[Paper Review] Generalized Degrees of Freedom of the Symmetric Gaussian $K$ User Interference Channel

Syed A. Jafar, Sriram Vishwanath|arXiv (Cornell University)|Apr 28, 2008

Wireless Communication Security TechniquesEngineering2 references17 citations

TL;DR

This paper characterizes the generalized degrees of freedom (GDOF) for the symmetric K-user Gaussian interference channel, showing that the GDOF per user is independent of K except at α=1, where it drops to 1/K. The analysis uses a deterministic channel model and nested lattice coding to enable interference alignment across signal levels, achieving optimal GDOF across all α regimes.

ABSTRACT

We characterize the generalized degrees of freedom of the $K$ user symmetric Gaussian interference channel where all desired links have the same signal-to-noise ratio (SNR) and all undesired links carrying interference have the same interference-to-noise ratio, ${INR}={SNR}^α$. We find that the number of generalized degrees of freedom per user, $d(α)$, does not depend on the number of users, so that the characterization is identical to the 2 user interference channel with the exception of a singularity at $α=1$ where $d(1)=\frac{1}{K}$. The achievable schemes use multilevel coding with a nested lattice structure that opens the possibility that the sum of interfering signals can be decoded at a receiver even though the messages carried by the interfering signals are not decodable.

Motivation & Objective

• To extend the generalized degrees of freedom (GDOF) characterization from the 2-user to the K-user symmetric Gaussian interference channel.
• To determine whether interference management regimes identified for the 2-user case generalize to K-user scenarios with constant channel coefficients.
• To investigate the role of structured coding and signal-level alignment in interference channels with more than two users.
• To establish the sum capacity scaling in the high-SNR regime for symmetric K-user interference channels.

Proposed method

• Uses a deterministic channel model to approximate the Gaussian interference channel, enabling analysis of signal-level alignment.
• Applies nested lattice coding with multilevel structure to allow decoding of the sum of interfering signals even when individual messages are not decodable.
• Derives outer bounds on sum capacity using the fact that at α=1, all receivers see statistically equivalent signals, leading to a multiple access channel sum capacity.
• Establishes inner bounds via achievable schemes that align interference in signal levels, with power constraints managed through qit-based signal construction.
• Uses asymptotic analysis in SNR to derive GDOF, comparing inner and outer bounds across different α regimes.
• Employs a quantized signal model with base Q, where symbols are represented as qits, and analyzes achievable rates as Q→∞.

Experimental results

Research questions

• RQ1 Does the GDOF characterization for the 2-user interference channel extend to the K-user symmetric case, excluding α=1?
• RQ2 What is the impact of the number of users K on the generalized degrees of freedom when interference-to-noise ratio scales as SNR^α?
• RQ3 Can interference alignment be achieved in K-user symmetric Gaussian interference channels using signal-level alignment and structured coding?
• RQ4 Why does a singularity occur at α=1, and how does it affect the degrees of freedom per user?
• RQ5 How do different interference regimes (noisy, weak, moderately weak, strong, very strong) scale in terms of GDOF for K≥3?

Key findings

• The generalized degrees of freedom per user, d(α), is independent of K for all α ≠ 1, matching the 2-user case.
• At α=1, d(α) = 1/K due to full symmetry, where all receivers can decode all messages, reducing the sum capacity to a multiple access channel.
• For 0 ≤ α ≤ 1/2 (noisy interference), d(α) = 1 - α, achieved by treating interference as noise with Gaussian codebooks.
• For 1/2 ≤ α ≤ 2/3 (weak interference), d(α) = α, achieved via signal-level alignment using nested lattice codes and qit-based power control.
• For 2/3 ≤ α < 1 (moderately weak interference), d(α) = 1 - α/2, achieved by aligning interference in the lower signal levels while preserving decodable data in the upper levels.
• For α ≥ 2 (very strong interference), d(α) = 1, as interference is strong enough to be decoded and canceled, allowing full multiplexing gain.