Precoding Matrix Indicator in the 5G NR Protocol: A Tutorial on 3GPP Beamforming Codebooks

The evolution of 5G-advanced (5G-A) systems relies heavily on advanced beamforming technologies to achieve high spectral efficiency and network capacity. Although abundant theoretical research has been devoted to optimizing beamforming based on full channel state information (CSI), a major gap remains with the codebook-based beamforming defined in 3rd Generation Partnership Project (3GPP) standards for practical deployment. This disconnect is exacerbated by the often complex and elusive nature of the protocol descriptions themselves. This paper bridges this critical gap by providing a systematic examination of the beamforming codebook technology, i.e., precoding matrix indicator (PMI), in the 5G NR from theoretical, standardization, and implementation perspectives. We begin by introducing the background of beamforming in multiple-input multiple-output (MIMO) systems and the signaling procedures for codebook-based beamforming in practical 5G systems. Then, we establish the fundamentals of regular codebooks and port-selection codebooks in 3GPP standards. Next, we provide rigorous technical analysis of 3GPP codebook evolution spanning Releases 15-18, with particular focus on: 1) We elucidate the core principles underlying codebook design, 2) provide clear physical interpretations for each symbolic variable in the codebook formulas, summarized in tabular form, and 3) offer intuitive visual illustrations to explain how codebook parameters convey information. These essential pedagogical elements are almost entirely absent in the often-obscure standardization documents. Through mathematical modeling, performance benchmarking, feedback comparisons, and scenario-dependent applicability analysis, we provide researchers and engineers with a unified understanding of beamforming codebooks in real-world systems. Furthermore, we identify future directions and other beamforming scenarios for ongoing research and development efforts. This work serves not only as a tutorial but also as a reference book for simulation and a guide for future research, facilitating more effective collaboration between academia and industry in advancing wireless communication technologies.