Interference Alignment With Asymmetric Complex Signaling—Settling the Høst-Madsen–Nosratinia Conjecture

Abstract

It has been conjectured by Høst-Madsen and Nosratinia that complex Gaussian interference channels with constant channel coefficients have only one degree-of-freedom regardless of the number of users. While several examples are known of constant channels that achieve more than 1 degree of freedom, these special cases only span a subset of measure zero. In other words, for almost all channel coefficient values, it is not known if more than 1 degree-of-freedom is achievable. In this paper, we settle the Høst-Madsen-Nosratinia conjecture in the negative. We show that at least 1.2 degrees-of-freedom are achievable for all values of complex channel coefficients except for a subset of measure zero. For the class of linear beamforming and interference alignment schemes considered in this paper, it is also shown that 1.2 is the maximum number of degrees of freedom achievable on the complex Gaussian 3 user interference channel with constant channel coefficients, for almost all values of channel coefficients. To establish the achievability of 1.2 degrees of freedom we introduce the novel idea of asymmetric complex signaling -i.e., the inputs are chosen to be complex but not circularly symmetric. It is shown that unlike Gaussian point-to-point, multiple-access and broadcast channels where circularly symmetric complex Gaussian inputs are optimal, for interference channels optimal inputs are in general asymmetric. With asymmetric complex signaling, we also show that the 2 user complex Gaussian X channel with constant channel coefficients achieves the outer bound of 4/3 degreesof-freedom, i.e., the assumption of time-variations/frequency-selectivity used in prior work to establish the same result, is not needed. * The ordering of authors is alphabetical. arXiv:0904.0274v1 [cs.IT] 1 Apr 2009 1. Signal Vector Space Alignment Schemes -Linear transmitter precoding and receiver combining operations are used to transform the interference channel into multiple non-interfering

On Alternate Relaying With Improper Gaussian Signaling

“…IGS has been shown to be beneficial in various interferencelimited systems [8]- [11]. Firstly, it has been proven in [8] that by employing IGS, we achieve higher degrees of freedom for the 3-user single-input single-output interference channel (SISO-IC) than its PGS counterpart.…”

“…Firstly, it has been proven in [8] that by employing IGS, we achieve higher degrees of freedom for the 3-user single-input single-output interference channel (SISO-IC) than its PGS counterpart. The Pareto boundary of the achievable rate region for the 2-user SISO-IC was characterized in [9].…”

Interference MAC: Impact of improper Gaussian signaling on the rate region Pareto boundary

“…Merging the constraints (14a) into the objective function, problem (14) can be expressed as a maximization problem with a weighted Chebyshev objective function [22],…”

“…Compared to proper signaling, improper signaling enables improving the achievable rates of the PIMAC since it enjoys the additional freedom of designing the pseudo-variance in addition to the variance of the transmit signal. For instance, improper signaling was proposed in [14] as a means to improve the degrees 2 of freedom (DoF) in the 3-user IC. In comparison, for the 2-user IC improper signaling does not enhance the DoF, yet improper signaling is useful in the low and moderate SNR (signal to noise power ratio) regime, as it improves the signalto-interference-plus-noise ratio (SINR) as shown in [15], [16].…”

Cognitive Interference Alignment for Spectral Coexistence

“…Since we usually deal with the complex numbers for channel coefficients, transmitted and received symbols as well as the noise, phase rotations can be exploited to find distinct rotations at each receiver. This can be realized as rotations in two dimensional real-imaginary plane and this is the main concept behind asymmetric complex signalling method [8,29].…”