# Extremely large-scale Array Systems: Near-Filed Codebook Design and
Performance Analysis

Feng Zheng

(March 2023)

###### Abstract

Extremely large-scale Array (ELAA) promises to deliver ultra-high data rates
with more antenna elements. Meanwhile, the increase of antenna elements leads
to a wider realm of near-field, which challenges the traditional design of
codebooks. In this paper, we propose novel codebook design schemes which
provide better quantized correlation with limited overhead. First, we analyze
the correlation between codewords and channel vectors uniform linear array
(ULA) and uniform planar array (UPA). The correlation formula for the ULA
channel can be expressed as an elliptic function, and the correlation formula
for the UPA channel can be represented as an ellipsoid formula. Based on the
analysis, we design a uniform sampling codebook to maximize the minimum
quantized correlation and a dislocation ULA codebook to reduce the number of
quantized bits further. Besides, we give a better sampling interval for the
codebook of the UPA channel. Numerical results demonstrate the appealing
advantages of the proposed codebook over existing methods in quantization bit
number and quantization accuracy.

###### Index Terms:

ELAA, codebook, correlation fitting formula, near-field, quantization bits

##  I Introduction

Massive multiple-input multiple-output (MIMO) technology is a vital component
of fifth-generation (5G) mobile communication networks. MIMO involves the
utilization of multiple antennas to concentrate signal power within a limited
area, contributing to enhanced energy efficiency and spectral efficiency [1,
2]. However, with the explosive increase of demand on data rates in the
forthcoming sixth-generation (6G) mobile communication networks, current
massive MIMO cannot meet the requirement because congregating enough signal
power with limited antennas is difficult. To address this issue, ELAA
technology, comprising hundreds or thousands of antennas, is considered as a
crucial enabling technology for next generation communication [3]. ELAA
enables efficient multiplexing of multiple user equipment (UE) on the same
time-frequency resource, thereby improving spectral efficiency and data rates.
Additionally, the deployment of ELAA’s high beamforming gain facilitates
enhanced spatial resolution and compensates for significant path loss
experienced in the terahertz frequency bands [4].

Although more antenna elements in ELAA bring benefits in spectral efficiency
and data rates, side effects on wireless channel characteristics brought by
the increasing antenna numbers also demand attention. The electromagnetic
field is generally divided into far-field and near-field, and their boundaries
can be determined by the Rayleigh distance $2{D^{2}}/\lambda$, where $D$
represents the array size and $\lambda$ represents the wavelength [6]. Due to
the deployment of ELAA and high-frequency carriers, the Rayleigh distance is
extended to tens to hundreds of meters, so UE is more likely to locate in the
near-field [7]. For example, for a $128\times 128$ UPA operating at $100$ GHz,
the near-field region extends to 490 meters.

The high beamforming gain of the ELAA system heavily relies on the accurate
channel state information (CSI) at the transmitter [9]. Especially in the
radiating near-field, the non-negligible distance factor poses a significant
challenge to the precise alignment of the beams [10]. In the time division
duplex (TDD) system, the uplink and downlink have reciprocity, and downlink
CSI can be obtained through uplink channel estimation. However, for the
frequency division duplex (FDD) system, the uplink and downlink work on
different frequencies, which makes the channel reciprocity very weak, so it is
difficult to deduce the downlink CSI from the uplink CSI [11]. That is, CSI
can only be obtained through dedicated feedback provided by UE over signaling
channels of limited capacity [12].

Currently，there are two typical categories of CSI acquisition methods, the
explicit CSI acquisition and the implicit CSI acquisition. Explicit feedback
schemes directly report an element-wise quantized channel vector. They allow
for more flexible transmission or reception methods, which can achieve a
higher scheduling gain. However, explicit feedback requires a higher overhead
than implicit feedback, so implicit feedback can enable more accurate link
adaptation [13]. A mainstream technique in implicit feedback is the codebook-
based approach, which feeds back an index of a quantized CSI in a pre-designed
codebook to the transmitter. For codebook-based feedback, the quantization
accuracy of CSI depends on the codebook structure and the allowed number of
feedback bits [14]. The existence of massive antennas and non-negligible
distance dimension in ELAA leads to an unexpected increase in pilot overhead.
Therefore, it is crucial to design a codebook to achieve accurate quantization
of CSI with limited feedback considering the near-field channel
characteristics of ELAA.

###  I-A Related Works

Extensive researches have focused on the design of far-field codebooks. The
far-field electromagnetic wave can be considered a plane wave, so the phase
changes linearly with the antenna index. The 5G new radio (NR) standard
adopted a discrete Fourier transform (DFT) codebook for the ULA system, and
the two-dimensional DFT (2D-DFT) codebook was introduced for the UPA system
[15]. Moreover, to enable more accurate CSI acquisition, the NR standard
supported codebook oversampling and the linear combination of multiple
codewords for feedback [16]. IEEE 802.16 m standard adopted an adaptive
codebook structure, such as the skewed codebook, and a differential codebook
structure, such as a Polar-Cap codebook [17]. Besides, in the case of low
pilot overhead, the hierarchical codebook [18], angle-of-departure (AoD)
adaptive subspace codebook [19], and compressed sensing (CS) [20] methods
could also be utilized to quantify CSI accurately. Among them, the codebook
feedback schemes based on CS utilize the sparsity of the channel in the angle
domain to achieve the goal of low feedback overhead.

Only some studies have focused on codebook design for large near-field ELAA
systems. [21] designed a codebook for the near-field UPA, which uniformly
samples in real space. However, significant quantization errors exist in
quantizing the channel in real space. Besides, [22] derived a one-dimensional
(1D) near-field beamforming codebook design, which locates the codeword
quantization point based on the Lloyd-Max algorithm. The scheme in [22]
improved the near-field beamforming gain, but this scheme still needed to give
a theoretical analysis of the codebook quantization performance.

Meanwhile, near-field CSI quantization and feedback have been studied in some
research. [23] designed a new polarization-domain codebook for near-field ULA
and a CS feedback method based on the sparsity of the polar domain.
Considering both angle and distance information, a new polarization-domain
codebook for near-field ULA and a CS feedback method based on the sparsity of
the channel in the polar domain was designed. This method used the on-grid
polar-domain simultaneous orthogonal matching pursuit (P-SOMP) algorithm and
the off grid polar-domain simultaneous iterative gridless weighted (P-SIGW)
algorithm CS algorithm to feedback near-field channel information. In [24], a
hierarchical codebook was designed by projecting the near-field channel into
the angle and slope domains, considering the incomplete coverage and overlap
of spatial chirp beams, further designing a hierarchical codebook via manifold
optimization and alternative minimization.

To our knowledge, no studies have analyzed the correlation between near-field
channels. The correlation is sensitive to angle and distance in the near-field
polarization domain. Correlation performs distinctive change patterns in near-
field and far-field. Unfair sampling methods will make codewords redundant and
reduce quantization accuracy. In addition, there are few studies on the
codebook and channel feedback scheme of the UPA channel. The UPA channel has
one more angle parameter than the ULA channel, significantly increasing the
codeword overhead. Therefore, a codebook suitable for near-field channels
needs to be carefully designed.

###  I-B Contributions

To fill in this gap, in this paper, we analyze the quantization performance in
theory and propose codebook design schemes for the ULA channel and the UPA
channel in the ELAA system. Our main contributions are summarized as follows

  * $\bullet$

We provide a theoretical analysis of the relationship between codeword
quantization performance and quantization region in ULA and UPA channels.
Specifically, we derive a polynomial form of the correlation formula between
the codeword and the channel. The correlation formula for the ULA channel can
be expressed as an elliptic function, and the correlation formula for each
codeword remains constant, which indicates its stability. Besides, the
correlation formula for the UPA channel can be represented as an ellipsoid
formula, and the correlation formulas for different codewords vary, indicating
non-stationarity. The polynomial expression of the correlation formula
facilitates the design of sampling between codewords.

  * $\bullet$

We propose two codebook design methods for ULA and UPA, respectively. Based on
the correlation formula of the ULA codebook, a uniform sampling codebook
scheme with maximized minimum correlation is given. Furthermore, we propose an
improved dislocation sampling scheme that reduces the number of quantized
codewords. Considering the non-stationary correlation in the UPA channel, we
propose a sampling scheme that achieves an upper bound on the codeword
quantization performance. Analytical results show that oversampling in the
angle domain can achieve higher quantization performance. In near-field
channels, the number of quantized bits of CSI is nonlinearly proportional to
the number of BS antennas.

###  I-C Organization and Notation

The remainder of the paper is organized as follows. Section II presents the
system model. Section III analyzes the characteristic of correlation, and
describes the polynomial fitting formula in both ULA and UPA model. In Section
IV, the near-field optimal uniformly codebook and uniformly dislocation
codebook of ULA channel are proposed. Section V presents the optimal codebook
design method of UPA channel and evaluates the proposed codebook. Simulation
results are provided in Section VI, and conclusions are drawn in Section VII.

_Notations_ : Vectors are denoted by lowercase bold letters, while matrices
are denoted by uppercase bold letters; $\otimes$ denotes the Kronecker
product; $(\cdot)^{H}$ and $(\cdot)^{T}$ denotes the conjugate-transpose and
diagonal operations, respectively; $\Upsilon_{(m,n)}$ denotes the $(m,n)$-th
entry of the matrix $\mathbf{\Upsilon}$; $\mathbf{\Upsilon}_{n}$ denotes the
$n$-th column of the matrix $\mathbf{\Upsilon}$; ${\upsilon}_{n}$ denotes the
$n$-th element of the vector $\mathbf{\upsilon}$.

##  II System Model

In this section, we describe the system model of the near-field ELAA
communication system. First, we introduce the spherical wave model of ELAA
system. Next, we present a CSI quantization feedback model, and formulate the
design of the codebook as an optimization problem.

###  II-A ULA Near-field Channel Model

Consider a downlink narrow-band ELAA system, where the BS is equipped with a
ULA to serve a single antenna UE. As shown in Fig. 1, the $N$-antenna array is
placed along the $y$-axis. The antenna spacing is $d=\lambda/2$, where
$\lambda$ is the electromagnetic wavelength. The coordinate of the $n$-th
antenna is given by $\mathbf{t}_{n}=\left({0,y_{n}}\right)$, where
$y_{n}=\;\left({n-\frac{N+1}{2}}\right)d$ with $n=1,2,\ldots,N$. Meanwhile,
the UE is located at
$\mathbf{u}=\left({r{\mathit{\cos}\theta},r{\mathit{\sin}\theta}}\right)$,
where $r$ and $\theta$ represent the distance and angle between UE and array
center, respectively.

The line-of-sight (LoS) channel is considered because this paper only focuses
on the quantization feedback problem of the near-field codebook. According to
the spherical wave model [23], the distance determines the signal phase, and
the near-field channel vector ${\mathbf{h}}$ can be expressed as

${\bf{h}}=\sqrt{N}g{\bf{b}}\left({r,\theta}\right),$ (1)

where $g={{\sqrt{\eta}}}{e^{-jkr}}/{r}$ is the complex-valued channel gain
with $\eta$ and $k=2\pi/\lambda$ denoting the reference channel gain at a
distance of $1m$ and wave number, respectively.
$\mathbf{b}\left({r,\theta}\right)$ denotes the beam focusing vector, which is
given by

${\mathbf{b}}\left({r,\theta}\right)=\frac{1}{{\sqrt{N}}}\left[{e^{-jk\left({{r_{1}}-r}\right)}},{e^{-jk\left({{r_{2}}-r}\right)}},\ldots,{e^{-jk\left({{r_{N}}-r}\right)}}\right]^{T},$
(2)

where $r_{n}=\left\|{\mathbf{t}_{\mathbf{n}}-\mathbf{u}}\right\|$ represents
the distance between the $n$-th antenna at the BS and the UE. Furthermore,
according to the second order Taylor series expansion
$\sqrt{1+x}=1+{x}/{2}-{x^{2}}/{8}+\mathcal{O}\left(x^{3}\right)$ , $r_{n}$ can
be approximated as

$\displaystyle{r_{n}}$
$\displaystyle=\sqrt{{{(r\sin\theta-{y_{n}})}^{2}}+{{(r\cos\theta)}^{2}}}$ (3)
$\displaystyle\approx
r-\sin\theta{y_{n}}+\;\frac{{{{\cos}^{2}}\theta}}{{2r}}y_{n}^{2}.$

Therefore, the $n$-th element of near-field beam focusing vector
${\mathbf{b}}$ can be simplified as

${b_{n}}=\frac{1}{\sqrt{N}}{e^{-jk\left(-\sin\theta{y_{n}}+\;\frac{{{{\cos}^{2}}\theta}}{{2r}}y_{n}^{2}\right)}}.$
(4)

###### Remark 1

When the $r$ is sufficiently large, the ${\cos^{2}\theta}/{2r}$ term can be
omitted, and $\mathbf{b}\left({r,\theta}\right)$ is simplified as

${\mathbf{a}}\left(\theta\right)=\frac{1}{{\sqrt{N}}}{\left[{1,{e^{j\pi\sin\theta}},\ldots,{e^{j\pi\left({N-1}\right)\sin\theta}}}\right]^{T}},$
(5)

which is equivalent to the conventional far-field beam steering vector for the
ULA. And the DFT codebook is adopted to quantify the far-field channel vector.
Therefore, to be more precise, the concept of “near-field” in this paper does
not exclude far-field as well.

Figure 1: Near-field channel model for ULA communication system.

###  II-B UPA Near-field Channel Model

As shown in Fig. 2, the BS employs a UPA, which is located on the $\rm{xOy}$
plane and the center of the array is located at the coordinate origin.
$N\times N$ uniformly spaced antenna elements are placed in both horizontal
and vertical directions, with a spacing of $d=\lambda/2$. The Cartesian
coordinate of the $(m,n)$-th antenna element of the UPA can be expressed as
$\mathbf{t}_{(m,n)}=\left({x_{m},y_{n},0}\right)$ with
$x_{m}=\left({m-\frac{N+1}{2}}\right)d$,
$y_{n}=\left({n-\frac{N+1}{2}}\right)d$, $m=1,...,N$, $n=1,\ldots,N$.
Meanwhile, we assume the coordination of UE is
${\bf{u}}=\left({r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta}\right)$,
where $r$, $\theta$ and $\phi$ represent the distance, elevation angle and
azimuth angle of UE relative to the UPA center, respectively. Therefore, the
beam focusing vector for UPA can be obtained based on the spherical wave
propagation model as

${\mathbf{b}}\left({r,\theta,\phi}\right)=\frac{1}{{N}}{\left[{e^{-jk\left({{r_{\left({1,1}\right)}}-r}\right)}},\ldots,{e^{-jk\left({{r_{\left({N,N}\right)}}-r}\right)}}\right]^{T}},$
(6)

where $r_{({m,n})}=\left\|{\mathbf{t}_{({m,n})}-\mathbf{u}}\right\|$
represents the distance between the $(m,n)$-th antenna at the BS and the UE,
which can be approximated as

$\displaystyle\begin{aligned} r_{(m,n)}&\approx
r-\sin\theta\cos\phi{x_{m}}-\sin\theta\sin\phi{y_{n}}\\\
&\quad+\frac{{1-{{\sin}^{2}}\theta{{\cos}^{2}}\phi}}{{2r}}x_{m}^{2}+\frac{{1-{{\sin}^{2}}\theta{{\sin}^{2}}\phi}}{{2r}}y_{n}^{2}\\\
&\quad-\frac{{{{\sin}^{2}}\theta\cos\phi\sin\phi}}{r}{x_{m}}{y_{n}}.\end{aligned}$
(7)

As a result, the $(m,n)$-th element of the LoS channel can be represented as

${b_{(m,n)}}=\frac{{1}}{N}{e^{-jk{(r_{(m,n)}-r)}}}.$ (8)

###### Remark 2

When the $r$ is sufficiently large, the last 3 terms in (7) can be omitted,
and $\mathbf{b}\left({r,\theta,\phi}\right)$ is simplified as

$\begin{split}\mathbf{a}\left(\theta,\phi\right)=&\frac{1}{N}\left[1,\ldots,e^{j\pi(m\sin\theta\cos\phi+n\sin\theta\sin\phi)},\dots,\right.\\\
&\left.\quad
e^{j\pi((N-1)\sin\theta\cos\phi+(N-1)\sin\theta\sin\phi)}\right]^{T},\end{split}$
(9)

which is equivalent to the conventional far-field beam steering vector for the
UPA, and the 2D-DFT codebook is adopted for the CSI feedback. Since the phase
of (9) can be decoupled into two parts in terms of $x$ and $y$, the 2D-DFT
codebook can be expressed in the form of the Kronecker product of the DFT
vectors, that is

${\bf{a}}={{\bf{a}}_{x}}\otimes{{\bf{a}}_{y}},$ (10)

where
${{\bf{a}}_{x}}=\frac{1}{{\sqrt{N}}}\left[{1,{e^{j\pi\sin\theta\cos\phi}},\ldots,{e^{j\pi(N-1)\sin\theta\cos\phi}}}\right]^{T}$
and
${{\bf{a}}_{y}}=\frac{1}{{\sqrt{N}}}\left[{1,{e^{j\pi\sin\theta\sin\phi}},\ldots,{e^{j\pi(N-1)\sin\theta\sin\phi}}}\right]^{T}$.
However, the cross-term
${{{{\sin}^{2}}\theta\cos\phi\sin\phi}}{x_{m}}{y_{n}}/r$ in (7) prevents it
from being decoupled as $\mathbf{a}\left({\theta,\phi}\right)$. Therefore, the
near-field codebook for the UPA cannot be directly constructed based on the
ULA codebook, which will be explained in detail later.

Figure 2:  Near-field channel model for UPA communication system.

###  II-C CSI Quantization Feedback Model

For FDD communication systems, the BS can obtain the CSI through the UE’s
feedback. Specifically, the pilot signal
$\mathbf{X}=\text{diag}\big{(}{x_{1},\ldots,x_{\widetilde{N}}}\big{)}$ is
transmitted at first, where $x_{n}$ denotes the pilot symbol for the $n$-th
antenna, and $\widetilde{N}$ represents the size of either ULA or UPA. At the
UE side, the received signal is given by

${\bf{y}}={\bf{Xh}}+{\bf{n}},$ (11)

where n is the additive Gaussian white noise (AWGN) with variance
$\sigma^{2}$. Based on this, the estimation of channel vector, denoted as
$\hat{\mathbf{h}}$, can be obtained by methods such as least squares (LS)
[25]. Since this paper mainly focuses on the codebook design for the near-
field communication, the perfect CSI estimation is assumed, i.e.,
$\hat{\mathbf{h}}=\mathbf{h}$.

To inform the BS of CSI with limited feedback, the channel vector is quantized
based on a predefined codebook
$\mathbf{W}=\left[\mathbf{W}_{1},\ldots,\mathbf{W}_{S}\right]$, which contains
S codewords and satisfies $\left\|\mathbf{W}_{s}\right\|=1$. The UE selects
the ideal codeword from the codebook and feeds back its index
$s^{\mathrm{*}}=\underset{s}{arg\max}\left|{\mathbf{h}^{H}\mathbf{W}_{s}}\right|^{2}$
to the BS. Finally, the BS can determine the transmission scheme based on the
CSI feedback from the UE. For example, the codeword
$\mathbf{W}_{s^{\mathrm{*}}}$ can be utilized as the beamforming weight.

In the above quantization feedback model, the codebook design affects the
channel quantization accuracy, which in turn affects the performance of the
communication system. Obviously, all the channel vectors in the region of
interest have different correlations with the codewords. This paper considers
the max-min correlation criterion, and assuming
$\mathcal{W}=\left\\{W_{1},\dots,W_{s}\right\\}$, the near-filed codebook
design problem can be formulated as

$\displaystyle\begin{aligned} \mathop{{\rm{max}}}\limits_{\mathbf{W}}&{\mkern
1.0mu}\mathop{{\rm{min}}}\limits_{{\bf{h}}\in{\cal H}}{\mkern
1.0mu}\mathop{{\rm{max}}}\limits_{s}|{{\bf{h}}^{\rm{H}}}{{\bf{W}}_{s}}|\;\\\
{\rm{s}}{\rm{.t}}{\rm{.}}&\left|{\mathcal{W}}\right|=S.\end{aligned}$ (12)

or

$\displaystyle\begin{array}[]{l}\mathop{\min}\limits_{\mathcal{W}}\;{\mkern
1.0mu}\left|{\mathcal{W}}\right|\\\
{\rm{s}}{\rm{.t}}{\rm{.}}\mathop{\;{\rm{min}}}\limits_{{\bf{h}}\in{\cal
H}}{\mkern 1.0mu}\mathop{{\rm{max}}}\limits_{s}{\mkern
1.0mu}|{{\bf{h}}^{\rm{H}}}{{\bf{W}}_{s}}|>c.\end{array}$ (15)

${\cal H}$ represents the set of LoS channel vectors within the region of
interest, and $c$ represents the correlation between codewords and channel
vectors.

In the above quantization feedback model, the codebook design affects the
channel quantization accuracy, which in turn affects the performance of the
communication system. Obviously, all the channel vectors in the region of
interest have different correlations with the codewords. This paper considers
the max-min correlation criterion, and assuming
$\mathcal{W}=\left\\{W_{1},\dots,W_{s}\right\\}$, the near-filed codebook
design problem can be formulated as

##  III Codeword Quantization Performance Analysis

This section first investigates the correlation between the codewords and the
near-field channel vectors. The transform domain perspective for analyzing
correlation function is proposed, demonstrating many desired mathematical
properties. Second, this section gives the fitting formula of the quantization
performance of the codeword to the channel vector, which inspires the codebook
design.

###  III-A Correlation Function for ULA systems

When codewords are selected from the beam focusing vectors, i.e.,
${\mathbf{W}_{s}}={\mathbf{b}}({r_{s},\theta_{s}})$, they can be viewed as a
LoS channel vectors at specific locations. For the ULA systems, the
correlation between the codeword $\mathbf{b}({{r_{s}},{\theta_{s}}})$ and the
beam focusing vector ${\mathbf{b}}({{r_{q}},{\theta_{q}}})$ can be calculated
as

$\displaystyle\tau\big{(}{{r_{s}},{\theta_{s}};{r_{q}},{\theta_{q}}}\big{)}=$
$\displaystyle\big{|}{{{\mathbf{b}}^{H}}\big{(}{{r_{s}},{\theta_{s}}}){\mathbf{b}}({{r_{q}},{\theta_{q}}}\big{)}}\big{|}$
(16) $\displaystyle=$
$\displaystyle\frac{1}{N}\Bigg{|}\sum\limits_{n=1}^{N}{\exp}\Bigg{(}-j\frac{2\pi}{\lambda}\bigg{(}\big{(}{\sin{\theta_{q}}-\sin{\theta_{s}}}\big{)}{y_{n}}$
$\displaystyle+\Big{(}{\frac{{{{\cos}^{2}}{\theta_{s}}}}{{2{r_{s}}}}-\frac{{{{\cos}^{2}}{\theta_{q}}}}{{2{r_{q}}}}}\Big{)}y_{n}^{2}\bigg{)}\Bigg{)}\Bigg{|}.$

Let $\alpha_{i}={\lambda\cos^{2}\theta_{i}}/{4r_{i}}$,
$\beta_{i}=\sin\theta_{i}$ with $i=s,q$, (16) can be simplified as

$\begin{split}f(\alpha_{s},\beta_{s};\alpha_{q},\beta_{q})\\\
=\frac{1}{N}\Bigg{|}\sum_{n=1}^{N}&\exp\Bigg{(}-j\pi\bigg{(}\big{(}\alpha_{s}-\alpha_{q}\big{)}n^{2}+\Big{(}\big{(}\beta_{q}-\beta_{s}\big{)}\\\
-\big{(}\alpha_{s}-&\alpha_{q}\big{)}\big{(}N+1\big{)}\Big{)}n\bigg{)}\Bigg{)}\Bigg{|}.\\\
\end{split}$ (17)

Further, let $\delta_{\alpha}$ and $\delta_{\beta}$ respectively represent as
the location difference between $\mathbf{b}(r_{q},\theta_{q})$ and
$\mathbf{W}_{s}$, which can be expressed as

$\begin{split}&\delta_{\alpha}=\alpha_{q}-\alpha_{s},\\\
&\delta_{\beta}=\beta_{q}-\beta_{s}.\end{split}$ (18)

Then, (17) can be simplified as

$\begin{split}f\left(\delta_{\alpha},\delta_{\beta}\right)=\frac{1}{N}\Bigg{|}\sum_{n=1}^{N}\exp\Bigg{(}-j\pi\bigg{(}-\delta_{\alpha}n^{2}\\\
+\Big{(}\delta_{\beta}+\delta_{\alpha}\big{(}N+1\big{)}\Big{)}\bigg{)}\Bigg{)}\Bigg{|}.\end{split}$
(19)

Without loss of generality, the correlation between the codeword and the
channel vector always satisfies
$f\left(\delta_{\alpha},\delta_{\beta}\right)\leq 1$.
$f\left(\delta_{\alpha},\delta_{\beta}\right)=1$ if and only if
$\delta_{\alpha}=0$ and $\delta_{\beta}=0$. Consequently, there is always a
certain quantization error when a codeword quantizes a channel other than
itself.

To consider the influence of wavelength and number of antennas on the
correlation function, we set
${\tilde{\delta}_{\alpha}}={\delta_{\alpha}}{N^{2}}$ and
${\tilde{\delta}_{\beta}}={\delta_{\beta}}{N}$. For the antenna with a
sufficiently large of $N$, the above formula can be approximated to

$\tilde{f}\left({\tilde{\delta}_{\alpha}},{{\tilde{\delta}_{\beta}}}\right)\approx\left|{\int_{-1/2}^{1/2}{\exp}\left({-j\pi\left({{{\tilde{\delta}_{\beta}}}t-{{\tilde{\delta}_{\alpha}}}{t^{2}}}\right)}\right)dt}\right|.$
(20)

(20) converts the coordinate from $(\delta_{\alpha},\delta_{\beta})$ to
$({\tilde{\delta}_{\alpha}},{{\tilde{\delta}_{\beta}}})$. The change in the
number of antennas $N$ does not affect the expression of
$f({\tilde{\delta}_{\alpha}},{{\tilde{\delta}_{\beta}}})$. Thus, (20) shows
that the channel correlation function is suitable for different antenna
numbers $N$ and wavelengths $\lambda$, in other words, (19) can be used to
describe the correlation between codeword and channel in both the far-field
and near-field. Under different correlations $c$, the difference between the
locations satisfying the condition is distributed in contour lines on the
$\alpha$-$\beta$ domain, as shown in Fig. 3. The correlation between the
codeword and the channel vector is only related to the difference in position.
Under different correlations $c$, the difference between the locations
satisfying the condition is distributed in contour lines.

Similar to the far-field DFT beam pattern in the angle domain, the correlation
function (16) can also be understood as the normalized beam pattern of the
near-field beam in the $\alpha$-$\beta$ domain. In Fig. 3, the yellow ellipse
in the center reflects the beamwidth under a specific correlation requirement.
The beamwidth on the $\alpha$ domain is inversely proportional to $N^{2}$, and
the beamwidth on the $\beta$ domain is inversely proportional to $N$.

Unfortunately, the exponential term in (19) is relatively complicated. Most
existing literature deals with the exponential term based on the Fresnel
integral [23]. However, it is still difficult to directly obtain the numerical
characteristic for the correlation function approximated by the Fresnel
integral. Inspired by this, we further explore the correlation properties of
codewords. We can deduce the following properties based on the correlation
(19).

######  Property 1 (Stationarity)

The correlation between the codeword and channel vector is only related to
$\delta_{\alpha}$ and $\delta_{\beta}$. For different codeword
$\mathbf{W}_{s}$ and $\mathbf{W}_{s^{{}^{\prime}}}$, the correlation’s
distribution of codeword with the channel within its quantization region is
always the same. The stationarity in the ULA channel can be formulated as

$\begin{split}f\left(\alpha_{s},\beta_{s};\right.&\left.\alpha_{s}+\delta_{\alpha},\beta_{s}+\delta_{\beta}\right)\\\
&=f\left(\alpha_{s^{{}^{\prime}}},\beta_{s^{{}^{\prime}}};\alpha_{s^{{}^{\prime}}}+\delta_{\alpha},\beta_{s^{{}^{\prime}}}+\delta_{\beta}\right).\end{split}$
(21)

######  Property 2 (Symmetry)

Similar to the symmetry of the DFT beam in the angle domain, the correlation
distribution of the near-field channel also satisfies symmetry. In the
quantization area of the codeword, the distribution of the correlation between
the codeword and the channel is symmetrical about the codeword in the $\alpha$
and $\beta$ domain, which can be expressed as

$\begin{split}f\left(\delta_{\alpha},\delta_{\beta}\right)=f\left(\delta_{\alpha},-\delta_{\beta}\right)&=f\left(-\delta_{\alpha},\delta_{\beta}\right)=f\left(-\delta_{\alpha},-\delta_{\beta}\right).\end{split}$
(22)

###### Proof 1

(19) can be rewritten as

$f\left(\delta_{\alpha},\delta_{\beta}\right)=\frac{1}{N}\left|\sum_{n=1}^{N}\exp\left(-j\frac{2\pi}{\lambda}\left(\delta_{\beta}y_{n}-\frac{2}{\lambda}\delta_{\alpha}y_{n}^{2}\right)\right)\right|.$
(23)

Set $\delta_{\beta}^{\prime}=-\delta_{\beta}$, the above formula can be
calculated as

$f\left(\delta_{\alpha},\delta_{\beta}^{\prime}\right)=\frac{1}{N}\left|\sum_{n=1}^{N}\exp\left(-j\frac{2\pi}{\lambda}\left(-\delta_{\beta}y_{n}-\frac{2}{\lambda}\delta_{\alpha}y_{n}^{2}\right)\right)\right|.$
(24)

$y_{n}$ is symmetric about $y_{n}=0$ in $\left[-(N-1)d/{2},(N-1)d/{2}\right]$,
hence there always exists $y_{N-n+1}=-y_{n}$. Thus, formula (24) can be
written as

$\begin{split}&f\left(\delta_{\alpha},-\delta_{\beta}\right)\\\
&=\frac{1}{N}\left|\sum_{n=1}^{N}\exp\left(-j\frac{2\pi}{\lambda}\left(\delta_{\beta}y_{N-n+1}-\frac{2}{\lambda}\delta_{\alpha}y_{N-n+1}^{2}\right)\right)\right|,\end{split}$
(25)

which indicates that
$f\left(\delta_{\alpha},\delta_{\beta}\right)=f\left(\delta_{\alpha},-\delta_{\beta}\right)$.
It is evident that
$f(\delta_{\alpha},\delta_{\beta})=f(-\delta_{\alpha},-\delta_{\beta})$ by
central symmetry. Therefore, we can easily deduce that
$f(\delta_{\alpha},\delta_{\beta})=f(-\delta_{\alpha},\delta_{\beta})$.
$\blacksquare$

Figure 3: Contour distribution between codeword quantization correlation and
position difference with $f=3\mathrm{GHz}$ and $N=512$.

According to these two properties, we can obtain the correlation
characteristics of any codewords by analyzing the correlation of a specific
codeword. These features provide convenience for designing the sampling
interval between codewords to ensure an ideal design. Next, we will use a
polynomial function to approximate the correlation function in Proposition 1.

###### Proposition 1

When $\delta_{\alpha}\neq 0$ and $\delta_{\beta}\neq 0$, the polynomial
fitting formula of $f(\delta_{\alpha},\delta_{\beta})$ can be expressed as

$f\left({{\delta_{\alpha}},{\delta_{\beta}}}\right)\approx{p_{\alpha}}{\delta_{\alpha}}^{2}{N^{4}}+{p_{\beta}}{\delta_{\beta}}^{2}{N^{2}}+1,$
(26)

where

$\begin{split}p_{\alpha}=-0.025983670363830,\\\
p_{\beta}=-0.391749735984250.\end{split}$ (27)

To sum up, if the correlation between the codeword $\mathbf{W}_{s}$ and the
beam focusing vector $\mathbf{b}(\alpha_{q},\beta_{q})$ is
$f(\delta_{\alpha},\delta_{\beta})=c\in(0,1)$, the distribution satisfied by
$\delta_{\alpha}$ and $\delta_{\beta}$ can be equivalent to

$p_{\alpha}\delta_{\alpha}^{2}N^{4}+p_{\beta}\delta_{\beta}^{2}N^{2}=c-1.$
(28)

This formula can be further simplified into the form of the following formula

$\frac{p_{\alpha}\delta_{\alpha}^{2}N^{4}}{c-1}+\frac{p_{\beta}\delta_{\beta}^{2}N^{2}}{c-1}=1.$
(29)

Evidently, the correlation fitting formula in Proposition 1 is an elliptic
function. The ellipse formula always centers around $(\alpha_{s},\beta_{s})$.
Moreover, the ellipse is the quantization boundary of $\mathbf{W}_{s}$ when
the quantization accuracy of the codeword satisfies $c$. The correlation $c$
affects the major and minor axes of the ellipse formula. The larger $c$ is,
the shorter the axial length of the ellipsoid is, and the smaller the codeword
quantization area is. Conversely, the minor $c$ is, the axis length of the
ellipse is smaller, and the lower the quantization accuracy of the codeword
is. Further, the axis length of the ellipse is also decided by $N$. The axis
length of the ellipse in the $\alpha$ and $\beta$ domains are inversely
proportional to $N^{2}$ and $N$, respectively. The formula in Proposition 1 is
more concise and easier to calculate than the approximated Fresnel integral
function and provides strong theoretical support for the codebook design
scheme in this paper. In the next section, we will design ULA codebooks based
on the above Proposition 1.

For $\mathbf{W}_{0}=\mathbf{b}(0,0)$, $f(\delta_{\alpha},\delta_{\beta})$ can
be written as $f(\alpha_{q},\beta_{q})$. At this time, (29) is the
quantization boundary of $\mathbf{W}_{0}$ and gives the channel vector
$\mathbf{b}(\alpha_{q},\beta_{q})$ which meet the conditions of
$\left|\mathbf{W}_{0}^{H}\mathbf{b}(\alpha_{q},\beta_{q})\right|=c$. For the
codeword $\mathbf{W}_{0}$ with the minimum correlation as $c$, the possible
channel vectors $\mathbf{b}(\alpha_{q},\beta_{q})$ always distribute in the
ellipse interior, which can be formulated as

$\Omega=\left\\{\mathbf{b}\left(\alpha_{q},\beta_{q}\right)\Big{|}\frac{{p_{\alpha}N^{4}}\alpha_{q}^{2}}{{c-1}}+\frac{{p_{\beta}N^{2}}\beta_{q}^{2}}{{c-1}}\leq
1\right\\}.$ (30)

Considering the stationarity characteristics, the correlation between any
codeword and its quantization channel vector can be described using the
contour lines shown in Fig. 3.

###  III-B Correlation Function for UPA systems

In the UPA system, the correlation between codeword
$\mathbf{W}_{s}=\mathbf{b}(r_{s},\theta_{s},\phi_{s})$ and beam focusing
vector $\mathbf{b}(r_{q},\theta_{q},\phi_{q})$ can be calculated as

$\tau\left({r_{s}},{\theta_{s}},{\phi_{s}};{r_{q}},{\theta_{q}},{\phi_{q}}\right)=\left|{{{\bf{b}}^{\rm{H}}}\left({{r_{s}},{\theta_{s}},{\phi_{s}}}\right){\bf{b}}\left({{r_{q}},{\theta_{q}},{\phi_{q}}}\right)}\right|.$
(31)

We replace the difference of location between the codeword and channel vector
as follows

$\begin{split}\delta_{\psi_{s}}&=\sin\theta_{s}\cos\phi_{s}-\sin\theta_{q}\cos\phi_{q},\\\
\delta_{\varphi_{s}}&=\sin\theta_{s}\sin\phi_{s}-\sin\theta_{q}\sin\phi_{q},\\\
\delta_{\rho_{s}}&=\frac{1}{r_{s}}-\frac{1}{r_{q}}.\end{split}$ (32)

Then, (31) can be written as

$\begin{split}&f\left(\delta_{\psi_{s}},\delta_{\varphi_{s}},\delta_{\rho_{s}}\right)\\\
&=\frac{1}{N^{2}}\Bigg{|}\sum_{n=1}^{N}\sum_{m=1}^{N}\exp\Bigg{(}-j\frac{2\pi}{\lambda}\bigg{(}x_{m}\delta_{\psi_{s}}+y_{n}\delta_{\varphi_{s}}\\\
&\quad-\frac{\Big{(}1-\big{(}\delta_{\psi_{s}}\big{)}^{2}\Big{)}\delta_{\rho_{s}}}{2}x_{m}^{2}-\frac{\Big{(}1-\big{(}\delta_{\varphi_{s}}\big{)}^{2}\Big{)}\delta_{\rho_{s}}}{2}y_{n}^{2}\\\
&\quad+\big{(}1-\delta_{\psi_{s}}\delta_{\varphi_{s}}\big{)}x_{m}y_{n}\bigg{)}\Bigg{)}\Bigg{|}.\end{split}$
(33)

Figure 4: Correlation distribution plot at $\mathbf{W}_{0}$.

The cross-term $x_{m}y_{n}$ contained in the UPA channel vector makes it
impossible to use the Kronecker product to decouple the channel vector. If the
cross-term $x_{m}y_{n}$ is ignored, it will cause a significant performance
error due to missing more CSI. Before giving the method to solve the above
problems, the characteristics of UPA channel correlation shown in the property
3 and property 4 are firstly given.

######  Property 3 (non-stationarity)

The non-stationarity manifests in that the quantized areas of different
codewords are not fixed under the minimum correlation $c$. Therefore,
codebooks in UPA channels do not exhibit stationarity. For two codewords
$\mathbf{W}_{s}$ and $\mathbf{W}_{s^{{}^{\prime}}}$, the non-stationarity
manifests can be formulated as

$\begin{split}f\Big{(}\psi_{s},&\varphi_{s},\rho_{s};\psi_{s}\\!+\\!\delta_{\psi_{s}},\varphi_{s}\\!+\\!\delta_{\varphi_{s}},\rho_{s}\\!+\\!\delta_{\rho_{s}}\Big{)}\\\
&\neq\\!f\left(\psi_{s^{{}^{\prime}}},\varphi_{s^{{}^{\prime}}},\rho_{s^{{}^{\prime}}};\psi_{s^{{}^{\prime}}}\\!+\\!\delta_{\psi_{s}},\varphi_{s^{{}^{\prime}}}\\!+\\!\delta_{\varphi_{s}},\rho_{s^{{}^{\prime}}}\\!+\\!\delta_{\rho_{s}}\right).\end{split}$
(34)

######  Property 4 (symmetry)

Symmetry similar to ULA, it is easy to prove that

$\begin{split}f\left(\delta_{\psi_{s}},\delta_{\varphi_{s}},\delta_{\rho_{s}}\right)&=f\left(-\delta_{\psi_{s}},\delta_{\varphi_{s}},\delta_{\rho_{s}}\right)=f\left(\delta_{\psi_{s}},-\delta_{\varphi_{s}},\delta_{\rho_{s}}\right)\\\
&=f\left(-\delta_{\psi_{s}},-\delta_{\varphi_{s}},\delta_{\rho_{s}}\right).\end{split}$
(35)

Unlike the ULA model, the non-stationarity of the UPA model brings challenges
to the codeword design of UPA. In proposition 2, we give the UPA channel
correlation fitting formula different from the ULA fitting formula.

###### Proposition 2

When $\delta_{\psi_{s}}\neq 0$, $\delta_{\varphi_{s}}\neq 0$ and
$\delta_{\rho_{s}}\neq 0$, the correlation between channels in the UPA channel
model can be better fitted by a polynomial function, and the fitting formula
can be written as

$\begin{split}&f\left(\delta_{\psi_{s}},\delta_{\varphi_{s}},\delta_{\rho_{s}}\right)\\\
&=p_{\psi}\delta_{\psi_{s}}^{2}N^{2}+p_{\varphi}\delta_{\varphi_{s}}^{2}N^{2}+p_{\rho}\delta_{\rho_{s}}^{2}N^{4}+1.\end{split}$
(36)

If $f(\delta_{\psi_{s}},\delta_{\varphi_{s}},\delta_{\rho_{s}})=c$ and
$c\in(0,1)$, the above formula can be converted into the following ellipsoid
formula

$\frac{{p_{\psi}N^{2}}\delta_{\psi_{s}}^{2}}{{c-1}}+\frac{{p_{\varphi}N^{2}}\delta_{\varphi_{s}}^{2}}{{c-1}}+\frac{{p_{\rho}N^{4}}\delta_{\rho_{s}}^{2}}{{c-1}}=1.$
(37)

The ellipsoid fitting formula in Proposition 2 gives the quantization boundary
of the codeword $\mathbf{W}_{s}$ with the minimized correlation is $c$. At a
fixed position, the smaller $c$ is, the smaller the axial length of the
ellipsoid will be. It should be noted that due to the non-stationary
characteristics of the UPA channel, the values of the coefficients
${p_{\psi}},{p_{\varphi}}$ and ${p_{\rho}}$ in (36) are related to the
specific position of the codeword sampling point and the minimum correlation
$c$. Thus, the ellipsoid fitting formula has different axial lengths for
different codewords. The correlation fitting formula in Proposition 2 provides
strong support for the codebook design of near-field UPA.

Below we give an example to verify the accuracy of Proposition 2. We plot the
actual beam focusing vector whose correlation with the codeword
$\mathbf{W}_{0}=\mathbf{b}(0,0,0)$ satisfies
$f(0,0,0;\psi_{q},\varphi_{q},\rho_{q})=0.95$. We calculate the appropriate
fitting coefficients from the simulation and plot the corresponding ellipsoid
images in Fig. 4, which shows that the actual beam focusing vector satisfying
the condition points to the fitted ellipsoid curve. For the codeword
$\mathbf{W}_{0}$ with the minimum correlation as $c$, the quantized channel
vectors always is distributed in the ellipsoid interior, which can be
formulated as

$\begin{split}\Omega=\left\\{\mathbf{b}\left(\psi_{q},\varphi_{q},\rho_{q}\right)\Big{|}\frac{p_{\psi}N^{2}\psi_{q}^{2}}{c-1}\\!+\\!\frac{p_{\varphi}N^{2}\varphi_{q}^{2}}{c-1}\\!+\\!\frac{p_{\rho}N^{4}\rho_{q}^{2}}{c-1}\\!\leq\\!1\right\\}.\end{split}$
(38)

This example shows that the fitting formula of the UPA channel can more
accurately describe the channel correlation of UPA.

##  IV  Near-field ULA Codebook Design

We have analyzed the effect of the angle and distance between the UE and the
antenna on the quantization accuracy of codewords. In this section, we
describe two quantized codebook design schemes of the ULA channel to ensure
the optimal beam gain according to the ULA channel correlation distribution
characteristics presented in Section III-A. First, an optimal uniform codebook
scheme is proposed, which achieves the maximum quantization range under a
minimum correlation. In addition, we redesign a codebook scheme with constant
offset in the transform domains to reduce the quantization overhead of the
codebook further.

###  IV-A Uniform Quantization Codebook Design Scheme

The most commonly used way to obtain the quantified location of the codewords
is to perform uniform sampling on the $\alpha$-$\beta$ domain, as shown in
Fig. 5. $\alpha$ domain and $\beta$ domain are sampled with $S_{\alpha}$ and
$S_{\beta}$ points, respectively. The number of sampling points is
$S=S_{\alpha}\mathrm{*}S_{\beta}$, and the index of the codeword is $s$. Let
$\Xi^{L}$ denote the collection of codeword sampling points, which can be
represented as

$\displaystyle\Xi^{L}=\Big{\\{}\big{(}\alpha_{s},\beta_{s}\big{)}\Big{|}\alpha_{s}$
$\displaystyle=\alpha_{min}\\!+\\!\frac{\Delta\alpha}{2},\alpha_{min}\\!+\\!\frac{3\Delta\alpha}{2},\dots,\alpha_{max};$
(39) $\displaystyle\beta_{s}$
$\displaystyle=\beta_{min}\\!+\\!\frac{\Delta\beta}{2},\beta_{min}\\!+\\!\frac{3\Delta\beta}{2},\dots,\beta_{max}\Big{\\}}.$

where $\left[{\alpha_{min},\alpha_{max}}\right]$ and
$\left[{\beta_{min},\beta_{max}}\right]$ are the total quantization intervals
of the codeword on the $\alpha$ domain and the $\beta$ domain respectively.
$\Delta\alpha$ and $\Delta\beta$ represent the sampling steps on the $\alpha$
and $\beta$ domain respectively.

Figure 5: Uniform codewords on the $\alpha$-$\beta$ domain.

Next, we present an in-depth analysis of the distribution performance of the
codewords in Fig. 6 to explore the optimal codebook design scheme. The blue
triangle is the intersection of the quantization intervals of adjacent
codewords, located on the same correlation contour line for each codeword. The
correlation contour line where the blue triangle is situated is the outermost
correlation contour line of the four codewords, representing the quantization
boundary of the codeword. Extending the distribution characteristics to the
entire $\alpha$-$\beta$ domain, as shown in Fig. 6, the spacing between
adjacent codewords is always symmetrical about the minimum correlation point
of the codeword.

Furthermore, we reveal the relationship between codeword quantization regions
and codeword quantization accuracy for individual codewords in the
$\alpha$-$\beta$ domain with the constant minimum quantization error. As shown
in Fig. 7, the quantization area of the codeword is rectangular with uniform
sampling points. The vertices of the rectangular region are located on the
minimum correlation contour line. Multiple layouts of rectangular vertices on
the minimum correlation contour line constitute quantization schemes with
different areas. In order to improve the accuracy of codeword quantization
CSI, the minimum correlation of each codeword quantization is always expected
to be as large as possible. Therefore, the target can be mapped as the area of
the rectangle enclosed by the four quantization boundaries is the largest on a
specific ellipse contour line.

According to Cauchy’s inequality, for any point on the ellipse (29), when its
coordinates and axis length satisfy the relationship
$\delta_{\alpha}\sqrt{(c-1)/{(p_{\beta}N^{2})}}=\delta_{\beta}\sqrt{{(c-1)}/{(p_{\alpha}N^{4})}}$,
the area of the rectangle surrounded by the points is the largest. Therefore,
when the channel correlation is $c$, the sampling interval of the achievable
maximum quantization area on the $\alpha$-$\beta$ domain can be calculated as

$\Delta\alpha=\frac{1}{N^{2}}\sqrt{\frac{2\left({c-1}\right)}{p_{\alpha}}},\quad\Delta\beta=\frac{1}{N}\sqrt{\frac{2\left({c-1}\right)}{p_{\beta}}}.$
(40)

Figure 6: Correlation distribution of four adjacent uniform sampling points.
Figure 7: Different codebook schemes under minimum quantization correlation
$c$.

Consider that the user distribution is within a range of
$r\in\big{[}\sqrt{{0.62D^{3}}/{\lambda}},\infty\big{)}$ distance and angle of
$\theta\in\left[-{\pi}/{2},{\pi}/{2}\right]$. The maximum quantization range
on $\alpha$ and $\beta$ domain can be respectively calculated as

$Q_{\alpha}=\sqrt{\frac{\lambda}{2.48D^{3}}}\approx\frac{1}{N\sqrt{N}},Q_{\beta}=2.$
(41)

Then, the number of codewords in the $\alpha$ domain and $\beta$ domain are
given by

$S_{\alpha}=\frac{Q_{\alpha}}{\Delta{\alpha}}=\sqrt{\frac{Np_{\alpha}}{2\left({c-1}\right)}},\quad
S_{\beta}=\frac{Q_{\beta}}{\Delta{\beta}}=N\sqrt{\frac{2p_{\beta}}{\left({c-1}\right)}}.$
(42)

Thus, the total number of codewords to achieve the minimum number of feedback
bits can be calculated as

$S_{ULA}=S_{\alpha}S_{\beta}=\frac{N\sqrt{Np_{\alpha}p_{\beta}}}{\left({1-c}\right)}.$
(43)

At this time, the maximum quantization area of each codeword is

$R_{max}=\frac{2(1-c)}{N^{3}}\sqrt{\frac{1}{p_{\alpha}p_{\beta}}}.$ (44)

Therefore, for all sampling points of the ULA codebook, the coordinates of the
$s_{\alpha}$-th codeword in the $\alpha$ domain can be reformulated as

$\alpha\left(s_{\alpha}\right)=\left(s_{\alpha}-\frac{1}{2}\right)\Delta{\alpha},\text{~{}~{}}s_{\alpha}=1\ldots
S_{\alpha}.$ (45)

And the $s_{\beta}$-th codeword in the $\beta$ domain can be reformulated as

$\beta\left(s_{\beta}\right)=-1+\left({s_{\beta}-\frac{1}{2}}\right)\Delta{\beta},\quad
s_{\beta}=1\ldots,S_{\beta}.$ (46)

(43) shows that the number of quantized bits of the codeword is only related
to the channel correlation $c$ and the number $N$ of antennas but has nothing
to do with the frequency. When the channel correlation $c$ is constant, the
number of codewords in the $\alpha$ domain is proportional to $\sqrt{N}$, and
the number of codewords in the $\beta$ domain is proportional to N. Moreover,
if the number of antennas $N$ remains unchanged, the channel correlation $c$
increase can lead to an increase in the number of codebook quantization
vectors.

It should be noted that, in the design of this scheme, for a fixed codeword
quantization correlation, the number of codewords in the $\beta$ domain is
always far greater than the number of codewords in the $\alpha$ domain.
Therefore, under the same feedback bits, dense sampling in the $\beta$ domain
will be more conducive to improving the quantization performance of the
codeword. In addition, if the minimum correlation is the same, quantizing
$r\in\left[8,\infty\right)$ can save 1 bit of feedback than
$r\in\left[4,\infty\right)$.

###### Example 1

In the case of the same channel correlation and different antenna array sizes,
the optimal codeword numbers for the $\alpha$ domain and $\beta$ domain are
summarized in Table I, respectively. As the table shows, as the number of
antennas and codewords on the $\alpha$ domain increase sequentially, the
number of codewords on the $\beta$ domain does not change much.

Table I: The number of codewords of the $\alpha$ domain and the $\beta$ domain
when the minimum correlation of codeword is 0.95.

| 64 | 128 | 256 | 512
---|---|---|---|---
$\alpha$ domain | 4 | 7 | 8 | 11
$\beta$ domain | 254 | 507 | 1014 | 2027

###  IV-B Dislocation Quantization Codebook Design Scheme

In the uniform quantization codebook design, the quantization region of each
codeword is distributed in a rectangular shape. To further improve the
accuracy of codeword quantized CSI, we propose a codebook design with
dislocation. The quantization area of each codeword is distributed in a
hexagon, which is symmetric about the $a$ and $b$ domains.
$\overline{\Delta{\alpha}}$ and $\overline{\Delta{\beta}}$ are the sampling
intervals of the offset codebook sampling scheme in the $\alpha$ and $\beta$
domains, respectively. It should be noted that the dislocation here means that
the codewords of the even or odd rows on the $\beta$ domain are collectively
shifted to the right by $\overline{\Delta\alpha}/2$ along the $\alpha$ domain,
as shown in Fig. 8.

Figure 8: Dislocation codewords on the $\alpha$-$\beta$ domain.  Figure 9:
Different codebook schemes with minimum quantization correlation $c$.

Based on the hexagon’s symmetry, the quantization area of a single codeword is
always twice that of the inscribed triangle in Fig. 9. Therefore, the problem
of maximizing the quantization area of a dislocation codeword can be
transformed into the problem of finding the giant inscribed triangle of an
ellipse. As mentioned above, based on Cauchy inequality, the codeword sampling
steps in the $\alpha$ domain and $\beta$ domain can be calculated as

$\overline{\Delta{\alpha}}=\frac{3}{N^{2}}\sqrt{\frac{\left({c-1}\right)}{{p}_{\alpha}}},\quad\overline{\Delta{\beta}}=\frac{1}{N}\sqrt{\frac{3\left({c-1}\right)}{{p}_{\beta}}}.$
(47)

Under the maximum quantization area, the corresponding vertex of the
quantization area of the codeword $\mathbf{W}_{0}$ is

$\begin{split}&(\alpha_{1},\beta_{1})=\left(-\sqrt{\frac{c-1}{p_{\alpha}N^{4}}},0\right),\\\
&(\alpha_{2},\beta_{2})=\left(\frac{1}{2}\sqrt{\frac{c-1}{p_{\alpha}N^{4}}},\frac{1}{2}\sqrt{\frac{3(c-1)}{p_{\beta}N^{2}}}\right),\\\
&(\alpha_{3},\beta_{3})=\left(\frac{1}{2}\sqrt{\frac{c-1}{p_{\alpha}N^{4}}},-\frac{1}{2}\sqrt{\frac{3(c-1)}{p_{\beta}N^{2}}}\right).\end{split}$
(48)

Currently, the codeword is located at the center of gravity of the triangular
quantization area. If codeword $\mathbf{W}_{0}$ is selected and used as the
origin, then the position of the point with the minor channel correlation from
the codeword under the two schemes satisfies the relationship as

$f\left(-\frac{2}{3}\overline{\Delta\alpha},0\right)=f\left(\frac{1}{6}\overline{\Delta\alpha},\frac{1}{2}\overline{\Delta\beta}\right).$
(49)

Therefore, the maximum quantization area of a single codeword is

$\overline{R}_{max}=\frac{3(1-c)}{2N^{3}}\sqrt{\frac{3}{{p}_{\alpha}{p}_{\beta}}}.$
(50)

The number of the sampling points in the $\alpha$ and $\beta$ domain is

${\overline{S}_{\alpha}}=\frac{1}{3}\sqrt{\frac{Np_{\alpha}}{\left({c-1}\right)}}~{},\quad{\overline{S}_{\beta}}=2N\sqrt{\frac{p_{\beta}}{3\left(c-1\right)}}.$
(51)

And the number of codewords can be calculated as

$\overline{S}_{ULA}=2{\overline{S}_{\alpha}}{\overline{S}_{\beta}}=\frac{4N}{3(1-c)}\sqrt{\frac{Np_{\alpha}p_{\beta}}{3}}.$
(52)

In the collection of ULA codewords, the coordinates of the $s$-th codeword in
the $\alpha$ domain can be reformulated as

${\alpha}\left(\overline{s}_{\alpha}\right)=\left\\{\begin{matrix}{\left(\overline{s}_{\alpha}-1\right)\overline{\Delta{\alpha}},\quad
odd~{}row}&\\\
{\frac{\overline{\Delta\alpha}}{6}+\left(\overline{s}_{\alpha}-1\right)\overline{\Delta{\alpha}},\quad
even~{}row}\end{matrix}\right.$ (53)

where $\overline{s}_{\alpha}=1\ldots\overline{S}_{\alpha}$ and the $t$-th
sampling point of the $\beta$ domain can be reformulated as

${\beta}\left(\overline{s}_{\beta}\right)=\left\\{\begin{matrix}&{-1+\left({\overline{s}_{\beta}-1}\right)\overline{\Delta{\beta}},\quad
odd~{}column}\\\
&{-1+\frac{\overline{\Delta\beta}}{2}+\left({\overline{s}_{\beta}-1}\right)\overline{\Delta{\beta}}},\quad
even~{}column\end{matrix}\right.$ (54)

where $\overline{s}_{\beta}=1\ldots,{\overline{S}}_{\beta}$.

(43) and (52) provide the number of codewords under uniform and dislocation
sampling. The number of dislocation sampling codewords is only $75\%$ of the
number of uniform sampling codewords. Therefore, in the same space, the
dislocation quantization scheme can achieve the goal of low codebook
quantization overhead by expanding the sampling interval of codewords.

##  V Near-field UPA Codebook Design

The non-stationary characteristic of the UPA channel correlation results in a
different distribution for each codeword in 3D space. This section proposes
the concept of a reference ellipsoid and assumes that the correlation formula
for each codeword is always the same as the reference ellipsoid. Based on this
assumption of stationarity, uniformly sampled codewords can be obtained. In
this scheme, each codeword in the non-stationary UPA channel can meet the
minimum correlation requirement.

###  V-A Definition of Reference Ellipsoid

Proposition 1 shows that for any codeword, the pointing position of the
channel vector that satisfies the minimum quantization correlation of $c$ is
always uniformly distributed on an ellipsoid centered around the quantization
center of the codeword. It should be noted that due to the non-stationary
characteristics of UPA channels, the size of the ellipsoid enclosed by
different codewords at the positions that meet the conditions under the same
minimum quantization correlation is always different. Therefore, when
designing the optimal sampling interval between UPA codewords, the correlation
feature of any codeword cannot be directly regarded as the correlation feature
of all codewords, just like in ULA codebooks. In order to solve the problem
caused by non-stationary features when designing the optimal sampling interval
for codebooks, we hope to find a reference ellipsoid to describe the
quantization features of any codeword in space. This reference ellipsoid
provides the maximum allowable space, ensuring that all codewords can
guarantee the minimum quantization correlation $c$ at this volume. Below, we
define a reference ellipsoid.

###### Definition 1

Consider sampling $S_{\psi}$, $S_{\varphi}$ and $S_{\rho}$ points on the
$\psi$, $\varphi$ and $\rho$ domains, respectively. Using the channel vector
pointed at by the sampling point as the codeword, a total of
$S^{*}=S_{\psi}S_{\varphi}S_{\rho}$ can be obtained. From Proposition 2, it
can be concluded that when the quantization performance of all codewords
satisfies the minimum correlation $c^{\mathrm{*}}$, the sets of ellipsoidal
axis lengths enclosed by the quantization boundaries of all codewords are
respectively represented as $\mathbf{L}_{\psi}=\left\\{l_{\psi,1},\ldots
l_{\psi,{S_{\psi}}}\right\\}$,
$\mathbf{L}_{\varphi}=\left\\{l_{\varphi,1},\ldots
l_{{\varphi,S_{\varphi}}}\right\\}$,
$\mathbf{L}_{\rho}=\left\\{l_{\rho,1},\ldots l_{{\rho,S_{\rho}}}\right\\}$.
Meanwhile, ${l}_{\psi}^{\mathrm{*}}=\mathit{\min}\ \mathbf{L}_{\psi}$,
${l}_{\varphi}^{\mathrm{*}}=\mathit{\min}\ \mathbf{L}_{\varphi}$ and
${l}_{\rho}^{\mathrm{*}}=\mathit{\min}\ \mathbf{L}_{\rho}$ are used as the
axial length of the reference ellipsoid, then the formula of the reference
ellipsoid can be written as

$\frac{\delta_{\psi_{s}}^{2}}{\big{(}{l}_{\psi}^{\mathrm{*}}\big{)}^{2}}+\frac{\delta_{\varphi_{s}}^{2}}{\big{(}{l}_{\varphi}^{\mathrm{*}}\big{)}^{2}}+\frac{\delta_{\rho_{s}}^{2}}{\big{(}{l}_{\rho}^{\mathrm{*}}\big{)}^{2}}=1.$
(55)

According to the fitting formula given in Proposition 2, the fitting
coefficient can be calculated as

$\begin{split}{p}_{\psi}^{*}=\frac{\mathrm{c}-1}{({l}_{\psi}^{*}N)^{2}},\
\quad{p}_{\varphi}^{*}=\frac{\mathrm{c}-1}{({l}_{\varphi}^{*}N)^{2}},\
\quad{p}_{\rho}^{*}=\frac{\mathrm{c}-1}{({l}_{\rho}^{*}N)^{2}}.\end{split}$
(56)

The formula for the reference ellipsoid can be completed as

$\frac{\delta_{\psi_{s}}^{2}}{\frac{\mathrm{c}-1}{{p}_{\psi}^{\mathrm{*}}N^{2}}}+\frac{\delta_{\varphi_{s}}^{2}}{\frac{\mathrm{c}-1}{{p}_{\varphi}^{\mathrm{*}}N^{2}}}+\frac{\delta_{\rho_{s}}^{2}}{\frac{\mathrm{c}-1}{{p}_{\rho}^{\mathrm{*}}N^{4}}}=1.$
(57)

It is worth noting that some codewords may not achieve the expected minimum
quantization correlation $c$ when considering the ellipse whose axis length is
longer than the axis length of the reference ellipse to represent the
quantized region of the codeword. Therefore, the reference ellipsoid is the
lower bound that can describe the quantization region of the codeword. The
reference ellipsoid is a virtual ellipsoid reconstructed by taking the minimum
value of the quantized region of all codewords, which may not exist.

Figure 10: Sampling points distribution in 3D space.

###  V-B Uniform Codebook Quantization Scheme

For the UPA channel model, UE is located at
$r\in\big{[}\sqrt{0.62D^{3}/\lambda},\infty\big{)}$ in distance,
$\left[-{\pi}/{2},{\pi}/{2}\right]$ in elevation angle and
$\left[0,\pi\right]$ in azimuth angle. At this moment, the UE can be
considered uniformly distributed in 3D space of $\psi\in\left[-1,1\right]$,
$\varphi\in\left[-1,1\right]$ and
$\rho\in\big{[}\sqrt{\lambda/0.62D^{3}},\infty\big{)}$. Assuming that the
codeword sampling step in the $\psi$ domain, $\varphi$ domain, and $\rho$
domain is $\Delta{\psi}$, $\Delta{\varphi}$, and $\Delta{\rho}$. Respectively,
the collection of sampling points can be represented as

$\begin{split}\Xi^{A}=\Big{\\{}\big{(}&\psi_{s},\varphi_{s},\rho_{s}\big{)}\Big{|}\psi_{s}=\psi_{min}+\frac{\Delta{\psi}}{2},\psi_{min}+\frac{3\Delta{\psi}}{2},\ldots,\\\
&\psi_{max};\varphi_{s}=\varphi_{min}+\frac{\Delta{\varphi}}{2},{\varphi}_{min}+\frac{3\Delta{\varphi}}{2},\ldots,{\varphi}_{max};\\\
&\rho_{s}=\rho_{min}+\frac{\Delta{\rho}}{2},\rho_{min}+\frac{3\Delta{\rho}}{2},\ldots,\rho_{max}\Big{\\}}.\end{split}$
(58)

Next, we provide a design for the UPA codebook. Considering a codebook design
scheme for uniform sampling in the transform domain, as shown in Fig. 10.
Among them, the red rectangle represents the codeword, and the solid black
line represents the dividing line of the codeword quantization area. Assuming
that the UPA channel is stationary, with a minimum correlation of $c$ for all
codewords, the quantization regions of all codewords are distributed in an
ellipsoid of the same shape centered on the codeword, as shown in Fig. 11. For
adjacent eight codewords, the boundaries of the quantization region of the
codeword always intersect at one point under the same minimum quantization
correlation $\mathrm{c}$. Moreover, the intersection point is located at the
center of a cuboid surrounded by eight adjacent codewords.

By distributing the features described in Fig. 11 throughout the entire space,
it can be concluded that the quantization area of each codeword is enclosed in
a cuboid. Moreover, the eight vertices of the quantized cuboid are located on
the ellipsoidal boundary enclosed by the codeword under fixed correlation
$\mathrm{c}$.

Figure 11: Correlation distribution of eight adjacent uniform sampling points.

When designing a codebook, higher quantization accuracy and less quantization
overhead are always expected. With a fixed minimum quantization correlation,
the larger the cuboid quantization area of the codeword in Fig. 11, the less
quantization cost. For the enclosed cuboid of an ellipsoid, the maximum volume
can be obtained when
$\frac{\mathrm{c}-1}{{p}_{\rho}^{\mathrm{*}}N^{4}}{\delta_{\psi_{s}}^{2}}=\frac{\mathrm{c}-1}{{p}_{\psi}^{\mathrm{*}}N^{2}}{\delta_{\rho_{s}}^{2}}$
and
$\frac{c-1}{p_{\rho}^{\mathrm{*}}N^{4}}{\delta_{\varphi_{s}}^{2}}=\frac{\mathrm{c}-1}{{p}_{\varphi}^{\mathrm{*}}N^{2}}{\delta_{\rho_{s}}^{2}}$
is satisfied. And the optimal sampling steps on the three domains can be
calculated as

$\begin{split}\Delta{\psi}=\frac{2\sqrt{3}}{3N}\sqrt{\frac{\mathrm{c}-1}{{p}_{\psi}^{\mathrm{*}}}},\\\
\Delta{\varphi}=\frac{2\sqrt{3}}{3N}\sqrt{\frac{\mathrm{c}-1}{{p}_{\varphi}^{\mathrm{*}}}},\\\
\Delta{\rho}=\frac{2\sqrt{3}}{3N^{2}}\sqrt{\frac{\mathrm{c}-1}{{p}_{\rho}^{\mathrm{*}}}}.\end{split}$
(59)

Therefore, the number of codewords in 3D space can be calculated as

$S_{\psi}=\sqrt{\frac{3{p}_{\psi}^{\mathrm{*}}}{c-1}}N,\quad
S_{\varphi}=\sqrt{\frac{3{p}_{\varphi}^{\mathrm{*}}}{c-1}}N,\quad
S_{\rho}=2.4\sqrt{\frac{N{p}_{\rho}^{\mathrm{*}}}{c-1}}.$ (60)

The positions represented by the $s_{\psi}$-th, $s_{\varphi}$-th and
$s_{\rho}$-th sampling points in $\psi$, $\varphi$ and $\rho$ domain can be
expressed as

$\begin{split}\psi\left(s_{\psi}\right)&=-1+\left(s_{\psi}-\frac{1}{2}\right)\Delta{\psi},s_{\psi}=1,\ldots,S_{\psi},\\\
\varphi\left(s_{\varphi}\right)&=-1+\left(s_{\varphi}-\frac{1}{2}\right)\Delta{\varphi},s_{\varphi}=1,\ldots,S_{\varphi},\\\
\rho\left(s_{\rho}\right)&=\left(s_{\rho}-\frac{1}{2}\right)\Delta{\rho},s_{\rho}=1,\ldots,S_{\rho}.\end{split}$
(61)

For the proposed codebook scheme, the sampling step is only related to the
number of antennas and the minimum correlation. The larger the number of
antennas, the smaller the sampling step. However, for the maximum quantization
error, the sampling step will increase with its value, and the number of
codeword sampling points in a specific space will decrease.

When the correlation of UPA channels is regarded as approximately stationary
for sampling, the designed codeword accuracy is always lower than the accuracy
using other points as reference positions. Because under the same sampling
interval, the actual minimum correlation of each codeword is more significant
than the minimum correlation of the reference position, its quantization
accuracy is also more significant than the quantization area of the reference
codeword. Thus, under this reference ellipsoid, when the quantization error is
$c$, the quantization accuracy of the approximately stationary UPA sampling
method is the lower bound among all uniform sampling methods. The quantization
region of the codeword approximated by stationarity is always smaller than or
equal to the actual maximum quantization region of the codeword under the
minimum quantization correlation $c$. Therefore, a single codeword’s
achievable minimum quantization correlation is always greater than or equal to
$c$.

###### Example 2

When the channel correlation is $0.95$, the table below illustrates the number
of sampling points for codewords in the 3D space (refer to Table II). Notably,
the number of codewords in the $\psi$ and $\varphi$ domains consistently
exceed the number in the $\rho$ domain. Additionally, the number of sampling
points in the angular domain is larger than the number of antennas.
Incorporating oversampling in the angle domain during the design of the UPA
codebook can effectively enhance the quantization accuracy of codewords.

Table II: The number of codewords in 3D space when the correlation of codeword
is 0.95.

| 8*8 | 12*12 | 16*16 | 32*32
---|---|---|---|---
$\psi/\varphi\quad\mathrm{domain}$ | 39*39 | 59*59 | 78*78 | 156*156
$\rho\textit{\quad}\mathrm{domain}$ | 2 | 3 | 3 | 4

##  VI SIMULATION RESULTS

In this section, we provide the simulation results to illustrate the
performance of the proposed codebook schemes for ULA and UPA systems. The
central frequency is $f=3\mathrm{GHz}$. Define the signal-to-noise ratios
(SNR) of the system as

$SNR=\frac{P\eta N}{r^{2}\sigma^{2}}.$ (62)

$P$ is the transmit power and $\sigma^{2}$ is the nosie power. The achievable
rate is given by

$R=\log_{2}\left(1+\frac{P\eta
N\left|\mathbf{W}^{H}\mathbf{b}\left(r,\theta\right)\right|^{2}}{r^{2}\sigma^{2}}\right).$
(63)

The simulation results are the average results of 1000 randomly distributed
UE. Firstly, we analyze the codebook of the ULA channel. The number of BS
antennas is set as $N=512$. The range of UE locates randomly in the space
spanned
$\big{(}r,\theta\big{)}\in\big{[}\sqrt{0.62D^{3}/\lambda},\infty\big{)}\times\big{[}-\pi/2,\pi/2\big{]}$.
Next, we evaluate the performance of the proposed UPA channel codebook.
Considering the system model in Fig. 12, the BS is configured with UPA, whose
number of antenna is $16\times 16$. The elevation angle and azimuth angle of
the UE is $\theta\in\left[-{\pi}/{2},{\pi}/{2}\right]$ and
$\phi\in\left[0,{\pi}\right]$, respectively. The distance between the BS and
the UE distributes in $r\in\big{[}\sqrt{0.62D^{3}/\lambda},\infty\big{)}$.

Fig. 12 illustrates the cumulative probability function (CDF) of quantized
correlation with various codebooks. In order to provide a comparative
analysis, we conduct simulations using several codebooks. Firstly, we consider
a codebook with identical sampled points in the $\alpha$ and $\beta$ domains.
Additionally, we evaluate a codebook optimized using the Lloyd algorithm. The
proposed dislocation codebook and uniform codebook are both quantized with
$B=15$ bits. It is worth noting that, with an equal number of quantization
vectors, the dislocation codebook consistently outperforms the uniform
codebook. The proposed uniform codebook comprised 2617 sampling points in the
$\alpha$ domain and 14 sampling points in the $\beta$ domain.

On the other hand, the dislocation quantization codeword contains 1912
sampling points in the $\alpha$ domain and 18 sampling points in the $\beta$
domain. The number of sampling points in the $\alpha$ domain is significantly
higher than in the $\beta$ domain. This phenomenon highlights the robustness
of near-field beamforming in the $\beta$ domain, and the denser sampling of
the $\beta$ domain enhances codeword quantization accuracy. It is observed
that the performance of the codebook employing $N$ sampled points in both
domains is considerably inferior to the proposed codebooks, even when
employing a more significant number of quantization vectors. Furthermore, the
proposed schemes demonstrate significant superiority over the Lloyd algorithm
optimization scheme, which validates the effectiveness of our proposed
codeword design.

Figure 12: CDF of the codebook quantification correlation in ULA channel.
Figure 13: Comparisons of average sum-rate against SNR for several codeword
schemes in ULA channel, with $N=512$ and $f=3\mathrm{GHz}$.

Fig. 13 illustrates the achievable rate for two scenarios: the ideal case of
perfect channel state information (CSI) and the case where the precoding
matrix is selected based on channel quantization. The beamforming scheme with
perfect CSI represents the theoretical upper limit. In this comparison, we
consider the same codebook scheme shown in Fig. 12. The proposed codebooks and
the ideal case of perfect CSI exhibit remarkably similar performance. Notably,
the dislocation codebook significantly enhances the achievable rate compared
to the uniform codebook. When employing the same quantization bits, we observe
that the achievable rate of the proposed uniform and dislocation codebook
consistently outperforms the rate achieved using the Lloyd framework,
particularly as the receiver SNR increases. Furthermore, the proposed codebook
outperforms the codebook that employs the same sampling points in the $\alpha$
and $\beta$ domains. The average sum rate demonstrates an improvement of
approximately 0.4 $\mathrm{bits/s/Hz}$ compared to the codebook with the same
sampling points in the transformed domain.

Figure 14: Quantization Overhead consumption against the correlation of
codebook.

In Fig. 14, we compare the number of quantization channel vectors of the
proposed quantization schemes. The quantization vector’s number of the
proposed uniform codebook and dislocation codebook are respectively set
according to (43) and (52). We observe that as the requirement for codebook
quantization accuracy increases, the number of vectors required to quantize
the channel also increases. The result shows that the dislocation codeword can
approximately reduce the number of quantization bits by $25\%$ compared with
the uniform codeword, which is consistent with our analysis in Section IV.

Figure 15: CDF of the codeword quantification correlation in UPA channel, with
$N_{BS}=16\times 16$, $f_{c}=3GHz$. Figure 16: Comparisons of average sum rate
against SNR for several codebooks in UPA channel, with $N_{BS}=16\times 16$,
$f_{c}=3\mathrm{GHz}$.

We compare the proposed UPA codebook with two other schemes. In the first
scheme, the codewords are uniformly sampled at $N$ points in the $\psi$,
$\varphi$, and $\rho$ domains. The second scheme involves designing codewords
with optimal sampling points based on the Lloyd algorithm in the $\psi$,
$\varphi$, and $\rho$ domains. Fig. 15 depicts the CDF of quantized
correlation for the different codebooks. The quantized correlation achieved by
the proposed uniform codebook is significantly superior to the other schemes.
Following the codeword design scheme described in Section V, we assume a
minimum codeword quantization correlation of $c=0.95$. The simulation results
consistently demonstrate that the quantized correlation achieved by the
proposed codebook consistently exceeds 0.95. The result confirms that the
proposed codebook is designed based on the lower bound of quantization
correlation for all sampled codewords. Moreover, the proposed codebook
exhibits more sampling points in the angle domain than in the distance domain.
The results also reveal that the achievable rate performance of our proposed
codebook consistently outperforms schemes with an equal number of samples in
the $\psi$, $\varphi$, and $\rho$ domains. Therefore, the near-field codebook
should be oversampled in the angle domain. Considering the same quantization
overhead, our proposed scheme achieves superior quantization correlation
compared to the optimal codebook designed using the Lloyd Max algorithm.

We evaluate the average sum rate achieved by different codebooks when utilized
as the beamforming matrix at the BS in the UPA system. Fig. 16 illustrates the
results at various SNR. The outcomes demonstrate that our proposed UPA
codebook scheme significantly outperforms the other two schemes and closely
approaches the performance achieved with perfect CSI. Compared to the uniform
codeword scheme, which employs the same number of samples in both the angle
and distance domains, our proposed scheme exhibits an improvement of nearly
$0.5$ bits/s/Hz. Furthermore, our proposed codebook outperforms the codebook
designed based on the Lloyd algorithm by more than $0.1$ bits/s/Hz.

##  VII Conclusion

This paper introduces a novel codebook design to maximize the minimum
quantization correlation for quantized ELAA channels. We analyze the
correlation between codewords and channel vectors and derive a suitable
formula for this correlation. Based on these insights, we propose two ULA
codebooks: uniform sampling and dislocation sampling. Notably, the uniform
offset codebook achieves the same quantization performance as the uniform
codebook while requiring fewer quantization bits. Furthermore, our study
reveals that the channel observed by the UPA exhibits non-stationarity. To
address this, we propose a lower-bound scheme for optimally sampled codebooks.
Additionally, we emphasize the robustness of the angle domain in ELAA systems
and the advantages of oversampling the angle domain for codebook design with
higher precision. Simulation results confirm that the proposed codebook
achieves minimal quantization bits while maintaining the desired minimum
quantization correlation.

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