1 When does vectored Multiple Access Channels (MAC) optimal power allocation converge to an FDMA solution? Vincent Le Nir, Marc Moonen, Jan Verlinden, Mamoun Guenach Abstract—Vectored Multiple Access Channels (MAC) have attracted a lot of interest during the past few years. The optimal structure of vectored MAC in the uplink is based on the Successive Interference Canceller-Minimum Mean Square Error (SIC-MMSE). For equal weights and zero Signal to Noise Ratio (SNR) gap, the optimal transmit covariance matrices are found by iterative waterfilling. However, practical scenarios include non- zero SNR gap required to achieve the target probability of error at the desired data rate. With non-zero SNR gap, it is possible to achieve a weighted rate sum higher than iterative waterfilling by means of MAC-Optimal Spectrum Balancing (MAC-OSB). Moreover, it has been shown in the literature that the optimal power allocation for single-carrier flat scalar MAC with non- zero SNR gap is given by a Frequency Division Multiple Access (FDMA) type solution. In this paper, we investigate the problem of optimal power allocation for multi-carrier vectored MAC with non-zero SNR gap. Simulation results are given for VDSL2 channels and wireless channels. We confirm by simulations that FDMA is indeed the optimal power allocation in multi-carrier scalar MAC systems. When extending to multi-carrier vectored MAC systems, the optimal power allocation will tend towards a shared spectrum solution or a FDMA type solution depending on the level of crosstalk, the SNR, the SNR gap and the power constraints. I. I NTRODUCTION Vectored Multiple Access Channels (MAC) have attracted a lot of interest during the past few years [1], [2]. The optimal receiver structure of vectored MAC in the uplink is based on the the Successive Interference Canceller-Minimum Mean Square Error (SIC-MMSE) [1]. For equal weights and zero Signal to Noise Ratio (SNR) gap, the optimal transmit covari- ance matrices are found by iterative waterfilling [2]. However, practical scenarios include non-zero SNR gap required to achieve the target probability of error at the desired data rate. With non-zero SNR gap, it is possible to achieve a weighted rate sum higher than iterative waterfilling by means of MAC- Optimal Spectrum Balancing (MAC-OSB) [3]. In a recent paper, it has been shown that the optimal power allocation for single-carrier flat scalar Multiple Access Channels (MAC) V. Le Nir and M. Moonen are with the SISTA/ESAT laboratory, Katholieke universiteit of Leuven (KUL), Leuven, Belgium. E-mail: vin- cent.lenir@esat.kuleuven.be marc.moonen@esat.kuleuven.be J. Verlinden and M. Guenach are with Alcatel-Lucent, Antwerpen, Bel- gium. E-mail: jan.vj.verlinden@alcatel-lucent.be mamoun.guenach@alcatel- lucent.be This research work was carried out in the frame of the FWO project ’Design and evaluation of DSL systems with common mode signal exploitation’ and the IWT project 060207: ’SOPHIA, Stabilization and Optimization of the Physical layer to Improve Applications’. The scientific responsibility is assumed by its authors. with non-zero SNR gap is given by an Frequency Division Multiple Access (FDMA) type solution [4]. In this paper, we investigate the problem of optimal power allocation for multi-carrier vectored MAC with non-zero SNR gap. Therefore we extend the single-carrier flat scalar MAC with non-zero SNR gap to the multi-carrier vectored MAC with non-zero SNR gap. In the vectored MAC case, it is shown that the crosstalk plays an important role in the optimal power allocation. The optimal power allocation can tend towards a FDMA type solution or a shared solution of each tone between users, depending on the level of crosstalk scenario, the SNR, the SNR gap and the power constraints. Using the Lagrange multipliers in the vectored MAC objective function, we study the convergence of the multi-carrier vectored MAC with non- zero SNR gap toward these two solutions depending on these different parameters. In section II, we first recall the comparison between the weighted rate sums for single-carrier systems FDMA, scalar MAC and vectored MAC with non-zero SNR gap. In section III, we extend the different weighted rate sums from the single-carrier case to the multi-carrier case. In fact, the primal vectored MAC capacity optimization problem subject to per- modem total power constraints and non-zero SNR gap is transformed into a collection of per-tone unconstrained vec- tored MAC capacity optimization problems using Lagrangian parameters. We derive optimal receiver structures in combina- tion with optimal transmit covariance matrices which achieve vectored MAC channel capacity. Simulation results are given for VDSL2 channels and wireless channels. We confirm by simulations that FDMA is indeed the optimal power allocation in multi-carrier scalar MAC systems. When extending to multi- carrier MIMO MAC systems, the optimal power allocation will tend towards a shared spectrum solution or a FDMA type solution depending on the level of crosstalk, the SNR, the SNR gap and the power constraints. We consider a vectored MAC with T receivers and K users, each user k having a single transmitter in an uplink scenario and using Discrete Multi-Tone (DMT) modulation with a cyclic prefix longer than the maximum delay spread of the channels. The transmission on tone i can then be modelled as: v i = H H i u i + w i where H H i =  h H i1 ... h H iK  (1) where N c is the number of subcarriers, v i is the received