Contract No U-BROAD D2.1 Statistical Characterization and Modelling of the Copper Physical Channel

Transcription

1 Contract No U-BROAD D2.1 Statistical Characterization and Modelling of the Copper Physical Channel Prepared by: Telecommunication System Institute (TSI) - Greece Bar Ilan Univesity (BIU) - Israel Metalink - Israel France Telecom R&D - France OTE - Greece Abstract: This document describes the measurement campaign and statistical data analysis and model validation efforts undertaken by the partners involved in WP02 of U-BROAD to assess the VDSL copper physical channel. As per the technical annex, the focus is on short VDSL loops, extending from 75 to 600 meters, and on measurements up to 30 MHz, which is the viable bandwidth for these loop lengths. FRANCE TELECOM R&D conducted the measurement campaign, while OTE provided partial measurements to be used for comparison purposes. The remaining partners were primarily involved with the data-analytic aspects of the objective. The sought data were collected and analyzed. The results indicate that, with few mild exceptions, the log-normal model fits well for both NEXT and FEXT. This conclusion further qualifies earlier log-normal modeling and validation results for bandwidths in the order of a few MHz. The results drive R&D efforts under WP03, and will have an impact on several other U-BROAD WPs; they are also timely for the VDSL community-at-large. Keyword list: DSL channel measurement, short VDSL loops, statistical modelling and characterization, lognormal distribution, Gaussian distribution, phase, frequency covariance, spatial covariance, exploratory data analysis, regression, model fitting

2 Executive Summary This deliverable describes the results of an extensive measurement campaign and associated data analysis effort orchestrated by the U-BROAD partners under WP02 in order to characterize the copper physical channel for short DSL loops. Target architectures include fibre to the basement (FTTB), fibre to the curb/cabinet (FTTC), and fibre to the exchange (FTTX) scenarios. Loop lengths considered were in the meter range, while measurements were taken up to 30 MHz. The emphasis was on measuring and analyzing crosstalk channels. Both near-end crosstalk (NEXT) and far-end crosstalk (FEXT) were considered. Measurements of insertion loss were also taken. There had been considerable prior work concerning the characterization of NEXT, but this was limited to much narrower bandwidths (in the order of 1-3 MHz). This was due to the fact that for the longer DSL loops currently in use (in the order of 2-3 km) the viable bandwidth of the direct (insertion loss) channel was limited to a few MHz, whereas we aim for shorter loops which support transmission over a ten-fold wider bandwidth. In the case of shorter loops, FEXT becomes a more important issue: for very short loops (e.g. 75 meters) FEXT coupling should be similar to NEXT. This intuition has been confirmed by analysis of our measurements. The first part of the work leading to this deliverable was the measurement campaign. Full measurements were conducted by FRANCE TELECOM R&D, while side-measurements were conducted by OTE for comparison purposes. FRANCE TELECOM R&D measured the complex frequency response of approximately 3200 crosstalk channels for various loop lengths. This was a laborious and time-consuming process that also involved continuous feedback from the remaining partners in order to specify and adapt, as necessary, the measurement process. The second part of the work involved statistical data analysis, modelling, and characterization of the measured data. TSI-TUC and BIU were responsible for this part of the work. In particular, the log-normal model with frequency-dependent mean was tested to see if it captures well the statistics of NEXT and FEXT for the given loop lengths over the full 30 MHz bandwidth. Different models for the frequency-dependent mean were tested, and the log-power frequency covariance and spatial covariance structures were assessed. The phase distribution was measured and compared to the uniform model. Finally, the often-made symmetry assumption, that geometrically equivalent pair configurations in the binder should give rise to approximately equivalent crosstalk channels, was also tested, and refuted. Our key results can be summarized as follows. The log-normal model is well-justified for NEXT and FEXT over the 30 MHz bandwidth. The uniform phase model held up quite well for both NEXT and FEXT. The log-power frequency covariance structure of FEXT revealed relatively stronger correlations than expected between frequencies that are rather far apart. Finally, the symmetry assumption (that geometrically equivalent pair configurations in the binder give rise to approximately equivalent crosstalk channels) is a risky one: comparing the results of the analysis of the full measurement data, and the symmetry-based completion of the partial data, we observed that the former is much more accurate. Our results have important ramifications for the next deliverable of U-BROAD/WP02, the theoretical developments under U-BROAD/WP03, and also impact the other U-BROAD WPs. Furthermore, the results are of considerable interest to the VDSL technical community-at-large, due to the scarcity of published literature on the characterization of short copper loops. Page 2-57

3 Table Of Contents EXECUTIVE SUMMARY... 2 TABLE OF CONTENTS INTRODUCTION.4 2. DESCRIPTION OF THE CHANNEL MEASUREMENT PROCESS AND APPARATUS FRANCE TELECOM R&D DATA OTE DATA MODELLING OF COPPER LINES INSERTION LOSS NEAR-END CROSSTALK (NEXT) Empirical Single Channel Models The Multichannel NEXT Noise Model FAR-END CROSSTALK (FEXT) MACROSCOPIC MESH PLOTS OF NEXT AND FEXT PARTIAL DATA (26/378 CHANNELS MEASURED) FULL DATA (ALL 378 CHANNELS MEASURED) DENSITY ESTIMATES: TESTING THE LOG-NORMAL MODEL FOR SHORT VDSL LOOPS UP TO 30 MHZ PARTIAL DATA (26/378 CHANNELS MEASURED) FULL DATA (ALL 378 CHANNELS MEASURED) TESTING THE UNIFORM PHASE ASSUMPTION FOR SHORT VDSL LOOPS UP TO 30MHZ PARTIAL DATA (26/378 CHANNELS MEASURED) FULL DATA (ALL 378 CHANNELS MEASURED) FREQUENCY COVARIANCE SPATIAL COVARIANCE INSERTION LOSS PLOTS CONCLUSIONS VOCABULARY REFERENCES Page 3-57

4 1 Introduction The ever-increasing demand for bandwidth in the access network can be addressed in a number of ways. The expensive solution is the deployment of new fibre to the home infrastructure (FTTH). Less expensive alternatives include hybrid fibre-copper solutions, which use copper over the last segment of the access network, as in fibre to the basement (FTTB) and fibre to the curb/cabinet (FTTC) architectures. With these architectures, wherein the copper infrastructure is used only over the last mile, VDSL and its extensions are the solutions of choice. In this case, the transmission systems typically use frequency division duplex (FDD) where upstream and downstream transmissions are done in separate bands. In this case the main impairment is far end crosstalk (FEXT). In other cases, one wants to provide high bandwidth directly from the central office (fibre to the exchange FTTX). In the latter scenario, the band 0-1 MHz is the most useful due to the extreme signal attenuation at higher frequencies; in which case one needs to resort to the use of multiple copper pairs, as is done in the enhanced SHDSL. To provide high bandwidth in a business environment, where interference from adjacent legacy E1 and SDSL system dominates the crosstalk noise, one has to use an echo-cancellation technology which uses the same bandwidth for both upstream and downstream transmission. In this case, near-end crosstalk (NEXT) is the major impairment and NEXT cancellation techniques are most important. For these reasons, we find that both NEXT and FEXT characterization and modelling are important for developing next generation access technology. This deliverable describes the measurement campaign and statistical analysis undertaken under WP02 of U-BROAD, with the aim of understanding and characterizing the copper physical channel for short (75m-600m) loops. The emphasis is on the frequency response of NEXT and FEXT, for frequencies going up to 30MHz. We develop statistical models that enable us to accurately assess the capacity of both single-channel and (multichannel) coordinated transmission systems. Page 4-57

5 2 Description Of The Channel Measurement Process And Apparatus 2.1 FRANCE TELECOM R&D Data Insertion Loss, NEXT and FEXT have been measured for different line lengths on a S88 cable constituted of 14 quads. The measured lengths are those available in the laboratory of France Telecom R&D. These lengths are 75m, 150m, 300m, and 600m. Figure 1 FRANCE TELECOM R&D measurement setup A Network Analyser (NA) is used to in the measurement process. A power splitter divides the power from the NA. Half of the power is injected in the cable; the other part of the power is injected in the R input of the NA and is used as a reference. Insertion Loss, NEXT and FEXT are measured on the cable via input A of the NA and accessed by extracting the ratio A/R. In order to connect the measured pairs with the measurement device, an impedance transformer (balun) is used. The reference for the baluns is North Hills 0302BB (10kHz 60MHz), except for FEXT and insertion loss for 300m and 590m, for which the reference is North Hills 413BF (100kHz 100MHz). Figure 2 Close-up photograph of the balun used in the FRANCE TELECOM R&D experiments In a first stage, it is important to calibrate the device by connecting the 2 baluns directly together in order to cancel their influence and the influence of the coaxial cables. Different types of equipment were used to measure crosstalk for different loop lengths, as specified next. NEXT and FEXT for 75m are measured using: HP8753ES, Resolution Bandwidth = 20Hz NEXT and FEXT for 150m are measured using: HP8751A, Resolution Bandwidth = 20Hz Page 5-57

6 NEXT (and FEXT) for 300m and 600m (resp. 590m) are measured using: HP4395A, Resolution Bandwidth = 100Hz For all the measurements, the set up is as follows: Source Power = 15 dbm Start Freq = 10 khz Stop Freq = 30 MHz Number of points = 801 Frequency sweep scale = Log 15 dbm represents the maximum source power available in the lab. The different types of measurement equipment used by FRANCE TELECOM R&D are shown in the figures below. Figure 3 HP8753ES Figure 4 HP8751A Figure 5 HP4395A Page 6-57

7 Due to the fact that measurements are taken in log frequency scale, and four specific lengths (75, 150, 300, and 600m) are measured, there is a need to interpolate the measured data to a linear frequency scale, and in-between measured channel lengths. For this reason, FRANCE TELECOM R&D has developed an interface that performs log-to-linear frequency interpolation, and cable 28 length interpolation. For each measured length, all possible, i.e. = 378, crosstalk channels in 2 the binder were actually measured (henceforth called full binder data). Furthermore, in order to test the validity of approximating all 378 channels in the full binder dataset using just 26 measured channels and symmetry-based binder completion (based on geometric configuration considerations), the said interface can also generate 378 channels from 26 measured basic channels, essentially replicating each basic channel as many times as the associated geometric configuration appears in the binder (henceforth called partial data). Figure 6 FRANCE TELECOM R&D data interpolation interface An example of (cubic-spline) frequency interpolation as implemented by FRANCE TELECOM R&D is shown in the next figure. Figure 7 Illustration of interpolation for a typical channel (FEXT 7_19, 75m) Page 7-57

8 2.2 OTE Data OTE measurements were performed using a network spectrum analyzer HP4195A along with a reflection/transmission device HP 35676B. The control as well as the measurement data I/O is performed with the use of a GPIB (IEEE 488) board from National Instruments (model AT- GPIB/TNT) installed in a Personal Computer running Microsoft Windows 2000 operating system. The whole procedure is semi-automated by issuing macro-commands to the Spectrum Analyser via the GPIB interface and reading the result of the measurements in ASCII form in a IEEE 64 bit floating point format. The devices under test (DUT) were copper pairs (twisted pairs) cables having lengths 50m, 100m, 200m, 300m, 400m and 500m. The cables consisted of 3 groups each of 8 copper pairs (twisted pairs) giving a total of 24 pairs where each wire had a 0.4mm diameter. These cables are regularly used in OTE regional access telecommunication networks and local computer networks. More specifically these cables are used by OTE S.A. between customer premises and Optical Network Units of its Optical Access Network. They constitute a transmission medium in the form of a symmetrical pair with parameters influenced particularly by material and diameter of the core, insulation, length of twist, and shielding. Figure 8 The HP4195 spectrum analyzer with HP35676B reflection/transmission device along with the measuring cables (foreground and picture on the right), and the spools with the cables used in the measurements (background). Impedance matching 50 Ω to 75 Ω was used between the HP4195A and the HP 35676B to match the 50 Ω input impedance of the HP4195A to the 75 Ω impedance of the HP 35676B. This system is able to measure various transmission parameters, including phase characteristics and attenuation characteristics, for near-end crosstalk (NEXT), far-end crosstalk (FEXT), and insertion loss. Balance transformers (baluns) which are regularly used at the ISDN backbone network were used to match the 75 Ω system impedance to the 120 Ω impedance of the twisted pairs and change the asymmetric mode of the network analyzer to the symmetric mode of the twisted pair. The input and output wires of the cables were attached to a Kronen Mini Distribution Frame which is the same as the ones used by OTE in Distribution Frames for POTS, ISDN and ADSL in ONUs. Page 8-57

9 Figure 9 Picture of the baluns with the Kronen Mini Distribution Frame where the different cable lengths are being cross-connected where the input and output signals are interfaced The baluns employed for the OTE measurements have better response in low frequencies, and fairly good response up to approximately 15 MHz. OTE has ordered new baluns that cover the entire frequency range of interest for U-BROAD, that is up to 30 MHz. Measurements will be repeated upon receipt of these new baluns. Measurements of insertion loss and FEXT/NEXT response (caused by capacitive and inductive coupling), were made for all the cable lengths mentioned above. The 1dBm output of the analyzer is split into two paths by the external splitter, one output of which is used as a reference signal (R), and the other is used as the test signal (T). The receiver detects the two input signals R and T and the real and complex amplitude ratios between the selected test input and the selected reference input are measured in db. These data and then stored into the above mentioned PC by using the GPIB port. Frequency sweeping is performed all the way from 10 Hz to 30 MHz. Data points are acquired in 4.25 khz increments. The number of data points for each full measurement should then be approximately 30 MHz / 4.25 khz = Since the analyzer can only record 401 frequency points at a time (in each sweep), a full frequency range sweep was completed by doing 18 individual sweeps (each covering approximately 1.7 MHz range = 401 * 4.25 khz). Therefore 1.7 MHz * 18 = 30.6 MHz. The starting frequency for each sweep is the ending frequency of the previous one. This way a total of 7201 complex data points were collected in each full measurement (18*401 complex data points). While FRANCE TELECOM R&D conducted full-binder but log frequency-sweep measurements, OTE R&D Labs conducted linear frequency-sweep measurements of one sample channel of each type (NEXT, FEXT, insertion loss) for each length. Thus FRANCE TELECOM R&D measured many channels but with fewer points per channel, while OTE R&D measured a few channels but with many points per channel. Due to linear frequency sweeping, OTE R&D did not need to perform any interpolation of the data points, and thus the raw measurements were provided. Page 9-57

10 3 Modelling Of Copper Lines 3.1 Insertion Loss The most important property of a twisted pair is its ability to deliver a transmitted signal. The design of any communication system over twisted pairs needs to take into account the frequency selective behaviour of the channel. A very good overview of the modelling of twisted pair channels can be found in [5]. Generally a simple model can be used by fitting a transfer function of the form () l f H( f, l) = e α where l is the length of the channel, and f is the frequency in Hz. 3.2 Near-End Crosstalk (NEXT) In this section we describe the semi-empirical NEXT single channel models and their verification on real data up to 30 MHz. We use the model described in [6][1], adapted to various types of binders. We begin with a single channel statistical characterization of NEXT. We then generalize it to a multidimensional statistical model that enables us to estimate the multidimensional system capacity. Typically the behaviour of NEXT is independent of loop length and can be modelled by the following approximation 2 h( f) = Kf α where α =3/2 is a typical value and K is a log-normal random variable. Equivalently we can present it in logarithmic scale by ( ) 20log h( f) = 10log ( K) + 10α log f Since K is a log-normal random variable 10log 10( K ) is a normal random variable and 20log 10 h( f ) is a normal variable with mean that linearly depends on the power law constant α Empirical Single Channel Models We present the results of Kerpez [6] for the NEXT coupling of two copper pairs. The power distribution of the NEXT attenuation is log-normal distributed with frequency-dependent mean µ p ( f ) = log( f ) where f is measured in Hz, and standard deviation σ = 9.2dB. Therefore, for a real signal we shall assume that the amplitude is a product of a log-normal variable with µ ( f ) = log( f ) α and 4.6dB 1,1 that represents the sign of the crosstalk coupling. Similarly in a QAM-based system, we need to assume a uniformly distributed random phase (see also Lin [2]). σ =, and a uniform random variable on { } Page 10-57

11 3.2.2 The Multichannel NEXT Noise Model In order to extend the work of Kerpez [6] into the multidimensional case we make several simplifying assumptions. Our first assumption is that the system s pairs are randomly located within the binder. This yields independence of the amplitudes of the NEXT of a pair towards the various pairs in the multichannel system. The second assumption is that the propagation delay of the crosstalk signal towards each of the pairs in the system manifests as a phase shift in the complex case or a random sign in the real case. This is justified by the randomness of the twisting of the pairs in the binder. Under the above assumptions, it follows that the cross spectral matrix of the interferers at frequency f is given by: where R r 11 1p L T n( f ) = = glfglf l= 1 rp 1 r pp r p is the number of channels, L is the number of NEXT disturbers and each g = [ g, g ] T is a vector of independent log-normal random variables (each with a random lf l1 f lpf sign) of the attenuation from interferer number l towards the various pairs in the system at frequency f. The model is still not completely specified, since we have not yet defined the relation between R ( f1), and R ( f2) for f1 f2. There are two extreme cases: n n g lif is independent of f, i.e., there is an envelope of the crosstalk and the attenuation varies by the amplitude; or g lif has some smooth dependence on f. The second case is more realistic. Therefore, we suggest that the generation of a crosstalk response should be done by picking randomly g on a grid of frequencies, and then smoothing it by lif interpolation. The number of grid points is determined by the smoothness of the NEXT attenuation for the given type of cable. pair 1 pair 2 Figure 10 NEXT is correlated between pairs The previous analysis assumed a random set of pairs chosen among all pairs in the binder. To provide some insight into the case of closely spaced pairs where we would expect the NEXT to be more correlated between the pairs we propose the following generalization of the previous model. Let g [ 1, ] T lf = gl f glpf be as before. Let g 0 f be a log-normal random variable independent of the components of g lf. Define 2 g ˆ lkf = g0 f ρlk + glkf 1 ρlk where ρlk ρ lm is the correlation between the channels km, for interference coming from the l th interfering pair. Again, we can construct the noise covariance matrix in a similar fashion. Page 11-57

12 3.3 Far-End Crosstalk (FEXT) Similarly to NEXT the model for FEXT has the following dependence [5]: 2 α hij ( f, l) = Kf H( f, l) For a given loop length l we can model H( f, l ) as a constant times the loop attenuation. This reduces to (, ) 2 hij f l = Kf α e β f where again K is a random variable that depends on the line. We have tested the possibility of modelling K as a log-normal random variable, resulting in a frequency dependent log-normal model for 20log 10 hij ( f, l ) : 20log h ( f, l) = 10log ( K) + 10αlog ( f) + γ + δ f 10 ij The constant γ makes 10log 10( K ) a zero mean normal random variable. Page 12-57

13 4 Macroscopic Mesh Plots Of NEXT And FEXT In this section, we present macroscopic mesh plots of the frequency response of the various measured channels, for both NEXT and FEXT. This allows the reader to visually assess the general characteristics of NEXT and FEXT for various loop lengths, before delving into more detailed analysis. All plots are presented in db scale, as 2-D functions of (channel, frequency); that is, we plot 10 [ ] 20log Hif (, ) db where f ranges from 40KHz to 30 MHz, and i ranges from 1 to 378. Two sets of measurements are presented. The first includes 26 measured channels, while the remaining = 352 channels are predicted based on symmetry considerations (channel pairs that are equivalent in terms of geometry in the binder are expected to exhibit similar crosstalk characteristics). The second set contains independent measurements of all 378 possible pairs in the binder. Analyzing both sets allows us to assess the validity of the often-assumed symmetry-induced equivalence of crosstalk responses. As we will see, there is often significant variability among equivalent pairs, which affects the statistics of crosstalk, but also the quality of sample average estimates. This variability can be attributed to various factors, including random twisting. The conclusion is that symmetry-induced equivalence is often a risky assumption. Plots for the partial data (including the symmetry-derived channels) are given first, followed by the ones for the full data. The grouping of the results is as follows. Results for NEXT are presented first, in order of increasing loop length; FEXT follows, again in order of increasing loop length. For the symmetry-completed partial data, we only show results for 75 meter NEXT and FEXT, in order to keep the length of this document manageable. As expected, the general characteristics of NEXT (levels, frequency dependence) remain stable across all measured loop lengths. NEXT generally varies between -120 and -20 db. FEXT attenuation is strongly dependent on loop length. Page 13-57

14 4.1 Partial Data (26/378 Channels Measured) Figure 11 Figure 12 Page 14-57

15 4.2 Full Data (All 378 Channels Measured) Figure 13 Figure 14 Page 15-57

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19 5 Density Estimates: Testing The Log-normal Model For Short VDSL Loops Up To 30 MHz In this section, we present the results of our analysis to test the log-normal model for short loops up to 30 MHz. As before, plots for the partial data (including the symmetry-derived channels) are given first, followed by plots for the full data. Results for NEXT are presented first, in order of increasing loop length; FEXT follows, again in order of increasing loop length. For the symmetrycompleted partial data, we again show results only for 75 meter NEXT and FEXT. For each type of channel (NEXT or FEXT) and each loop length, four individual plots are presented. The first shows the measured mean power of all available channels of the given type, and the associated fitted model. As per Section 3, we use the following parametric model for the mean NEXT signal power in db scale: E 20log 10 H( i, f) c1 + c2 log 10( f) The constants are fitted to the model as follows. First, E 20log 10 H( f) is replaced by its sample estimate, i.e. s ( f ) : mean ( 20log 10 Hif (, ) ) µ = ; then, the sought parameters are fitted to µ ( f ) in a Least-Squares (LS) sense. That is, c 1 and c 2 are chosen to minimize s f ( ) 2 µ ( f ) c + c log ( f) s which is a standard linear LS problem. The fitted curve is plotted along with µ ( f ) in the first of the group of four plots corresponding to each type of channel. The second figure shows the variance of the channel s power response, as a function of frequency. The result of constant regression along with the value predicted by the Kerpez model is also shown in the second figure. After frequency-dependent mean removal ( centering or de-trending ), using the fitted parametric mean model, the resulting frequency samples of the associated channels power should behave like zero-mean normal random variables, if our model is correct. The next two figures in the group of four assess the normality of the ensuing mean-centered channels. In particular, the third figure in each group shows a histogram of the mean-centered magnitude responses, accumulated across all channels of the given type and across all frequencies. A Gaussian probability density function is also fitted to the said data (not the histogram per se), and overlaid on top of the same plot. Gaussian fitting is performed in the Maximum Likelihood (ML) sense, which boils down to using the sample estimate of the variance of the centered data. This figure helps to assess (deviation from) normality, however tail inconsistencies are relatively hard to detect this way. For this reason, we also include a fourth so-called normal probability plot figure, which is produced using Matlab s normplot routine. The purpose of a normal probability plot is to graphically assess whether the data could come from a normal distribution. If the data are normal the plot will be linear. Other distribution types will introduce curvature in the plot. The normal probability plot helps better assess deviations from normality, especially in the tails of the distribution. Other normality tests can also be used (e.g., using the kurtosis, higher-order statistics, Kolmogorov tests, etc), and in fact have been tried, with similar results: with few notable exceptions (mainly regarding the tails of the distribution), the normal assumption is upheld. The situation is similar for FEXT, except that this time the parametric mean regression model is E 20log 10 H( i, f) c1 + c2 log 10( f) + c3 f as per the associated discussion in Section 3. The three model parameters are again fitted using LS regression, and the resulting problem is a simple linear LS problem. The rest of the procedure follows as for NEXT. s Page 19-57

20 5.1 Partial Data (26/378 Channels Measured) Figure 21 Figure 22 Page 20-57

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24 5.2 Full Data (All 378 Channels Measured) Figure 29 Figure 30 Page 24-57

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40 6 Testing The Uniform Phase Assumption For Short VDSL Loops Up To 30 MHz In this section, we test the uniform phase model. The results are presented in the usual order: plots for the partial data (including the symmetry-derived channels) are given first, followed by plots for the full data. Results for NEXT are presented first, in order of increasing loop length; FEXT follows, again in order of increasing loop length. Again, for the symmetry-completed partial data, we only show results for 75 meter NEXT and FEXT. Uniformity of phase is assessed by histogram plots, accumulated over all pertinent channels in the measurement set and over all frequencies. From these plots, it is evident that the partial data set does not exhibit enough variation to gauge whether the phase is uniform or not; the associated histograms suggest considerable deviation from uniformity. The corresponding histograms for the full data, however, are much closer to a uniform distribution. In the context of assessing the capacity of single-loop (non-vectored) transmission, the phase of crosstalk plays no role. For vectored transmission, however, the marginal distribution of phase is important. Although a complete statistical characterization of the phase process is not needed for capacity purposes, and is therefore beyond the scope of our work package, we note the following: we have observed that the phase response of very short channels tends to be linear; whereas deviations from linear phase are encountered, particularly at high frequencies, for relatively longer loop lengths. Linearity (or quasi-linearity) implies that the phase process is strongly correlated, as the full phase response can be predicted (respectively, estimated) from two phase samples. Page 40-57

41 6.1 Partial Data (26/378 Channels Measured) Figure 61 Figure 62 Page 41-57

42 6.2 Full Data (All 378 Channels Measured) Figure 63 Figure 64 Page 42-57

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46 7 Frequency Covariance As we have demonstrated the NEXT coupling function has a log-normal distribution law. The same holds for FEXT coupling. Therefore in order to fully characterize the process across frequencies we need to characterize its covariance function. This would completely characterize the amplitude behaviour. We demonstrate the covariance properties of the NEXT and FEXT channels. We compute the covariance matrix (after de-trending) of the data ( ) C( f1, f2) = E log 10 h( f1) Elog 10 h( f1) log 10 h( f2) Elog 10 h( f2) where the inner expectation can be estimated by either the fitted parametric mean model, or per frequency using the sample mean across spatial channels. In these plots we have used the sample mean. The results show that, while the NEXT covariance matrix is indeed diagonally dominated and decays quickly with frequency separation, this is not quite the case for FEXT. The partial data were not sufficient for accurately estimating frequency correlation; we thus only report frequency correlation results for the full data. Note that this is exactly a point where the symmetry assumption brakes down and the small fluctuations dominate the behaviour of the covariance function. Page 46-57

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50 8 Spatial Covariance We now examine spatial correlations in the data. These correlations form the basis for interference mitigation techniques. To that end we have studied the covariance of the amplitudes. Let and define where Finally let log 10 h1( fmin) 10log 10 h1( fmax) X = log 10 h378( fmin ) 10log 10 h378( fmax ) Y = X 1µ 1 = [ 1,,1] T, = [ µ ( f ),, µ ( f )] Z= YY * / N f, where f * µ min max, and f ) 10Eij log ( ij ( f) ) T ( ) µ( = X. N is the total number of frequencies for which measurements were taken. We study the eigen-structure of the matrix Z for various loops and different types of crosstalk. Let λ λ be the singular values of Z. We use singular value decomposition 1 Κ because it automatically orders the singular values. We plot λ1,, λκ for all loops. We observe that they all have very similar behaviour. The first figure depicts the 10 strongest singular values of Z, while the second shows all 378 singular values. We can clearly see a dominant singular value followed by two more, that are much higher than the rest. This implies a strong spatial correlation of the channel amplitudes that is due to the geometry of the binder, c.f. schematic in page 7. The strongest singular value can be attributed to the quad structure. The other significant singular values are less obvious to interpret, but the low-rank eigen-pattern that permeates all spatial covariances has important ramifications in the context of crosstalk cancellation via spatial processing. Page 50-57

51 Figure 77 Figure 78 Page 51-57

52 9 Insertion Loss Plots In this section, we provide insertion loss and phase plots for direct channels of various lengths, measured during the FRANCE TELECOM R&D campaign. The db power frequency response (insertion loss) plots for the four measured lengths appear in the next figure, followed by four phase plots, one for each individual channel length. Notice that the usable bandwidth indeed extends to 30 MHz for the shortest (75m) loop, but is effectively limited to about 7.5 MHz for the longest (590m) loop considered. At that point, the loop s insertion loss drops under - 50 db. For 300m and 590m, the average db power of 28 channels is shown. For 75m and 150m, insertion loss is far more stable. Insertion loss plots were generated using linear interpolation in the log-power domain from the 801 measured samples. It is also interesting to observe the phase responses in the four figures that follow. Note that for the first two loop lengths (75m and 150m) the (un-wrapped) phase is almost exactly linear. For the two longer loop lengths (300m and especially 590m), the un-wrapped phase deviates from linearity at higher frequencies. Figure 79 Page 52-57

53 Figure 80 Figure 81 Page 53-57

54 Figure 82 Figure 83 Page 54-57

55 10 Conclusions Our conclusions can be summarized as follows: The log-normal model is well-justified for NEXT and FEXT over the 30 MHz bandwidth The uniform phase assumption is valid for both NEXT and FEXT. The analysis of log-power frequency covariance for NEXT confirmed that the covariance matrix is diagonally dominated, and thus the magnitude of NEXT is approximately uncorrelated in the log domain as a function of frequency. The analysis of log-power frequency covariance for FEXT revealed stronger than expected correlations between frequencies that are rather far apart. The often-made symmetry assumption (that geometrically equivalent pair configurations in the binder should give rise to approximately equivalent crosstalk channels) is a risky one: comparing the results of the analysis of the full measurement data, and the symmetry-based completion of the partial data, we observed that the former is much more accurate. A final note is due regarding interpolation. While the cubic interpolation employed herein works reasonably well in most cases (see, e.g., Figure 7), it can occasionally produce ringing or ripplelike artifacts. This comes up in the analysis of 300m and 590m FEXT (see Figures 53, 57, and also 19), at higher frequencies. These artifacts are due to a number of reasons. First, due to the logfrequency sampling, sampling density is lower at higher frequencies. Second, we need to interpolate a complex sequence. Linear interpolation of a complex sequence yields second-order interpolation of the magnitude, which can produce ringing; cubic interpolation of a complex sequence yields higher-order interpolation of the magnitude. The level of ringing turned out being similar for both cubic and linear interpolation of the complex data. On the other hand, one may think about, e.g., linear interpolation of the magnitude sequence; but phase is discontinuous due to warping, thus phase interpolation is problematic. Summarizing, interpolation of a complex sequence from irregularly-spaced samples is an ill-posed problem. Different solutions may be pursued, but most reasonable solutions give similar results. While the occasional ringing artifacts are undesirable, their effect is relatively minor, as measured by deviation from the presumed (log-normal) statistical model. Page 55-57

56 Vocabulary ADSL db DSL FDD FEXT FTTB FTTC FTTH FTTX HDSL HFC INSERTION LOSS ISDN LS ML NEXT POTS QAM SDSL VDSL Asymmetric Digital Subscriber Line Decibels Digital Subscriber Line Frequency Division Duplex Far-End cross (X) Talk Fibre To The-Basement Fibre To The Curb/Cabinet Network where an optical fibre runs from the telephone switch to a curb-side distribution point close to the subscriber where it is converted to copper pair Fibre To The Home - Network where an optical fibre runs from the telephone switch to the subscriber's premises Fibre To The exchange High bit-rate Digital Subscriber Line - Modems on either end of one or more twisted wire pair that deliver T1 speeds. At present, this requires two lines Hybrid Fibre-Coax Frequency response of direct channel Integrated Services Digital Network line Least Squares Maximum Likelihood parameter estimation Near-End cross (X) Talk Plain Old Telephone Service Quadrature Amplitude Modulation Symmetrical Digital Subscriber Line - HDSL plus POTS over a single telephone line. Very high bit-rate Digital Subscriber Line Page 56-57

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