[Paper Review] PSK Precoding in Multi-User MISO Systems

Andreas Noll, Hela Jedda+1 more|arXiv (Cornell University)|Jun 27, 2017

Advanced MIMO Systems Optimization8 references3 citations

TL;DR

This paper proposes a constant-envelope (CE) precoding scheme for multi-user MISO systems using PSK modulation to enable energy-efficient transmission via power amplifiers in saturation. It formulates a symbol-wise minimum squared error (SMSE) optimization problem and evaluates three algorithms—GDM, QGDM, and GPM—showcasing that QGDM achieves near-optimal performance with low complexity, especially at 3-bit quantization, enabling high spectral efficiency with minimal SNR penalty.

ABSTRACT

We consider the downlink scenario of multiuser multiple-input-single-output (MU-MISO) communication systems with constant envelope (CE) signals emitted from each antenna. This results in energy efficient power amplifiers (PAs). We propose a holistic CE precoding scheme based on the symbol-wise minimum squared error (SMSE) criterion. Additionally, we analyze the distortions introduced by low-resolution quantization to PSK for higher energy efficiency reasons. We present three solution algorithms and examine their performance to decide for the best pick for different quantization resolutions. Our results show that good performance can be achieved with minimal loss compared to an ideal unquantized case. Finally, we analyze and discuss the results and consider the overall complexity of the precoder as well as implementation issues.

Motivation & Objective

• To design an energy-efficient downlink MU-MISO system using constant-envelope (CE) signals to enable saturation-mode operation of power amplifiers.
• To address the performance degradation caused by low-resolution digital-to-analog converters (DACs) in CE precoding systems.
• To develop and compare efficient solution algorithms for SMSE-based precoding under PSK signal constraints.
• To evaluate the trade-off between performance, complexity, and implementation feasibility in practical systems with finite DAC resolution.
• To enable practical deployment by analyzing algorithmic complexity and proposing hardware-aware design strategies for real-time implementation.

Proposed method

• Formulates a symbol-wise minimum squared error (SMSE) criterion to optimize the precoder, minimizing the error between the desired signal and the effective received signal.
• Introduces a holistic CE precoding scheme where the transmit vector is constrained to have unit magnitude per antenna, ensuring constant envelope signals.
• Applies a quantization block to model finite-resolution DACs, mapping continuous-phase vectors to 2^B-PSK constellations.
• Develops three solution algorithms—Gradient Descent Method (GDM), Quasi-Newton Gradient Descent Method (QGDM), and Generalized Projected Method (GPM)—to solve the non-convex SMSE optimization problem.
• Uses step size adaptation and line search techniques to improve convergence, with performance evaluated under varying quantization resolutions (B=1 to B=3).
• Leverages rotational symmetry of QPSK to reduce computational load by computing precoders for only 1/4 of input vectors, reducing complexity from 4^M to 4^{M-1} per channel realization.

Experimental results

Research questions

• RQ1 How does low-resolution quantization (B=1 to B=3) affect the bit error rate (BER) performance of CE precoding in MU-MISO systems?
• RQ2 Which of the three proposed algorithms—GDM, QGDM, or GPM—offers the best trade-off between BER performance and computational complexity?
• RQ3 Can the SMSE-based CE precoding scheme achieve near-optimal performance with minimal SNR penalty compared to an ideal unquantized system?
• RQ4 How does the number of iterations and step size adaptation influence convergence speed and stability across different algorithms and quantization levels?
• RQ5 What are the practical implementation challenges related to memory and processing load, and how can they be mitigated in real-time systems?

Key findings

• For B=3, the QGDM algorithm achieves an SNR of 3.86 dB at BER=10^-3, with only a 0.5 dB degradation compared to the best-performing GDM and GPM algorithms.
• The GDM and GPM algorithms achieve nearly identical performance (3.39 dB and 3.37 dB SNR at BER=10^-3, respectively), indicating strong robustness under SMSE optimization.
• The QGDM algorithm is the fastest, requiring only 22 average iterations and 17 step size halvings, making it the most efficient for real-time implementation.
• The number of iterations is nearly invariant to the initial step size μ₀, indicating stable convergence behavior across different starting points.
• Despite exponential theoretical complexity in M (number of users), practical implementation can reduce the number of required precoder computations by exploiting QPSK rotational symmetry, cutting the load to 4^{M-1} per channel realization.
• The study concludes that the proposed precoding scheme is a strong candidate for real-world deployment due to its high energy efficiency, low complexity, and near-optimal performance with low-resolution DACs.