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.

翻译：本文研究了在$N$个并行高斯信道（例如OFDM子载波）上的功率分配问题，其中每个信道具有期望的目标频谱效率。给定信道增益与噪声系数$a_i>0$以及各信道目标$T_i\ge 0$，我们在总功率约束$\sum_i P_i \le P_{\mathrm{tot}}$和非负性约束$P_i \ge 0$下，最小化总平方速率偏差$\sum_{i=1}^{N}(\log_2(1+a_iP_i)-T_i)^2$。我们证明，最优分配方案从不超过任何目标，并且在所有目标联合可行时可能剩余功率未使用，这一结构与经典的注水算法有本质区别。利用KKT条件，我们推导出关于活跃集的、以Lambert~W函数表示的每信道闭式解，并将剩余计算简化为对偶变量的一维单调二分搜索。所得算法的时间复杂度为$O(N\log(1/\varepsilon))$，在$N=1024$个信道时，相比通用数值求解器可实现高达1{,}890倍的加速。在瑞利衰落信道上的数值实验证实，该闭式解与数值优化结果在机器精度上一致，并且在一系列运行条件下，相比注水算法、均匀分配和比例公平策略，展现出更优的目标跟踪性能。