# Successive Bayesian Reconstructor for
FAS Channel Estimation

Zijian Zhang1, Jieao Zhu1, Linglong Dai1, Fellow, IEEE, and Robert W. Heath
Jr.2, Fellow, IEEE 1Department of Electronic Engineering, Tsinghua University,
Beijing 100084, China
1Beijing National Research Center for Information Science and Technology
(BNRist), Beijing 100084, China
2Department of Electrical and Computer Engineering, University of California,
San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
E-mails: {zhangzj20<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Fluid antenna systems (FASs) can reconfigure their locations freely within a
spatially continuous space. To keep favorable antenna positions, the channel
state information (CSI) acquisition for FASs is essential. While some
techniques have been proposed, most existing FAS channel estimators require
several channel assumptions, such as slow variation and angular-domain
sparsity. When these assumptions are not reasonable, the model mismatch may
lead to unpredictable performance loss. In this paper, we propose the
successive Bayesian reconstructor (S-BAR) as a general solution to estimate
FAS channels. Unlike model-based estimators, the proposed S-BAR is prior-
aided, which builds the experiential kernel for CSI acquisition. Inspired by
Bayesian regression, the key idea of S-BAR is to model the FAS channels as a
stochastic process, whose uncertainty can be successively eliminated by
kernel-based sampling and regression. In this way, the predictive mean of the
regressed stochastic process can be viewed as the maximum a posterior (MAP)
estimator of FAS channels. Simulation results verify that, in both model-
mismatched and model-matched cases, the proposed S-BAR can achieve higher
estimation accuracy than the existing schemes.

##  I Introduction

In recent years, fluid antenna systems, also called fluid antennas or movable
antennas, are proposed to achieve higher diversity and multiplexing gains than
conventional multiple-input multiple-output (MIMO) systems [1, 2, 3].
Different from MIMO with fixed antennas, FAS introduces a structure where a
few fluid antennas can freely switch their locations within a given space [4].
In this way, the spacing of the available locations (referred to as “ports”)
for fluid antennas can be arbitrarily small. This almost continuously movable
feature allows FASs to keep fluid antennas at favorable positions, thus
promising to achieve high diversity and multiplexing gains with very few
antennas [1, 2, 3].

Despite these encouraging prospects, the expected gains of FASs are hard to
achieve in practice. In specific, the transmission performance of FASs heavily
relies on the positions of fluid antennas [5, 6, 7]. To ensure favorable
antenna placements, the channel state information (CSI) knowledge of available
locations is essential [8, 9, 10]. However, the channel estimation for FASs is
challenging. The reason is that, the allowed locations (i.e., the ports) of
fluid antennas are densely deployed, leading to very high-dimensional port
channels [4]. Thereby, it requires an unacceptable number of pilots to acquire
the channels. Besides, limited by the hardware structure of FASs, only a few
ports can be connected to radio frequency (RF) chains for channel measurements
within the coherence time, which exacerbates the difficulty of channel
estimation. To address the high-dimensional FASs channels, pilot-reduced
channel estimators have been investigated in [8, 9, 10]. However, most
existing channel estimators rely on some channel assumptions, such as the slow
variation [8], angle-domain sparsity [9], and known angles-of-arrival (AoAs)
[10]. When these assumptions are not reasonable, the model mismatch will lead
to an unpredictable performance loss.

In this paper, we propose the successive Bayesian reconstructor (S-BAR) as a
general solution to estimate FAS channels. Different from the existing model-
based estimators relying on channel assumptions, the proposed S-BAR builds the
experiential kernel of FASs channels for CSI acquisition. Specifically,
inspired by the Bayesian regression [11], the key idea of S-BAR is to model
the FAS channels as a stochastic process with an experiential kernel, which
characterizes the inherent correlation of FAS channels. Then, the uncertainty
of the stochastic process can be successively eliminated by kernel-based
sampling and regression. Particularly, the proposed S-BAR is a two-stage
scheme. In the first stage, the measured channels are determined by following
the principle of maximum posterior variance. In the second stage, the channel
measurements are combined with the experiential kernel for process regression.
Then, the mean of the regressed stochastic process is exactly the maximum a
posterior (MAP) estimator of FAS channels. Simulation results reveal that, in
both model-mismatched and model-matched cases, the proposed S-BAR can achieve
higher estimation accuracy than the existing schemes based on channel
assumptions.

The rest of this paper is organized as follows. In Section II, the system
model of an FAS is introduced, and the problem of channel estimation is
formulated. In Section III, the general S-BAR is proposed for FAS channel
estimation. In Section IV, simulation results are presented to evaluate the
estimation performance. Finally, conclusions are drawn in Section V.

Notation: ${[\cdot]^{-1}}$, ${[\cdot]^{*}}$, ${[\cdot]^{\rm T}}$, and
${[\cdot]^{\rm H}}$ denote the inverse, conjugate, transpose, and conjugate-
transpose operations, respectively; ${\bf x}(i)$ denotes the $i$-th entry of
vector ${\bf x}$; ${\bf X}({i,j})$, ${\bf X}({j,:})$ and ${\bf X}({:,j})$
denote the $(i,j)$-th entry, the $j$-th row, and the $j$-th column of matrix
${\bf X}$, respectively; ${\rm Tr}(\cdot)$ denotes the trace of its argument;
${\mathsf{E}}\left(\cdot\right)$ is the expectation of its argument; ${\rm
dim}(\cdot)$ is the dimensional of its argument;
$\mathcal{C}\mathcal{N}\\!\left({\bm{\mu}},{\bf\Sigma}\right)$ and
$\mathcal{G}\mathcal{P}\\!\left({\bm{\mu}},{\bf\Sigma}\right)$ respectively
denote the complex Gaussian distribution and complex Gaussian process, with
mean ${\bm{\mu}}$ and covariance ${\bf\Sigma}$; $\mathbf{0}_{L}$ is an all-
zero vector or matrix with dimension $L$.

##  II System Model

In this paper, we consider the narrowband channel estimation in an uplink FAS,
which consists of an $N$-port base station (BS) equipped with $M$ fluid
antennas and a single-antenna user. The $N$ feeding ports are uniformly
distributed along a linear dimension at the receiver. The $M$ fluid antennas
can be repositioned to the $M$ locations of $N$ available ports ($M\ll N$),
and each antenna is connected to an RF chain. Let ${\bf h}\in{\mathbb{C}}^{N}$
denote the channels of $N$ ports, and let $P$ denote the number of transmit
pilots within a coherence time frame. To characterize the locations of $M$
fluid antennas in timeslot $p$, we introduce the definition of switch matrix
as follows:

######  Definition 1 (Switch Matrix)

Binary indicator ${\bf S}_{p}\in\\{0,1\\}^{M\times N}$ is defined as the
switch matrix of multiple fluid antennas in timeslot $p$. The $(m,n)$-th entry
being 1 (or 0) means that the $m$-th antenna is (or not) located at the $n$-th
port. Since $M$ of $N$ ports are selected in each timeslot, each row of ${\bf
S}_{p}$ has one entry of 1, and all entries of 1 in ${\bf S}_{p}$ are not in
the same column, i.e., $\|{\bf S}_{p}({m,:})\|=1$ for all
$m\in\\{1,\cdots,M\\}$, $\|{\bf S}_{p}({:,n})\|\in\\{0,1\\}$ for all
$n\in\\{1,\cdots,N\\}$, and ${\bf S}_{p}{\bf S}_{p}^{\rm H}={\bf I}_{M}$.

Utilizing Definition 1, the signal vector ${\bf y}_{p}\in{\mathbb{C}}^{M}$
received at the BS in timeslot $p$ can be modeled as

${\bf y}_{p}={\bf S}_{p}{\bf h}s_{p}+{\bf z}_{p},$ (1)

where $s_{p}$ is the pilot transmitted by the user and ${\bf
z}_{p}\sim\mathcal{C}\mathcal{N}\\!\left({\bf 0}_{M},\sigma^{2}{\bf
I}_{M}\right)$ is the additive white Gaussian noise (AWGN) at $M$ selected
ports. Without loss of generality, we assume that $s_{p}=1$ for all
$p\in\\{1,\cdots,P\\}$. Considering the total $P$ timeslots for pilot
transmission, we arrive at

${\bf y}={\bf S}{\bf h}+{\bf z},$ (2)

where ${\bf y}:=\left[{\bf y}_{1}^{\rm T},\cdots,{\bf y}_{P}^{\rm
T}\right]^{\rm T}$, ${\bf S}:=\left[{\bf S}_{1}^{\rm T},\cdots,{\bf
S}_{P}^{\rm T}\right]^{\rm T}$, and ${\bf z}:=\left[{\bf z}_{1}^{\rm
T},\cdots,{\bf z}_{P}^{\rm T}\right]^{\rm T}$. Our goal is to reconstruct the
$N$-dimensional channel $\bf h$ according to the $PM$-dimensional noisy pilot
${\bf y}$. Since fluid antennas move almost continuously, $N$ is much larger
than $PM$ ($N\gg PM$). Besides, due to the zero-one distribution of ${\bf S}$,
most elements of $\bf h$ cannot be observed directly or indirectly. As a
result, the channel estimation of FASs is usually challenging.

##  III Proposed Successive Bayesian Reconstructor

In this section, based on the Bayesian regression, we propose the S-BAR as a
general solution to realize FAS channel estimation. Specifically, in
Subsection III-A, the classical Bayesian regression is introduced. Then, in
Subsection III-B, the proposed S-BAR scheme is illustrated. Finally, in
Subsection III-C, the kernel selection of S-BAR is discussed.

###  III-A Bayesian Regression

Without making any prior assumptions, the attempt to recover the function
$f({\bf x})$ from a few samples appears to be a challenging endeavor.
Fortunately, by building the experiential kernel of $f({\bf x})$, Bayesian
regression can determine the sampling strategy and reconstruct $f({\bf x})$
with a few samples in a non-parametric way. Under this framework, Gaussian
process regression (GPR) has become a popular solution [11]. Specifically,
function $f({\bf x})$ can be modeled as a sample of Gaussian process
${\mathcal{GP}}\left(\mu\left({\bf x}\right),k\left({\bf x},{\bf
x}^{\prime}\right)\right)$. It is completely specified by its mean
$\mu\left({\bf x}\right)$ and its kernel $k\left({\bf x},{\bf
x}^{\prime}\right)$, which encodes the smoothness of regressed $f({\bf x})$.
In timeslot $t$, consider a prior ${\mathcal{GP}}\left(\mu\left({\bf
x}\right),k\left({\bf x},{\bf x}^{\prime}\right)\right)$ over $f({\bf x})$.
Let ${\bm{\gamma}}^{t}:=[\gamma^{1},\cdots,\gamma^{t}]^{\rm T}$ denote $t$
noisy measurements for points in ${\cal A}^{t}:=\\{{\bf x}^{1},\cdots,{\bf
x}^{t}\\}$, where $\gamma^{i}=f({\bf x}^{i})+n_{i}$ with
$n_{i}\sim{\cal{CN}}\left(0,\delta^{2}\right)$. It is easy to prove that,
given ${\bm{\gamma}}^{t}$, the posterior over $f({\bf x})$ is also a Gaussian
process whose mean and covariance are

$\displaystyle{\mu^{t}\left({\bf x}\right)}$ $\displaystyle=\mu\left({\bf
x}\right)+\left({\bf k}^{t}({\bf x})\right)^{\rm H}\\!\left({\bf
K}^{t}+\delta^{2}{\bf
I}_{t}\right)^{-1}\\!\left({\bm{\gamma}}-{\bm{\mu}}^{t}\right),$ (3)
$\displaystyle{k^{t}\left({\bf x},{\bf x}^{\prime}\right)}$
$\displaystyle=k\left({\bf x},{\bf x}^{\prime}\right)-\left({\bf k}^{t}({\bf
x})\right)^{\rm H}\\!\left({\bf K}^{t}+\delta^{2}{\bf
I}_{t}\right)^{-1}\\!{\bf k}^{t}({\bf x}^{\prime}),$ (4)

where ${\bf k}^{t}({\bf x}):=\left[k\left({\bf x}^{1},{\bf
x}\right),\cdots,k\left({\bf x}^{t},{\bf x}\right)\right]^{\rm T}$;
${\bm{\mu}}^{t}:=\left[\mu\left({\bf x}^{1}\right),\cdots,\mu\left({\bf
x}^{t}\right)\right]^{\rm T}$ ; and the $(i,j)$-th entry of ${\bf
K}^{t}\in{\mathbb{C}}^{t\times t}$ is $k\left({\bf x}^{i},{\bf x}^{j}\right)$,
for all $i,j\in\\{1,\cdots,t\\}$.

Then, the next candidate point to be sampled, i.e., ${\bf x}^{t+1}$, can be
determined based on the updated posterior. For successive sampling, sampling
the point with the maximum posterior variance can obtain the most information.
By assuming that ${\bf x}\in{\cal S}$, ${\bf x}^{t+1}$ can be chosen according
to

$\displaystyle{\bf x}^{t+1}=\mathop{\arg\max}\limits_{{\bf x}\in{\cal S}/{\cal
A}^{t}}~{}{k^{t}\left({\bf x},{\bf x}\right)},$ (5)

where $/$ is the set difference. By letting $t\to\infty$, the value of
variance $k^{t}\left({\bf x},{\bf x}\right)$ decreases asymptotically, which
means that the uncertainty of $f({\bf x})$ is reduced. After reaching the
tolerance threshold, the posterior mean ${\mu^{t}\left({\bf x}\right)}$ can be
viewed as a MAP estimator of $f({\bf x})$ [11].

###  III-B Proposed S-BAR Scheme

In each pilot timeslot, $M$ fluid antennas move positions and measure
channels, thus the channel estimation of FASs is similar to a successive
sampling process. Since the port spacing is short, the FAS channels are highly
correlated. These features inspire us to recover $\bf h$ through Bayesian
regression. To reconstruct FAS channels based on experiential kernel, we model
$\bf h$ as a sample of Gaussian process ${\mathcal{GP}}\left({\bf
0}_{N},{\bm{\Sigma}}\right)$. Semidefinite Hermitian matrix
${\bm{\Sigma}}\in{\mathbb{C}}^{N\times N}$ is called the kernel or prior
covariance, of which the selection will be introduced in Subsection III-C.
Then, the proposed S-BAR scheme is summarized in Algorithm 1. For clarity, the
basic principle of S-BAR is firstly introduced as follows.

Algorithm 1 Proposed Successive Bayesian Reconstructor

0: Number of pilots $P$, kernel ${\bm{\Sigma}}$.

0: Reconstructed FAS channel $\hat{\bf h}$.

1: # Stage 1 (Offline Design):

2: Initialization: $\Omega=\varnothing$, ${\bf S}_{p}={\bf 0}_{M\times N}$ for
all $p\in\\{1,\cdots,P\\}$

3: for $p\in\\{1,\cdots,P\\}$ do

4: for $m\in\\{1,\cdots,M\\}$ do

5: Posterior covariance update: Calculate ${\bm{\Sigma}}_{\Omega}$ by (7)

6: Candidate selection:
$n^{\star}=\mathop{\arg\max}\limits_{n\in\\{1,\cdots,N\\}/\Omega}~{}{{\bm{\Sigma}}_{\Omega}(n,n)}$

7: Switch matrix update: ${\bf S}_{p}(m,n^{\star})=1$

8: Sequence update: $\Omega=\Omega\cup\\{n^{\star}\\}$

9: end for

10: end for

11: Merge switch matrices: ${\bf S}:=\left[{\bf S}_{1}^{\rm T},\cdots,{\bf
S}_{P}^{\rm T}\right]^{\rm T}$

12: Weight calculation: ${\bf w}=({\bm{\Sigma}}(\Omega,\Omega)+\sigma^{2}{\bf
I}_{PM})^{-1}{\bm{\Sigma}}(\Omega,:)$

13: # Stage 2 (Online Regression):

14: Employ the designed switch matrix ${\bf S}$ at the BS, and then obtain the
received pilot: ${\bf y}={\bf S}{\bf h}+{\bf z}$

15: Channel reconstruction: $\hat{\bf h}={\bf w}^{\rm H}{\bf y}$

16: return Reconstructed FAS channel $\hat{\bf h}$

####  III-B1 Algorithmic Principle

At some moment, let $\Omega$ denote the index sequence of the measured
channels and let ${\bf y}_{\Omega}\in{\mathbb{C}}^{{\rm dim}(\Omega)}$ denote
the corresponding received pilots, which is from
${{\bf{y}}_{\Omega}}={\bf{h}}\left(\Omega\right)+{{\bf{z}}_{\Omega}}$ with
${\bf{z}}_{\Omega}\sim{\cal CN}({\bf 0}_{{\rm dim}(\Omega)},\sigma^{2}{\bf
I}_{{\rm dim}(\Omega)})$ being the AWGN. For given ${{\bf{y}}_{\Omega}}$, the
posterior mean ${\bm{\mu}}_{\Omega}$ and posterior covariance
${\bm{\Sigma}}_{\Omega}$ of $\bf h$ can be calculated by:

$\displaystyle{\bm{\mu}}_{\Omega}$
$\displaystyle={\bm{\Sigma}}(:,\Omega)\left({\bm{\Sigma}}(\Omega,\Omega)+\sigma^{2}{\bf
I}_{{\rm dim}(\Omega)}\right)^{-1}{\bf y}_{\Omega},$ (6)
$\displaystyle{\bm{\Sigma}}_{\Omega}$
$\displaystyle={\bm{\Sigma}}-\left({\bm{\Sigma}}(\Omega,:)\right)^{\rm
H}\left({\bm{\Sigma}}(\Omega,\Omega)+\sigma^{2}{\bf I}_{{\rm
dim}(\Omega)}\right)^{-1}{\bm{\Sigma}}(\Omega,:).$ (7)

For given $\Omega$, the next candidate channel to be measured can be
determined by finding the index associated with the largest posterior
variance, i.e.,

$\displaystyle
n^{\star}=\mathop{\arg\max}\limits_{n\in\\{1,\cdots,N\\}/\Omega}~{}{{\bm{\Sigma}}_{\Omega}(n,n)}.$
(8)

Subsequently, we can update $\Omega$ by $\Omega\cup\\{n^{\star}\\}$ and repeat
the above process until the posterior mean ${\bm{\mu}}_{\Omega}$ can well
approximate $\bf h$.

####  III-B2 Observations

From the above equations, we obtain the following three observations.

  * •

(6) indicates that, the posterior mean ${\bm{\mu}}_{\Omega}$ is the linear
weighted sum of pilot ${\bf y}_{\Omega}$, i.e., ${\bm{\mu}}_{\Omega}={\bf
w}^{\rm H}{\bf y}_{\Omega}$, wherein the weight ${\bf
w}:=\left({\bm{\Sigma}}(\Omega,\Omega)+\sigma^{2}{\bf I}_{{\rm
dim}(\Omega)}\right)^{-1}{\bm{\Sigma}}(\Omega,:)$ only relies on the kernel
${\bm{\Sigma}}$.

  * •

(7) shows that posterior covariance ${\bm{\Sigma}}_{\Omega}$ only relies on
kernel ${\bm{\Sigma}}$ and is unrelated to the received pilot ${\bf
y}_{\Omega}$.

  * •

(8) suggests that the next channel to be measured only relies on the posterior
covariance ${\bm{\Sigma}}_{\Omega}$.

These observations reveal that, the switch matrix $\bf S$ and the weight $\bf
w$ are unrelated to the received pilot $\bf y$, thus they can be designed
offline and then deployed online to reduce the complexity. Thereby, the
proposed S-BAR can be realized in the following two stages.

####  III-B3 Stage 1 (Offline Design)

Since index sequence $\Omega$ is determined by the posterior covariance
${\bm{\Sigma}}_{\Omega}$, and ${\bm{\Sigma}}_{\Omega}$ only relies on the
kernel ${\bm{\Sigma}}$. The switch matrix ${\bf S}\in{\\{0,1\\}}^{PM\times N}$
and the weight ${\bf w}\in{\mathbb{C}}^{PM}$ for recovering ${\bf
h}\in{\mathbb{C}}^{N}$ can be designed offline at the first stage. By updating
${\bm{\Sigma}}_{\Omega}$ in (7) and $n^{\star}$ in (8) alternatingly until
${\rm dim}(\Omega)=PM$, sequence $\Omega$ can collect all required indexes of
the $M$ selected ports in $P$ pilot timeslots.

Then, recall that we have ${\bf{h}}\left(\Omega\right)={\bf S}{\bf{h}}$. To
achieve the conversion from $\Omega$ to ${\bf S}$, we can initialize $\bf S$
as an all-zero matrix and then fill in an one at the position associated with
the selected index in each of its row. Note that, this operation naturally
satisfies $\|{\bf S}({m,:})\|=1$ for all $m\in\\{1,\cdots,M\\}$, $\|{\bf
S}({:,n})\|\in\\{0,1\\}$ for all $n\in\\{1,\cdots,N\\}$, and ${\bf S}{\bf
S}^{\rm H}={\bf I}_{PM}$. These properties ensure that the designed $\bf S$ is
practically implementable in FASs. After obtaining $\Omega$, the weight for
reconstructing $\bf h$ can be obtained by

${\bf w}=\left({\bm{\Sigma}}(\Omega,\Omega)+\sigma^{2}{\bf
I}_{PM}\right)^{-1}{\bm{\Sigma}}(\Omega,:).$ (9)

####  III-B4 Stage 2 (Online Regression)

Since Stage 1 is realized offline, the switch matrix ${\bf S}$ and weight
${\bf w}$ can be designed and saved at the BS in advance. In Stage 2, the
scheme is then employed online for channel measurements. The $M$ fluid
antennas of the BS will move and receive pilots according to the designed
${\bf S}$, arriving at the noisy pilot $\bf y$. Finally, according to the MAP
estimator in (6), channel $\bf h$ can be reconstructed by $\hat{\bf h}={\bf
w}^{\rm H}{\bf y}$, which completes the proposed S-BAR.

####  III-B5 Computational Complexity

The proposed S-BAR incorporates a hybrid offline and online implementation
process, thereby substantially reducing its computational complexity in
practical applications. Specifically, the signal processing of S-BAR is
composed of two stages. In Stage 1, the computational complexity is dominated
by the calculation of posterior covariance ${\bm{\Sigma}}_{\Omega}$, which is
updated $PM$ times. According to (7), the complexity of Stage 1 is ${\cal
O}\left({P^{2}}{M^{2}}\left({{P^{2}}{M^{2}}+NPM+{N^{2}}}\right)\right)$. In
Stage 2, the computational complexity is from the weighted sum of received
pilot $\bf y$, i.e., $\hat{\bf h}={\bf w}^{\rm H}{\bf y}$, thus the
computational complexity is ${\cal O}\left(N\right)$. Note that, although the
complexity of Stage 1 is high, Stage 1 can be implemented offline in advance.
From the perspective of practical employment, the effective complexity of
S-BAR scheme is only linear to the number of ports $N$.

###  III-C Kernel Selection for S-BAR Scheme

The selection of kernel $\bm{\Sigma}$ determines the shape and flexibility of
the proposed S-BAR, which in turn affects its ability to capture patterns and
make accurate reconstruction. Considering the localized correlation property
of FAS channels, an appropriate kernel should assign higher similarity to
nearby ports and decrease influence rapidly with distance. Let ${\bf x}_{n}$
denote the position of the $n$-th port. Three kernel selections are
recommended as follows.

####  III-C1 Exponential Kernel

The exponential kernel ${\bm{\Sigma}}_{\rm exp}$ is a popular choice in
regression, given by

${\bm{\Sigma}}_{\rm exp}(n,n^{\prime})=\alpha^{2}e^{-\frac{\|{\bf x}_{n}-{\bf
x}_{n^{\prime}}\|^{2}}{\eta^{2}}}$ (10)

for all $n,n^{\prime}\in\\{1,\cdots,N\\}$, where $\alpha$ and $\eta$ are
adjustable hyperparameters. Compared with the other kernels, the exponential
kernel is less sensitive to outliers, which makes it suitable to recover
channels without obvious regularity.

####  III-C2 Bessel Kernel

The Bessel kernel ${\bm{\Sigma}}_{\rm bes}$ is well-suited for capturing and
modeling complex-valued data with oscillatory or periodic patterns, given by

${\bm{\Sigma}}_{\rm bes}(n,n^{\prime})=\alpha^{2}J_{v}\left(\frac{\|{\bf
x}_{n}-{\bf x}_{n^{\prime}}\|}{\eta}\right)$ (11)

for all $n,n^{\prime}\in\\{1,\cdots,N\\}$, wherein $J_{v}$ is the $v$-order
Bessel function of the first kind. It has the flexibility to adapt to data
that exhibits regular and repeating fluctuations, thus ${\bm{\Sigma}}_{\rm
bes}$ is suitable to reconstruct the channels with periodic patterns.

####  III-C3 Covariance Kernel

An ideal approach is to use the real covariance of $\bf h$ as the kernel for
reconstruction, i.e., ${\bm{\Sigma}}_{\rm cov}={\mathsf{E}}\left({\bf h}{\bf
h}^{\rm H}\right)$. Since ${\bm{\Sigma}}_{\rm cov}$ is unknown in practice, we
can train an approximated ${\bm{\Sigma}}_{\rm cov}$ before employing S-BAR,
given by

${\bm{\Sigma}}_{\rm
cov}\approx\frac{1}{T}\sum\limits_{t=1}^{T}{{{\bf{h}}_{t}}{\bf{h}}_{t}^{\rm
H}},$ (12)

where ${\bf{h}}_{t}$ is the channel at the $t$-th training timeslot and $T$ is
the number of training timeslots. Since the channel covariance
${\mathsf{E}}\left({\bf h}{\bf h}^{\rm H}\right)$ does not change so
frequently as channels, ${\bm{\Sigma}}_{\rm cov}$ is only updated in a large
timescale.

##  IV Simulation Results

In this section, simulation results are provided to verify the effectiveness
of the proposed S-BAR scheme. Since we have assumed the normalized transmit
power, the receiver signal-to-noise ratio (SNR) is defined as ${\rm
SNR}=\frac{{\mathsf{E}}\left(\|{\bf h}\|^{2}\right)}{\sigma^{2}}$, of which
the default value is set to $20$ dB. Let $\hat{\bf h}$ denote the estimated
value of channel $\bf h$. The performance is evaluated by the normalized mean
square error (NMSE), i.e., ${\rm NMSE}={\mathsf{E}}\left(\frac{\|{\bf
h}-\hat{\bf h}\|^{2}}{\|{\bf h}\|^{2}}\right)$.

(a) $P=1$, $M=1$. (b) $P=2$, $M=2$. (c) $P=3$, $M=3$. (d) $P=4$, $M=4$.

(e) $P=1$, $M=1$. (f) $P=2$, $M=2$. (g) $P=3$, $M=3$. (h) $P=4$, $M=4$.

Figure 1:  An illustration of employing S-BAR scheme to estimate FAS channel
$\bf h$. (a)-(d) provide the real part of $\bf h$ versus the index of ports.
(e)-(h) provide the imaginary part of $\bf h$ versus the index of ports.
Particularly, the curve “Truth” denotes the real channel $\bf h$, and the
circle marks denote the sampled (measured) channels. The dotted line “Mean”
denotes the posterior mean of Bayesian regression ${\bm{\mu}}_{\Omega}$, i.e.,
the estimated channel $\hat{\bf h}$. The highlighted shadows in the figures
represent the confidence intervals of $\bf h$, defined as
$[{\bm{\mu}}_{\Omega}(n)-3{{\bm{\Sigma}}_{\Omega}(n,n)},{\bm{\mu}}_{\Omega}(n)+3{{\bm{\Sigma}}_{\Omega}(n,n)}]$
for the $n$-th port.

####  IV-1 Simulation Setup

The simulations are provided based on both the QuaDRiGa channel model and the
spatially-sparse clustered (SSC) channel model. For existing model-based
estimators, these two models can be viewed as the matched case and mismatched
case, respectively. Otherwise particularly specified, the system parameters
are set as: $N=256$, $M=4$, $P=10$. The carrier frequency is set to
$f_{c}=3.5$ GHz, and the length of the fluid antenna array is set to
$W=10\lambda$. For the QuaDRiGa channel model, all parameters are generated
according to Table 7.7.1-2 in 3GPP TR 38.901. For the SSC channel model, the
number of clusters is set to $C=9$ and that of rays is set to $R=100$. Both
models have assumed the maximum angle spread to be 5∘. For kernel settings,
the hyperparameters are set as $\alpha=1$ and
$\eta=\sqrt{\frac{\lambda}{2\pi}}$ to generate the exponential kernel
${\bm{\Sigma}}_{\rm exp}$ and Bessel kernel ${\bm{\Sigma}}_{\rm bes}$ [11].
Inspired by the covariance model in [2, 1], the order of Bessel function in
${\bm{\Sigma}}_{\rm bes}$ is set to $\nu=0$. To account for an ideal baseline,
the number of training timeslots is set to $T=100$ to train the covariance
kernel ${\bm{\Sigma}}_{\rm cov}$.

####  IV-2 Simulation Schemes

We consider the following three schemes for simulations. 1) FAS-OMP: Assuming
that the FAS channels are spatially sparse, the scheme in [9] is modified and
employed at the BS to explicitly estimate $\bf h$. 2) SeLMMSE:  The SeLMMSE
proposed in [8] is adopted to estimate channel $\bf h$, which can be achieved
by sequentially measuring channels of $PM$ equally-spaced ports and then using
zero-order interpolation to reconstruct $\bf h$. 3) Proposed S-BAR: Given a
kernel $\bm{\Sigma}$, the proposed S-BAR scheme, i.e., Algorithm 1, is
employed to estimate $\bf h$. Particularly, due to the lack of obvious
regularity, the exponential kernel ${\bm{\Sigma}}_{\rm exp}$ is selected as
the input of S-BAR for QuaDRiGa channels. Due to their periodic patterns in
the spatial domain, the Bessel kernel ${\bm{\Sigma}}_{\rm bes}$ is selected as
the input of S-BAR for SSC channels. To provide an ideal baseline, the pre-
trained covariance kernel ${\bm{\Sigma}}_{\rm cov}$ is considered for both
channel models.

####  IV-3 Simulation Results

Figure 2: Model-mismatched case: The NMSE as a function of the number of
pilots $P$ under the assumption of QuaDRiGa channel model.

To better understand the working principle of the proposed S-BAR, we plot Fig.
1 to intuitively show its behavior, where the QuaDRiGa channel model is
considered and the covariance kernel ${\bm{\Sigma}}_{\rm cov}$ is used to
enable S-BAR. From this figure, we have two observations. Firstly, as the
number of samples increases, the confidence interval is gradually reduced. It
indicates that more pilots or antennas can better eliminate the uncertainty of
FAS channels. Secondly, one can note that the sample spacing is usually large.
The reason is that, for each sampling, the proposed S-BAR samples the channel
with the largest posterior variance. When a port is selected and measured, the
channel uncertainty of its nearby ports will decrease, which reduces the trend
of selecting them as samples.

Then, we plot the NMSE as a function of the number of pilots $P$ in Fig. 2 for
QuaDRiGa model and Fig. 3 for SSC model, respectively. From these two figures,
we have the following observations. Firstly, the proposed S-BAR achieves the
highest estimation accuracy in both cases. The reason is that, the existing
methods do not fully utilize the channel prior for estimation. For FAS-OMP,
due to the non-ideal port selection, the information provided by the randomly
measured channels may not be sufficient to capture all channel patterns. For
SeLMMSE, the unmeasured channels are directly obtained by zero-order
interpolation, while their potential estimation errors are not considered. In
contrast, the proposed S-BAR incorporates the effect of prior correlation into
its estimator, which naturally considers the potential estimation errors of
all channels. Through kernel-based sampling and regression, S-BAR can
eliminate the uncertainty of many channels with a few pilots. Secondly, the
S-BAR enabled by the experiential kernels ${\bm{\Sigma}}_{\rm exp}$ and
${\bm{\Sigma}}_{\rm bes}$ can achieve similar performance as that enabled by
covariance kernel ${\bm{\Sigma}}_{\rm cov}$. Recall that ${\bm{\Sigma}}_{\rm
exp}$ and ${\bm{\Sigma}}_{\rm bes}$ are generated by experiential parameters,
while ${\bm{\Sigma}}_{\rm cov}$ is trained from real channel data. This
observation indicates that, even if the real channel covariance
${\mathsf{E}}\left({\bf h}{\bf h}^{\rm H}\right)$ is unknown, experiential
parameters still allow S-BAR to achieve considerable performance.

Figure 3: Model-matched case: The NMSE as a function of the number of pilots
$P$ under the assumption of SSC channel model.

##  V Conclusions

In this paper, we have proposed S-BAR as a general solution to estimate
channels in FASs. Different from the existing channel estimators relying on
channel assumptions, the general S-BAR utilizes the experiential kernel to
acquire CSI in a non-parametric way. Inspired by the Bayesian regression, the
proposed S-BAR can select a few informative channels for measurement and
combine them with experiential kernel to reconstruct high-dimensional FAS
channels. Simulation results reveal that, in both model-mismatched and model-
matched cases, the proposed S-BAR can achieve much higher estimation accuracy
than the existing schemes.

## Acknowledgment

This work was supported in part by the National Natural Science Foundation of
China (Grant No. 62031019).

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  *[MIMO]: multiple-input multiple-output
  *[FAS]: fluid antenna system
  *[FASs]: fluid antenna system
  *[CSI]: channel state information
  *[RF]: radio frequency
  *[S-BAR]: successive Bayesian reconstructor
  *[BS]: base station
  *[AWGN]: additive white Gaussian noise
  *[SNR]: signal-to-noise ratio
  *[NMSE]: normalized mean square error