Target-Rate Least-Squares Power Allocation over Parallel Channels

TL;DR

This paper introduces a novel power allocation method for parallel Gaussian channels that minimizes squared rate deviation from targets, offering a closed-form solution with significant computational efficiency and improved target tracking over traditional methods.

Contribution

It develops a new target-rate least-squares power allocation algorithm with a closed-form solution and demonstrates its efficiency and effectiveness compared to classical waterfilling.

Findings

Achieves up to 1890x speedup over numerical solvers.

Matches numerical optimization to machine precision.

Outperforms waterfilling and other heuristics in target tracking.

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.