# Fiber radio frequency transfer using bidirectional frequency division
multiplexing dissemination

Qi Li, Liang Hu, Jinbo Zhang, Jianping Chen, and Guiling Wu Manuscript
received xxx xxx, xxx; revised xxx xxx, xxx. This work was supported by the
National Natural Science Foundation of China (NSFC) (61627817, 61905143).
(Corresponding author: Liang Hu.)The authors are with the State Key Laboratory
of Advanced Optical Communication Systems and Networks, Shanghai Institute for
Advanced Communication and Data Science, Shanghai Key Laboratory of Navigation
and Location-Based Services, Department of Electronic Engineering, Shanghai
Jiao Tong University, Shanghai 200240, China (email<EMAIL_ADDRESS>
###### Abstract

We report on the realization of a novel fiber-optic radio frequency (RF)
transfer scheme with the bidirectional frequency division multiplexing (FDM)
dissemination technique. Here, the proper bidirectional frequency map used in
the forward and backward directions for suppressing the backscattering noise
and ensuring the symmetry of the bidirectional transfer RF signals within one
telecommunication channel. We experimentally demonstrated a 0.9 GHz signal
transfer over a 120 km optical link with the relative frequency stabilities of
$2.2\times 10^{-14}$ at 1 s and $4.6\times 10^{-17}$ at 20,000 s. The
implementation of phase noise compensation at the remote site has the
capability to perform RF transfer over a branching fiber network with the
proposed technique as needed by large-scale scientific experiments.

###### Index Terms:

Radio frequency transfer, backscattering noise, bidirectional frequency
division multiplexing, metrology.

##  I Introduction

Ultra-stable radio frequency (RF) dissemination plays a significant role in
numerous areas, such as radio astronomy, accurate navigation and basic
physical measurement [1, 2, 3]. Compared with satellite links, fiber-optic
links have the advantages of low attenuation, wide bandwidth and immunity to
electromagnetic interference, etc [4], which enable ultra-long-haul and high-
precision frequency dissemination [5, 6, 7, 8, 9, 10, 11, 12]. Unfortunately,
the temperature variations and mechanical perturbations along the fiber links
will change the propagation delay of the fiber links and hence introduce
additional phase noise into the transferred RF signal [13]. In order to
suppress this part of phase noise, various fiber-optic RF transfer schemes
based on active [5, 8, 6, 7] and passive phase noise compensation schemes [9,
10, 11, 12] have been proposed by several research groups over the last two
decades.

The limits of the short-term stability for the single-fiber bidirectional
fiber-optic RF transfer system are mainly decided by system’s signal-to-noise
ratio (SNR) [13]. The deterioration of the system SNR is generally caused by
the backscattering noise such as Fresnel reflection, Rayleigh and Brillouin
scattering for single-fiber bidirectional dissemination systems. The
bidirectional wavelength division multiplexing (WDM) technique at the local
and remote sites has been widely adopted to largely suppress the
backscattering noise at the cost of breaking the wavelength symmetry in both
directions [10, 11]. However, the telecommunication channels in the
metropolitan network are scarce resources, this WDM method may occupy an
additional telecommunication channel. Alternatively, the bidirectional
frequency division multiplexing (FDM) technique has drawn widely attention for
the fiber-optic RF transfer application [14, 8, 9], in which only one
telecommunication channel (e.g.100 GHz) is used for bidirectional
transmission. Lopez et al. have proposed an active phase noise compensation
scheme based on the variable optical delay lines (VODLs), which adopted the
different frequencies and same wavelength in the bidirectional directions to
largely suppress the backscattering noise [14]. However, the used VODLs are
cumbersome and have the limited compensation range. Huang et al. proposed a
passive compensation scheme by using frequency dividing and filtering [9].
Although this scheme can effectively eliminate crosstalk and partially
suppress the backscattering noise with the same wavelength at both sides, the
two modulated RF signals used at the local site have the multiple
relationship, such as 4 times or 5 times. Zhang et al. have proposed an active
electrical compensation scheme using a carrier suppressed double-sideband
(CSDSB) signal [8]. Although the CSDSB signal used at the local site can
efficiently suppress the backscattering noise, the modulator has to be biased
at the $V_{\pi}$ point, which is easily affected by the temperature
fluctuations. In this configuration, the bidirectional RF signals have a
factor of 2 difference and the RF signal with the lower power will be obtained
through mutual-beating between double-sidebands, causing the lower SNR and
limiting the system performance, in particular the short-term stability, for
long-haul fiber link dissemination. We can see that the existing bidirectional
FDM dissemination based fiber-optic RF transfer schemes have the multiple
relationship between the bidirectional RF signals, resulting in that the
performance of system in terms of the relative stability will be determined by
the lowest frequency signal among the bidirectional RF signals [9]. One
intriguing question is how to distribute a reference RF to the user by using
bidirectional FDM dissemination with the improved performance in a cost-
effective and robust way as optical frequency transfer does [15, 16].

In this article, we propose a novel active phase noise compensation scheme
with the bidirectional FDM technique. The effect of the backscattering noise
is effectively suppressed by properly allocating the forward and backward RF
signals. The bidirectional optical signals transmitted over the same fiber
within one telecommunication channel guarantee the bidirectional propagation
symmetry. Moreover, the active compensation devices placed at the remote site
can greatly simplify the configuration of the local site and have the
capability to expand into the tree topology for multiple-point stable RF
optical transmission [17, 18]. We experimentally demonstrate the scheme by
transferring a 0.9 GHz RF signal via 120 km single-mode fibers (SMFs),
achieving the relative frequency stabilities of approximately $2.2\times
10^{-14}$ at 1s and $4.6\times 10^{-17}$ at 20,000s.

##  II Principle

Figure 1: (a) Schematic diagram of the proposed fiber-optic RF transfer scheme
based on the bidirectional frequency division multiplexing (FDM) transmission.
FS: frequency standard, PD: photodetector, MZM: Mach-Zehnder modulator, CIR:
optical circulator, VCO: voltage-controlled oscillator, MIX: frequency mixer,
PDRO: phase-locked dielectric resonant oscillator, BPF: band-pass filter,
EDFA: erbium-doped fiber amplifier, CWL: continuous wave laser, PLL: phase-
lock loop. (b) Diagram view of the bidirectional FDM technique to suppress the
backscatteringnoise.

Figure 1 illustrates a schematic diagram of our proposed stable RF transfer
system using bidirectional FDM dissemination. The local site (LS) and remote
site (RS) are connected via a fiber-optic link. At the LS, the frequency
standard (FS) can be denoted as a cosine function without considering its
amplitude [11],

$E_{1}\propto\cos(\omega_{r0}t+\varphi_{r0}),$ (1)

where $\omega_{r0}$ and $\varphi_{r0}$ are its angular frequency and initial
phase, respectively. To improve the stability of the RF transfer system, the
frequency of $E_{1}$ is multiplied to $E_{2}$ by a phase-locked dielectric
resonant oscillator (PDRO). At the same time, $E_{1}$ is multiplied to $E_{3}$
as an assistant signal for achieving the bidirectional FDM dissemination with
another PDRO. $E_{2}$ and $E_{3}$ can be expressed as,

${E_{2}}\propto\cos({\omega_{r1}}t+{\varphi_{r1}}),$ (2)
${E_{3}}\propto\cos({\omega_{r2}}t+{\varphi_{r2}}).$ (3)

$E_{2}$ is modulated onto the continuous wave laser (CWL1) by a Mach-Zehnder
intensity modulator (MZM) biased at the quadrature point and subsequently
propagated through the optical fiber with the length of $L$ to the
photodetector (PD2) at the RS. The recovered signal has a form of,

${E_{4}}\propto\cos[{\omega_{r1}}(t-{\tau_{LR}}(t))+{\varphi_{r1}}],$ (4)

where $\tau_{LR}(t)$ represents the fiber-delay variations induced by the
environmental perturbations propagating from the LS to the RS. The detected RF
signal is split two parts. One part is mixed with the $E_{5}$ signal produced
by a voltage-controlled oscillator (VCO) with $E_{5}\propto
cos(\omega_{vco}t+\varphi_{c})$, where $\omega_{vco}$ and $\varphi_{c}$
respectively, are the angular frequency and initial phase of the VCO. To
satisfy the bidirectional FDM transmission requirements, we set
$\omega_{r2}=2\omega_{vco}$. By mixing $E_{4}$ with $E_{5}$, the lower side-
band signal is selected by a band-pass filter and can be expressed as,

${E_{6}}\propto\cos[{\omega_{r1}}(t-{\tau_{LR}}(t))-\frac{1}{2}{\omega_{r2}}t-{\varphi_{c}}+{\varphi_{r1}}].$
(5)

Subsequently, $E_{6}$ is divided into two branches. One part is used for the
user and the other part is modulated onto the CWL2 by another MZM biased at
the $V_{\pi/2}$ point and transmitted to the LS via the same SMFs. To get rid
of the interference from the backscattered optical signal carrier in the
bidirectional fiber-optic link, we can set the wavelengths of the CWL1 and
CWL2 with the slight difference. The RF signal recovered by the PD1 at the LS
can be depicted as,

$\begin{split}{E_{7}}\propto\cos[{\omega_{r1}}(t-{\tau_{RL}}(t)-{\tau_{LR}}(t))\\\
-\frac{1}{2}{\omega_{r2}}(t-{\tau_{RL}}(t))-{\varphi_{c}}+{\varphi_{r1}}],\end{split}$
(6)

where $\tau_{RL}(t)$ represents the fiber-delay variations from the RS to the
LS. The upper side-band signal obtained by mixing $E_{3}$ and $E_{7}$ can be
expressed as,

$\begin{split}{E_{8}}\propto\cos[{\omega_{r1}}(t-{\tau_{RL}}(t)-{\tau_{LR}}(t)){\rm{}}\\\
-\frac{1}{2}{\omega_{r2}}(t-{\tau_{RL}}(t))+{\omega_{r2}}t-{\varphi_{c}}+{\varphi_{r1}}+{\varphi_{r2}}].\end{split}$
(7)

Then $E_{2}$ and $E_{8}$ are simultaneously modulated onto the CWL1 by the MZM
at the LS and transmitted to the RS via the fiber link, the signal recovered
by the PD3 can be denoted as,

$\begin{split}{E_{9}}\propto\cos[{\omega_{r1}}(t-{\tau_{RL}}(t)-2{\tau_{LR}}(t))+{\omega_{r2}}(t-{\tau_{LR}}(t))\\\
{\rm{}}-\frac{1}{2}{\omega_{r2}}(t-{\tau_{RL}}(t)-{\tau_{LR}}(t))-{\varphi_{c}}+{\varphi_{r1}}+{\varphi_{r2}}].\end{split}$
(8)

By mixing $E_{4}$ with $E_{9}$, the lower side-band signal is obtained and can
be given by,

$\begin{split}{E_{10}}\propto\cos[\frac{1}{2}\omega_{r2}t-({\omega_{r1}}-\frac{1}{2}{\omega_{r2}}){\tau_{RL}}(t)\\\
{\rm{}}-({\omega_{r1}}+\frac{1}{2}{\omega_{r2}}){\tau_{LR}}(t)-{\varphi_{c}}+{\varphi_{r2}}]\end{split}.$
(9)

In Eq. 9, the DC error signal carrying the fiber link phase noise is obtained
by mixing $E_{5}$ and $E_{10}$, resulting in ,

${V_{error}}\propto({\omega_{r1}}-\frac{1}{2}{\omega_{r2}}){\tau_{RL}}(t){\rm{+}}({\omega_{r1}}+\frac{1}{2}{\omega_{r2}}){\tau_{LR}}(t)+2{\varphi_{c}}-{\varphi_{r2}}.$
(10)

Assuming the fiber-delay variations introduced by the environmental
perturbations slower than the triple transmission time of the fiber link, we
can have the relationship of $\tau_{LR}(t)=\tau_{RL}(t)=\tau(t)$. As the first
and second terms indicated in the right side of Eq. 10, the phase noise of the
fiber-optic link caused by forward and backward transfer is centrally
symmetric about the angular frequency $\omega_{r1}$ as indicated by Fig. 1(b).
Equation 10 can be further rewritten as $V_{error}\propto
2\omega_{r1}\tau(t)+2\varphi_{c}-\varphi_{r2}.$ When the phase-locked loop
(PLL) is active, the fiber link is stabilized by tuning the frequency of the
VCO, i.e., $V_{err}\rightarrow 0$. We obtain the relationship of,

${\varphi_{c}}=\frac{{{\varphi_{r2}}}}{2}-{\omega_{r1}}\tau(t).$ (11)

By substituting the Eq. 11 into Eq. 5, we can find that a stable RF signal can
be recovered at the RS, which is independent of the fiber-optic induced phase
fluctuations. The principle of the proposed scheme to suppress the
backscattering noise can be seen in Fig. 1(b). As the frequencies of the RF
signals transmitted through the fiber-optic link are different on the
bidirectional directions, and these RF signals are symmetric about the center
of $\omega_{r1}$, the backscattering noise can be suppressed by electrical
filters.

##  III Experimental apparatus and results

The experiment setup based on the proposed scheme is shown in Fig. 1. At the
LS, the frequency standard (Rigol Inc., DSG 830) with $\omega_{r0}=100\
\rm{MHz}$ is multiplied to $\omega_{r1}=1\ \rm{GHz}$ and $\omega_{r2}=200\
\rm{MHz}$ by the PDROs. The LS and the RS are connected by 120 km SMFs
(consisting of 10, 30, 50, 30 km single-mode fiber spools), placed in a normal
laboratory with the peak-to-peak temperature fluctuations about 3 ∘C [19, 20].
To suppress the impact of the chromatic dispersion, the proper dispersion
compensating fibers (DCFs) are used [19, 20]. We set the wavelengths of the
CWLs with the 0.4 nm difference at the both sites around 1550 nm. The
wavelength difference is determined by the minimum adjustment step size of the
lasers used in our experiment. Each transmitted RF signal is carried on the
corresponding optical carrier by a LiNbO3 MZM (iXblue Inc., MXAN-LN-10) biased
at the $V_{\pi/2}$ point and detected by the PD with 2 GHz bandwidth at each
site. With this configuration, we have $\omega_{r1}=1\ \rm{GHz}$,
$\omega_{r1}-\omega_{r2}/2=0.9\ \rm{GHz}$ and $\omega_{r1}+\omega_{r2}/2=1.1\
\rm{GHz}$. The PID controller (Newport Inc., LB1005) as the PLL control unit
is used and we set the corner frequency of approximately 200 Hz. Moreover, a
bidirectional erbium-doped fiber amplifier (Bi-EDFA) [14] is employed between
the 40 km and 80 km fiber spools, to boost the fading bidirectional optical
signals. The optical gain of Bi-EDFA at each direction is less than 20 dB to
suppress the stimulated Brillouin scattering and amplified spontaneous
emission (ASE) noise. To evaluate the performance of the proposed scheme, the
standard signal at the LS and the duplicated 0.9 GHz signal at RS are
converted to 10 MHz by a dual-mixer time difference method [7]. The outputs
are measured by a phase noise measurement instrument (Symmetricom Inc.,
TSC5120A).

Figure 2: Measured phase noise PSDs for different system configurations: the
120 km free running link (blue curve), the 120 km proposed scheme (red curve),
the back-to-back system noise floor (green curve) and the 120 km compensated
link with the backscattering noise (black curve).

The measured phase noise power spectral densities (PSDs) of the system under
different configurations are illustrated in Fig. 2. As a comparison, a RF
transfer scheme with the backscattering noise by adopting the same 1 GHz
signal for the bidirectional transmission is measured. The configuration is
similar to Ref [21]. The back-to-back system noise floor is also measured by
connecting two sites of the proposed scheme with a 1 m fiber link. The 120 km
free running link is evaluated by transferring a 0.9 GHz signal to the RS
without round-trip phase correction. As depicted in Fig. 2, the phase noise of
the 120 km free running link is obviously higher than the 120 km proposed
scheme (about -117 dBc at 0.1 Hz, -129 dBc at 1 Hz) at the offset frequency of
less than 1 Hz, due to the fiber delay fluctuations caused by the temperature
variations. In the range of $0.002-0.1\ \rm{Hz}$, the 120 km proposed
compensated link is almost consistent with the system noise floor, which
indicates that the phase noise in this low frequency range induced by the
fiber-optic link can be effectively suppressed. Compared with the 120 km
compensated scheme with the backscattering noise (about -116 dBc at 0.1 Hz,
-121 dBc at 1 Hz), the phase noise of the 120 km proposed scheme is
significantly lower within $0.1-100\ \rm{Hz}$, demonstrating that the
backscattering noise induced by the optical reflections including connectors
and the Rayleigh backscattering has a significant effect on the system
performance [19]. All the measurements have the sharp bump in the range of
$0.001-0.003\ \rm{Hz}$, which are mainly affected by the outside loop phase
noise induced by the temperature fluctuations. The phase noise of all
compensated configurations has a bump observed at about 200 Hz. This part
noise is mainly determined by the PLL parameters, the phase noise of the RF
signal and the propagation delay from the fiber [19]. At the high frequency
range ($>10^{3}$ Hz), the ripples with the same period can be observed in
different transfer configurations under the 120 km fiber link, which are
mainly caused by delayed self-interferometry [22].

Figure 3: Measured Allan deviations in different system configurations: the
120 km free running link (blue curve), the 120 km proposed scheme (red curve),
the back-to-back system noise floor (green curve) and the 120 km compensated
link with the backscattering noise (black curve).

Figure 3 shows the measured Allan deviations for different system
configurations with a 5 Hz measurement bandwidth. It is seen that, the long-
term stability of $4.6\times 10^{-17}$ at the averaging time of 20,000 s is
achieved for the 120 km proposed scheme, which is improved more than four
orders of magnitude in contrast with the 120 km free running link. Compared
with the 120 km compensated scheme with the backscattering noise, the short-
term of the 120 km proposed scheme is improved from $7.0\times 10^{-14}$ to
$2.2\times 10^{-14}$ at the averaging time of 1 s. Moreover, this value is
close to the back-to-back system noise floor, illustrating the fact that the
backscattering noise in the fiber link is largely suppressed by the proposed
scheme. Whereas, the Allan deviation of the 120 km proposed scheme is slightly
larger than the system noise floor, due to the noise of the source signals and
RF components, the ASE noise introduced by the EDFA and so on [23, 20]. As the
outside loop of the system is easily susceptible to the temperature
fluctuations, the Allan deviations of the different compensated systems
decrease slowly within $10^{2}-10^{3}\ \rm{s}$, which correspond to the sharp
bump at the range of $0.001-0.003\ \rm{Hz}$ in Fig. 2 [24].

##  IV Conclusion

In conclusion, we proposed and demonstrated a novel fiber-optic RF transfer
scheme with the bidirectional FDM transmission technique. In the bidirectional
fiber-optic link, the frequencies of the RF signals transmitted on both
directions are different and have a proper relationship, which can efficiently
suppress the effect of the backscattering noise and settle out the issues that
the system performance is limited to the lowest transfer frequency signal in
the conventional FDM transmission schemes with the multiple relationship for
the bidirectional frequency signals. We experimentally demonstrated a 0.9 GHz
signal transfer based on the proposed scheme over a 120 km fiber-optic link.
The stable 0.9 GHz signal transferred to the remote site has the relative
frequency stabilities of less than $2.2\times 10^{-14}$ at 1 s and $4.62\times
10^{-17}$ at 20,000 s. The phase noise compensation devices placed at the
remote site, which are compatible with the tree topology, provide a guidance
for developing multiple-point RF transfer as required by many scientific
applications.

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# Distance-Aware Precoding for Near-Field
Capacity Improvement

Zidong Wu, _Student Member, IEEE_ , Mingyao Cui, _Student Member, IEEE_ ,
Zijian Zhang, _Student Member, IEEE_ , Linglong Dai, _Fellow, IEEE_ All
authors are with the Beijing National Research Center for Information Science
and Technology (BNRist) as well as the Department of Electronic Engineering,
Tsinghua University, Beijing 100084, China (e-mails: {wuzd19, cmy20,
<EMAIL_ADDRESS>daill@tsinghua.edu.cn).

###### Abstract

Extremely large-scale MIMO (XL-MIMO) is a promising technology to improve the
capacity for future 6G networks. With a very large number of antennas, the
near-field property of XL-MIMO systems becomes dominant. Unlike the classical
far-field line-of-sight (LoS) channel with only one available data stream, the
significantly increased degrees of freedom (DoFs) are available in the near-
field LoS channel. However, limited by the small number of radio frequency
(RF) chains, the existing hybrid precoding architecture widely used for 5G is
not able to fully exploit the extra DoFs in the near-field region. In this
paper, the available DoFs and the capacity of the near-field LoS channel are
theoretically analyzed at first. Then, to exploit the near-field effect as a
new possibility for capacity improvement, the distance-aware precoding (DAP)
scheme is proposed. We develop the DAP architecture, where a dedicated
selection circuit is inserted to connect phase shifters and RF chains.
Moreover, each RF chain can be flexibly configured to active or inactive
according to the distance-related DoFs in the proposed DAP architecture. Based
on the developed DAP architecture, a DAP algorithm is proposed to optimize the
number of activated RF chains and precoding matrices to match the increased
DoFs in the near-field region. Finally, simulation results verify that, the
proposed DAP scheme can efficiently utilize the extra DoFs in the near-field
region to improve the spectrum efficiency and the energy efficiency as well.

###### Index Terms:

Extremely large-scale MIMO (XL-MIMO), hybrid precoding, near-field, spectrum
efficiency, energy efficiency.

##  I Introduction

Due to the explosively growing demand for communication capacity, spatial
multiplexing is regarded as a key technology to significantly increase the
spectrum efficiency [1]. By exploiting the spatial degrees of freedom (DoFs)
with spatial multiplexing, massive multiple-input multiple-output (MIMO) is
the key enabler to increase the capacity by orders of magnitude for 5G. To
further exploit the promising potential of spatial DoFs, massive MIMO for 5G
is evolving to extremely large-scale MIMO (XL-MIMO) for future 6G
communications [2].

The evolution from massive MIMO to XL-MIMO not only implies a significant
increase in the number of antennas, but also leads to a fundamental change of
the characteristics in the electromagnetic field. Specifically, since the
number of antennas of massive MIMO systems is usually not very large, the
classical _far-field_ MIMO channel model is widely adopted, which is based on
planar wave assumptions [3]. By contrast, with the dramatically increasing
number of antennas in XL-MIMO communication systems, receivers may lie in the
_near-field_ region of the transmitter, which is defined by the area within
the Rayleigh distance [4]. For example, if we consider an XL-MIMO system with
1000 linear antennas with half wavelength spacing, the Rayleigh distance
reaches 500 m at 300 GHz, which covers a large area of a standard cell. Unlike
the classical _far-field_ channel based on planar wave assumptions, the _near-
field_ channel should be modeled based on spherical wave assumptions in XL-
MIMO communications, leading to significant near-field effect [5]. Thus,
traditional far-field transmission techniques may suffer from a severe
performance loss due to the mismatch between _far-field_ assumptions and
_near-field_ channel property.

###  I-A Prior Works

Most of existing works on near-field communications focus on overcoming the
performance degradation caused by the near-field effect [6, 7, 5]. To be
specific, the steering beams are only related to the user angles under the
planar wave assumptions. By contrast, due to the property of spherical waves,
the beams are related to both the user angle and distance [6]. The initial
work [7] has studied the near-field beamforming techniques by designing the
angle- and distance-dependent beams for narrow-band communications. To achieve
the near-field beamforming in wideband systems, by exploiting the varied
spherical wavefronts at different frequencies, a phase-delay focusing method
was proposed in [5].

To acquire high-gain beamforming, channel estimation is essential for XL-MIMO
communications. Several works [8, 9, 10, 11] have studied how to overcome the
performance loss of channel estimation in the near-field region. In [8], a
near-field channel estimation method was proposed to capture the sparsity in
the joint angle-distance (polar) domain caused by spherical waves. Moreover, a
more realistic hybrid-field channel model characterizing both far-field and
near-field paths was investigated in [10], and the corresponding channel
estimation method was also proposed. Existing works on near-field
communications indicate that the near-field effect only induces performance
degradation for wireless communications which is caused by the mismatch of the
near-field channel model and classical far-field-based beamforming or channel
estimation methods in the physical layer.

Nevertheless, the recently developed electromagnetic information theory (EIT)
indicates that, wireless communications can also benefit from the near-field
effect [12]. Specifically, for far-field channels, due to the high attenuation
of millimeter-wave and terahertz bands, the rank-one line-of-sight (LoS) path
becomes predominant in the XL-MIMO channels, so only very limited spatial
multiplexing gain can be achieved [13]. Compared with the far-field LoS path
with _planar_ wavefronts determined by a single spatial angle, the near-field
LoS path with _spherical_ wavefronts contains a range of angles, which brings
significantly increased spatial DoFs (or transmission modes) [14, 15]. Owing
to the increased spatial DoFs, it is expected that the spectrum efficiency can
be naturally enhanced in the near-field region.

Unfortunately, the widely considered hybrid precoding architecture for XL-MIMO
can hardly benefit from the increased DoFs provided by the near-field LoS
path. In the far-field MIMO channel, due to the limited DoFs, the reduced
number of RF chains for the classical hybrid precoding is still larger than
the spatial DoFs. Thus, the classical hybrid precoding with a much reduced
number of RF chains can fully utilize the spatial DoFs to achieve the near-
optimal spectrum efficiency in the far-field region [16]. By contrast, this
reduced number of RF chains is much smaller than the increased DoFs in the
near-field region, resulting in a very limited number of data streams. That is
to say, the classical hybrid precoding architecture designed for far-field
communications can hardly utilize the increased spatial DoFs in the near-field
region. To the best of our knowledge, this important problem of how to exploit
the extra spatial DoFs brought by the near-field effect to improve the
capacity has not been studied in the literature.

###  I-B Our Contributions

To fill in this gap, in this paper, the theoretical capacity of the near-field
channel is analyzed, and the precoding architecture for near-field MIMO
communications is proposed111Simulation codes are provided to reproduce the
results presented in this paper:
http://oa.ee.tsinghua.edu.cn/dailinglong/publications/publications.html.. The
main contributions of this paper can be summarized as follows:

  * •

We introduce the eigenproblem in the electromagnetic theory to investigate the
communication capacity of the near-field LoS channel. First, the increased
DoFs are estimated based on the eigenproblem of prolate spheroidal wave
functions. The effectiveness of the approximation of the singular values of
the near-field LoS channel is verified by numerical simulations. Then, the
channel capacity of the near-field LoS channel is analyzed, which reveals the
channel capacity improvement in the near-field region.

  * •

We propose a distance-aware precoding (DAP) architecture, which provides a new
possibility to improve the capacity by utilizing the distance-related DoFs in
the near-field region. The key idea is to adaptively adjust the number of RF
chains to match the DoFs in the near-field region. Compared with the classical
hybrid precoding architecture where the number of RF chains is fixed and
limited, extra RF chains are equipped in the proposed DAP architecture, where
each RF chain can be flexibly configured to active or inactive. To realize
this structure, a dedicated selection circuit is inserted to connect phase
shifters to activated RF chains, which controls the selection pattern between
the RF chains and array elements.

  * •

Based on the DAP architecture, a DAP algorithm is proposed to improve the
spectrum efficiency. In the proposed DAP algorithm, the number of activated RF
chains is adaptively optimized according to the DoFs of the channel. Since the
eigenvalues are close to each other in the near-field region, a near-field
subarray partitioning algorithm is performed to optimize the selection matrix
which represents the linking pattern between the RF chains and phase shifters
to get a balanced subarray structure. Then, the corresponding digital precoder
and analog precoder can be obtained utilizing the sub-connected feature.
Finally, simulation results verify that, the proposed DAP scheme can
efficiently utilize the extra DoFs to improve the spectrum efficiency and
energy efficiency in the near-field region.

###  I-C Organization and Notation

The remainder of the paper is organized as follows. Section II introduces the
XL-MIMO communication system model in the near-field region, and compares the
relationship between the near-field LoS channel and far-field LoS channel.
Section III theoretically investigates the DoFs of the near-field LoS channel
and provides an approximation method for the channel capacity. In Section IV,
the DAP architecture is proposed. In Section V, the DAP algorithm is
illustrated in detail. Simulation results are provided in Section VI, and
conclusions are drawn in Section VII.

Notations: $\mathbb{C}$ denotes the set of complex numbers; ${[\cdot]^{-1}}$,
${[\cdot]^{T}}$, ${[\cdot]^{H}}$ and ${\rm diag}(\cdot)$ denote the inverse,
transpose, conjugate-transpose and diagonal operations, respectively;
$\|\cdot\|_{F}$ denotes the Frobenius norm of the matrix; $\angle[\cdot]$
denotes the angle of its complex argument; $\mathbf{I}_{N}$ is an $N\times N$
identity matrix; $[a]^{+}$ represents $\max\\{0,a\\}$; finally
$\mathbf{1}_{L}$ indicates an $L$-length vector with all elements being 1.

##  II System Model

In this paper, we consider a single-user XL-MIMO communication system. The
transmitter and receiver are equipped with $N_{\rm{t}}$ and $N_{\rm{r}}$
antennas, respectively. To enable the transmission of $N_{\rm{s}}$ data
streams, the transmitter and receiver contain $N_{\rm{t}}^{\rm{RF}}$ and
$N_{\rm{r}}^{\rm{RF}}$ RF chains, satisfying $N_{\rm{s}}\leq
N_{\rm{t}}^{\rm{RF}}\leq N_{\rm{t}}$ and $N_{\rm{s}}\leq
N_{\rm{r}}^{\rm{RF}}\leq N_{\rm{r}}$. We assume that the number of RF chains
equals the number of data streams for both transmitter and receiver. Thus, the
$N_{\rm{s}}\times 1$ symbol vector ${\bf{s}}$ is followed by a
$N_{\rm{s}}\times N_{\rm{s}}$ digital precoder ${\bf{F}}_{\rm{D}}$ and
$N_{\rm{t}}\times N_{\rm{s}}$ analog precoder ${\bf{F}}_{\rm{A}}$. The
received signal is expressed as

$\displaystyle{\bf{y}}={\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{D}}{\bf{s}}+{\bf{n}},$
(1)

where ${\bf{y}}$ denotes the $N_{\rm{r}}\times 1$ received vector, ${\bf{H}}$
is the $N_{\rm{r}}\times N_{\rm{t}}$ channel matrix, and ${\bf{n}}$ represents
the noise vector ${\bf{n}}=\left[n_{1},n_{2},\cdots,n_{N_{\rm{r}}}\right]$,
whose entities are mutually independent and follow the complex Gaussian
distribution $\mathcal{CN}(0,\sigma_{\rm{n}}^{2})$.

Due to the high attenuation of millimeter-wave and terahertz bands, LoS path
plays a dominant role in the XL-MIMO channel. Therefore, the classical far-
field channel can be approximated by [3]

$\displaystyle{\bf{H}}$
$\displaystyle=\sum\limits_{l=1}^{L}\alpha_{l}\Lambda_{\rm{r}}(\phi_{l}^{\rm{r}})\Lambda_{\rm{t}}(\phi_{l}^{\rm{t}}){\bf{a}}_{\rm{r}}(\phi_{l}^{\rm{r}}){\bf{a}}_{\rm{t}}^{H}(\phi_{l}^{\rm{t}})$
(2)
$\displaystyle\approx\alpha_{\rm{LoS}}\Lambda_{\rm{r}}(\phi_{\rm{LoS}}^{\rm{r}})\Lambda_{\rm{t}}(\phi_{\rm{LoS}}^{\rm{t}}){\bf{a}}_{\rm{r}}(\phi_{\rm{LoS}}^{\rm{r}}){\bf{a}}_{\rm{t}}^{H}(\phi_{\rm{LoS}}^{\rm{t}}),$

where $\alpha_{\rm{LoS}}$ $(\alpha_{l})$ is the gain of the LoS $(l^{th})$
path, $\phi_{\rm{LoS}}^{\rm{t}}$ $(\phi_{l}^{\rm{t}})$ and
$\phi_{\rm{LoS}}^{\rm{r}}$ $(\phi_{l}^{\rm{r}})$ are the azimuth angles of
departure and arrival (AoDs/AoAs) for the LoS $(l^{th})$ path, respectively.
The functions $\Lambda_{\rm{t}}(\phi_{\rm{LoS}}^{\rm{t}})$ and
$\Lambda_{\rm{r}}(\phi_{\rm{LoS}}^{\rm{r}})$ represent the antenna element
gain for the transmitter and receiver at the corresponding angle,
respectively. The antenna array response vectors ${\bf{a}}_{\rm{t}}$ and
${\bf{a}}_{\rm{r}}$ are introduced to capture the spatial correlation
characteristics for MIMO communications. For an $N$-element uniform linear
array (ULA), the far-field array response vector ${\bf{a}}(\phi)$ can be
written as

$\displaystyle{\bf{a}}(\phi)=\frac{1}{\sqrt{N}}[1,e^{j\frac{2\pi}{\lambda}d{\rm{sin}}\phi},\cdots,e^{j(N-1)\frac{2\pi}{\lambda}d{\rm{sin}}\phi}]^{T},$
(3)

where $d$ denotes the element spacing, and $\lambda$ denotes the carrier
wavelength. Note that the far-field channel in (2) has a low rank, i.e., it
has very limited DoFs. Therefore, with much fewer RF chains than the number of
antennas, the classical hybrid precoding architecture can fully utilize the
limited spatial DoFs to achieve the near-optimal spectrum efficiency.

Figure 1: The near-field geometric channel model of two non-parallel arrays.

However, with the extremely large number of antennas in both the transmitter
and receiver in XL-MIMO, receivers may lie in the near-field region of the
transmitter. Then, the far-field LoS channel based on planar wave assumptions
in (2) is no longer accurate to describe the LoS channel in the near-field
region, where the channel has to be modeled by spherical wave assumptions. As
shown in Fig. 1, based on the geometric free space assumption [4], the near-
field LoS channel can be modeled as

${\bf{H}}=\left[\begin{matrix}\alpha_{11}e^{-j\frac{2\pi}{\lambda}r_{11}}&\cdots&\alpha_{1N_{\rm{t}}}e^{-j\frac{2\pi}{\lambda}r_{1N_{\rm{t}}}}\\\
\vdots&\ddots&\vdots\\\
\alpha_{N_{\rm{r}}1}e^{-j\frac{2\pi}{\lambda}r_{N_{\rm{r}}1}}&\cdots&\alpha_{N_{\rm{r}}N_{\rm{t}}}e^{-j\frac{2\pi}{\lambda}r_{N_{\rm{r}}N_{\rm{t}}}}\\\
\end{matrix}\right],$ (4)

where $r_{pq}$ and $\alpha_{pq}$ denote the distance and normalized path gain
between $p^{th}$ receiver antenna and $q^{th}$ transmitter antenna,
respectively.

It is worth noting that ULA is considered in this paper for simplicity, and it
is straightforward to deploy the more realistic planar arrays and the main
results remain the same222We also assume that the number of antennas for both
the transmitter and receiver is large. This scenario can happen in backhauls
or internet of vehicles in future 6G..

###### Proposition 1

The far-field channel model in (2) is a particular case of the near-field
channel in (4) when the array apertures $(N_{\rm{r}}-1)d$ and
$(N_{\rm{t}}-1)d$ can be negligible compared with the communication distance
$r$.

###### Proof:

The proof is provided in Appendix A. ∎

Moreover, it can be obtained that the rank of the far-field LoS channel in (2)
is only $1$. As a contrast, the rank of the near-field LoS channel in (4) can
be huge, or the channel can even be full rank, which indicates a significantly
increased multiplexing gain compared with the far-field LoS channel. To
investigate the capacity gain in the near-field region, the available DoFs
need to be approximated.

##  III DoFs and Capacity Analysis in Near-Field

In this section, the theoretical analysis of DoFs and channel capacity in the
near-field region is provided. In Section III-A, we analyze the increased
available DoFs by introducing eigenproblems in the electromagnetic wave
theory. In Section III-B, to avoid the complicated process of channel
decomposition, an estimation of the channel capacity in the near-field region
is provided. Finally, we investigate the capacity curve in the near-field
region in Section III-C.

###  III-A DoFs Estimation of the Near-field LoS Channel

It is difficult to directly analyze the DoFs, to be precise, the singular
values, of the near-field LoS channel. It has been proved that, through the
Nyquist space sampling theorem, the half-wavelength spaced antennas can be
equivalently modeled as continuous apertures without loss of accuracy.
Therefore, we could generalize the classical discrete antenna arrays to
continuous apertures. By incorporating the analysis method in electromagnetic
field theory, we are able to estimate the DoFs in the near-field region.

In [14], the DoFs of a pair of _parallel_ positioned continuous linear arrays
are researched. However, the assumption of parallel arrays is hard to be
guaranteed in real communication systems. Therefore, the DoFs of _non-
parallel_ linear arrays are needed to be analyzed. In this paper, we consider
a pair of _non-parallel_ positioned continuous linear arrays333We assume that
the arrays are modeled in 3D space, where the linear array is along with the
y-direction with the angle of $\theta$ or $\phi$. The width of the array in
other directions is small and can be neglected. Therefore, the current on the
antenna array is not restricted along with the array. as in Fig. 1. The angle
between the array and vertical line is denoted by $\theta$ and $\phi$ for
transmitter and receiver, respectively. The centers of the two arrays are
positioned on the same level, with separation distance denoted by $r$. We
consider a location vector denoted by ${\bm{r}}$. The area within the
transmitter and receiver is denoted by $S_{T}$ and $S_{R}$, respectively.
Since the arrays are modeled as continuous apertures, we can assume the
monochromatic source $\psi({\bm{r}}_{\rm{T}})$ on the transmitter Tx that
generate waves $\phi({\bm{r}}_{\rm{R}})$ on the receiver array Rx, which
derives from Green’s function as

$\displaystyle\phi({\bm{r}})=\int_{S_{T}}G({\bm{r}},{\bm{r}}_{\rm{T}})\psi({\bm{r}}_{\rm{T}})d{\bm{r}}_{\rm{T}},$
(5)

where $G({\bm{r}},{\bm{r}}_{\rm{T}})$ denotes the corresponding Green’s
function. Based on the spherical wave assumptions in the near-field region,
the possible waves resulting from a single point source ${\bm{r}}_{1}$ in free
space can be modeled as

$\displaystyle
G({\bm{r}},{\bm{r}}_{1})=\frac{\exp(-jk|{\bm{r}}-{\bm{r}}_{1}|)}{4\pi|{\bm{r}}-{\bm{r}}_{1}|},$
(6)

where the nagative sign inside the exponential represents the outgoing
directions. Similar to singular value decomposition (SVD), we introduce the
function as

$\displaystyle
K({\bm{r}}_{T}^{\prime},{\bm{r}}_{T})=\int_{S_{R}}G^{*}({\bm{r}}_{R},{\bm{r}}_{T}^{\prime})G({\bm{r}}_{R},{\bm{r}}_{T})d{\bm{r}}_{R}.$
(7)

Therefore, following the formulation in [14], we could derive a series of
eigenproblems as

$\displaystyle\upsilon\psi({\bm{r}}_{T})=\int_{S_{T}}K({\bm{r}}_{T}^{\prime},{\bm{r}}_{T})\psi({\bm{r}}_{T}^{\prime})d{\bm{r}}_{T}^{\prime},$
(8)

where $\upsilon$ denotes the eigenvalue corresponding to the function
$\psi(\cdot)$.

###### Proposition 2

The solution to the eigenproblem in (8) could be further simplified as

$\displaystyle\upsilon_{n}\psi_{n}(c_{y},\xi_{T})=\int_{-1}^{1}\frac{{\rm{sin}}[c_{y}(\xi_{T}-\xi_{T}^{\prime})]}{\pi(\xi_{T}-\xi_{T}^{\prime})}\psi_{n}(c_{y},\xi_{T}^{\prime})d\xi_{T}^{\prime},$
(9)

where $\psi_{n}(c_{y},\xi_{T})$ represents a series of complete orthonormal
basis sets, in which $\xi_{T}$ and $\xi_{T}^{\prime}$ represent the scaled
length with given $\xi_{T}=\eta_{T}/((N_{t}-1)d\cos\theta)$. The dimensionless
parameter $c_{y}$ represents the equivalent bandwidth along the array aperture
defined as

$\displaystyle
c_{y}=\frac{\pi(N_{t}-1)(N_{r}-1)d^{2}\cos\theta\cos\phi}{2\lambda r}.$ (10)

###### Proof:

The proof is provided in Appendix B. ∎

Based on the equation in (9), it can be proved that the prolate spheroidal
wave functions (PSWFs) are the solutions to the eigenproblem [17]. The
normalized distinct eigenvalues follow the inequation as

$\displaystyle 1\geq\upsilon_{0}>\upsilon_{1}>\upsilon_{2}>\cdots>0.$ (11)

Figure 2: Calculated and estimated singular values for a pair of parallel 256
ULA antennas. Colored solid lines represent the calculated singular values
based on SVD on different distances, while black dashed lines are derived with
eigenvalues of PSWFs of equal channel power for corresponding distance.

In the electromagnetic field theory, the eigenvalues represent the coupling
coefficients between orthogonal basis functions, which are equivalent to the
square of singular values through SVD in communications. Therefore, the
singular values of the near-field LoS channel can be approximated with the
eigenvalues of PSWFs. If we consider a pair of parallel arrays, the
eigenvalues of PSWFs are consistent with the singular values derived from SVD
for a pair of parallel ULA as in Fig. 2. It indicates that the DoFs can be
estimated without complicated SVD when the size of channel matrix ${\bf{H}}$
becomes huge as the number of antennas scales up. It is worth noting that, in
the beginning, the eigenvalues fall off slowly. However, there is a critical
value after which the eigenvalues of PSWFs (or singular values of the channel)
fall off exponentially, which can also be derived with the PSWFs as

$\displaystyle
N_{\rm{DoF}}(r)\approx\frac{2}{\pi}c_{y}=\frac{(N_{\rm{t}}-1)(N_{\rm{r}}-1)d^{2}\cos\theta\cos\phi}{\lambda
r}.$ (12)

As illustrated in (12), as the transmitter-receiver distance decreases, the
critical value increases, which enables the communication systems to transmit
multiple data streams. For special parallel cases, when setting $\theta=0$ and
$\phi=0$, the estimated DoFs can be simplified as

$\displaystyle
N_{\rm{DoF}}(r)\approx\frac{(N_{\rm{t}}-1)(N_{\rm{r}}-1)d^{2}}{\lambda r},$
(13)

which is consistent with the results in [14]. Note that
$(N_{\rm{t}}-1)d\cos\theta$ represents the effective length of the transmitter
array projected along the y-axis. Therefore, a pair of non-parallel arrays is
equivalent to the parallel arrays with the effective length projected
perpendicularly to the line of array centers for DoFs estimation.

It has been theoretically proved that, unlike classical far-field LoS channel,
the spatial DoFs significantly scales up as the distance decreases in the
near-field LoS channel. It is expected that channel capacity can be enhanced
by utilizing the increasing DoFs for spatial multiplexing in near-field
communications.

###  III-B Channel Capacity Analysis

As mentioned above, the near-field channel differs from the far-field channel
in that the DoFs significantly increase within the LoS path. The channel
capacity can also benefit from the increased DoFs. If we consider a MIMO
system model as in (1), the channel capacity can be formulated as

$\displaystyle C$
$\displaystyle=\mathop{{\rm{max}}}\limits_{{\bf{F}}}~{}{\rm{log}}_{2}\left|{\bf{I}}+\frac{1}{\sigma_{n}^{2}}{\bf{H}}{\bf{F}}{\bf{F}}^{H}{\bf{H}}^{H}\right|$
(14)
$\displaystyle=\sum_{i=1}^{\min(N_{t},N_{r})}{\rm{log}}_{2}\left(1+\frac{p_{i}}{\sigma_{n}^{2}}\lambda_{i}^{2}\right),$

where $\lambda_{i}$ denotes the $i^{th}$ singular value of channel matrix,
$p_{i}$ denotes the power allocated to $i^{th}$ sub-channel according to the
classical water-filling method [18]. ${\bf{F}}$ denotes the fully-digital
precoding matrix with the power constraints.

To find more insights, we consider a case where the array apertures are
parallel positioned. Through perfect transmission power control, we could
assume the large-scale fading can be ignored, thus the Frobenius norm of the
channel matrix share the same value over different distances $r$. According to
the estimated singular values of PSWFs in (9), the capacity of the near-field
LoS channel can be rewritten as

$\displaystyle C$
$\displaystyle\mathop{\approx}\limits^{(a)}\sum_{i=1}^{N_{\rm{DoF}}}{\rm{log}}_{2}\left(1+\frac{p_{i}}{\sigma_{n}^{2}}\lambda_{i}^{2}\right)$
(15)
$\displaystyle\mathop{\approx}\limits^{(b)}N_{\rm{DoF}}{\rm{log}}_{2}\left(1+\frac{P_{\rm{tot}}P_{\rm{H}}}{\sigma_{n}^{2}N_{\rm{DoF}}^{2}}\right)$
$\displaystyle\approx\frac{(N_{\rm{t}}-1)(N_{\rm{r}}-1)d^{2}}{\lambda r}$
$\displaystyle\times\log_{2}\left(1+\frac{P_{\rm{tot}}P_{\rm{H}}\lambda^{2}r^{2}}{\sigma_{n}^{2}(N_{\rm{t}}-1)^{2}(N_{\rm{r}}-1)^{2}d^{4}}\right),$

where $N_{\rm{DoF}}$ denotes the DoFs in the near-field LoS channel as in
(12), $P_{\rm{H}}$ denotes the power of the channel. Approximation (a) is
obtained by assuming that the first $N_{\rm{DoF}}$ sub-channels take up the
main power of all the sub-channels. Approximation (b) is obtained by assuming
the singular values fall off slowly within the dominant $N_{\rm{DoF}}$ sub-
channels and thus equal power is allocated to each sub-channel. According to
Fig. 2, the assumptions above are reasonable for a pair of parallel ULA. Here
we have three ways to obtain the channel capacity in the near-field region, by
precisely SVD and water-filling method, by assuming an equal power allocation
as in (15) and by water-filling method using eigenvalues of PSWFs. The
simulation results are illustrated in Fig. 3.

Figure 3: Calculated and estimated channel capacity in the near-field region.
Orange line represents the calculated channel capacity, which the purple and
red lines denote the estimated channel capacity with DoFs and eigenvalues of
PSWFs repectively. The total transmitted SNR is set to be 15 dB.

It is shown from Fig. 3 that, the capacity significantly increases as the
transmission distance decreases. Compared with the classical estimation of
DoFs as in [14], the estimation with PSWFs eigenvalues approximates better to
the channel capacity for large transmission distance $r$. It reveals a
theoretical method to fastly estimate the channel capacity in the near-field
region without calculation-consuming SVD process. Although the simulation is
carried out under the strict assumptions that the arrays have to be parallel
positioned, the method can also be generalized to non-parallel arrays which
removes the degeneracy of eigenvalues as illustrated in [19]. The capacity
analysis of non-parallel linear arrays in the near-field region is also one
direction for future research.

###  III-C Discussions on Channel Capacity

Based on the approximation results in Fig. 3, it is worth noting that channel
capacity does not monotonically decrease as the distance scales up. We explain
it with the approximation result in equation (15). The denominator within the
logarithm contains square of DoFs while the logarithm is also multiplied by
DoFs. By derivation, the capacity is maximized when DoFs achieve
$N_{\rm{DoF}}^{*}$ which satisfies

$\displaystyle\left(\frac{N_{\rm{DoF}}^{*2}\sigma_{n}^{2}}{P_{\rm{tot}}P_{\rm{H}}}+1\right){\rm{log}}_{2}\left(1+\frac{P_{\rm{tot}}P_{\rm{H}}}{N_{\rm{DoF}}^{*2}\sigma_{n}^{2}}\right)=\frac{2}{{\rm{ln}}2}.$
(16)

Through numerical calculations, the solution to the transcendental equation in
(16) can be obtained by
$N_{\rm{DoF}}^{*}\approx\sqrt{0.255\frac{P_{\rm{tot}}P_{\rm{H}}}{\sigma_{n}^{2}}}$.
The transmitter-receiver distance $r$ can also be estimated with
$N_{\rm{DoF}}^{*}$. The special phenomenon originates from the spread of the
channel power. For near-field communications, although the DoFs scale up as
the distance decreases, the energy of the channel is also averagely allocated
to more orthogonal sub-channels, reducing the equivalent signal-noise ratio
(SNR) within each sub-channel. Therefore, the capacity for each sub-channel
decreases as the DoFs increase, resulting in non-monotonic increasing capacity
as distance decreases444It is worth noting that, here we assume the large-
scale fading is not considered in our model. When large-scale fading is
incorporated, the capacity will be monotone increasing as the distance
decreases..

From Fig. 2 and Fig. 3, as the communication distance decreases, the capacity
of near-field LoS channel dramatically (but not monotonically) increases.
Therefore, it is expected to utilize the increase DoFs to improve the spectrum
efficiency in the near-field region.

##  IV Proposed DAP Architecture

With the small DoFs of the far-field LoS channel, the classical far-field
hybrid precoding with a reduced number of RF chains could achieve the near-
optimal spectrum efficiency. However, with the increased DoFs of the near-
field LoS channel, classical hybrid precoding with limited RF chains is unable
to approach the optimal spectrum efficiency. According to our best knowledge,
an architecture that can flexibly change the number of RF chains to match the
increased DoFs in the near-field region is still a blank.

To efficiently utilize the DoFs of the near-field LoS channel, fully-digital
precoding is able to achieve the optimal spectrum efficiency regardless of in
the far-field region or near-field region. Nevertheless, fully-digital
precoding requires the number of RF chains equal to the number of transmitter
antennas, which leads to the unaffordable power consumption in millimeter-wave
and terahertz bands. It is worth noting that the mobile receiver may sometimes
be located in the far-field region and be located in the near-field region at
other times, resulting in varying distance-related DoFs of the MIMO channel as
illustrated in (12). Actually, the extra activated RF chains more than the
DoFs of the channel can be turned off without significant loss of spectrum
efficiency. Motivated by this observation, we propose a DAP architecture that
can flexibly control the number of activated RF chains to match the varying
DoFs, which can significantly increase the spectrum efficiency in the near-
field region. The proposed DAP architecture is shown in Fig. 4.

Figure 4: Comparison of the precoding architecture: (a) Classical sub-
connected hybrid precoding architecture; (b) Proposed DAP architecture.

Specifically, the sub-connected architecture is adopted in the proposed DAP
architecture to save unnecessary power consumptions as in [20]. In the DAP
architecture, each antenna element only connects to one RF chain. To enable a
variable number of RF chains, extra RF chains are equipped in the transmitter
and receiver and each RF chain can be configured to active or inactive. In
addition, each antenna element can dynamically select one arbitrary RF chain
through a selection circuit. The RF chains without any connection can be
regarded as inactive, thus enabling the control of the number of RF chains.

The advantage of introducing the selection circuit is two-fold. First, the
selection pattern between RF chains and antennas enables the choice of the
active RF chain numbers, which can be adaptively adjusted according to the
available channel DoFs. Second, the selection pattern also provides a new
dimension to design the precoding, which can further enhance the spectrum
efficiency.

Based on the proposed DAP structure, the received signal in equation (1) can
be rewritten as

$\displaystyle{\bf{y}}={\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}}{\bf{F}}_{\rm{D}}{\bf{s}}+{\bf{n}},$
(17)

where the $N_{\rm{t}}\times N_{\rm{s}}$ selection matrix ${\bf{F}}_{\rm{S}}$
represents the adjustable selection matrix between array elements and
activated RF chains. Each entity of ${\bf{F}}_{\rm{S}}$ is assigned either $0$
or $1$. Since each antenna element can only select one RF chain, each row of
${\bf{F}}_{\rm{S}}$ contains one element that equals $1$. Besides, since the
analog precoder is implemented by analog phase shifters, ${\bf{F}}_{\rm{A}}$
has an extra constant norm constraint. Different from the classical hybrid
precoding architecture, the introduction of selection matrix changes the size
of analog precoder, which is a $N_{\rm{t}}\times N_{\rm{t}}$ diagonal matrix
with constant norm. The feasible set of the analog precoder can be written as

$\displaystyle{\cal{F}}\triangleq\left\\{{\rm{diag}}(\widetilde{\theta}_{1},\cdots,\widetilde{\theta}_{N_{\rm{t}}})~{}\Big{|}~{}\widetilde{\theta}_{i}\in\mathbb{C},~{}\left|\widetilde{\theta}_{i}\right|=1,~{}\forall
i\right\\}.$ (18)

###### Remark 1

Unlike the classical hybrid precoding architecture where the number of data
streams $N_{\rm{s}}$ is fixed, the proposed DAP architecture could flexibly
adjust the number of data streams $N_{\rm{s}}$. Therefore, the optimization of
the number of RF chains $N_{\rm{s}}$ should also be taken into account.

When considering the precoding algorithm to maximize the spectrum efficiency,
we can assume that an optimal combiner is employed at the receiver side [3].
Thus, the spectrum efficiency can be written as

$\displaystyle
R=\log_{2}\left(\left|{\bf{I}}+\frac{1}{\sigma_{\rm{n}}^{2}}{\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}}{\bf{F}}_{\rm{D}}{\bf{F}}_{\rm{D}}^{H}{\bf{F}}_{\rm{S}}^{H}{\bf{F}}_{\rm{A}}^{H}{\bf{H}}^{H}\right|\right).$
(19)

Unlike the previous work which assumes an equal power allocation over
different sub-channels [20], we consider a power allocation process according
to the classical water-filling method, which will be detailed described in the
following section.

##  V Proposed DAP Algorithm

In this section, we propose a DAP algorithm for the proposed DAP structure to
improve the spectrum efficiency for near-field XL-MIMO communications. In
Section V-A, we provide an overview of the DAP algorithm. The design of
digital precoder, selection matrix, and analog precoder are introduced from
Section V-B to Section V-E.

###  V-A Overview of the DAP Algorithm

Based on the proposed DAP structure, the key problem is to determine the
number of data streams $N_{\rm{s}}$. As $N_{\rm{s}}$ increases, the proposed
DAP architecture approaches the fully-digital precoding architecture, which is
considered to be ideal in terms of spectrum efficiency with high power
consumption. According to the analysis in Section III, the DoFs can be well
estimated in the near-field region. Similar to the hybrid precoding scheme in
the far-field region, a reduced number of RF chains can be deployed to match
the DoFs in the near-field region to reduce power consumption as well as
achieve the near-optimal spectrum efficiency. With fixed data streams
$N_{\rm{s}}$, the optimization can be formulated as

$\displaystyle{{\cal P}:}~{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{{\bf{F}}_{\rm{A}},{\bf{F}}_{\rm{S}},{\bf{F}}_{\rm{D}}}~{}R({\bf{F}}_{\rm{A}},{\bf{F}}_{\rm{S}},{\bf{F}}_{\rm{D}})$
$\displaystyle~{}~{}~{}\,{\rm{s.t.}}~{}C_{1}:\|{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}}{\bf{F}}_{\rm{D}}\|_{F}^{2}\leq
P_{\rm{tot}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\,C_{2}:{\bf{F}}_{\rm{A}}\in\cal{F}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\,C_{3}:({\bf{F}}_{\rm{S}})_{ij}\in\\{0,1\\},~{}\forall
i,j$ (20)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\,C_{4}:{\rm{diag}}({\bf{F}}_{\rm{S}}{\bf{F}}_{\rm{S}}^{H})={\bf{1}}_{N_{t}},$

where $\cal{F}$ is the feasible set for analog precoder in (18). The precoding
algorithm can be summarized into three stages. First, the number of data
streams as well as the number of activated RF chains $N_{\rm{s}}$ is
determined according to the DoFs for the near-field LoS channel. Second, we
determine the selection matrix ${\bf{F}}_{\rm{S}}$ by optimizing the
partitioning pattern of the arrays, which aims to choose the best pattern of
subarrays that maximizes the spectrum efficiency. Third, we can obtain the
corresponding analog precoder ${\bf{F}}_{\rm{A}}$ and digital precoder
${\bf{F}}_{\rm{D}}$ that satisfy the constraints in $C_{1}$ and $C_{2}$. The
proposed DAP algorithm can be summarized in Algorithm 1.

Algorithm 1 Proposed DAP Alogorithm.

0: Channel ${\bf{H}}$, $r$, $N_{\rm{t}}$ and $N_{\rm{r}}$.

0: Optimized data streams $N_{\rm{s}}^{\rm{opt}}$, digital precoder
${\bf{F}}_{\rm{D}}^{\rm{opt}}$, analog precoder ${\bf{F}}_{\rm{A}}^{\rm{opt}}$
and selection matrix ${\bf{F}}_{\rm{S}}^{\rm{opt}}$

1: Obtain the eigenvalues of PSWFs in (9) with distance $r$;

2: Set the data streams $N_{\rm{s}}^{\rm{opt}}$ by $N_{\rm{DoF}}$ calculated
by water-filling method with eigenvalues of PSWFs;

3: Obtain subarray sets
$\mathcal{S}_{1},\mathcal{S}_{2},\cdots,\mathcal{S}_{N_{\rm{s}}^{\rm{opt}}}$
by Algorithm 2;

4: Calculate ${\bf{F}}_{\rm{S}}^{\rm{opt}}$ according to (26)

5: Obtain ${\bf{F}}_{\rm{A}}^{\rm{opt}}$ by (35);

6: Obtain ${\bf{F}}_{\rm{D}}^{\rm{opt}}$ by (22);

7: return $N_{\rm{s}}^{\rm{opt}}$, ${\bf{F}}_{\rm{A}}^{\rm{opt}}$,
${\bf{F}}_{\rm{S}}^{\rm{opt}}$, ${\bf{F}}_{\rm{D}}^{\rm{opt}}$.

###  V-B Estimation of Available Data Streams $N_{\rm{s}}$

The analysis of channel DoFs is provided in Section III-A. In (12) and (13),
the DoFs can be estimated well for small distance $r$. However, the
eigenvalues of PSWFs offer a better approximation of the singular values of
the near-field LoS channel. Therefore, we adopt a water-filling process
utilizing the eigenvalues of PSWFs to obtain the DoFs. Precisely,
$N_{\rm{DoF}}$ is equal to the number of non-zero power allocated to the sub-
channels in the water-filling process. After obtaining $N_{\rm{DoF}}$, similar
to the classical hybrid precoding architectures, we set
$N_{\rm{s}}=N_{\rm{DoF}}$ to fully utilize the spatial multiplexing gain for
capacity improvement in the near-field region at minimum energy consumption.

###### Remark 2

The choice of $N_{\rm{s}}=N_{\rm{DoF}}$ is not the optimal choice, since the
increased number of RF chains could further approach to the fully-digital
precoding scheme. Yet it is an essential condition that $N_{\rm{s}}\geq
N_{\rm{DoF}}$ to utilize the spatial multiplexing gain. In the following
sections, we will compare the performance with the different number of data
streams to reveal the rationality of the choice of $N_{\rm{s}}$.

###  V-C Optimization of Digital Precoder ${\bf{F}}_{\rm{D}}$

With fixed number of data streams $N_{\rm{s}}$, to simplify the formulation in
(20), we define the effective channel
${\bf{H}}_{\rm{e}}\triangleq{\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}}$. The
optimization object can be rewritten as

$\displaystyle
R=\log_{2}\left(\left|{\bf{I}}+\frac{1}{\sigma_{\rm{n}}^{2}}{\bf{H}}_{\rm{e}}{\bf{F}}_{\rm{D}}{\bf{F}}_{\rm{D}}^{H}{\bf{H}}_{\rm{e}}^{H}\right|\right).$
(21)

By assuming the SVD of effective channel
${\bf{H}}_{\rm{e}}={\bf{U}}_{\rm{e}}{\bf{\Sigma}}_{\rm{e}}{\bf{V}}_{\rm{e}}^{H}$,
we can obtain the solution of digital precoder

$\displaystyle{\bf{F}}_{\rm{D}}^{\rm{opt}}={\bf{V}}_{\rm{e}}{{\bf{\Gamma}}},$
(22)

where ${\bf{\Gamma}}$ denotes a diagonal power allocation matrix
${\bf{\Gamma}}={\rm{diag}}(\sqrt{p_{1}},\sqrt{p_{2}},\cdots,\sqrt{p_{N_{s}}})$.
Then the spectrum efficiency can be rewritten as

$\displaystyle
R=\sum_{i=1}^{N_{\rm{s}}}\log_{2}\left(1+\frac{\lambda_{i}^{2}({\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}})p_{i}}{\sigma_{\rm{n}}^{2}}\right),$
(23)

where $\lambda_{i}^{2}(\cdot)$ denotes the $i^{th}$ singular value of the
argument matrix. And the elements of ${\bf{\Gamma}}$ can be calculated through
the water-filling process in which

$\displaystyle
p_{i}=\left(\frac{1}{\mu}-\frac{\sigma_{\rm{n}}^{2}}{\lambda_{i}^{2}({\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}})}\right)^{+}.$
(24)

The auxiliary variable $\mu$ is determined by the power constraint that
$\sum_{i}p_{i}=P_{\rm{tot}}$. Therefore, the digital precoder
${\bf{F}}_{\rm{D}}$ can be easily obtained when analog precoder
${\bf{F}}_{\rm{A}}$ and selection matrix ${\bf{F}}_{\rm{S}}$ are fixed.

###  V-D Optimization of Selection Matrix ${\bf{F}}_{\rm{S}}$

The selection matrix ${\bf{F}}_{\rm{S}}$ represents the linking pattern of the
RF chains and array elements. Here we denote the set of array indices that
connect to the $i^{th}$ RF chain as

$\displaystyle\mathcal{S}_{i}=\left\\{n_{i1},n_{i2},\cdots,n_{|\mathcal{S}_{i}|}\right\\},$
(25)

where $|\mathcal{S}_{i}|$ denotes the number of entities in $\mathcal{S}_{i}$.
The selection matrix can be directly derived from $\mathcal{S}_{i}$ as

$[{\bf{F}}_{\rm{S}}]_{j,i}=\left\\{\begin{aligned}
~{}~{}1,~{}~{}~{}j\in\mathcal{S}_{i},\\\
~{}~{}0,~{}~{}~{}j\notin\mathcal{S}_{i}.\end{aligned}\right.$ (26)

To get the optimal partitioning of subarrays, according to Jensen’s
inequality, the formulation of spectrum efficiency can be further rewritten as

$\displaystyle R$
$\displaystyle=\sum_{i=1}^{N_{\rm{s}}}\log_{2}\left(1+\frac{\lambda_{i}^{2}({\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}})p_{i}}{\sigma_{\rm{n}}^{2}}\right)$
(27)
$\displaystyle\leq{N_{\rm{s}}}\log_{2}\left(1+\frac{1}{{N_{\rm{s}}}}\sum_{i=1}^{N_{\rm{s}}}\frac{\lambda_{i}^{2}({\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}})p_{i}}{\sigma_{\rm{n}}^{2}}\right).$

As illustrated in Section III-A, the singular values of the near-field LoS
channel are close to each other if $N_{\rm{s}}$ satisfies $N_{\rm{s}}\leq
N_{\rm{DoF}}$, which is consistent with our settings. Thus the approximation
is tight for near-field XL-MIMO communication systems. Then, the optimization
of selection matrix ${\bf{F}}_{\rm{S}}$ and analog precoder
${\bf{F}}_{\rm{A}}$ can be simplified as

$\displaystyle({\bf{F}}_{\rm{A}}^{\rm{opt}},{\bf{F}}_{\rm{S}}^{\rm{opt}})=$
$\displaystyle\mathop{{\rm{argmax}}}\limits_{{\bf{F}}_{\rm{A}},{\bf{F}}_{\rm{S}}}\sum_{i=1}^{N_{\rm{s}}}\lambda_{i}^{2}({\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}}).$
(28)

Utilizing the feature of ${\bf{F}}_{\rm{S}}$ that all elements in each row of
${\bf{F}}_{\rm{S}}$ are equal $0$ except one element, ${\bf{F}}_{\rm{S}}$ can
be reformulated as

$\displaystyle{\bf{F}}_{\rm{S}}={\bf{P}}_{\rm{S}}\widetilde{{\bf{F}}}_{\rm{S}},$
(29)

where ${\bf{P}}_{\rm{S}}$ is a permutation matrix and
$\widetilde{{\bf{F}}}_{\rm{S}}$ follows a block diagonal pattern of

$\widetilde{{\bf{F}}}_{\rm{S}}=\left[\begin{matrix}{\bf{1}}_{\mathcal{S}_{1}}&0&\cdots&0\\\
0&{\bf{1}}_{\mathcal{S}_{2}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&{\bf{1}}_{\mathcal{S}_{N_{\rm{s}}}}\\\ \end{matrix}\right],$ (30)

where ${\bf{1}}_{\mathcal{S}_{i}}$ denotes a vector with $|\mathcal{S}_{i}|$
elements that are equal to 1. Due to the property of permutation matrix, we
can switch the order of the diagonal analog precoder ${\bf{F}}_{\rm{A}}$ and
permutation matrix ${\bf{P}}_{\rm{S}}$ as

$\displaystyle{\bf{H}}{\bf{F}}_{\rm{A}}{\bf{P}}_{\rm{S}}\widetilde{{\bf{F}}}_{\rm{S}}={\bf{H}}{\bf{P}}_{\rm{S}}\widetilde{{\bf{F}}}_{\rm{A}}\widetilde{{\bf{F}}}_{\rm{S}},$
(31)

where $\widetilde{{\bf{F}}}_{\rm{A}}$ denotes the diagonal analog precoder
with a permutation operation. A brief proof is provided in Appendix C.

Due to the block diagonal feature of $\widetilde{{\bf{F}}}_{\rm{S}}$,
$\widetilde{{\bf{H}}}={\bf{H}}{\bf{P}}_{\rm{S}}$ can be classified by columns
as submatrices
$\widetilde{{\bf{H}}}=[{\bf{H}}_{\mathcal{S}_{1}},\cdots,{\bf{H}}_{\mathcal{S}_{\rm{Ns}}}]$.
Therefore, the optimization object in (28) can be rewritten as

$\displaystyle~{}~{}~{}\mathop{{\rm{max}}}\limits_{{\bf{F}}_{\rm{A}},{\bf{F}}_{\rm{S}}}\sum_{i=1}^{N_{\rm{s}}}\lambda_{i}^{2}({\bf{H}}{\bf{F}}_{\rm{A}}{\bf{F}}_{\rm{S}})$
(32)
$\displaystyle=\mathop{{\rm{max}}}\limits_{{\bf{F}}_{\rm{A}},{\bf{F}}_{\rm{S}}}\sum_{i=1}^{N_{\rm{s}}}\lambda_{i}^{2}({\bf{H}}{\bf{P}}_{\rm{S}}\widetilde{{\bf{F}}}_{\rm{A}}\widetilde{{\bf{F}}}_{\rm{S}})$
$\displaystyle=\mathop{{\rm{max}}}\limits_{{\bf{F}}_{\rm{A}},{\bf{F}}_{\rm{S}}}\left\|{\bf{H}}{\bf{P}}_{\rm{S}}\widetilde{{\bf{F}}}_{\rm{A}}\widetilde{{\bf{F}}}_{\rm{S}}\right\|_{F}^{2}$
$\displaystyle=\mathop{{\rm{max}}}\limits_{{\bf{F}}_{\rm{A}},{\bf{F}}_{\rm{S}}}\left\|\left[{\bf{H}}_{\mathcal{S}_{1}}{\bf{f}}_{1},\cdots,{\bf{H}}_{\mathcal{S}_{N_{\rm{s}}}}{\bf{f}}_{N_{\rm{s}}}\right]\right\|_{F}^{2}$
$\displaystyle\mathop{\approx}\limits^{(a)}\sum_{i=1}^{N_{\rm{s}}}\lambda_{1}({\bf{H}}_{\mathcal{S}_{i}}^{H}{\bf{H}}_{\mathcal{S}_{i}}),$

where ${\bf{f}}_{i}$ denotes non-zero elements of the $i^{th}$ colomn of
$\widetilde{{\bf{F}}}_{\rm{A}}\widetilde{{\bf{F}}}_{\rm{S}}$. The
approximation (a) is derived by omitting the constant modulus constraints of
${\bf{F}}_{\rm{A}}$. Therefore, the optimization object can be formulated as
the summation of the largest singular value of submatrices. However, if we
perform exhaustive searching to find all the sets $\mathcal{S}_{i}$ that
maximize the spectrum efficiency, the complexity will be unaffordable.
Therefore, we propose a low-complexity algorithm to optimize the selection
matrix ${\bf{F}}_{\rm{S}}$ by utilizing the normalized Minkowski
$\ell_{1}$-norm.

It has been proved that Minkowski $\ell_{1}$-norm gives a good approximation
of the largest singular value [21] as

$\displaystyle\lambda_{1}({\bf{R}}_{\mathcal{S}})\approx\hat{\lambda}_{1}({\bf{R}},{\mathcal{S}})=\frac{1}{|\mathcal{S}|}\sum_{i\in\mathcal{S}}\sum_{j\in\mathcal{S}}|[{\bf{R}}_{i,j}]|.$
(33)

Therefore, we could perform a low-complex greedy searching process to maximize
the average Minkowski $\ell_{1}$-norm for each partitioning set
$\mathcal{S}_{i}$. The proposed near-field subarray partitioning algorithm is
summarized in Algorithm 2.

Algorithm 2 Near-Field Subarray Partitioning Alogorithm.

0: Channel ${\bf{H}}$, $N_{\rm{s}}$, $N_{\rm{bound}}$ and $N_{\rm{t}}$.

0: $\mathcal{S}_{1},\mathcal{S}_{2},\cdots,\mathcal{S}_{N_{\rm{s}}}$

1: ${\bf{R}}={\bf{H}}^{H}{\bf{H}}$, $\mathcal{S}_{\rm{sel}}=\varnothing$,
$n_{\rm{group}}=\lfloor\frac{N_{\rm{t}}}{N_{\rm{s}}}\rfloor$

2: Initialize $\mathcal{S}_{i}=\\{i\cdot n_{\rm{group}}\\}$,
$\mathcal{S}_{\rm{sel}}\leftarrow\mathcal{S}_{\rm{sel}}\cup\\{i\cdot
n_{\rm{group}}\\}$, for $i=1,2,\cdots,N_{\rm{s}}$

3: for $k=1:N_{\rm{t}}-N_{\rm{s}}$ do

4:
$\\{i_{k},j_{k}\\}=\mathop{{\rm{argmax}}}\limits_{i\in\mathcal{S}_{\rm{sel}},j\notin\mathcal{S}_{\rm{sel}}}|[{\bf{R}}]_{i,j}|$

5:
$\hat{r}=\mathop{{\rm{argmax}}}\limits_{r\in\\{1,\cdots,N_{\rm{s}}\\}}\hat{\lambda}_{1}({\bf{R}},\mathcal{S}_{r}\cup\\{j_{k}\\})-\hat{\lambda}_{1}({\bf{R}},\mathcal{S}_{r})$

6: $\mathcal{S}_{\rm{sel}}\leftarrow\mathcal{S}_{\rm{sel}}\cup j_{k}$,
$\mathcal{S}_{\hat{r}}\leftarrow\mathcal{S}_{\hat{r}}\cup j_{k}$

7: if $|\mathcal{S}_{\hat{r}}|\geq N_{\rm{bound}}$ then

8:
$\hat{m}=\mathop{{\rm{argmin}}}\limits_{m\in\mathcal{S}_{\hat{r}}}\sum_{n\in\mathcal{S}_{\hat{r}}}|{\bf{R}}_{m,n}|$

9:
$\hat{r}^{\prime}=\mathop{{\rm{argmax}}}\limits_{r^{\prime}\neq{\hat{r}}}\hat{\lambda}_{1}({\bf{R}},\mathcal{S}_{r^{\prime}}\cup\hat{m})-\hat{\lambda}_{1}({\bf{R}},\mathcal{S}_{r^{\prime}})$

10: $\mathcal{S}_{\hat{r}}\leftarrow\mathcal{S}_{\hat{r}}\backslash\hat{m}$,
$\mathcal{S}_{\hat{r}^{\prime}}\leftarrow\mathcal{S}_{\hat{r}^{\prime}}\cup\hat{m}$

11: end if

12: end for

13: for $l=1:N_{\rm{s}}$ do

14:
$\hat{m}=\mathop{{\rm{argmin}}}\limits_{m\in\mathcal{S}_{l}}\sum_{n\in\mathcal{S}_{l}}|{\bf{R}}_{m,n}|$

15:
$\hat{r}^{\prime}=\mathop{{\rm{argmax}}}\limits_{r^{\prime}}\hat{\lambda}_{1}({\bf{R}},\mathcal{S}_{r^{\prime}}\cup\hat{m})-\hat{\lambda}_{1}({\bf{R}},\mathcal{S}_{r^{\prime}})$

16: $\mathcal{S}_{\hat{r}}\leftarrow\mathcal{S}_{\hat{r}}\backslash\hat{m}$,
$\mathcal{S}_{\hat{r}^{\prime}}\leftarrow\mathcal{S}_{\hat{r}^{\prime}}\cup\hat{m}$

17: end for

18: return $\mathcal{S}_{1},\mathcal{S}_{2},\cdots,\mathcal{S}_{N_{\rm{s}}}$

Different from other dynamic subarray design algorithms in [22, 23, 24], we
aim to obtain subarrays with similar scales. As illustrated in Section III-A,
the singular values of the near-field LoS channel are close to each other.
Thus the scale of subarrays should be similar to generate equal beamforming
gain. First, we initialize all the sets ${\mathcal{S}}$ with uniformly
distributed antennas. Then, we add the antennas to sets one by one. We select
the $j^{th}$ antenna with the largest absolute value of $|{\bf{R}}_{ij}|$ in
which $j^{th}$ antenna is not added yet. If the cardinal of sets exceeds the
maximum scale $N_{\rm{bound}}$, we remove the $m^{th}$ antenna with the least
contribution and add it to the set with largest improvement. Finally, we check
all the sets one by one to remove the least contributor, which aims to
eliminate the influence of the manual initialization.

###### Remark 3

It is worth noting that, as the number of data streams $N_{\rm{s}}$ increases,
the heuristic dynamic subarray searching algorithm proposed in [22, 23] may
result in a situation where some subarrays are totally empty. The degradation
of subarrays causes inefficient usage of spatial DoFs in the near-field
region, making the spectrum efficiency far less than optimal. Another greedy
searching algorithm proposed in [24] aims to maximize the largest singular
value of each submatrix one by one. In this way, the submatrices can hardly
remain balanced to maximize the sum rate of all sub-channels. Therefore, the
proposed partitioning algorithm is more suitable for near-field communication
scenarios.

Since in each iteration, one antenna is added into one set, the convergence of
the proposed algorithm is guaranteed. In the worst case where one set quickly
achieves the maximum size, the complexity of the algorithm is estimated as
$\mathcal{O}(N_{\rm{t}}^{2}+N_{\rm{t}}N_{\rm{bound}})$ as for comparing
operations and $\mathcal{O}(N_{\rm{s}}N_{\rm{t}}N_{\rm{bound}})$ as for add
operations.

###  V-E Optimization of Analog Precoder ${\bf{F}}_{\rm{A}}$

As illustrated in (32), the optimal analog precoder corresponding to the
$i^{th}$ subarray can be formulated as

$\displaystyle\angle({\bf{f}}_{{\rm{A}},\mathcal{S}_{i}}^{\rm{opt}})=\angle({\bf{v}}_{\mathcal{S}_{i}}),$
(34)

where ${\bf{v}}_{\mathcal{S}_{i}}$ is the singular vector corresponding to the
largest singular value of ${\bf{H}}_{\mathcal{S}_{i}}$. Therefore, each array
${\bf{f}}_{{\rm{A}},\mathcal{S}_{i}}^{\rm{opt}}$ can be determined satisfying
the constraints on the norm. Finally, the formulation of the analog precoder
can be written as

$\displaystyle{\bf{F}}_{\rm{A}}^{\rm{opt}}={\bf{P}}_{\rm{S}}{\rm{diag}}\left(\left[({\bf{f}}_{{\rm{A}},\mathcal{S}_{1}}^{\rm{opt}})^{T},\cdots,({\bf{f}}_{{\rm{A}},\mathcal{S}_{\rm{Ns}}}^{\rm{opt}})^{T}\right]\right){\bf{P}}_{\rm{S}}^{T}.$
(35)

It is worth noting that, due to the constant norm constraints of
${\bf{F}}_{\rm{A}}$, the close-form solution solution in (32) is an
approximation of the optimal solution without constraints. However, the
approximation often induces little performance degradation.

##  VI Simulation Results

In this section, to verify the effectiveness of the proposed DAP architecture,
we evaluate the performance of spectrum efficiency as well as the energy
efficiency compared with classical sub-connected hybrid precoding and
classical fully-connected precoding algorithms. Moreover, we investigate the
performance for different choices of data streams $N_{\rm{s}}$ in the near-
field XL-MIMO communication systems.

In the simulations, we consider a single-user XL-MIMO communication scenario.
Both the transmitter and receiver are equipped with ULA with $256$ elements
spacing of half wavelength. The carrier wave is set to $100$ GHz, with wave
length of $\lambda=3$ mm. It can be calculated that the Rayleigh distance is
around $100$ m under the settings. The transmission distance varies within the
Rayleigh distance in our simulations. For the wireless channel, as mentioned
before, we adopt the deterministic LoS channel in (4) and assume that the
large-scale channel fading can be ignored through perfect power control.

###  VI-A Evaluating the Performance of Spectrum Efficiency

The spectrum efficiency for the proposed DAP architecture over different
distances is shown in Fig. 5. The SNR $=P_{\rm{tot}}/\sigma_{n}^{2}$ is set to
$30$ dB. The baselines include fully-digital precoding, fully-connected hybrid
precoding with 8, 4 RF chains using an alternating minimization algorithm[25],
and sub-connected hybrid precoding with 8, 4 RF chains using the SIC-based
algorithm[20]. When the transmitter-receiver distance is small, the proposed
scheme can achieve about $40\%$ increase in the achievable spectrum efficiency
compared to classical hybrid precoding schemes with 8 RF chains. This is
because the proposed architecture can benefit from the extra DoFs in the near-
field region. When the transmitter-receiver distance grows, the fully-
connected hybrid precoding outperforms our proposed architecture. The reason
lies in that there still exists a gap between the fully-connected architecture
and sub-connected architecture with the same number of RF chains, as
illustrated in Section VI-C. When the transmitter-receiver distance grows
larger, the DoFs reduce and our proposed architecture could no longer obtain
the multiplexing gain from the extra RF chains.

Figure 5: The comparison of spectrum efficiency over different distances.
Figure 6: The comparison of spectrum efficiency over different SNRs.

Another simulation is carried out over different SNRs in Fig. 6. The SNR
ranges from $20$ dB to $30$ dB. From Fig. 6, the proposed DAP architecture
outperforms other precoding architectures with fixed RF chains.

###  VI-B Evaluating the Performance of Energy Efficiency

Figure 7: The comparison of energy efficiency over different distances.

With the increasing $N_{\rm{s}}$, the proposed DAP is approaching the
classical fully-digital precoding scheme. The advantage of the DAP
architecture lies in that it significantly increases the spectrum efficiency
utilizing the enlarging DoFs in the near-field region. Also, the scalable RF
chains can match the DoFs to save unnecessary energy consumption. Therefore,
another critical performance factor, energy efficiency, needs to be considered
in the simulations. For simplicity, we claim a downlink power consumption
model at the base station, and the energy efficiency can be formulated as

$\displaystyle\eta=\frac{R}{P_{\rm{T}}+N_{\rm{s}}P_{\rm{RF}}+N_{\rm{PS}}P_{\rm{PS}}+N_{\rm{SW}}P_{\rm{SW}}+N_{\rm{t}}P_{\rm{PA}}},$
(36)

where $P_{\rm{T}}$ is the constant circuit power, $P_{\rm{RF}}$, $P_{\rm{PS}}$
and $P_{\rm{SW}}$ denote the power consumption for each RF chain, phase
shifter and switch respectively, $N_{\rm{PS}}$ and $N_{\rm{SW}}$ denote the
number of phase shifters and switches respectively. $P_{\rm{T}}=2500$ mW,
$P_{\rm{RF}}=160$ mW, $P_{\rm{PS}}=10$ mW, $P_{\rm{SW}}=10$ mW and
$P_{\rm{PA}}=30$ mW are assumed. The baselines also include the fully-digital
precoding, fully-connected and sub-connected hybrid precoding schemes. In
fully-connected hybrid precoding architectures, the number of phase shifters
is equal to $N_{\rm{PS}}^{\rm{FC}}=N_{\rm{t}}N_{\rm{RF}}$. While in sub-
connected hybrid precoding architectures, the number of phase shifters equals
to the number of antennas $N_{\rm{PS}}^{\rm{SC}}=N_{\rm{t}}$. Classical fully-
connected and sub-connected hybrid architectures are implemented without
switches, while in the proposed DAP architectures $N_{\rm{SW}}=N_{\rm{s}}$.
The simulation results of energy efficiency are shown in Fig. 7. We can
observe from Fig. 7 that, the proposed DAP architecture also outperforms all
other precoding architectures, which reveals the superiority of the proposed
DAP architecture in terms of the energy consumption for transmitting the same
amount of data.

###  VI-C Evaluating the Effect of Different $N_{\rm{s}}$

In the proposed DAP algorithm, we have assumed that $N_{\rm{s}}$ is equal to
$N_{\rm{DoF}}$. To verify the rationality of choice, we compare different
choices of $N_{\rm{s}}$ with $N_{\rm{s}}=N_{\rm{DoF}},12,8,4$. The
transmission distance is $2$ m. The precoding baselines considered include the
proposed DAP architecture and fully-connected precoding architecture with
alternating precoding algorithm [25]. As shown in Fig. 8(a), the spectrum
efficiency increases with increasing $N_{\rm{s}}$. With different
$N_{\rm{s}}$, the performance of fully-connected precoding scheme outperforms
our proposed sub-connected DAP architecture, which is consistent with previous
works. When $N_{\rm{s}}$ is chosen to $N_{\rm{DoF}}$, the fully-connected
precoding scheme approaches the optimal fully-digital precoding scheme, which
reveals that the choice of $N_{\rm{s}}$ can be considered optimal. It is worth
noting that, there is still a gap between the fully-connected precoding and
proposed sub-connected DAP precoding scheme. However, as illustrated in Fig.
8(b), the proposed DAP architecture with $N_{\rm{s}}=N_{\rm{DoF}}$ outperforms
all other settings in terms of energy efficiency. Therefore, the proposed DAP
architecture can achieve spectrum efficiency and energy efficiency
improvements with $N_{\rm{s}}$ equal to $N_{\rm{DoF}}$.

(a) Spectrum efficiency

(b) Energy efficiency

Figure 8: The comparison of different $N_{\rm{s}}$ over spectrum efficiency
and energy efficiency. The number of RF chains is chosen within
$N_{\rm{DoF}}$, 12, 8, 4. The precoding architectures considered include
fully-digital precoding architecture, proposed DAP architecture, fully-
connected precoding architecture. The SNR varies from 30 dB to 40 dB.

##  VII Conclusions

In this paper, the significantly increased DoFs and the channel capacity in
the near-field region are theoretically analyzed. To utilize the increased
spatial DoFs, a DAP architecture and the corresponding precoding algorithm
have been proposed for the XL-MIMO communication scenario, in which the number
of activated RF chains can be flexibly adjusted to match the increased DoFs in
the near-field region. Simulations results show that, compared with the
classical sub-connected hybrid precoding architecture, the proposed DAP
architecture can efficiently utilize the increased DoFs to significantly
improve the spectrum efficiency in the near-field region. Different from
alleviating problems brought by the near-field effect, this paper has revealed
that the near-field property can be exploited to improve the performance of
XL-MIMO communications, which may provide another methodology to meet the
increasing demand of communication capacity. Our future research will focus on
the capacity enhancement employing reconfigurable intelligent surface (RIS)
considering the near-field effect.

## Appendix A

As shown in Fig. 1, according to the near-field channel in (4), the difference
of the distance between $j$th and ($j+n$)th transmitter antenna with $i$th
receiver antenna is written as

$\displaystyle r_{pq}-r_{p(q+n)}$
$\displaystyle=r_{pq}\left(1-\sqrt{1+(\frac{nd}{r_{pq}})^{2}-2\frac{nd}{r_{pq}}\sin\phi_{pq}}\right)$
(37) $\displaystyle\mathop{\approx}\limits^{(a)}nd\sin\phi_{pq},$

where $\phi_{pq}$ denotes the azimuth angle between the $p^{th}$ receiver
antenna and the $q$th transmitter antenna. Approximation (a) is derived based
on the Taylor Series expansion that $\sqrt{1-x}\approx 1-\frac{1}{2}x$ when
$x=2\frac{nd}{r_{ij}}\sin\phi_{ij}$ is small enough. The approximation holds
when $r_{ij}\gg nd$, which means that the size of the receiver antenna can be
neglected compared with the communication distance. In the far-field range,
the azimuth of different elements and the normalized path loss in the same
antenna can be regarded as the same. Then, the $p^{th}$ row of $\bf{H}$ can be
approximated as

$\displaystyle{\bf{H}}_{p,:}=\gamma_{\rm{LoS}}\alpha
e^{-j\frac{2\pi}{\lambda}r_{p1}}[1,e^{j\frac{2\pi}{\lambda}d\sin\phi_{t}},\cdots,e^{j(N_{t}-1)\frac{2\pi}{\lambda}d\sin\phi_{t}}],$
(38)

where $\phi_{\rm{t}}$ and $\phi_{\rm{r}}$ represents the azimuth angle for all
elements of the transmitter and receiver. The $q^{th}$ column of $\bf{H}$ can
be formulated similarly. Thus, the near-field channel can be approximated as

$\displaystyle{\bf{H}}=\gamma_{\rm{LoS}}\alpha
e^{-j\frac{2\pi}{\lambda}r_{11}}{\bf{a}}_{r}(\phi_{\rm{r}}){\bf{a}}_{t}(-\phi_{\rm{t}})^{H}.$
(39)

Finally, by assuming $\alpha_{\rm{LoS}}=\alpha
e^{-j\frac{2\pi}{\lambda}r_{11}}$ , $\phi_{\rm{LoS}}^{\rm{r}}=\phi_{\rm{r}}$
and $\phi_{\rm{LoS}}^{\rm{t}}=-\phi_{\rm{t}}$, the near-field LoS channel in
(4) is degraded to LoS channel model in (2).

## Appendix B

Figure 9: Non-parallel continuous linear arrays in the near-field region.

As shown in Fig. 9, we consider a pair of non-parallel continuous arrays in
which the center of both transmitter and receiver are in the same horizontal
line. The length of the transmitter and receiver is $L_{\rm{t}}$ and
$L_{\rm{r}}$ respectively. And the slant angle for transmitter and receiver is
$\theta$ and $\phi$. Starting from the definition in (7), we aim to maximize
the power of the derived wave function

$\displaystyle|g|^{2}$
$\displaystyle=\int_{S_{R}}\phi^{*}({\bm{r}}_{\rm{R}})\phi({\bm{r}}_{\rm{R}})d{\bm{r}}_{\rm{R}}$
(40)
$\displaystyle=\int_{S_{R}}\int_{S_{T}}\int_{S_{T}}G^{*}({\bm{r}}_{\rm{R}},{\bm{r}}_{\rm{T}}^{\prime})\psi^{*}({\bm{r}}_{\rm{T}}^{\prime})G({\bm{r}}_{\rm{R}},{\bm{r}}_{\rm{T}})$
$\displaystyle~{}~{}~{}~{}\times\psi({\bm{r}}_{\rm{T}})d{\bm{r}}_{\rm{T}}^{\prime}d{\bm{r}}_{\rm{T}}d{\bm{r}}_{\rm{R}}$
$\displaystyle=\int_{S_{T}}\psi^{*}({\bm{r}}_{T}^{\prime})\int_{S_{T}}K({\bm{r}}_{T}^{\prime},{\bm{r}}_{T})\psi({\bm{r}}_{T})d{\bm{r}}_{T}^{\prime}d{\bm{r}}_{T}.$

Then, to maximize the value of $|g|^{2}$, $\psi({\bm{r}}_{T})$ has to be
chosen as the eigenfunction corresponding to the largest eigenvalue [14]. The
eigenfunction can be defined as

$\displaystyle|g|^{2}\psi({\bm{r}}_{T})$
$\displaystyle=\int_{S_{T}}K({\bm{r}}_{T}^{\prime},{\bm{r}}_{T})\psi({\bm{r}}_{T}^{\prime})d{\bm{r}}_{T}^{\prime}$
(41)
$\displaystyle=\int_{S_{T}}\int_{S_{R}}\frac{{\rm{exp}}(jk|{\bm{r}}_{R}-{\bm{r}}_{T}|){\rm{exp}}(-jk|{\bm{r}}_{R}-{\bm{r}}_{T}^{\prime}|)}{(4\pi)^{2}|{\bm{r}}_{R}-{\bm{r}}_{T}||{\bm{r}}_{R}-{\bm{r}}_{T}|}$
$\displaystyle~{}~{}~{}~{}\times
d{\bm{r}}_{R}\psi({\bm{r}}_{T}^{\prime})d{\bm{r}}_{T}^{\prime}.$

It is worth noting that, the green function $G({\bm{r}}_{1},{\bm{r}}_{2})$ is
determined by the spherical wave assumption, which is also a basic assumption
in the near-field communications. Then, since
$|{\bm{r}}_{R}-{\bm{r}}_{T}||{\bm{r}}_{R}-{\bm{r}}_{T}|$ in the denominator is
only a scalar factor, it can be viewed as $r^{2}$ if the transmission distance
is relatively large. Then we concentrate on the nominator which contains a
product of two phase items. Consider the variable $\eta_{\rm{t}}$ and
$\eta_{\rm{r}}$ denote the distance from the center of transmitter and
receiver respectively. Then we have

$\displaystyle|{\bm{r}}_{R}-{\bm{r}}_{T}|=$ (42)
$\displaystyle~{}~{}\sqrt{(r+\eta_{T}\sin\theta-\eta_{R}\sin\phi)^{2}+(\eta_{T}\cos\theta-\eta_{R}\cos\phi)^{2}}.$

Here we adopt the paraxial assumption that the distance between the two arrays
are relatively large compared with their apertures. The first three items in
Taylor series $\sqrt{1+x}\approx 1+\frac{1}{2}x-\frac{1}{8}x^{2}$ are adopted.
Then we have

$\displaystyle|{\bm{r}}_{R}-{\bm{r}}_{T}|$ $\displaystyle\approx
r+\eta_{T}\sin\theta-\eta_{R}\sin\phi$ (43)
$\displaystyle~{}~{}~{}~{}+\frac{1}{2r}(\eta_{T}\cos\theta-\eta_{R}\cos\phi)^{2}.$

The product of the two phase items are formulated as

$\displaystyle~{}~{}~{}\exp(jk|{\bm{r}}_{R}-{\bm{r}}_{T}|){\rm{exp}}(-jk|{\bm{r}}_{R}-{\bm{r}}_{T}^{\prime}|)$
(44)
$\displaystyle=\exp\big{\\{}jk\big{(}(\eta_{T}-\eta_{T}^{\prime})\sin\theta+\frac{\cos^{2}\theta}{2r}(\eta_{T}^{2}-\eta_{T}^{{}^{\prime}2})$
$\displaystyle~{}~{}~{}~{}-\frac{\cos\theta\cos\phi}{r}(\eta_{T}-\eta_{T}^{\prime})\eta_{R}\big{)}\big{\\}}.$

We assume a focusing function as

$\displaystyle
F_{T}(\eta_{T})=\exp\left\\{jk\left(\eta_{T}\sin\theta+\frac{\cos^{2}\theta}{2r}\eta_{T}^{2}\right)\right\\}.$
(45)

Then we have

$\displaystyle\psi(\eta_{T})=F_{T}(\eta_{T})\alpha_{T}(\eta_{T}),$ (46)

where $\alpha_{T}(\eta_{T})$ are the solution of the formulation below

$\displaystyle|g|^{2}(4\pi
r)^{2}\alpha_{T}(\eta_{T})=\int_{-L_{T}/2}^{L_{T}/2}\int_{-L_{R}/2}^{L_{R}/2}$
(47)
$\displaystyle~{}~{}~{}~{}\exp\left\\{jk\frac{\cos\theta\cos\phi}{r}(\eta_{T}-\eta_{T}^{\prime})\eta_{R}\right\\}d\eta_{R}\alpha_{T}(\eta_{T}^{\prime})d\eta_{T}^{\prime}.$

It can be easily derived that the integral

$\displaystyle~{}~{}~{}\int_{-L_{R}/2}^{L_{R}/2}{\rm{exp}}\left\\{jk\frac{\cos\theta\cos\phi}{r}(\eta_{T}-\eta_{T}^{\prime})\eta_{R}\right\\}d\eta_{R}$
(48) $\displaystyle=\frac{\lambda
r}{\cos\theta\cos\phi}\cdot\frac{\sin\Omega_{T}(\eta_{T}-\eta_{T}^{\prime})}{\pi(\eta_{T}-\eta_{T}^{\prime})},$

where $\Omega_{T}=\frac{kL_{R}\cos\theta\cos\phi}{2r}$. Thus, the
eigenfunction can be rewritten as

$\displaystyle|g|^{2}(4\pi r)^{2}\alpha_{T}(\eta_{T})$
$\displaystyle=\int_{-L_{T}/2}^{L_{T}/2}\frac{\lambda r}{\cos\theta\cos\phi}$
(49)
$\displaystyle~{}~{}~{}~{}\times\frac{\sin\Omega_{T}(\eta_{T}-\eta_{T}^{\prime})}{\pi(\eta_{T}-\eta_{T}^{\prime})}\alpha_{T}(\eta_{T}^{\prime})d\eta_{T}^{\prime}.$

Compared with the definition of prolate spheroidal wave functions in (9),
parameter $c_{y}=\Omega_{T}\frac{L_{T}}{2}=\frac{\pi
L_{T}L_{R}\cos\theta\cos\phi}{2\lambda r}$ is proportional to
$\cos\theta\cos\phi$. Since $N_{\rm{DoF}}$ can be approximated as
$N_{\rm{DoF}}=\frac{2}{\pi}c_{y}$. Thus, the formulation in (12) can be
obtained. Noting that the derivative process is under paraxial assumptions.
When the arrays are non-parallel, the paraxial degeneracy will be removed,
resulting different eigenvalues $\upsilon_{i}$ in (9) when
$i\leq\frac{2}{\pi}c_{y}$. However, since the eigenvalues $\upsilon_{i}$ fall
off rapidly in an exponential manner for $i>\frac{2}{\pi}c_{y}$, the DoFs
still remain unchanged for $\theta$ and $\phi$ far smaller than $\pi/2$.

## Appendix C

For a diagonal matrix ${\bf{F}}_{\rm{A}}$ and permuation matrix
${\bf{P}}_{\rm{S}}$, since it has been proved that for any matrix ${\bf{A}}$,
${\bf{B}}={\bf{P}}_{\rm{S}}^{T}{\bf{A}}{\bf{P}}_{\rm{S}}$ has the same
diagonal entities as ${\bf{A}}$ (but may with different orders) [26]. Thus,
for any diagonal matrix ${\bf{F}}_{\rm{A}}$, there exists a diagonal matrix
$\widetilde{\bf{F}}_{\rm{A}}$ that has the same entities with
${\bf{F}}_{\rm{A}}$ subjecting to

$\displaystyle{\bf{F}}_{\rm{A}}{\bf{P}}_{\rm{S}}=$
$\displaystyle~{}~{}{\bf{P}}_{\rm{S}}\left[{\bf{P}}_{\rm{S}}^{T}{\bf{F}}_{\rm{A}}{\bf{P}}_{\rm{S}}\right]={\bf{P}}_{\rm{S}}\widetilde{\bf{F}}_{\rm{A}}.$

Therefore, the order of the permuation matrix ${\bf{P}}_{\rm{S}}$ and diagonal
analog precoder ${\bf{F}}_{\rm{A}}$ can be switched for further processings.

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