# Channel Estimation and Reconstruction in Fluid Antenna System: Oversampling
is Essential

Wee Kiat New, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, Hao Xu, Member, IEEE,
Farshad Rostami Ghadi, Member, IEEE, Ross Murch, Fellow, IEEE, and Chan-Byoung
Chae, Fellow, IEEE The work of W. K. New, K. K. Wong, H. Xu, and F. R. Ghadi
is supported by the Engineering and Physical Sciences Research Council (EPSRC)
under grant EP/W026813/1.The work of R. Murch is supported by the Hong Kong
Research Grants Council Area of Excellence grant AoE/E-601/22-R.The work of
C.-B. Chae is supported by the IITP/NRF grant funded by the Ministry of
Science and ICT (MSIT), Korea (No. 2021-0-02208, No. 2022R1A5A1027646).W. K.
New (${\rm a.new@ucl.ac.uk}$), K.-K. Wong (${\rm
kai\text{-}kit.wong@ucl.ac.uk}$), H. Xu (${\rm hao.xu@ucl.ac.uk}$), and F. R.
Ghadi (${\rm<EMAIL_ADDRESS>are with the Department of Electronic
and Electrical Engineering, University College London, London, WC1E 6BT,
United Kingdom. K.-K. Wong is also affiliated with Yonsei Frontier Lab.,
Yonsei University, Seoul 03722, Korea.R. Murch ($\rm<EMAIL_ADDRESS>is with
the Department of Electronic and Computer Engineering and Institute for
Advanced Study (IAS), The Hong Kong University of Science and Technology,
Clear Water Bay, Hong Kong SAR, China.C.-B. Chae ($\rm<EMAIL_ADDRESS>are with the School of Integrated Technology, Yonsei University, Seoul, 03722,
Korea.Corresponding author: Kai-Kit Wong.

###### Abstract

Fluid antenna system (FAS) has recently surfaced as a promising technology for
the upcoming sixth generation (6G) wireless networks. Unlike traditional
antenna system (TAS) with fixed antenna location, FAS introduces a flexible
component where the radiating element can switch its position within a
predefined space. This capability allows FAS to achieve additional diversity
and multiplexing gains. Nevertheless, to fully reap the benefits of FAS,
obtaining channel state information (CSI) over the predefined space is
crucial. In this paper, we explore the interaction between a transmitter
equipped with a traditional antenna and a receiver with a fluid antenna over
an electromagnetic-compliant channel model. We address the challenges of
channel estimation and reconstruction using Nyquist sampling and maximum
likelihood estimation (MLE) methods. Our analysis reveals a fundamental
tradeoff between the accuracy of the reconstructed channel and the number of
estimated channels, indicating that half-wavelength sampling is insufficient
for perfect reconstruction and that oversampling is essential to enhance
accuracy. Despite its advantages, oversampling can introduce practical
challenges. Consequently, we propose a suboptimal sampling distance that
facilitates efficient channel reconstruction. In addition, we employ the MLE
method to bound the channel estimation error by $\epsilon$, with a specific
confidence interval (CI). Our findings enable us to determine the minimum
number of estimated channels and the total number of pilot symbols required
for efficient channel reconstruction in a given space. Lastly, we investigate
the rate performance of FAS and TAS and demonstrate that FAS with imperfect
CSI can outperform TAS with perfect CSI.

###### Index Terms:

6G, fluid antenna system, channel estimation, channel reconstruction,
electromagnetic field, Nyquist sampling.

##  I Introduction

Fluid antenna system (FAS) represents the new paradigm of shape-and-position-
flexible antenna technologies that benefit from the new forms of
reconfigurable antennas such as liquid-based antennas [1, 2], reconfigurable
radio-frequency (RF) pixels [3, 4] or movable mechanical antenna structures
[5]. The concept of FAS was originally brought into wireless communication
systems in 2020 [6] and its application to the upcoming sixth-generation (6G)
was discussed in [7, 8, 9]. Unlike a traditional antenna system (TAS), where
the antenna location is fixed, FAS refers to any antenna structures that can
intelligently adapt their shape and position to reconfigure the gain,
radiation patterns, and other radiation characteristics to enhance
communication performance.

Recent efforts have treated FAS as a new degree of freedom (DoF) which is
dictated by its size and the number of flexible positions (a.k.a. ports).
Early attempts have been to investigate the fundamental performance limits of
FAS, revealing considerable gains over TAS [10, 11, 12, 13, 14, 15]. Most
recently, the diversity and multiplexing gain for the multiple-input multiple-
output (MIMO) FAS setup was also understood [16]. Promising results by
optimizing FAS in different setups can be found in [17, 18, 19, 20].
Furthermore, there has been great interest in combining FAS with other
technologies, such as reconfigurable intelligent surfaces [21], physical layer
security [22, 23], non-orthogonal multiple access [24, 25], full-duplex radios
[26], and index modulation [27, 28]. A more complete list of recent works is
included in [29].

However, most existing studies rely on full channel state information (CSI) to
fully exploit the benefits of FAS. Recent advancements in CSI acquisition have
greatly benefited from sophisticated signal processing methods such as array
processing [30], machine learning [31], the Bayesian learning framework [32],
and compressive sensing [33]. Exploitation of channel statistics, sparsity,
and correlation has also enhanced channel estimation performance [34, 35, 36].
These techniques, nonetheless, focus on scenarios with fixed radiating element
locations, thereby often limiting the number of channel estimations. In
contrast, in FAS scenarios, acquiring the CSI of a predefined space (i.e.,
full CSI) becomes challenging.

To obtain the full CSI, latest research has delved into the channel estimation
and reconstruction problem for FAS. For example, in [26], the authors devised
a sequential linear minimum mean square error (MMSE)-based channel estimation
method for large-scale fluid antenna-aided full-duplex cellular networks,
while [37] employed deep learning techniques to reconstruct the channels of
the unobserved ports based on the estimated channels from a few known ports.
Also, [38] used recurrent neural networks to address the channel
reconstruction and port selection issues in FAS. In scenarios where the
channel is sparse, [39] illustrated that low-sample-size sparse channel
reconstruction techniques are effective for obtaining the full CSI.
Additionally, [40] proposed a successive Bayesian reconstructor for estimating
the FAS channels and compressed sensing methods were utilized in [41].

Collectively, these studies indicate that traditional channel estimation
schemes designed for TAS are inadequate for FAS, as they would incur
unbearable pilot overhead due to the possibility of positioning the radiating
element at near-infinitely many locations. To reduce the pilot overhead,
state-of-the-art methods typically only estimate the channels at a few
locations and reconstruct (or extrapolate) the FAS channel over a finite space
based on these initial estimates. This can be achieved by leveraging the
spatial correlation of the channels, their sparsity, or any other implicit
structural characteristics. Despite considerable efforts, the minimum number
of estimated channels and the minimum distance between these estimated
channels required to perfectly reconstruct the FAS channel over a given space
remain unknown. This challenge persists partly because the analysis is
exceptionally complex, especially for the continuous FAS case (i.e., when the
number of ports is infinite). Note that most literature on FAS has observed
that increasing the number of estimated channels or decreasing the minimum
distance between the estimated channels typically enhances the accuracy of
channel reconstruction.

However, recent findings in holographic MIMO and electromagnetic information
theory present a contrasting perspective. Specifically, the functional DoF
represents the minimum number of samples required to reconstruct a given
electromagnetic field [42]. This metric is closely tied to the transmission
capacity of an electromagnetic system, as the maximum number of complex values
that can be transmitted within a single electromagnetic channel is equivalent
to the minimum number of required samples to reconstruct a given
electromagnetic field [42, 43]. It was thus argued in [42, 43, 44, 45] that
half-wavelength sampling suffices to perfectly reconstruct an arbitrary
electromagnetic field with any specified precision in the far-field because
the electromagnetic field is band-limited. This argument is supported by
analyses in the space-wave number domain or through Petersen-Middleton’s
sampling theorem, both of which are closely related to the Nyquist sampling
theorem. In the near-field, however, sampling at intervals smaller than half a
wavelength is considered beneficial [45].

In this paper, our aim is to bridge the research gap between the two
contrasting findings. Specifically, we consider a scenario in which a
transmitter equipped with a traditional antenna serves a receiver equipped
with a fluid antenna over an electromagnetic-compliant channel model. We
address the channel estimation and reconstruction problem in FAS using Nyquist
sampling and maximum likelihood estimation (MLE) methods. Through these
methods, we reveal a fundamental tradeoff between the accuracy of the
reconstructed channel and the number of estimated channels, highlighting that
oversampling is essential in FAS, even in far-field propagation. Furthermore,
we identify the minimum number of estimated channels and the suboptimal
distance between these estimated channels necessary for efficient
reconstruction of the FAS channel over a finite space. Then, we investigate
the rate of FAS with channel estimation and reconstruction problem. The key
contributions of this paper are summarized as follows:

  * •

We have developed an electromagnetic-compliant channel model for FAS for the
first time, covering both one-dimensional (1D) and two-dimensional (2D) fluid
antenna surfaces of arbitrary size. This model is applicable to RF pixel-based
fluid antennas, where the radiating elements are adjusted in discrete manner,
as well as to movable antennas or liquid-based fluid antennas, where the
number of ports approaches infinity, allowing the radiating element to be
repositioned continuously.

  * •

Utilizing the Nyquist sampling theorem, we demonstrate a fundamental tradeoff
between the accuracy of the reconstructed channel and the number of estimated
channels. This finding underscores the necessity of oversampling in FAS to
ensure accurate channel reconstruction. Furthermore, by employing the MLE
method, we show that the error of the estimated channels can be bounded within
a specific confidence interval (CI).

  * •

To strike a balance between the accuracy of the reconstructed channel and the
number of estimated channels, we propose a suboptimal sampling distance. This
strategy, combined with MLE, allows us to determine the minimum number of
estimated channels needed for a given space and obtain the number of pilot
symbols required to efficiently estimate and reconstruct the FAS channel.

  * •

We present comprehensive simulation results that demonstrate the practicality
and efficiency of estimating and reconstructing the FAS channel on both 1D and
2D fluid antenna surfaces. Additionally, our results highlight the
relationships between various parameters, such as sampling distance, number of
pilot symbols per sub-block, signal-to-noise ratio (SNR) and CI. More
importantly, we show that FAS with imperfect CSI can outperform TAS with
perfect CSI in a point-to-point scenario.

Note that FAS can be conceptually viewed as a holographic MIMO system with
antenna selection. The reconstructed electromagnetic field can be interpreted
as the reconstructed FAS channel (i.e., channels over a continuous space with
infinitely many ports), while the sampling points in the electromagnetic field
can be seen as the estimated channels in FAS. However, it is crucial to
distinguish that an electromagnetic field is proven to be band-limited (i.e.,
half-wavelength sampling sufficient for perfect reconstruction) and the
estimated FAS channels are affected by noise. Leveraging ports that are less
than half a wavelength apart can also be interpreted as oversampling.

The remainder is organized as follows. Section II presents the
electromagnetic-compliant channel model for FAS. In Section III, we discuss
the channel estimation and reconstruction methods while deriving important
parameters, relationships, tradeoffs, and rates. Simulation results are
provided in Section IV. Finally, the conclusion is drawn in Section V.

##  II Electromagnetic-Complaint Channel Model

This paper considers a point-to-point scenario in which a transmitter serves a
receiver. We assume that the transmitter is equipped with a traditional fixed
antenna, while the receiver employs a fluid antenna. Unless otherwise stated,
we assume that the fluid antenna surface is 2D and has a size of
$X\lambda\times Y\lambda$ where $\lambda$ denotes the carrier wavelength.
Furthermore, the fluid antenna has only one active radiating element. In a
liquid-based fluid antenna [2], the radiating element can move to any location
on the 2D fluid antenna surface. In the case of a pixel-based antenna [4],
this means that certain pixels can be activated to form an active radiating
element. Compared to existing works, this paper considers an electromagnetic-
compliant channel model to gain better understanding into the channel
estimation and reconstruction problem.111It is worth highlighting that [46,
47] are two pioneering works that established electromagnetic-compliant
channel models for holographic MIMO. Motivated by these remarkable
contributions, this paper adopts a similar approach to develop an
electromagnetic-compliant channel model for FAS, applicable to both 1D and 2D
surfaces.

According to [46], the electromagnetic nature of small-scale fading requires
that each realization channel, $h\left(x,y,z\right)$, satisfies the scalar
Helmholtz equation. This stipulates that

$\left(\nabla^{2}+\kappa^{2}\right)h\left(x,y,z\right)=0,$ (1)

where (1) is a second-order linear partial differential equation with constant
coefficients, $\nabla^{2}$ is the Laplace operator, and $\kappa$ is the
wavenumber. Based on their Fourier properties, it is evident from (1) that the
following condition must be met:

$k_{x}^{2}+k_{y}^{2}+k_{z}^{2}-\kappa^{2}=0,$ (2)

where we have used the definition of wave vector,
$\boldsymbol{\kappa}=k_{x}\hat{\bf x}+k_{y}\hat{\bf y}+k_{z}\hat{\bf z}$.
Given $k_{x}$ and $k_{y}$, we know that

$k_{z}=\pm
k_{z}\left(k_{x},k_{y}\right)=\pm\sqrt{\kappa^{2}-k_{x}^{2}-k_{y}^{2}}.$ (3)

By superposition, the general solution to (1) is given by [48]

$h\left(x,y,z\right)=\underset{\mathcal{D}\left(\kappa\right)}{\iint}H^{+}\left(k_{x},k_{y}\right)e^{j\left(k_{x}x+k_{y}y+k_{z}\left(k_{x},k_{y}\right)z\right)}\\\
+H^{-}\left(k_{x},k_{y}\right)e^{j\left(k_{x}x+k_{y}y-k_{z}\left(k_{x},k_{y}\right)z\right)}dk_{x}dk_{y},$
(4)

where $H^{+}\left(k_{x},k_{y}\right),H^{-}\left(k_{x},k_{y}\right)$ are
unknown constants and
$\mathcal{D}\left(\kappa\right)=\left\\{\left(k_{x},k_{y}\right)\in\mathbb{{R}}\;|\;k_{x}^{2}+k_{y}^{2}\leq\kappa^{2}\right\\}$.222By
assuming that $k_{x}^{2}+k_{y}^{2}\leq\kappa^{2}$, we are implicitly focusing
on the far-field propagation. In the near-field, where
$k_{x}^{2}+k_{y}^{2}>\kappa^{2}$, it is evident that oversampling is useful
[45].

The spatial autocorrelation function is given as

$c_{h}\left(x,y,z\right)\\\ =\iiint
h\left(x+x^{\prime},y+y^{\prime},z+z^{\prime}\right)h^{*}\left(x^{\prime},y^{\prime},z^{\prime}\right)dx^{\prime}dy^{\prime}dz^{\prime}.$
(5)

Consequently, the power spectral density in the wavenumber domain can be
derived as

$S_{h}\left(k_{x},k_{y},k_{z}\right)\\\ =\iiint
c_{h}\left(x,y,z\right)e^{-j\left(k_{x}x+k_{y}y+k_{z}z\right)}dxdydz.$ (6)

Using the inverse Fourier transform, the power spectral density in the
wavenumber domain can be converted back into the spatial autocorrelation
function in the spatial domain by

$c_{h}\left(x,y,z\right)=\frac{1}{\left(2\pi\right)^{3}}\iiint
S_{h}\left(k_{x},k_{y},k_{z}\right)\times\\\
e^{j\left(k_{x}x+k_{y}y+k_{z}z\right)}dk_{x}dk_{y}dk_{z}.$ (7)

Due to (1) and the translation-invariant property, applying the Helmholtz
operators to both sides of (5) results in

$\left(\nabla^{2}+\kappa^{2}\right)c_{h}\left(x,y,z\right)=0.$ (8)

By applying the Fourier transform to both sides of (8), then we obtain

$\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}-\kappa^{2}\right)S_{h}\left(k_{x},k_{y},k_{z}\right)=0.$
(9)

This implies that $S_{h}\left(k_{x},k_{y},k_{z}\right)=0$ except at the
wavenumber support, which is defined as

$S=\left\\{\left(k_{x},k_{y},k_{z}\right)\in\mathbb{{R}}^{3}\;|\;k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=\kappa^{2}\right\\}.$
(10)

Since $k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=\kappa^{2}$ and $k_{z}$ is determined by
$k_{x}$ and $k_{y}$, as shown in (2) and (3), respectively, we can simplify
$S_{h}\left(k_{x},k_{y},k_{z}\right)$ as

$\displaystyle S_{h}\left(k_{x},k_{y}\right)=$ $\displaystyle
A^{2}\left(k_{x},k_{y}\right)\delta\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}-\kappa^{2}\right),$
(11)

where $A^{2}\left(k_{x},k_{y}\right)$ is the non-negative real function. Using
$(\ref{eq:3})$, we can define

$A^{2}\left(k_{x},k_{y}\right)=\begin{cases}A_{+}^{2}\left(k_{x},k_{y}\right),k_{z}\left(k_{x},k_{y}\right)\geq
0,\\\
A_{-}^{2}\left(k_{x},k_{y}\right),k_{z}\left(k_{x},k_{y}\right)<0,\end{cases}\left(k_{x},k_{y}\right)\in\mathcal{{D}}.$
(12)

According to [49], the Dirac delta function can be split into the sum of two
delta functions as

$\delta\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}-\kappa^{2}\right)=\frac{\delta\left(k_{z}-k_{z}\left(k_{x},k_{y}\right)\right)}{2k_{z}\left(k_{x},k_{y}\right)}\\\
+\frac{\delta\left(k_{y}+k_{z}\left(k_{x},k_{y}\right)\right)}{2k_{z}\left(k_{x},k_{y}\right)}.$
(13)

Substituting (12) and (13) into (11), and then taking the inverse Fourier
transform with respect to (w.r.t.) $k_{z}$, we obtain

$\displaystyle\frac{1}{2\pi}\int$ $\displaystyle
S_{h}\left(k_{x},k_{y}\right)e^{j\left(k_{z}z\right)}dk_{z}$ (14)
$\displaystyle=\int_{k_{z}\geq
0}\frac{A_{+}^{2}\left(k_{x},k_{y}\right)e^{+j\left(k_{z}\left(k_{x},k_{y}\right)z\right)}}{4\pi
k_{z}\left(k_{x},k_{y}\right)}dk_{z}$
$\displaystyle+\int_{k_{z}<0}\frac{A_{-}^{2}\left(k_{x},k_{y}\right)e^{-j\left(k_{z}\left(k_{x},k_{y}\right)z\right)}}{4\pi
k_{z}\left(k_{x},k_{y}\right)}dk_{z}$ (15) $\displaystyle=\iint
c_{h}\left(x,y,z\right)e^{-j\left(k_{x}x+k_{y}y\right)}dxdy.$ (16)

Note that in (15), we can define

$S_{h}^{\pm}\left(k_{x},k_{y}\right)=\frac{A_{\pm}^{2}\left(k_{x},k_{y}\right)}{4\pi
k_{z}\left(k_{x},k_{y}\right)},\leavevmode\nobreak\
\left(k_{x},k_{y}\right)\in\mathcal{D}\left(\kappa\right),$ (17)

as the plane-wave spectrum.

According to [46], the random small-scale fading effect can be modeled as

$\displaystyle h\left(x,y,z\right)=$ $\displaystyle\frac{1}{2\pi}\iint
H\left(k_{x},k_{y}\right)$ $\displaystyle\times
e^{j\left(k_{x}x+k_{y}y+k_{z}\left(k_{x},k_{y}\right)z\right)}dk_{x}dk_{y},$
(18) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\iint\sum_{i=\pm}\sqrt{S_{h}^{i}\left(k_{x},k_{y}\right)}W_{i}\left(k_{x},k_{y}\right)$
$\displaystyle\times e^{j\left(k_{x}x+k_{y}y\pm
k_{z}\left(k_{x},k_{y}\right)z\right)}dk_{x}dk_{y}$ (19) $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\pi\kappa}}\underset{\mathcal{\mathcal{D}}\left(\kappa\right)}{\iint}\frac{e^{j\left(k_{x}x+k_{y}y\right)}}{\left(\kappa^{2}-k_{x}^{2}-k_{y}^{2}\right)^{1/4}}W_{+}\left(k_{x},k_{y}\right)$
$\displaystyle\times e^{+k_{z}\left(k_{x},k_{y}\right)z}dk_{x}dk_{y}$
$\displaystyle+\frac{1}{2\sqrt{\pi\kappa}}\underset{\mathcal{\mathcal{D}}\left(\kappa\right)}{\iint}\frac{e^{j\left(k_{x}x+k_{y}y\right)}}{\left(\kappa^{2}-k_{x}^{2}-k_{y}^{2}\right)^{1/4}}W_{-}\left(k_{x},k_{y}\right)$
$\displaystyle\times e^{-k_{z}\left(k_{x},k_{y}\right)z}dk_{x}dk_{y},$ (20)

in which (20) is obtained by considering a three-dimensional (3D) isotropic
channel and assuming that $h\left(x,y,z\right)$ has a unit power, which yields
$A_{\pm}\left(k_{x},k_{y}\right)=\frac{2\pi}{\sqrt{\kappa}}$. Also,
$W_{i}\left(k_{x},k_{y}\right)$ in (19) and (20) is the white Gaussian noise
field with unit variance, to model the stochastic nature of the
propagation.333In (20), it is evident that the $z$-direction only introduces a
phase difference. Therefore, no additional functional DoF is introduced, even
if a 3D fluid antenna surface is considered. With this, we can express the
unknown constants as

$H^{\pm}\left(k_{x},k_{y}\right)=\begin{cases}\sum_{i=\pm}\sqrt{{\scriptstyle
S_{h}^{i}\left(k_{x},k_{y}\right)}}W_{i}{\scriptstyle\left(k_{x},k_{y}\right)},&{\scriptstyle\left(k_{x},k_{y}\right)\in\mathcal{D}\left(\kappa\right),}\\\
0,&{\rm{otherwise}}.\end{cases}$ (21)

If we let $z=0$, then (20) can be rewritten as

$h\left(x,y,0\right)=\frac{1}{2\pi}\iint\sum_{i=\pm}\frac{\sqrt{\frac{\pi}{k}}}{\left(\kappa^{2}-k_{x}^{2}-k_{y}^{2}\right)^{1/4}}W_{i}\left(k_{x},k_{y}\right)\\\
\times e^{j\left(k_{x}x+k_{y}y\right)}dk_{x}dk_{y}.$ (22)

This implies

$S_{h}^{\pm}\left(k_{x},k_{y}\right)=\frac{\frac{\pi}{k}}{\left(\kappa^{2}-k_{x}^{2}-k_{y}^{2}\right)^{1/2}},\leavevmode\nobreak\
\left(k_{x},k_{y}\right)\in\mathcal{D}\left(\kappa\right),$ (23)

and we can further simplify (22) as

$h\left(x,y\right)=\frac{1}{2\pi}\iint\sum_{i=\pm}\sqrt{S_{h}^{i}\left(k_{x},k_{y}\right)}W_{i}\left(k_{x},k_{y}\right)\\\
\times e^{j\left(k_{x}x+k_{y}y\right)}dk_{x}dk_{y}.$ (24)

To compute $h\left(x,y\right)$, we divide the integration interval of (24)
uniformly with frequency spacing $\Delta_{k_{x}}=\frac{2\pi}{L_{x}}$ and
$\Delta_{k_{y}}=\frac{2\pi}{L_{y}}$ as

$h\left(x,y\right)=\sum_{\bar{y}=\frac{\kappa L_{y}}{2\pi}}^{\frac{\kappa
L_{y}}{2\pi}-1}\sum_{\bar{x}=\frac{\kappa L_{x}}{2\pi}}^{\frac{\kappa
L_{x}}{2\pi}-1}H_{\bar{x},\bar{y}}^{\pm}e^{j2\pi\left(\frac{\bar{x}}{L_{x}}x+\frac{\bar{y}}{L_{y}}y\right)},$
(25)

where

$H_{\bar{x},\bar{y}}^{i}=\\\
\int_{\frac{2\pi\bar{y}}{Ly}}^{\frac{2\pi\left(\bar{y}+1\right)}{L_{y}}}\int_{\frac{2\pi\bar{x}}{L_{x}x}}^{\frac{2\pi\left(\bar{x}+1\right)}{L_{y}}}\sum_{i=\pm}\sqrt{\frac{S_{h}^{i}\left(k_{x},k_{y}\right)}{2\pi}}W_{i}\left(k_{x},k_{y}\right)dk_{x}dk_{y}.$
(26)

Let us define

$s_{\bar{x},\bar{y}}^{i}\left(k_{x},k_{y}\right)=\begin{cases}\sqrt{\frac{S_{h}^{i}\left(k_{x},k_{y}\right)}{2\pi}},&\left(k_{x},k_{y}\right)\in\mathcal{\mathcal{D}\left(\kappa\right)},\\\
0,&{\rm{otherwise}}.\end{cases}$ (27)

Then (27) has an energy of

$\int\int
s_{\bar{x},\bar{y}}^{i}\left(k_{x},k_{y}\right)s_{x^{\prime},y^{\prime}}^{i}\left(k_{x},k_{y}\right)dk_{x}dk_{y}\\\
=\sigma_{\bar{x},\bar{y}}^{2}\delta\left(\bar{x}-x^{\prime}\right)\delta\left(\bar{y}-y^{\prime}\right).$
(28)

The variance, $\sigma_{\bar{x},\bar{y}}^{2}$, can be computed as

$\displaystyle\sigma_{\bar{x},\bar{y}}^{2}$
$\displaystyle=\frac{1}{4\pi}\int_{\frac{\bar{y}\lambda}{L_{y}}}^{\frac{\left(\bar{y}+1\right)\lambda}{L_{y}}}\int_{\frac{\bar{x}\lambda}{L_{x}}}^{\frac{\left(\bar{x}+1\right)\lambda}{L_{x}}}\frac{\boldsymbol{1}_{\mathcal{\mathcal{D}}\left(1\right)}\left(k_{x},k_{y}\right)}{\left(1-k_{x}{}^{2}-k_{y}{}^{2}\right)^{1/2}}dk_{x}dk_{y},$
(29)

for $\left(\bar{x},\bar{y}\right)\in\mathcal{E},$ where

$\mathcal{E}=\left\\{\left(\bar{x},\bar{y}\right)\in\mathbb{Z}^{2}\left|\left(\frac{\bar{x}\lambda}{L_{x}}\right)^{2}+\left(\frac{\bar{y}\lambda}{L_{y}}\right)^{2}\leq
1\right.\right\\}.$ (30)

It is known that when a 3D isotropic channel is observed over an infinite
line, the electromagnetic field is 2D. We can use similar steps to show
that444Note that $A_{\pm}\left(k_{x}\right)=2\sqrt{\pi}$ when
$h\left(x,y\right)$ has a unit power.

$h\left(x\right)=\sum_{l=\frac{\Omega L}{2\pi}}^{\frac{\Omega
L}{2\pi}-1}\sum_{i=\pm}H_{l}^{i}e^{j\frac{2\pi l}{L}x},$ (31)

where

$H_{l}^{i}=\int_{\frac{2\pi
l}{L}}^{\frac{2\pi\left(l+1\right)}{L}}\sqrt{\frac{S_{h}^{i}\left(k_{x}\right)}{2\pi}}W_{i}\left(k_{x}\right)dk_{x},$
(32)

and

$s_{l}^{i}\left(k_{x}\right)=\begin{cases}\sqrt{\frac{S_{h}^{i}\left(k_{x}\right)}{2\pi}},&\frac{2\pi
l}{L}\leq k_{x}\leq\frac{2\pi\left(l+1\right)}{L},\\\
0,&{\rm{otherwise}}.\end{cases}$ (33)

Note that (33) has an energy of

$\int
s_{l}^{i}\left(k_{x}\right)s_{m}^{i}\left(k_{x}\right)dk_{x}=\frac{\sigma_{l}^{2}}{2}\delta\left(l-m\right),$
(34)

and the variance, $\sigma_{l}^{2}$, can be computed as

$\sigma_{l}^{2}=\frac{1}{\pi}\left[\arcsin\left(\frac{l+1}{L}\right)-\arcsin\left(\frac{l}{L}\right)\right].$
(35)

##  III Channel Estimation and Reconstruction in FAS using Nyquist Sampling
and MLE Methods

Figure 1: A schematic of the proposed channel estimation and reconstruction
process within a coherence time.

As shown in Fig. 1, we consider a slow, flat fading channel within a coherence
time of $Z\gg 0$ digital symbols such that $Z=Z_{p}+Z_{q}$, where $Z_{p}$ and
$Z_{q}$ are the total number of symbols used for channel estimation and data
transmission, respectively. The total number of pilot symbols, $Z_{p}$, are
further divided into $N_{d}^{\star}$ sub-blocks and each sub-block is
dedicated to estimating the channel at a predetermined location represented by
$\left(n_{x}D_{x},n_{y}D_{y}\right)$, where $n_{w}\in\mathbb{Z}$ and $D_{w}$
represents the sampling distance in $w\in\left\\{x,y\right\\}$ dimension. Each
sub-block includes $z_{p}$ pilot symbols for channel estimation purposes
(e.g., using MLE). For brevity, we focus our analysis on the real part of the
channel in this section as similar methodology can be applied to the imaginary
part in parallel.

Without loss of generality, let us treat $h_{{\rm FAS}}\left(x,y\right)$ as
the real part of the channel of the FAS receiver at position
$\left(x,y\right)$. As it is infeasible to estimate $h_{{\rm
FAS}}\left(x,y\right)$ for $-\frac{X}{2}\lambda\leq x\leq\frac{X}{2}\lambda$
and $-\frac{Y}{2}\lambda\leq y\leq\frac{Y}{2}\lambda$ at every possible
location, we leverage the Nyquist sampling method for channel reconstruction.
Using this method, we can then determine the minimum value of $N_{d}^{\star}$
to efficiently reconstruct $\hat{h}_{{\rm FAS}}\left(x,y\right)$ from
$\hat{h}_{{\rm FAS}}\left(n_{x}D_{x},n_{y}D_{y}\right),\forall n_{x,}n_{y}$.
Furthermore, we adopt the MLE method to estimate $\hat{h}_{{\rm
FAS}}\left(n_{x}D_{x},n_{y}D_{y}\right)$. This technique ensures that the
estimation error for $\hat{h}_{{\rm FAS}}\left(n_{x}D_{x},n_{y}D_{y}\right)$
remains within a bound of $\varepsilon$, with a specific CI. This approach
ultimately enables us to determine the total number of pilot symbols required
for channel estimation and reconstruction in FAS. Subsequently, we can analyze
the effects of channel estimation and reconstruction on the rate of FAS.

###  III-A 2D Nyquist Sampling Method

To reconstruct $\hat{h}_{{\rm FAS}}\left(x,y\right)$, we employ the 2D Nyquist
sampling method. For ease of expositions, let us denote the spatial spectrum
of $h\left(x,y\right)$ as $H\left(k_{x},k_{y}\right)$. From the uncertainty
principles [50], it is known that

$H\left(k_{x},k_{y}\right)\in\mathcal{D}\left(\kappa\right)\Rightarrow
h\left(x,y\right)\in\mathbb{{R}}^{2}.$ (36)

In other words, a band-limited signal cannot be simultaneously a spatial-
limited signal. According to the Nyquist sampling theorem [51], if
$h\left(x,y\right)$ consists of no frequency higher than $\kappa$ in $k_{x}$
or $k_{y}$ dimensions, then $h\left(x,y\right)$ can be completely
reconstructed by uniformly sampling the signal with
$D_{x},D_{y}\leq\frac{2\pi}{2\kappa}=\frac{\lambda}{2}$ distance.

Practically speaking, nevertheless, a FAS receiver can only observe
$h\left(x,y\right)$ over a 2D fluid antenna surface, which has a finite space
of $X\lambda\times Y\lambda$. This is equivalent to multiplying
$h\left(x,y\right)$ by a 2D rectangular window function

$u\left(x,y\right)=\begin{cases}1,&-\frac{X}{2}\lambda\leq
x\leq\frac{X}{2}\lambda,-\frac{Y}{2}\lambda\leq y\leq\frac{Y}{2}\lambda,\\\
0,&{\rm{otherwise}.}\end{cases}$ (37)

As a result, the channel experienced at the FAS receiver is

$h_{{\rm FAS}}\left(x,y\right)=h\left(x,y\right)u\left(x,y\right).$ (38)

In the frequency domain, the spectrum of $h_{{\rm FAS}}\left(x,y\right)$ can
be expressed as555Note that the terms “frequency domain” and “wavenumber
domain” actually refer to the same conceptual domain. More concretely, the
term “frequency domain” is commonly used in signal processing, whereas
“wavenumber domain” is more frequently used in the context of physics.

$H_{{\rm FAS}}\left(k_{x},k_{y}\right)=H\left(k_{x},k_{y}\right)\circledast
U\left(k_{x},k_{y}\right),$ (39)

where $\circledast$ is the 2D convolution operator. It can be easily verified
that

$U\left(k_{x},k_{y}\right)=\frac{\sin\left(\frac{k_{x}}{2}X\lambda\right)}{\frac{k_{x}}{2}}\frac{\sin\left(\frac{k_{y}}{2}Y\lambda\right)}{\frac{k_{y}}{2}},$
(40)

where the domain of $U\left(k_{x},k_{y}\right)$ spans over the entire
$\mathbb{R}^{2}$.

Now, substituting (40) into (39) suggests that $H_{{\rm
FAS}}\left(k_{x},k_{y}\right)$ is no longer band-limited. This means that
perfect reconstruction of $h_{{\rm FAS}}\left(x,y\right)$ is impossible when
the FAS receiver only observes $h\left(x,y\right)$ over a finite space of
$X\lambda\times Y\lambda$. In other words, oversampling is essential in FAS to
improve the accuracy of the reconstructed channel.

This raises a fundamental tradeoff between the accuracy of reconstructed
channel and the number of estimated channels. Increasing the latter can
introduce hardware challenges and rate degradation. To strike a balance, we
propose a suboptimal method for channel reconstruction that aims to capture
most of the energy of the mainlobe of $H_{{\rm FAS}}\left(k_{x},k_{y}\right)$.
In particular, the spectral leakage of the mainlobe of
$U\left(k_{x},k_{y}\right)$ in the $k_{x}$\- and $k_{y}$-dimensions are
$\frac{2\pi}{X\lambda}$ and $\frac{2\pi}{Y\lambda}$, respectively [52].
Therefore, the spectral leakage of the mainlobe of $H_{{\rm
FAS}}\left(k_{x},k_{y}\right)$ should extend to $\kappa+\frac{2\pi}{X\lambda}$
and $\kappa+\frac{2\pi}{Y\lambda}$, respectively. This means that the
suboptimal sampling distance can be derived as

$\displaystyle D_{x,0}^{\star}$
$\displaystyle=\frac{2\pi}{2\left(\kappa+\frac{2\pi}{X\lambda}\right)}=\frac{\lambda}{2+\frac{2}{X}},$
(41) $\displaystyle D_{y,0}^{\star}$
$\displaystyle=\frac{2\pi}{2\left(\kappa+\frac{2\pi}{Y\lambda}\right)}=\frac{\lambda}{2+\frac{2}{Y}}.$
(42)

Intuitively, we can improve the channel reconstruction by further capturing
the spectrum energy up to the $d$-th sidelobe at a cost of smaller sampling
distance. Since the spectral spreading of the $d$-th sidelobe in the $k_{x}$\-
and $k_{y}$-dimensions are simply $\left(d+1\right)\frac{2\pi}{X\lambda}$ and
$\left(d+1\right)\frac{2\pi}{Y\lambda}$, respectively [52], the $d$-th
suboptimal sampling distance can be expressed as

$\displaystyle D_{x,d}^{\star}$
$\displaystyle=\frac{\lambda}{2+\frac{2}{X}\left(d+1\right)},$ (43)
$\displaystyle D_{y,d}^{\star}$
$\displaystyle=\frac{\lambda}{2+\frac{2}{Y}\left(d+1\right)}.$ (44)

Because the size of the 2D fluid antenna surface is fixed (e.g.,
$X\lambda\times Y\lambda$), we can determine the number of estimated channels
or sampling points for a given space, denoted as
$N_{d}^{\star}=N_{x,d}^{\star}N_{y,d}^{\star}$, where

$\displaystyle N_{x,d}^{\star}$
$\displaystyle=\left\lfloor\frac{X\lambda}{D_{x,d}^{\star}}\right\rfloor,$
(45) $\displaystyle N_{y,d}^{\star}$
$\displaystyle=\left\lfloor\frac{Y\lambda}{D_{y,d}^{\star}}\right\rfloor.$
(46)

Very importantly, $N_{x,d}^{\star}$ and $N_{y,d}^{\star}$ are the minimum
numbers of estimated channels that include the spectral leakage of the
mainlobe or $d$-th sidelobe of the spectrum in the $x$ and $y$ dimensions,
respectively.

The sampled signal can be expressed as

$h_{{\rm
FAS}}^{s}\left(x,y\right)=\sum_{n_{x}\in\mathbb{Z}}\sum_{n_{y}\in\mathbb{Z}}\hat{h}_{{\rm
FAS}}\left(n_{x}D_{x,d}^{\star},n_{y}D_{y,d}^{\star}\right)\\\
\times\delta\left(x-n_{x}D_{x,d}^{\star}\right)\delta\left(y-n_{y}D_{y,d}^{\star}\right),$
(47)

where $\hat{h}_{{\rm FAS}}(n_{x}D_{x,d}^{\star},n_{y}D_{y,d}^{\star})$
represents the estimated channel at location
$(n_{x}D_{x,d}^{\star},n_{y}D_{y,d}^{\star})$. In the frequency domain, we
have

$\displaystyle H_{{\rm FAS}}^{s}\left(k_{x},k_{y}\right)$ $\displaystyle=$
$\displaystyle\sum_{n_{x}\in\mathbb{Z}}\sum_{n_{y}\in\mathbb{Z}}\hat{h}_{{\rm
FAS}}\left(n_{x}D_{x,d}^{\star},n_{y}D_{y,d}^{\star}\right)e^{-j\left(k_{x}n_{x}D_{x,d}^{\star}+k_{t}n_{y}D_{y,d}^{\star}\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{D_{x,d}^{\star}D_{y,d}^{\star}}\sum_{l_{x}\in\mathbb{Z}}\sum_{l_{y}\in\mathbb{Z}}H_{{\rm
FAS}}\left(k_{x}-\frac{2\pi}{D_{x,d}^{\star}}l_{x},k_{y}-\frac{2\pi}{D_{y,d}^{\star}}l_{y}\right).$
(48)

Following the Nyquist sampling method, $\hat{h}_{{\rm FAS}}\left(x,y\right)$
can be reconstructed using a low-pass filter as

$\hat{H}_{{\rm FAS}}\left(k_{x},k_{y}\right)=H_{{\rm
FAS}}^{s}\left(k_{x},k_{y}\right)f\left(k_{x},k_{y}\right),$ (49)

where

$f\left(k_{x},k_{y}\right)=\begin{cases}1,&-\frac{2\pi}{D_{x,d}^{\star}}\leq
k_{x}\leq\frac{2\pi}{D_{x,d}^{\star}},-\frac{2\pi}{D_{y,d}^{\star}}\leq
k_{y}\leq\frac{2\pi}{D_{y,d}^{\star}},\\\ 0,&{\rm{otherwise}.}\end{cases}$
(50)

Finally, we can obtain $\hat{h}_{{\rm FAS}}\left(x,y\right)$ for
$-\frac{X}{2}\lambda\leq x\leq\frac{X}{2}\lambda$ and $-\frac{Y}{2}\lambda\leq
y\leq\frac{Y}{2}$ using the inverse Fourier transform as

$\hat{h}_{{\rm FAS}}\left(x,y\right)=\frac{1}{2\pi}\int\int\hat{H}_{{\rm
FAS}}\left(k_{x},k_{y}\right)e^{-j\left(k_{x}x+k_{y}y\right)}dxdy.$ (51)

The normalized mean square error (NMSE) between $\hat{h}_{{\rm
FAS}}\left(x,y\right)$ and $h_{{\rm FAS}}\left(x,y\right)$ can be computed as

${\rm{NMSE}}=\frac{\int_{-\frac{Y}{2}\lambda}^{\frac{Y}{2}\lambda}\int_{-\frac{X}{2}\lambda}^{\frac{X}{2}\lambda}\left(\hat{h}_{{\rm
FAS}}\left(x,y\right)-h_{{\rm
FAS}}\left(x,y\right)\right)^{2}dxdy}{\int_{-\frac{Y}{2}\lambda}^{\frac{Y}{2}\lambda}\int_{-\frac{X}{2}\lambda}^{\frac{X}{2}\lambda}\left(h_{{\rm
FAS}}\left(x,y\right)\right)^{2}dxdy}.$ (52)

The 1D counterpart can be derived in a similar fashion as elaborated in
Appendix I.

###  III-B MLE

For brevity, let us denote
$\left(\boldsymbol{D}\boldsymbol{n}\right)=\left(n_{x}D_{x},n_{y}D_{y}\right)$.
Suppose the radiating element of the FAS receiver is located at
$\left(\boldsymbol{D}\boldsymbol{n}\right)$. Then the received signal at
symbol $z$ can be expressed as

$r_{z}\left(\boldsymbol{D}\boldsymbol{n}\right)=h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)q_{z}+\omega_{z}\left(\boldsymbol{D}\boldsymbol{n}\right),\leavevmode\nobreak\
z=1,\ldots,z_{p},$ (53)

where $h_{{\rm FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$ is the unknown
constant channel coefficient within the coherence time, while $q_{z}$ and
$\omega_{z}\left(\boldsymbol{D}\boldsymbol{n}\right)$ is, respectively, the
pilot symbol and additive white Gaussian noise (AWGN) during symbol $z$. For
simplicity, we assume that $\left|q_{z}\right|^{2}=\frac{P}{2}$ and
$\omega_{z}\left(\boldsymbol{D}\boldsymbol{n}\right)$ is an independent
Gaussian random variable with zero mean and variance of $\frac{N_{0}}{2}$,
where $P$ and $N_{0}$ is the transmit power and noise level, respectively.

To proceed, we can rewrite (53) as

$\boldsymbol{r}\left(\boldsymbol{D}\boldsymbol{n}\right)=\boldsymbol{q}h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)+\boldsymbol{\omega}\left(\boldsymbol{D}\boldsymbol{n}\right).$
(54)

where

$\left\\{\begin{aligned}
\boldsymbol{r}\left(\boldsymbol{D}\boldsymbol{n}\right)&=\left[r_{1}\left(\boldsymbol{D}\boldsymbol{n}\right)\leavevmode\nobreak\
\cdots\leavevmode\nobreak\
r_{z_{p}}\left(\boldsymbol{D}\boldsymbol{n}\right)\right]^{T},\\\
\boldsymbol{q}&=\left[q_{1},\ldots,q_{z_{p}}\right]^{T}\\\
\boldsymbol{\omega}\left(\boldsymbol{D}\boldsymbol{n}\right)&=\left[\omega_{1}\left(\boldsymbol{D}\boldsymbol{n}\right),\ldots,\omega_{z_{p}}\left(\boldsymbol{D}\boldsymbol{n}\right)\right]^{T}.\end{aligned}\right.$
(55)

In (54), $\boldsymbol{\omega}\left(\boldsymbol{D}\boldsymbol{n}\right)$ is a
Gaussian random vector with zero vector mean and covariance of
$\frac{N_{0}}{2}\boldsymbol{I}$. Thus, the joint pdf of
$\boldsymbol{\omega}\left[n\right]$ is

$\displaystyle f_{\boldsymbol{\omega}}\left(w\right)$
$\displaystyle=\frac{1}{\pi
N_{0}^{\frac{z_{p}}{2}}}\exp\left\\{-\sum_{z=1}^{z_{p}}\frac{\left(r_{z}\left(\boldsymbol{D}\boldsymbol{n}\right)-q_{z}h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)\right)^{2}}{N_{0}}\right\\}.$
(56)

Note that (56) can be interpreted as the likelihood function of the unknown
parameter $h_{{\rm FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$. Thus, the
log likelihood of $h_{{\rm FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$ can
be found as

$\mathcal{{L}}\left(\boldsymbol{r}\left(\boldsymbol{D}\boldsymbol{n}\right)|h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)\right)\\\
=-\frac{z_{p}}{2}\ln\left(\pi
N_{0}\right)-\left[\sum_{z=1}^{z_{p}}\frac{\left(r_{z}\left(\boldsymbol{D}\boldsymbol{n}\right)-q_{z}h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)\right)^{2}}{N_{0}^{z_{p}}}\right].$
(57)

The optimal condition for maximizing (57) w.r.t. $h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$ (i.e., the MLE) is given
by666Due to the electromagnetic properties, we assume that the distribution of
$h_{{\rm FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$ is not known as a
priori. If the distribution of $h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$ is known, one can resort to
the MMSE approach to further improve the channel estimation performance.

$\displaystyle\hat{h}_{{\rm FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$
$\displaystyle=\frac{\boldsymbol{q}^{H}\boldsymbol{r}\left(\boldsymbol{D}\boldsymbol{n}\right)}{\left\|\boldsymbol{q}\right\|^{2}},$
$\displaystyle=h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)+\hat{\omega}_{z}\left(\boldsymbol{D}\boldsymbol{n}\right),$
(58)

where
$\hat{\omega}_{z}\left(\boldsymbol{D}\boldsymbol{n}\right)=\frac{\boldsymbol{q}^{H}}{\left\|\boldsymbol{q}\right\|^{2}}\boldsymbol{\omega}\left(\boldsymbol{D}\boldsymbol{n}\right)$.
It can be easily verified that $\hat{h}_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$ is also a Gaussian random
variable. Specifically, it has a mean of $h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$ and a variance of
$\left(z_{p}\;{\rm SNR}\right)^{-1}$. This implies that the estimator is
unbiased and the average mean square error is $\left(z_{p}\;{\rm
SNR}\right)^{-1}$. Using statistics, we can ensure that the estimation error
of $\hat{h}_{{\rm FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)$ is less than
$\varepsilon>0$ with a CI of

$\mathbb{{P}}\left\\{\left|\hat{h}_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)-h_{{\rm
FAS}}\left(\boldsymbol{D}\boldsymbol{n}\right)\right|<\varepsilon\right\\}={\rm{erf}}\left(\varepsilon\sqrt{\frac{z_{p}\;{\rm
SNR}}{2}}\right),$ (59)

where ${\rm{erf}\left(\cdot\right)}$ is the error function. Since the error
function is an increasing function, this suggests that we can increase the CI
either by increasing $\varepsilon$, $z_{p}$ or ${\rm SNR}$. For some fixed
$\varepsilon$, ${\rm SNR}$, and $\gamma$-CI, it is worth noting that we can
find the minimum value via a 1D searching method as

$z_{p}^{*}=\left\\{\min
z_{p}\left|{\rm{erf}}\left(\varepsilon\sqrt{\frac{z_{p}\;{\rm
SNR}}{2}}\right.\right)\geq\gamma\right\\}.$ (60)

Thus, the total number of pilot symbols required for channel estimation and
reconstruction in FAS can be obtained as

$Z_{p}=z_{p}N_{d}^{\star}.$ (61)

Note that if $Z$ is not significantly larger than $Z_{p}$ (e.g., fast-fading
channel), adjustments can be made to parameters such as $\varepsilon$, ${\rm
SNR}$, and $\gamma$-CI to reduce the required symbols. Other options include
employing other state-of-the-art channel estimation methods or fixing the
antenna location (i.e., reducing FAS to TAS for a robust communication
performance).

###  III-C Data Transmission

To maximize the data rate, we assume that the radiating element of FAS is
activated at the location with the maximum amplitude. In the case of perfect
CSI, the optimal position is

$\left(x^{*},y^{*}\right)=\arg\max_{\left(x,y\right)}\left\\{\left|h_{{\rm
FAS}}\left(x,y\right)\right|^{2}\right\\}.$ (62)

The received signal is expressed as

$r_{{\rm{perfect}}}=h_{{\rm FAS}}\left(x^{*},y^{*}\right)q+\omega,$ (63)

where $q$ is the random information signal and $\omega$ is the AWGN. The
capacity of the FAS receiver is given as

$R_{{\rm FAS}}^{*}=B\log_{2}\left(1+\left|h_{{\rm
FAS}}\left(x^{*},y^{*}\right)\right|^{2}\frac{{\rm SNR}}{B}\right),$ (64)

where $B$ is the system bandwidth and ${\rm SNR}=\frac{P}{N_{0}}$. In the case
of imperfect CSI, it is necessary to estimate the channel using the proposed
Nyquist sampling and MLE methods. After that, the FAS receiver can select its
port based on the estimated and reconstructed channel. The suboptimal position
is

$\left(x^{\star},y^{\star}\right)=\arg\max_{\left(x,y\right)}\left\\{\left|\hat{h}_{{\rm
FAS}}\left(x^{\star},y^{\star}\right)\right|^{2}\right\\}.$ (65)

With imperfect CSI, we have the received signal [53]

$r_{{\rm{imperfect}}}=\hat{h}_{{\rm
FAS}}\left(x^{\star},y^{\star}\right)q+\omega,$ (66)

and the achievable rate can be obtained as [54]

$R_{{\rm
FAS}}^{\star}=B\left(1-\frac{N_{d}^{\star}z_{p}}{Z}\right)\log_{2}\left(1+\frac{\left|h_{{\rm
FAS}}\left(x^{\star},y^{\star}\right)\right|^{2}{\rm
SNR}}{\left|\epsilon\right|^{2}{\rm SNR}+B}\right),$ (67)

where the additional pre-log factor accounts for the remaining symbols that
can be used for data transmission per coherence time, and $\epsilon=h_{{\rm
FAS}}\left(x^{\star},y^{\star}\right)-\hat{h}_{{\rm
FAS}}\left(x^{\star},y^{\star}\right)$ accounts for the estimation error.
Clearly, (67) shows that there is a fundamental tradeoff between high data
rate and accurate channel estimation and reconstruction.

In addition, we can further consider FAS where the position of the radiating
element is fixed. This is equivalent to TAS, and the capacity of TAS with
perfect CSI is [53]

$R_{{\rm TAS}}=B\log_{2}\left(1+\left|h_{{\rm
FAS}}\left(x,y\right)\right|^{2}\frac{{\rm SNR}}{B}\right),$ (68)

where $\left(x,y\right)$ is an arbitrary fixed position throughout the time.

(a)  (b)

Figure 2: The fundamental effects of different parameters on a 1D channel
estimation and reconstruction using: a) $D_{0}^{\star}$ sampling distance; and
b) $D=\frac{\lambda}{2}$ sampling distance.

##  IV Results and Discussion

In this section, we evaluate the performance of the proposed Nyquist sampling
and MLE methods for both channel estimation and reconstruction, as well as
rate. For simplicity, we consider a symmetric setting; therefore, we denote
$W$ as the size and $D$ as the sampling distance per dimension. Unless
otherwise stated, we assume that $Z=1200$, $B=30\leavevmode\nobreak\ {\rm
KHz}$, $z_{p}=7$, ${\rm{SNR}=20\leavevmode\nobreak\ dB},$ $L_{x}=L_{y}=32$,
$W=X=Y=2$, and $D=D_{0}^{\star}=D_{x,0}^{\star}=D_{y,0}^{\star}$.

(a)  (b)

Figure 3: The 2D channel estimation and reconstruction using: a)
$D_{0}^{\star}$ sampling distance; b) $D=\frac{\lambda}{2}$ sampling distance.

We first explore the fundamental effects of different parameters on the 1D
channel estimation and reconstruction case, with $X=4$. For the sake of
brevity, we focus only on the real part, as similar behaviors are observed in
the imaginary part. As we can see in Fig. 2, the SNR influences the variation
of the received signals, while $z_{p}$ determines the amount of pilot
training. When the SNR is low, a higher $z_{p}$ can improve the accuracy of
estimated channels or sampling points by averaging out the noise effects with
MLE. However, the accuracy of the reconstructed channel strongly depends on
both the accuracy of MLE at sampling points and the sampling distance. As
observed in Fig. 2(b), if the sampling distance is too large, the inaccuracies
in the reconstructed channel could be much more significant than those caused
by the SNR, (i.e., the variation of the received signals). This suggests that
the sampling distance is equally, if not more, important than the accuracy of
MLE at the sampling points. Moreover, this result clearly highlights that a
sampling distance of $D=\frac{\lambda}{2}$ is insufficient to perfectly
reconstruct the channel.

(a)  (b)

Figure 4: The 2D power spectrum density of $\hat{h}_{{\rm
FAS}}\left(x,y\right)$ using: a) $D_{0}^{\star}$ sampling distance; b)
$D=\frac{\lambda}{2}$ sampling distance.

Fig. 3 shows the 2D channel estimation and reconstruction of $h_{{\rm
FAS}}\left(x,y\right)$ using different values of $D$. Similarly, we only focus
on the real part as similar behaviors can be observed in the imaginary part.
As it is illustrated, the FAS channel can be well estimated at each sampling
point and efficiently reconstructed using the suboptimal sampling distance
$D_{0}^{\star}$. This is because the effect of noise can be averaged out
through the MLE, while the suboptimal sampling distance is capable of
capturing most of the energy of the spectrum. The latter can be verified by
further analyzing the power spectrum density as shown in Fig. 4. When using
the sampling distance $D_{0}^{\star}$, it is observed that the sampled and
reconstructed spectrum are able to capture the spectral leakage of the
mainlobe. Nevertheless, a crucial part of spectral leakage of the mainlobe
cannot be captured when using sampling distance of $D=\frac{\lambda}{2}$,
leading inaccurate channel reconstruction. For example, due to severe
aliasing, we can see that $D=\frac{\lambda}{2}$ sampling distance has more
compact repetition in the sampled spectrum than $D_{0}^{\star}$ sampling
distance. Moreover, the reconstructed spectrum of $D=\frac{\lambda}{2}$
sampling distance does not fit well with the FAS spectrum.

Figure 5: The tradeoff between the average NMSE and sampling distance, $D$.

Fig. 5 presents the tradeoff between the average NMSE and sampling distance,
$D$. The sampling distance is closely related to the number of estimated
channels, as revealed in (45) and (46). Since
$\left|H_{{\rm{FAS}}}\left(k_{x},k_{y}\right)\right|$ spans over the entire
domain, the average NMSE generally decreases as the sampling distance
decreases. However, if the MLE is inaccurate, decreasing the sampling distance
may not decrease the average NMSE as shown in the low SNR case. Moreover, it
is seen that if the MLE is accurate, then the decrease of the average NMSE is
almost concave in a log-log scale when the sampling distance is larger than
$D_{0}^{\star}$ and, in contrast, the decrement of the average NMSE is
approximately convex when the sampling distance is smaller than
$D_{0}^{\star}$. Although smaller sampling distance decreases the average
NMSE, it is worth noting that a smaller sampling distance comes with
implementation complexity since it could lead to frequent position switching
for liquid-based fluid antenna or strong mutual coupling for pixel-based fluid
antenna. Moreover, it would also cause significant rate degradation. Thus, we
suggest using the suboptimal sampling distance $D_{0}^{\star}$ as it is the
largest sampling distance that at least covers the mainlobe of
$\left|H_{{\rm{FAS}}}\left(k_{x},k_{y}\right)\right|$.

(a)  (b)

Figure 6: The sensitivity of sampling distance: a) the average NMSE versus
$W$; b) $D_{0}^{\star}$ versus $W$.

Fig. 6(a) highlights the sensitivity of the sampling distance over different
values of $W$ in terms of the average NMSE. As anticipated, the average NMSE
of $D=\frac{\lambda}{2}$ is evidently higher than that of $D_{0}^{\star}$ when
$W$ is small. We may think that the average NMSE of $D=\frac{\lambda}{2}$ may
converge to that of $D_{0}^{\star}$ when $W$ is large. Nevertheless, our
results show that there is still a large gap between $D_{0}^{\star}$ and
$D=\frac{\lambda}{2}$ even when $W$ is increased to $10$. The reason is that
$D_{0}^{\star}$ only converges to $\frac{\lambda}{2}$ at a rate proportional
to $\frac{1}{W}$, as seen in Fig. 6(b). Thus, extremely large $W$ is still
required to make $\frac{\lambda}{2}$ sufficient for an efficient channel
reconstruction. This suggests that a minor variation in the sampling distance
can affect the average NMSE substantially when $W$ is large. Note that when
$W$ is extremely large, the FAS receiver might be located at the near-field
region. In that case, a smaller sampling distance is always useful [45].

Fig. 7 demonstrates the effects of $z_{p}$, ${\rm SNR}$, and $\varepsilon$. As
it is seen, the average NMSE always decreases as $z_{p}$ or SNR increases.
However, the effect of $z_{p}$ is less substantial than the SNR. This is
because in practice $z_{p}$ could only be adjusted linearly while the ${\rm
SNR}$ usually ranges in the logarithmic scale. Furthermore, the numerical CI
closely matches the analytical CI, and thus we can guarantee that the channel
estimation error is bounded by $\varepsilon$ with a tractable CI, $\gamma$,
implying that we can determine the value of $z_{p}^{*}$ given some fixed
$\varepsilon$, ${\rm SNR}$, and $\gamma$-CI.

(a)  (b)  (c)

Figure 7: The effects of MLE based on different values of: a) $z_{p}$; b)
${\rm SNR}$; and c) $\varepsilon$.

Fig. 8 evaluates the rates of FAS and TAS over different values of $z_{p}$. To
highlight the tradeoff between high data rate and accurate channel estimation
and reconstruction, we fix the SNR to $10\leavevmode\nobreak\ {\rm dB}$. For
ease of exposition, we represent the proposed Nyquist sampling and MLE methods
as FAS with imperfect CSI. As observed, FAS with perfect CSI delivers the
highest data rate, followed by FAS with imperfect CSI, and finally TAS with
perfect CSI. This result shows that, on one hand, FAS can significantly
outperform TAS while considering the channel estimation and reconstruction
problem. On the other hand, there is an optimal value of $z_{p}$ that
maximizes the rate of FAS with imperfect CSI. Nevertheless, the optimal
$z_{p}$ that maximizes the achievable rate intricately depends on the system
parameters $D,W,Z,$ and ${\rm SNR}$.

Figure 8: The rates of FAS and TAS versus $z_{p}$.

######  Remark 1.

Our studies discover that the initial discrepancies stem from subtle
differences in the research focus. In fields such as holographic MIMO or
electromagnetic information theory, the primary concern is maximizing
transmission performance. This focus often leads to the utilization of the
entire continuous antenna aperture for transmission, enabling researchers to
concentrate exclusively on the electromagnetic field, as the channel
properties become the limiting factor for transmission efficiency. In
contrast, FAS typically does not employ the entire antenna aperture for
transmission. However, obtaining the full CSI is crucial to optimizing the
performance of FAS. Since a FAS receiver only observes the electromagnetic
field across a finite space, the received signal is not band-limited, which
makes oversampling incredibly beneficial for channel reconstruction.

##  V Conclusion

In this paper, we investigated the problem of channel estimation and
reconstruction in FAS using the Nyquist sampling and MLE methods. In contrast
to prior studies, we incorporated an electromagnetic-compliant channel model
to enhance our understanding in this problem. Our analysis indicated that
using a half-wavelength sampling distance, though fundamental, proved
inefficient in terms of NMSE due to the limited spatial reception of the FAS
receiver. This spatial constraint necessitated oversampling to achieve
accurate channel reconstruction, establishing a critical tradeoff between the
accuracy of the reconstructed channel and the number of estimated channels. To
strike a balance between these metrics, we derived closed-form expressions for
the suboptimal sampling distance and the total number of estimated channels
needed for a given space. This suboptimal approach was designed to capture the
majority of the energy from the mainlobe or $d$-th sidelobe of the channel
spectrum. Additionally, we employed MLE to estimate the channel at
predetermined locations, ensuring that the channel estimation error remained
within a bound of $\varepsilon$, with a tractable CI. Our analysis further
revealed the relationships between $\varepsilon$, $z_{p}$, ${\rm SNR}$, and
$\gamma$-CI. Utilizing these techniques allowed us to effectively determine
the total number of pilot symbols required for the channel estimation and
reconstruction process in FAS. Finally, we demonstrated that FAS with
imperfect CSI can significantly outperform TAS with perfect CSI using the
proposed Nyquist sampling and MLE methods.

## Appendix I: 1D Nyquist Sampling Method

Let us denote the spatial spectrum of $h\left(x\right)$ as
$H\left(k_{x}\right)$. In a 1D fluid antenna surface, the receiver can only
observe $h\left(x\right)$ over a finite length of $X\lambda$. Thus, it is
necessary to consider the following 1D rectangular window function

$u\left(x\right)=\begin{cases}1,&-\frac{X}{2}\lambda\leq
x\leq\frac{X}{2}\lambda,\\\ 0,&{\rm{otherwise}.}\end{cases}$ (69)

The channel observed at the FAS receiver is given by

$h_{{\rm FAS}}\left(x\right)=h\left(x\right)u\left(x\right).$ (70)

In the frequency domain, the spectrum of $h_{{\rm FAS}}\left(x\right)$ can be
expressed as

$H_{{\rm FAS}}\left(k_{x}\right)=H\left(k_{x}\right)\circledast
U\left(k_{x}\right),$ (71)

where

$U\left(k_{x}\right)=\frac{\sin\left(\frac{k_{x}}{2}X\lambda\right)}{\frac{k_{x}}{2}}.$
(72)

Apparently, $H_{{\rm FAS}}\left(k_{x}\right)$ is no longer band-limited. Using
similar steps as the 2D Nyquist sampling method, we can obtain the $d$-th
efficient sampling distance and the minimum number of estimated channels as

$D_{d}^{\star}=\frac{\lambda}{2+\frac{2}{X}\left(d+1\right)}.$ (73)

and

$N_{d}^{\star}=\left\lfloor\frac{X\lambda}{D_{d}^{\star}}\right\rfloor,$ (74)

respectively.

The sampled signal in the 1D case can be expressed as

$h_{{\rm FAS}}^{s}\left(x\right)=\sum_{n=0}^{N_{d}^{\star}}\hat{h}_{{\rm
FAS}}\left(nD_{d}^{\star}\right)\delta\left(x-nD_{d}^{\star}\right).$ (75)

In the frequency domain, we hence have

$\displaystyle H_{{\rm FAS}}^{s}\left(k\right)=$
$\displaystyle\sum_{n=0}^{N_{d}^{\star}}\hat{h}_{{\rm
FAS}}\left(nD_{d}^{\star}\right)e^{-jknD_{d}^{\star}}$ (76) $\displaystyle=$
$\displaystyle\frac{1}{D_{d}^{\star}}\sum_{n=0}^{N_{d}^{\star}}H_{{\rm
FAS}}\left(k-nk_{s}^{\star}\right),$ (77)

where $k_{s}^{\star}=\frac{2\pi}{D_{d}^{\star}}$ denotes the sampling
frequency. Furthermore, $\hat{h}_{{\rm FAS}}\left(x\right)$ can be
reconstructed using a low-pass filter as

$\hat{H}_{{\rm FAS}}\left(k\right)=H_{{\rm
FAS}}^{s}\left(k\right)f\left(k\right),$ (78)

where

$f\left(k\right)=\begin{cases}1,&-\frac{2\pi}{D_{d}^{\star}}\leq
k\leq\frac{2\pi}{D_{d}^{\star}},\\\ 0,&{\rm{otherwise}.}\end{cases}$ (79)

Finally, we can obtain $\hat{h}_{{\rm FAS}}\left(x\right)$ for
$-\frac{X}{2}\lambda\leq x\leq\frac{X}{2}\lambda$ using the inverse Fourier
transform as

$\hat{h}_{{\rm FAS}}\left(x\right)=\frac{1}{2\pi}\int\hat{H}_{{\rm
FAS}}\left(k\right)e^{-jk_{x}x}dx.$ (80)

The NMSE between $\hat{h}_{{\rm FAS}}\left(x\right)$ and $h_{{\rm
FAS}}\left(x\right)$ is simplified as

${\rm{NMSE}}=\frac{\int_{-\frac{X}{2}\lambda}^{\frac{X}{2}\lambda}\left(\hat{h}_{{\rm
FAS}}\left(x\right)-h_{{\rm
FAS}}\left(x\right)\right)^{2}dx}{\int_{-\frac{X}{2}\lambda}^{\frac{X}{2}\lambda}\left(h_{{\rm
FAS}}\left(x\right)\right)^{2}dx}.$ (81)

## References

  * [1] S. Dash, C. Psomas, and I. Krikidis, “Selection of metallic liquid in sub-6 GHz antenna design for 6G networks,” Scientific Reports, vol. 13, no. 1, p. 20551, Nov. 2023.
  * [2] Y. Shen, K.-F. Tong, and K.-K. Wong, “Radiation pattern diversified single-fluid-channel surface-wave antenna for mobile communications,” in Proc. IEEE-APS Topical Conf. Antennas & Propag. Wireless Commun., pp. 49–51, 5-9 Sept. 2022, Cape Town, South Africa.
  * [3] B. Cetiner et al., “Multifunctional reconfigurable MEMS integrated antennas for adaptive MIMO systems,” IEEE Commun. Mag., vol. 42, no. 12, pp. 62–70, Dec. 2004.
  * [4] L. Jing, M. Li, and R. Murch, “Compact pattern reconfigurable pixel antenna with diagonal pixel connections,” IEEE Trans. Antennas & Propag., vol. 70, no. 10, pp. 8951–8961, Oct. 2022.
  * [5] S. Basbug, “Design and synthesis of antenna array with movable elements along semicircular paths,” IEEE Antennas & Wireless Propag. Lett., vol. 16, pp. 3059–3062, Oct. 2017.
  * [6] K.-K. Wong, K.-F. Tong, Y. Zhang, and Z. Zheng, “Fluid antenna system for 6G: When Bruce Lee inspires wireless communications,” Elect. Lett., vol. 56, no. 24, pp. 1288–1290, Nov. 2020.
  * [7] K.-K. Wong, K.-F. Tong, Y. Shen, Y. Chen, and Y. Zhang, “Bruce Lee-inspired fluid antenna system: Six research topics and the potentials for 6G,” Frontiers Commun. Netw., vol. 3, no. 853416, Mar. 2022.
  * [8] A. Shojaeifard et al., “MIMO evolution beyond 5G through reconfigurable intelligent surfaces and fluid antenna systems,” Proc. IEEE, vol. 110, no. 9, pp. 1244–1265, Sept. 2022.
  * [9] J. Zheng et al., “Flexible-position MIMO for wireless communications: Fundamentals, challenges, and future directions,” IEEE Wireless Commun., early access, doi:10.1109/MWC.011.2300428, 2024.
  * [10] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna systems,” IEEE Trans. Wireless Commun., vol. 20, no. 3, pp. 1950–1962, Mar. 2021.
  * [11] M. Khammassi, A. Kammoun, and M.-S. Alouini, “A new analytical approximation of the fluid antenna system channel,” IEEE Trans. Wireless Commun., vol. 22, no. 12, pp. 8843–8858, Dec. 2023.
  * [12] P. Ramirez-Espinosa, D. Morales-Jimenez, and K.-K. Wong, “A new spatial block-correlation model for fluid antenna systems,” arXiv preprint, arXiv:2401.04513, 2024.
  * [13] W. K. New, K.-K. Wong, H. Xu, K.-F. Tong, and C.-B. Chae, “Fluid antenna system: New insights on outage probability and diversity gain,” IEEE Trans. Wireless Commun., vol. 23, no. 1, pp. 128–140, Jan. 2024.
  * [14] P. D. Alvim et al., “On the performance of fluid antennas systems under $\alpha$-$\mu$ fading channels,” IEEE Wireless Commun. Lett., vol. 13, no. 1, pp. 108–112, Jan. 2024.
  * [15] J. D. Vega-Sánchez, A. E. López-Ramírez, L. Urquiza-Aguiar, and D. P. M. Osorio, “Novel expressions for the outage probability and diversity gains in fluid antenna system,” _IEEE Wirel. Commun. Lett._ , vol. 13, no. 2, pp. 372–376, Feb. 2024
  * [16] W. K. New, K.-K. Wong, H. Xu, K.-F. Tong, and C.-B. Chae, “An information-theoretic characterization of MIMO-FAS: Optimization, diversity-multiplexing tradeoff and $q$-outage capacity,” IEEE Trans. Wireless Commun., early access, doi:10.1109/TWC.2023.3327063, 2024.
  * [17] Y. Chen, S. Li, Y. Hou, and X. Tao, “Energy-efficiency optimization for slow fluid antenna multiple access using mean-field game,” IEEE Wireless Commun. Lett., vol. 13, no. 4, pp. 915–918, Apr. 2024.
  * [18] L. Zhu, W. Ma, B. Ning, and R. Zhang, “Movable-antenna enhanced multiuser communication via antenna position optimization,” IEEE Trans. Wireless Commun., early access, doi:10.1109/TWC.2023.3338626, 2023.
  * [19] H. Xu et al., “Capacity maximization for FAS-assisted multiple access channels,” arXiv preprint, arXiv:2311.11037, 2023.
  * [20] C. Wang et al., “Fluid antenna system liberating multiuser MIMO for ISAC via deep reinforcement learning,” IEEE Trans. Wireless Commun., early access, doi:10.1109/TWC.2024.3376800, 2024.
  * [21] F. R. Ghadi et al., “On performance of RIS-aided fluid antenna systems,” arXiv preprint, arXiv:2402.16116, 2024.
  * [22] B. Tang et al., “Fluid antenna enabling secret communications,” IEEE Commun. Lett., vol. 27, no. 6, pp. 1491–1495, Jun. 2023.
  * [23] J. D. Vega-Sánchez, L. Urquiza-Aguiar, H. R. C. Mora, N. V. O. Garzón and D. P. M. Osorio, “Fluid antenna system: Secrecy outage probability analysis,” IEEE Trans. Veh. Technol., early access, doi:10.1109/TVT.2024.3376475, 2024.
  * [24] L. Tlebaldiyeva, S. Arzykulov, T. A. Tsiftsis, and G. Nauryzbayev, “Full-duplex cooperative NOMA-based mmWave networks with fluid antenna system (FAS) receivers,” in Proc. Inter. Balkan Conf. Commun. Netw., 5-8 Jun. 2023, Istanbul, Turkey.
  * [25] W. K. New et al., “Fluid antenna system enhancing orthogonal and non-orthogonal multiple access,” IEEE Commun. Lett., vol. 28, no. 1, pp. 218–222, Jan. 2024.
  * [26] C. Skouroumounis and I. Krikidis, “Fluid antenna-aided full duplex communications: A macroscopic point-of-view,” IEEE J. Select. Areas Commun., vol. 41, no. 9, pp. 2879–2892, Sept. 2023.
  * [27] J. Zhu et al., “Index modulation for fluid antenna-assisted MIMO communications: System design and performance analysis,” IEEE Trans. Wireless Commun., early access, doi:10.1109/TWC.2024.3364712, 2024.
  * [28] Y. Chen, and T. Xu, “Fluid antenna index modulation communications,” IEEE Wireless Commun. Lett., vol. 13, no. 4, pp. 1203–1207, Apr. 2024.
  * [29] L. Zhu and K. K. Wong, “Historical review of fluid antenna and movable antenna,” arXiv preprint, arXiv:2401.02362v2, 2024.
  * [30] M. Wang, F. Gao, S. Jin, and H. Lin, “An overview of enhanced massive MIMO with array signal processing techniques,” IEEE J. Select. Topics Sig. Process., vol. 13, no. 5, pp. 886–901, Sept. 2019.
  * [31] D. Neumann, T. Wiese, and W. Utschick, “Learning the MMSE channel estimator,” IEEE Trans. Sig. Process., vol. 66, no. 11, pp. 2905–2917, Jun. 2018.
  * [32] L. Cheng, F. Yin, S. Theodoridis, S. Chatzis, and T.-H. Chang, “Rethinking Bayesian learning for data analysis: The art of prior and inference in sparsity-aware modeling,” IEEE Sig. Process. Mag., vol. 39, no. 6, pp. 18–52, Nov. 2022.
  * [33] M. F. Duarte and Y. C. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans. Sig. Process., vol. 59, no. 9, pp. 4053–4085, Sept. 2011.
  * [34] G. Zhou, C. Pan, H. Ren, P. Popovski, and A. L. Swindlehurst, “Channel estimation for RIS-aided multiuser millimeter-wave systems,” IEEE Trans. Sig. Process., vol. 70, pp. 1478–1492, Mar. 2022.
  * [35] G. Zhou, C. Pan, H. Ren, K. Wang, and A. Nallanathan, “A framework of robust transmission design for IRS-aided MISO communications with imperfect cascaded channels,” IEEE Trans. Sig. Process., vol. 68, pp. 5092–5106, Aug. 2020.
  * [36] Z. Peng et al., “Two-stage channel estimation for RIS-aided multiuser mmWave systems with reduced error propagation and pilot overhead,” IEEE Trans. Sig. Process., vol. 71, pp. 3607–3622, Sept. 2023.
  * [37] H. Zhang et al., “Learning-induced channel extrapolation for fluid antenna systems using asymmetric graph masked autoencoder,” IEEE Wireless Commun. Lett., early access, doi:10.1109/LWC.2024.3386153, 2024.
  * [38] N. Waqar, K.-K. Wong, K.-F. Tong, A. Sharples, and Y. Zhang, “Deep learning enabled slow fluid antenna multiple access,” IEEE Commun. Lett., vol. 27, no. 3, pp. 861–865, Mar. 2023.
  * [39] H. Xu et al., “Channel estimation for FAS-assisted multiuser mmWave systems,” IEEE Commun. Lett., vol. 23, no. 3, pp. 632–636, Mar. 2024.
  * [40] Z. Zhang, J. Zhu, L. Dai, and R. W. Heath Jr, “Successive Bayesian reconstructor for channel estimation in fluid antenna systems,” in Proc. IEEE Wireless Commun. Netw. Conf., 21-24 Apr. 2024, Dubai, United Arab Emirates.
  * [41] W. Ma, L. Zhu, and R. Zhang, “Compressed sensing based channel estimation for movable antenna communications,” IEEE Commun. Lett., vol. 27, no. 10, pp. 2747–2751, Oct. 2023.
  * [42] J. Zhu, Z. Wan, L. Dai, M. Debbah, and H. V. Poor, “Electromagnetic information theory: Fundamentals, modeling, applications, and open problems,” IEEE Wireless Commun., early access, doi:10.1109/MWC.019.2200602, 2024.
  * [43] E. Björnson et al., “Towards 6G MIMO: Massive spatial multiplexing, dense arrays, and interplay between electromagnetics and processing,” arXiv preprint, arXiv:2401.02844, 2024.
  * [44] A. Pizzo, A. d. J. Torres, L. Sanguinetti, and T. L. Marzetta, “Nyquist sampling and degrees of freedom of electromagnetic fields,” IEEE Trans. Sig. Process., vol. 70, pp. 3935–3947, Jun. 2022.
  * [45] M. Di Renzo and M. D. Migliore, “Electromagnetic signal and information theory–on electromagnetically consistent communication models for the transmission and processing of information,” arXiv preprint, arXiv:2311.06661, 2023.
  * [46] T. L. Marzetta, “Spatially-stationary propagating random field model for massive MIMO small-scale fading,” in Proc. IEEE Int. Symp. Inform. Theory, pp. 391–395, 17-22 Jun. 2018, Vail, Colorado, USA.
  * [47] A. Pizzo, T. L. Marzetta, and L. Sanguinetti, “Spatially-stationary model for holographic MIMO small-scale fading,” IEEE J. Select. Areas Commun., vol. 38, no. 9, pp. 1964–1979, Sept. 2020.
  * [48] F. B. Hildebrand, Advanced calculus for applications, Pearson, 2nd Edition, 1962.
  * [49] V. Balakrishnan, “All about the Dirac delta function (?),” Resonance, vol. 8, no. 8, pp. 48–58, 2003.
  * [50] H. Landau, “An overview of time and frequency limiting,” Fourier Tech. & Appl., pp. 201–220, 1985.
  * [51] C. Shannon, “Communication in the presence of noise,” Proc. IRE, vol. 37, no. 1, pp. 10–21, 1949.
  * [52] B. P. Lathi and R. A. Green, Linear systems and signals, vol. 2, Oxford University Press New York, 2005.
  * [53] M. Medard, “The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Trans. Inform. Theory, vol. 46, no. 3, pp. 933–946, May 2000.
  * [54] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Information-theoretic and communications aspects,” IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998.