Target-Rate Least-Squares Power Allocation over Parallel Channels

2026-03-06

AI Summary

This work addresses power allocation over parallel Gaussian channels—such as OFDM subcarriers—under a total power constraint, aiming to minimize the sum of squared deviations between achieved spectral efficiencies and prescribed targets. By analyzing the KKT conditions, the study reveals a novel structural property: the optimal solution never overshoots the target spectral efficiencies and may leave part of the total power unused, thereby departing from the classical water-filling paradigm. A closed-form solution is derived using the Lambert W function, and the associated dual variable is efficiently computed via a one-dimensional monotonic bisection method with complexity O(N log(1/ε)). Numerical experiments demonstrate that, for N = 1024, the proposed algorithm achieves machine-precision accuracy and is up to 1890× faster than generic numerical solvers, while significantly outperforming water-filling and other baselines in target tracking performance.

Technology Category

• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study power allocation over $N$ parallel Gaussian channels, such as OFDM subcarriers, when each channel has a desired target spectral efficiency. Given channel gain-to-noise coefficients $a_i>0$ and per-channel targets $T_i\ge 0$, we minimize the total squared rate deviation $\sum_{i=1}^{N}(\log_2(1+a_iP_i)-T_i)^2$ subject to a sum-power constraint $\sum_i P_i \le P_{\mathrm{tot}}$ and nonnegativity $P_i \ge 0$. We prove that the optimal allocation never overshoots any target and may leave power unused when all targets are jointly feasible, a structure fundamentally different from classical waterfilling. Using the KKT conditions, we derive a per-channel closed-form solution in terms of the Lambert~W function on the active set and reduce the remaining computation to a one-dimensional monotone bisection for the dual variable. The resulting algorithm runs in $O(N\log(1/\varepsilon))$ time and achieves up to 1{,}890$\times$ speedup over general-purpose numerical solvers at $N=1024$ channels. Numerical experiments over Rayleigh fading channels confirm that the closed-form solution matches numerical optimization to machine precision and demonstrate superior target-tracking performance compared to waterfilling, uniform allocation, and proportional fairness across a range of operating conditions.

Problem

• power allocation
• parallel channels
• target spectral efficiency
• sum-power constraint
• rate deviation

Innovation

• target-rate power allocation
• Lambert W function
• parallel Gaussian channels
• closed-form solution
• KKT conditions