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

Boyu Ning, Haifan Yin, Sixu Liu, Hao Deng, Songjie Yang, Yuchen Zhang, Weidong Mei, David Gesbert, Jaebum Park, Robert W. Heath, Emil Björnson

arXiv:2601.05092v1cs.IT

TL;DR

The paper addresses the difficulty of understanding 5G NR beamforming codebooks and their practical implementation. It presents a tutorial spanning beamforming fundamentals, CSI-RS measurements, PMI feedback, and 3GPP Releases 15–18, while analyzing codebook evolution, performance, overhead, and applicability. The resulting framework clarifies how standardized codebooks support flexible CSI acquisition and reporting for network-resource optimization and user experience.

  • Problem

    Understanding 5G NR beamforming codebooks requires navigating complex protocol specifications and their constraints on feedback flexibility and adaptability.

  • Method

    The paper develops an accessible tutorial covering beamforming, CSI-RS measurements, PMI feedback, regular and port-selection codebooks, and 3GPP codebook evolution from Release 15 to Release 18.

  • Results

    The analysis shows progressive codebook performance enhancement from R15 to R18 through spectral compression, optimized port selection, temporal compression, and scenario-specific tradeoffs.

  • Takeaways & Limitations

    The unified framework clarifies how 5G NR codebooks enable flexible and robust CSI acquisition and reporting for dynamic network-resource optimization and improved user experience.

Abstract

from arXiv · show

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 as both an informative tutorial and a guidance for future research, facilitating more effective collaboration between academia and industry in advancing wireless communication technologies.

I. INTRODUCTION

The paper addresses the gap between theoretical beamforming-codebook research and elusive 5G NR PMI standardization by connecting MIMO fundamentals, protocol procedures, and practical codebook design. It motivates codebook-based beamforming as a practical approach for CSI acquisition while identifying continuing research challenges in standardized feedback.

  • Motivation: Massive MIMO uses spatial processing to transmit multiple data streams and improve spectral efficiency, capacity, and user experience.Its large base-station antenna arrays support spatial diversity and simultaneous multiuser transmission.
  • Motivation: TDD reciprocity can reduce downlink CSI-acquisition overhead, whereas FDD requires explicit UE feedback because uplink and downlink bands are asymmetric.Alternative approaches may also be needed in TDD under calibration errors, low uplink SNR, or rapidly changing environments.
  • Motivation: PMI codebooks provide the base station with information about the preferred precoding matrix while limiting feedback, but conventional codebooks constrain beamforming flexibility.3GPP Release 18 extends earlier frameworks through more flexible reconstruction using spatial, spectral, and temporal bases.
  • Research gap: The paper targets a significant theory–practice gap: 5G NR PMI descriptions are difficult to interpret, and no comprehensive tutorial holistically connects codebook theory with implementation.The stated goal is to support clearer collaboration between academia and industry on future beamforming-codebook development.
  • Contributions: Its contributions cover beamforming fundamentals, standardized CSI-RS and PMI procedures, codebook structures, and analysis of 3GPP evolution from Releases 15 to 18.The paper also compares codebooks by compact modeling, performance, feedback overhead, and applicable scenarios.
  • Beamforming background: Downlink beamforming must match channel characteristics, multiplex spatial streams, and limit inter-user and inter-stream interference.The paper frames these objectives through linear beamforming and achievable-rate analysis in downlink OFDM MIMO systems.
  • Open challenges: Codebook-indicator feedback remains underexplored academically despite widespread 3GPP adoption, alongside challenges from far-field, near-field, wideband, and hybrid architectures.The paper identifies beam-pattern, beam-squint, constant-modulus, and multistage-training issues as distinct research challenges.

IV. CODEBOOK FUNDAMENTALS

3GPP codebooks construct beamforming matrices from spatial, frequency, and temporal bases, with the PEB and array topology determining the spatial basis and its dimensions. Regular and port-selection codebooks support different PEB structures while serving the same transmission role.

  • Codebook bases: 3GPP codebooks build beamforming matrices from spatial, frequency, and temporal bases representing beams, delay taps, and Doppler shifts.Frequency and temporal bases are DFT vectors; the spatial basis uses a Kronecker product of two vectors shaped by the PEB and array topology.
  • PEB and codebook selection: The PEB determines whether the UE uses a regular or port-selection codebook, while array topology sets the dimensions of the spatial basis vectors.Sub-connected PEBs lead to logical antenna-domain measurements and regular-codebook feedback; full-connected PEBs use port-selection feedback.
  • Regular codebook: Regular-codebook spatial bases combine horizontal and vertical DFT steering vectors through a Kronecker product to form two-dimensional beams.The horizontal and vertical vectors index different angular directions, and their combination creates the spatial basis used by the codebook.
  • Regular codebook: Oversampling creates OHOV orthogonal beam groups, each containing NHNV basis vectors, allowing more spatial beams than the number of orthogonal beams.The oversampling factors OH and OV expand the horizontal and vertical beam grids beyond the basic DFT basis.
  • Regular codebook: Type II regular codebooks select L beam bases from one orthogonal DFT group and linearly combine them across two polarization directions.The coefficients a1,i and a2,i weight each selected beam on the first and second polarization directions, while β denotes power scaling.
  • Port-selection codebook: Port-selection codebooks select combinations of DFT beams within a full-connected PEB, which maps logical CSI-RS ports to physical antenna ports.Regular codebooks generate beam combinations directly toward antenna ports, whereas port-selection codebooks select combinations inside the PEB.

V. 5G CODEBOOK EVOLUTION

Release 15 established standardized Type I and Type II codebooks, with Type I distinguishing feedback modes by subband beam consistency and Type II supporting higher-precision beam combinations.

  • Release 15 introduced standardized Type I and Type II codebooks, while Type I excluded port-selection codebooks.
  • R15 Type I Codebook: Type I selects one beam and uses Mode 1 when it remains consistent across the BWP, otherwise using Mode 2.
  • R15 Type I Codebook: Type I supports 1–8 data streams, with dedicated tables and collective Mode 1-2 terminology above three streams.
  • R15 Type I Codebook: Mode 2 divides beam selection into wideband beam-group selection through i1 and subband beam and polarization-phase selection through i2.
  • R15 Type II Codebook: Type II forms beamforming vectors by weighted combinations of L ∈ {2, 3, 4} DFT bases, allowing linearly independent vectors across polarizations.
  • R15 Type II Codebook: The Type II feedback representation requires B2 to use 8N1N2 bits after each restriction requires 2N1N2 bits.

D. R16 Enhanced Type II Regular Codebook

Release 16 Enhanced Type II extends Type II codebooks into the frequency domain by combining spatial and spectral bases, selecting delay taps, and compressing amplitude and phase feedback under an overhead limit.

  • Spectral representation: Enhanced Type II combines spatial bases with a spectral basis to represent precoding matrices across frequency-domain subbands.
  • Spectral representation: The spectral basis contains N3 orthogonal DFT vectors, from which Mv taps are selected for each layer.
  • Tap selection: For a 100 MHz BWP with 15 kHz spacing and subband size 16, 18 subbands result and Mv can be 3 or 5.
  • Tap selection: The UE identifies the strongest delay-domain tap, remaps it to index zero, and selects Mv−1 additional taps from the remaining candidates.
  • Coefficient feedback: Amplitude and phase indicators describe coefficients across selected taps and beams, while bitmaps identify which coefficients are reported.
  • Feedback compression: The parameter K0 limits the number of reported nonzero weight entries across taps and beams on each layer.

E. R17 Further Enhanced Type II Port-Selection Codebook

Release 17 Further Enhanced Type II Port-Selection modifies Enhanced Type II by allowing flexible port selection and simplifying delay-tap indication while retaining spatial and frequency-domain compression.

  • R17 evolution: The Further Enhanced codebook remains highly similar in essence to the Enhanced Type II Port-Selection Codebook.
  • Port and tap selection: Spatial compression selects L beams from CSI-RS ports, while frequency compression selects M delay-domain taps from N3 available taps.
  • Port selection: The UE can freely indicate any L ports instead of selecting consecutive ports.
  • Tap indication: The selected-tap count is reduced to M ∈ {1, 2}, and the two-level indication method is canceled.
  • Configuration: The codebook supports CSI-RS port configurations from 4 through 32 ports, including 16 ports, and derives L through α rather than configuring it directly.
  • R17 evolution: The i1 index structure changes because arbitrary port combinations replace consecutive-beam indication and two-level tap indication is no longer needed.
  • Tap indication: The selected-tap count is unified to M for single-stream and multi-stream transmission, eliminating the separate Mv distinction.

F. R18 Enhanced Type II Codebook for Predicted PMI

Release 18 extends the Enhanced Type II codebook with temporal compression for predicted PMI, representing time-domain variation through selected Doppler shifts alongside spatial and frequency-domain bases. The resulting feedback structure adds shift selection and compresses reported coefficients through configured parameters, bitmaps, and strongest-coefficient indicators.

  • Temporal compression: R18 adds temporal compression by representing N4 time-domain dimensions with Q temporal bases, alongside spatial beams and frequency-domain taps.The temporal bases are called shifts or Doppler shifts; Q is currently fixed at 2.
  • Temporal compression: The temporal basis z_τ uses selected shifts indexed by τ, with τ ranging from 0 to 1 because Q=2.The basis can be interpreted as DFT vectors over the N4 time-domain dimension.
  • Feedback compression: The codebook controls feedback overhead by configuring the numbers of beams, compression factors, and reported nonzero coefficients through higher-layer parameters and bitmaps.Parameters β, L, and pv are configured for coefficient reporting, while nonzero bitmap bits identify reported coefficients.
  • Coefficient reporting: The protocol remaps indices around strongest coefficients and omits remaining indicators to limit the number of reported feedback entries.The strongest beam, tap, shift, and polarization direction determine which coefficient indicators are retained.
  • New PMI indication: R18 predicted PMI reports an additional indicator i1,10,l to identify selected shifts, while the other indicator content remains unchanged.The new shift indicator distinguishes the predicted-PMI codebook from the earlier Enhanced Type II codebook.

VI. DISCUSSION OF 5G CODEBOOKS

The discussion section distills the detailed 3GPP codebooks into simplified models, compares their performance, evaluates feedback overhead, and summarizes where each codebook applies.

  • Discussion of 5G codebooks: The paper uses simplified codebook models to analyze performance disparities, feedback overhead, and application scenarios.This discussion abstracts the detailed protocol formulations into models suitable for comparison.

A. Compact Model

The compact model abstracts secondary protocol details so the principal structure and features of 3GPP codebooks can be represented more clearly.

  • Compact model: Ignoring secondary factors such as scaling, normalization, quantization, and bitmaps yields compact expressions that expose codebook structure.The simplified expressions are intended to clarify the key features and structural makeup of the codebooks.
  • Compact model: The compact model defines full and effective bases for the space, frequency, and time domains.These bases provide the modeling framework for subsequent codebook representations.

1) Bases in the Space Domain:

Spatial-domain bases differ between regular and port-selection codebooks: regular codebooks use oversampled, dual-polarization DFT beams, whereas port-selection codebooks use selected standard basis vectors.

  • Bases in the Space Domain: The spatial basis uses an oversampled DFT matrix to represent beams, with dimensions determined by antenna geometry and oversampling factors.Dual polarization makes the spatial-domain representation more complex than the frequency- and time-domain bases.
  • Bases in the Space Domain: The DFT matrix provides N1N2 orthogonal beams, from which L beams are selected for each precoding vector.An orthogonal beam group is determined by the codebook index i1,1.
  • Bases in the Space Domain: Regular-codebook full bases duplicate the DFT beam matrix across the two polarization directions.The resulting full-basis matrix has dimension PCSI-RS × PCSI-RS, with PCSI-RS = 2N1N2.
  • Bases in the Space Domain: The effective-basis matrix contains the selected beams reported through PMI rather than the complete spatial beam set.Selected beam indices are linked to the PMI parameters i1,1 and i1,2.
  • Bases in the Space Domain: Port-selection codebooks replace DFT beams with standard basis vectors and form effective bases by selecting L columns from an identity matrix.The truncated identity matrix represents the selected port basis.

2) Bases in the Frequency Domain:

The frequency-domain construction distinguishes full bases from effective bases and forms the latter from selected delay taps whose indices are determined by PMI feedback.

  • The full frequency-domain bases are defined first for n = 0, 1, . . . , N3 −1.
  • The effective frequency-domain basis matrix is ˆ Wf = [y0, y1, . . . , yMv−1] ∈CN3×Mv.
  • Each effective basis vector yn is a delay tap defined previously in the codebook construction.
  • The selected taps correspond to taps in the full-basis representation through an index relation, with their values determined by i1,6,l.

3) Bases in the Temporal Domain:

The temporal-domain construction parallels the frequency-domain formulation by distinguishing full and effective bases, selecting shifts through feedback, and using unified spatial notation across codebook types.

  • The full temporal-domain bases are defined over n = 0, 1, . . . , N4 −1, while effective bases retain selected shifts.
  • The effective temporal basis matrix is ˆ Wt = [z0, z1, . . . , zQ−1] ∈CN4×Q.
  • Each shift zn is defined previously, and the selected shifts are related to the full temporal basis through an index-selection relation.
  • Regular and port-selection codebooks share the compact matrix notation for spatial bases, but their spatial bases remain different.

4) Compact Model for R15 Type I:

The codebook models evolve from single-beam and independently reported subband precoding toward compressed, jointly represented space-frequency-time precoding.

  • R15 Type I: R15 Type I selects one beam shared by both polarizations, with a polarization phase difference encoded in the precoding vector.
  • R15 Type II: R15 Type II selects L shared beams, while polarization and data-stream combinations use distinct coefficients and each subband reports its own matrix.
  • R15 Type II: The R15 Type II compact model can use effective or full bases; only the full-basis representation captures beam selection completely in wPMI.
  • R16 Type II: R16 Type II jointly compresses N3 subband matrices onto Mv spectral bases, with shared wideband amplitudes for the L beams under each polarization.
  • R17 Type II: R17 Type II retains space-frequency compression while allowing freely selected ports, with K1 beam selections and M tap selections; its spatial bases are standard vector bases.
  • R18 Type II: R18 Type II jointly feeds back space, frequency, and time through spatial, spectral, and temporal bases, represented as a Tucker-decomposition tensor.

B. Codebook Performance

3GPP codebooks trade precoding accuracy against feedback overhead, with successive releases adding compression, port-selection optimization, and temporal prediction for broader deployment conditions. The paper also identifies future codebook directions and application-specific challenges beyond current standards.

  • Performance trade-off: Codebook performance balances precoding-matrix accuracy against feedback overhead rather than equaling ideal beamforming performance.Reporting ideal precoding matrices maximizes beamforming performance but creates substantial feedback overhead.
  • Release evolution: R15 Type I supports single-stream, single-beam transmission for low-complexity and low-power scenarios, while R15 Type II improves precision for higher-rate eMBB use cases.R15 Type II supports single-stream multi-beam transmission and higher-order MCS modes but has greater structural complexity.
  • Release evolution: R16 and R17 Type II introduce frequency-domain compression to reduce feedback overhead, while R18 adds temporal compression and channel prediction for mobile users.The later releases target scenarios including NTN, ITS, high-speed trains, vehicles, eMBB, and URLLC.
  • Future frameworks: Future framework research emphasizes unified codebooks and AI-enabled designs that can adapt bases or compensate for model inaccuracies across hardware, channel, mobility, and traffic conditions.The paper discusses tensor-structured reporting, standardized neural compression, site-specific bases, and trainable parameters.
  • Application challenges: Hierarchical codebooks could reduce beam scans in sparse channels, but user-specific measurements increase feedback rounds, alignment latency, and associated overhead.Current standards lack hierarchical codebooks using multi-stage measurements and feedback for rapidly establishing a satisfactory precoding matrix.
  • Future codebook bases: Existing regular codebooks exploit sparse DFT-based spatial, spectral, and temporal representations, but these bases are poorly matched to near-field spherical-wave propagation.Polar-domain dictionaries are identified as a potential way to represent near-field channels more efficiently.

APPENDIX A DETAILS OF FULL CSI-BASED BEAMFORMING

The appendix surveys full-CSI beamforming methods for SU-MIMO and MU-MIMO, then describes power-allocation strategies and their principal trade-offs.

  • SU-MIMO beamforming: SVD decomposes the channel into singular values and vectors, enabling eigenmode transmission, power allocation, and interference management in SU-MIMO.
  • SU-MIMO beamforming: MRT maximizes SNR by aligning transmission with the dominant channel direction but does not address inter-stream interference.
  • SU-MIMO beamforming: ZF suppresses inter-stream interference through channel inversion, but ill-conditioned channels can create excessive antenna weights and require power reduction.
  • SU-MIMO beamforming: RZF uses a regularization factor to balance noise amplification, approaching MRT as ξ →∞, ZF when ξ = 0, and MMSE at the stated operating point.
  • MU-MIMO beamforming: MU-MIMO methods extend interference management across users; block diagonalization achieves perfect suppression but has high complexity and requires K ≤Nt.
  • Power allocation: Water-filling maximizes capacity by favoring stronger subchannels, whereas harmonic-mean allocation limits stream imbalance to improve joint decoding robustness.

APPENDIX B RESOURCE ALLOCATION FOR CSI-RS

CSI-RS resource allocation configures reference-signal placement, timing, transmission structure, and reporting behavior through coordinated 5G NR parameters.

  • CSI-RS resource configuration: CSI-RS configuration maps symbols onto resource elements, configures frequency-domain RB locations, and sets time-domain periodicity and offset.
  • CSI-RS signal construction: CSI-RS symbols use pseudo-random sequences, orthogonal cover codes, and power scaling to support channel estimation across multiple antenna ports.
  • CSI-RS resource mapping: CSI-RS placement varies with port count, density, and CDM type; Release 18 defines 18 mapping schemes, with eight presented in the appendix.
  • CSI reporting configuration: CSI-RS resources are organized through Resource, ResourceSet, CSI-ResourceConfig, and CSI-ReportConfig, which respectively identify locations, group resources, activate transmission, and select measurement configurations.
  • CSI reporting modes: The standard supports periodic, aperiodic, and semi-persistent CSI reporting, with periodic reports carried on PUCCH and aperiodic reports triggered by DCI on scheduled PUSCH.
  • Reporting granularity: To reduce feedback overhead, PMI reporting uses subbands rather than independent feedback for every OFDM subcarrier, with configurable subband sizes across BWP configurations.

APPENDIX D PORT-SELECTION CODEBOOKS IN R15-16

R15–R16 port-selection codebooks adapt Type II codebook feedback to full-connected PEB architectures by selecting CSI-RS ports that represent external beams.

  • Codebook rationale: Type II port-selection codebooks target full-connected PEBs, which transform antenna-domain channels into sparse beam-domain channels while hiding the physical array from the UE.
  • Port versus beam selection: Unlike Type II regular codebooks, which select L beams per layer, port-selection codebooks select L CSI-RS ports per layer to construct each precoding vector.
  • Port versus beam selection: With a PEB, selecting CSI-RS ports is equivalent to selecting external beams, making the resulting port-selection precoding vector functionally equivalent to the regular-codebook vector.
  • R15 Type II port-selection codebook: The R15 spatial basis uses standard basis vectors with one nonzero element, and supported CSI-RS port counts are 4, 8, 12, 16, 24, and 32.
  • R15 Type II port-selection codebook: The sampling size d is configured from {1, 2, 3, 4} and is constrained by d ≤ min(PCSI-RS/2, L).
  • R16 enhanced codebook: R16 Enhanced Type II port-selection codebooks retain the regular codebook’s compression mechanisms while replacing implicit spatial parameters with an explicit i1,1 indication.
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