Interference Alignment with Asymmetric Complex Signaling - Settling the Host-Madsen-Nosratinia Conjecture (0904.0274v1)

Abstract: It has been conjectured by Host-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 Host-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 degrees-of-freedom, i.e., the assumption of time-variations/frequency-selectivity used in prior work to establish the same result, is not needed.

• The paper introduces asymmetric complex signaling to achieve at least 1.2 degrees-of-freedom in nearly every three-user Gaussian interference channel configuration.
• It proves that conventional circularly symmetric Gaussian inputs are suboptimal, highlighting the advantage of asymmetric complex signaling in interference environments.
• The study’s findings pave the way for improved interference alignment strategies and enhanced spectral efficiency in dense wireless networks.

Interference Alignment with Asymmetric Complex Signaling: Implications for Degrees of Freedom in Complex Gaussian Channels

The paper "Interference Alignment with Asymmetric Complex Signaling - Settling the Høst-Madsen-Nosratinia Conjecture," authored by Viveck Cadambe, Syed A. Jafar, and Chenwei Wang, addresses the conjecture proposed by Høst-Madsen and Nosratinia regarding the degrees of freedom (DoF) of complex Gaussian interference channels. Specifically, they theorized that such channels, despite the number of users, possess only a singular degree-of-freedom when characterized by constant channel coefficients. This work counters the conjecture by demonstrating that 1.2 degrees-of-freedom can be consistently achieved across nearly all configurations of channel coefficients, with the exception being a subset of measure zero.

Key Contributions

This investigation's substantial contribution lies in introducing the asymmetric complex signaling concept, which plays a crucial role in achieving more than a single degree-of-freedom in interference channels. This method deviates from the standard optimal circularly symmetric input distributions common in point-to-point and multiple-access channels, demonstrating instead that asymmetric complex inputs offer notable advantages within interference networks.

Highlights of the results include:

• Evidence that at least 1.2 degrees of freedom can be achieved in a three-user complex Gaussian interference channel for almost any channel coefficient value.
• Proof that conventional signaling methods, such as circularly symmetric Gaussian inputs, are suboptimal for interference channels.
• Establishing that the straightforward beamforming (in conjunction with asymmetric complex signaling) achieves 1.2 degrees of freedom while users treat interference as noise.

Implications and Innovation

This paper's findings have broad implications for theoretical and practical aspects of communication networks. Primarily, the concept of asymmetric complex signaling introduces a new paradigm into communication theory, offering a robust method for crossing traditional DoF boundaries in Gaussian interference channels.

From a theoretical perspective, this research not only refutes the Høst-Madsen-Nosratinia conjecture but also enriches the understanding of interference alignment in static environments—a topic historically constrained by the difficulties presented by non-varying channel matrices. The comprehensive alignment strategy elucidated here greatly expands the potential for signal design in real-world applications.

Practically, this discovery holds promise for enhanced spectral efficiency in wireless networks, particularly under high interference scenarios. As wireless technology continues to evolve toward denser and more user-intensive deployments, the additional degrees of freedom offered by asymmetric complex signaling could significantly optimize network throughput and reliability.

Future Directions

The implications open several avenues for future exploration and development in the field of wireless communications:

• Exploring the boundaries for degrees-of-freedom achievable through combinations of signal-level alignment schemes, which focus on channel magnitudes, and vector-space schemes, emphasizing phase information.
• Investigating the utility of asymmetric complex signaling within other standard communication model paradigms, such as the cognitive radio or MIMO extensions.
• Designing practical algorithms to incorporate asymmetric complex signaling in adaptive modulation and coding schemes for real-time network enhancements.

In conclusion, this work not only settles previously hypothesized limits of Gaussian interference channels but also carves a novel path for improved interference management strategies through the innovative use of asymmetric signaling. The detailed methodology and results elucidated here establish a strong foundation for subsequent explorations into communications under constraints of constant channels, with anticipation of considerable advancements in both theoretical constructs and tangible applications.