⚡ electrical engineering

Energy-Efficient Resource Allocation for PA Distortion-Aware M-MIMO OFDM System

This paper proposes an energy-efficient resource allocation framework for downlink MU-MIMO-OFDM systems that jointly optimizes user transmit power and active antenna count while explicitly accounting for nonlinear power amplifier distortion, demonstrating significant energy efficiency gains over existing baselines through an alternating optimization approach.

Original authors: Siddarth Marwaha, Pawel Kryszkiewicz, Eduard Jorswieck

Published 2026-05-27

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CC BY 4.0

Original authors: Siddarth Marwaha, Pawel Kryszkiewicz, Eduard Jorswieck

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive wireless network as a giant orchestra trying to play music for hundreds of people (the users) at once. The conductor is the Base Station (the "BS"), and the musicians are the antennas. In a "Massive MIMO" system, there are hundreds of these musicians.

The goal of this paper is to make this orchestra energy-efficient. They want to play the loudest, clearest music possible while using the least amount of electricity.

Here is the problem the authors solved, explained through simple analogies:

1. The "Distorted Amplifier" Problem

In a perfect world, if you turn up the volume on a speaker, the music gets louder and clearer. But in the real world, the speakers (called Power Amplifiers or PAs) have a limit.

The Analogy: Imagine a singer trying to hit a high note. If they push too hard, their voice cracks and distorts.
The Paper's Insight: Most previous studies assumed the speakers were perfect or only slightly imperfect. This paper says, "Let's assume the speakers will crack if we push them too hard." When they crack, they waste energy and create "noise" (distortion) that ruins the music.

2. The Old Way vs. The New Way

The Old Way (Fixed Settings): Imagine a conductor who decides, "We will always use 32 musicians and always sing at 60% of our maximum volume." This is safe, but it's wasteful. If the audience is close, you don't need 32 musicians; you could get away with 5. If the audience is far, 60% volume might be too quiet, or pushing to 100% might crack the singers' voices.
The New Way (DEEP-DEAL): The authors created a smart system (named DEEP-DEAL) that acts like a genius conductor who constantly adjusts two things:
1. How many musicians are playing? (Antenna count)
2. How loud is each person singing? (Power allocation)

3. The "Sweet Spot" (The Core Discovery)

The paper found a surprising truth: To save energy, you often want to push the speakers right up to the edge of cracking, but not quite break them.

The Analogy: Think of a car engine. You don't always drive at 100 mph (too much fuel) or 10 mph (too slow). There is a specific speed where you get the most miles per gallon.
The Paper's Finding: The best energy efficiency happens when the system operates near the "saturation" point (where distortion starts). If you stay too far below this point, you are wasting energy on circuitry (cooling, electronics) that isn't needed. If you go too far above, the distortion ruins the signal, and you have to waste even more power to fix it.

4. How the System Works (The "Alternating" Dance)

The math behind this is very complex, but the logic is like a dance with two partners:

Step A (DEEP): "Okay, we have 50 musicians. How much total power should we use, and how do we split it among the users?" The system calculates the perfect split.
Step B (DEAL): "Okay, we have that power. Do we actually need 50 musicians? Maybe 40 is enough? Or maybe we need 60?" The system adds or removes musicians to find the perfect number.
Repeat: They keep swapping roles, tweaking the power and the number of musicians, until they find the absolute best combination where the energy efficiency is at its peak.

5. The Results

The authors tested this with computer simulations:

The Gain: In scenarios with many users, their new method used 3 times less energy to do the same job compared to the old "fixed" methods.
The Flexibility: If a user is close to the tower, the system turns off most antennas and whispers (saving huge amounts of power). If a user is far away, it turns on more antennas and shouts, but carefully manages the volume so the speakers don't crack.

Summary

This paper teaches us that to save energy in 5G and future networks, we shouldn't just turn up the volume or add more antennas blindly. Instead, we need a smart system that knows exactly how much "distortion" the hardware can handle and adjusts the number of active antennas and the power levels in real-time to find the perfect, energy-saving balance.

Technical Summary: Energy-Efficient Resource Allocation for PA Distortion-Aware M-MIMO OFDM Systems

Problem Statement
The paper addresses the challenge of maximizing Energy Efficiency (EE) in downlink multi-user (MU) massive multiple-input multiple-output (M-MIMO) systems employing Orthogonal Frequency-Division Multiplexing (OFDM). While M-MIMO is a cornerstone of 5G and beyond, its energy consumption is heavily driven by hardware components, particularly Power Amplifiers (PAs), which can account for up to 65% of a base station's (BS) total energy consumption.

Existing resource allocation strategies often fail to accurately model the nonlinear distortion introduced by practical PAs. Most prior works either assume ideal linear PAs or model hardware impairments as additive Gaussian distortion with a constant Signal-to-Distortion Ratio (SDR). These simplifications are valid only over limited power ranges and fail to capture the complex interactions between transmit power, nonlinear distortion, and power consumption, especially near the PA saturation region. Consequently, these models often require externally imposed transmit-power constraints and yield suboptimal EE designs. The paper identifies a gap in jointly optimizing transmit power, per-user power allocation, and the number of active antennas under realistic, wide-range nonlinear PA models for wideband M-MIMO OFDM systems.

Methodology
The authors propose a system model for a downlink MU M-MIMO OFDM system using Zero-Forcing (ZF) precoding over independent Rayleigh fading channels. Key modeling choices include:

PA Model: A soft-limiter model is adopted to represent the PA. This model linearly amplifies signals below a saturation power ( $P_{max}$ ) and clips them otherwise. It captures the dependence of both signal distortion and power consumption on the Input Back-Off (IBO).
Power Consumption: The total BS power consumption ( $P_{tot}$ ) includes static power, per-antenna RF chain power, and PA power. The PA power is modeled for both "Class B" (practical) and "Perfect" (idealized) architectures, showing how consumption scales with the IBO.
Performance Metric: The Signal-to-Noise-and-Distortion Ratio (SNDR) is derived using Bussgang decomposition, treating nonlinear distortion as noise-like interference. The EE is defined as the ratio of the sum achievable rate to the total consumed power.

Optimization Framework
The authors formulate an EE maximization problem that jointly optimizes the total transmit power ( $P$ ), the power allocation fractions per user ( $\omega$ ), and the number of active antennas ( $M$ ). This results in a non-convex mixed-integer fractional optimization problem. To solve this, they develop an Alternating Optimization (AO) framework named DEEP-DEAL (Distortion-aware Energy Efficient Power allocation and Antenna aLlocation):

DEEP (Power Allocation): For a fixed antenna count, the problem is decoupled into:
- Total Power Optimization: A non-convex single-variable problem. The authors prove the existence of a stationary point by analyzing the asymptotic behavior of the EE derivative. They employ a bisection-based root-finding algorithm with a function-specific starting point search to find the optimal total power.
- Per-User Power Distribution: A multi-variable problem shown to be concave. The optimal solution follows a water-filling structure, solvable via standard bisection methods on the Lagrange dual variable.
DEAL (Antenna Allocation): For fixed power and user fractions, the number of active antennas ( $M$ ) is optimized. Although $M$ is an integer, the authors relax it to a continuous variable. They prove the existence of a unique root for the EE derivative with respect to $M$ and use a bisection search to find the continuous optimum. The optimal integer solution is then selected by evaluating the EE at the floor and ceiling of the continuous root.
Convergence: The algorithm iterates between these blocks. The authors prove that the sequence of EE values is monotonically non-decreasing and converges to a stationary point. The computational complexity scales linearly with the number of users and logarithmically with numerical precision.

Key Contributions

Realistic Modeling: The paper introduces an EE maximization formulation that utilizes a soft-limiter PA model, capturing the nonlinear relationship between transmit power, distortion, and power consumption across the full operating range, eliminating the need for arbitrary external power constraints.
Joint Optimization: It is the first work to jointly optimize total transmit power, per-user power fractions, and the number of active antennas for wideband M-MIMO OFDM systems under nonlinear PA distortion.
Algorithmic Solution: The authors propose the DEEP-DEAL framework, which decomposes the complex mixed-integer non-convex problem into tractable sub-problems solvable via efficient bisection and water-filling techniques, avoiding the high complexity of nested iterative procedures like Dinkelbach's method combined with Successive Convex Approximation (SCA).
Theoretical Guarantees: The paper provides rigorous proofs for the existence of stationary points for the power and antenna sub-problems and establishes the convergence properties of the overall AO algorithm.

Simulation Results
Numerical simulations compare the proposed DEEP-DEAL framework against baselines including fixed IBO (REF-E), fixed SDR models (FixedSDR), and power-only optimization (DEEP).

EE Improvements: In large multi-user scenarios (e.g., 60 users), DEEP-DEAL achieves median EE improvements of up to 3× compared to fixed power and antenna allocation baselines.
Operating Points: The results demonstrate that the EE-optimal operating point often occurs at finite transmit power and a finite number of active antennas. Operating PAs near saturation (low IBO) combined with antenna scaling is shown to be EE-optimal, particularly in high path-loss conditions.
Heterogeneous Channels: In scenarios with heterogeneous path losses, the framework dynamically allocates more power to users with stronger channels and adjusts the antenna count and IBO accordingly, outperforming fixed-SDR approximations by up to 2× in extreme asymmetry cases.
Convergence: The algorithm exhibits fast empirical convergence, typically stabilizing within 5 outer iterations.

Significance
The paper claims that its findings underscore the necessity of balancing distortion-aware power control with antenna scaling to maximize EE in M-MIMO systems. By moving beyond simplified linear or constant-SDR models, the work provides a more accurate and efficient design methodology for future sustainable wireless networks. The authors note that the optimal strategy often involves operating PAs closer to saturation while dynamically adjusting the number of active antennas, a trade-off that traditional models fail to capture. The study suggests that ignoring the nonlinear nature of PAs leads to significant suboptimality in energy-constrained network designs.