Target-Rate Least-Squares Power Allocation over Parallel Channels

TL;DR

It is proved 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.

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.

Figures (9)

• Figure 1: Objective $J^*$ vs. total power budget $P_{\mathrm{tot}}$, comparing the Lambert W closed form with SLSQP numerical optimization. The two solutions agree to machine precision.
• Figure 2: Per-channel power allocation for $N = 8$ channels. The target-rate allocation respects per-channel caps (dashed lines) and distributes power to minimize rate deviations. Waterfilling concentrates power on the strongest channels.
• Figure 3: Achieved rates per channel. Target-rate allocation never exceeds the target $T = 3$ (no overshoot). Waterfilling overshoots on strong channels and falls short on weak ones.
• Figure 4: Total power $S(\lambda) = \sum_i P_i^*(\lambda)$ as a function of the dual variable $\lambda$. The optimal $\lambda^*$ satisfies $S(\lambda^*) = P_{\mathrm{tot}}$. Monotonicity guarantees unique convergence via bisection.
• Figure 5: Optimal objective vs. power budget. The objective reaches zero at $\sum_i \bar{P}_i \approx 16.8$ (dashed line), beyond which power goes unused.

Theorems & Definitions (15)

• Theorem 4.1: No Overshoot
• proof
• Proposition 4.2: Per-Channel Cap
• Proposition 4.3: Convexity
• proof
• Theorem 4.4: Regime Characterization
• proof
• Remark 4.1
• Corollary 4.5: Monotonicity
• Lemma 5.1: Monotonicity of $P_i(\lambda)$