Contract No U-BROAD D2.2 Analysis of Multiuser Capacities and Capacity Regions

Transcription

1 U-BROAD D2.2 Contract No U-BROAD D2.2 Analysis of Multiuser Capacities and Capacity Regions Prepared by: Telecommunication System Institute (TSI) - Greece Bar Ilan University (BIU) - Israel Abstract: This document describes the results of statistical analysis of capacity for short DSL loops, based on actual insertion loss and NEXT/FEXT channel measurement data provided by France Telecom R&D. The channel measurement process and the resulting data have been described in detail in Deliverable D2.1. The focus of the present document is on assessing the capacity of both single-line and coordinated (vectored) DSL transmission systems, using the aforementioned data. Since the selection of direct (vectored) loops and interferers in the binder is random, the resulting capacity is a random variable. For this reason, statistical analysis is appropriate. The pertinent metrics are the mean, minimum, maximum, and standard deviation of capacity, as well as capacity outage. These are all derived from capacity histograms, which are also provided herein. Both exhaustive (enumeration-based) computation, and Monte-Carlo sampling have been considered, for short DSL loop lengths and bandwidth up to 3 MHz. We find that capacities in the order of several hundred Megabits per second per loop are attainable, and the ratio of minimum to mean capacity is often quite high. We also find that transmitreceive coordination yields a significant boost in average per-loop capacity only when the number of coordinated loops is higher than the number of alien interferers. In situations wherein the number of alien interferers exceeds the number of coordinated loops, coordination improves the capacity spread. The results drive R&D efforts under WP3, and will have an impact on several other U-BROAD WPs; they are also very timely for the VDSL community-at-large. Keyword list: DSL, VDSL, channel capacity, outage, coordinated (vectored) transmission, analysis of measurements, experimental capacity estimates for short DSL loops

2 Executive Summary This document describes the results of statistical analysis of capacity for short DSL loops, based on actual insertion loss and NEXT/FEXT channel measurement data provided by France Telecom R&D. The channel measurement process and the resulting data have been described in detail in Deliverable D2.1, [1]. The focus of the present document is on assessing the capacity of both single-line and coordinated (vectored) DSL transmission systems, using the aforementioned data. Since the selection of direct (vectored) loops and interferers in the binder is random, the resulting capacity is a random variable. For this reason, statistical analysis is appropriate. Pertinent metrics include the mean, minimum, maximum, and standard deviation of capacity, as well as capacity outage. These are all derived from capacity histograms, which are also provided herein. Both exhaustive (enumeration-based) computation and Monte-Carlo sampling have been considered, for short DSL loop lengths and bandwidth up to 3 MHz. We find that: Capacities in the order of several hundred Megabits per second per loop are attainable, and the ratio of minimum to mean capacity is often quite high; Coordination yields a more significant boost in average per-loop capacity only when the number of coordinated loops is higher than the number of alien interferers. When the number of coordinated loops is smaller than the number of alien interferers, there is usually less than 25% increase in the average per-loop capacity; however: Coordination significantly reduces the capacity spread, even when only a small fraction of loops in the binder are coordinated. This is important because it reduces the ratio of minimum to average capacity, which is a measure of outage. The results drive R&D efforts under WP3, and will have an impact on several other U-BROAD WPs; they are also very timely for the VDSL community-atlarge. 1

3 Contents 1 Introduction & Roadmap Capacity calculations and related assumptions Capacity Statistics via Full Combinatorial Calculation Capacity versus number of disturbers Light vs. Full crosstalk: Capacity versus number of coordinated pairs Capacity Statistics via Reduced Monte-Carlo Sampling Average Capacity Capacity CDFs Conclusions & Coordination Benefit Assessment 31 5 Appendix A: Capacity Histograms and Moments via Full Combinatorial Calculations FEXT plus NEXT FEXT only Appendix B: Capacity Statistics via Reduced Monte-Carlo Sampling for 3m and 6m loops Average Capacity Capacity CDFs

4 1 Introduction & Roadmap This document is structured as follows. We begin with a short introduction to capacity calculations for single-loop and coordinated DSL transmission subject to crosstalk. For coordinated Multiple-Input Multiple-Output (MIMO) transmission, we start from the general MIMO capacity formula, and work our way through the various application-specific assumptions and simplifications. In particular, we briefly discuss self-next echo-cancellation, and self-fext transmit precompensation, and their effect on the crosstalk covariance. Since capacity depends on the specific configuration of interferers (not only on the number thereof), it is important to calculate the capacity distribution and associated summary metrics (minimum, mean, maximum capacity; plus standard deviation and outage) as a function of the number of coordinated loops and the type and number of interferers. For this purpose, one can either opt for full enumeration-based computation of the capacity distributions, or computationally simpler Monte-Carlo sampling-based estimates. We decided to pursue both. The TUC team worked using the enumeration-based approach, and the BIU team worked using the Monte-Carlo approach. Section 2 contains the results of enumeration-based capacity calculations, whereas Section 3 contains those of Monte-Carlo sampling-based capacity calculations. 1.1 Capacity calculations and related assumptions For general background on capacity, we refer the reader to Cover & Thomas [1]. On the capacity of Multiple-Input Multiple-Output (MIMO) systems, see [2,3]. References [4,5] are good background on DSL system capacity. One of the earliest references on coordinated transmission over a pair of DSL lines is [6]; see Ginis & Cioffi [7] for an overview of the current state-of-art on coordinated transmission for DSL. Crosstalk cancellation techniques for DSL systems are discussed in [8]. Let there be N loops in the binder (N =28for France Telecom s data), out of which p loops are employed for coordinated (vectored) transmission. Let L NEXT (L FEXT ) denote the number of NEXT (respectively, FEXT) loops that interfere with the p coordinated loops. Let σ 2 denote the additive white Gaussian noise (AWGN) power spectral density (PSD) - typically at -14 dbm/hz (or 1 17 W/Hz in absolute scale) for DSL systems. For a single direct loop (p = 1), and a certain configuration of interfering loops, the Shannon capacity is given by [1,4] C = log 2 (1 + SINR(f))df (1) BW where BW denotes the available bandwidth, and the Signal to Interference plus 3

5 Noise Ratio is given by H IL (f) 2 p(f) SINR(f) = σ 2 + L NEXT i=1 H NEXT,i (f) 2 p NEXT (f)+ L FEXT j=1 H FEXT,j (f) 2 p FEXT (f) (2) Here, H IL (f) is the insertion loss (frequency response) of the direct channel, H NEXT,i (f) (H FEXT,j (f)) is the frequency response of the i-th NEXT (resp. j-th FEXT) channel, p(f) is the PSD (or, spectral mask) employed for the direct channel, while p NEXT (f) (p FEXT (f)) is the PSD of NEXT (resp. FEXT) interferers. Assuming that all spectral masks are equal equation (2) simplifies to H IL (f) 2 SINR(f) = σ 2 + L NEXT p(f) i=1 H NEXT,i (f) 2 + L FEXT (3) j=1 H FEXT,j (f) 2 In the vectored case (p >1) the capacity of the coordinated system is given by [2,3] C = log 2 det(i + p(f)h(f)r 1 nn(f)h (f))df (4) BW where H(f) is the p p input-output MIMO channel transfer matrix at frequency f, denotes Hermitian (conjugate) transpose, and R nn (f) is the p p interference plus noise covariance matrix at the output of the MIMO subsystem at frequency f: R nn = p NEXT (f)g NEXT (f)g NEXT(f)+p FEXT (f)g FEXT (f)g FEXT(f)+σ 2 I (5) where G NEXT (f) is a p L NEXT crosstalk transfer matrix, whose (m, l)-element is the complex coupling coefficient from the l-th NEXT disturber to the m-th loop in the vectored subsystem at frequency f; and similarly for the p L FEXT FEXT coupling matrix, G FEXT (f). The matrix H is diagonally-dominated in DSL systems, wherein insertion loss is usually 2 or more db higher than interference. For this reason, it is possible to pre-equalize H at the transmitter s side, without a significant penalty in terms of transmission power. Thus, upon pre-multiplication by H 1 diag ([H IL,1 (f),,h IL,p (f)]), the effective MIMO channel transfer matrix becomes diag ([H IL,1 (f),,h IL,p (f)]) (note that we do not invert the direct insertion loss channels; that would entail a significant power penalty, esp. at higher frequencies). 4

6 If we further assume that all insertion loss channels of the vectored subsystem are approximately equal [9] (this is well-justified, for insertion loss primarily depends on length, termination and bridge taps), then the capacity expression further simplifies to C = log 2 det(i + H IL (f) 2 p(f)r 1 nn(f))df (6) BW A few remarks are in order: Equation (6) effectively assumes that self-fext (from within the vectored subsystem) has been pre-compensated at the transmitter. External (often called alien) FEXT, from the remaining loops in the binder, is accounted for in R nn (f) (the p FEXT (f)g FEXT (f)g FEXT(f) term). Self-NEXT at the receiver can be mitigated by employing echo cancellation techniques, which essentially amount to subtracting self-next interference to a given loop from other loops in the coordinated subsystem, taking into account the associated frequency-dependent coupling factor. If echo cancellation is employed at the receiver, then there is only alien NEXT, if any. If upstream-downstream Frequency Division Duplex (FDD) is further employed, then alien NEXT is effectively suppressed as well. In this case, only alien FEXT remains. In non-fdd systems, however, alien NEXT is the performance-limiting factor, for FEXT is usually much lower than NEXT, even for relatively short loops (one significant exception is very short loops, under 1 meters, wherein FEXT looks much like NEXT, for obvious reasons). Because interference is generally correlated (R nn (f) in Equation (6) is not diagonal), the capacity in Equation (6) implicitly assumes that the coordinated system s receiver employs multiuser detection. Capacity in Equation (4) depends on the particular configuration of coordinated loops and interferers in the binder. Even under the simplifying assumptions leading to Equation (6), capacity depends not only on the number, but also on the particular configuration of interferers, through the coupling coefficients in G NEXT (f) and G FEXT (f). For this reason, we are interested in assessing the capacity distribution for a given number of FEXT interferers, as is appropriate for echo-cancelled, FDD, transmit precompensated coordinated subsystems of order p. We are interested in measuring the per-loop capacity as a function of p, and L FEXT. The per-loop capacity is the MIMO capacity of the coordinated subsystem divided by p. Furthermore, in the case of non-fdd systems, we are interested in the 5

7 distribution of C/p as a function of p, L NEXT, and L FEXT. This will allow us to assess the capacity benefit afforded by vectoring, i.e., the impact of coordinated transmission on the per-loop capacity as a function of the size of the coordinated subsystem and the number, type, and configuration of disturbers. For each p, L NEXT, and L FEXT, we will thus obtain a capacity distribution. This distribution can be summarized by means of a few statistics: minimum, mean, maximum, standard deviation, and outage capacity. We will compute those for a range of configurations of interest, and plot as a function of the parameters p, L NEXT, and L FEXT. Capacity also depends on the spectral mask, p(f). Depending on whether or not frequency planning is used (e.g., FDD and coarse frequency-dependent power-loading), p(f) may vary with f. We will thus consider two cases: one in which p(f) is set to -6 dbm/hz (1 9 W/Hz in absolute scale) for all f (this is a typical PSD level for DSL modems, and maintaining it across the 3 MHz bandwidth will yield slightly optimistic capacity results); but also use a standard FDD frequency plan (VDSL 998) and its extension to frequencies up to 3 MHz (extended VDSL 998). In the first case (constant p(f) at -6 dbm/hz), since we are interested in assessing theoretical bounds on performance, we will also ignore implementation issues like modulation loss, noise margin, and coding gain. In the second case (VDSL 998 / extended VDSL 998), we will incorporate these practical considerations. Pursuing both scenarios will allow us to gauge how far practical implementations will be from theoretical capacity. 2 Capacity Statistics via Full Combinatorial Calculation Assuming, as we did, that the insertion loss for all coordinated loops is identical, the clear-cut way of calculating the capacity distribution is to go over all possible configurations of (L NEXT, L FEXT ) alien interferers, calculate the capacity for each configuration using Equation (6), and then generate a histogram of the results. The difficulty is that certain combinations of p, L NEXT, L FEXT generate an immense number of possible configurations; e.g., for a total of N =28loops in the binder, p =4, and L NEXT = L FEXT =12, generates about 246 million possibilities, for each of which the integral in Equation (6) must be approximated. It is easy to get into situations wherein generating the capacity distribution for a given p, L NEXT, L FEXT takes about 1 week in a state-of-art PC. For this reason, we restrict ourselves 6

8 to looking at carefully selected combinations of p, L NEXT, L FEXT, which provide sufficient insights at a relatively moderate computational cost. To further reduce complexity, we assume that whenever p is even, quads (i.e., pairs of loops in the same quad) are chosen for vectoring, as is likely to be the case in practice. In Section 3 we will complement our results with Monte-Carlo sampling-based estimates of the capacity distribution, wherein loops are drawn in random from the binder. Monte-Carlo techniques yield an estimate of the capacity distribution, but allow us to consider cases that are essentially intractable using the enumerationbased approach. In order to keep this document manageable, we chose to present statistical summary results (minimum, mean, and maximum capacity; standard deviation of capacity) for various scenarios (p, L NEXT, L FEXT ) in this section. All these quantities are derived from capacity histograms. Histograms convey further information (e.g., allow the calculation of the of outage events), but since there are numerous such histograms, we preferred to defer them to Appendix A. 7

9 2.1 Capacity versus number of disturbers 7 x Minimum capacity for 75m loop p = 1 p = 2 p = 4 p = 6 p = 8 p = 14 p = 2 p = Number of disturbers L NEXT = L FEXT = L Figure 1: Minimum capacity per loop; 75m 7.5 x Maximum capacity for 75m loop p = 1 p = 2 p = 4 p = 6 p = 8 p = 14 p = 2 p = Number of disturbers L NEXT = L FEXT = L Figure 2: Maximum capacity per loop; 75m 8

10 7 x Average capacity for 75m loop p = 1 p = 2 p = 4 p = 6 p = 8 p = 14 p = 2 p = Number of disturbers L NEXT = L FEXT = L Figure 3: Average capacity per loop; 75m 7 x Standard deviation from average capacity for 75m loop p = 1 p = 2 p = 4 p = 6 p = 8 p = 14 p = 2 p = Number of disturbers L = L = L NEXT FEXT Figure 4: Standard deviation of capacity per loop; 75m 9

11 7 x Minimum capacity for 15m loop p = 1 p = 2 p = 4 p = 6 p = 8 p = 14 p = 2 p = Number of disturbers L NEXT = L FEXT = L Figure 5: Minimum capacity per loop; 15m 6.5 x Maximum capacity for 15m loop p = 1 p = 2 p = 4 p = 6 p = 8 p = 14 p = 2 p = Number of disturbers L = L = L NEXT FEXT Figure 6: Maximum capacity per loop; 15m 1

12 6.5 x Average capacity for 15m loop p = 1 p = 2 p = 4 p = 6 p = 8 p = 14 p = 2 p = Number of disturbers L NEXT = L FEXT = L Figure 7: Average capacity per loop; 15m 8 x Standard deviation from average capacity for 15m loop p = 1 p = 2 p = 4 p = 6 p = 8 p = 14 p = 2 p = Number of disturbers L = L = L NEXT FEXT Figure 8: Standard deviation of capacity per loop; 15m 11

13 2.2 Light vs. Full crosstalk: Capacity versus number of coordinated pairs 7 x 18 Minimum capacity m, Lnext = Lfext = 1 75m, Lnext = Lfext = 27 p 15m, Lnext = Lfext = 1 15m, Lnext = Lfext = 27 p Number p of coordinated pairs Figure 9: Minimum capacity per loop; FEXT and NEXT 8 x 18 Maximum capacity m, Lnext = Lfext = 1 75m, Lnext = Lfext = 27 p 15m, Lnext = Lfext = 1 15m, Lnext = Lfext = 27 p Number p of coordinated pairs Figure 1: Maximum capacity per loop; FEXT and NEXT 12

14 7 x 18 Average capacity m, Lnext = Lfext = 1 75m, Lnext = Lfext = 27 p 15m, Lnext = Lfext = 1 15m, Lnext = Lfext = 27 p Number p of coordinated pairs Figure 11: Average capacity per loop; FEXT and NEXT 8 x 17 7 Standard deviation from average capacity 75m, Lnext = Lfext = 1 75m, Lnext = Lfext = 27 p 15m, Lnext = Lfext = 1 15m, Lnext = Lfext = 27 p Number p of coordinated pairs Figure 12: Standard deviation of capacity per loop; FEXT and NEXT 13

15 7.5 x 18 Minimum capacity, FEXT only m, Lfext = 1 75m, Lfext = 27 p 15m, Lfext = 1 15m, Lfext = 27 p Number p of coordinated pairs Figure 13: Minimum capacity per loop; FEXT only 7.5 x 18 Maximum capacity, FEXT only m, Lfext = 1 75m, Lfext = 27 p 15m, Lfext = 1 15m, Lfext = 27 p Number p of coordinated pairs Figure 14: Maximum capacity per loop; FEXT only 14

16 7.5 x 18 Average capacity, FEXT only m, Lfext = 1 75m, Lfext = 27 p 15m, Lfext = 1 15m, Lfext = 27 p Number p of coordinated pairs Figure 15: Average capacity per loop; FEXT only 8 x 17 7 Standard deviation from average capacity, FEXT only 75m, Lfext = 1 75m, Lfext = 27 p 15m, Lfext = 1 15m, Lfext = 27 p Number p of coordinated pairs Figure 16: Standard deviation of capacity per loop; FEXT only 15

17 3 Capacity Statistics via Reduced Monte-Carlo Sampling In this section, we assess the capacity performance of multichannel DSL systems, using Monte-Carlo sampling to randomly select loops from the binder. This is a computationally simpler alternative to enumeration-based full-binder capacity calculations, presented in the previous section. We assume that a coordinated system with p channels is employed, along with L other alien interferers drawn from the same binder. Unlike the results presented in the previous section (which assumed a flat transmission power spectrum at -6 dbm/hz), in this section we employ two frequency band plans, and associated spectral masks. The first is VDSL 998 with downstream frequency bands as MHz, MHz, and 12-2 MHz; while the second plan is called extended VDSL L998 with frequency bands as MHz, MHz, and 12-3 MHz. In the first two frequency bands, the transmitter has -6 dbm/hz, while in the third one, the transmitter has -7 dbm/hz, as shown in figure 17, and figure 18. The noise is assumed to be AWGN with PSD of -14 dbm/hz. For the FEXT-limited case, we focus herein on loop lengths of 75m and 15m, for which France Telecom s FEXT channel measurements are reliable. As concluded in Deliverable D2.1, associated FEXT measurements for longer loop lengths are not reliable (especially at higher frequencies), due to the noise floor effect. For this reason, we defer the corresponding plots to Appendix B of this document. NEXT channels are essentially invariant to varying loop length, and the corresponding measurements are reliable all the way up to 6m. For the NEXTlimited case, we therefore include results for all loop lengths considered (75m, 15m, 3m, and 6m) in this section. Attaining Shannon capacity requires Gaussian signaling and long codes. In reality, there is an effective SINR loss due to the use of practical modulation schemes and the need to guarantee a certain noise margin. This SINR penalty is alleviated to a certain extent by the use of advanced coding techniques. The combined effect of modulation loss, coding gain, and noise margin, can be captured by replacing SINR by SINR/Γ, where the gap Γ := l mod + m noise g coding. Here, l mod is the modulation loss (9.8 db for QAM at BER of 1 7 ), m noise is the noise margin, and g coding is the coding gain. For coding gain and noise margin, we use three choices: [g coding, m noise ]= [3.8, 6], [6.8, 6], or [9.8, ] db, which result in a gap Γ =12, 9, and db, respectively, at BER 1 7. As before, the channel insertion loss, NEXT and FEXT come from the France Telecom R&D data. 16

18 5 VDSL downstream PSD PSD [dbm/hz] f [khz] Figure 17: VDSL 998 PSD mask. 5 VDSL + downstream PSD PSD [dbm/hz] f [khz] Figure 18: Extended VDSL 998 PSD mask. 17

19 3.1 Average Capacity We first present average total capacity results for all p coordinated loops. We begin with p =8, and L {1, 2, 4, 8, 12, 16}. Figures 19, 2, depict the average capacity from 1 independent runs for loop length of 75 m, and 15 m, respectively, where both self cross-talk and alien interference are due to FEXT. The three curves in the upper part of each figure follow the extended VDSL 998 frequency band, with difference choices for the coding gain and the noise margin, while the rest three curves, in the lower part, follow the VDSL 998 frequency band Average of 1 independent runs for 8 channels when line length is 75m [3.8 6] db [7.8 6] db [9.8 ] db [3.8 6] db [7.8 6] db [9.8 ] db Data rate (M) Number of alien noise (fext 75) Figure 19: Average capacity of 1 independent runs vs. the number of disturbers for 75m loop. The self cross-talk and alien interferences are all from FEXT. 18

20 2 18 Average of 1 independent runs for 8 channels when line length is 15m [3.8 6] db [7.8 6] db [9.8 ] db [3.8 6] db [7.8 6] db [9.8 ] db 16 Data rate (M) Number of alien noise (fext 15) Figure 2: Average capacity of 1 independent runs vs. the number of disturbers for 15m loop. The self cross-talk and alien interferences are all from FEXT. We now assume that p =12, and L {1, 2, 4, 8, 12, 16}. Figures 21, and 22, depict the average capacity of 1 independent runs for loop length of 75 m, and 15 m respectively, when the self cross-talk and alien interferences are all from FEXT. Again, the three curves in the upper part of each figure follow the extended VDSL 998 frequency band, with difference choices for the coding gain and the noise margin, while the rest three curves, in the lower part, follow the VDSL 998 frequency band. 19

21 Average of 1 independent runs for 12 channels when line length is 75m(coding gain=3.8 db, margin=6 db) VDSL 998 Extended VDSL Data rate (M) Number of alien noise (fext 75) Figure 21: Average capacity of 1 independent runs vs. the number of disturbers for 75m loop. The self cross-talk and alien interferences are all from FEXT. Average of 1 independent runs for 12 channels when line length is 15m(coding gain=3.8 db, margin=6 db) 22 2 VDSL 998 Extended VDSL Data rate (M) Number of alien noise (fext 15) Figure 22: Average capacity of 1 independent runs vs. the number of disturbers for 15m loop. The self cross-talk and alien interferences are all from FEXT. 2

22 3.2 Capacity CDFs We now present estimated Cumulative Distribution Functions (CDFs) for capacity. Again, the total capacity of all p coordinated loops is shown. Firstly, the case where p =8is considered. Figures 23, 24, and 25 depict capacity CDFs from 1 independent runs for different choices of coding gain and noise margin when the loop length is 75 m. The self cross-talk and alien interferences are all from FEXT. The frequency band follows VDSL 998. Figures 26, 27, and 28 contain the corresponding results for the extended VDSL 998 frequency band. 1 Capacity distribution of 8 pair systems(coding gain=3.8 db, margin=6 db).9.8 prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 8 Figure 23: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 3.8 db, margin 6 db. The loop length is 75 m. Self cross-talk and alien noise are from FEXT. The frequency band follows VDSL

23 1 Capacity distribution of 8 pair systems(coding gain=7.8 db, margin=6 db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 8 Figure 24: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 7.8 db, margin 6 db. The loop length is 75 m. Self cross-talk and alien noise are from FEXT. The frequency band follows VDSL Capacity distribution of 8 pair systems(coding gain=9.8 db, margin= db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 25: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 9.8 db, margin db. The loop length is 75 m. Self cross-talk and alien noise are from FEXT. The frequency band follows VDSL

24 1 Capacity distribution of 8 pair systems(coding gain=3.8 db, margin=6 db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 26: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 3.8 db, margin 6 db. The loop length is 75 m. Self cross-talk and alien noise are from FEXT. The frequency band follows extended VDSL Capacity distribution of 8 pair systems(coding gain=7.8 db, margin=6 db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 27: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 7.8 db, margin 6 db. The loop length is 75 m. Self cross-talk and alien noise are from FEXT. The frequency band follows extended VDSL

25 1 Capacity distribution of 8 pair systems(coding gain=9.8 db, margin= db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 28: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 9.8 db, margin db.the loop length is 75 m. Self cross-talk and alien noise are from FEXT. The frequency band follows extended VDSL 998 Figures 29, 3, and 31 depict the capacity CDF from 1 independent runs for different choices of coding gain and noise margin, when the loop length is 15 m. The self cross-talk and alien interferences are all from FEXT. The frequency band follows VDSL 998. Figures 32, 33, and 34 depict the corresponding results for the extended VDSL 998 frequency band. 24

26 1 Capacity distribution of 8 pair systems(coding gain=3.8 db, margin=6 db).9.8 prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 8 Figure 29: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 3.8 db, margin 6 db. The loop length is 15 m. Self cross-talk and alien noise are from FEXT. The frequency band follows VDSL Capacity distribution of 8 pair systems(coding gain=7.8 db, margin=6 db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 8 Figure 3: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 7.8 db, margin 6 db. The loop length is 15 m. Self cross-talk and alien noise are from FEXT. The frequency band follows VDSL

27 1 Capacity distribution of 8 pair systems(coding gain=9.8 db, margin= db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 8 Figure 31: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 9.8 db, margin db. The loop length is 15 m. Self cross-talk and alien noise are from FEXT. The frequency band follows VDSL Capacity distribution of 8 pair systems(coding gain=3.8 db, margin=6 db).9.8 prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 32: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 3.8 db, margin 6 db. The loop length is 15 m. Self cross-talk and alien noise are from FEXT. The frequency band follows extended VDSL

28 1 Capacity distribution of 8 pair systems(coding gain=7.8 db, margin=6 db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 33: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 7.8 db, margin 6 db. The loop length is 15 m. Self cross-talk and alien noise are from FEXT. The frequency band follows extended VDSL Capacity distribution of 8 pair systems(coding gain=9.8 db, margin= db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 34: Capacity distribution of 8 pair systems from 1 independent runs for coding gain 9.8 db, margin db. The loop length is 15 m. Self cross-talk and alien noise are from FEXT. The frequency band follows extended VDSL

29 Now the case where p =12is considered. Figures 35, 36 depict the capacity CDF for the frequency band of VDSL 998, and extended VDSL 998, respectively, from 1 independent runs when the loop length is 75 m. Here the coding gain is 3.8 db and noise margin is 6 db. The self cross-talk and alien interferences are all from FEXT. 1 Capacity distribution of 12 pair systems(coding gain=3.8 db, margin=6 db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 35: Capacity distribution of 12 pair systems from 1 independent runs for coding gain 3.8 db, margin 6 db. The loop length is 75 m. Self cross-talk and alien noise are from FEXT. The frequency band follows VDSL

30 1 Capacity distribution of 12 pair systems(coding gain=3.8 db, margin=6 db).9.8 prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 36: Capacity distribution of 12 pair systems from 1 independent runs for coding gain 3.8 db, margin 6 db. The loop length is 75 m. Self cross-talk and alien noise are from FEXT. The frequency band follows Extended VDSL 998 Figures 37, 38 depict the capacity CDF for the frequency band of VDSL 998, and extended VDSL 998, respectively, from 1 independent runs when the loop length is 15 m. Here the coding gain is 3.8 db and noise margin is 6 db. The self cross-talk and alien interferences are all from FEXT. 29

31 1 Capacity distribution of 12 pair systems(coding gain=3.8 db, margin=6 db).9.8 prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 37: Capacity distribution of 12 pair systems from 1 independent runs for coding gain 3.8 db, margin 6 db. The loop length is 15 m. Self cross-talk and alien noise are from FEXT. The frequency band follows VDSL Capacity distribution of 12 pair systems(coding gain=3.8 db, margin=6 db) prob(rate<x) L=1 L=2 L=4 L=8 L=12 L= Total rate () x 1 9 Figure 38: Capacity distribution of 12 pair systems from 1 independent runs for coding gain 3.8 db, margin 6 db. The loop length is 15 m. Self cross-talk and alien noise are from FEXT. The frequency band follows Extended VDSL 998 3

32 4 Conclusions & Coordination Benefit Assessment Our conclusions can be summarized as follows: Rates in the order of several hundred M per loop are attainable over very short DSL loops. Sum rates in the order of several G are attainable over a 28-loop binder, even if practical limitations, like modulation loss, are taken into account. Coordinated transmission can boost the per-loop capacity by as much as 8% when a significant fraction of the loops in the binder are coordinated. Coordinating a few loops can still afford a 2-3% increase in per-line capacity, but only if most of the remaining loops in the binder are silent. Even though a 2% boost in per-loop capacity may not appear impressive, it yields one more line for every five lines. This can be important in certain situations. Furthermore, Coordination appears to improve the ratio of minimum capacity to mean capacity. This can be quickly seen from the standard deviation plots in Section 2, and it is important from the operator s perspective as it limits outage. 31

33 5 Appendix A: Capacity Histograms and Moments via Full Combinatorial Calculations 32

34 5.1 FEXT plus NEXT.12.1 Capacity per line histogram of 75m loop for p=1 and Lnext=2, Lfext=2 Min = 2.63E+8 Mean = 3.96E+8 Max = 5.81E+8 Std = 5.21E+7 Min/Mean = x 1 8 Figure 39: Capacity histogram of 75m loop for uncoordinated transmission and 2 disturbers Capacity per line histogram of 15m loop for p=1 and Lnext=2, Lfext=2 Min = 1.88E+8 Mean = 3.16E+8 Max = 5.14E+8 Std = 5.8E+7 Min/Mean = x 1 8 Figure 4: Capacity histogram of 15m loop for uncoordinated transmission and 2 disturbers 33

35 .12.1 Capacity per line histogram of 75m loop for p=1 and Lnext=3, Lfext=3 Min = 2.55E+8 Mean = 3.69E+8 Max = 5.61E+8 Std = 4.3E+7 Min/Mean = x 1 8 Figure 41: Capacity histogram of 75m loop for uncoordinated transmission and 3 disturbers.12.1 Capacity per line histogram of 15m loop for p=1 and Lnext=3, Lfext=3 Min = 1.79E+8 Mean = 2.87E+8 Max = 4.92E+8 Std = 4.82E+7 Min/Mean = x 1 8 Figure 42: Capacity histogram of 15m loop for uncoordinated transmission and 3 disturbers 34

36 Capacity per line histogram of 75m loop for p=1 and Lnext=25, Lfext=25 Min = 2.25E+8 Mean = 2.57E+8 Max = 3.31E+8 Std = 1.92E+7 Min/Mean = x 1 8 Figure 43: Capacity histogram of 75m loop for uncoordinated transmission and 25 disturbers Capacity per line histogram of 15m loop for p=1 and Lnext=25, Lfext=25 Min = 1.48E+8 Mean = 1.73E+8 Max = 2.56E+8 Std = 1.72E+7 Min/Mean = x 1 8 Figure 44: Capacity histogram of 15m loop for uncoordinated transmission and 25 disturbers 35

37 Capacity per line histogram of 75m loop for p=2 and Lnext=2, Lfext=2 Min = 3.13E+8 Mean = 4.18E+8 Max = 5.81E+8 Std = 4.65E+7 Min/Mean = x 1 8 Figure 45: Capacity per line histogram of 75m loop for 2 coordinated pairs and 2 disturbers Capacity per line histogram of 15m loop for p=2 and Lnext=2, Lfext=2 Min = 2.34E+8 Mean = 3.45E+8 Max = 5.9E+8 Std = 5.14E+7 Min/Mean = x 1 8 Figure 46: Capacity per line histogram of 15m loop for 2 coordinated pairs and 2 disturbers 36

38 .12.1 Capacity per line histogram of 75m loop for p=2 and Lnext=3, Lfext=3 Min = 2.96E+8 Mean = 3.87E+8 Max = 5.58E+8 Std = 3.91E+7 Min/Mean = x 1 8 Figure 47: Capacity per line histogram of 75m loop for 2 coordinated pairs and 3 disturbers Capacity per line histogram of 15m loop for p=2 and Lnext=3, Lfext=3 Min = 2.12E+8 Mean = 3.1E+8 Max = 4.85E+8 Std = 4.42E+7 Min/Mean = x 1 8 Figure 48: Capacity per line histogram of 15m loop for 2 coordinated pairs and 3 disturbers 37

39 .12.1 Capacity per line histogram of 75m loop for p=2 and Lnext=24, Lfext=24 Min = 2.45E+8 Mean = 2.7E+8 Max = 3.42E+8 Std = 1.87E+7 Min/Mean = x 1 8 Figure 49: Capacity per line histogram of 75m loop for 2 coordinated pairs and 24 disturbers Capacity per line histogram of 15m loop for p=2 and Lnext=24, Lfext=24 Min = 1.59E+8 Mean = 1.88E+8 Max = 2.65E+8 Std = 2.3E+7 Min/Mean = x 1 8 Figure 5: Capacity per line histogram of 15m loop for 2 coordinated pairs and 24 disturbers 38

40 Capacity per line histogram of 75m loop for p=4 and Lnext=1, Lfext=1 Min = 5.27E+8 Mean = 5.8E+8 Max = 6.58E+8 Std = 2.25E+7 Min/Mean = x 1 8 Figure 51: Capacity per line histogram of 75m loop for 4 coordinated pairs and 1 disturber Capacity per line histogram of 15m loop for p=4 and Lnext=1, Lfext=1 Min = 4.55E+8 Mean = 5.8E+8 Max = 5.85E+8 Std = 2.41E+7 Min/Mean = x 1 8 Figure 52: Capacity per line histogram of 15m loop for 4 coordinated pairs and 1 disturber 39

41 .12.1 Capacity per line histogram of 75m loop for p=4 and Lnext=2, Lfext=2 Min = 3.35E+8 Mean = 4.48E+8 Max = 5.96E+8 Std = 3.35E+7 Min/Mean = x 1 8 Figure 53: Capacity per line histogram of 75m loop for 4 coordinated pairs and 2 disturbers Capacity per line histogram of 15m loop for p=4 and Lnext=2, Lfext=2 Min = 2.82E+8 Mean = 3.85E+8 Max = 5.29E+8 Std = 3.52E+7 Min/Mean = x 1 8 Figure 54: Capacity per line histogram of 15m loop for 4 coordinated pairs and 2 disturbers 4

42 .12.1 Capacity per line histogram of 75m loop for p=4 and Lnext=3, Lfext=3 Min = 3.15E+8 Mean = 4.2E+8 Max = 5.38E+8 Std = 2.8E+7 Min/Mean = x 1 8 Figure 55: Capacity per line histogram of 75m loop for 4 coordinated pairs and 3 disturbers Capacity per line histogram of 15m loop for p=4 and Lnext=3, Lfext=3 Min = 2.47E+8 Mean = 3.32E+8 Max = 4.62E+8 Std = 3.8E+7 Min/Mean = x 1 8 Figure 56: Capacity per line histogram of 15m loop for 4 coordinated pairs and 3 disturbers 41

43 .12.1 Capacity per line histogram of 75m loop for p=4 and Lnext=23, Lfext=23 Min = 2.52E+8 Mean = 2.74E+8 Max = 3.12E+8 Std = 1.26E+7 Min/Mean = x 1 8 Figure 57: Capacity per line histogram of 75m loop for 4 coordinated pairs and 23 disturbers.12.1 Capacity per line histogram of 15m loop for p=4 and Lnext=23, Lfext=23 Min = 1.7E+8 Mean = 1.92E+8 Max = 2.39E+8 Std = 1.39E+7 Min/Mean = x 1 8 Figure 58: Capacity per line histogram of 15m loop for 4 coordinated pairs and 23 disturbers 42

44 .12.1 Capacity per line histogram of 75m loop for p=6 and Lnext=1, Lfext=1 Min = 5.93E+8 Mean = 6.23E+8 Max = 6.74E+8 Std = 1.23E+7 Min/Mean = x 1 8 Figure 59: Capacity per line histogram of 75m loop for 6 coordinated pairs and 1 disturber Capacity per line histogram of 15m loop for p=6 and Lnext=1, Lfext=1 Min = 5.19E+8 Mean = 5.47E+8 Max = 6.1E+8 Std = 1.34E+7 Min/Mean = x 1 8 Figure 6: Capacity per line histogram of 15m loop for 6 coordinated pairs and 1 disturber 43

45 .12.1 Capacity per line histogram of 75m loop for p=6 and Lnext=2, Lfext=2 Min = 4.57E+8 Mean = 5.16E+8 Max = 6.14E+8 Std = 1.82E+7 Min/Mean = x 1 8 Figure 61: Capacity per line histogram of 75m loop for 6 coordinated pairs and 2 disturbers.12.1 Capacity per line histogram of 15m loop for p=6 and Lnext=2, Lfext=2 Min = 3.93E+8 Mean = 4.47E+8 Max = 5.45E+8 Std = 1.95E+7 Min/Mean = x 1 8 Figure 62: Capacity per line histogram of 15m loop for 6 coordinated pairs and 2 disturbers 44

46 Capacity per line histogram of 75m loop for p=6 and Lnext=21, Lfext=21 Min = 2.62E+8 Mean = 2.8E+8 Max = 3.17E+8 Std = 9.92E+6 Min/Mean = x 1 8 Figure 63: Capacity per line histogram of 75m loop for 6 coordinated pairs and 21 disturbers.12.1 Capacity per line histogram of 15m loop for p=6 and Lnext=21, Lfext=21 Min = 1.78E+8 Mean = 1.98E+8 Max = 2.41E+8 Std = 1.11E+7 Min/Mean = x 1 8 Figure 64: Capacity per line histogram of 15m loop for 6 coordinated pairs and 21 disturbers 45

47 Capacity per line histogram of 75m loop for p=8 and Lnext=1, Lfext=1 Min = 6.28E+8 Mean = 6.47E+8 Max = 6.85E+8 Std = 7.95E+6 Min/Mean = x 1 8 Figure 65: Capacity per line histogram of 75m loop for 8 coordinated pairs and 1 disturber.12.1 Capacity per line histogram of 15m loop for p=8 and Lnext=1, Lfext=1 Min = 5.51E+8 Mean = 5.7E+8 Max = 6.9E+8 Std = 8.78E+6 Min/Mean = x 1 8 Figure 66: Capacity per line histogram of 15m loop for 8 coordinated pairs and 1 disturber 46

48 Capacity per line histogram of 75m loop for p=8 and Lnext=19, Lfext=19 Min = 2.68E+8 Mean = 2.87E+8 Max = 3.2E+8 Std = 8.32E+6 Min/Mean = x 1 8 Figure 67: Capacity per line histogram of 75m loop for 8 coordinated pairs and 19 disturbers Capacity per line histogram of 15m loop for p=8 and Lnext=19, Lfext=19 Min = 1.85E+8 Mean = 2.6E+8 Max = 2.46E+8 Std = 9.58E+6 Min/Mean = x 1 8 Figure 68: Capacity per line histogram of 15m loop for 8 coordinated pairs and 19 disturbers 47

49 Capacity per line histogram of 75m loop for p=14 and Lnext=1, Lfext=1 Min = 6.75E+8 Mean = 6.81E+8 Max = 7.2E+8 Std = 3.43E+6 Min/Mean = x 1 8 Figure 69: Capacity per line histogram of 75m loop for 14 coordinated pairs and 1 disturber Capacity per line histogram of 15m loop for p=14 and Lnext=1, Lfext=1 Min = 5.94E+8 Mean = 6.2E+8 Max = 6.22E+8 Std = 3.87E+6 Min/Mean = x 1 8 Figure 7: Capacity per line histogram of 15m loop for 14 coordinated pairs and 1 disturber 48

50 .12.1 Capacity per line histogram of 75m loop for p=14 and Lnext=13, Lfext=13 Min = 3.1E+8 Mean = 3.19E+8 Max = 3.49E+8 Std = 6.5E+6 Min/Mean = x 1 8 Figure 71: Capacity per line histogram of 75m loop for 14 coordinated pairs and 13 disturbers Capacity per line histogram of 15m loop for p=14 and Lnext=13, Lfext=13 Min = 2.25E+8 Mean = 2.46E+8 Max = 2.84E+8 Std = 7.86E+6 Min/Mean = x 1 8 Figure 72: Capacity per line histogram of 15m loop for 14 coordinated pairs and 13 disturbers 49

51 Capacity per line histogram of 75m loop for p=2 and Lnext=1, Lfext=1 Min = 6.94E+8 Mean = 6.97E+8 Max = 7.7E+8 Std = 2.7E+6 Min/Mean = x 1 8 Figure 73: Capacity per line histogram of 75m loop for 2 coordinated pairs and 1 disturber Capacity per line histogram of 15m loop for p=2 and Lnext=1, Lfext=1 Min = 6.12E+8 Mean = 6.16E+8 Max = 6.28E+8 Std = 2.35E+6 Min/Mean = x 1 8 Figure 74: Capacity per line histogram of 15m loop for 2 coordinated pairs and 1 disturber 5

52 Capacity per line histogram of 75m loop for p=2 and Lnext=2, Lfext=2 Min = 6.5E+8 Mean = 6.56E+8 Max = 6.76E+8 Std = 3.2E+6 Min/Mean = x 1 8 Figure 75: Capacity per line histogram of 75m loop for 2 coordinated pairs and 2 disturbers Capacity per line histogram of 15m loop for p=2 and Lnext=2, Lfext=2 Min = 5.7E+8 Mean = 5.77E+8 Max = 6.E+8 Std = 3.41E+6 Min/Mean = x 1 8 Figure 76: Capacity per line histogram of 15m loop for 2 coordinated pairs and 2 disturbers 51

53 Capacity per line histogram of 75m loop for p=2 and Lnext=3, Lfext=3 Min = 6.7E+8 Mean = 6.15E+8 Max = 6.37E+8 Std = 3.89E+6 Min/Mean = x 1 8 Figure 77: Capacity per line histogram of 75m loop for 2 coordinated pairs and 3 disturbers Capacity per line histogram of 15m loop for p=2 and Lnext=3, Lfext=3 Min = 5.28E+8 Mean = 5.39E+8 Max = 5.63E+8 Std = 4.35E+6 Min/Mean = x 1 8 Figure 78: Capacity per line histogram of 15m loop for 2 coordinated pairs and 3 disturbers 52

54 Capacity per line histogram of 75m loop for p=2 and Lnext=6, Lfext=6 Min = 4.85E+8 Mean = 4.98E+8 Max = 5.24E+8 Std = 5.89E+6 Min/Mean = x 1 8 Figure 79: Capacity per line histogram of 75m loop for 2 coordinated pairs and 6 disturbers Capacity per line histogram of 15m loop for p=2 and Lnext=6, Lfext=6 Min = 4.12E+8 Mean = 4.29E+8 Max = 4.55E+8 Std = 6.49E+6 Min/Mean = x 1 8 Figure 8: Capacity per line histogram of 15m loop for 2 coordinated pairs and 6 disturbers 53

55 .8.7 Capacity per line histogram of 75m loop for p=2 and Lnext=7, Lfext=7 Min = 4.47E+8 Mean = 4.62E+8 Max = 4.87E+8 Std = 6.53E+6 Min/Mean = x 1 8 Figure 81: Capacity per line histogram of 75m loop for 2 coordinated pairs and 7 disturbers Capacity per line histogram of 15m loop for p=2 and Lnext=7, Lfext=7 Min = 3.78E+8 Mean = 3.94E+8 Max = 4.21E+8 Std = 7.15E+6 Min/Mean = x 1 8 Figure 82: Capacity per line histogram of 15m loop for 2 coordinated pairs and 7 disturbers 54

56 Capacity per line histogram of 75m loop for p=22 and Lnext=1, Lfext=1 Min = 6.98E+8 Mean = 7.E+8 Max = 7.8E+8 Std = 1.82E+6 Min/Mean = x 1 8 Figure 83: Capacity per line histogram of 75m loop for 22 coordinated pairs and 1 disturber.12.1 Capacity per line histogram of 15m loop for p=22 and Lnext=1, Lfext=1 Min = 6.16E+8 Mean = 6.19E+8 Max = 6.28E+8 Std = 2.7E+6 Min/Mean = x 1 8 Figure 84: Capacity per line histogram of 15m loop for 22 coordinated pairs and 1 disturber 55

57 Capacity per line histogram of 75m loop for p=22 and Lnext=2, Lfext=2 Min = 6.58E+8 Mean = 6.62E+8 Max = 6.77E+8 Std = 2.72E+6 Min/Mean = x 1 8 Figure 85: Capacity per line histogram of 75m loop for 22 coordinated pairs and 2 disturbers Capacity per line histogram of 15m loop for p=22 and Lnext=2, Lfext=2 Min = 5.77E+8 Mean = 5.83E+8 Max = 6.1E+8 Std = 3.8E+6 Min/Mean = x 1 8 Figure 86: Capacity per line histogram of 15m loop for 22 coordinated pairs and 2 disturbers 56

58 Capacity per line histogram of 75m loop for p=22 and Lnext=3, Lfext=3 Min = 6.18E+8 Mean = 6.25E+8 Max = 6.41E+8 Std = 3.51E+6 Min/Mean = x 1 8 Figure 87: Capacity per line histogram of 75m loop for 22 coordinated pairs and 3 disturbers.12.1 Capacity per line histogram of 15m loop for p=22 and Lnext=3, Lfext=3 Min = 5.39E+8 Mean = 5.48E+8 Max = 5.67E+8 Std = 3.95E+6 Min/Mean = x 1 8 Figure 88: Capacity per line histogram of 15m loop for 22 coordinated pairs and 3 disturbers 57

59 .12.1 Capacity per line histogram of 75m loop for p=22 and Lnext=4, Lfext=4 Min = 5.8E+8 Mean = 5.88E+8 Max = 6.5E+8 Std = 4.26E+6 Min/Mean = x 1 8 Figure 89: Capacity per line histogram of 75m loop for 22 coordinated pairs and 4 disturbers Capacity per line histogram of 15m loop for p=22 and Lnext=4, Lfext=4 Min = 5.3E+8 Mean = 5.13E+8 Max = 5.32E+8 Std = 4.77E+6 Min/Mean = x 1 8 Figure 9: Capacity per line histogram of 15m loop for 22 coordinated pairs and 4 disturbers 58

60 Capacity per line histogram of 75m loop for p=22 and Lnext=5, Lfext=5 Min = 5.43E+8 Mean = 5.52E+8 Max = 5.7E+8 Std = 5.E+6 Min/Mean = x 1 8 Figure 91: Capacity per line histogram of 75m loop for 22 coordinated pairs and 5 disturbers Capacity per line histogram of 15m loop for p=22 and Lnext=5, Lfext=5 Min = 4.68E+8 Mean = 4.79E+8 Max = 4.98E+8 Std = 5.57E+6 Min/Mean = x 1 8 Figure 92: Capacity per line histogram of 15m loop for 22 coordinated pairs and 5 disturbers 59

61 5.2 FEXT only.8.7 Capacity per line histogram of 75m loop for p=1 and Lnext=, Lfext=2 Min = 3.26E+8 Mean = 4.78E+8 Max = 6.42E+8 Std = 5.99E+7 Min/Mean = x 1 8 Figure 93: Capacity histogram of 75m loop for uncoordinated transmission and 2 disturbers.7.6 Capacity per line histogram of 15m loop for p=1 and Lnext=, Lfext=2 Min = 2.87E+8 Mean = 4.41E+8 Max = 5.95E+8 Std = 6.2E+7 Min/Mean = x 1 8 Figure 94: Capacity histogram of 15m loop for uncoordinated transmission and 2 disturbers 6

62 Capacity per line histogram of 75m loop for p=1 and Lnext=, Lfext=25 Min = 2.99E+8 Mean = 3.33E+8 Max = 4.36E+8 Std = 2.18E+7 Min/Mean = x 1 8 Figure 95: Capacity histogram of 75m loop for uncoordinated transmission and 25 disturbers Capacity per line histogram of 15m loop for p=1 and Lnext=, Lfext=25 Min = 2.57E+8 Mean = 2.91E+8 Max = 4.5E+8 Std = 2.33E+7 Min/Mean = x 1 8 Figure 96: Capacity histogram of 15m loop for uncoordinated transmission and 25 disturbers 61