Revealing the evanescent components in Kronecker-product based codebooks: insights and implications

TL;DR

This work reveals that Kronecker-product based codebooks for UPAs in NR harbor evanescent codewords, which correspond to non-physical, high-spatial-frequency EM components. Through mathematical derivations, EM insights, beam-pattern analyses, and system-level simulations, the authors demonstrate that these evanescent codewords are redundant and should be excluded without sacrificing throughput. They introduce a practical codebook compression method that removes evanescent codewords and reduces CBSR signaling overhead while preserving NR feedback structure, and show that Fresnel near-field channels do not support evanescent components. The study also extends to near-field and Rayleigh channel modeling, proposing a method to generate evanescent-free Rayleigh channels, thereby guiding future wideband and ELAA codebook design and standardization.

Abstract

The orthogonal bases of discrete Fourier transform (DFT) has been recognized as the standard spatial-domain bases for Type I, Type II and enhanced Type II codewords by the 3rd Generation Partnership Project (3GPP). For uniform planar arrays, these spatial-domain bases are derived as the Kronecker product of one-dimensional DFT bases. Theoretically, each spatial basis corresponds to a beam directed towards a specific angle of departure and the set of bases represent the orthogonal beams that cover the front hemisphere of an array. While the Kronecker-product based precoding scheme facilitates the concise indexing of a codeword in the codebooks through precoding matrix indicators (PMIs) in channel state information feedback, it introduces redundant spatial beams characterized by high spatial-frequency components. This paper investigates the presence of codewords representing high spatial-frequency components within the Kronecker-product based codebooks. Through theoretical analysis and simulations, we confirm the redundancy of these codewords in MIMO communications, advocating for their removal from the codebooks to enhance system performance. Several topics relevant to the high spatial components are also involved in the discussion. Practical suggestions regarding future standard design are provided based on our theoretical analysis and simulation results.

Figures (11)

• Figure 1: (a) The wave vector $\boldsymbol{k}$ in the reference Cartesian and spherical coordinate systems. (b) Illustration of the square confined by the indices of the codewords and the ellipse confined by inequality \ref{['eq-hf-area']}.
• Figure 2: The distribution of propagating and evanescent codewords in the index space for $N_1=N_2=8$, $O_1=O_2=4$ and $\alpha_1=\alpha_2=0.5$. The dark purple patches indicate the evanescent codewords.
• Figure 3: Beam patterns of codewords $\boldsymbol{v}_{4,10}$ (the left column) and $\boldsymbol{v}_{14,16}$ (the right column) from $\mathbf{CB}_{8,8}^{4,4}$. Subfigures (a) and (b) are based on array synthesis while (c) and (d) are results of full-waveform simulations.
• Figure 4: (a) Illustration of the spatial frequencies supported by a rectangular grid. (b) Maximum spatial frequency supported by a UPA with $\alpha_1=\alpha_2=0.5$ (solid line), along with the spatial frequencies of the outermost codewords shown in Fig. \ref{['f-hf-area']}b (orange "x").
• Figure 5: Beam patterns of precodings (a) $\mathbf{w}_1$ and (b) $\mathbf{w}_2$; Beam patterns of precodings with the same phase gradient as the evanescent codeword $\boldsymbol{v}_{14,16}$ on UPAs with (c) $\alpha_1=\alpha_2=0.4$ and (d) $\alpha_1=\alpha_2=0.25$. Isotropic radiation pattern is assumed.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Proposition 1
• Corollary 1
• Corollary 2
• Lemma 1
• Corollary 3
• Proposition 2