Profile detection in multiuser digital subscriber line ... - Semantic Scholar

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 5, JUNE 2002

Profile Detection in Multiuser Digital Subscriber Line Systems Atul A. Salvekar, Member, IEEE, Jérôme Louveaux, Carlos Aldana, Jeannie Lee Fang, Elisabeth de Carvalho, and John M. Cioffi, Fellow, IEEE

Abstract—Multiuser transmission methods for digital subscriber line (DSL) systems have become of interest with the potential for increased data rate and loop reach. These methods often assume that the set of crosstalk interferers, called the crosstalk profile, and their associated channel responses are known. For DSL systems, the interferers are often uncoordinated, so that in a dynamic environment where DSL transmitters can energize and deenergize, the crosstalk profile cannot be transmitted to the user of interest. While the crosstalk channel estimation problem in a dynamic environment can be intractable for general transmission systems, channel and crosstalk analysis can make use of the specific DSL environment. Namely, the physical channels in a DSL system do not change rapidly, and hence estimates of the crosstalk channel can be saved for future reference. For this reason, we introduce the concept of a channel profile. We develop several algorithms to detect the crosstalk profile and investigate the asymptotic behavior of the new algorithms. Simulations show that for typical crosstalk interference scenarios, the observation time to determine the correct crosstalk profile at probability of error less than 10 3 can be less than 2 ms. Index Terms—Crosstalk, crosstalk profile, digital subscriber line (DSL), multiuser detection (MUD), multiuser transmission (MUT), profile detection.

I. INTRODUCTION

D

IGITAL subscriber lines (DSLs) have become a major source of broadband data transmission for households in the United States because of the ubiquitous nature of the existing twisted pair infrastructure. While the number of users is increasing rapidly, so is the demand for bit rate. In order to accommodate the new users and the additional demand for bit rate, there is a critical need to improve DSL technology. As such, multiuser transmission (MUT) methods in DSL systems has become a very important topic with promises of increased data rate Manuscript received November 10, 2001; revised December 20, 2001. This work was supported by Alcatel, by France Telecom, by IBM, by Samsung, by Telcordia, by Fujitsu and by Intel. The work of J. Louveaux was supported by the Belgian F.N.R.S. This paper was presented in part at the 2001 IEEE Conference on Communications (ICC’01), Helsinki, Finland, June 2001. A. Salvekar was with the Electrical Engineering Department, Stanford University, Stanford, CA 94305 USA. He is now with Intel Communications Group, Sacramento, CA 95827 USA (e-mail: [email protected]). C. Aldana was with the Electrical Engineering Department, Stanford University, Stanford, CA 94305 USA. He is now with Solaflare Communications Inc. Irvine, CA 92618 USA (e-mail: [email protected]). J. L. Fang, and J. M. Cioffi are with the Electrical Engineering Department, Stanford University, Stanford, CA 94305 USA (email: jeannie@dsl. stanford.edu; [email protected]). J. Louveaux is with the Universite Catholique de Louvain, UCL/TELE, B-1348 Louvain-la-Neuve, Belgium (e-mail: [email protected]). E. de Carvalho is with the Virata Corporation, Santa Clara, CA 95051 USA (e-mail: [email protected]). Publisher Item Identifier S 0733-8716(02)05385-4.

Fig. 1. Block diagram of a multiuser detector.

and loop reach. The main innovation in multiuser DSL transmission is that interference caused by neighboring DSL lines, also known as the crosstalk interference, is modeled as a multiple-access channel. In this case, multiuser transmission uses an analysis of the crosstalk scenario and often views the interferers as signals whose spectra or format may be optimized. For example, a general multiuser detector is shown in Fig. 1 with as the output sequence, as the input sequences (sequences since there can be several inputs), as the additive noise, represents all the channel impulse responses, and represents all the assumed or estimated channel responses. A good review is provided in [1], [2]. In previous DSL systems, crosstalk was treated like noise because the crosstalk channels are long, the users are uncoordinated, and there are no code sequences to separate the users. These DSL systems were effectively treated as single-user systems. However, recent advances may allow the use of multiuser methods in DSL systems. New low-complexity algorithms, such as soft iterative decoding [3] and oversampled detection [4] are able to increase data rates significantly. Other MUT schemes that can dramatically increase the data rate are described in [5], [6]. All these algorithms assume some level of knowledge of the set of crosstalk interferers present and sometimes of their channel transfer functions. From either the single or multiuser perspective, the DSL system is crosstalk limited. Current DSL systems transmit continuously, but new proposals suggest that in the future bursty transmission will be possible. In particular, DSL transmitters could energize (turn on) when information is being sent and deenergize (turn off) when information is not being sent [7], [8]. This would save money for the transmitter in the form of power

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SALVEKAR et al.: PROFILE DETECTION IN MULTIUSER DIGITAL SUBSCRIBER LINE SYSTEMS

reduction and allow rate adaptive systems to increase their data rates when the neighboring crosstalk is deenergized. For a DSL system to take advantage of a bursty crosstalk environment, it must be able to quickly determine which crosstalkers are present and adapt its parameters. However, as a result of the process of unbundling [9], the different xDSL services can act in an uncoordinated manner even if they are the same type of service. So, crosstalk information that can be useful for transmission purposes in a coordinated system cannot always be used [2]. Obtaining the crosstalk information necessary for adjusting the system parameters can be extremely difficult if it is assumed that the crosstalk information is completely new each time a crosstalk signal energizes or deenergizes. If treated as a completely new transmission environment, the issues of long channel lengths, possible lack of transmission coordination, and no use of code sequences, significantly complicate the estimation of the channel responses of several interferers simultaneously. However, in contrast to the generic transmission system environment which is time-varying, the DSL physical channels do not change quickly. So, if a DSL modem were to monitor the crosstalk interference intermittently, it could acquire crosstalk channel responses by using the crosstalk interferer’s synchronization patterns over a long period of time while the signal of interest were inactive. A method to provide even better channel estimates is to use a network maintenance center [10]. Once the DSL crosstalk channels are estimated, they can be stored and used to detect which crosstalk interferers are transmitting or “present.” This will be called crosstalk profile detection. There are slow variations in the transmission media, but for the remainder of the paper a static physical channel is assumed. If the crosstalk channels have been stored from recent measurements, then except for the timing offsets that the crosstalk channels have with respect to some nominal reference, the multiuser transmission scheme can be loaded appropriately just by detecting the presence of the crosstalker profile. Timing issues can be addressed separately and are left considered in future work [11]. So, a detection scheme that is independent of timing issues is desired. For this purpose, we devise several algorithms that under very mild conditions, can detect the crosstalk profile without knowledge of the timing information. The outline of the paper is as follows. Section II describes the concept of channel profiles which are used in Section III to detect the crosstalk scenario (crosstalk profile). Sections IV and V introduce and analyze methods for crosstalk profile detection. Simulation results are provided in Section VI. Section VII concludes the paper. II. CHANNEL PROFILES A channel profile describes the physical state of the transmission environment. An example of a channel profile is a set of possible crosstalk interferers, also called a crosstalk profile. To account for the influence they have on the physical system, crosstalk profiles could consist of the possible scenarios that most influence the performance of the DSL link or occur most often. For instance, the set of interferers that have the widest

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Fig. 2. Channel profile state transition diagram.

or most intense power spectra will generally have the most influence on system performance. This set of interferers is referred to as the dominant crosstalkers. When the dominant crosstalkers are active, they can be detected without regard to smaller nondominant crosstalkers. When dominant crosstalkers are not active, it is desirable to determine the presence of the nondominant crosstalkers since, in this case, they determine system performance. In addition to the presence of crosstalkers, in some DSL systems, the user of interest’s channel can change in a deterministic way. For instance in splitterless ADSL, when a phone goes on-hook/off-hook, the channel characteristics change in a welldefined manner [12]. This deterministic channel gain change can also be included in the channel profile definition. More abstractly, a system is assumed to be in one of pro, where no two such states can file “states,” , occur simultaneously. The entire set of possible profiles shall . A set of profiles can be modeled by be represented as a state transition diagram as shown in Fig. 2. In this example, the system of interest is a splitterless ADSL service where one phone is attached to the DSL port and there is one potential through crosstalker. There are four possible configurations . In this particular state diagram, only one feature can change at any one time. It is assumed that all the states of the system are known and held in system memory. Once the state of the system is ascertained, then the corresponding transmission parameters can be loaded. To determine the system’s state, we describe the general profile detection problem, first studied in [12], [13]. Consider where a block of sampled output data is the th sample of the output, is the block size, and represents transpose. The current profile is denoted by . For profile detection, the maximum likelihood (ML) detection rule is given by (1) denotes the conditional probability of given where the profile . A maximum a posteriori (MAP) rule could be used, but the prior probabilities of every profile state may be difficult to determine. Because the timing offset can be determined independently of the crosstalk scenario and detection of on-hook and off-hook

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events have been extensively studied, we investigate the problem of determining which crosstalkers are present. This will be known as crosstalk profile detection. III. CROSSTALK PROFILE DETECTION To determine the crosstalk interferers that are present, we first describe the crosstalk interference scenario. The DSL crosstalk interference scenario consists of several different services that can interfere with the line of interest. These services include high-speed digital subscriber lines (HDSL), asymmetric digital subscriber lines (ADSL), symmetric digital subscriber lines (SDSL), integrated services digital network (ISDN), and T1 services. Typically, the user of interest has access to only one output, so this environment can be described by a multiple-input single-output (MISO) model. For the remainder of the paper, we concentrate on the MISO crosstalk channel environment, which is typical for DSL systems. The different symbol rates of the different services leads to some additional complexity in the system model. Assume there lines in the model. There are crosstalkers into each are be the samples sent on line (or equivalently sent line. Let . by user ). The input symbol rate of this user is denoted by The crosstalk channel response from line into is denoted by . This response includes both the shaping filter of user and the receive filter of user . For the remainder of the paper, will also be known as the transmit pulse from crosstalker to user . The crosstalk interference is given by (2) At the receiver, this signal is

sampled and becomes (3) (4)

accounts for the timing offset between user and user where . Without loss of generality, the subscript 0 is the desired sampled aggregate signal. For the user of interest, the crosstalk signal can be written as

where

is the crosstalk channel matrix depending on , is the column vector of input samples , and is a column vector of noise samples. For synchronization and estimation purposes, the input samples may contain not only data but also some known sequences such as synchronization or training sequences. For example, in ADSL, a fixed synchronization sequence is sent every 69 discrete multitone (DMT) symbols. Equation (6) can be separated into the contributions of the known symbols (synchronization sequence) with the subscript and the unknown symbols (the data) with the subscript . The set of received samples can be rewritten as

(7) By concatenating the input symbols of all users into bigger vecand and by grouping the crosstalk channels tors and , where stands for “all,” into bigger matrices we get (8) For notational convenience, every configuration has been whose binary expansion is represented by an integer . Every such binary expansion refers to a different crosstalk channel profile. Similarly, refers to the set of all timing informations . is The determination of the conditional probabilities not straightforward due to the possibly different sampling rates used by the different users and due to the unknown timing offsets between users. There are various ML criteria based on different assumptions for the input data [11], [13], [14]. In our problem, the simplest of these criteria is the Gaussian maximum likelihood (GML) detector, which treats input symbols as Gaussian random variables, so that the output waveform is a Gaussian random variable. Since the channels are long and the transmit constellations are multilevel, treating the output to be Gaussian is acceptable [15]. Then, from (8), under the th profile, (9) represents a normal random vector with mean where and covariance matrix and

(5) if crosstalker is active and 0 otherwise and where are the samples of the noise, assumed to be additive white does Gaussian noise (AWGN) with variance . Notice that not include the user signal. The user’s signal is separated from the crosstalk signal because later in the paper, the line is monitored at start-up or in decision directed mode, when the user’s signal can be cleanly subtracted and only the crosstalk remains. can be written in matrix notation A set of output samples as where (6)

(10) is the expectation operator, and is the identity mawhere trix. Then, the GML test can be written as:

(11) is the determinant of matrix . where While GML is less complex than the other ML criteria, it is still very complex due to the effects of the timing information.

SALVEKAR et al.: PROFILE DETECTION IN MULTIUSER DIGITAL SUBSCRIBER LINE SYSTEMS

The timing offsets influence the mean of the output sample vector through the position of the synchronization sequences and affect the covariance matrix through the sampling phase . Since this paper is interested of the crosstalk channels in the presence or absence of the crosstalk interferers, timing information is not relevant. Therefore, we can simplify the detection with the following observations. First, we ignore the presence of the synchronization sequences which are typically designed to have low cross-correlation properties and have average energy equal to the expected input symbol energy of the interfering DSL. Second, we note that practical multiuser transmission schemes will sample the output waveform at a rate at least as high as the largest effective bandwidth of any crosstalker. This allows for the design of timing offset independent crosstalk profile detectors. To design the profile detectors, we consider two bandlimiting cases. 1) In the “ideal” case all the crosstalkers have an effective bandwidth smaller than their symbol rate, i.e., for crosstalker , the bandwidth is within , hence the sampled output process is stationary. Since the crosstalk is stationary, a simple maximum likelihood detector can be designed. Because of excess bandwidth, this bandlimiting scenario is not met in practice. However, these detectors form a basis for the detectors in the following more realistic scenario. 2) In a more realistic scenario, the crosstalk interference will have excess bandwidth making the crosstalk response cyclostationary. A multiuser transmission system will likely oversample the crosstalk signal. In particular, the crosstalkers are effectively bandlimited to a bandwidth , where is the smaller than sampling rate of the user of interest. An analysis of this crosstalk scenario will yield crosstalk profile detectors. IV. PROFILE DETECTION IN BANDLIMITING CASE 1) A. Bandlimited Maximum Likelihood Detection In bandlimiting case 1), ignoring the effects of the synchrois not a function of since the autocornization sequences, relation is not a function of , as shown in Appendix A. The bandlimited GML detector becomes

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sequential detection method are analyzed in Section IV-A1 and Section IV-A2. 1) Fixed Block Length Detection: This section roughly analyzes the number of data blocks, , used in (12) needed to attain a specific probability of error, , where an error denotes that the incorrect profile has been chosen. We note that a Toeplitz matrix is asymptotically circulant, and then apply the central limit theorem to achieve a given performance level. is any other profile, then If is the current profile, and an error occurs if

(13) For large enough blocks of data , the Toeplitz autocorrelation matrix is approximately circulant, and hence, can nearly be diagonalized by the discrete Fourier transform (DFT) matrix, . Making the transformation, , (13) can be approximated by

(14) denotes conjugate transpose and where matrix whose th diagonal entry is given by

is a diagonal (15)

, is the th element of the binary where is the th element of the DFT expansion of , and . Using the central limit theorem [16], of the sequence (16) (17) Hence, using the properties of Gaussian random variables, the , where, if probability of (14) can be written as blocks are averaged,

(12) (18) is given by (10) when there is no known sequence. where We shall call this bandlimited maximum likelihood detection (BMLD). While BMLD looks complex, it is in reality simple. For fixed and can be precomputed. block lengths , Then, the complexity of the detection scheme is only a matrix multiply. Furthermore, simulations show that by using several such estimates together, even small block sizes can be used to reliably detect the crosstalkers present, thus making the complexity reasonable. The detector performance can be analyzed for either a fixed number of blocks, known as the fixed block length detector, or a sequential method. The fixed block detection method and the

Using the union bound, the number of blocks needed for a particular , is given by (19) This analysis is somewhat rough since the noise is not exactly Gaussian and the autocorrelation matrix is not exactly Circulant. 2) Sequential Detection: Sequential detection consists of selecting a crosstalk profile after a given performance metric is met, for instance the probability of misclassification [17]. In this case, the number of blocks is not fixed. Sequential methods are of interest since typically they have smaller wait times then

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fixed methods for a given performance level. Moreover, our performance analysis for the sequential detection method is more exact since besides the assumption that the crosstalk is Gaussian, no other approximations are made. denote the negative log-likelihood of crosstalk proLet . This is given by file given the th data vector

periodograms, the central limit theorem can be invoked, leading to the new detection rule

(25) (20) If the blocks of data are spaced far enough apart, the loglikelihood can be summed, as the likelihood functions are effectively from independent samples. So, the log-likelihood of are given by blocks 1 through (21) When the difference between the maximum log-likelihood function and all other log-likelihood functions is greater than , the largest likelihood is chosen as the some threshold is the smallest negative likelihood crosstalk profile. So, if function (largest likelihood) at step , then a decision is made at time , when

is the expected periodogram under profile . where Using other windows can increase performance by reducing the variance per sample [19]. C. Discussion The BMLD and DFT methods developed in this section are for crosstalk interferers whose symbol rates are greater than the effective bandwidth of the transmit pulse. Because the transmit pulse has excess bandwidth, this bandlimiting scenario is not met in practice. However, in multiuser systems, the crosstalk will likely be oversampled. In the next section new detectors will be developed for this case. Some of the new detectors will use the detectors of Section IV as a starting point. V. PROFILE DETECTION IN BANDLIMITING CASE 2)

(22) . The decision as to the crosstalk profile is then , this is known as an Armitage scheme and the If theory of sequential analysis can help in choosing a threshold for [18]. Using an Armitage scheme, the probaany particular bility of misclassification is given by the Armitage bound: misclassification

(23)

Practical multiuser systems will oversample the crosstalk interference. Using this fact, two methodologies are used to determine the crosstalk scenario. In Section V-A, detection schemes based on the principles of Section IV are argued to work in the oversampled case with minor adjustments even when bandlimiting case 1 is not satisfied. Then, in Section V-B, a detection scheme based only on the assumption that the crosstalk interference is oversampled is developed. The methods of Section V are used for simulations in Section VI. A. Approximate Methods

B. DFT Method Many of the high speed DSL modems that could potentially use multiuser transmission schemes already employ a discrete Fourier transform (DFT) for signal processing. Examples include ADSL, ADSL-lite, and DMT-VDSL. As such, the squared output of the DFT, also known as a periodogram , where are equally spaced points between 1/2 and 1/2, can be used for profile detection. The periodogram is a biased estimate of the spectrum of the signal of interest and has mean value

(24) is the Fourier transform of the Bartlett window where is the power spectrum of the sampled of size and crosstalker output [19]. and is Using the fact that the covariance of about equal to 0 [19] and recognizing that asymptotically the samples of the periodogram are chi-square random variables formed from a circularly symmetric Gaussian random process is , then by averaging so that the variance of

While the methods in Section IV cannot be directly applied to the DSL crosstalk scenarios due to the excess bandwidth of the crosstalk interference, it can be argued that the blocks of data, , used for detection are similar to blocks of data from the time-shifted version of the crosstalk interference. s are formed from output samples sufficiently sepIf the arated in time, then these blocks are independent. Hence, assuming ergodicity, block by block processing corresponds to the case of a time-shifted process. is the crosstalk output, then the timeAs a review, if , where is a uniform random shifted version is and is the least common multiple variable between of the s. As the time-shifted process is stationary, the GML criterion can be applied to this process. The power spectral density (PSD) is given by [20]: of (26) is the frequency response of the crosstalk channel where and is an indicator variable of whether the crosstalker is present. For the time-shifted process all the methods in

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Section IV can be applied, except that the autocorrelation that is used would be the inverse Fourier transform of (26). It is assumed that the crosstalk channel responses are known up to can be calculated for any crosstalk a timing offset so that scenario. Even if the crosstalk channel responses are not known, it is shown in Appendix B that the measured autocorrelation function asymptotically corresponds to the autocorrelation of the time-shifted version of the crosstalk interference process. can be calculated from the waveform itself without Hence, the assumption of known channel responses. Thus, the methods in Section IV can be used in the presence of practical crosstalk provided that the time-shifted PSD is used. These adjusted methods will be called the modified bandlimited maximum likelihood detector (MBMLD) and the modified DFT (MDFT) method. B. Sample Autocorrelation Matching

Fig. 3. The power spectral density of the first crosstalk scenario.

In bandlimiting case 2, the ML detector is no longer simple is no longer wide because the autocorrelation function of sense stationary. In general, it is cyclostationary, so that is now a function of . However, even though is a function of , it can be shown (Appendix B) that the expected limiting sample , defined as autocorrelation function,

(27) is a constant independent of the timing offset. The measured sample autocorrelation function is given by

(28) Fig. 4.

If the process is covariance ergodic (29) The asymptotic behavior of the autocorrelation estimate is complicated. So, instead of defining a GML criterion, we simply minimize the mean square error between the expected limiting sample autocorrelation and the measured sample autocorrelation. This is written as (30) is the number of autocorrelation terms used and is the th autocorrelation lag of (27) given is the current profile. We call this method the sample autocorrelation matching (SAM) method. A degenerate version of this algorithm would be to match the measured power to the closest expected power. This shall be referred to as power matching (PM). We turn to Monte Carlo simulations to show that for reasonable observation lengths the current profile can be determined by the SAM method.

where

The power spectral density of the second crosstalk scenario.

VI. SIMULATION RESULTS For the simulations, the signal of interest is assumed to be an ADSL modem with sampling rate 2.208 MHz. We assume two possible crosstalk scenarios. In scenario one, there are three near end crosstalkers (NEXT) whose symbol rate (392 kHz), constellation (4-PAM), and spectral mask are defined as in the HDSL standard [21]. Their individual crosstalk power spectral densities (PSDs) are given in Fig. 3. These PSDs are representative of the actual characteristics of HDSL for typical transmission lines. Scenario two has one NEXT interferer whose symbol rate, constellation size, and spectral mask are defined as in the HDSL standard and one NEXT interferers whose symbol rate (80 kHz), constellation size (4-PAM), and spectral mask are defined as in the ISDN standard [22]. They have crosstalk PSDs as seen in Fig. 4. In this case there are two dominant crosstalkers, and a nondominant one; crosstalk three in Fig. 4 is nondominant since its PSD is dominated by the PSD of crosstalkers one and two. We, therefore, define a profile such that only the set of dominant crosstalkers needs to be correctly identified when at least one dominant crosstalker is active. When a dominant crosstalker is not active, a profile is defined as the nondominant crosstalker.

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Probability of error versus block size for a total of 1024 samples. Fig. 7. Probability of error versus total number of symbols for 1 HDSL and 2 ISDN crosstalkers.

Fig. 6. Probability of error versus total number of symbols for 3 HDSL crosstalkers.

For scenarios one and two, there is background AWGN noise of 140 dBm/Hz. The excess bandwidth of the crosstalk preclude directly using the detectors of bandlimiting case 1, but the strong dropoff in the spectral energy upto the ADSL sampling rate make the detectors of bandlimiting case 2) applicable. In order to choose the block sizes for the various methods, we examine the performance of the detectors versus block size in an example where a total of 1024 samples are used. Fig. 5 shows that for scenario one, a block length of 100 samples provides nearly maximum performance for the SAM and MBMLD detectors. We therefore choose block lengths of 100 for all simulations. For the MDFT method, we choose a DFT size of 512 which corresponds to the DFT size for traditional ADSL [23]. For testing purposes, the minimum total number of samples was chosen to be 512 so that a fair comparison with the MDFT method was possible. The results of the simulations in scenario one is given in Fig. 6. As expected, the modified bandlimited maximum likelihood detector performs best out of all the detection methods. The MDFT method is not as good as the SAM method. One

possible reason for this is that the covariance terms that were assumed to be zero were actually significant. As expected the power method has the least detection capability. Recall that an error denotes that the incorrect profile has been chosen. For a probability of error of 10 , within 10 ms, the PM method excluded, all the methods can detect the crosstalk profile and for MBMLD, the detection time is less than 2 ms. In scenario two, as shown in Fig. 7, the MBMLD detection scheme again performs extremely well. In fact, the probability of error is less than 10 for total number of samples greater than 512. Hence there is no line for the MBMLD method in Fig. 7. The MDFT method also has extremely good performance, since in this case, the two dominant crosstalk profiles have very different power spectral densities, so this method can easily distinguish between the different possible sets of crosstalkers. Notice that the SAM method is also able to quickly determine the crosstalk scenario. All the methods, excluding the PM method, can detect the crosstalk profile with probability of error less than 10 in less than 2 ms. Next, we examine the error rate versus threshold level for the MBMLD detector used in the first scenario. This is shown in Fig. 8. Note the important quality that as the threshold level is increased, the error rate decreases exponentially as would be expected since a linear increase in the log-likelihood function implies an exponential decrease in the probability of error. This is expressed in (23). So, the error rate can be made arbitrarily close to zero by increasing the threshold, at the cost of increased latency. At the expense of complexity, the MBMLD method provides the best performance among the detectors. This method has complexity where is the block length. The SAM order method and the MDFT methods do not have any clear performance distinction between them and both have low comand log respectively, where is the number plexity, of autocorrelation terms. Finally, the simplest and least reliable method is the power matching method which has order complexity. Therefore, there is a tradeoff between complexity and performance.

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Now, we examine the autocorrelation of the output process:

(35) (36) (37) where (37) can be rewritten as

and

. Equation (38)

Fig. 8. Probability of error versus threshold for detection of 3 HDSL crosstalkers.

VII. CONCLUSION The main conclusion to be drawn is that while the channel response of each individual channel may be difficult, or even impossible to estimate in a short period of time, once a channel has been estimated, it can be stored as a profile. Once the presence of the channel has been sensed, the previously measured channel response can be loaded into the multiuser transmission system. Using the MBMLD method, the MDFT method, or the SAM method, the set of crosstalk interferers can be quickly determined. For two sets of examples corresponding to real world crosstalk configurations, it has been shown that the crosstalk scenario could be determined with less than 0.001 probability of incorrect detection in less than 2 ms.

and are the discrete-time Fourier where and respectively. Because transforms of the channel is bandlimited, there is no aliasing beand . It is easy to show that between tween and and , where is the continuous-time Fourier transform of , so that

(39) (40) (41) (42) Thus, the output covariance only depends on the time delay, and is wide-sense stationary. the process

APPENDIX A We would like to show under the conditions in the following theorem that the autocorrelation function is independent of the sampling instant. We consider a single crosstalker. The result can be simply extended to multiple crosstalkers by using the fact that the crosstalkers are mutually independent. This same result is found in [24], in an elegant, but somewhat less straightforward proof. Theorem 1: Given a bandlimited transmit pulse with total , and PAM signaling with bandwidth less than (31) (32) (33) is wide sense stationary. the output process Proof: We examine the mean of the output process: (34) Thus,

is a constant.

APPENDIX B In this appendix, we consider a multiuser received signal written as (see Section III) (43) and with some set of timing offsets sampled with spacing with respect to the different users. The limiting sample autocorrelation function is defined as (44) , The signal is assumed to be bandlimited to , i.e., the crosstalk is oversampled. It where is proved that, if the crosstalk is oversampled, the expected limiting sample autocorrelation function (45) is independent of the set of timing offsets and equals the autocorrelation of the time-shifted version of the crosstalk. Then,

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by assuming, the sample autocorrelation function converges to the expected limiting sample autocorrelation function. This result basically means that the average over all available samples provides the same effect as the expectation over even though the timing offset is all possible timing offsets fixed in the real system. This is obviously not true if because, in this case, becomes independent of but depends on (unless the bandwidth is limited to ). However, in the actual system, the received signal is generally oversampled in order to catch all the information on the signal. Thus, we can apply the above stated reand are irreciprocal sult. This result is intuitive when because the average over will “scan” all the possible sampling positions. This is less obvious for typical values such as . A proof for discrete time random processes that does not include the effects of timing offsets is shown in [25]. We present the proof hereunder. Proof: First develop

Now, the expected sample autocorrelation function becomes

(52) The behavior of (52) as a function of the timing offsets depends on the bandwidth of the transmitted signals. Let us assume that the signal transmitted by user is bandlimited to . For , the integral can be nonzero for , therefore, if , i.e., if the receiver oversamples the output, then for these (53)

(46) symbols are independent (in time and beassuming all tween users) with variance . We denote by

, and hence, The only terms that remains corresponds to also vanishes. The expected limiting the dependence upon sample autocorrelation function becomes

(47) the Fourier transform of the sequence for and for fixed . The correlation given by (46) may then be rewritten as (48) Expressing the discrete-time Fourier transform in terms of the continuous-time Fourier transform, we have

(54) (55) which is also the expected autocorrelation for uniformly distributed timing offsets , i.e., the autocorrelation for the timeshifted version. Now, we assume ergodicity, and conclude that the measured sample autocorrelation is asymptotically independent of the timing offset and converges to the expected limiting sample autocorrelation function. ACKNOWLEDGMENT

(49) is the Fourier transform of the corresponding where channel. We get

The authors would like to thank the anonymous reviewers for their constructive criticism and comments, which greatly improved the quality of the paper. REFERENCES

(50)

(51) which is a function of

.

[1] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [2] J. Cioffi, G. Ginis, W. Yu, and C. Zeng, “Example improvements of dynamic spectrum management,” ANSI Contribution, Los Angeles, CA, T1E1.4/2001-089, Feb. 2001. [3] K. W. Cheong, “Multiuser detection for DSL applications,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 2000. [4] C. Zeng and J. M. Cioffi, “Crosstalk cancellation in xDSL systems,”. [5] G. Ginis and J. Cioffi, “Vectored DMT: A FEXT-cancelling modulation scheme for coordinating users,” in Proc. ICC, vol. 1, 2001, pp. 305–309. [6] W. Yu, G. Ginis, and J. Cioffi, “An adaptive multiuser power control algorithm for VDSL,” in Proc. Globecom, vol. 1, 2001, pp. 394–398. [7] A. A. Salvekar, “State detection techniques for DSL systems,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 2002. [8] C. H. Aldana, “Interference estimation in multicarrier systems,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 2002.

SALVEKAR et al.: PROFILE DETECTION IN MULTIUSER DIGITAL SUBSCRIBER LINE SYSTEMS

[9] P. Odling, B. Mayr, and S. Palm, “The technical impact of the unbundling process and regulatory action,” IEEE Commun. Mag., vol. 38, pp. 74–80, May 2000. [10] C. Zeng, C. Aldana, A. Salvekar, and J. M. Cioffi, “Crosstalk identification in xDSL systems,” IEEE J. Select. Areas Commun., vol. 19, pp. 1488–1496, Aug. 2001. [11] A. Salvekar, C. Aldana, E. de Carvalho, and J. Cioffi, “Crosstalk profile detection for use in multiuser detection,” in Proc. ICC, vol. 7, 2001, pp. 2171–2175. [12] A. Salvekar, C. Aldana, J. Tellado, and J. Cioffi, “Channel gain change detection and channel profile selection in a multicarrier system,” in Proc. Globecom, vol. 2, 1999, pp. 1133–1138. [13] C. Aldana, A. Salvekar, J. Tellado, and J. Cioffi, “MAP crosstalk profile matching for multicarrier systems,” in Proc. ICT, 2001. [14] L. Tong and S. Perreau, “Multichannel blind identification: From subspace to maximum likelihood methods,” Proc. IEEE, vol. 86, pp. 1951–1968, Oct. 1998. [15] K. J. Kerpez, “Near end crosstalk is almost Gaussian,” IEEE Trans. Commun., vol. 41, pp. 670–672, May 1993. [16] S. Ross, Stochastic Processes, 2 ed. New York: Wiley, 1996. [17] D. Siegmund, Sequential Analysis: Tests and Confidence Intervals. New York: Springer-Verlag, 1985. [18] P. Armitage, “Sequential analysis with more than two alternative hypothesis,” J. Roy. Stat. Soc. B, vol. 12, pp. 137–144, 1950. [19] S. M. Kay, Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1988. [20] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3 ed. New York: McGraw-Hill Inc., 1991. [21] “High bit rate Digital Subscriber Line (HDSL) transceivers,”, G.991.1, Oct. 1998. [22] ISDN-basic access interface for use on metallic loops for application on the network side of the NT (layer 1 specification), ANSI T1.601, 1999. [23] “Asymmetric Digital Subscriber Line (ADSL) transceivers,”, ITU-T Rec. G.992.2. [24] W. A. Gardner and L. E. Franks, “Characterization of cyclostationary random signal processes,” IEEE Trans. Inform. Theory, vol. IT–21, pp. 4–14, May 1975. [25] G. B. Giannakis, “Cyclostationary signal analysis,” in Statistical Signal Processing Section of Digital Signal Processing Handbook, V. K. Madisetti and D. Williams, Eds. Boca Raton, FL: CRC Press, 1998.

Atul Salvekar (S’98–M’02) received the B.S. degree in electrical engineering from the California Institute of Technology, Pasadena, CA in 1996, the M.S. degree in electrical engineering in 1998, the M.S. degree in statistics in 2001, and the Ph.D. degree in electrical engineering in 2002, all from Stanford University, Stanford, CA. He is currently working for Intel Corporation. His research interests lie in the digital communication and signal processing.

Jérôme Louveaux was born in Belgium in 1974. He received the electrical engineering degree and the Ph.D. degree from the Universite catholique de Louvain (UCL, Belgium) in 1996 and 2000, respectively. From 2000 to 2001, he has been a visitor at the Electrical Engineering Department, Stanford University, CA. He is currently a Postdoctoral researcher at the UCL funded by the Belgian National Funds for Scientific Research (F.N.R.S.). His research interests include signal processing for digital communications, mainly high bit rate multicarrier transmission by means of filter banks, synchronization and MIMO systems. Dr. Louveaux was a co-recipient of the 2000 Biennial Siemens Award from the Belgian NSF.

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Carlos Aldana received the B.S. degree in electrical engineering from the California Institute of Technology, Pasadena, CA, and is working toward the Ph.D. degree at Stanford University, Stanford, CA. He is currently working for Solarfare Communications Inc. His current research interests include multiuser channel identification, statistical signal processing, multiuser detection, nonlinear systems, and adaptive algorithms.

Jeannie Lee Fang received the B.S. degree in electrical engineering (summa cum laude) from the University of Texas at Austin in 1995 and the M.S. degree in electrical engineering from Stanford University, Stanford, CA, in 1997. She is working toward the Ph.D. degree in electrical engineering at Stanford University. Her research interests lie in the areas of digital communications and signal processing, including multicarrier systems, channel modeling, and channel identification.

Elisabeth de Carvalho received the M.Sc. degree in electrical engineering from the National Institute of Telecommunications (INT), Evry, France, in 1994 and the Ph.D. degree in electrical engineering from the Ecole Nationale Superieure des Telecommunications (Telecom Paris), Paris, France, in 1999. From 1995 to 1998, she was with Laboratoires d’Electronique PHILIPS supported by a CIFRE scholarship. Her Ph.D. work was on receiver algorithms for wireless communications, with emphasis on blind and semi-blind equalization. From October 1999 to June 2001, she was a postdoctoral fellow at Star Lab, Stanford University in the group of Prof. J. M. Cioffi, working on signal processing issues for DSL communications. From October 1999 to September 2000, she was the recipient of a Lavoisier scholarship from the French Ministry of Foreign Affairs. Since July 2001, she has been with Virata Corporation, Santa Clara, CA. Her research interests are the area of signal processing for digital communications, synchronization, equalization for wireless and DSL communications.

John M. Cioffi (S’77–M’78–SM’90–F’96) received the B.S.E.E. degree from the University of Illinois, Urbana-Champaign, in 1978, and the Ph.D.E.E. degree from Stanford University, Stanford, CA, in 1984. He was with Bell Laboratories, Holmdel, NJ, from 1978 to 1984 and IBM Research, San Jose, CA, from 1984 to 1986. He has been with Stanford University as an Electrical Engineering Professor from 1986 to the present. He founded Amati Communications Corporation, Palto Alto, CA, in 1991 (it was purchased by Texas Instruments, Incorporated in 1997) and was Officer/Director from 1991 to 1997. He is currently on the boards or advisory boards of BigBand Networks, Coppercom, GoDigital, Ikanos, Ionospan, Ishoni, IteX, Marvell, Kestrel, Charter Ventures, and Portview Ventures. He is a Member of the U.S. National Research Council’s CSTB. His specific interests are in the area of high-performance digital transmission. Dr. Cioffi has received the following awards: the National Academy of Engineering in 2001, IEEE Kobayashi Medal in 2001, IEEE Millennium Medal in 2000, IEE JJ Tomson Medal in 2000, 1999 University of Illinois Outstanding Alumnus, 1991 IEEE COMMUNICATIONS MAGAZINE best paper; 1995 ANSI T1 Outstanding Achievement Award, and NSF Presidential Investigator from 1987 to 1992. He became a member of the National Academy of Engineering in 2001. He has published over 200 papers and holds over 40 patents, most of which are widely licensed, including basic patents on DMT, VDSL, and V-OFDM.