also been developed using the shield of the binder as a reference point [26]. In [27], achievable throughput using this model has been investigated in the presence of radio ingress. It is noted that no experimental data are available for the transfer functions of any of the CM channel suggested so far.
In this study, we use earth as a common reference and develop a MIMO CM channel model by incorporating the MIMO DM stochastic channel model of [23], and the singlepair model of CM channel proposed in [18]. As discussed in [23], the stochastic DM channel H^{(d)}is given by
where f is the frequency in Hz, l is the looplength in meters, ${\mathbf{H}}_{\mathsf{\text{diag}}}^{\left(d\right)}=\mathsf{\text{diag}}\left({\mathbf{H}}^{\left(d\right)}\right)$ is a diagonal matrix of the DM channel H^{(d)}, I is the N × N identity matrix, and C^{(d)}is a zerodiagonal matrix that reflects the crosstalk coupling dispersion (and extend the 99% crosstalk model) as per the existing standard, including the phase. The phase of elements of the matrix C^{(d)}is assumed to be random, and uniformly distributed [23]. The dispersion (excluding the phase) is modeled by lognormal random variables as
where X(f) (that models the magnitude variation) is a Gaussian random variable in dB with mean μ_{dB} = 2.33 σ_{dB} and variance σ_{dB}. The model presented in [23] is found to perform well for a large range of crosstalk situations. Particularly, this model is valid as long as μ_{dB} = 2.33 σ_{dB}, which is typically observed at lower frequencies. However, for higher frequencies, this pair of μ_{dB} and σ_{dB} values, tends to overestimate the crosstalk. For higher frequencies, the applicable values of these quantities have been extensively measured upto 30 MHz (and were made available to us by Conexant Systems Inc.), which have been used in the simulations/numerical values presented in Section 7.
Using the physical model of the singlepair CM channel presented in [18] along with the DM to CM leakage factor B(f) of [28], we develop a CM stochastic MIMO channel model. Specifically, we model H^{(c)}as:
where ${\mathbf{H}}_{\mathsf{\text{diag}}}^{\left(c\right)}=\mathsf{\text{diag}}\left({\mathbf{H}}^{\left(c\right)}\right)$ is the diagonal matrix whose ii th element is given as ${H}_{ii,\mathsf{\text{diag}}}^{\left(c\right)}={H}_{\mathsf{\text{line}}}^{\left(c\right)}/B\left(f\right)$. Here, ${H}_{\mathsf{\text{line}}}^{\left(c\right)}$ is the line insertionloss transfer function of the CM channel which can be obtained in terms of its twoport RLCG parameters. The RLCG parameters of CM channel listed in [18] are used. The CM dispersion factor C^{(c)}can be expressed as ${\mathbf{C}}^{\left(c\right)}=B\left(f\right){H}_{ii}^{\left(d\right)}{\mathbf{C}}^{\left(d\right)}$. The frequencydependent leakage factor B(f) is an unbalance function [28] while ϕ(f) is the differential to common phase function [18]. These are defined as
and
Unbalance function B(f) of lines is usually a function of frequency and determines the conversion of signals from differentialmode to commonmode. ϕ(f) compensates the slower line propagation of CM compared with that of the DM. It is observed from (6) that the coupling and direct path components H^{(c)}can be obtained from the corresponding components of DM channel H^{(d)}by using the leakage factor B(f).
In the frequency domain, it has been shown that the DM channel exhibits a strong CWDD property [7, 8]. This means that the crosstalk channel ${H}_{ji}^{\left(d\right)}$ from a disturber i into a victim j is always weaker than the direct channel ${H}_{ii}^{\left(d\right)}$. The degree of CWDD can be characterized by the parameter ${\alpha}^{(d)}$ of the DM channel:
This CWDD property ${\alpha}^{(d)}$ is a function of the loop length and frequency. The characteristics of the DM channel matrix are well documented, and utilized by most of the crosstalk cancelation algorithms proposed to date [7–9]. Similarly, we define the CWDD parameter ${\alpha}^{(c)}$ of the CM channel as
Using (4), and substituting ${\mathbf{C}}_{ji}^{\left(c\right)}=B\left(f\right){H}_{ii}^{\left(d\right)}{\mathbf{C}}_{ji}^{\left(d\right)}$ in (6), we can find the ratio η^{(c/d)}of CM to DM CWDD parameters as
We observe from (11) that the CWDD characteristic of the CM channel is also dependent on frequency and loop length. In particular, the CM channel is more diagonally dominant than the DM channel at high frequencies and/or longer looplengths. This is due to the fact that both B(f) and $\left{H}_{ii}^{\left(d\right)}\right$ decrease with frequency while the latter also decreases with an increase in the loop length.
A proper understanding of the CWDD characteristic of the CM channel can assist in understanding the potential gains due to joint CMDM processing. To this end, we performed computer simulations to investigate the behavior of CWDD characteristic of the CM channel, and compare it with that of the DM channel. The ratios of CWDD parameters of both channels in three upstream bandwidths over various loop lengths are plotted in Figure 1. The frequency ranges of 3.75 to 5.2 MHz, 8.5 to 12 MHz, and 18.1 to 30 MHz are referred to as a lowfrequency band (LFB), midfrequency band (MFB) and highfrequency band (HFB), respectively. Unlike in all previous studies, we also study characteristics of the channel in the HFB, where the characteristics are quite different from those of the LFB and MFB. Figure 1a shows that the CM channel is not as CWDD as the DM channel in the LFB for shorter loop lengths. However, the CWDD property improves with an increase in frequency and loop length as shown in Figure 1b,c. The main conclusion of these studies is that the CM channel is more CWDD than the DM channel at higher frequencies and/or longer looplengths, a fact which is consistent with the analytical expression in (11). In the HFB, the CWDD nature of the DM matrix is quite weak as ${\alpha}^{(d)}$ increases with frequency and loop length. However, the variation of ${\alpha}^{(c)}$ with frequency and loop length is opposite to that of ${\alpha}^{(d)}$. Thus, the CM MIMO channel is well conditioned (in as far as the CWDD property is concerned) at a higher bandwidth and/or longer looplength (as compared to DM MIMO channel), which motivates the use of CM signals along with DM signals for crosstalk cancelation.
4 SingleTP CMDM processing
In this section, we employ CMDM joint processing by linearly combining the CM and DM signals at each TP to improve the SNR of each user while treating all the crosstalk as interference. Vectored crosstalk cancelation scheme based on such a processing is studied in the following section. We first assume the presence of white noise only, while spatially correlated noise due to the alien crosstalk is discussed in Section 6.
To facilitate interference cancelation with a linear combination of the CM and DM signals, we obtain signals after frequencydomain equalization (FEQ) (${Z}_{i}^{\left(d\right)}={Y}_{i}^{\left(d\right)}/{H}_{ii}^{\left(d\right)}$ and ${Z}_{i}^{\left(c\right)}={Y}_{i}^{\left(c\right)}/{H}_{ii}^{\left(c\right)}$) as :
where ${V}_{i}^{\prime \left(d\right)}={V}_{i}^{\left(d\right)}/{H}_{ii}^{\left(d\right)}$, and ${V}_{i}^{\prime \left(c\right)}={V}_{i}^{\left(c\right)}/{H}_{ii}^{\left(c\right)}$ are noise terms after FEQ. We assume that the central limit theorem holds for the second term of each of the two equations in (12), so that it can be regarded as a circular symmetric Gaussian random variable. This is reasonable, since the far end crosstalk (FEXT) component consists of the weighted sum of independent symbols. With this, ${\sum}_{j=1,j\ne i}^{N}\frac{{H}_{ij}^{\left(d\right)}}{{H}_{ii}^{\left(d\right)}}{X}_{j}$ is Gaussian with mean zero and variance equal to ${\mathsf{\text{FEXT}}}_{i}^{\left(d\right)}={\sum}_{j=1,j\ne i}^{N}{\left\frac{{H}_{ij}^{\left(d\right)}}{{H}_{ii}^{\left(d\right)}}\right}^{2}{\sigma}_{x,j}^{2}$ giving SNR without FEXT cancelation for the i th user as ${\mathsf{\text{SNR}}}_{i,\mathsf{\text{fext}}}^{\left(d\right)}=\frac{{\sigma}_{x,i}^{2}}{{\mathsf{\text{FEXT}}}_{i}^{\left(d\right)}+{\sigma}_{{\upsilon}^{\prime},i}^{2\left(d\right)}}$. Here, ${\sigma}_{{\upsilon}^{\prime},i}^{2\left(d\right)}$ is the variance of the noise ${V}_{i}^{\prime \left(d\right)}$ after FEQ. Similarly, the FEXT component of the CM signal has zero mean and variance ${\mathsf{\text{FEXT}}}_{i}^{\left(c\right)}={\sum}_{j=1,j\ne i}^{N}{\left\frac{{H}_{ij}^{\left(c\right)}}{{H}_{ii}^{\left(c\right)}}\right}^{2}{\sigma}_{x,j}^{2}$, thus resulting in an SNR of ${\mathsf{\text{SNR}}}_{i,\mathsf{\text{fact}}}^{\left(c\right)}=\frac{{\sigma}_{x,i}^{2}}{{\mathsf{\text{FEXT}}}_{i}^{\left(c\right)}+{\sigma}_{{\upsilon}^{\prime},i}^{2\left(c\right)}}$. Similar to the DM system, ${\sigma}_{{\upsilon}^{\prime},i}^{2\left(c\right)}$ is the variance of the noise ${V}_{i}^{\prime \left(c\right)}$ after FEQ. It is noted that in the presence of a large number of disturbers with high power spectral densities (PSD), ${\mathsf{\text{SNR}}}_{i,\mathsf{\text{fact}}}^{\left(d\right)}$ is quite low, necessitating the cancelation of crosstalk. When the FEXT is canceled, crosstalkfree SNR (bit loading is based on this quantity) ${S}_{i}^{\left(d\right)}=\frac{{\sigma}_{x,i}^{2}}{{\sigma}_{{\upsilon}^{\prime},i}^{2\left(d\right)}}$ can be obtained. As stated earlier, we use the CM signals for the cancelation of interferences to improve the performance of the DM channel.
Define 1 = [1, 1]^{T}. With this, the ML estimate ${\widehat{X}}_{i}$ can be obtained by maximization of the likelihood function $p\left({Z}_{i}^{\left(d\right)},{Z}_{i}^{\left(c\right)}/{X}_{i}\right)$ as:
We can obtain a noise covariance matrix R_{ i }of ${\mathbf{z}}_{i}^{\left(cd\right)}={\left[{Z}_{i}^{\left(d\right)},{Z}_{i}^{\left(c\right)}\right]}^{T}$ as
We note that the estimator in (13) bases its estimate on a weighted sum of the CM and DM signals. Therefore, we simplify (13) to get
where the combined signal ${Z}_{i}^{\left(cd\right)}$ is given by:
and coefficient K_{ i }can be obtained as
By assuming equal transmit power ${\sigma}_{x,j}^{2}={\sigma}_{x,i}^{2}$ and using (14) in (17), we can represent K_{ i }as
The exact value of K_{ i }can readily be determined with channel knowledge. However, to obtain an understanding of the SNR and CWDD behavior of the combined signal, an approximated value of K_{ i }can be obtained. The last term ${\sum}_{j\ne i}\frac{{H}_{ij}^{\left(d\right)}{H}_{ij}^{*\left(c\right)}}{{H}_{ii}^{\left(d\right)}{H}_{ii}^{*\left(c\right)}}$ in the numerator (and a similar term in the denominator) of (18) is very small for large N because of small magnitudes (due to CWDD property) and the independent phase factors in $\frac{{H}_{ij}^{\left(d\right)}}{{H}_{ii}^{\left(d\right)}}$ and $\frac{{H}_{ij}^{\left(c\right)}}{{H}_{ii}^{\left(c\right)}}$. Thus, their influence on K_{ i }can be ignored to get
It can be shown that the SNR after the CMDM combining using K_{ i }as implied by (13) is always greater than the SNR that can be obtained using the DM signals alone. To do so we define: ${e}_{i}^{\left(d\right)}={\sum}_{j=1,j\ne i}^{N}\frac{{H}_{ij}^{\left(d\right)}}{{H}_{ii}^{\left(d\right)}}{X}_{j}+{V}_{i}^{\prime \left(d\right)}$ (the sum of FEXT and noise in the DM signal after FEQ), ${e}_{i}^{\left(c\right)}={\sum}_{j=1,j\ne i}^{N}\frac{{H}_{ij}^{\left(c\right)}}{{H}_{ii}^{\left(c\right)}}{X}_{j}+{V}_{i}^{\prime \left(c\right)}$ (the sum of FEXT and noise in the CM signal after FEQ), and ${\mathbf{e}}_{i}^{\left(cd\right)}={\left[{e}_{i}^{\left(d\right)},{e}_{i}^{\left(c\right)}\right]}^{T}$. With this, we can rewrite (12) as:
Now consider a Minimum Output Energy (MOE) type of receiver that takes the weighted sum of the components of ${\mathbf{z}}_{i}^{\left(cd\right)}$ such that ${\mathbf{w}}_{i}^{H}1=1$ while minimizing the output energy $E\left\{{\left{\mathbf{w}}_{i}^{H}{\mathbf{z}}_{i}^{\left(cd\right)}\right}^{2}\right\}$. It can be shown in a straightforward manner that such a receiver is again given by (13). Since such receivers are known to have SNR maximization property, it is clear that the weighted sum of CM and DM signals implied by (13) results in an SNR that is higher than when using only DM signals. We now get an expression of SNR using joint CMDM processing for the i th user as
where ${H}_{ii}^{\left(cd\right)}={H}_{ii}^{\left(d\right)}+{K}_{i}{H}_{ii}^{\left(c\right)},\phantom{\rule{2.77695pt}{0ex}}{H}_{ij}^{\left(cd\right)}={H}_{ij}^{\left(d\right)}+{K}_{i}{H}_{ij}^{\left(c\right)}$ and ${\sigma}_{\upsilon ,i}^{2\left(cd\right)}$ is the variance of ${V}_{i}^{\left(d\right)}+{K}_{i}{V}_{i}^{\left(c\right)}$. Simulation results plotted (for a frequency tone of 12 MHz, loop length of 500 m, and 10 users) in Figure 2 clearly shows the increase in SNR due to such CMDM combining at each TP. We can find the capacity ${\mathcal{R}}_{i}^{\left(cd\right)}$ of the singleTP CMDM processing with SNRgap to capacity Γ as
where ${\overline{\mathbf{h}}}_{i}^{\left(cd\right)}={\left[{H}_{ii}^{\left(d\right)},{H}_{ii}^{\left(c\right)}\right]}^{T}$, and ${\overline{\mathbf{R}}}_{i}$ is the covariance matrix of total noise (white noise and FEXT) before FEQ. We can deduce from Figure 2 that the joint singleTP CMDM processing achieves capacity close to the theoretical capacity in the presence of a few crosstalker only. Similar observations have been made in [20, 22]. This motivates the use of vectoring of the CM signals for the cancelation of a number of disturbers.
5 Vectored CMDM processing
As such, a joint signallevel coordination among TPs can be realized by concatenating ${\alpha}^{(d)}$ the vectored DM and CM signals of (3) as:
where is 2N × 1 received vector, is a 2N × N matrix of channel coefficients, and is the vector of noise samples. To show the benefit of CMaided processing, we obtain the SUB of the 2Ndimensional vectored CMDM model of (23). SUB is the capacity achieved when a single user transmits and all the receivers of the vectored system are used to detect the transmitted signal via multipaths, i.e., based on the signals received on all the TP's (direct line and crosstalk paths) [8]. The SUB of the concatenated vectored CM and DM signals can be expressed as
where ${\gamma}_{i,\upsilon}^{2\left(cd\right)}={\sigma}_{i,\upsilon}^{2\left(c\right)}/{\sigma}_{i,\upsilon}^{2\left(d\right)}$ is the ratio of CM and DM noise variances. This ratio is generally greater than unity as CM port is usually more noisy than the DM port. We define ${\beta}_{i}^{\left(cd\right)}=\left\frac{{H}_{ii}^{\left(c\right)}}{{H}_{ii}^{\left(d\right)}}\right$ as the ratio of the frequency response of diagonal channel coefficients of the CM and DM channels. This ratio is greater than one when $\left{H}_{ii}^{\left(c\right)}\right>\left{H}_{ii}^{\left(d\right)}\right$, which holds at high frequency tones and/or longer loop lengths. This has also been observed in [5] for the CM channel using the binder sheath.
We use (24) and CWDD parameters of the CM and DM channels, to obtain an upper bound on the SUB of the 2Ndimensional CMDM system as
From (25), we can express the SUB with only DM processing as [8]
From (25) and (26), we can see that when $\frac{{\beta}_{i}^{2\left(cd\right)}}{{\gamma}_{i,\upsilon}^{2\left(cd\right)}}$ is high (which occurs at high frequency and/or longer loop lengths as noted earlier). This shows that a substantial rate enhancement may be realized with the vectored CMDM processing at longer looplengths.
Although the above mentioned model can effectively improve the rate performance, it imposes additional complexity in implementing the existing (ZF or DFE) receivers, since the size of the effective channel matrix is large (2N × N). As discussed earlier, the vectored model in [5] has the resultant channel matrix of size 2N × 2N and thus increases the complexity further for the implementation of equalizers. Therefore, instead of the step involved in (23), we formulate a new Ndimensional CMDM system model by exploiting (16) for each TP and vectorizing the resultant signals as
where ij th element of H^{(cd)}is ${H}_{ij}^{\left(d\right)}+{K}_{i}{H}_{ij}^{\left(c\right)}$ and the i th sample of v^{(cd)}is given by ${V}_{i}^{\left(d\right)}+{K}_{i}{V}_{i}^{\left(c\right)}$. With such a vectoring approach, the size of effective channel H^{(cd)}remains the same and thereby receivercomplexity for the CMDM system is comparable to that of the conventional, Ndimensional vector DM processing. Note that there is only one signal at each TP, so that the vectored channel matrix again becomes of size N × N at each tone. In fact, only $\mathcal{O}\left(N\right)$ multiplications are required per tone for the operation of linear combinations of the CM and DM signals. Hereafter we refer to this N × N equivalent channel as the Ndimensional CMDM vectored channel at each tone. Studying this lower dimensional CMDM joint processing in the vectoring context is important since the CMDM combining at the TP influences the channel matrix (the CWDD parameter and the SNR), and hence the attainable rate with vectoring. It is important to understand the structure of this matrix well so that performance of the standard crosstalk cancelers on the equivalent signal after the TP level CMDM combining can be assessed. We show later that the application of ZF canceler on Ndimensional CMDM vectored system results in performance close to the SUB of the same system. In what follows, we study the theoretical capacity of the considered CMDM system by using the CWDD property of the resultant channel H^{(cd)}.
5.1 CWDD parameter of the CMDM channel and SUB
In this subsection, we show that the new Ndimensional vectored channel H^{(cd)}in (27) using the combined CMDM signals is more strongly CWDD than the channel matrix with only vectored DM signals, especially at higher frequencytones and/or longer looplengths. The CWDD parameter (denoted by ${\alpha}^{(cd)}$ of the resultant channel H^{(cd)}can be upper bounded in terms of CWDD parameters (${\alpha}^{(d)}$ and ${\alpha}^{(c)}$) of the individual channel as
Using ${\eta}^{(c/d)}={\alpha}^{(c)}/{\alpha}^{(d)}$ in (28), we can find an upper bound on the ratio ${\eta}^{(cd/d)}={\alpha}^{(cd)}/{\alpha}^{(d)}$ of CMDM to DM CWDD parameters as
It is obvious from (29) that η^{(cd/d)}< 1 if η^{(c/d)}< 1. Hence, we can say that if ${\alpha}^{(c)}<{\alpha}^{(d)}$, then the vectored CMDM channel obtained after combining the CM and DM signals at each TP has a CWDD parameter that is smaller (that is, ${\alpha}^{(cd)}<{\alpha}^{(d)}$). We have already illustrated in Figure 1 that the condition η^{(c/d)}< 1 is usually satisfied at higher frequencies and/or longer looplengths. We also plot the ratio of the CWDD parameters of CMDM and DM channel in Figure 3. It can be seen from this figure that the resultant channel H^{(cd)}is more CWDD at higher frequencies and/or longer looplengths. We now find an upper bound on the SUB for the CMDM processing as
where ${\mathbf{h}}_{i}^{\left(cd\right)}$ denotes the i th column of H^{(cd)}, ${S}_{i}^{\left(cd\right)}=\frac{{\sigma}_{x,i}^{2}}{{\sigma}_{{\upsilon}^{\prime},i}^{2\left(cd\right)}}$, and ${\sigma}_{{\upsilon}^{\prime},i}^{2\left(cd\right)}=\frac{{\sigma}_{\upsilon ,i}^{2\left(cd\right)}}{{\left{H}_{ii}^{\left(cd\right)}\right}^{2}}$ denote the crosstalkfree SNR and noise after FEQ for the CMDM system, respectively.
The values of the rates achievable in these cases of concatenated CMDM, the N dimensional vectored CMDM and the vectored DM, are taken up later in Section 7, dealing with simulations and numerical results. It will be shown there that while the concatenated system produces the best results in all situations, the rate performance of the lower complexity Ndimensional CMDM processing is nearly identical to that of the concatenated system at larger frequencies or loop lengths. Next, we describe the proposed CMDM processing with the ZF receiver in order to achieve the SUB.
5.2 ZF receiver for CMDM processing
The ZF receiver (that performs close to the SUB) is a near optimal canceler for the vectored DM processing because of limited noise enhancement in well conditioned DSL channels [8]. Since CWDD property worsens with frequency and loop length, there may be a gap between the rate performance of ZF receiver and the SUB rate at higher frequency tones and/or longer looplengths. It is shown in the previous subsection that the Ndimensional CMDM processing offers superior CWDD characteristics than the corresponding DM processing, in particular at higher frequencies and/or longer looplengths. Hence, the Ndimensional CMDM processing can provide better rate performance for the crosstalk cancelation with the ZF receiver. On applying the ZF receiver to the Ndimensional CMDM system, we obtain the data rate as
where ${\psi}_{i}^{\left(cd\right),\mathsf{\text{zf}}}$ is the noise enhancement factor defined as [8]
and A_{ N }denotes an N × N matrix whose diagonal elements are 1 and magnitude of nondiagonal elements is upper bounded by ${\alpha}^{(cd)}$. Also, the structure of the matrix B_{ N }is same as that of A_{ N }except that the magnitude of the last diagonal element is also upper bounded by ${\alpha}^{(cd)}$. The reader is referred to [8] for details. We now derive an upper bound on ${\psi}_{i}^{\left(cd\right),\mathsf{\text{zf}}}$ in the following Lemma 1 for the case of a ZF receiver applied on the vectored CMDM combined signal.
Lemma 1 If$(N1){\alpha}^{(cd)}<1$, the noise enhancement factor of ZF receiver with CMDM processing can be upper bounded by
Proof By applying Hadamard's inequality as $\left\text{det}\left({\mathbf{A}}_{N1}\right)\right\le {\left(\prod _{i=1}^{N1}\sum _{j=1}^{N1}{\left{a}_{ij}\right}^{2}\right)}^{\frac{1}{2}}$, we can obtain upper bounds on det(A_{N1})^{2} and det(B_{N1})^{2} as follows:
Also by applying Ostrowski's inequality as $\left\text{det}\left({\mathbf{A}}_{N}\right)\right\ge \prod _{i=1}^{N}\left[\left{a}_{ii}\right\sum _{j=1,j\ne i}^{N}\left{a}_{ij}\right\right]$, we can lower bound det(A_{ N })^{2} as
For the lower bound in (35) to hold, it is required that ${\sum}_{j=1,j\ne i}^{N}\left{a}_{ij}\right<1$, which in turn can be expressed as
The condition $(N1){\alpha}^{(cd)}<1$ is readily satisfied for typical values of N since ${\alpha}^{(cd)}$ normally takes a very small value. By substituting (34) and (35) in (32), we obtain
The condition in (36) together with (37) completes the proof of Lemma 1.
It has been shown in [8] that noise enhancement factor of the ZF receiver for the DM system is close to unity due to the CWDD property of the DM channel. Since the Ndimensional CMDM vectored channel possesses stronger CWDD characteristic than the DM channel, ${\psi}_{i}^{\left(cd\right),\mathsf{\text{zf}}}$ is also close to unity. This implies that the ZF receiver can achieve performance close to the SUB of this lowerdimensional CMDM system. Cendrillon et al. [8] have also presented an upper bound on the noise enhancement of the ZF canceler for the DM system. However, the upper bound in [8] is conditioned to exist for a limited number of users (N ≤ 25) and bandwidth (upto 12 MHz), whereas the upper bound in (33) is applicable with the condition $(N1){\alpha}^{(cd)}<1$, which is typical for a large number of users and frequencies. Hence, the derived bound presented here can also be used to predict accurately the performance of highbandwidth VDSL2 (newest standard of xDSL) system. A lower bound on ${\psi}_{i}^{\left(cd\right),\mathsf{\text{zf}}}$ is also presented in Appendix 1.
The ZF receiver requires channel matrix inversion at each tone which causes an increase in its computational complexity. Such matrix inversions are required frequently due to changes in user status, or variations in crosstalk characteristics [14]. To avoid this, a loworder truncated series approximation of the inverse channel matrix was considered for the DM channel in the downstream transmissions [29]. We extend this concept for the vectored CMDM processing in the upstream transmissions by applying a first order approximation of inverse of the combined CMDM channel as
where ${\mathbf{H}}_{\mathsf{\text{diag}}}^{\left(cd\right)}=\mathsf{\text{diag}}\left({\mathbf{H}}^{\left(cd\right)}\right)$ is a diagonal matrix of the CMDM channel and C^{(cd)}is a zerodiagonal matrix whose ij th element denotes the normalized FEXT coupling coefficients from j th disturber to the i th victim. Here, the normalization refers to the division of each matrix row by its associated diagonal entry. The point to be noted is that this algorithm, referred to as approximate ZF (AZF), does not involve inversion of matrix H^{(cd)}but requires the inversion of the diagonal matrix ${\mathbf{H}}_{\mathsf{\text{diag}}}^{\left(cd\right)}$ only. By using the approximate inverse (38) as a canceler, we find the estimate of i th symbol as
By considering the worst case by assuming all nondiagonal elements of the matrix C^{(cd)}as ${\alpha}^{(cd)}$ and assuming equal transmit PSD and thermal noise, we simplify (39) to find an upper bound on noise enhancement of the AZF receiver for the i th user as
Note that with all combined CMDM values replaced by DM channel values, (40) also holds for the vectored DM case. It can be seen from (40) that the AZF receiver can perform better for the CMDM system than the DM signals at higher frequencies and/or longer looplengths (since ${\alpha}^{(cd)}\le {\alpha}^{(d)}$ and ${\left{H}_{ii}^{\left(cd\right)}\right}^{2}\ge {\left{H}_{ii}^{\left(d\right)}\right}^{2}$). Thus, the rates of the ZF receiver with the vectored CMDM processing will clearly be higher than those with DM processing alone, because the effective channel matrix after combining is more strongly CWDD in the former case. In general, it can be seen that the performance of the linear receivers can be improved with the vectored CMDM processing.
6 CMDM processing for alien crosstalk cancelation
In the absence of alien crosstalk, noise at different TPs is spatially white. This assumption was used in the analysis for CMDM processing in the preceding sections. In the presence of the alien crosstalk originating from the external noise sources, noise among different TPs experiences a high spatial correlation. This fact has been emphasized in many publications [3, 10, 30, 31]. Alien crosstalk can be particularly troublesome because it originates from other transmitters or channels that are not part of the system under design. Sources for alien crosstalk are, for example, RFI ingress, and signals from other DSL services. Cancelation of alien crosstalk is more involved and different techniques can be applied for its mitigation. One way is to exploit the spatial correlation by incorporating a whitening filter prior to applying cancelation algorithms. The presence of noise correlation clearly helps in mitigating the alien crosstalk. As such, a unity correlation (perfect correlation) completely eliminates the alien noise whereas there is no improvement if noise is uncorrelated. It is known that the nonwhite component of the noise (i.e., alien noise and crosstalk coupling from adjacent pairs) in the CM channel is correlated with the corresponding noise components in the DM [15]. In case of alien noise originating from a single or small number of noise sources, the correlation between DM and CM signals is strong. This high correlation can be used effectively to mitigate the alien noise in the DM channel. In what follows, we first extend the optimal linear combination of CM and DM signals at each TP in the presence of alien crosstalk. Later, we coordinate the singleTP CMDM processing to obtain Ndimensional CMDM system and study its capacity.
6.1 SingleTP CMDM processing for alien crosstalk
In the presence of alien crosstalk, ${V}_{i}^{\prime \left(d\right)}$ and ${V}_{i}^{\prime \left(c\right)}$ of (12) also contains alien component ${A}_{i}^{\left(d\right)}$ and ${A}_{i}^{\left(c\right)}$, respectively. Therefore, we can write DM and CM signals after FEQ as
where ${V}_{a,i}^{\prime \left(d\right)}=\frac{{V}_{i}^{\left(d\right)}}{{H}_{ii}^{\left(d\right)}}+\frac{{A}_{i}^{\left(d\right)}}{{H}_{ii}^{\left(d\right)}}$ and ${V}_{a,i}^{\prime \left(c\right)}=\frac{{V}_{i}^{\left(c\right)}}{{H}_{ii}^{\left(c\right)}}+\frac{{A}_{i}^{\left(c\right)}}{{H}_{ii}^{\left(c\right)}}$. The subscript "a" is used to highlight that alien crosstalk has been considered. We denote the variance of ${A}_{i}^{\left(d\right)}$ and ${A}_{i}^{\left(c\right)}$ by ${\sigma}_{a,i}^{\left(d\right)}$ and ${\sigma}_{a,i}^{\left(c\right)}$, respectively. Now, we can represent noise covariance matrix R_{ a,i }of ${\mathbf{z}}_{a,i}^{\left(cd\right)}={\left[{Z}_{a,i}^{\left(d\right)},{Z}_{a,i}^{\left(c\right)}\right]}^{T}$ in terms of R_{ i }of (14) as
where ρ is the correlation coefficient between CM and DM alien noise of the i th TP. Similar to the white noise case, we combine CM and DM signals after FEQ in the presence of alien crosstalk to get ${Z}_{a,i}^{\left(cd\right)}={Z}_{a,i}^{\left(d\right)}+{K}_{a,i}{Z}_{a,i}^{\left(c\right)}$. By applying an approach similar to that used in (18), we can obtain coefficient K_{ i,a }as
From (17) and (43), it can be easily shown that K_{ i,a }= K_{ i }for uncorrelated noise (ρ = 0). Similar to white noise, we can show that the linear combination approach maximizes the SNR in the presence of alien crosstalk.
6.2 Capacity of the vectored CMDM processing in the presence of alien crosstalk
Similar to (27), we can coordinate signals by combining the received CM and DM signals using K_{ i,a }to obtain an Ndimensional CMDM vectored system as
where ${\mathbf{y}}_{a}^{\left(cd\right)}={\left[{Y}_{a,1}^{\left(cd\right)},{Y}_{a,2}^{\left(cd\right)},\dots ,{Y}_{a,N}^{\left(cd\right)}\right]}^{T}$ with ${Y}_{a,i}^{\left(cd\right)}={Y}_{a,i}^{\left(d\right)}+{K}_{i}{Y}_{a,i}^{\left(c\right)}$, ∀i. It can be seen that ${Y}_{a,i}^{\left(d\right)}$ and ${Y}_{a,i}^{\left(c\right)}$ are received DM and CM signals at the i th TP in the presence of alien noise. Similarly, ${\mathbf{v}}_{a}^{\left(cd\right)}=\left[{V}_{a,1}^{\left(cd\right)},{V}_{a,2}^{\left(cd\right)},\dots ,{V}_{a,N}^{\left(cd\right)}\right]$ with ${V}_{a,i}^{\left(cd\right)}={V}_{a,i}^{\left(d\right)}+{K}_{i}{V}_{a,i}^{\left(c\right)}$, ∀i. In the presence of alien crosstalk, noise term ${\mathbf{v}}_{a}^{\left(cd\right)}$ is correlated among different TPs with a covariance matrix W^{(cd)}. This noise correlation can be exploited to mitigate alien noise. To do so, we whiten the noise vector ${\mathbf{v}}_{a}^{\left(cd\right)}$ by a whitening filter W^{1} where W (= W^{(cd)1/2}) is a square root of the covariance matrix. In upstream transmissions, we can use the whitening filter W^{1} prior to the application of an equalizer. The received Ndimensional signal y^{(cd)}of (44) can be processed by the whitening filter W^{1} to yield
It can be seen that resultant noise ${\mathbf{W}}^{1}{\mathbf{v}}_{\mathbf{a}}^{\left(\mathbf{c}d\right)}$ is spatially uncorrelated with identity covariance matrix. We define an absolute operation  ·  on a matrix A with elements a_{ ij }denoted by A whose ij th element is a_{ ij }. We now take absolute value of each element of the equivalent channel ${\stackrel{\u0303}{\mathbf{H}}}^{\left(cd\right)}={\mathbf{W}}^{1}{\mathbf{H}}^{\left(\mathbf{c}d\right)}$ of (45) to get $\left{\stackrel{\u0303}{\mathbf{H}}}^{\left(cd\right)}\right\le \left{\mathbf{W}}^{1}\right\left{\mathbf{H}}^{\mathbf{c}d}\right$, where the inequality is due to use of the relation $\left{\stackrel{\u0303}{H}}_{ij}^{\left(cd\right)}\right\le \left{\mathbf{W}}^{1}\left(i,:\right)\right\left{\mathbf{H}}^{\left(cd\right)}\left(:,j\right)\right$, ∀i, j. We now use the definition of matrix inverse to find an upper bound on the ij th element of W^{1} as $\left{\left[{\mathbf{W}}^{1}\right]}_{ii}\right\le \frac{\left\text{det}\left({\overline{\mathbf{W}}}_{p}\right)\right}{\left\text{det}\left(\mathbf{W}\right)\right}\phantom{\rule{1em}{0ex}}\forall i$, and $\left{\left[{\mathbf{W}}^{1}\right]}_{ij}\right\le \frac{\left\text{det}\left({\overline{\mathbf{W}}}_{np}\right)\right}{\left\text{det}\left(\mathbf{W}\right)\right}\phantom{\rule{1em}{0ex}}\forall i\ne j$, where $\left\text{det}\left({\overline{\mathbf{W}}}_{p}\right)\right$ and $\left\text{det}\left({\overline{\mathbf{W}}}_{np}\right)\right$ are, respectively, maximum absolute values of principal and nonprincipal minors (of sizes N  1 × N  1) of the matrix W. Using upper bounds on diagonal [W^{1}]_{ ii } and nondiagonal [W^{1}]_{ ij } elements of W^{1}(i,:) in $\left{\stackrel{\u0303}{\mathbf{H}}}^{\left(cd\right)}\right\le \left{\mathbf{W}}^{1}\right\left{\mathbf{H}}^{\mathbf{c}d}\right$, we can obtain an upper bound on the diagonal and nondiagonal elements of $\left{\stackrel{\u0303}{\mathbf{H}}}^{\left(cd\right)}\right$ as
Using (46) and (47), an upper bound on CWDD parameter of the modified channel ${\stackrel{\u0303}{\mathbf{H}}}^{\left(cd\right)}$ can be obtained as
We can obtain the SUB of Ndimensional CMDM system in the presence of alien crosstalk as
where selfcrosstalkfree SNR ${\stackrel{\u0303}{S}}_{i,a}^{\left(cd\right)}$ after noise whitening can be upper bounded using (46) and ${\stackrel{\u0303}{\sigma}}_{\upsilon ,a,i}^{2\left(cd\right)}=1$ as
Since CM signals are highly correlated with the DM signals, ${\stackrel{\u0303}{S}}_{i,a}^{\left(cd\right)}$ for Ndimensional CMDM system can be significantly higher than the DM counterpart. Consequently, the Ndimensional CMDM processing gives higher data rates than the DM processing. However, it is noted that the whitening procedure destroys the CWDD characteristic $\left({\stackrel{\u0303}{\alpha}}^{\left(cd\right)}>>{\alpha}^{\left(cd\right)}\right)$ of DSL channels and thus the ZF receiver does not achieve the SUB. Nonlinear receiver ZFDFE [10] and SAGE based iterative receiver proposed in [9] are alternative candidates for the mitigation of alien crosstalk even with reduced CWDD characteristics of the DSL channels.
7 Simulation and numerical results
In this section, we investigate the performance of the vectored CMDM processing using computer simulations and numerical results based on derived bounds. We consider Annex C band plan [1] which corresponds to three upstream frequency bands. We assume 10 TPs of equal length in a binder while other simulation parameters as per the ITUT standards [1] are listed in Table 1. We adopt the stochastic DMchannel model of [23] and use the proposed CM channel model in (6) for the simulations. We simulate the capacity of the proposed vectored CMDM processing and compare with the DM processing in all the three upstream bands. For the white noise case, we demonstrate the rate performance of the ZF receiver for the Ndimensional CMDM system model considered here. We also plot the data rates to verify the upper bound on the noise enhancement of ZF receiver in (33) for the Ndimensional CMDM system in the absence of alien noise.
Figure 4 shows that concatenating the vectored CM and DM signals provides higher datarate than the DM signals alone over a broad ranges of frequencies and loop lengths. This is consistent with the observations made by Magesacher et al. [4]. We also observe that when we combine the CM and DM signals optimally before vectoring, there is always a rate improvement, though its extent depends on the loop length and frequency. While the concatenated CMDM performs best, as expected, at higher looplengths the performance of the Ndimensional CMDM system seems to be nearly identical. (This is because the CM signal dominates at higher looplengths). It is clear that there is a large gap between the capacities achieved with combined Ndimensional CMDM and DM processing, which increases significantly with increase in the bandwidth and/or loop length. For lower frequencies and shorter loop lengths, however, the Ndimensional CMDM processing achieves the same maximum data rate (SUB) as that of the DM processing. This is because the DM channel is much cleaner (less noisy) in this region. It can be further seen from Figure 4 that the ZF receiver achieves the SUB of the Ndimensional CMDM processing. We also study the influence of the higher CM noise level on the capacity gain using the CM signals at high frequency tones in Figure 5. This figure depicts that there is a significant rate gain in the HFB using the Ndimensional CMDM processing for longer looplengths even when the CM noise level is higher than the DM noise. Hence, in the absence of alien crosstalk, we can achieve higher datarates at longer looplengths.
To illustrate the effect of alien crosstalk, we consider noise correlation values of 0.99 and 0.50 and equal CM and DM alien power of 100 dBm/Hz. Figure 6 shows the capacity of the vectored Ndimensional CMDM and DM signals after whitening the correlated noise. An upper bound on the SUB (indicated in (49)) is also presented. It can be seen that the datarate performance improves with increase in noise correlation of the alien crosstalk. It can also be observed that the Ndimensional CMDM processing is of much greater importance in the presence of alien crosstalk. As such, data rates of the Ndimensional CMDM is greater than that of DM with pronounced improvements at longer looplengths and higher frequencies as shown in Figure 6.
8 Conclusion
We have investigated the use of CM signals to increase the rate and reach of the VDSL systems in upstream transmissions. A vectored system consisting of both CM and DM signals has been proposed to enable the effective cancelation of crosstalk. We have shown that simply combining the CM and DM signals optimally results in rate improvement without significant increase in complexity. Further, vectoring such combined signals enables ZF type receivers to achieve the capacity due to the increased diagonal dominance of the resultant Ndimensional CMDM channel matrix. We have shown that the rate gain with CMDM joint processing is a strong function of the loop length and frequency. In the presence of alien crosstalk, use of the CM signals has been shown to increase the data rate significantly.
Appendix 1: A lower bound on noise enhancement of zf receiver
Similar to finding an upper bound, we use (32) to get a lower bound on ${\psi}_{i}^{\left(cd\right),\mathsf{\text{zf}}}$. To facilitate this, we use Hadamard's inequality to find an upper bound on det(A_{ N })^{2} as
If $(N2){\alpha}^{(cd)}<1$, Ostrowski's inequality is applied to obtain a lower bound on det(A_{N1})^{2} as
We can use absolute value of minimum and maximum eigenvalues $\left({\lambda}_{b}^{l},{\lambda}_{b}^{u}\right)$ of the matrix B_{N}1 to find a lower bound on det B_{N1}^{2} as [32],
where
Absolute values of mean ${\mu}_{{\lambda}_{b}}=\frac{\mathsf{\text{tr}}\left({\mathbf{B}}_{N1}\right)}{N1}$ and variance ${\sigma}_{{\lambda}_{b}}^{2}$ of eigenvalues of B_{N1}are given as
We now use the result of [33] to find magnitude of minimum eigenvalue of the matrix B_{N1}as
Substituting (55) and (56) in (57), we get
We use Geršgorin theorem [34] to find magnitude of maximum eigenvalue of the matrix B_{N1}as
We substitute (54), (58), and (59) in (53) to obtain a lower bound on det B_{N1}^{2}. Then substituting (51), (52), and (53) in (32), a lower bound on ${\psi}_{i}^{\left(cd\right),\mathsf{\text{zf}}}$ can be obtained.
Abbreviations
 VDSL:

Very highspeed digital subscriber lines
 CM:

common mode
 DM:

differential mode
 MIMO:

mutiinput multioutput
 TP:

twisted pair
 ZF:

zero forcing
 SUB:

single user bound
 CO:

central office
 CPE:

customer premise equipment
 DMT:

discrete multi tone
 DFE:

decision feedback equalizer
 SAGE:

space alternating generalized expectation maximization
 MMSE:

minimum mean square error
 RFI:

radio frequency interference
 SNR:

signaltonoise ratio
 CWDD:

column wise diagonally dominant
 LFB:

lowfrequency band
 MFB:

midfrequency band
 HFB:

highfrequency band
 FEQ:

frequencydomain equalization
 FEXT:

far end crosstalk
 PSD:

power spectral density
 ML:

maximum likelihood
 MOE:

minimum output energy
 AZF:

approximate zero forcing
 ITUT:

international telecommunication uniontelecommunication (standardization sector).
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Acknowledgements
This study was supported by research grants from Conexant Systems Inc. USA, and IIT Delhi. An earlier preliminary version of this study consisting of discussion on the CM channel model was presented at the Annual IEEE Indicon Conference, INDICON 2008, IIT Kanpur, India, in December 2008. The authors would like to thank the anonymous reviewer for useful comments which helped to improve the quality of the article.
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Zafaruddin, S., Prakriya, S. & Prasad, S. Performance analysis of a commonmode signal based lowcomplexity crosstalk cancelation scheme in vectored VDSL. EURASIP J. Adv. Signal Process. 2012, 63 (2012). https://doi.org/10.1186/16876180201263
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Keywords
 alien crosstalk
 commonmode signal
 crosstalk cancelation
 diagonal dominance
 digital subscriber lines (DSL)
 discrete multitone (DMT)
 farend crosstalk (FEXT)
 twisted pairs; vectoring