Target-Rate Least-Squares Power Allocation over Parallel Channels

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.