As already has been stated in 4.3, the ANSI standardising body selected DMT as t

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17:40 Aug 10, 2002

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English term or phrase:As already has been stated in 4.3, the ANSI standardising body selected DMT as t

As already has been stated in 4.3, the ANSI standardising body selected DMT as the modulation scheme for ADSL. It is a special form of implementation of multicarrier modulation, based on the discrete Fourier transform that can conveniently be implemented in a fully digital way. The main advantages of DMT in comparison with other modulation schemes are basically the same as the ones that have already been pointed out as the advantages of multicarrier modulation in general in 4.2.2. The advantage of DMT, compared with other multicarrier modulation (MCM) approaches, is the possibility of fully digital implementation and (compared with other MCM schemes) relatively low complexity as will be pointed out in the following sections. The achievable rate in such a system can be calculated using equation 26. The formula itself is very similar to the one for the calculation of the maximally achievable bit rate in a QAM-system [19]. In fact, the only difference is the range of the integral. In this case it only includes frequencies for that the integrand is at least equal to one, meaning that at least one bit can be transmitted over that subchannel, while in case of single carrier QAM the frequency range expands to all frequencies where the integrand is non-zero. In practice the difference implied by the different spectral coverage is negligible and on severely distorted channels, as for example telephone lines of realistic lengths, this disadvantage is compensated by the spectral flexibility of DMT that will be treated in 6.1.1. The use of the integral sign in equation 26 is an approximation that holds for high numbers of subchannels or, in other words, low bandwidths for each one of them. The formula, shown in equation 26 is in this form only valid for uncoded DMT without the use of a cyclic prefix, which is a special form of guard space in the time domain (see 6.1.3). Since the calculation of the inverse Q-function is somewhat complicated and requires advanced software, for a fixed bit error rate (BER, in this case 10-7) the then constant term 3/(Q-1(*))2 can be replaced by the performance gap, -9,8 dB (for this BER). This leads to the simplified formula, shown as equation 27. This one also includes the loss due to a cyclic prefix (see 6.1.3), although the impact in realistic approaches is only up to about 0,4 dB and therefore relatively small [19], [45], and the possible coding gain of a trellis code. In 6.1 the DMT-system will be presented in a step by step fashion, emphasising on the elements that have not already been sufficiently explained in 4.2.2. After that some specific topics, as for example synchronisation (see 6.2), coding (see 6.4) and echo cancellation (see 6.3.3) will be treated.

with
Rmax Maximal bit rate for uncoded DMT in bit/s
fup Upper frequency limit for that the integrand is still D 1
flox Lower frequency limit for that the integrand is still D 1
BER Bit error rate
SNR(f) Signal to noise ratio (power) at frequency f
Equation 26: Achievable bit rates for uncoded DMT [19]

with
Rmax, fup, flow see equation 26
SNRdB(f) Signal to noise ratio at frequency f in dB
9,8 dB Standard SNR-gap for uncoded DMT and BER = 10-7
2*N Block length of the DMT-system (see 6.1), N is the number of subchannels
CP Length of the cyclic prefix (see 6.1.3)
ã Coding gain in dB
Equation 27: Simplified formula for the maximal achievable bit rate in a DMT-system, considering cyclic prefix and coding gain

6.1 Structure and make-up of the DMT-system
Figure 49 shows the block-diagram of the DMT-system in a basic make-up. Additional elements in order to realise equaliser adaptation (see 6.5), trellis codes (see 6.4) or echo cancellation (see 6.3.3) have been omitted for simplicity of presentation. In the following subsections each block will be explained, emphasising special features of DMT that distinguish this system from other multicarrier schemes (see 4.2.2 and chapter 5).

Figure 49: Block-diagram of a DMT-system

6.1.1 Serial to parallel converter and block encoder
In this block the incoming serial bit-stream is converted into parallel data and grouped into blocks. Herein each block has as many bits as one symbol is able to carry. Due to the flexible structure of DMT this quantity is not fixed, but determined during the start-up procedure (see 6.5). The bits in a block again are sub-grouped into the bits that will be carried by each subchannel. This number is not fix either, but is also determined during initialisation [19], [45], [48], [49], [17], [13], [32]. The process of assigning flexible numbers of bits to the different subchannels is called bit loading. Bit loading is based on the idea of assigning the optimal number of bits and the optimal amount of power to each subchannel. Depending on the environmental conditions and other design constraints, bit loading can be optimised towards different parameters. These can be the bit error rate (BER), transmit power or total bit rate. Usually two of them are fixed and then the bit power allocation is optimised for the third. In the specific case of ADSL, two optimisations are possible with the BER fixed to 10-7 as a design constraint. So, the system can be designed for maximal rate or for minimal transmit power. Which one of these schemes is to be applied depends on the conditions under which the line is supposed to work. The pilot systems that are presently used in field-testing in many countries are typically designed for one or two fixed data rates. Therefore only optimisation toward minimal transmit power, in order to achieve the fixed rate and the fixed BER, is possible. It could be asked for the sense of minimising the transmit power, even when it was already below the maximum permitted level. The answer is the crosstalk spectrum, produced by the ADSL system, disturbing other services in the same binder group of the cable. The lower the transmit power, the lower also the disturbance for the other services. For the rate-adaptive ADSL (RADSL), envisioned for the future, the situation is somehow different, as there the total permitted transmit power will be used in order to achieve the maximal bit rate. For the bit power allocation itself there exist various possibilities, each having its advantages and disadvantages. In the following, two schemes will be presented and the resulting bit allocations, as well as the spectral usage and power distribution will be computed. In these allocation schemes a common point of departure is the assumption that the bit power allocation can only be optimal, if the BER in all the subchannels are about equal [19], [49]. This relationship has never been proofed theoretically, but it is intuitionally understandable, as the error function, shown in equation 26 (Q-function), is highly non-linear. An increase in SNR yields less improvement in terms of BER than reduction by the same amount of SNR yields in degradation of BER. Equation 26 therefore can be used as a starting point for all the bit power allocation schemes. Replacing the integral by a sum yields the adaptation to the finite multicarrier system. At this point, however, the total achievable bit rate is not so much of interest, but rather the number of bits, assigned to each subchannel. This leads to equation 28.

bn Number of bits assigned to subchannel n
Q-1, BER See equation 26
SNRn Averaged signal to noise ratio (SNR) of subchannel n
Equation 28: Bit assignation to subchannels, dependent on BER and SNR
Similar to equation 27, in equation 28 the term 3/(Q-1[*])2 can be replaced by the SNR-gap, valid for a BER of 10-7, 9,8 dB, and the SNR-loss due to a cyclic prefix (see 6.1.3). The coding gain of a trellis code (see 6.4) can be included as well. Moreover, in equation 29 the term SNRn has been replaced by magnitudes that have already been calculated in 2.5 and that therefore may be used directly for further computation. From equation 29 it can be seen that there is only one variable, influencing the bit allocation, that can be manipulated directly by the designer of the system. This is the PSD-function (power spectral density) of the transmitted signal. In all the previous computations (especially in chapter 3) always white transmit spectra have been assumed and will therefore be assumed here as well for a first set of computations. A certain problem is produced by the fact that in uncoded DMT-systems only integer numbers of bits can be transmitted in each subchannel. In coded multidimensional systems also certain fractions of bits can be transmitted, as will be shown in 6.4. Without any further treatment this granularity would lead to a noticeable loss in achievable data rate, as, in order to maintain the projected BER, always the next lower value would have to be chosen. Allowing the PSD of the transmitted signal to vary slightly around the white spectrum value can solve this problem. This can be done by first calculating the bit allocation with equation 29 and then rounding the bit values to the next integer values. In order to achieve the projected BER with these new numbers of bits, the PSD for the subchannels has to be computed, based on the rounded numbers of bits for the respective subchannels. This can be done by rearranging equation 29 to get equation 30. The great advantage of this, lets call it "small optimisation", is the fact that the PSD of the transmitted signal is always near to the value of a flat spectrum. That means that it produces approximately the same crosstalk spectrum as would be produced by a spectrally white (but unrealisable because of the granularity of the usable numbers of bits) signal. The total power of the transmitted signal can be calculated by summing up the powers of all the subchannels, using equation 31. Herein only the real bandwidth of the subchannel (equal to the symbol rate), not counting the guard space that is being created by the cyclic prefix (see 6.1.3), is being considered, as the energy loss, produced by the cyclic prefix, had already been taken into account as a part of Ã (equation 29). Table 17 shows the total transmit powers, computed for the models that had been defined in 2.5.4.1. The sub-optimality of this scheme can already be seen by the fact that for many cases the used power is considerably below the maximally permitted average power. This is due to the fact that, especially on the worse channels, the SNR in the higher frequency regions does not even allow the allocation of a single bit, therefore leaving parts of the usable frequency spectrum completely without power assignation. This explains quite well the relatively low power levels for the longer loops, especially the ones with 0, 4 mm and worst case model, but it does not explain the surprisingly low power levels on the best loops. In these cases, especially the ones without crosstalk and length of 2 km, the low power levels are due to an additional limitation included in this computation. In order not to let increase the signal constellations of the subchannels without bounds, the maximal number of bits on one subchannel has been limited to 16 bit [49]. Therefore in this case in the lower frequency portion there could not be assigned all the power, because the 16 bits in these cases can be achieved with less power. This additional limitation of the maximal size of signal constellations does not lead to noticeable reductions in the achievable bit rates. Even if there was a certain rate reduction, it would not matter too much, as ADSL is only defined for a maximal rate of 8 Mbit/s and most of the loops that suffer from the restriction can still with it carry more than 10 Mbit/s. The limitation to 16 bit/symbol is somewhat arbitrary and could be handled with some flexibility, as the sizes of constellations have relatively low impact on complexity (compare 4.2.1.1). Table 18 shows the total bit rates, achievable with this kind of system, for the different models, defined in 2.5.4.1. The system was assumed to have a block length 2*N=512 and a cyclic prefix of length CP=40 [49]. It can be seen that for the long distances (3 and 4 km) in the worst case model with ISDN overlay the presently required 2 Mbit/s can not be achieved. For the case without ISDN and the worst case model in 4 km twisted pair cable (0,5 mm) the system also fails to transport the minimum bit rate. It has to be noted that all these loops are inside the specified range for ADSL, while some loops in the real world model exceed the specified ranges, but achieve to transport the minimum bit rate. This shows clearly that not only the length and the wire diameter, but also the loop population and therefore the crosstalk environment decide whether an ADSL system works properly or not. The worst case model, however, is somewhat unrealistic, but shows clearly that telephone companies should, wherever possible, take some care in order not to create loop populations that approach this worst case model. As it had to be expected, the values shown in table 18 are slightly lower than the ones in table 15 and table 16 (see 3.3). This is due to the fact that the gap between capacity and DMT is 9,8 dB (instead of 9 dB assumed for the safety margin) and, moreover, an additional loss of about 0,35 dB had to be added to take into account the cyclic prefix. The performance, shown in table 15 and table 16, however, can be achieved, even leaving an operational safety margin of slightly more than 3 dB, using a powerful trellis code with a coding gain of about 5 dB (see 6.4).

with

bn Number of bits assigned to subchannel n
PSDsignal(n) PSD of the transmitted signal in subchannel n
|H(n)|2 Averaged channel power transfer function in subchannel n
PSDnoise(n) PSD of noise in subchannel n
Ã SNR-gap toward capacity, including cyclic prefix and coding gain
ã, N, CP See equation 27
Equation 29: Simplified bit assignation to the subchannels

Variable list, see equation 29
Equation 30: PSD of the transmitted signal for given bit allocation

Ptotal Total average power of the ADSL signal
fsymbol Symbol rate in Hz
N, PSDsignal(n) See equation 29
Equation 31: Total power of the ADSL signal

Table 17: Total transmit power for the different channel models, using the white spectrum optimisation

Table 18: Achievable data rates for the different channel models, using the flat spectrum optimisation

The sub-optimality of the scheme, shown above, inspires further thought in order to find a scheme to make better use of the available transmit power. From theory it is clear that the optimal distribution of transmit power resembles the water pouring solution of Gallager [31], [19]. However, a disadvantage of every other scheme, but the flat spectrum one, is the higher complexity, as the bit power allocation following the flat spectrum optimisation is in fact very easily calculated. An approach that is not water pouring in the classical sense, but achieves about maximal performance, uses a bit by bit optimisation scheme, directed by a power difference matrix. This algorithm is relatively easy to implement although it means some computational complexity with a fairly high amount of operations that is proportional to the number of subchannels and the number of bits to be assigned. The scheme is based on a matrix of N (number of subchannels) columns and as many rows as bits can maximally be assigned. In the cells of the first row there are written the powers that are needed in order to transmit one bit per symbol on the respective subchannels. To the cells of the following rows there are assigned the amounts of additional powers that are needed in order to transmit an additional bit in the respective subchannel. The columns of subchannels, where no bits are to be assigned, are filled with high values (higher than the maximally permitted transmit power). The matrix is schematically depicted in figure 50 for 7 subchannels and maximum constellations of 5 bits (32 signal points). The formula to calculate the power differences, to be filled into the matrix, can be derived easily from equation 30 and is shown as equation 32. The difference in the definition of Ã in this case is done in order to simplify the algorithm itself, as the additional energy, needed by the cyclic prefix, will be taken into account in a later step. The algorithm now searches the first row for the lowest value of power cost (for an additional bit). Once the minimum is found, the algorithm checks if the total power, including this additional bit, will exceed the maximally available (permitted) transmit power, multiplied with a factor 2N/(2N+CP). This factor is used in order to take the power loss, due to the cyclic prefix, into account. If the power limit is not exceeded, the number of bits of the corresponding subchannel (index n) is increased by one and the column of that subchannel is shifted one step upwards. In the lowest row, that is now empty because of the shifting, a value, higher than the total available power is filled in. The sum of used power also has to be updated by adding the power cost of the recently added bit. A flow diagram of the algorithm is depicted in figure 51. In the form, as the algorithm is depicted in the flow chart, it is optimising toward maximum bit rate but the same algorithm can also be used to optimise for fixed rate and minimum power. In this case there can be implemented a loop with a fixed number of returns (equal to the number of bits per symbol). The power sum is not necessary for that optimisation although it seems sensible to calculate it as well for control purposes. The total power of a bit power allocation, computed with the matrix optimisation, can also be calculated, using equation 30 and equation 31, but the slightly different definition for Ã has to be considered (the additional factor in this case is (CP+2*N)/(2*N)). Table 19 shows the average transmit powers for the DMT-system, using the matrix optimisation for the models of 2.5.4.1. Already at first sight it can be seen that for all the cases, except the 2 km loops without crosstalk in 0,5 mm wires, the transmit powers are very near to the 19,2 dBm of maximal permitted power. This means that this system makes use of all the available power. Moreover, it can be seen comparing table 20 (achievable bit rates for the matrix optimisation) with table 18 that even in the cases, where the flat spectrum system makes use of almost all the available power, still the rates are lower than in the matrix optimised system. This shows that it is not only important to assign all the power, but also to assign it efficiently. The effectiveness of the matrix optimisation can be seen very well comparing the tables 20, 15 and 16. While the bit rate for the matrix optimised system in the most cases is slightly lower than in the calculations with safety margin (because of higher SNR-gap and loss due to cyclic prefix), especially for the worst case model in the longer loops the matrix optimisation performs better. This means that the matrix optimised system is in these cases able to compensate for a disadvantage of about 1,2 dB in terms of power. However, a certain error was committed in the computations for the real world model, assuming that the matrix method was only used by the disturbed system, while the disturbers where using a flat spectrum. As ADSL only suffers from FEXT of other ADSL downstream systems, the error may be assumed to be negligible. Since the ADSL upstream channel can support higher bit rates than it has to, use of matrix optimisation seems not to be necessary there. This would mean that the NEXT interference of the upstream would remain unchanged, limiting the error to FEXT.

with
ÄP Power difference for an additional bit
b Number of bits
n Subchannel index
fsymb Symbol rate
PSDnoise,|H(n)|2 See equation 29
Equation 32: Elements of the power difference matrix for the matrix optimisation algorithm

Figure 50: Optimisation matrix for 7 subchannels and maximum 5 bits per symbol

Figure 51: Flow-chart of the matrix optimisation algorithm

In the figures 52a, 52b, 52c, 52d, 52e, 52f the bit allocations for ADSL systems with POTS overlay in twisted pair cables have been depicted. The curves for ADSL systems with ISDN overlay look basically the same, with the only main difference being the higher low band-edge, and have therefore not been depicted. As in quad cables the crosstalk interference is slightly lower, the curves for these cables again look like the ones for twisted pair cables, only with the steps towards the next lower number of bits being slightly transferred to the right (higher subchannel index, therefore higher frequency). For this reason only the curves for twisted pair cables have been depicted, somehow representing a worst case scenario, as it is the cable with the higher crosstalk coupling (with the exception of the disturber being situated in the same quad, see 2.5.4). In the figures 53a, 53b, 53c, 53d, 53e, 53f the power spectral densities for the signals, for which the bit allocations had been shown in figures 52a, 52b, 52c, 52d, 52e, 52f are depicted in dependence of the subchannel index. In these diagrams a line, corresponding to the PSD of an ADSL signal (POTS overlay) without any optimisation (flat spectrum), is also shown as a point of reference. As in general for the short loops there is no big difference between the PSD curve of the "small optimisation" to that of the matrix optimisation, only the curves for the longer loops (3 km for worst case in 0,4 mm wires and 4 km for all the others) are shown. Only exception is figure 53a, where it can be seen, comparing with figure 52a, how the optimisation scheme even yields some advantage in this almost optimal case. While the matrix optimised system is able to transmit whole 16 bits over all the subchannels in the 2 km loop without crosstalk, the flat spectrum system has to reduce the number of bits on some subchannels that are affected by residual echo noise. While the matrix optimised system is able to use power, that is not used in the higher frequency portion, to cope with the residual echo noise, the flat spectrum system is constrained to maintain its output power approximately constant over the used frequency bandwidth. In this case the matrix optimisation only produces a small peak in transmit power, but in general, the smaller the usable frequency band (where bits are assigned), the higher the excess power in the used frequency portion. In the figures 53a, 53b, 53c, 53d, 53e, 53f this effect can be observed with increasing strength as the usable bandwidth decreases. The figures 52a, 52b, 52c, 52d, 52e, 52f also indicate clearly, in which cases an increase in the sample rate (equal to transmit bandwidth) could still yield an improvement in terms of data rate. It can be seen that for the shorter loops (2 km) there is still some reserve that may be exploited in future (probably in form of the VDSL system). For the longer loops the number of bits, transmitted in the higher subchannels is zero or at least near to it. Therefore an increase in bandwidth would not yield increase in data rate, showing clearly, how much future VDSL systems will depend on hybrid topologies (see 2.3), in order to shorten the lengths of twisted copper pair drop lines.

Table 19: Total transmit power for ADSL in the pre-defined models with matrix optimisation

Table 20: Achievable bit rates for ADSL in the pre-defined models with matrix optimisation

mt2 2 km loop (0,5 mm) with flat spectrum optimisation
mt4 4 km loop (0,5 mm) with flat spectrum optimisation
omt2 2 km loop (0,5 mm) with matrix optimisation
omt4 4 km loop (0,5 mm) with matrix optimisation
Figure 52a: Bit allocation for an ADSL system (POTS overlay) without crosstalk noise in twisted pair cable

rt2 2 km loop (0,5 mm) with flat spectrum optimisation
rt4 4 km loop (0,5 mm) with flat spectrum optimisation
ort2 2 km loop (0,5 mm) with matrix optimisation
ort4 4 km loop (0,5 mm) with matrix optimisation
Figure 52b: Bit allocation for an ADSL system (POTS overlay) with the real world model in twisted pair cable

wt2 2 km loop (0,5 mm) with flat spectrum optimisation
wt4 4 km loop (0,5 mm) with flat spectrum optimisation
owt2 2 km loop (0,5 mm) with matrix optimisation
owt4 4 km loop (0,5 mm) with matrix optimisation
Figure 52c: Bit allocation for an ADSL system (POTS overlay) with the worst case model in twisted pair cable

tmt2 2 km loop (0,4 mm) with flat spectrum optimisation
tmt4 4 km loop (0,4 mm) with flat spectrum optimisation
otmt2 2 km loop (0,4 mm) with matrix optimisation
otmt4 4 km loop (0,4 mm) with matrix optimisation
Figure 52d: Bit allocation for an ADSL system (POTS overlay) without crosstalk noise in twisted pair cable

trt2 2 km loop (0,4 mm) with flat spectrum optimisation
trt4 4 km loop (0,4 mm) with flat spectrum optimisation
otrt2 2 km loop (0,4 mm) with matrix optimisation
otrt4 4 km loop (0,4 mm) with matrix optimisation
Figure 52e: Bit allocation for an ADSL system (POTS overlay) with the real world model in twisted pair cable

twt2 2 km loop (0,4 mm) with flat spectrum optimisation
twt3 3 km loop (0,4 mm) with flat spectrum optimisation
otwt2 2 km loop (0,4 mm) with matrix optimisation
otwt3 3 km loop (0,4 mm) with matrix optimisation
Figure 52f: Bit allocation for an ADSL system (POTS overlay) with the worst case model in twisted pair cable

mt2p 2 km loop (0,5 mm) with flat spectrum optimisation
mt4p 4 km loop (0,5 mm) with flat spectrum optimisation
omt2p 2 km loop (0,5 mm) with matrix optimisation
omt4p 4 km loop (0,5 mm) with matrix optimisation
PSDadsll Flat transmit spectrum of an ADSL signal (POTS overlay) without bit-granularity
Figure 53a: PSD of an ADSL signal (POTS overlay) without crosstalk noise in twisted pair cable

rt4p 4 km loop (0,5 mm) with flat spectrum optimisation
ort4p 4 km loop (0,5 mm) with matrix optimisation
PSDadsll Flat transmit spectrum of an ADSL signal (POTS overlay) without bit-granularity
Figure 53b: PSD of an ADSL signal (POTS overlay) with the real world model in twisted pair cable

wt4p 4 km loop (0,5 mm) with flat spectrum optimisation
owt4p 4 km loop (0,5 mm) with matrix optimisation
PSDadsll Flat transmit spectrum of an ADSL signal (POTS overlay) without bit-granularity
Figure 53c: PSD of an ADSL signal (POTS overlay) with the worst case model in twisted pair cable

tmt4p 4 km loop (0,4 mm) with flat spectrum optimisation
otmt4p 4 km loop (0,4 mm) with matrix optimisation
PSDadsll Flat transmit spectrum of an ADSL signal (POTS overlay) without bit-granularity
Figure 53d: PSD of an ADSL signal (POTS overlay) without crosstalk in twisted pair cable

trt4p 4 km loop (0,4 mm) with flat spectrum optimisation
otrt4p 4 km loop (0,4 mm) with matrix optimisation
PSDadsll Flat transmit spectrum of an ADSL signal (POTS overlay) without bit-granularity
Figure 53e: PSD of an ADSL signal (POTS overlay) with the real world model in twisted pair cable

twt3p 3 km loop (0,4 mm) with flat spectrum optimisation
otwt3p 3 km loop (0,4 mm) with matrix optimisation
PSDadsll Flat transmit spectrum of an ADSL signal (POTS overlay) without bit-granularity
Figure 53f: PSD of an ADSL signal (POTS overlay) with the worst case model in twisted pair cable

rt4pn 4 km loop (0,5 mm) with flat spectrum optimisation
ort4pn 4 km loop (0,5 mm) with matrix optimisation
PSDnexttpadsl NEXT interference of a flat ADSL signal (POTS overlay)
Figure 54a: NEXT interference spectra of ADSL signals (POTS overlay) with the real world model in twisted pair cable

wt4pn 4 km loop (0,5 mm) with flat spectrum optimisation
owt4pn 4 km loop (0,5 mm) with matrix optimisation
PSDnexttpadsl NEXT interference of a flat ADSL signal (POTS overlay)
Figure 54b: NEXT interference spectra of ADSL signals (POTS overlay) with the worst case model in twisted pair cable

trt4pn 4 km loop (0,4 mm) with flat spectrum optimisation
otrt4pn 4 km loop (0,4 mm) with matrix optimisation
PSDnexttpadsl NEXT interference of a flat ADSL signal (POTS overlay)
Figure 54c: NEXT interference spectrum of ADSL signals (POTS overlay) with the real world model in twisted pair cable

twt3pn 3 km loop (0,4 mm) with flat spectrum optimisation
otwt3pn 3 km loop (0,4 mm) with matrix optimisation
PSDnexttpadsl NEXT interference of a flat ADSL signal (POTS overlay)
Figure 54d: NEXT interference spectrum of ADSL signals (POTS overlay) with the worst case model in twisted pair cable

In the figures 53a, 53b, 53c, 53d, 53e, 53f another typical property of the matrix optimisation can be observed. Although the algorithm is not clearly water pouring, the form of the envelope of the transmit spectrum approaches the form of the SNR-distribution. Especially in figure 53f it can be seen well that the assigned power (proportional to the PSD) in the higher frequency region, where the SNR is lower, is reduced. Another interesting point in this context is that the frequency band, used by the matrix method, is narrower than the one used by the flat spectrum approach. Therefore it may be stated that the matrix optimisation algorithm tends to concentrate the available power in the frequency portion with the highest SNR. In figure 53e it can be seen that this is not only true for the outer band-edges of the signal, but also in the case that there is a strong depression in the SNR distribution within the frequency band. Both optimisation schemes spare out an inner part of the transmit band in this case, but it can be seen that the bandwidth, spared out by the matrix method is bigger. This example also shows the ability of DMT to cope with disjoint frequency bands that are separated either by spectral nulls of the channel or by strong disturbers, which also may lead to a strong decay in SNR [49]. The major drawback of the matrix method can be seen in the figures 54a, 54b, 54c, 54d. Here the crosstalk spectra, with that the flat spectrum and the matrix optimised systems interfere into other services in the same binder group, are depicted. As point of reference there has also been included the crosstalk that would be caused by an ADSL system (POTS overlay) with a perfectly flat spectrum. It can be seen that the crosstalk interference of the flat spectrum optimised system just lingers around the crosstalk spectrum of the ideally flat system. This may be assumed not to cause major trouble for other systems, since single carrier or baseband systems anyway average the noise over the whole bandwidth and whiten it by their equalisers (see 4.1.1.2) and multicarrier systems can adapt themselves to this situation. This could be done by assigning more bits on frequencies, where the crosstalk interference is lower, and less, where it is higher. In case of the matrix optimised system, however, the situation is totally different. Since it concentrates the power in the frequency portion with the highest SNR, there it also produces NEXT that is much stronger than that produced by the flat spectrum system. The difference can be up to several dB as can be estimated from figure 54d.The difference between both optimisation methods with respect to crosstalk decreases with increasing quality of the channel. In figure 54a it can be seen for a channel with relatively good characteristics (0,5 mm, 4 km, real world model) that the level of crosstalk interference is hardly distinguishable between both systems. The increase in crosstalk interference, especially in the low frequency portion, does not affect very much the ADSL systems themselves as they only suffer from FEXT from each other and it is assumed that the ADSL upstream uses flat spectrum optimisation. Other systems, however, especially the ones that use lower bandwidths, might be affected seriously by the higher level of crosstalk noise. So, the most threatened systems by the matrix optimisation method are HDSL and ISDN basic access. A more extensive study about the cross-effects of spectral shaping in DMT and crosstalk into other systems has been done in [49]. There it is stated that with proper optimisation of the transmit spectrum of DMT, up to 5 dB performance gain may be achieved in environments with severe crosstalk. Although the different ADSL systems in one binder group only affect each other by FEXT, the problem of instabilities due to cross-adaptations should be considered. If the bit power allocation is fixed once in the start-up procedure (see 6.5) and then maintained equal, there will be no problems as long as the security margin, included in the calculation of the bit loading, is high enough. However, if continuous adaptation is used, a co-ordinated strategy has to be found in order to avoid continuous cross-adaptations. Cross-adaptation in this context means that one transceiver adapts to the environment that was created by the adaptation of another one, thereby causing the first one to adapt again to the conditions, created by the second. Without a co-ordination for all the transceivers in one binder group such cross-adaptations are relatively likely to happen and can be a serious impairment for reliable data transmission [19]. In this subsection only the bit power allocation itself had been considered, assuming that the channel transfer function and the spectral noise distribution are known. In practice this is normally not the case. In real world applications this information has to be obtained by measurement during the start-up procedure. For further treatment see 6.5. It should also be noted at the end of this subsection that the optimisation schemes, that have been presented, offer the possibility of very small step rate-adaptation. Theoretically adaptation in steps of the symbol rate, therefore 4000 bit/s in case of ADSL, is possible. Much will depend on the transmission mode and the protocols in the higher layers, whether this flexibility can be fully exploited. Another innate advantage of the bit loading technique, in concatenation with full digital multicarrier modulation, is the relaxed requirement on correct choice of the sampling rate. As DMT has the flexibility not to use subchannels with a low SNR, an increase of the sample rate may increase the cost without increasing data rates, but at least it will not degrade the achievable rates as in case of PAM and QAM (see 4.1.1 and 4.2.1) [13].

6.1.2 Inverse fast Fourier transform (IFFT)
In the DMT-system modulation is realised by a 2N-point IFFT. One might be surprised at the first moment about the inverse transformation in the transmitter. This make-up is due to the fact that the data is assigned to sub-symbols, represented by constellation points in parallel independent subchannels in the frequency domain. In order not to need the filter banks and banks of modulators, that had been necessary in the classical multicarrier systems (see 4.2.2) in order to create the signal in the frequency domain, in DMT the transmit signal is created in the time domain. This is the reason for the use of the inverse Fourier transformation, transforming from the frequency domain to the time domain. The inputs to the 2N-point transform are the constellation points of the N subchannels. In order to achieve a real valued output, Hermitian symmetry has to be maintained at the input of the IFFT. The symmetry rule is shown as equation 33. In a DMT-system the Fourier transform and it's inverse is usually performed by the well-known fast Fourier transform (FFT) algorithm [28]. By this means the computational complexity can be reduced considerably. While the normal discrete Fourier transform (DFT) needs about M2 operations (with M being the total length of the transform), the FFT only needs about 1,5*M*log(M) for lengths beyond 128, as they are usual for DMT [48]. There also exists a small, but for DMT not very important, disadvantage of the application of the FFT-algorithm. It only performs optimally for transformation lengths that are integer powers of 2. As the block length (equal to the transformation length) for a fixed sample rate (and fixed length of the cyclic prefix, see 6.1.3) defines the bandwidth of the subchannels, it has an impact on performance. Since the basic idea is to create subchannels that are virtually distortionless and only impaired by white noise, the bandwidth of each subchannel would have to tend toward zero. In this limit the subchannels would be free of distortion and no equalisation would be needed [13]. In reality, however, the number of subchannels (proportional to the block length, see equation 27) can not be chosen arbitrarily high. It was shown that an increase in block length beyond 512 does not yield significant improvement in performance, but it does increase complexity and throughput delay [17], [48]. A longer delay, although undesirable in order to maintain compatibility with a maximum number of services that might be transported over the ADSL platform, eases the requirements on equalisation and computational speed. In practical systems a combination is made between a small amount of equalisation (see 6.1.5) and a large amount of multicarrier modulation. The relationship between sample rate, block length, symbol rate (= subchannel bandwidth), symbol period, subchannel distance and length of the cyclic prefix (see 6.1.3) is shown in equation 34. There it can also be seen that the subchannel bandwidth is smaller than the spacing between the subchannels, which means a certain loss in bandwidth efficiency. This guard band is introduced by the cyclic prefix (see 6.1.3). The modulation by the IFFT corresponds to modulation of the respective carrier frequencies with a rect-pulse. The use of a rect-pulse results in a sinc-spectrum in the frequency domain, causing one subchannel to overlap with many of its neighbours. As long as the channel is non-dispersive, orthogonality is maintained and there do neither occur interchannel (ICI) nor interblock (IBI) interference (see figure 48). On a dispersive channel, however, additional measures have to be taken in order to maintain orthogonality, as it will be shown later in this section. In recent years there was a growing interest in applying other pulse shapes in order to increase the spectral containment of the subchannels in their own frequency range [45], [15] (see also 5.1). With the rect-pulse the first side lobe is only about 13,6 dB lower than the main lobe and the strength of the further side lobes decreases as f-2[45]. One of the great advantages of the present DMT-system, using the IFFT for modulation, is its relatively low complexity and it's ease of implementation. The whole system can be built in a single chip fashion. A great improvement could be made in future by designing an FFT peripheral in order not to have to perform all these operations in the main signal processor [17], [19], [48].

for
Xk Signal point in the constellation of subchannel k
(Xk)* The star denotes complex conjugate
2*N Block length of the transformation
Equation 33: Symmetry rule for the IFFT in order to receive a real valued time signal

and
Äf Subchannel spacing
fsymbol Symbol rate
T Symbol period
fsample Sample rate
N Number of subchannels (=1/2 block length)
CP Length of the cyclic prefix (in samples)
Equation 34: Relationship between the basic time and frequency magnitudes of a DMT-system

6.1.3 Parallel to serial conversion and addition of the cyclic prefix
In this block the parallel output (2N values) of the IFFT is converted into a serial bit-stream and the so-called cyclic prefix is added. The principle of the cyclic prefix is shown in figure 55. The last CP samples are taken from the end of the block and copied to the beginning of it. This makes the transmitted signal look somehow like a periodic signal [13], [45], [48], [50]. The effect of the cyclic prefix is twofold. The most obvious one is to create a guard space between neighbouring transmit symbols in the time domain. Instead of zeros this guard space is filled with the cyclic extension as shown in figure 55, but it has basically the same effect and therefore combats ISI efficiently in the time domain (see also figure 48) by concentrating it in the cyclic prefix that is later discarded in the receiver. ISI, however, is not the only internal interference problem that can affect a DMT-system. The other problem to be coped with is ICI. By the periodicity of the transmitted signal (due to the cyclic prefix) a cyclic convolution, instead of a linear convolution, between the channel impulse response and the transmitted signal is efficiently simulated. This means that the effect of the channel is reduced to a element by element multiplication between the Fourier transforms of the channel impulse response and the transmitted signal, therefore introducing only different gains and delays on each subchannel [45]. These different gains and phases can be coped with by one-tap per channel equalisers (see 4.2.2, figure 46a, figure 46b and 6.1.7) and no ICI is produced. As could be seen in equation 34, the cyclic prefix also introduces a difference between the subchannel bandwidth and the spacing of the subchannels and can therefore be understood as a guard band in the frequency domain. Moreover, the cyclic prefix enhances the spectral containment of the main lobe of the subchannels' transmit spectra, although only slightly. The additional difference between the main lobe and the first side lobe of the subchannels' spectra is calculated with equation 35 [15]. However, the inclusion of a cyclic prefix also yields some disadvantages. In first place it brings a certain loss in SNR due to the fact that energy is used in order to transmit redundant contents. Depending on the length of the cyclic prefix, in typical ADSL systems the loss due to cyclic prefixing is about 0,35 dB, therefore a relatively low price to pay for the relaxed requirements on equalisation (see 6.1.5). The exact loss can be calculated with the second part of equation 27. The cyclic prefix, however, only works in the described manner, if its length (CP) is at least as long or longer than the channel impulse response. As the impulse response of typical ADSL channels may have lengths of several hundred samples (see 2.5.2, figures 17a, 17b, 17c, 17d, 17e, 17f), the block length would have to be about 13 times longer than the longest impulse response of a typical ADSL channel. Only this could maintain the previously mentioned additional loss of about 0,35 dB [13]. Block lengths of several thousand samples, however, are not realisable in practice, firstly, because of the increased complexity and, secondly, because of the large additional delay that would be introduced. Therefore in practical systems a combination is being done between cyclic prefixing and a small amount of time domain equalising (see 6.1.5) in order to restrict the length of the equalised impulse response to the length of the cyclic prefix [13]. A shortening of the channel impulse response, however, is much easier to achieve than zero ISI, as it is necessary in baseband and single carrier modulation schemes (see 4.1.1 and 4.2.1). For ADSL systems lengths of cyclic prefixes of 8 [17] and more recently 32 [51] and 40 [49] have been proposed. All computations in this work assume a cyclic prefix of 40 samples, therefore somehow representing a worst case, as it is the one with the highest power loss. Although the inclusion of a cyclic prefix is, after all a clearly sub-optimal solution, the additional power loss is a low price to pay for maintaining the orthogonality between the subchannels and avoiding ISI between subsequent symbols. Moreover, the cyclic prefix facilitates synchronisation and timing recovery as will be shown in 6.2[45].

ÄL Additional suppression of the side lobes
N Number of subchannels
CP Length of the cyclic prefix
Equation 35: Additional suppression of the subchannels' side lobes by inclusion of a cyclic prefix [15]

Figure 55: Make-up of the cyclic prefix

6.1.4 D/A-, A/D-converters and transmit and receive filters
The A/D- and D/A-converters in a DMT-system have to be more sophisticated than in single carrier or baseband systems (see 4.1.1 and 4.2.1). This is mainly due to the many possible values in the transmitted signal, due to the addition of many (in case of ADSL 256) components, each having complex Gaussian amplitude probability [13]. The converters also have to offer a great dynamic range to cope with the high peak to average ratio of multicarrier systems (see 4.2.2). The analog filter in the transmitter is not included for pulse shaping as it may be done in single carrier and baseband systems, but only to remove the higher frequency copies of the transmit signal that are created due to the fully digital make-up of the DMT-system. At the input of the receiver there is to be found the same kind of filter, here used in order to exclude the noise of the higher frequency portion. It is also needed for clear band limitation or, in other words, as an anti-aliasing filter previous to sampling. It should be noted at this point that the POTS/ISDN-splitter was assumed to be part of the channel and has therefore neither been depicted, nor been described in this chapter. For further information about this device refer to [36].

6.1.5 Time domain equaliser (TEQ) and removal of the cyclic prefix
As it had already been mentioned in subsection 6.1.3, the DMT-system requires some sort of equalisation although it is much less than in comparable single carrier or passband systems (see 4.1.1.2). The task of the time domain equaliser is to shorten the impulse response of the channel to a length, shorter or equal to the one of the cyclic prefix. This task is much easier to achieve than to create zero ISI. The equaliser itself is typically a short tapped delay line of between 10 [48] and 64 [50] taps. To find the tap-settings there exist different possibilities. The most straightforward is to measure the channel impulse response and to derive a pole zero model of the form shown in equation 36. There the response is represented in the polynomial form in the D-domain. If a(D) has as its maximal length the one of the cyclic prefix and b(D) has maximally the length of the TEQ, then b(D) is the optimal setting for the equaliser taps. If in the derived model b(D) is shorter than the TEQ, the rest of the taps can be filled with zeros. With the equaliser taps set equal to b(D), a(D) becomes the impulse response of the equalised channel. In the typical ADSL environment there are slightly more poles than zeroes, which means in other words that b(D) will be slightly longer than a(D) [48]. Even on bad loops and up to data rates of 6 Mbit/s the channel response can be shortened to 10-30 samples [13], requiring a TEQ of maximally about 32 taps of length. This form of calculation of the tap-settings, however, requires relatively complex mathematical operations after channel estimation. This causes additional complexity for the start-up procedure (see 6.5). Other schemes work in an iterative manner, applying some form of least mean square (LMS) algorithm in order to converge the tap-settings. This does not require any complex mathematical operations and therefore does not increase computational complexity unduly, but has the disadvantage of long convergence times [50]. For common DMT-equalisers it takes several millions of iterations to converge the equaliser, resulting in considerable delays in the start-up procedure. An approach to shorten the necessary time-span without increasing prohibitively computational complexity has been made by Lee et al. [50]. They observed that a DMT-system with cyclic prefix and TEQ could be understood as a DFE where the TEQ corresponds to the feedforward filter and the cyclic prefix to the feedback filter. Their method is therefore based on the fast adaptation algorithm for the DFE. Channel estimation is used in order to measure the frequency transfer function of the channel and the spectral noise distribution. Depending on the optimisation scheme that is being used in the system, a part or all of this information is also needed for the calculation of the bit power allocation and therefore this measurement does not mean much additional complexity. The frequency transfer function of the channel is then transferred into the time domain by an inverse FFT. Now the output of the channel with TEQ is compared to a desired impulse response (see figure 56) and the equaliser is adapted in order to match both responses. The desired response is calculated based on the measured channel data and optimising towards highest possible data rate. In this calculation a simplification is used in order to avoid matrix inversion and to replace it by FFT operations. As the FFT has to be implemented anyway in the DMT-transceiver, this function can be shared and does not yield additional complexity. Due to the simplification a slight error is being made, resulting in a slightly sub-optimal performance. The performance gap is about 0,2 to 0,3 dB for usual ADSL loops and TEQ lengths, but approaches zero with increasing filter lengths (up to 64) or decreasing loop lengths. Even including the channel estimation, this algorithm still needs much less iterations than a normal LMS scheme as only a few thousand iterations are needed for the channel estimation and another few thousand for the optimisation of the equaliser tap-settings (see also 6.5). After initial convergence with the fast algorithm, a LMS optimisation may be used for continuous tracking. In this case the performance gap for the fast algorithm diminishes, causing little cost in additional complexity. It has to be checked more profoundly, however, if the about two tenths of a dB, achievable by concatenation with the LMS algorithm, are worth the additional complexity. After equalisation the cyclic prefix is removed and simply discarded, but before its removal it had already been used for synchronisation and timing recovery that are not shown in figure 49 (see also 6.2).

h(D) Channel impulse response, modelled by the pole-zero model
a(D) Zero-polynomial of the pole-zero model
b(D) Pole-polynomial of the pole-zero model
Equation 36: Polynomial equation for the pole-zero model of the channel impulse response

Figure 56: TEQ for ADSL application, using an optimised desired impulse response

6.1.6 Serial to parallel conversion and fast Fourier transform (FFT)
After equalisation and removal of the cyclic prefix the incoming serial stream of samples is converted into blocks of parallel data with 2N parallel values. These are fed into a 2N-point real to complex FFT, therefore transferring the time domain signal again into the frequency domain. The transfer into the frequency domain also means the separation of the N parallel independent subchannels whose contents can now be further processed on a per subchannel base. Of the 2N outputs of the FFT, only N are used for further processing due to the Hermitian symmetry that had already been discussed in 6.1.2.

6.1.7 Frequency domain equaliser (FEQ)
The FEQ is a set of complex one-tap per subchannel equalisers that are included in order to cope with the different gains and phase delays that the equalised channel introduces on the different subchannels. By multiplication with a complex number, the amplitude levels and the phase positions are readjusted as was already shown in figure 46a and figure 46b. The initial tap-values for the FEQ are learned during the start-up procedure (see 6.5). First, the channel frequency transfer function of the equalised channel is estimated and then it is inverted in order to receive the tap values for the FEQ. If during the start-up procedure the impulse response of the equalised channel is determined, then the initial tap-settings are the inverse FFT of it [48]. During the data mode the equaliser taps are continuously updated in a decision directed manner. Therefore, even in case of the use of a Viterbi receiver (see 6.4) to decode trellis coded modulation, a decision element is included to make decisions at the output of the FEQ. The output of this decision device is then compared with its input for each subchannel and the FEQ-taps are then adjusted accordingly. As the decisions, made without a Viterbi-receiver in systems with trellis coded modulation, are not very reliable, the step size for the tap-adaptation has to be chosen small in order to avoid problems due to error propagation, leading to adjustments into the wrong direction due to wrong decisions [50].

6.1.8 Decision device and parallel to serial conversion
In the decision device the symbols of the subchannels are detected, using the knowledge about the bit power allocation, and therefore the sub-blocks of bits that had been assigned to the subchannels are recovered. Once the sub-blocks are detected, the data is re-converted into a serial bit stream in a block by block fashion. If trellis coded modulation is used, the decision device in the main datastream is replaced by a Viterbi receiver in order to make maximum likelihood decisions on the transmitted sequences. Further information about the concatenation of coding and DMT is presented in 6.4.

6.2 Synchronisation and timing recovery
A major drawback off all digitally synthesised multicarrier systems is their high sensitivity to timing and synchronisation errors. This problem is somehow even more severe due to the fact that in the DMT-system no coherent carrier, that would be helpful for synchronisation and timing recovery, is transmitted. Three kinds of synchronisation have to be distinguished in this context:
· Frequency synchronisation
· Block synchronisation
· Sample instant synchronisation

In case of DMT for ADSL, the synchronisation is already somewhat relaxed compared with, for example, radio frequency applications as in that case the transmitted signal has to be modulated onto another carrier, which has to be synchronised as well. Since the multicarrier signal in case of DMT over telephone lines is transmitted in baseband, no carrier frequency is used and therefore no synchronisation for it needed. Block synchronisation, however, plays an important role in ADSL-systems. Basically there exist two possibilities to realise block synchronisation. Firstly, it can be done by using pilot tones that are transmitted in a certain time and frequency scheme in the transmitted signal. This method has the disadvantage of further reducing the achievable data rate since some of the subchannels are occupied permanently by pilots, while others are at least occupied for a certain part of the time, being therefore not available for data transmission. For this reason this method is not considered for the ADSL application. Secondly, block synchronisation can be done by using the cyclic prefix. The basic idea is to observe the difference between a received sample and the one that had been transmitted 2N samples ago. If the latter was one of the cyclic prefix (see 6.1.3) of the symbol that the former corresponds to, the difference r(k)-r(k+N) should be zero. Due to noise this difference will never be exactly zero and due to statistical processes of probability this difference also can take values near zero for any other arbitrary moment. Therefore the square of the above-specified difference is observed, using a window function of the length of the cyclic prefix, and averaging the result over several symbol periods is performed. Over a longer time of averaging, a clear minimum should occur once in every symbol interval, marking clearly the beginning of each new block. The scheme, described above, only works well for medium and high SNR, as it is typically found in the ADSL environment. For low SNR maximum likelihood estimation has to be integrated into the scheme. The last kind of synchronisation, that has to be done in a DMT-system, is the one of the sampling instants. It would also be possible to use unsynchronised sampling, but this would require more complexity for post-sampling processing in the digital domain and is therefore discarded [45]. If the block lengths are not excessively long, synchronisation of the sampling instants can be done by optimal symbol synchronisation and division of the symbol period through the number of samples per symbol (including the cyclic prefix). This scheme is suitable for the block lengths typically used in ADSL applications (512). For longer block lengths the performance of this system degrades due to the increasingly rare synchronisation moments as the cyclic prefixes are increasingly separated. Therefore the possibility for the receiver clock to deviate from the transmitter clock in the interval between the synchronisation instants, including the latency due to the averaging process in the symbol synchronisation, increases. In typical DMT-systems the synchronisation is done in two steps [45]. In a first coarse approach, synchronisation is achieved within +/- 0,5 Tsample (sampling period). In a second step the synchronisation is optimised. This two-step procedure is easier to implement, since in this case during the second step the timing error already may be assumed to be small. Errors in the synchronisation of the sampling instants cause additional attenuation and phase rotation as well as loss of orthogonality between the subchannels, resulting in ICI. Deviations in the block synchronisation lead to phase shifts on the subchannels that increase towards the band-edges. If the sum of the timing error plus channel impulse response is smaller than the cyclic prefix, no ISI is introduced and if this timing error is constant or at least very slowly varying, it can be viewed as an additional phase rotation, introduced by the channel. In this case the additional phase rotations in the subchannels can be compensated by the one-tap per channel equalisers, described in 6.1.7 [45].

6.3 Duplex schemes
A duplex scheme is always needed, if transmission in both directions at the same time is desired. "At the same time" in this context does not necessarily mean exactly the same instant, but at least virtually at the same time, meaning that the differences are minimal, as will be discussed later in this section. For duplex transmission there basically exist three different possibilities as is shown also in the figures 57a, 57b and 57c. The signals in both directions can be separated in time, in frequency or simply by their direction of propagation. In the following subsections the three duplex schemes are briefly presented and their applicability for ADSL as well as their advantages and disadvantages for this application are discussed.

Figure 57a: Frequency division Figure multiplexing (FDM)

Figure 57b: Time division multiplexing (TDM)

Figure 57c: Transmission with echo cancellation

6.3.1 Frequency division multiplexing (FDM)
The most obvious choice for an asymmetric service, such as ADSL, would be FDM (see figure 57a), as the narrow upstream signal does not occupy a wide frequency band and still allows a lot of usable bandwidth for the downstream in the higher frequency region. Another point in favour of FDM is its ease of implementation, as only splitting filters are needed in order to divide the frequency band into an upstream and a downstream section. FDM totally avoids the problem of self-NEXT, as there is no transmission in the frequency band where the receiver is receiving and vice versa. In fact, ADSL already uses FDM in order to realise the overlay facility for POTS and ISDN. In the first standard for ADSL, FDM was defined as the duplex scheme and it is still to the decision of the manufacturer to implement his equipment with this scheme [6]. The big disadvantage, however, is that the low frequency portion, where in case of FDM only the upstream signal is transmitted, is not fully used (in both directions) and therefore valuable transport capacity for the wideband downstream is lost. Anyway, DMT has the flexibility to work as an FDM-system, even if echo cancellation is used. This can be done by the bit power allocation, not allocating bits for the downstream where already bits for the upstream are allocated and vice versa. Finally it may be said that FDM is a cost-attractive, but definitely sub-optimal choice for the ADSL system.

6.3.2 Time division multiplexing (TDM)
TDM has never been considered as a strong candidate for ADSL, mainly because the line bit rate has to be increased considerably in order to transmit the data in the two directions in different time slots (see figure 57b). In this case the line bit rate not only has to take the value of the sum of the upstream and the downstream rate, but there also has to be considered a certain guard time. This is needed for the impulse response and the echoes of the transmission in one direction to linger off before transmission in the other direction is started. The overhead due to the guard space depends on the maximum permitted line length and also on the length of the bursts in each direction. The more often the transmission direction is changed (the shorter the bursts or blocks and therefore the transmission delay) the more often this guard space is needed and the higher will be the overhead at the end. In a thumb rule it may be said that the overhead will be approximately 1/8 of sum of upstream and downstream rate [27]. TDM, however, can also be used to suppress NEXT, but only in the case that all the transceivers in a binder group are synchronised so that all the receivers at one end of the line burst together and afterwards all the transceivers of the other end of the line. Due to the necessary overhead, that is considerably higher than the one, created by the cyclic prefix of the presently defined DMT (see 6.1.3), and that would have to be added with it, TDM has not been chosen for ADSL and is not a strong candidate either for VDSL. Only for symmetric services TDM might be a better choice at very high bit rates, because in that case FDM would cause almost the same reduction in possible transmission rate due to the necessary guard space in the frequency domain and echo cancellers would lead to prohibitive complexity. But as VDSL is planned to be realised in symmetric and asymmetric versions and it is not likely that the different versions will be implemented with different duplex schemes, even for symmetric VDSL TDM is not likely to be the duplex scheme.

6.3.3 Echo cancellation (EC)
EC is the second possibility that the present standard for ADSL offers to the choice of the manufacturer (together with FDM, see 6.3.1). In this scheme the signals are only separated by their direction of propagation inside the cable. However, due to discontinuities and bridged taps (see also 2.5) there occur reflections on the loop that cause a part of the signal energy to travel back to its respective transmitter. On the way back it mixes with the signal that is being transmitted from the opposite side of the line and has therefore to be seen as additive noise at the receiver. But as this noise is correlated with the previous transmit signal, this knowledge may be used in order to cancel this undesirable echoes out of the received signal. To achieve this, a so-called echo canceller is being used in the transceivers. A block-diagram of a transceiver with hybrid and echo canceller is shown in figure 58. The hybrid is the element that matches the 2-wire loop to the 4-wire structure inside the transceiver, separating the transmit path from the receive path. Originally hybrids were made of transformers, but nowadays they are realised as active circuits, achieving between 10 and 20 dB of echo suppression [51]. As a total of about 70 dB of echo rejection has to be achieved, the remaining 50 to 60 dB have to be yielded by the echo canceller that is placed on the 4-wire side in the transceiver. Basically the task of the echo canceller is to model the echo, caused by the transmit signal, and to subtract it from the received signal. Two basic forms of echo cancellers have to be distinguished, signal driven and