Achieving the DoF Limits of the SISO X Channel with Imperfect-Quality CSIT Jingjing Zhang Mobile Communications Department EURECOM, Sophia Antipolis, France Email: jingjing.zhang@eurecom.fr Dirk Slock Mobile Communications Department EURECOM, Sophia Antipolis, France Email: dirk.slock@eurecom.fr Petros Elia Mobile Communications Department EURECOM, Sophia Antipolis, France Email: petros.elia@eurecom.fr Abstract—In the setting of the two-user single-input single- output X channel, recent works have explored the degrees-of- freedom (DoF) limits in the presence of perfect channel state information at the transmitter (CSIT), as well as in the presence of perfect-quality delayed CSIT. Our work shows that the same DoF-optimal performance — previously associated to perfect- quality current CSIT — can in fact be achieved with current CSIT that is of imperfect quality. The work also shows that the DoF performance previously associated to perfect-quality delayed CSIT, can in fact be achieved in the presence of imperfect-quality delayed CSIT. These follow from the presented sum-DoF lower bound that bridges the gap — as a function of the quality of delayed CSIT — between the cases of having no feedback and having delayed feedback, and then another bound that bridges the DoF gap — as a function of the quality of current CSIT — between delayed and perfect current CSIT. The bounds are based on novel precoding schemes that are presented here and which employ imperfect-quality current and/or delayed feedback to align interference in space and in time. Index Terms—degrees of freedom, SISO X channel, CSIT. I. I NTRODUCTION We consider the two-user Gaussian single-input single- output (SISO) X channel (XC), with two single-antenna transmitters and two single-antenna receivers, where each transmitter has an independent message for each of the two receivers. The corresponding channel model takes the form y t = h (1) t x (1) t + h (2) t x (2) t + m t z t = g (1) t x (1) t + g (2) t x (2) t + n t (1) where at any time t, h (i) t , g (i) t denote the scalar fading coefficients of the channel from transmitter i to receiver 1 and 2 respectively, where m t ,n t denote the unit-power AWGN noise at the two receivers, and where x (i) t , i =1, 2 denotes the transmitted signals at transmitter i, satisfying a power constraint E(|x (i) t | 2 ) ≤ P . Naturally each x (i) t may include some private information – originating from transmitter i – intended for receiver 1, and some private information intended for receiver 2. In this setting, for a quadruple of achievable rates R ij , i,j =1, 2 corresponding to communication from trans- mitter i to receiver j , we adopt the high-SNR degrees-of- freedom (DoF) approximation d ij = lim P →∞ R ij log P , i, j =1, 2 to describe the limits of performance over the XC, particularly focusing on the sum DoF measure d Σ := d 11 +d 21 +d 12 +d 22 . In this context, the challenge originates from the fact that each transmitter is both an interferer as well as an intended transmitter to both receivers. Crucial in addressing Rx 1 Rx 2 Tx 1 Tx 1 W 11 ,W 12 W 21 ,W 22 y z Fig. 1. 2-user SISO X channel. this challenge is the role of feedback — and specifically of channel state information at the transmitters (CSIT) — which can allow for separation, at each receiver, of the intended and the interfering signals. In particular, while the optimal sum DoF without CSIT has been shown to be d Σ =1 (cf. [1]), the DoF increases to d Σ = 6 5 in the presence of perfect-quality delayed CSIT (see [2] which proved that this performance is optimal over all linear schemes), and the DoF further increases to an optimal sum-DoF of d Σ = 4 3 (see [3]) in the presence of perfect-quality and instantaneously available CSIT (perfect current CSIT). A. Feedback quality model Motivated by practical settings of limited feedback links, we here consider the case where feedback can be of imperfect- quality, and potentially also delayed. Towards this, let ˆ h (i) t , ˆ g (i) t denote the current CSIT estimates of channels h (i) t ,g (i) t re- spectively, and let ˜ h (i) t = h (i) t - ˆ h i t , ˜ g (i) t = g (i) t - ˆ g (i) t (2) be the estimation errors, modeled here as having i.i.d Gaussian entries. Similarly for delayed CSIT, along the same lines as in [4], [5], let ˇ h (i) t , ˇ g (i) t denote the delayed estimates of channels h (i) t ,g (i) t , where these estimates are obtained sometime after the channel elapses, and let ¨ h (i) t = h (i) t - ˇ h (i) t , ¨ g (i) t = g (i) t - ˇ g (i) t (3) be the associated CSIT errors, modeled here as having i.i.d Gaussian entries. In our context, motivated by the approach in [6]–[9], we consider α = - lim P →∞ log E[|| ˜ h (i) t || 2 ] log P = - lim P →∞ log E[|| ˜ g (i) t || 2 ] log P