Interimpulse-delay generation 3. Impulse amplitude scaling

Figure 3.4. Impulse generation mechanism.

The waveform storage may have measured impulses or mathematically modeled impulses stored, perhaps several of them. (Alternatively, the waveforms may be statistically generated.) Each stored waveform may have a certain probability of being selected when the generator extracts the next impulse from storage. The time of extraction is measured from the termination of the last impulse generated and is determined by the interimpulse-delay generator or "delay to next impulse" in Figure 3.4. The delay generator is reset each time the last impulse is finished transmitting. The impulse waveform is scaled by the amplitude generator. The amplitude is selected randomly from a distribution of amplitude or possibly fixed at some constant, depending on the specifics of the impulses that the telephone company desires to test (or that the ADSL designer desires to analyze).

Sometimes a random amplitude is selected for each stored sample of the impulse in addition to a gross gain applied to the entire impulse.

Impulse Amplitude Distribution

The amplitude box in Figure 3.4 simply scales the impulse waveform by some (usually randomly chosen) constant. Of interest is the distribution for the constant, which is often randomly selected each time a sample of an impulse is generated.

France Telecom [6] provides histograms of various impulse-noise measurements. In particular, Figure 3.5 (repeated from [6]) illustrates a histogram of impulse energy. Small-energy impulses have the highest probability, whereas very large impulses tend to be less probable. One can see that about 90 percent of impulses have a duration of less than 250 µs (see also Figure 3.7) and have amplitudes of less than about 10 mV. However, there are a small fraction of impulses that may also have very high amplitudes. Thus a theoretical model that perhaps somewhat conservatively overestimates the probability of large peak impulses may be useful in making theoretical projections based on the energy histogram in Figure 3.5. Impulses with amplitudes greater than 200 mV are very unlikely. For impulse sizes below 10 mV, the probability grows roughly linearly with the log of the peak voltage (or energy, assuming the logarithm of the two are linearly related).

Figure 3.5. Impulse energy histogram (horizontal axis is dB in Watts/Hz). Courtesy France Telecom [6].

Figure 3.7. Impulse duration histogram (horizontal axis is μs, to 50 ms spanned). Courtesy France Telecom [6].

The author has empirically guessed a distribution for the energy from the histogram in dBm/Hz.

which has a discontinuity at –75 dBm/Hz but projects 90 percent of impulses below this energy level and 10 percent above. A distribution such as the one above, even given it is not perfect, allows

projection of ADSL error statistics theoretically. This particular model does not predict a large number of tiny impulses. It is useful only for a gross gain per generated impulse and not for each sample of an impulse. This model would be used when a set of a few impulses is stored and generated as in Figure 3.4.

A more elegant approach to impulse energy characterization has been provided by the work of Henkel and colleagues [24],[25]. Two distributions are provided that are shown [25] to model well measurements by British Telecom and Deutsche Telekom. The first model is a simple exponential distribution^[5] given by

[5] One can verify the coefficient 240 by using the substitution in the integral and then using the unit-variance one-sided odd-moment Gaussian distribution formula

.

which has mean zero and variance . Typical values for the parameter "A" range between 10 nV and 10 μV with the lower part of the range being more typical. The tail of this distribution above 10 nanovolts is plotted in Figure 3.6 for A = 30 nV. One can see an enormous concentration of probability at very low impulse amplitudes. Impulses that exceed 1 mV in

amplitude have a probability of occurrence less than 0.1 percent. This is somewhat of an artifact of the impulse model, but experts [24], [25] have found that the tails of this distribution tend to model well the probabilities of large impulse occurrences. They also show that this distribution is

dominated most of the time by typical Gaussian noise so the overweighting of small impulses is inconsequential. The difficulty in generating this random variable from other more available distributions like uniform or Gaussian motivated Mann et al. [25] to find another distribution with more mathematically pleasant properties, most notably the Weibull distribution that is related to the gamma distribution family and can be generated with specified autocorrelation properties according to a recent method of Tough and Ward [8]. However, the authors [25] did find greater error with the Weibull distribution. The author of this chapter notes instead that generation of any random variable with density p(a) from a uniform random variable essentially involves solving the differential

equation g'(a) = p(a) and then inverting the function to obtain a = g^-1(u) where u is a uniform random variable. The differential equation corresponds to a family for the exponential distribution p(a) that must be evaluated numerically to form a table look up for . The impulse

amplitude scaling apparatus inverts that table (which is just another look-up table) to get a = g^-1(u).

The latter alternative may be preferred for many implementation reasons in generating impulses for actual laboratory measurement or simulation. Clearly this same method of table look up can be used for the empirical impulse-energy distribution above also. These impulse distributions apply (unlike P (E) above) to each sample of a transmitted impulse.

Figure 3.6. Exponential probability distribution for impulses.

The alternative Weibull distribution offered by Mann et al. [25] is

This distribution also overemphasizes small values excessively, but well approximates the tail of the impulse distribution because of the same exponential dependence. Generation of this distribution is described in [8] and can be tailored to provide a given autocorrelation function also. However, the generator in Figure 3.5 (unlike the structures in [25] and considered by ETSI) does not need

specification of the autocorrelation function of the impulse because several impulses can simply be stored and then selected each with some individual probability, following more closely the impulse classification in [7] into a handful of representative shapes.

Various phone companies have measured the values for A in the exponential distribution and α,b in the Weibull distribution. We list a few here, partially repeated from [7]:

α in Volts for Parameters in This Table Weibull Exponential

North American phone company impulse measurements were not available at time of writing.

Impulse Duration Distribution

Figure 3.7 illustrates a histogram of measured impulse durations. Clearly some impulses are very long, but have low probability of occurrence. Indeed, many impulses are very short, less than 100 microseconds in duration. These short impulses will most often have small amplitudes, and indeed it is possible for many small impulses to follow each other closely in time without disruption of

service. The approach of standards groups so far has been to define models that are accurate in the tails of probability distributions, that is, that big impulse samples occur infrequently and last a long time whereas small impulse samples of little consequence can occur often. If the tails are modeled correctly by testing or analysis, then the frequent small impulses may not significantly alter performance projections (and indeed match measurements anyway as evident from Figure 3.7).

Extremely long impulses typically have energy concentrated in a small number of frequency bands.

Specifically in the model of Figure 3.4, the impulses are separated by a delay. Figure 3.8 illustrates a model of the generation of the inter-impulse delay. Basically, the generator contains a 2-state

machine (Markov model) with one of the states being that the next impulse will occur in less than 1 ms and the other state being that the next impulse will occur after more than 1 ms of delay. Each time an impulse is sent, the state machine can make a transition, either to stay in the same state or to change states.

Figure 3.8. State transition diagram for interimpulse delay.

The probability that a short delay is followed by another short delay is written as p_s/s whereas the probability of a long delay following a short delay is p_l/s. The probabilities can be summarized in the matrix

α b A

BT (CPE) .263 4.77 9.12μV

DT (CPE) .486 44.40 23.3nV

DT (CO) .216 12.47 30.67nV

and the so-called Markov distribution for the probability of short delays and long delays is determined by

There are two distributions for the interimpulse delay τ:

with typical λ = .16 and

with typical α = 1.5 and τ in ms. The first distribution produces with small probability delays that can exceed 1 ms, and these delays are discarded and the random selection procedure cycled again until a delay less than 1 ms occurs. The second distribution exists from 1 ms to infinity. Both can be easily generated from a uniform random variable u on the unit interval [0,1] by the transformations

and

respectively.

A 2x2 matrix is then provided with upper left entry corresponding to the probability that a short interarrival time is followed by a second short interarrival, the lower right corresponding to long followed by long. For instance, ETSI uses a model based on British measurements:

which (e.g., for the default) means that 80 percent of the time a short interarrival occurs, the next interarrival is also short, and the other 20 percent of the time the next impulse is more than 1 ms away and has the probability of interarrival given by the long distribution. Similarly, 60 percent of long-interarrival impulses are followed by long interarrival impulses. The default matrix provides that one-third of the impulses are separated by at least 1 ms, whereas the other two-thirds are very close together. Most samples of any impulse are again small, so even very many samples of those

impulses with long interarrival times between them are small. The possibility of two large impulses in a row is thus very small, but not zero. France Telecom has found yet another model that is more characteristic of their network and is

Impulse Waveforms

One particularly in-depth study by C. Collobert and colleagues at France Telecom [7] has found after looking at enormous volumes of ADSL impulse data that impulses can be classified through the use of neural networks into categories that have similar profiles in terms of a cumulative spectral density.

FT was able to find five classes that well characterized all their impulses. Each ADSL class is then characterized by a single impulse for test purposes, but the amplitude of that impulse may be nominal (0 dB) or increased to 6 dB larger, and decreased -6, -12, -18, and -24 dB lower in

amplitude. An impulse used in one of the ADSL classes derived by FT appears in Figure 3.9. Here the amplitude of the impulse is very large, corresponding to a scaling with rare occurrence in the model of Figure 3.4. The associated frequency spectrum is also plotted in Figure 3.9. Note the impulse is large and long and could be expected to cause difficulty for transmission. With only a few classes, one could evaluate the probability of the different classes empirically (or perhaps adaptively in real time) and then complete the model of Figure 3.4by storing representative impulse waveforms in the impulse-waveform-storage box. By contrast, in statistical generation using the exponential or Weibull distributions, the autocorrelation of classes of impulses may be stored, and the impulses are generated using these autocorrelations [25].

Figure 3.9. France Telecom's "Imp" class impulse (horizontal axes in μs and ADSL DMT tone index, respectively, vertical axes in mV and dBm/Hz).

Others have tried to create mathematical models for the impulse. For instance, Reference [1]

describes impulses and a characterization published by BT known as the Cook Pulse. The Cook Pulse has a discrete-time model when sampled at interval T, or equivalently at times kT, that is, zero at time k = 0, and otherwise is

Approximately 45 percent of the impulse energy is in the peak sample and the rest decays with time.

Longer impulses tend to have larger peak voltages, and the peak voltage also increases with increased bandwidth use (sampling rate, 1/T) of the DSL system, roughly as

In recent years, many phone companies have further studied impulse noise and found the Cook pulse

to be insufficient in characterization. Thus other models have been attempted. One such model was proposed in [25] for ETSI specification. This model essentially models the impulse as an

exponentially windowed sinusoid with the exponential decay of the window and the frequency of the sinusoid each determined by random selection from a Guassian distribution. The duration of the window is also random and determined by selection from a convex sum of two log-normal distributions.

The model for the autocorrelation function of the impulse is thus

where g and β are the Gaussian parameters. The parameter g is selected to be one of 3 Gaussian random variables as shown in Table 3.1, whereas the window parameter β is selected from one of four Gaussians characterized by the randomly selected length of the impulse as shown in Table 3.2.

The distribution for the length of the impulse (equivalently the length of the exponential window) is

where the customer premises parameters are B = 1, s₁ = 1.15, t₁ = 18 µs, and thus the second term above is zero, and the CO parameters are B = .25, s₁ = .75, t₁ = 8 µs, s₂ = 1.0, t₂ = 125 µs.This model is likely less representative, and certainly much more complex to implement and understand, than Collobert's representative waveforms in [7]. ETSI is now evaluating the France Telecom impulse-class model.

Probability of Error Analysis with Impulse Noise

The authors would especially like to thank Dr. Wei Yu and Mr. Daniel Gardan for inputs to this section.

ADSL uses Reed-Solomon codes to provide coding gain and also to mitigate impulse noise

disturbance. The RS code is a block code of block size up to 255 based on GF(2⁸) (or bytes). An RS code word can therefore be up to 255 bytes long (in ADSL systems). The RS code word boundary is aligned with the DMT symbol boundary. Because of the rate adaptive nature of a DMT system, the number of bits in each DMT symbol varies depending on the data rate. ADSL [3] allows a variable number (1, 2, 4, 8, or 16) of DMT symbols for each RS code word. ADSL also allows an even number of parity bytes (up to 16) to be included in an RS code word. The decoder is able to correct up to half as many error bytes.

Table 3.1. g Values

Probability of Gaussian Mean (ns) Standard Deviation (ns)

.24 115 25.4

.56 486 178.7

.20 640 9.3

In order to take full advantage of the error-protecting ability from RS code, it is important to correctly choose the number of DMT symbols per RS code word and the number of parity bytes included in each code word. From a pure error-correcting point of view, for the same amount of parity overhead, it is always the best to have as many DMT symbols in an RS code word as possible.

For example, a system that includes 8 DMT symbols and 16 bytes of parity for each RS code word performs better than a system that includes 4 DMT symbols and 8 bytes of parity for each RS code word. This is because the former system can correct at least as many errors as the latter system. A longer RS code word means longer delay, and possibly more decoder complexity. The analysis here concentrates on reducing the impact of impulses as best as is possible, and will therefore include as many DMT symbols per RS code word as possible for impulse protection, necessarily enduring delay. Then it only remains to decide how many parity bytes to include in each RS code word.

In most cases, the maximum error-correcting protection against Gaussian noise when the system operates near capacity is obtained when the parity overhead is approximately 6 to 10 percent [18], which is about 16 parity bytes in 128–256 bytes. The RS code word length depends on the number of DMT symbols per RS code word; thus, it depends on the system data rate. (A framing overhead of 128 kbps is assumed here.) For example, at 608 kbps, each DMT symbol is about (608 + 128)/4 = 184 bits = 23 bytes long. So, up to 8 DMT symbols can be grouped into an RS code word. Assuming 16 bytes of parity are inserted, the resulting RS code word is 23 · 8 + 16 = 200 bytes long. At 1.216 Mbps, each DMT symbol is about (1216 + 128)/4 = 336 bits = 42 bytes long. So, up to 4 DMT symbols can be grouped into an RS code word. Again assuming 16 bytes of parity, the code word length is 42 · 4 + 16 = 184 bytes long. At 2.048 Mbps, each DMT symbol is (2048 + 128)/4 = 68 bytes long. So, 2 DMT symbols are grouped into an RS code word, resulting in a code word length of 68 · 2 + 16 = 152 bytes.

Assuming 16 bytes of parity, the RS code is able to correct up to 8 bytes of error. When more than 8 errors occur, the Reed-Solomon decoder will either decode to a false code word, in which case nearly all the bytes (or half of the bits) are incorrect; or the decoder will declare decoding failure, in which case the original code word is unchanged so the output has the same number of error bytes as before. The following calculation shows that decoder failure is a much more likely possibility:

The RS code operates on GF(256). The total number of length -256 valid code words is 256²⁴⁰. (This is because the 16-byte parity is to be added to any 240-byte message.) For a correctable error to occur, at most 8 bytes can be in error. These errors can be located in possible byte

locations. In each error location, the number of ways that an error can occur is 255. So, for each code

Table 3.2. β Values

word, there are "neighbors" that a decoder can perfectly correct. But, this is a small fraction of the total number of errors, as , where 256²⁵⁶ is the total number of possible 256-byte strings. So, when an impulse hits and an uncorrectable error occurs, the probability that the uncorrectable error will fall in the neighborhood of some other code words is small. So, the RS decode is likely to declare decoding failure, thus keeping the number of error bytes same as before.

Many ADSL receivers use erasures for impulses. A simple erasure mechanism that works very well is to compare the sum of squared differences between tone decoder slicers' outputs and inputs.

Nominally this sum is nearly zero if there is no impulse. When this sum exceeds a threshold, all bytes in the DMT symbol are "erased" (marked). The RS decoder can then correct erased bytes (twice as much). Even in this case, failed RS decoding is easily detected.

The following analyzes the coding gain of an RS code in an AWGN channel. In an AWGN channel (or a channel with a well-designed equalizer), the probability of error for each byte is approximately the same. Let Pe denote the probability of a byte error. When Pe is small, the probability of code word error is closely approximated by the probability that 9 byte errors occur. When 9 byte errors occur, the Reed-Solomon decoder will declare decoding failure, so the total number of error bytes will be 9. For each byte error, the most probable channel defect is that a single bit is likely to be wrong. So the probability of bit error in the code word is 9/200/8 = 4.4 · 10^-3, when a decoding error occurs. Because the required probability of bit error is 10^-7, the required probability of code word error is 10^-7/4.4 · 10^-3. In this case, a 608 kbps ADSL system with 200 bytes per RS code word needs to satisfy:

Thus the uncoded system needs to have Pe = 0.0065 for each byte to attain an overall probability of bit error 10^-7. Since the probability of byte error is 0.0065, the uncoded probability of bit error is then^[6] 0.0065/8 = 8.1 · 10^-4. The gap for QAM at P_b = 8.1 · 10^-4 is found by noticing 2Q(10.5dB) = 8.1 · 10^-4. So instead of requiring the argument of the Q-function to be 14.5dB, with coding, only 10.5dB is needed. In other words, coding provided a gain of 14.5dB - 10.5dB = 4.0dB. This is the coding gain for a 608 kbps system with 8 DMT symbols per RS code word and 16 parity bytes. This (raw) coding gain does not take the extra parity overhead into account. In reality, having 16 bytes of parity for every 8 DMT symbols translates to an extra 64kbps coding overhead. The loss in dB for

Thus the uncoded system needs to have Pe = 0.0065 for each byte to attain an overall probability of bit error 10^-7. Since the probability of byte error is 0.0065, the uncoded probability of bit error is then^[6] 0.0065/8 = 8.1 · 10^-4. The gap for QAM at P_b = 8.1 · 10^-4 is found by noticing 2Q(10.5dB) = 8.1 · 10^-4. So instead of requiring the argument of the Q-function to be 14.5dB, with coding, only 10.5dB is needed. In other words, coding provided a gain of 14.5dB - 10.5dB = 4.0dB. This is the coding gain for a 608 kbps system with 8 DMT symbols per RS code word and 16 parity bytes. This (raw) coding gain does not take the extra parity overhead into account. In reality, having 16 bytes of parity for every 8 DMT symbols translates to an extra 64kbps coding overhead. The loss in dB for