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Optimal Discrete Bit-Loading for DMT-Based DSL Systems with Equal-Length Loops Jungwon Lee, Member, IEEE, Seong Taek Chung, Member, IEEE, and John M. Cioffi, Fellow, IEEE
Abstract— This letter examines a discrete bit-loading problem of upstream and downstream for discrete multi-tone (DMT) digital subscriber lines (DSL) with echo cancellation. Both far-end crosstalk (FEXT) and near-end crosstalk (NEXT) are taken into account. An optimal discrete bit-loading algorithm is developed when the loop lengths of all same-service users in a common binder are the same. Simulation results show that the optimal algorithm achieves a substantially higher data rate than existing suboptimal schemes. Index Terms— Power allocation, bit loading, spectrum management, digital subscriber lines (DSL), discrete multi-tone (DMT)
I. I NTRODUCTION Due to the demand for high-speed data transmission over a twisted-pair channel, there has been growing interest in multicarrier modulation technique, commonly known as discrete multi-tone (DMT) modulation [1]. Digital subscriber lines (DSL) systems using the DMT modulation can achieve a high data rate by allocating the rate and power to subchannels wisely. The data rate can be increased further by taking into account the far-end crosstalk (FEXT) and the near-end crosstalk (NEXT) explicitly and by jointly allocating the rate and power of different users. This letter investigates the rate and power allocation of the upstream and downstream of all same-service users whose twisted-pair loops are of equal length when the echo cancellation is employed. As in [2] and [3], to simplify the allocation problem, it is assumed that the power allocations of all signals flowing in the same direction are the same. Under this assumption, the performance of a system is barely degraded when FEXT is relatively small compared to other noises [3]. Then the rate and power allocation for upstream and downstream signals of only one user needs to be determined. In [2], the optimal algorithm has been found when only NEXT is considered. In [3], the result in [2] is extended to the case when FEXT is also considered. However, in [2] and [3], the rate and power allocation problem is considered under the condition that the data rate can be any non-negative real number. In this letter, the rate and power allocation problem This paper was presented in part at the IEEE International Conference on Communications, Anchorage, Alaska, May 11 - 15, 2003 J. Lee was with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305. He is now with Marvell Semiconductor, Inc., Santa Clara, CA 95054 USA (email:
[email protected]). S. T. Chung was with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305. He is now with Assia Inc., Redwood City, CA 94065 USA (email:
[email protected]). J. M. Cioffi is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (email:
[email protected]).
is solved with a discrete-rate constraint such that the data rate in each subcarrier can take discrete values. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION The signal-to-interference-plus-noise ratio (SINR) of the upstream signal in subchannel is expressed as
(1) where , , and are the upstream transmit power,
the downstream transmit power, and the noise power in subchannel . When we consider the crosstalk from other services, includes this crosstalk noise power in addition to the background noise power. , , and are the squared magnitude of the direct channel gain, the FEXT channel gain, and the NEXT channel gain in subchannel , respectively. When the echo cancellation is not perfect, is the sum of the squared magnitude of the residual echo transfer function and the squared magnitude of the NEXT channel gain. The SINR of the downstream signal, is similarly determined. When the background noise and the crosstalk are assumed to be Gaussian [4], the upstream data rate in subchannel , , should satisfy the following inequality:
(2)
where is signal-to-noise ratio (SNR) gap [1]. This SNR gap approximation holds only when the Gaussian assumption on the background noise and crosstalk is valid. The maximum downstream data rate can be similarly expressed. For notational convenience, let so that the inequality (2) can be represented as follows:
(3)
The problem of interest is minimizing the total power given a target data rate when the data rate in each subchannel is limited to non-negative integers: minimize subject to
target ½ ½ for
(4)
where target is a target rate-sum of the upstream and the downstream, ½ is the set of non-negative integers, and is the number of subchannels. As is shown in [5], this powerminimization problem is equivalent to the rate-maximization
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problem. So, the optimal algorithm in Section III can also be used to solve the rate-maximization problem. III. O PTIMAL R ATE AND P OWER A LLOCATION The rate and power allocation problem in (4) is a two dimensional allocation problem: the target rate target should be allocated to subchannels and streams. To solve this twodimensional allocation problem, we can use the suboptimal multi-user bit-loading algorithm in [6] that takes the greedy approach over both subchannels and streams. To develop an optimal algorithm, this letter takes the approach of dividing the two dimensional allocation problem into two one-dimensional allocation problems as is shown in Fig. 1. First, the target rate is optimally allocated to subchannels and then the allocated data rate in each channel is optimally distributed to the upstream and the downstream. For the allocation of the target rate to subchannels, a greedy algorithm is used, and it is shown that the greedy algorithm is optimal. On the other hand, for the distribution of the allocated rate to streams, the optimal rate and power distribution is analytically found depending on the rate sum in each subchannel. A. Optimal Rate and Power Distribution in Each Subchannel We examine the problem of allocating the rate and power to the upstream and downstream in a given subchannel such that the power sum is minimized for a given rate sum, i.e., minimize
½ ½
(5)
where is the rate sum in subchannel . To solve this problem, we first need to find the minimum power-sum that is necessary to transmit data at the rate of and for the upstream and the downstream, respectively. From (3) and the corresponding expression for , the following equations should hold:
where
and . When there is a nonnegative solution
(6)
that satisfies (6), the Pareto optimal solution, which minimizes both and , can be found [6]. Since all the elements of are nonnegative, it can be shown using Perron-Frobenius Theorem for nonnegative matrices that the Pareto optimal solution to (6) is
Rate allocation over the streams and the subchannels.
subject to
(7)
and matrix or vector inequality represents element-wise inequality. This requirement may not be necessarily satisfied
(8)
Thus, the minimum power-sum for the rate distribution and can be calculated by summing the elements of the Pareto optimal solution (8). When there is no nonnegative solution that satisfies (6), at least one of and obtained by (8) becomes negative. Thus, (8) can be used not only to find the minimum power, but also to determine whether there is a feasible solution. With the Pareto optimal solution, this letter has found the optimal rate distribution method of a rate sum to the upstream and the downstream, and the result is summarized in the following Proposition. Proposition 1: The power sum in (5) is minimized by the following rate distribution. 1) When , the optimal rate distribution is , where represents the largest integer that does not exceed
, and represents the smallest integer that is not less than . 2) When , the optimal rate distribution is . 3) When , the optimal rate distribution depends on . a) If th , the optimal rate distribution is , where th . b) Otherwise, . Proposition 1 shows that the rate sum should be distributed almost equally to the upstream and the downstream for the subchannels with a small NEXT gain. On the other hand, the rate sum should be distributed to either the upstream or the downstream, but not both, when the NEXT gain is large. For subchannels with intermediate levels of NEXT gains, the rate distribution depends on the rate sum. Once the rate distribution is determined by Proposition 1, the necessary power for each stream can be determined by (8). In this way, the optimal rate and power allocation in each subchannel is found easily when the rate sum in that subchannel is given. B. Optimal Rate and Power Allocation to Subchannels and Streams For the rate and power allocation to subchannels, we use a greedy algorithm [7]. Then the rate and power allocation algorithm can be described as follows: 1) Initialization: For each subchannel, calculate the cost to transmit one bit in that subchannel.
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where is the minimum power to transmit bits in subchannel . The minimum power for bits can be calculated easily from Proposition 1. At most two bitdistributions and need to be considered because of symmetry. For the subchannels with , can be calculated from (8) with a rate distribution of . For the subchannels with , can be calculated from (8) with a rate distribution of . For the other subchannels, can be calculated from or depending on and th . In Appendix, it is shown that the minimum power to transmit bits satisfies the following three properties sufficient for the optimality of a greedy algorithm [8]:
, is an increasing function of , and is an increasing function of . With these properties, the optimality of the greedy algorithm can be shown as follows. Assume that there is a rate distribution that requires less power than the rate distribution achieved by the greedy algorithm. Then the optimal rate distribution can be improved by subtracting one bit from a certain subchannel and adding it to another subchannel because the optimal rate distribution is different from the rate distribution achieved by the greedy algorithm. So, unless the optimal rate distribution is the one that is achieved by the greedy algorithm, the optimal distribution can be improved, which is contradictory to the fact that the distribution is optimal. Thus, the rate distribution achieved by the greedy algorithm is optimal. Alternatively, the optimality of a greedy algorithm can be explained using a matroid concept as in [9]. This algorithm has a computational complexity of target since the total number of bits to be allocated is target . The proposed algorithm can be extended to solve the powerminimization problem with a power mask and a bit cap. The method that distributes the rate sum to the upstream and the downstream also needs to be changed slightly, and the cost to increase one bit needs to be set as infinite if any of the power-mask and bit-cap constraints is violated. IV. S IMULATION R ESULTS In this section, the optimal discrete bit-loading algorithm is applied to DSL channels. For simulation, the channel model
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2) Bit-Loading Iterations: Repeat the following until the target rate target is achieved. a) Increase the data rate by one bit in subchannel which requires the minimum cost among all available subchannels. b) Update the cost to increase the data rate by one bit in subchannel . The cost is calculated using Proposition 1, and the optimal rate and power distribution to the upstream and the downstream is also determined. In the above algorithm, the cost to increase one bit in subchannel is defined as if (9) if
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Fig. 2. Channel gains, optimal PSD, optimal rate allocation, and optimal frequency band plan at 4 kft with dBm.
in Annex A of [10] is used, and it is assumed that there are 25 twisted pairs in one binder. The frequency band from 0 kHz to 2208 kHz is considered, and this band is divided into 512 subchannels. The effective SNR gap is chosen as 12 dB, which achieves the symbol error probability of with 3.8 dB coding gain and 6 dB noise margin. Fig. 2 shows the optimal power spectral density (PSD), rate allocation, and frequency band plan along with the channel dBm. As is shown in the gains at 4 kft with
figure, the direct channel gain decreases monotonically as the frequency increases, while the FEXT and NEXT channel gains increase. Thus rate decreases as the frequency increases. In addition, it is shown that the frequency band up to 440 kHz should be shared by the upstream and the downstream and the frequency band above 440 kHz should be divided to the upstream and the downstream. Fig. 3 illustrates the optimal frequency band plan with various loop lengths. As revealed by the figure, the optimal frequency band plan is highly sensitive to the length of the twisted pairs. Specifically, as the loop length increases, the cutoff frequency between the shared band and the monopolized band is decreased. We now compare the optimal rate and power allocation algorithm with two suboptimal schemes: the Equal PSD scheme in which the entire band is shared and the FDD scheme in which the upstream and the downstream use disjoint bands. In both the Equal PSD and FDD schemes, we allocate the power optimally given a frequency band plan. On the other hand, the optimal scheme does not assume any predetermined band plan. Fig. 4 shows the rate sums of the upstream and the downstream for the optimal, Equal PSD, and FDD schemes when the total power is 17.5 dBm. As shown in the figure, the optimal scheme outperforms both the Equal PSD and FDD schemes. In addition, the figure indicates that the FDD scheme is better than the Equal PSD scheme for long loops, whereas the Equal PSD scheme is better than the FDD scheme for short loops. This result can be explained as follows.
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A PPENDIX P ROPERTIES OF We show that the minimum power for bits satisfy the following three properties: , is an increasing function of , and is an increasing function of . Let and be the minimum power necessary for and for , respectively. The power sum then takes the form of or depending on , , , and . For subchannels with , . For subchannels with , . For the other subchannels, if th
otherwise
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if otherwise
It is trivial to show that satisfies the first and second properties. The incremental power is
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From (8) and some algebraic manipulation,
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which is an increasing function of . Similarly, it can be shown that and satisfy all three properties. It remains to show that for th th where the form of changes from to :
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Rate sum of the upstream and the downstream versus loop length.
The direct channel gains become smaller as the loop length increases, while the NEXT channel gain is constant regardless of the loop length. Thus, the FDD scheme, which avoids the NEXT, should perform better than the Equal PSD scheme with long loops. Conversely, the Equal PSD scheme should perform better than the FDD scheme with short loops, where the NEXT channel gains are relatively small compared to the direct channel gains.
V. C ONCLUSION This letter investigated the rate and power allocation problem of the upstream and the downstream when an echo canceller is used. An optimal discrete bit-loading algorithm was developed with both the FEXT and NEXT taken into account. This optimal algorithm is of low complexity and outperforms existing suboptimal schemes significantly.
[1] T. Starr, J. M. Cioffi, and P. J. Silverman, Understanding Digital Subscriber Line Technology, Upper Saddle River, NJ:Prentice Hall, 1999. [2] A. Sendonaris, V. V. Veeravalli, and B. Aazhang, “Joint signaling strategies for approaching the capacity of twisted-pair channels,” IEEE Trans. Commun., vol. 46, pp. 673-685, May 1998. [3] R. V. Gaikwad and R. G. Baraniuk, “Joint signaling techniques and spectral optimization for symmetric bit-rate communication over selfNEXT-dominated channels,” IEEE Trans. Commun., vol. 52, pp. 1080, July 2004. [4] K. J. Kerpez, “Near-end crosstalk is almost Gaussian,” IEEE trans. Commun., vol. 41, pp. 670 -672, Jan. 1993. [5] J. Lee, R. V. Sonalkar, and J. M. Cioffi, “A multi-user power control algorithm for digital subscriber lines”, IEEE Commun. Lett., vol. 9, March 2005. [6] J. Lee, R. V. Sonalkar, and J. M. Cioffi, “Multi-user discrete bitloading for DMT-base DSL systems,” in Proceedings of IEEE Global Telecommunications Conference (GLOBECOM ’02), 2002, pp. 12591263. [7] J. Campello, “Optimal discrete bit loading for multicarrier modulation systems,” International Symposium on Information Theory (ISIT), p. 193, 1998. [8] A. Fasano, G. Di Blasio, and E. Baccarelli, “Optimal discrete bit loading for DMT based constrained multicarrier systems,” inProceedings of International Symposium on Information Theory (ISIT) 2002, p. 243, 2002. [9] E. Baccarelli and M. Baigi, “Optimal integer bit-loading for multicarrier ADSL systems subject to spectral-compatibility limits,” Signal Processing, vol. 84, pp. 729-741 , 2004. [10] Spectrum Management for Loop Transmission Systems, American National Standard for Telecommunications, ANSI T1.417-2001.
