Power Amplifier-aware Power Allocation for Noise-limited and Distortion-limited Regimes

1. Problem Statement

Conventional power allocation strategies in Multi-Input Multi-Output (MIMO) systems, such as the Water-Filling (WF) algorithm, rely on the assumption that the Radio Frequency (RF) chain is linear. This assumption holds only when the Power Amplifier (PA) operates well below its saturation point.

However, as transmit power requirements increase, PAs enter a nonlinear saturation regime. In this regime:

• Signal-Dependent Distortion: Unlike thermal noise, distortion noise generated by PA nonlinearities scales with the transmit power.
• Capacity Ceiling: Increasing transmit power beyond a certain point does not improve the Signal-to-Noise Ratio (SNR); instead, it increases self-generated interference, potentially causing a collapse in channel capacity.
• Sub-optimality: Standard WF algorithms, which assume static noise floors, allocate excessive power to strong eigen-channels, driving them into deep saturation and maximizing distortion, thereby reducing overall system performance.

The paper addresses the need for a new power allocation strategy that explicitly accounts for PA nonlinearity to optimize capacity in both noise-limited and distortion-limited regimes.

2. Methodology

A. Statistical Linearization via Bussgang's Theorem

To incorporate PA nonlinearity into an information-theoretic framework, the authors employ the Bussgang theorem.

• Model: They model the PA as a memoryless hard-limiting (soft-limiter) nonlinearity.
• Decomposition: The nonlinear PA output is decomposed into a coherent signal component (attenuated by a linear gain $\alpha$) and an uncorrelated distortion noise term ($\eta_{PA}$).
• Derivation:
  • The equivalent linear gain $\alpha$ is derived as a function of transmit power ($P_T$) and supply voltage ($V_{CC}$), showing gain compression as $P_T$ increases.
  • The distortion variance $\sigma^2_{\eta}$ is calculated as the residual power, which grows rapidly once the input exceeds the saturation threshold.
• Vectorization: This scalar model is extended to a MIMO vector form where each antenna has an independent PA, resulting in a diagonal gain matrix $A(p)$ and a distortion noise vector $\eta_{PA}$.

B. System Capacity Formulation

The received signal model is updated to include the effective channel and effective noise:

• Effective Channel: $H_{eff} = H \cdot A(p)$, where the physical channel $H$ is weighted by the power-dependent PA gains.
• Effective Noise: The total noise at the receiver includes both thermal noise and the PA distortion noise, which is spatially colored by the channel matrix $H$.
• Capacity Equation: The ergodic capacity is formulated as a function of the power allocation vector $p$, the effective channel, and the dynamic noise covariance matrix $R_{neff}(p)$.

C. Optimization Algorithm

The goal is to maximize capacity subject to a total power constraint.

• Challenge: The objective function is non-convex because the noise covariance depends on the power allocation itself.
• Solution: The authors propose a Projected Gradient Descent (PGD) algorithm.
  • Gradient Calculation: The gradient of the capacity with respect to power is derived, revealing a "penalty term" proportional to the distortion gradient. This term forces the algorithm to reduce power on channels that contribute heavily to aggregate distortion, even if those channels have high gain.
  • Projection: An iterative projection operator ensures the solution respects the total power budget and non-negativity constraints (similar to a dynamic water-filling thresholding).
  • Spatial Back-off: The algorithm naturally identifies an optimal spatial back-off strategy, intentionally reducing power on high-gain eigen-channels to prevent them from dominating the distortion noise floor.

D. Regime Transition Threshold

The authors derive a closed-form thermal noise variance threshold ($\sigma^2_{n,th}$) that separates the two operating regimes:
$$\sigma^2_{n,th} = \frac{\sigma^2_{\eta}(P_{avg}) \|H\|_F^2}{N_R}$$

• Noise-Limited Regime: When thermal noise is dominant ($\sigma^2_n \gg \sigma^2_{n,th}$), standard water-filling is near-optimal.
• Distortion-Limited Regime: When distortion dominates ($\sigma^2_n \ll \sigma^2_{n,th}$), the proposed amplifier-aware strategy is mandatory. The threshold depends on the PA characteristics and the Frobenius norm of the channel matrix, indicating that stronger channels accelerate the onset of distortion limitations.

3. Key Contributions

1. Amplifier-Aware Power Allocation: A novel PGD-based algorithm that optimizes power distribution by balancing coherent signal gain against power-dependent distortion noise.
2. Spatial Back-off Strategy: The discovery that optimal performance in saturation regimes requires intentionally under-utilizing high-gain spatial channels to minimize aggregate distortion.
3. Closed-Form Regime Threshold: A mathematical derivation defining the exact boundary between noise-limited and distortion-limited regimes, linking it to channel spatial properties ($\|H\|_F$).
4. Theoretical Framework: A rigorous extension of MIMO capacity theory that integrates PA saturation effects via Bussgang decomposition, moving beyond the linear RF chain assumption.

4. Numerical Results

Simulations were conducted with $N_T = N_R = 32$ antennas.

• Power Utilization: In low-noise environments, the proposed strategy significantly reduces power utilization compared to standard WF to avoid saturation, whereas standard WF attempts to use the full budget, leading to inefficiency.
• Capacity Gains:
  • In the distortion-limited regime (low thermal noise), the proposed strategy provides significant capacity gains, exceeding 100% compared to standard water-filling in deep saturation.
  • In the noise-limited regime (high thermal noise), both strategies converge, as the PA nonlinearity becomes negligible relative to the thermal noise floor.
• Robustness: The proposed algorithm maintains higher capacity than standard WF even as the channel rank fluctuates over time, demonstrating robustness to channel variations.

5. Significance

This work fundamentally shifts the paradigm of power allocation in high-power MIMO systems. By acknowledging that noise is not static but a function of transmit power in the presence of nonlinear amplifiers, the paper provides a practical solution for:

• High-SNR Systems: Where traditional WF fails due to PA saturation.
• Energy Efficiency: Preventing the "waste" of power that converts into distortion rather than information.
• Future 6G/High-Frequency Systems: As systems push for higher spectral efficiency and power, managing the trade-off between signal gain and distortion becomes critical. The derived threshold offers a quick metric for system designers to determine if amplifier-aware optimization is necessary for their specific channel and hardware conditions.