⚡ electrical engineering

Optimal Analog Beamforming and Power Allocation for Multiuser TDMA Systems

This paper investigates the joint design of analog beamforming and power allocation for single-RF-chain multiuser TDMA systems under a max-min SNR criterion, deriving a closed-form optimal power allocation for fixed beamformers and developing globally optimal branch-and-bound algorithms for both discrete and continuous phase-shift optimization.

Original authors: Songnan Gu, Chongjun Ouyang, Hao Jiang, Xingqi Zhang

Published 2026-04-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Songnan Gu, Chongjun Ouyang, Hao Jiang, Xingqi Zhang

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 you are the conductor of a massive orchestra, but with a very strange twist: you only have one baton (one radio transmitter) to control hundreds of musicians (antennas). Your goal is to make sure every single musician in the audience (the users) hears the music clearly, even if they are sitting in different, noisy corners of the hall.

This paper is about finding the perfect way to wave that one baton and how to split the volume so that the quietest listener gets the best possible signal.

Here is the breakdown of the problem and the solution, using everyday analogies:

1. The Problem: The "One-Baton" Dilemma

In modern wireless networks (like 5G), we use many antennas to send signals. Usually, you'd want a separate "baton" (radio chain) for every user to talk to them all at once. But that's expensive and uses too much power.

So, engineers use a Single-Radio-Frequency (RF) Chain system. It's like having one megaphone connected to a wall of speakers.

The Constraint: You can't change the volume of each speaker individually. You can only change the timing (phase) of the sound waves coming out of them.
The Strategy: Since you can't talk to everyone at once with one baton, you use TDMA (Time-Division Multiple Access). This means you talk to User A, then User B, then User C, very quickly.
The Goal: You want to make sure the person with the worst connection gets the best possible signal. This is called "Max-Min" fairness. You don't want one person getting a crystal-clear signal while another gets static.

2. The Two-Step Solution

The authors realized they couldn't solve the whole puzzle at once, so they broke it down into two steps:

Step A: The Fairness Splitter (Power Allocation)

Imagine you have a fixed amount of water (power) to pour into a set of buckets (users). Some buckets have holes (bad signal channels), and some are solid (good channels).

The paper proves that for any specific way you aim your antenna, there is a mathematically perfect formula to split the water.
The Result: You give more water to the buckets with holes and less to the solid ones, so that every single bucket ends up with exactly the same water level. This ensures no one is left dry.

Step B: The Perfect Aim (Beamforming)

Now that we know how to split the power, the hard part is: How do we aim the antenna?
The antenna needs to point its "beam" in a direction that helps everyone. But the antenna can only point in specific directions (like a flashlight that clicks into fixed positions, or one that can slide smoothly).

The Challenge: This is like trying to find the perfect angle to throw a ball so it bounces off a wall and hits a target. There are millions of angles, and most are wrong. If you just guess and check (a method called "Alternating Optimization"), you might get stuck in a "local optimum"—a good spot, but not the best spot.

3. The "Branch-and-Bound" Detective (The Algorithm)

To find the absolute best angle, the authors invented a smart search method called Branch-and-Bound (BB).

Think of this like a detective searching for a lost diamond in a giant, dark warehouse:

Branching: The detective divides the warehouse into smaller and smaller rooms (splitting the possible angles).
Bounding (The Magic Trick): Before entering a room, the detective checks a map. The map says, "Even if the diamond is in the best possible spot in this room, it won't be as valuable as the diamond we already found in the kitchen."
Pruning: If the map says a room can't possibly hold the best diamond, the detective ignores that entire room. They don't waste time searching it.

By doing this, the computer can quickly eliminate millions of bad angles and zoom in on the one perfect angle that gives the best signal to everyone.

4. The Three Types of Antennas

The paper tested this detective method on three types of "flashlights":

Binary (On/Off): The flashlight can only point Left or Right. (Like a binary switch).
Discrete (Click-Click): The flashlight clicks into 4 or 8 fixed positions. (Like a dial with numbers).
Continuous (Smooth): The flashlight can point anywhere in a smooth circle. (Like a high-end laser pointer).

5. The Results: Why It Matters

The Gold Standard: The authors' "Branch-and-Bound" method found the true global optimum. It is the "Gold Standard" answer.
The Reality Check: They compared this perfect answer to the "cheap" method (Alternating Optimization) that most companies use today.
The Surprise: They found that the "cheap" method is actually very good in most cases! It gets 90-95% of the way to the perfect solution.
The Value: However, knowing the exact perfect answer is crucial. It tells engineers, "Hey, you are already doing great, but if you want that last 5% of performance, here is exactly how much effort you need to put in."

Summary

This paper is like a master chef who figured out the perfect recipe for a multi-course meal where everyone gets the same amount of food.

First, they calculated exactly how to slice the ingredients (Power Allocation).
Then, they used a super-smart search algorithm (Branch-and-Bound) to find the perfect cooking temperature and timing (Beamforming).
They proved that while the "quick and dirty" cooking method most people use is pretty good, knowing the perfect recipe helps us understand exactly how much better we could do if we tried harder.

In short: They solved a very hard math puzzle to find the "perfect" way to aim a wireless signal, providing a benchmark to see how well our current technology is actually performing.

1. Problem Statement

The paper addresses the joint design of analog beamforming and power allocation in a downlink multiuser Time-Division Multiple Access (TDMA) system equipped with a single Radio-Frequency (RF) chain.

System Constraints: The system uses a phased-array architecture where the beamforming vector is shared among all users. Due to the single RF chain, the beamforming weights are subject to constant-modulus (unit-modulus) constraints, making the optimization problem highly non-convex.
Objective: The goal is to maximize the minimum received Signal-to-Noise Ratio (SNR) among all $K$ users (max-min fairness criterion) under a total transmit power budget.
Challenge: The coupling between power allocation and phase shifts, combined with the non-convex unit-modulus constraint, makes finding a global optimum difficult. Existing methods (e.g., Alternating Optimization) typically provide only local optima or heuristic solutions, leaving the performance gap unquantified.

2. Methodology

The authors propose a two-stage approach to solve the problem:

A. Optimal Power Allocation (Closed-Form Derivation)

For any fixed analog beamforming vector, the authors derive the optimal power allocation in closed form using convex optimization and Karush-Kuhn-Tucker (KKT) conditions.

Key Insight: To maximize the minimum SNR, the optimal strategy equalizes the received SNR for all users.
Result: The original joint problem is reduced to a phase-shift optimization problem only. The objective function becomes minimizing the sum of the reciprocals of the effective channel gains:
$\min_{\theta} \sum_{k=1}^{K} \frac{1}{|h_k^H w|^2}$
where $w$ is the beamforming vector determined by phase shifts $\theta$ .

B. Globally Optimal Branch-and-Bound (BB) Algorithms

Since the reduced phase-shift problem remains non-convex, the authors develop globally optimal Branch-and-Bound (BB) algorithms for three distinct phase-shift scenarios:

Binary Phase Shifts ( $M=2$ ):
- Phases are restricted to $\{0, \pi\}$ (entries in $\{-1, +1\}$ ).
- Technique: The authors derive a lower bound (LB) for the objective function using the arithmetic-harmonic mean inequality and an upper bound on the quadratic form $w^T R w$ (where $R$ is a sum of channel correlation matrices).
- Pruning: Nodes in the search tree are pruned if their lower bound exceeds the current best objective value.
M-ary Discrete Phase Shifts ( $M > 2$ ):
- Phases are restricted to a finite set $\{2\pi m/M\}$ .
- Technique: A dual-layer lower bound is developed to maintain pruning efficiency as the search space grows ( $M^{N-1}$ $M^{N - 1}$ ).
  - Layer 1 (Individual): Uses the triangle inequality to bound individual user channel gains.
  - Layer 2 (Aggregate): Uses a system-level quadratic bound similar to the binary case.
- The combined lower bound is the maximum of these two layers.
Continuous Phase Shifts:
- Phases can take any value in $[0, 2\pi)$ .
- Technique: A Spatial Branch-and-Bound (SBB) algorithm is proposed.
  - Relaxation: The non-convex unit-modulus constraint is relaxed using Semidefinite Relaxation (SDR) and an epigraph reformulation.
  - Lifting: The problem is lifted into a higher-dimensional matrix space ( $\tilde{W} = \tilde{w}\tilde{w}^H$ ).
  - Bounding: Convex hull constraints are applied to phase sub-regions to generate valid lower bounds. Feasible upper bounds are constructed by projecting the relaxed solution back to the feasible set.
- The algorithm systematically branches the phase space until the gap between the lower and upper bounds is within a prescribed tolerance $\epsilon$ .

3. Key Contributions

Closed-Form Power Allocation: Proved that for any fixed beamformer, optimal power allocation equalizes user SNRs, reducing the problem dimensionality significantly.
Global Optimality: Developed the first globally optimal BB algorithms for this specific max-min fairness problem under binary, discrete, and continuous phase constraints.
Tight Lower Bounds: Derived novel, computationally efficient lower bounds (using matrix eigenvalues and harmonic means) that enable effective pruning of the search tree.
Benchmarking Framework: Provided a rigorous benchmark to quantify the "optimality loss" of low-complexity Alternating Optimization (AO) methods, which are commonly used in practice.

4. Numerical Results

The authors evaluated the algorithms using Rayleigh fading channels with $N$ antennas and $K$ users.

Performance vs. Antenna Count: As the number of antennas ( $N$ ) increases, the max-min SNR improves for all schemes. Continuous phase shifts perform best, followed closely by $M=4$ discrete shifts, with binary ( $M=2$ ) showing the largest performance loss.
Global vs. Local Optima:
- The proposed BB algorithms consistently achieved lower objective values (better SNR) than the Alternating Optimization (AO) baseline.
- Gap Analysis: The performance gap between the global optimum (BB) and the AO method is generally small, indicating that AO is a near-optimal, low-complexity solution for many practical scenarios.
- However, the gap is more pronounced in small-array regimes (small $N$ ), where the AO method is more likely to get trapped in local optima.
Continuous Phase: The SBB algorithm outperformed AO in continuous phase settings, particularly for small $N$ , but the gap narrowed as $N$ increased.

5. Significance

Theoretical Value: This work fills a critical gap in the literature by providing the global optimum for single-RF-chain multiuser TDMA systems, a problem previously solved only by heuristics.
Practical Insight: By quantifying the performance gap between global and local solutions, the paper validates the use of low-complexity AO methods in large-scale systems while highlighting scenarios (small arrays, strict hardware constraints) where global optimization is necessary.
Hardware Efficiency: The focus on binary and discrete phase shifts aligns with low-cost, energy-efficient hardware implementations (e.g., fixed wireless access, mmWave communications), offering design guidelines for trade-offs between hardware complexity and system performance.