Sphere Decoding Revisited

2025-12-11

AI Summary

To address the uncontrollable computational complexity and unclear trade-off mechanisms of sphere decoding (SD) for integer least-squares (ILS) problems, this paper proposes the Equivalent Sphere Decoding (ESD) framework. ESD introduces two explicit, tunable parameters—the initial search size $K$ and the bias factor $sigma$—and derives, for the first time, a tight analytical upper bound on SD complexity: $|S| < nK$. This enables predictable and adjustable balance between complexity and performance. By incorporating normalized weighting and candidate-set protection, ESD unifies optimal and suboptimal decoding paradigms within a single coherent framework. Extensive evaluation in massive MIMO detection demonstrates that ESD achieves linear complexity scaling with respect to $K$, attains bit-error-rate performance close to maximum-likelihood (ML) decoding, and significantly improves both decoding efficiency and robustness.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Search
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, the paradigm of sphere decoding (SD) for solving the integer least square problem (ILS) is revisited, where extra degrees of freedom are introduced to exploit the decoding potential. Firstly, the equivalent sphere decoding (ESD) is proposed, which is essentially the same with the classic Fincke-Pohst sphere decoding but characterizes the sphere radius $D>0$ with two new parameters named as initial searching size $K>1$ and deviation factor $σ>0$. By fixing $σ$ properly, we show that given the sphere radius $D riangleqσsqrt{2ln K}$, the complexity of ESD in terms of the number of visited nodes is upper bounded by $|S|<nK$, thus resulting in an explicit and tractable decoding trade-off solely controlled by $K$. To the best of our knowledge, this is the first time that the complexity of sphere decoding is exactly specified, where considerable decoding potential can be explored from it. After that, two enhancement mechanisms named as normalized weighting and candidate protection are proposed to further upgrade the ESD algorithm. On one hand, given the same setups of $K$ and $σ$, a larger sphere radius is achieved, indicating a better decoding trade-off. On the other hand, the proposed ESD algorithm is generalized, which bridges suboptimal and optimal decoding performance through the flexible choice of $K$. Finally, further performance optimization and complexity reduction with respect to ESD are also derived, and the introduced tractable and flexible decoding trade-off is verified through large-scale MIMO detection.

Problem

• Revisits sphere decoding to solve integer least squares problem
• Introduces equivalent sphere decoding with explicit complexity bound
• Enhances algorithm with mechanisms for better decoding trade-off

Innovation

• Equivalent sphere decoding with two new parameters
• Complexity upper bound explicitly controlled by K
• Normalized weighting and candidate protection enhancements