# Semantic Communications for Wireless Sensing: RIS-aided Encoding and Self-
supervised Decoding

Hongyang Du, Jiacheng Wang, Dusit Niyato, , Jiawen Kang, Zehui Xiong, Junshan
Zhang, , and Xuemin (Sherman) Shen H. Du, J. Wang and D. Niyato are with the
School of Computer Science and Engineering, Nanyang Technological University,
Singapore (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>dniyato@ntu.edu.sg).J. Kang is with the School of Automation, Guangdong
University of Technology, China. (e-mail: kavinkang@gdut.edu.cn)Z. Xiong is
with the Pillar of Information Systems Technology and Design, Singapore
University of Technology and Design, Singapore (e-mail:
zehui_xiong@sutd.edu.sg)J. Zhang is with the Department of Electrical and
Computer Engineering, University of California Davis, USA (e-mail:
jazh@ucdavis.edu)X. Shen is with the Department of Electrical and Computer
Engineering, University of Waterloo, Canada (e-mail<EMAIL_ADDRESS>
###### Abstract

Semantic communications can reduce the resource consumption by transmitting
task-related semantic information extracted from source messages. However,
when the source messages are utilized for various tasks, e.g., wireless
sensing data for localization and activities detection, semantic communication
technique is difficult to be implemented because of the increased processing
complexity. In this paper, we propose the inverse semantic communications as a
new paradigm. Instead of extracting semantic information from messages, we aim
to encode the task-related source messages into a hyper-source message for
data transmission or storage. Following this paradigm, we design an inverse
semantic-aware wireless sensing framework with three algorithms for data
sampling, reconfigurable intelligent surface (RIS)-aided encoding, and self-
supervised decoding, respectively. Specifically, on the one hand, we design a
novel RIS hardware for encoding several signal spectrums into one
MetaSpectrum. To select the task-related signal spectrums for achieving
efficient encoding, a semantic hash sampling method is introduced. On the
other hand, we propose a self-supervised learning method for decoding the
MetaSpectrums to obtain the original signal spectrums. Using the sensing data
collected from real-world, we show that our framework can reduce the data
volume by $95\%$ compared to that before encoding, without affecting the
accomplishment of sensing tasks. Moreover, compared with the typically used
uniform sampling scheme, the proposed semantic hash sampling scheme can
achieve $67\%$ lower mean squared error in recovering the sensing parameters.
In addition, experiment results demonstrate that the amplitude response matrix
of the RIS enables the encryption of the sensing data.

###### Index Terms:

Semantic communications, reconfigurable intelligent surface, wireless sensing,
self-supervised learning

##  I Introduction

With the evolution of the next-generation Internet and the proliferation of
wireless applications, the demand of network resources for data transmission,
storage, and computation has been increasing rapidly. In particular, the
maturity of technologies such as extended reality and digital twins
accelerates the realization of Metaverse and Web $3.0$ concepts. This
consequently leads to a growing demand for communication and computing
support. To meet stringent requirements such as low latency, high reliability,
and high immersion for next-generation Internet applications, the semantic
communication technique is proposed as one of the fundamental approaches for
the sixth generation wireless communications [1]. By transmitting only task-
related semantic information extracted from source messages, semantic
communications are believed to break the conventional Shannon communication
paradigm and bring higher quality of experience to users [2, 3].

While semantic communication techniques have demonstrated their significant
effectiveness in processing source data in multiple modes, e.g., audio [4],
image [5], video [6], and text [7], one of the most promising application
scenarios for semantic communication could be the processing of wireless
sensing data, which is not thoroughly studied yet. The sensing data is
important because that wireless signals are ubiquitous in our daily life, and
can be used to accomplish various tasks requested by service providers.
Specifically, wireless signals not only help users access the Internet more
efficiently, e.g., Metaverse, but also enable indoor positioning and
activities detection more effectively. The wireless sensing data also
facilitates the construction of virtual worlds such as digital twins. Unlike
on-body sensor-based solutions [8], wireless sensing does not require the user
to carry any devices and equipment, which is more practical and convenient.
Additionally, the wireless sensing method is more robust than camera-based
methods particularly in cases of occlusion or inadequate illumination, while
causing fewer privacy issues.

However, the wireless sensing technique has one major limitation. The
transmission and storage of the sensing data, such as signal amplitude and
phase spectrums, consumes a large number of resources [9]. In particular, the
development of communication technologies such as multiple-input multiple-
output and orthogonal frequency-division multiplexing (OFDM) improve the
sensing resolution in the spatial and time-frequency domains, which, however,
further increases the sensing data volume. Therefore, the semantic
communication technique is expected to achieve efficient sensing data
transmission or storage while achieving sensing tasks. This vision is more
meaningful for applications that require long-term storage of sensing data,
such as incremental learning for recognition [10], healthcare services [11]
and Internet-of-Things (IoT) systems and applications [12]. The reason that
semantic communications can “exceed” the Shannon limit is the “impairment” of
the transmitted data, i.e., an effective semantic encoder extracts only task-
independent semantic information from the source messages. However, a
potential pitfall here is that the well-trained semantic encoding and decoding
models for one specific task may fail when the source messages are needed to
accomplish several different tasks. As shown at the top of Fig. 1, instead of
transmitting an image, the semantic encoder can extract sentences describing
the content of the image. This greatly reduces the number of bits that are
required to be transmitted. However, semantic communications would not work
well when the task is not only to know the type and number of fruits in the
images, but also to know the spatial location. In this case, updated semantic
models are required to be re-trained. In a word, semantic communications
achieve efficiency transmission while introducing limitations. For the
wireless sensing data, if we extract only the semantic information used for
localization, the gesture detection task might not be accomplished.

Figure 1: The ideas of conventional, semantic, and inverse semantic
communications. Motivated by the inverse semantic-aware communication, we
propose an inverse semantic-aware encoding and decoding framework and show the
results. Specifically, we select $10$ original signal amplitude/phase spectrum
(Part I) by using our proposed Algorithm 2, and encode them into one
MetaSpectrum by using the RIS and Algorithm 1. After wireless transmission,
the reconstruction results (Part II) are obtained by decoding the MetaSpectrum
using Algorithm 3. The sensing data is collected by real experiments with an
IEEE 802.11ax based test platform [13].

To fill this gap in semantic communications, we propose an inverse semantic-
aware approach by treating the source messages as semantic information of a
hyper-source message. As shown in Fig. 1, the “inverse” means that the
processing of source messages is no longer to extract semantic information,
but to combine multiple source messages (Part I) into one hyper-source message
(Part III) for transmission or storage. Subsequently, by decoding, the
semantic information of the hyper-source message (Part II), i.e., source
messages, can be obtained to support multiple different tasks. Using the
inverse semantic-aware approach, we reduce the data volume for transmission or
storage, while avoiding the task limitations brought by semantic
communications. For the wireless sensing, the source messages are signal
amplitude and phase spectrums, and we call the hyper-source messages as
amplitude and phase MetaSpectrums, respectively. We use the reconfigurable
intelligent surface (RIS) to ensure efficient inverse semantic-aware encoding
and decoding111Our scheme can alternatively be achieved by using active
antennas and processors to simulate the same signal processing as the RIS.
However, higher hardware costs are introduced compared to the scheme using
RIS.. With the RIS’s superior ability to modulate signals, our scheme can be
implemented effectively by modifying a small number of elements on the RIS
without affecting the RIS-aided communications. Unlike most RIS research works
that consider only the phase response matrix of RIS, to the best of our
knowledge, this is the first paper to make full use of the amplitude response
matrix of RIS to help the system design for wireless sensing. The amplitude
response matrix is not only used to reduce the sensing data volume
significantly, but also to encrypt the sensing data because the amplitude
response matrix is inevitable in the decoding process. Visual representation
of the contributions of this paper is shown in Fig. 1, which are summarized as
follows:

  * •

Following the paradigm of the inverse semantic communications, we design a
novel RIS hardware, in which $L$-shaped active sensors are placed behind
transmissive elements, to achieve the inverse semantic-aware wireless sensing
that reduces the sensing data volume to 5% of the original data volume.

  * •

We develop the inverse semantic-aware encoding and decoding methods. The
amplitude response matrix of the RIS embeds prior knowledge in the encoded
sensing signals. The decoding method is based on self-supervised learning,
which can achieve high-quality recovery of the original sensing signals
without pre-training resource consumption.

  * •

We propose an effective semantic hash sampling algorithm to select the task-
related sensing signal spectrums for decoding. The mean squared error (MSE)
between the ground truth and the 2D angles-of-arrival (AoA) estimation results
obtained by the semantic hash sampling scheme is $67\%$ lower than that of
typically used uniform sampling scheme.

  * •

We build an IEEE 802.11ax based test platform [13] to collect the real-world
sensing data, and perform experiments to demonstrate the effectiveness of our
proposed framework.

The remainder of the paper is organized as follows. In Section II, we review
the related work in the literature. Section III introduces the system model,
which contains the novel RIS hardware and the sensing signal model. The
inverse semantic-aware encoding and decoding methods are proposed in Section
IV and Section V, respectively. Section VI presents the experiment results. In
Section VII, we present the conclusion and discuss some potential research
directions.

##  II Related Work

In this section, we provide a brief review of three related techniques, i.e.,
wireless sensing, RIS, and spectral snapshot compressive imaging.

###  II-A Wireless Sensing

Wireless signals such as WiFi [14] have been used for a variety of sensing
tasks, from large-scale intrusion detection and indoor localization, to small-
scale gesture recognition and breathing monitoring. Moreover, with the rapid
advancement of wireless sensing techniques, next-generation Internet service
providers (SPs) can construct digital models of the physical world (for
digital twin service) or conduct analysis of users’ behaviors (for Metaverse
services) [15, 16, 11]. We introduce a completed sensing process. First,
wireless IoT devices collect the sensing data. With the frequency conversion
and channel estimation, the channel state information (CSI) can be obtained as
the sampled version of channel frequency response (CFR), which is proven to be
one of the most effective signal sources for sensing tasks such as human
activities detection [17] and passive localization [18]. The CFR can be
expressed as a complex matrix, e.g., rows are sub-carriers frequencies and
columns are active sensors. For easy transmission and storage, the IoT devices
can decompose the CFR complex matrix into an amplitude spectrum and a phase
spectrum. By using a three-dimensional multiple signal classification
(3D-MUSIC) algorithm, 3D spectrum can be obtained using the amplitude and
phase spectrums, which contains the information of 2D AoA and time of flight
(ToF). The 2D AoA means the elevation and azimuth AoA as shown in Fig. 2 (Part
I). The obtained 3D spectrum can be then used to achieve several purposes,
e.g., physical world user localization [19], or activities detection. A
challenge in the above process is that the storage or transmission causes
excessive network resource consumption due to a large amount of sensing data.

###  II-B Reconfigurable Intelligent Surface

Significant developments in RIS-aided wireless communications have been
witnessed over the past $3$ years, from hardware and algorithms design to deep
integration with various technologies. One of the most important application
scenarios is to enhance wireless sensing [20], such as indoor localization
[21] and direction-of-arrival estimation [22]. However, the existing methods
typically aim to improve the sensing accuracy through signal enhancement by
the RIS. The signal control capability of the RIS is not fully utilized, and
most literature is limited in the study of reflective RIS that cannot achieve
complete coverage. Fortunately, with the deepening understanding of RIS
hardware, transmissive and refractive RISs are gaining more and more attention
[23, 24, 25]. Simultaneously transmitting and reflecting (STAR) RIS [24] and
intelligent omni-surface (IOS) [25] have been proposed as novel instances of
RIS to achieve full-dimensional communications. We believe that STAR RIS or
IOS can also bring further improvement to wireless sensing. In addition to
improving sensing performance by intuitively enhancing signals, adjustment to
the amplitude of transmissive signals can be used as prior knowledge to
achieve efficient compression of wireless sensing data, which will be
discussed in this paper.

###  II-C Spectral Snapshot Compressive Imaging

Capturing high dimension (HD) data is a long-term challenge in signal
processing and related fields [26]. With theoretical guarantees, snapshot
compressed imaging (SCI) uses two-dimensional (2D) detectors to capture HD,
e.g., 3D, data in snapshot measurements using novel optical design. Then,
reconstruction algorithms are applied to obtain the required HD data cubes
[27, 28]. SCI has been used in many fields such as hyper-spectral imaging,
video, holography, tomography, focal depth imaging, polarization imaging, and
microscopy [29]. However, there is no prior work discussing how to apply SCI
to compressed sensing signals in the time dimension. The reason is that the
highly dynamic nature of sensing signals brings difficulties to detector
hardware design, coded aperture structure, and decompression algorithms. To
fill this gap, in Section IV-A, we use the novel RIS hardware to perform one
kind of special SCI to the sensing data. Using our proposed inverse semantic-
aware encoding and decoding methods, the compression and self-supervised
decompression of the sensing data can be achieved on time scale. Note that our
design is different from compressive sensing (CS) methods in wireless sensing,
and in fact can be used to further improve the performance of wireless CS
systems.

One primary objective of this study is to solve an important problem of
overwhelming storage or transmission resources consumption in the wireless
sensing. Inspired by the SCI system, we propose an encoding and decoding
framework using the RIS to achieve inverse semantic-aware sensing, which
significantly reduces the data volume and does not affect the accomplishment
of various sensing tasks.

##  III System Models

Figure 2: The framework of the proposed inverse semantic-aware wireless
sensing system.

Wireless signals contain user information such as activities and walking
trajectories, and can preserve user privacy better than camera-based methods.
Thus, mobile application SPs can use wireless signals to provide better
services to users. For example, healthcare SPs can provide medical advice by
analyzing the user’s sleeping postures, and Metaverse SPs can customize
virtual traveling scenes by positioning the users. To meet the needs of
ubiquitous sensing data collection, we consider a 3D wireless indoor
communications scenario as an example. As shown in Fig. 2 (Part I), a multi-
antenna transmitter, e.g., IoT devices or WiFi router, transmits signals to
multiple users with the help of an RIS. Different from the conventional scheme
that uses RIS to improve sensing accuracy by enhancing signal strength, in
this section, we propose a novel RIS hardware design to enable RIS with
wireless sensing capability. Then, we analyze the mathematical formulas of the
received sensing signals.

###  III-A Novel Hardware of Reconfigurable Intelligent Surface

To enable RIS to sense the environment, a widely used solution is to replace
some reflecting elements on the RIS with active sensors, e.g., for channel
estimation using CS [30]. Thus, a part of the RIS elements can switch between
two operation modes, i.e., i) channel sensing mode that is used to estimate
the channels, ii) reflection mode that reflects the signal. However, we can
see that the RIS cannot assist communications in mode 1. We do not adopt
directly the aforementioned solution since our goal is not merely to estimate
the channel, but also constantly to sense the environment for the purposes of
localizing and detecting user activities. To enable the RIS to assist the
sensing function without affecting its communications auxiliary function, we
first integrate the RIS with a small number of simultaneous transmitting and
reflecting patches [24], which are called transmissive elements in this paper
for convenience. Specifically, as shown in Fig. 2 (Part II), $L$-shaped
$\left(M+N+1\right)$ transmissive elements are deployed on the RIS, and active
sensors are placed behind the transmissive elements to receive the signals
modulated by the RIS.

######  Remark 1.

The reason for using the $L$-shaped array is that such a structure has more
accurate 2D AoA estimation results than other structures, e.g., cross, linear,
and rectangular arrays. This conclusion can be obtained by comparing the
Cramer-Rao Bound metrics of different structures [31, 32].

Accordingly, the signal incident on the $q^{\rm th}$ transmissive element can
be transmitted and reflected as [24]

${\beta_{i,q}}{{\rm exp}{\left(j{\delta_{i,q}}\right)}},\qquad
i\in\left\\{{T,R}\right\\},$ (1)

where $i=T$ is for transmission coefficients and $i=R$ is for reflection
coefficients. Note that, for each element, the responses of the RIS for
transmission and reflection modes can be designed independently from each
other [33]. In the following, we focus on the sensing function that only uses
the transmitted signals. The reflection coefficients can be designed
independently, which is outside the scope of this paper. Thus, after one path
signal penetrates the $q^{\rm th}$ transmissive element on the RIS, the
amplitude of the signal is multiplied by ${\beta_{T,q}}$, and the phase is
added by ${\delta_{T,q}}$.

In much of the literature, the amplitude and phase response of each element on
the RIS is assumed to be constant over the signal bandwidth [34]. Although
this assumption is acceptable when the bandwidth is narrow, it may become
inaccurate when receiving multiple sub-carriers with different frequencies in
a large range [34]. In our system model, we consider that the transmitter
sends the wireless signals modulated by OFDM technology into $K$ sub-
carriers222The OFDM is a widely used modulation method, which makes our
analysis general. Moreover, OFDM can provide multi-carriers information, which
is useful for signal parameters estimation.. Because that $K$ might be large,
e.g., $2048$ OFDM sub-carriers are used to transmit data in the IEEE 802.11ax
protocol, we consider the practical case in which the element on the RIS has
different responses to signals with different frequencies. Thus, the amplitude
and phase response matrices of the $L$-shaped transmissive elements to $K$
sub-carriers at time $t$ can be expressed as

${\bf\Phi}_{A}^{\left(t\right)}=\\!\\!\left[\\!\\!{\begin{array}[]{*{20}{c}}{\beta_{{f_{1}}}^{[1,0]}}&\\!\\!\\!\cdots\\!\\!\\!&{\beta_{{f_{1}}}^{[M,0]}}&{\beta_{{f_{1}}}^{[0,0]}}&{\beta_{{f_{1}}}^{[0,1]}}&\\!\\!\\!\cdots\\!\\!\\!&{\beta_{{f_{1}}}^{[0,N]}}\\\
\\!\\!\\!\vdots\\!\\!\\!&\\!\\!\\!\ddots\\!\\!\\!&\\!\\!\\!\vdots\\!\\!\\!&\\!\\!\\!\vdots\\!\\!\\!&\\!\\!\\!\vdots\\!\\!\\!&\\!\\!\\!\ddots\\!\\!\\!&\\!\\!\\!\vdots\\!\\!\\!\\\
{\beta_{{f_{K}}}^{[1,0]}}&\\!\\!\\!\cdots\\!\\!\\!&{\beta_{{f_{K}}}^{[M,0]}}&{\beta_{{f_{K}}}^{[0,0]}}&{\beta_{{f_{K}}}^{[0,1]}}&\\!\\!\\!\cdots\\!\\!\\!&{\beta_{{f_{K}}}^{[0,N]}}\end{array}}\\!\\!\right]\\!,$
(2)

and

${\bf\Phi}_{P}^{\left(t\right)}=\\!\\!\left[\\!\\!{\begin{array}[]{*{20}{c}}{\delta_{{f_{1}}}^{[1,0]}}&{\\!\\!\\!\cdots\\!\\!\\!}&{\delta_{{f_{1}}}^{[M,0]}}&{\delta_{{f_{1}}}^{[0,0]}}&{\delta_{{f_{1}}}^{[0,1]}}&{\\!\\!\\!\cdots\\!\\!\\!}&{\delta_{{f_{1}}}^{[0,N]}}\\\
{\\!\\!\\!\vdots\\!\\!\\!}&{\\!\\!\\!\ddots\\!\\!\\!}&{\\!\\!\\!\vdots\\!\\!\\!}&{\\!\\!\\!\vdots\\!\\!\\!}&{\\!\\!\\!\vdots\\!\\!\\!}&{\\!\\!\\!\ddots\\!\\!\\!}&{\\!\\!\\!\vdots\\!\\!\\!}\\\
{\delta_{{f_{K}}}^{[1,0]}}&{\\!\\!\\!\cdots\\!\\!\\!}&{\delta_{{f_{K}}}^{[M,0]}}&{\delta_{{f_{K}}}^{[0,0]}}&{\delta_{{f_{K}}}^{[0,1]}}&{\\!\\!\\!\cdots\\!\\!\\!}&{\delta_{{f_{K}}}^{[0,N]}}\end{array}}\\!\\!\right],$
(3)

respectively, where $f_{i}$ denotes the frequency of the $i^{\rm th}$ sub-
carrier, and $\left[x,y\right]$ denotes the location of the activate element.
As shown in Fig. 2 (Part I), $0\leqslant x\leqslant M$ and $0\leqslant
y\leqslant N$ indicate the locations of transmissive elements that are in the
$X$-direction and $Y$-direction of the $L$-shape array, respectively.

In the following, we analyze the sensing signal model, and propose the
amplitude and phase response matrices design scheme.

###  III-B Sensing Signal Model

To better understand the phase differences of the incident signals from
different directions, we analyze the signals that impact the transmissive
elements at different positions separately, i.e., the origin, the
$X$-direction, and the $Y$-direction elements. At time $t$, the CFR of the
multipath signals corresponding to the $k$-th OFDM sub-carrier obtained by the
transmissive element located at the origin of the $L$-shaped array, can be
expressed as

$\displaystyle
h_{f_{k}}^{\left[{0,0}\right]}\left(t\right)=\sum\limits_{i=1}^{I}{\alpha_{i}^{\left[{0,0}\right]}{e^{-j2\pi{f_{k}}{\tau_{i}}}}},$
(4)

where $\left[{0,0}\right]$ represents the origin point, ${\alpha_{i}}$ and
${\tau_{i}}$ are the complex signal amplitude attenuation and time delay of
the $i$-th propagation path, respectively, ${f_{k}}$ is the frequency of the
$k$-th sub-carrier, and $I$ is the total number of multipaths. Since different
elements are placed at different positions in $L$-shaped array, the signal
needs to travel different distances to arrive each transmissive element.
Taking $h_{f_{k}}^{\left[{0,0}\right]}$ as a reference, therefore, the CFR
obtained by the $m$-th $X$-direction transmissive element can be expressed as

$\displaystyle
h_{f_{k}}^{\left[{m,0}\right]}\left(t\right)=\sum\limits_{i=1}^{I}{\alpha_{i}^{[m,0]}{e^{-j2\pi{f_{k}}\left({{\tau_{i}}+m\frac{{d\cos\left({{\theta_{i}}}\right)\sin\left({{\varphi_{i}}}\right)}}{c}}\right)}}},$
(5)

where $\alpha_{i}^{[m]}$ is the signal amplitude attenuation, $d$ is the
antenna spacing equals half wave length, ${\theta_{i}}$ and ${\varphi_{i}}$
represent the elevation angle and azimuth angle of the incident signal,
respectively, as shown in Fig. 2 (Part I), and $c$ is the signal propagation
speed in the air. Similarly, we can obtain the CFR of the $n$-th $Y$-direction
transmissive element as

$\displaystyle
h_{f_{k}}^{\left[{0,n}\right]}\left(t\right)=\sum\limits_{i=1}^{I}{\alpha_{i}^{[0,n]}{e^{-j2\pi{f_{k}}\left({{\tau_{i}}+n\frac{{d\sin\left({{\theta_{i}}}\right)\sin\left({{\varphi_{i}}}\right)}}{c}}\right)}}}.$
(6)

Therefore, at time $t$, the overall CFR obtained by the $L$-shaped
transmissive element array on the RIS can be expressed as a CFR matrix as
follows:

$\displaystyle{{\bm{H}}_{xoy}^{\left(t\right)}}=\left[{{{\bm{H}}_{x}^{\left(t\right)}}\;{{\bm{H}}_{0}^{\left(t\right)}}\;{{\bm{H}}_{y}^{\left(t\right)}}}\right]$
$\displaystyle=\\!\left[{\underbrace{\begin{array}[]{*{20}{c}}{h_{{f_{1}}}^{\left[{1,0}\right]}}&\\!\\!\\!\cdots\\!\\!\\!&{h_{{f_{1}}}^{\left[{M,0}\right]}}\\\
\vdots&\\!\\!\\!\ddots\\!\\!\\!&\vdots\\\
{h_{{f_{K}}}^{\left[{1,0}\right]}}&\\!\\!\\!\cdots\\!\\!\\!&{h_{{f_{K}}}^{\left[{M,0}\right]}}\end{array}}_{{{\bm{H}}_{x}}}\underbrace{\begin{array}[]{*{20}{c}}{h_{{f_{1}}}^{\left[{0,0}\right]}}\\\
\vdots\\\
{h_{{f_{K}}}^{\left[{0,0}\right]}}\end{array}}_{{{\bm{H}}_{0}}}\underbrace{\begin{array}[]{*{20}{c}}{h_{{f_{1}}}^{\left[{0,1}\right]}}&\\!\\!\\!\cdots\\!\\!\\!&{h_{{f_{1}}}^{\left[{0,N}\right]}}\\\
\vdots&\\!\\!\\!\ddots\\!\\!\\!&\vdots\\\
{h_{{f_{K}}}^{\left[{0,1}\right]}}&\\!\\!\\!\cdots\\!\\!\\!&{h_{{f_{K}}}^{\left[{0,N}\right]}}\end{array}}_{{{\bm{H}}_{y}}}}\right].$
(16)

Note that ${{\bm{H}}_{xoy}^{\left(t\right)}}$ is the original sensing data
that can support various sensing tasks. Because of the high available sampling
frequency of the sensing device, e.g., $300$ times in one second [16], and
novel services that require long-term sensing, a large amount of
${{\bm{H}}_{xoy}^{\left(t\right)}}$ would be collected. To reduce the
resources costed to store and transmit ${{\bm{H}}_{xoy}^{\left(t\right)}}$, we
propose the inverse semantic-aware encoding and decoding methods in the
following sections, respectively.

##  IV Inverse Semantic-aware Encoding

In this section, we introduce the inverse semantic-aware RIS-aided encoding
method to compress multiple signal spectrums into one. Two steps, i.e.,
differential encoding and shifting addition compression, are discussed.
Moreover, we propose a semantic hash sampling method to select the task-
related signal spectrum to record.

###  IV-A Encoding Method

Figure 3: The process of modulating the amplitude and phase spectrums through
RIS, and then performing differential encoding and shifting addition.

One can observe from (III-B) that every element in the CFR matrix is a complex
number, which denotes the amplitude and phase of the CFR. Taking
${{\bf{H}}_{x}^{\left(t\right)}}$ as an example, it can be further decomposed
into the amplitude and phase spectrums as

$\displaystyle{{\bm{H}}_{x}^{\left(t\right)}}\to\left\\{{{{\bm{H}}_{x_{a}}^{\left(t\right)}},{{\bm{H}}_{x_{p}}^{\left(t\right)}}}\right\\}$
$\displaystyle=\\!\\!\left\\{\\!{\underbrace{\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}{\left\|{h_{f_{1}}^{\left[{1,0}\right]}}\right\|}&\\!\\!\\!\\!\cdots\\!\\!\\!\\!&{\left\|{h_{f_{1}}^{\left[{M,0}\right]}}\right\|}\\\
\\!\\!\\!\\!\vdots\\!\\!\\!\\!&\\!\\!\\!\\!\ddots\\!\\!\\!\\!&\\!\\!\\!\\!\vdots\\!\\!\\!\\!\\\
{\left\|{h_{f_{K}}^{\left[{1,0}\right]}}\right\|}&\\!\\!\\!\\!\cdots\\!\\!\\!\\!&{\left\|{h_{f_{K}}^{\left[{M,0}\right]}}\right\|}\end{array}}\\!\\!\\!\right]}_{\textit{amplitude
matrix}},\underbrace{\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}{\angle
h_{f_{1}}^{\left[{1,0}\right]}}&\\!\\!\\!\\!\cdots\\!\\!\\!\\!&{\angle
h_{f_{1}}^{\left[{M,0}\right]}}\\\
\\!\\!\\!\\!\vdots\\!\\!\\!\\!&\\!\\!\\!\\!\ddots\\!\\!\\!\\!&\\!\\!\\!\\!\vdots\\!\\!\\!\\!\\\
{\angle h_{f_{K}}^{\left[{0,1}\right]}}&\\!\\!\\!\\!\cdots\\!\\!\\!\\!&{\angle
h_{f_{K}}^{\left[{M,0}\right]}}\end{array}}\\!\\!\\!\right]}_{\textit{phase
matrix}}}\\!\right\\}\\!,$ (23)

where $\cdot\\!\to\\!\cdot$ denotes the amplitude and phase extraction
operation, $\left\\{\cdot\right\\}$ represents the set of matrices,
$\left\|\cdot\right\|$ is the Euclidean norm operator, and $\angle
h_{f_{k}}^{\left[{m,0}\right]}$ denotes the signal phase of
$h_{f_{k}}^{\left[{m,0}\right]}$. Through the same way, ${{\bf{H}}_{0}}$ and
${{\bf{H}}_{y}}$ can be expressed as amplitude and phase spectrums,
respectively. Hence, the CFR extracted from the $L$-shaped transmissive
element array on the RIS can be expressed as

$\displaystyle{{\bf{H}}_{xoy}^{\left(t\right)}}=\left[{{{\bf{H}}_{x}^{\left(t\right)}}{\
}{{\bf{H}}_{0}^{\left(t\right)}}{\
}{{\bf{H}}_{y}^{\left(t\right)}}}\right]\to\left\\{{{\bf{H}}_{A}^{\left(t\right)}},{{\bf{H}}_{P}^{\left(t\right)}}\right\\},$
(24)

where
${{\bf{H}}_{A}^{\left(t\right)}}=\left[{{\bm{H}}_{x_{a}}}\;{{\bm{H}}_{0_{a}}}\;{{\bm{H}}_{y_{a}}}\right]$
and
${{\bf{H}}_{P}^{\left(t\right)}}=\left[{{\bm{H}}_{x_{p}}}\;{{\bm{H}}_{0_{p}}}\;{\bm{H}}_{y_{p}}\right]$
denote the overall amplitude and phase at time $t$, respectively. As shown in
Fig. 3 (Parts I and II), after being modulated by the transmissive elements,
we can express $T$ received amplitude and phase spectrums by the $L$-shaped
active sensor array in two sets as

${\bm{Y}}_{A}=\left\\{{\bm{H}}_{A}^{\left(1\right)}\circ{\bm{\Phi}}_{A}^{\left(1\right)},\ldots,{\bm{H}}_{A}^{\left(T\right)}\circ{\bm{\Phi}}_{A}^{\left(T\right)}\right\\},$
(25)

and

${\bm{Y}}_{P}=\left\\{{\bm{H}}_{P}^{\left(1\right)}+{\bm{\Phi}}_{P}^{\left(1\right)},\ldots,{\bm{H}}_{P}^{\left(T\right)}+{\bm{\Phi}}_{P}^{\left(T\right)}\right\\},$
(26)

respectively, where $\circ$ is the Hadamard product calculator, a.k.a.,
element-wise product. In the following, we encode the 3D data ${\bm{Y}}_{A}$
and ${\bm{Y}}_{P}$ onto 2D measurements, respectively. The encoding idea is
inspired by the SCI system that compresses several optical spectrums of an
object over multiple wavelengths into one spectrum, or several frames of a
high-speed video into one frame. Specifically, the 3D data is first modulated
by a coded aperture, and then spectrally dispersed by the dispersing element,
and finally integrated across the spectral dimension to a 2D measurement. For
the 3D sensing data ${\bm{Y}}_{A}$ and ${\bm{Y}}_{P}$, although the spectral
dispersion process can be performed by low-power computing elements, there are
several difficulties in adopting the compression scheme as in the SCI system:

  1. D1)

The fixed coded aperture in the SCI system is hard to be used in encoding
signal spectrums that change dramatically on the time scale. However, the
time-varying coded aperture scheme [35] increases the hardware cost and
consume more storage space to record the patterns.

  2. D2)

It is difficult for system designers to strike the balance between decoding
performance and resource consumption. The inverse decoding problem is hard to
be solved by traditional methods. The deep learning method, e.g.,
convolutional neural networks, requires expensive well-labeled dataset and a
long time training [36], making the aperture patterns cannot be changed
frequently.

  3. D3)

Signal spectrums are more sensitive than spectral images or video frames. We
find from the experiments that the decoded sensing signal spectrums may lead
to errors when performing some sensing tasks that are sensitive to the
deviations in signal phase values, e.g., localization.

To overcome the aforementioned difficulties, we rethink the SCI system from
hardware design to software algorithms. For (D1) and (D2), we can observe from
(25) that the amplitude response matrix of the RIS has potential to perform a
similar function as the coded aperture in the SCI system. It has been shown
that the reconfiguration time for the RIS to change the response matrix is
around $33$ ns [37]. Therefore, by changing the response matrix over time, the
low-cost transmissive elements on the RIS can encode the sensing signals. In
addition, the response values can be obtained from the hardware design
parameters. This saves the storage resources to record a large number of
original response values. Moreover, the amplitude and phase response values
are discrete numbers, which can be determined by the number of coding bits.
For example, $4$-bit coding bringse $16$ different available response values.
Following that, we propose a self-supervision decoding algorithm for arbitrary
RIS response matrices, which is discussed in Section V. Error negligible
decoding results are achieved without pre-training resource consumption.

To solve (D3), we compress the differential matrices of the amplitude and
phase spectrums instead of the original spectrums to ensure the sensing
performance. Unlike channel estimation, which focuses on accurately obtaining
the CSI to better perform channel equalization, wireless sensing focuses on
extracting information describing the physical environment from the CSI, e.g.,
2D AoA and time of flight. This information is hidden in the value difference
of amplitude and phase spectrums obtained by the sensors at different
locations. For example, the phase difference between active sensors supports
the signal AoA estimation. Another advantage to encode the differential
spectrum is that the differential spectrum tends to be smoother than the
original spectrum, due to the existence of correlation. This results in
improved decoding performance. We show that real images can also benefit from
the differential encoding in Fig. 10 using the dataset [38].

The differential encoding and shifting addition compression methods are
presented in the following.

####  IV-A1 Differential Encoding

We first focus on the amplitude spectrum set ${\bm{Y}}_{A}$. Let
${\bm{Y}}_{A}\left\\{i\right\\}$ denote the $i^{\rm th}$ matrix in
${\bm{Y}}_{A}$. Each column in ${\bm{Y}}_{A}\left\\{i\right\\}$ represents the
amplitude values of received signals at different frequencies by an active
sensor, after amplitude modulation by the transmissive element on the RIS. Let
${\bm{Y}}_{A^{\prime}}\left\\{i\right\\}$ denote the
${\bm{Y}}_{A}\left\\{i\right\\}$ after the differential encoding.
Specifically, we let the $j^{\rm th}$ column in
${\bm{Y}}_{A^{\prime}}\left\\{i\right\\}$ store the difference values of the
$j^{\rm th}$ column and $\left(j-1\right)^{\rm th}$ column in
${\bm{Y}}_{A}\left\\{i\right\\}$ as

${{\bm{Y}}_{A^{\prime}}}\left\\{i\right\\}\left[{j,:}\right]={{\bm{Y}}_{A}}\left\\{i\right\\}\left[{j,:}\right]-{{\bm{Y}}_{A}}\left\\{i\right\\}\left[{j-1,:}\right],$
(27)

where $j=2,\ldots,L$. The first columns in
${\bm{Y}}_{A^{\prime}}\left\\{i\right\\}$ and ${\bm{Y}}_{A}\left\\{i\right\\}$
are the same. Then, we have

${{\bm{Y}}_{A^{\prime}}}\left\\{i\right\\}\left[{1,:}\right]={{\bm{Y}}_{A}}\left\\{i\right\\}\left[{1,:}\right].$
(28)

Similar differential encoding method can be used for the received phase
spectrum set ${\bm{Y}}_{P}$. For the $i^{\rm th}$ matrix in ${\bm{Y}}_{P}$,
i.e., ${{\bm{Y}}_{P}}\left\\{i\right\\}$, we obtain the differential encoded
matrix ${{\bm{Y}}_{P^{\prime}}}\left\\{i\right\\}$ by

${{\bm{Y}}_{P^{\prime}}}\left\\{i\right\\}\left[{j,:}\right]={{\bm{Y}}_{P}}\left\\{i\right\\}\left[{j,:}\right]-{{\bm{Y}}_{P}}\left\\{i\right\\}\left[{j-1,:}\right],$
(29)

where $j=2,\ldots,L$. Considering that the phase response value of the RIS is
added to the signal phase value, we let the first column in
${\bm{Y}}_{P^{\prime}}\left\\{i\right\\}$ be the first column in
${\bm{Y}}_{P}\left\\{i\right\\}$ minus the phase response of the first
transmissive element as

${{\bm{Y}}_{P^{\prime}}}\left\\{i\right\\}\left[{1,:}\right]={{\bm{Y}}_{P}}\left\\{i\right\\}\left[{1,:}\right]-{\bm{\Phi}}_{P}^{\left(i\right)}\left[{1,:}\right].$
(30)

To use the amplitude response matrix of the RIS as the prior knowledge, we
multiply the amplitude response matrix of the RIS at the $i^{\rm th}$ moment
and ${\bm{Y}}_{P^{\prime}}\left\\{i\right\\}$ by elements as

${{\bm{Y}}_{P^{\prime}}}\left\\{i\right\\}={{\bm{Y}}_{P^{\prime}}}\left\\{i\right\\}\circ{\bm{\Phi}}_{A}^{\left(i\right)}.$
(31)

In addition to the steps of (27), (28), (29), (30), and (31), the transmissive
elements on the RIS should be designed by following Remark 2 to make the
amplitude response matrix of the RIS available as a special coded aperture,
i.e., prior knowledge used in decoding.

######  Remark 2.

To achieve differential encoding, we should let every transmissive element on
the RIS have the same hardware structure. Thus, different transmissive
elements have the same amplitude and phase response to the signals with the
same frequency, as shown in Fig. 3 (Part I). Specifically, every column in (2)
and (3) is the same. This ensures that each column of
${\bm{Y}}_{A^{\prime}}\left\\{i\right\\}$ can be represented as the signal
amplitude difference values multiplied by the amplitude response values of the
RIS as in (32).

Then, we can express ${\bm{Y}}_{A^{\prime}}\left\\{i\right\\}$ and
${\bm{Y}}_{P^{\prime}}\left\\{i\right\\}$ as

${\bm{Y}}_{A^{\prime}}\left\\{i\right\\}={\bm{H}}_{A^{\prime}}^{\left(i\right)}\circ{\bf\Phi}_{A}^{\left(i\right)},$
(32)

and

${\bm{Y}}_{P^{\prime}}\left\\{i\right\\}={\bm{H}}_{P^{\prime}}^{\left(i\right)}\circ{\bf\Phi}_{A}^{\left(i\right)},$
(33)

where ${\bm{H}}_{A^{\prime}}^{\left(i\right)}$ and
${\bm{H}}_{P^{\prime}}^{\left(i\right)}$ are the $i^{\rm th}$ differential
encoded amplitude and phase spectrums, respectively, and
${\bf\Phi}_{A}^{\left(i\right)}$ can be regarded as the corresponding
codebook.

Algorithm 1 The algorithm for inverse semantic-aware encoding.

Input: The received amplitude and phase spectrums in the active sensors:
${\bm{Y}}_{A}$ and ${\bm{Y}}_{P}$
Output: The amplitude and phase MetaSpectrums: $\bm{Z}_{A}$ and $\bm{Z}_{P}$

1:## Achieve differential encoding

2:for Every ${\bm{Y}}_{A}\left\\{i\right\\}$ in ${\bm{Y}}_{A}$ do

3: Obtain ${\bm{Y}}_{A^{\prime}}\left\\{i\right\\}$ according to (27) and (28)

4:for Every ${\bm{Y}}_{P}\left\\{i\right\\}$ in ${\bm{Y}}_{P}$ do

5: Obtain ${\bm{Y}}_{P^{\prime}}\left\\{i\right\\}$ according to (29), (30),
and (31)

6:## Achieve shifting addition compression

7:Use ${{\bm{Y}}_{A^{\prime}}}$ to obtain ${\bm{X}}_{A}$ according to (34)

8:Use ${{\bm{Y}}_{P^{\prime}}}$ to obtain ${\bm{X}}_{P}$ according to (37)

9:Obtain amplitude MetaSpectrum $\bm{Z}_{A}$ according to (35)

10:Obtain phase MetaSpectrum $\bm{Z}_{P}$ according to (36)

11:return $\bm{Z}_{A}$ and $\bm{Z}_{P}$

####  IV-A2 Shifting Addition

To replace the spatial shifting operation to the object spectrum that is
performed by a dispersing lens in the SCI system, we perform zero compensation
processing to the amplitude and phase spectrums as follows:

${{\bm{X}}_{A}}=\\!\left\\{\\!{\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}{{{\bm{Q}}_{1}}\left(1\right)}\\\
{{{\bm{Y}}_{A^{\prime}}}\left\\{1\right\\}}\\\
{{{\bm{Q}}_{2}}\left(1\right)}\end{array}}\\!\\!\\!\right]\\!\\!,\\!\ldots\\!,\\!\\!\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}{{{\bm{Q}}_{1}}\left(i\right)}\\\
{{{\bm{Y}}_{A^{\prime}}}\left\\{i\right\\}}\\\
{{{\bm{Q}}_{2}}\left(i\right)}\end{array}}\\!\\!\\!\right]\\!\\!,\\!\ldots\\!,\\!\\!\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}{{{\bm{Q}}_{1}}\left(T\right)}\\\
{{{\bm{Y}}_{A^{\prime}}}\left\\{T\right\\}}\\\
{{{\bm{Q}}_{2}}\left(T\right)}\end{array}}\\!\\!\\!\right]}\\!\right\\},$ (34)

where ${\bm{Q}_{1}}\left(i\right)\in{\mathbb{R}^{\left({i-1}\right)D\times
L}}$, ${\bm{Q}_{2}}\left(i\right)\in{\mathbb{R}^{\left({T-i}\right)D\times
L}}$, ${{\bm{X}}_{A}}\in{\mathbb{R}^{\left(D\left({T-1}\right)+K\right)\times
L}}$, every elements in both ${\bm{Q}_{1}}$ and ${\bm{Q}_{2}}$ is zero, and
$D$ is the unit displacement step.

Thus, the amplitude MetaSpectrum, $\bm{Z}_{A}$, can be obtained by

$\bm{Z}_{A}=\sum\limits_{i=1}^{T}{{{\bm{X}}_{A}}\left\\{i\right\\}},$ (35)

where $\bm{Z}_{A}\in{\mathbb{R}^{\left(K+\left({T-1}\right)D\right)\times L}}$
can be transmitted or stored. Similarly, the phase MetaSpectrum, $\bm{Z}_{P}$,
can be expressed as

$\bm{Z}_{P}=\sum\limits_{i=1}^{T}{{{\bm{X}}_{P}}\left\\{i\right\\}},$ (36)

where

${{\bm{X}}_{P}}=\\!\left\\{\\!{\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}{{{\bm{Q}}_{1}}\left(1\right)}\\\
{{{\bm{Y}}_{P^{\prime}}}\left\\{1\right\\}}\\\
{{{\bm{Q}}_{2}}\left(1\right)}\end{array}}\\!\\!\\!\right]\\!\\!,\\!\cdots\\!,\\!\\!\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}{{{\bm{Q}}_{1}}\left(i\right)}\\\
{{{\bm{Y}}_{P^{\prime}}}\left\\{i\right\\}}\\\
{{{\bm{Q}}_{2}}\left(i\right)}\end{array}}\\!\\!\\!\right]\\!\\!,\\!\cdots\\!,\\!\\!\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}{{{\bm{Q}}_{1}}\left(T\right)}\\\
{{{\bm{Y}}_{P^{\prime}}}\left\\{T\right\\}}\\\
{{{\bm{Q}}_{2}}\left(T\right)}\end{array}}\\!\\!\\!\right]}\\!\right\\}.$ (37)

The overall RIS-aided encoding method is in Algorithm 1, which has polynomial
complexity. After the RIS-aided encoding, we observe that the sensing data
volume is significantly reduced. To indicate the efficiency of data
compression, we define the data compression ratio, i.e., $\rho$, as the ratio
of the number of elements in the received amplitude and phase spectrums and
that in the coded MetaSpectrums. The analysis of $\rho$ is given in
Proposition 1.

######  Proposition 1.

The data compression ratio $\rho$ of our proposed inverse semantic-aware
coding method is ${1}/{T}$.

###### Proof:

The number of elements in the $T$ recorded signal amplitude and phase spectrum
is $2KLT$. The number of elements in $\bm{Z}_{A}$ and $\bm{Z}_{P}$ is
$2{\left({K+\left({T-1}\right)D}\right)\times L}$. Thus, $\rho$ can be
expressed as

$\rho=\frac{2{\left({K+\left({T-1}\right)D}\right)\times L}}{2{KLT}}{\text{ =
}}\frac{1}{T}+\left({1-\frac{1}{T}}\right)\frac{D}{K}.$ (38)

Since $D$ is small especially compared to $K$, e.g., $D=1$ in [28] and
$K=2048$ in the IEEE 802.11ax protocol,
$\left({1-\frac{1}{T}}\right)\frac{D}{K}$ in (38) can be ignored. Thus, the
value of $\rho$ is close to ${1}/{T}$, which completes the proof. ∎

Note that in the above discussion, we encode $T$ amplitude or phase spectrums
into one spectrum. However, the $T$ spectrums does not need and should not be
sensed continuously in time. The reason is that the wireless channel remains
stable during the channel coherence time. Specifically, as the moving or
action speed of people is limited, the CSI within the channel coherence time
can be considered as constant without loss of precision [39]. Considering that
the available maximal sensing frequency of the active sensors is much higher
than the required frequency, we next propose a sampling scheme that selects
the most relevant spectrum for the completion of sensing tasks over the
channel coherence time for recording and encoding.

###  IV-B Semantic Hash Sampling

Figure 4: The process of generating the resized matrices from the amplitude
and phase spectrums, and then obtaining the semantic hash fingerprint.

We divide the time into segments. Without loss of generality, we consider that
the active sensors can perform $T_{N}$ times sensing in one time segment. From
each time segment, one pair of amplitude and phase spectrums is selected to
record. In the $i^{\rm th}$ time segment, we express $T_{N}$ received
amplitude and $T_{N}$ phase spectrums as two sets, i.e.,
${\bm{S}}_{A_{i}}=\left\\{{\bm{H}}_{A_{i}}^{\left(k\right)}\circ{\bm{\Phi}}_{A_{i}}^{\left(k\right)}\right\\}$
$\left(k=1,\ldots,T_{N}\right)$ and
${\bm{S}}_{P_{i}}=\left\\{{\bm{H}}_{P_{i}}^{\left(k\right)}\circ{\bm{\Phi}}_{P_{i}}^{\left(k\right)}\right\\}$,
respectively. The recorded amplitude and phase spectrums that are selected
from the $i^{\rm th}$ time segment are ${\bm{Y}}_{A}{\left\\{i\right\\}}$
$\left(i=1,\ldots,T\right)$ and ${\bm{Y}}_{P}{\left\\{i\right\\}}$,
respectively.

To remove information that is not relevant to the task, the traditional method
is uniform sampling that selects the first pair of amplitude and phase
spectrums in each time segment to record. However, we cannot guarantee that
the first pair in every segment is always the most informative pair.
Therefore, a better solution is to use one indicator to judge the semantic
information richness of the pair of spectrums. As we discussed in Section
IV-A, the information related to sensing tasks is contained in the changes of
amplitude and phase spectrums. Therefore, we can select the pair of spectrums
that has the largest change compared to the previous signal spectrums in each
time segment. Note that the mean square error (MSE) is not recommended to be
used as the indicator to compare the difference between spectrums. The reasons
are given as follows:

  * •

The MSE is calculated using the absolute values of the signal amplitude.
However, the absolute values are not important for sensing tasks. The critical
information is in the changing process of the signal amplitude over time [14].

  * •

The results of MSE may be affected by several outliers, i.e., signal amplitude
fluctuation at a certain time caused by the interference.

  * •

Because the number of elements in the signal spectrum is large, calculating
MSE brings large resource consumption.

Therefore, we have to propose a new indicator to characterize the semantic
information richness in signal spectrums. Considering the success of the
perceptual image hashing method [40] in the field of image retrieval, we aim
to use a string of characters, i.e., fingerprints, to characterize the
amplitude and phase spectrums. Perceptual image hashing [40] is a family of
algorithms that generate content-based image hash fingerprints. Then, the
Hamming distance between two fingerprints can be used to quantify the
similarity of two images. The larger the Hamming distance is, the smaller the
similarity of the images have. Although the hash fingerprints can be
calculated efficiently with low energy cost, it cannot be applied directly to
the similarity detection of sensing data. The reason is that, at each moment,
we have one amplitude spectrum and one phase spectrum as shown in Fig. 4,
which are both required to achieve sensing tasks. Thus, we propose a novel
four-level semantic hash sampling method in Algorithm 2 to select task-related
signals spectrums for encoding, which is used before Algorithm 1.

As shown in Algorithm 2, to obtain the semantic hash matrices, the first step
is to resize the $T_{K}$ amplitude and phase spectrums. The purpose is to
produce a small data size, which hastens the processing time [41] and
preserves the features of the spectrums. Similar to the image pHash method
[42], we calculate the average values of the resized amplitude and phase
matrices. Different from the conventional hash method, we define four values,
i.e., 0, 1, 2, and 3, as values in the hash fingerprints. Thus, we perform the
operations as shown in lines $8-15$ of Algorithm 2 to convert the spectrums to
semantic hash fingerprints in polynomial complexity. For the $k^{\rm th}$ pair
of amplitude and phase spectrums, we use the Hamming distance, which measures
the number of different values, between the $k^{\rm th}$ hash fingerprint and
the $\left(k-1\right)^{\rm th}$ one to indicate the semantic information
richness of the $k^{\rm th}$ spectrums. Therefore, we can record the pair of
spectrums that have the largest Hamming distance to the previous pair of
spectrums.

Algorithm 2 The algorithm for semantic hash sampling

Input:

  * •

The received amplitude and phase spectrums sets in the $i^{\rm th}$ time
segment: ${\bm{S}}_{A_{i}}$ and ${\bm{S}}_{P_{i}}$

  * •

The dimensions of the resized matrices: $R_{x}$ and $R_{y}$

Output: The selected amplitude and phase spectrums:
${\bm{Y}}_{A}{\left\\{i\right\\}}$ and ${\bm{Y}}_{P}{\left\\{i\right\\}}$

1:## Obtain the semantic hash matrix set

2:Create an empty matrix set
${\bm{\mathcal{H}}}_{i}\in{\mathbb{R}}^{R_{x}\times{R_{y}}\times T_{K}}$ to
record the semantic hash values

3:for Every ${\bm{S}}_{A_{i}}{\left\\{k\right\\}}$ in ${\bm{S}}_{A_{i}}$ do

4: Obtain amplitude and phase spectrums ${{\bm{H}}}_{A_{i}}^{\left(k\right)}$
and ${\bm{H}}_{P_{i}}^{\left(k\right)}$ with the prior knowledge
${\bm{\Phi}}_{A_{i}}^{\left(k\right)}$ and
${\bm{\Phi}}_{P_{i}}^{\left(k\right)}$, respectively

5: Resize ${\bm{H}}_{A_{i}}^{\left(k\right)}$ and
${\bm{H}}_{P_{i}}^{\left(k\right)}$ into small matrices
${\bm{h}}_{A_{i}}^{\left(k\right)}\in{\mathbb{R}}^{R_{x}\times{R_{y}}}$ and
${\bm{h}}_{P_{i}}^{\left(k\right)}\in{\mathbb{R}}^{R_{x}\times{R_{y}}}$,
respectively

6: Calculate the average values of ${\bm{h}}_{A_{i}}^{\left(k\right)}$ and
${\bm{h}}_{P_{i}}^{\left(k\right)}$, denoted as
${{h}}_{A_{i}}^{\left(k\right)}$ and ${{h}}_{P_{i}}^{\left(k\right)}$,
respectively

7: for Every element pair in ${\bm{h}}_{A_{i}}^{\left(k\right)}$ and
${\bm{h}}_{P_{i}}^{\left(k\right)}$ do

8: if
${{\bm{h}}}_{A_{i}}^{\left(k\right)}\left[{x,y}\right]\geqslant{{h}}_{A_{i}}^{\left(k\right)}$
and
${{\bm{h}}}_{P_{i}}^{\left(k\right)}\left[{x,y}\right]\geqslant{{h}}_{P_{i}}^{\left(k\right)}$
then

9: Let ${\bm{\mathcal{H}}}_{i}{\left\\{k\right\\}}\left[{x,y}\right]\leftarrow
3$

10: else if
${{\bm{h}}}_{A_{i}}^{\left(k\right)}\left[{x,y}\right]\geqslant{{h}}_{A_{i}}^{\left(k\right)}$
and
${{\bm{h}}}_{P_{i}}^{\left(k\right)}\left[{x,y}\right]<{{h}}_{P_{i}}^{\left(k\right)}$
then

11: Let
${\bm{\mathcal{H}}}_{i}{\left\\{k\right\\}}\left[{x,y}\right]\leftarrow 2$

12: else if
${{\bm{h}}}_{A_{i}}^{\left(k\right)}\left[{x,y}\right]<{{h}}_{A_{i}}^{\left(k\right)}$
and
${{\bm{h}}}_{P_{i}}^{\left(k\right)}\left[{x,y}\right]\geqslant{{h}}_{P_{i}}^{\left(k\right)}$
then

13: Let
${\bm{\mathcal{H}}}_{i}{\left\\{k\right\\}}\left[{x,y}\right]\leftarrow 1$

14: else

15: Let
${\bm{\mathcal{H}}}_{i}{\left\\{k\right\\}}\left[{x,y}\right]\leftarrow 0$

16:## Calculate the Hamming distance

17:Create an empty vector ${\bm{\mathcal{D}}}\in{\mathbb{R}}^{1\times{T_{K}}}$
to record the Hamming distance values

18:for Every ${\bm{\mathcal{H}}}_{i}{\left\\{k\right\\}}$ in
${\bm{\mathcal{H}}}_{i}$ do

19: Create a temporary variable $d$

20: for Every element in ${\bm{\mathcal{H}}}_{i}{\left\\{k\right\\}}$ do

21: if The element value is different from the element value in the same
position in ${\bm{\mathcal{H}}}_{i}{\left\\{k-1\right\\}}$ then

22: $d\leftarrow d+1$

23: ${\bm{\mathcal{D}}}\left(k\right)\leftarrow d$.

24:## Select the spectrum and record the information richness

25:Find $k_{\max}$ that maximizes ${\bm{\mathcal{D}}}\left(k\right)$, i.e.,
${\bm{\mathcal{D}}}\left(k_{\max}\right)=\max\left\\{{\bm{\mathcal{D}}}\right\\}$

26:Record the value of ${\bm{\mathcal{D}}}\left(k_{\max}\right)$

27:Let $i=k_{\max}$

28:return ${\bm{Y}}_{A}{\left\\{i\right\\}}$ and
${\bm{Y}}_{P}{\left\\{i\right\\}}$

##  V Inverse Semantic-aware Decoding

In this section, we propose the inverse semantic-aware self-supervised
decoding method. We also introduce how to use the recovered signal spectrums
for 2D AoA and ToF estimation, which supports various sensing tasks.

###  V-A Objective Function

We first rewrite the amplitude and phase MetaSpectrums $\bm{Z}_{A}$ and
$\bm{Z}_{P}$ in the vectorized formulations, respectively. Let
$\rm{vec}\left(\cdot\right)$ denote the matrix vectorization operation that
concatenates columns into one vector, and $\rm{diag}\left({\bf{a}}\right)$
denote the operation of converting the vector $\bf{a}$ into a diagonal matrix
where the diagonal element is $\bf{a}$. As such, we rewrite the matrix
formulations (35) and (36) as

${\bf z}_{A}={\bf\Phi}{\bf x}_{A},$ (39)

and

${\bf z}_{P}={\bf\Phi}{\bf x}_{P},$ (40)

respectively, where $\bm{z}_{A}={\rm{vec}}\left({\bm{Z}}_{A}\right)$,
${\bm{z}}_{P}={\rm{vec}}\left({\bm{Z}}_{P}\right)$, $\bm{z}_{A}$ and
$\bm{z}_{P}\in{\mathbb{R}^{\left(K+\left({T-1}\right)D\right)L\times 1}}$,
$\bm{x}_{A}=\left[\bm{x}_{1,A}^{\rm T}\cdots{\bm{x}}_{i,A}^{\rm
T}\cdots\bm{x}_{T,A}^{\rm T}\right]^{\rm T}$,
$\bm{x}_{P}=\left[\bm{x}_{1,P}^{\rm T}\cdots{\bm{x}}_{i,P}^{\rm
T}\cdots\bm{x}_{T,P}^{\rm T}\right]^{\rm T}$,
$\bm{x}_{i,A}={\rm{vec}}\left({{{\bm{H}}_{A^{\prime}}^{\left(i\right)}}}\right)$,
$\bm{x}_{i,P}={\rm{vec}}\left({{{\bm{H}}_{P^{\prime}}^{\left(i\right)}}}\right)$,
${\bm{x}}_{A}$ and ${\bm{x}}_{P}\in{\mathbb{R}^{TKL\times 1}}$,

${\bf\Phi}=\left[{\begin{array}[]{*{20}{c}}{{{\bm{Q}}_{3}}\left(1\right)}&\cdots&{{{\bm{Q}}_{3}}\left(i\right)}&\cdots&{{{\bm{Q}}_{3}}\left(T\right)}\\\
{{\bm{\phi}}_{A}^{\left(1\right)}}&\ddots&{{\bm{\phi}}_{A}^{\left(i\right)}}&\ddots&{{\bm{\phi}}_{A}^{\left(T\right)}}\\\
{{{\bm{Q}}_{4}}\left(1\right)}&\cdots&{{{\bm{Q}}_{4}}\left(i\right)}&\cdots&{{{\bm{Q}}_{4}}\left(T\right)}\end{array}}\right],$
(41)

${{{\bm{Q}}_{3}}\left(1\right)}\in{\mathbb{R}^{\left(i-1\right)DL\times KL}}$,
${{{\bm{Q}}_{4}}\left(1\right)}\in{\mathbb{R}^{\left(T-1\right)DL\times KL}}$,
every element in both ${\bm{Q}_{3}}$ and ${\bm{Q}_{4}}$ is zero,
${\bm{\phi}}_{A}^{\left(i\right)}={\rm diag}\left({{\rm
vec}\left({{\bf\Phi}_{A}^{\left(i\right)}}\right)}\right)$,
${\bm{\phi}}_{A}^{\left(i\right)}\in{\mathbb{R}^{KL\times KL}}$, and
${\bf\Phi}\in{\mathbb{R}^{{\left(K+\left({T-1}\right)D\right)L}\times TKL}}$.

Note that ${\bf\Phi}$ can be obtained using (41) and the prior knowledge,
i.e., the amplitude response matrices of the RIS, and ${\bf z}_{A}$ and ${\bf
z}_{P}$ are the known vectorized amplitude and phase MetaSpectrums,
respectively. Our goal is to decode ${\bf x}_{A}$ and ${\bf x}_{P}$ from ${\bf
z}_{A}$ and ${\bf z}_{P}$, respectively. Although this problem is related to
CS, most theories that are developed for CS cannot be used because that the
matrix ${\bf\Phi}$ follows a very specific structure as (41). Fortunately,
solid theoretical proof has shown that both ${\bf x}_{A}$ in (39) and ${\bf
x}_{P}$ in (40) can be recovered even when $T>1$ [43].

The decoding objective function can be formulated as

$\mathop{\min}\limits_{{{\bm{x}}_{A}},{{\bm{x}}_{P}}}\\!\alpha_{1}\\!\left\|{{{\bm{z}}_{A}}\\!-\\!{{\bm{\Phi}}}{{\bm{x}}_{A}}}\right\|^{2}+\\!\alpha_{2}\\!\left\|{{{\bm{z}}_{P}}\\!-\\!{{\bm{\Phi}}}{{\bm{x}}_{P}}}\right\|^{2}\\!,$
(42)

where $\alpha_{1}$ and $\alpha_{2}$ are the balance parameters that can be
selected according to the specific wireless sensing task. For example,
heartbeat and breath detection requires higher accuracy for the phase spectrum
[16], and the amplitude spectrum is more significant in sensing tasks such as
intrusion or fall detection [15]. We propose the algorithm for solving (42) as
follows.

###  V-B Self-supervised Decoding Method

To solve (42), although different hand-crafted priors, e.g., total variation
and sparsity, can be added as the regularization term to improve the decoding
performance, it is hard to choose a suitable prior that fits the differential
encoded amplitude and phase spectrums ${{\bm{x}}_{A}}$ and ${{\bm{x}}_{P}}$.
Motivated by the success of deep convolutional neural networks (ConvNets) in
inverse problems such as single-image super-resolution [44] and denoising
[45], we use the implicit prior captured by the ConvNets, e.g., deep image
prior [46, 47], to achieve self-supervised decoding333Another solution is to
design a suitable explicit regularization term for decoding sensing signal
spectrums and use the explicit and implicit priors jointly [47]. This is left
for the future work..

By considering that the unknown amplitude and phase spectrums are the outputs
of neural networks, i.e.,
${{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)$ and
${{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{P}}}}\\!\\!\left({\bm{e}}\right)$,
respectively, the decoding problem (42) can be re-written as

$\begin{array}[]{*{20}{l}}{\mathop{\min}\limits_{{{\bm{\Theta}}_{A}},{{\bm{\Theta}}_{P}}}}&\\!\\!\\!\\!{{\alpha_{1}}\\!\left\|{{{\bm{z}}_{A}}\\!-\\!{{\bm{\Phi}}}{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)}\right\|^{2}\\!+\\!{\alpha_{2}}\\!\left\|{{{\bm{z}}_{P}}\\!-\\!{{\bm{\Phi}}}{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{P}}}}\\!\\!\left({\bm{e}}\right)}\right\|^{2},}\\\
{\>\>\>{\rm{s.t.}}}&\\!\\!\\!\\!{{{{\bm{\hat{x}}}}_{A}}={{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)},\\\
{}&\\!\\!\\!\\!{{{{\bm{\hat{x}}}}_{P}}={{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{P}}}}\\!\\!\left({\bm{e}}\right)},\end{array}$
(43)

where ${{\bm{\Theta}}_{A}}$ and ${{\bm{\Theta}}_{P}}$ are the parameters of
networks to be learned, and ${\bm{e}}$ is a random vector. Since the training
of ${{\bm{\Theta}}_{A}}$ and ${{\bm{\Theta}}_{P}}$ is part of the decoding
process, this procedure is self-supervised and no pre-training process is
required.

To solve the problem (43), we introduce two auxiliary variables ${\bm{t}_{1}}$
and ${\bm{t}_{2}}\in{\mathbb{R}}^{TKL}$, and corresponding weight parameters
$\beta_{1}$ and $\beta_{2}$. Then, the constraints can be turned into penalty
terms using the augmented Lagrangian method [48] as

$\begin{array}[]{*{20}{l}}{\mathop{\min}\limits_{{\bm{\Theta}}_{A},{\bm{\Theta}}_{P},{\bm{x}}_{A},{\bm{x}}_{P}}}&{\bm{\mathcal{F}}}_{1}\left({\bm{\Theta}}_{A},{\bm{x}}_{A}\right)+{\bm{\mathcal{F}}}_{2}\left({\bm{\Theta}}_{P},{\bm{x}}_{P}\right),\end{array}$
(44)

where

${\bm{\mathcal{F}}}_{1}\\!\\!=\\!{\alpha_{1}}\\!\\!\left(\\!\left\|{{{\bm{z}}_{A}}\\!-\\!{{\bm{\Phi}}}\\!{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)}\right\|^{2}\\!+\\!{\beta_{1}}\\!\left\|{{{\bm{x}}_{A}}\\!-\\!{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)\\!-\\!{\bm{t}}_{1}}\right\|^{2}\\!\right)\\!,$
(45)

and

${\bm{\mathcal{F}}}_{2}\\!\\!=\\!{\alpha_{2}}\\!\\!\left(\\!\left\|{{{\bm{z}}_{P}}\\!-\\!{{\bm{\Phi}}}{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{P}}}}\\!\\!\left({\bm{e}}\right)}\right\|^{2}\\!+\\!{\beta_{2}}\\!\left\|{{{\bm{x}}_{P}}\\!-\\!{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{P}}}}\\!\\!\left({\bm{e}}\right)\\!-\\!{\bm{t}}_{2}}\right\|^{2}\\!\right)\\!.$
(46)

With the help of the alternating direction method of multipliers (ADMM) [49],
the problem (44) can be solved by a sequential update of the six variables,
i.e., ${\bm{\Theta}}_{A}$, ${\bm{\Theta}}_{P}$, ${\bm{x}}_{A}$,
${\bm{x}}_{P}$, ${\bm{t}}_{1}$, and ${\bm{t}}_{2}$.

1) The update of ${\bm{\Theta}}_{A}$ while fixing other variables:

$\begin{array}[]{*{20}{l}}{\mathop{\min}\limits_{{\bm{\Theta}}_{A}}}&\\!\\!\\!\left\|{{{\bm{z}}_{A}}\\!-\\!{{\bm{\Phi}}}{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)}\right\|^{2}\\!+\\!{\beta_{1}}\\!\left\|{{{\bm{x}}_{A}}\\!-\\!{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)\\!-\\!{\bm{t}}_{1}}\right\|^{2}\\!,\end{array}$
(47)

which can be solved using the steepest descent and back-propagation
optimization methods [46]. Note that
${\beta_{1}}\\!\left\|{{{\bm{x}}_{A}}\\!-\\!{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)\\!-\\!{\bm{t}}_{1}}\right\|^{2}$
in (47) can be regarded as the denoising of
${{\bm{x}}_{A}}\\!-\\!{{\bm{t}}_{1}}$, which also serves as a proximity
regularization that forces
${{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)$ to
be close to ${{\bm{x}}_{A}}\\!-\\!{{\bm{t}}_{1}}$. This second term provides
additional stabilizing and robustifying effect to the back-propagation method.

2) The update of ${\bm{x}}_{A}$ while fixing other variables:

$\begin{array}[]{*{20}{l}}{\mathop{\min}\limits_{{\bm{x}}_{A}}}&\\!\\!\\!\left\|{{{\bm{x}}_{A}}\\!-\\!{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)\\!-\\!{\bm{t}}_{1}}\right\|^{2}\\!,\end{array}$
(48)

which can be regarded as a denoising problem for
${{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)+{\bm{t}}_{1}$.
Thus, we have

${\bm{\hat{x}}}_{A}={\bm{\mathcal{D}}}\left({{{\bm{\mathcal{T}}}_{{{\bm{\Theta}}_{A}}}}}\\!\\!\left({\bm{e}}\right)+{\bm{t}}_{1}\right),$
(49)

where ${\bm{\mathcal{D}}}\left(\cdot\right)$ represents the denoising operator
that could be well-studied plug-and-play algorithms [50] or a simpler
steepest-descent (SD) operator. We present the update equation for SD method
as

${\bm{x}}_{A}^{\left(j+1\right)}={\bm{x}}_{A}^{\left(j\right)}-s\left({\bm{x}}_{A}^{\left(j\right)}-{{{\bm{\mathcal{T}}}_{{{\bm{\Theta}}_{A}}}}}\\!\\!\left({\bm{e}}\right)-{\mathbf{t}}_{1}\right),$
(50)

where $s$ is the steepest-descent step size, and $j$ is the inner loop
iteration number.

3) The update of ${\bm{t}}_{1}$ while fixing other variables: Because
${\bm{t}}_{1}$ can be regarded as the Lagrange multipliers vector,
${\bm{t}}_{1}$ can be updated according to the augmented Lagrangian method
[48] as

${\bm{t}}_{1}^{\left(k+1\right)}={\bm{t}}_{1}^{\left(k\right)}+{{\bm{\mathcal{T}}}_{{{\bm{\Theta}}_{A}^{\left(k\right)}}}}\\!\\!\left({\bm{e}}\right)-{\mathbf{x}}_{A}^{\left(k\right)},$
(51)

where $k$ denotes the outer loop iteration number.

4) The update of ${\bm{\Theta}}_{P}$ while fixing other variables: Because the
network with parameter ${\bm{\Theta}}_{P}$ is trained independently, we can
update ${\bm{\Theta}}_{P}$ by solving

$\begin{array}[]{*{20}{l}}{\mathop{\min}\limits_{{\bm{\Theta}}_{P}}}&\\!\\!\\!\left\|{{{\mathbf{z}}_{P}}\\!-\\!{{\mathbf{\Phi}}}{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{P}}}}\\!\\!\left({\bm{e}}\right)}\right\|^{2}\\!+\\!{\beta_{2}}\\!\left\|{{{\mathbf{x}}_{P}}\\!-\\!{{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{P}}}}\\!\\!\left({\bm{e}}\right)\\!-\\!{\bm{t}}_{2}}\right\|^{2}\\!,\end{array}$
(52)

with the same method as in (47).

5) The update of ${\bm{x}}_{P}$ while fixing other variables: To minimize the
difference between ${\mathbf{x}}_{P}$ and
${{\bm{\mathcal{T}}}_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)+{\mathbf{t}}_{1}$,
we can update ${\mathbf{x}}_{P}$ as

${\mathbf{\hat{x}}}_{P}={\bm{\mathcal{D}}}\left({{{\bm{\mathcal{T}}}_{{{\bm{\Theta}}_{P}}}}}\\!\\!\left({\bm{e}}\right)+{\mathbf{t}}_{2}\right).$
(53)

where ${\bm{\mathcal{D}}}$ is the same kind of denoising operator as (49).

6) The update of ${\bm{t}}_{2}$ while fixing other variables: According to the
augmented Lagrangian method [48], ${\bm{t}}_{2}$ can be updated as

${\bm{t}}_{2}^{\left(k+1\right)}={\bm{t}}_{2}^{\left(k\right)}+{{\bm{\mathcal{T}}}_{{{\bm{\Theta}}_{P}^{\left(k\right)}}}}\\!\\!\left({\bm{e}}\right)-{\mathbf{x}}_{P}^{\left(k\right)}.$
(54)

Algorithm 3 The algorithm for inverse semantic-aware decoding

Input:

  * •

The weight parameters: $\beta_{1}$ and $\beta_{2}$

  * •

The number of inner iterations of the denoising operator for updating
${\bm{x}}_{A}$ and ${\bm{x}}_{P}$: $N_{J}$

  * •

The steepest-descent parameters for updating ${{\bm{\Theta}}_{A}}$ and
${{\bm{\Theta}}_{P}}$, respectively

Output: The original amplitude and phase spectrums, i.e.,
${\mathbf{H}}_{A}^{\left(i\right)}$ and ${\mathbf{H}}_{P}^{\left(i\right)}$
$\left(i=1,\ldots,T\right)$

1:## Reconstruction of the ${\bm{x}}_{A}$ and ${\bm{x}}_{P}$

2:Initialize the iteration number $k=0$

3:Set ${{\bm{\Theta}}_{A}}$ and ${{\bm{\Theta}}_{P}}$ randomly

4:while Not converged do

5: Update ${\bm{\Theta}}_{A}$ by solving (47) using steepest descent and back-
propagation methods

6: Update ${\bm{x}}_{A}$ according to (49)

7: Update ${\bm{t}}_{1}$ according to (51)

8: Update ${\bm{\Theta}}_{P}$ by solving (52) using steepest descent and back-
propagation methods

9: Update ${\bm{x}}_{P}$ according to (53)

10: Update ${\bm{t}}_{2}$ according to (54)

11: Let $k\leftarrow k+1$

12:Record ${\bm{x}}_{A}$ and ${\bm{x}}_{P}$ after converged

13:## Differential decoding

14:Recover ${\mathbf{H}}_{A^{\prime}}{\left\\{i\right\\}}$ and
${\mathbf{H}}_{P^{\prime}}{\left\\{i\right\\}}$ $\left(i=1,\ldots,T\right)$
according to the definition of ${\bm{x}}_{A}$ and ${\bm{x}}_{P}$, i.e., (39)
and (40)

15:for Every ${\mathbf{H}}_{A^{\prime}}{\left\\{i\right\\}}$ and
${\mathbf{H}}_{P^{\prime}}{\left\\{i\right\\}}$ do

16: Create empty ${\bm{H}}_{A}\left\\{i\right\\}$ and
${\bm{H}}_{P}\left\\{i\right\\}$ to record the decoded results

17: Obtain ${\bm{H}}_{A}\left\\{i\right\\}$ and
${\bm{H}}_{P}\left\\{i\right\\}$ according to (55), (56), (57), and (58)

18:return ${\mathbf{H}}_{A}{\left\\{i\right\\}}$ and
${\mathbf{H}}_{P}{\left\\{i\right\\}}$ $\left(i=1,\ldots,T\right)$

Algorithm 3 summarizes the steps to perform the aforementioned decoding
methods, and then recover the original amplitude and phase spectrums.
Specifically, after decoding, we obtain the estimated
${\bm{H}}_{A^{\prime}}\left\\{i\right\\}$ and
${\bm{H}}_{P^{\prime}}\left\\{i\right\\}$. Let the first column in
${\bm{H}}_{A}\left\\{i\right\\}$ and ${\bm{H}}_{P}\left\\{i\right\\}$ be the
same as that of ${\bm{H}}_{A^{\prime}}\left\\{i\right\\}$ and
${\bm{H}}_{P^{\prime}}\left\\{i\right\\}$ as

${{\bm{H}}_{A}}\left\\{i\right\\}\left[{1,:}\right]={{\bm{H}}_{A}^{\prime}}\left\\{i\right\\}\left[{1,:}\right],$
(55)

and

${{\bm{H}}_{P}}\left\\{i\right\\}\left[{1,:}\right]={{\bm{H}}_{P^{\prime}}}\left\\{i\right\\}\left[{1,:}\right].$
(56)

For the second to the last columns ($j=2,\ldots,L$), we have

${{\bm{H}}_{A}}\left\\{i\right\\}\left[{j,:}\right]={{\bm{H}}_{A^{\prime}}}\left\\{i\right\\}\left[{j-1,:}\right]+{{\bm{H}}_{A^{\prime}}}\left\\{i\right\\}\left[{j,:}\right],$
(57)

and

${{\bm{H}}_{P}}\left\\{i\right\\}\left[{j,:}\right]={{\bm{H}}_{P^{\prime}}}\left\\{i\right\\}\left[{j-1,:}\right]+{{\bm{H}}_{P^{\prime}}}\left\\{i\right\\}\left[{j,:}\right].$
(58)

Note that because of the independent iterative training of the two networks
and the use of the ADMM method, $\alpha_{1}$ and $\alpha_{2}$ have no effect
on the objective function. The running time is mainly taken in updating
${{\bm{\Theta}}_{A}}$ and ${{\bm{\Theta}}_{P}}$ since the inner denoising
operators work efficiently. In Section VI, we set the inner iteration numbers
of the denoising operators for updating ${\bm{x}}_{A}$ and ${\bm{x}}_{P}$ to
be both $600$, and the outer loop maximal iteration number is $18$, i.e., $18$
ADMM iterations. The average running time for decoding one MetaSpectrum, which
is obtained by encoding $20$ original amplitude spectrums, is about $1$ minute
with the experiment setting in Section VI. Although the self-supervised method
is not suitable for high real-time sensing signal data decoding, our method
can be used for sensing tasks that require large amounts of historical data
for analysis, i.e., healthcare monitoring, sleeping position detection,
historical intrusion or walking behavior analysis.

With the decoded ${\mathbf{H}}_{A}{\left\\{i\right\\}}$ and
${\mathbf{H}}_{P}{\left\\{i\right\\}}$ $\left(i=1,\ldots,T\right)$, the
original signal at each moment can be recovered to the form of complex
matrices. Then, the 2D AoA and ToF can be jointly estimated [31], which can be
used to complete a series of sensing tasks. Thus, the steering matrices of
$L$-shaped array in the $x$ and $y$ directions, which describe how the sensor
array uses each individual element to select a spatial path for the
transmission, can be expressed as

$\displaystyle{{\bf{A}}_{x}}\\!=\\!\\!\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}1&\\!\\!\\!\\!\\!\cdots\\!\\!\\!\\!\\!&1\\\
{{e^{-j2\pi\\!{f_{k}}d\cos\left({{\theta_{1}}}\\!\right)\sin\left({{\varphi_{1}}}\\!\right){\rm{}}}}}&\\!\\!\\!\\!\\!\cdots\\!\\!\\!\\!\\!&{{e^{-j2\pi\\!{f_{k}}d\cos\left({{\theta_{I}}}\\!\right)\sin\left({{\varphi_{I}}}\\!\right)}}}\\\
\\!\\!\\!\vdots\\!\\!\\!&\\!\\!\\!\vdots\\!\\!\\!&\\!\\!\\!\vdots\\!\\!\\!\\\
{{e^{-j2\pi\\!{f_{k}}md\cos\left({{\theta_{1}}}\\!\right)\sin\left({{\varphi_{1}}}\\!\right)}}}&\\!\\!\\!\\!\\!\cdots\\!\\!\\!\\!\\!&{{e^{-j2\pi\\!{f_{k}}md\cos\left({{\theta_{I}}}\\!\right)\sin\left({{\varphi_{I}}}\\!\right)}}}\end{array}}\\!\\!\\!\\!\right]\\!,$
(63)

and

$\displaystyle{{\bf{A}}_{y}}\\!=\\!\\!\left[\\!\\!\\!{\begin{array}[]{*{20}{c}}1&\\!\\!\\!\\!\\!\cdots\\!\\!\\!\\!\\!&1\\\
{{e^{-j2\pi\\!{f_{k}}d\sin\left({{\theta_{1}}}\\!\right)\sin\left({{\varphi_{1}}}\\!\right)}}}&\\!\\!\\!\\!\\!\cdots\\!\\!\\!\\!\\!&{{e^{-j2\pi\\!{f_{k}}d\sin\left({{\theta_{I}}}\\!\right)\sin\left({{\varphi_{I}}}\\!\right)}}}\\\
\\!\\!\\!\vdots\\!\\!\\!&\\!\\!\\!\vdots\\!\\!\\!&\\!\\!\\!\vdots\\!\\!\\!\\\
{{e^{-j2\pi\\!{f_{k}}nd\sin\left({{\theta_{1}}}\\!\right)\sin\left({{\varphi_{1}}}\\!\right)}}}&\\!\\!\\!\\!\\!\cdots\\!\\!\\!\\!\\!&{{e^{-j2\pi\\!{f_{k}}nd\sin\left({{\theta_{I}}}\\!\right)\sin\left({{\varphi_{I}}}\\!\right)}}}\end{array}}\\!\\!\\!\\!\right],$
(68)

respectively. Inspired by [51], here, we take multiple subcarriers into
consideration and extend the 2D AoA estimation into three dimensions, to
acheive joint 2D AoA and ToF estimation via the following Proposition 2:

######  Proposition 2.

The signal 2D AoA and ToF at time $t$ can be estimated using

$\displaystyle{P_{3D}}\left({\theta,\varphi,\tau,t}\right){\rm{=}}\frac{1}{{{\bf{A}}_{0x^{\prime}y^{\prime}}^{\rm{H}}{{\bf{E}}_{N}}\left(t\right){\bf{E}}_{N}^{\rm{H}}\left(t\right){{\bf{A}}_{0x^{\prime}y^{\prime}}}}},$
(69)

where $P_{3D}$ describes the signal magnitude for a given set of
$\left({\theta,\varphi,\tau}\right)$, the superscript H is the conjugate
transpose operator, ${{\bf{E}}_{N}}\left(t\right)$ is the noise subspace
obtained by decomposing the auto-correlation matrix of the smoothed original
signal at time $t$ [51], ${{\bf{A}}_{0x^{\prime}y^{\prime}}}$ is the steering
matrix that is obtained using (63) and (68) as

$\displaystyle{{\bf{A}}_{0x^{\prime}y^{\prime}}}={\left[{{{\bf{A}}_{0}}{\rm{}}{{\bf{A}}_{x^{\prime}}}{\rm{}}{{\bf{A}}_{y^{\prime}}}}\right]^{\rm{T}}}$
$\displaystyle=\\!\\!{\left[\\!{\underbrace{\begin{array}[]{*{20}{c}}\\!1\\!\\\
\\!\vdots\\!\\\ \\!1\\!\\\ \\!\vdots\\!\\\ \\!1\\!\\\ \\!\vdots\\!\\\
\\!1\\!\end{array}}_{{{\mathbf{A}}_{0}}}\underbrace{\begin{array}[]{*{20}{c}}{{e^{-j2\pi{f_{1}}d\cos\left(\theta\right)\sin\left(\varphi\right)}}}\\\
\vdots\\\
{{e^{-j2\pi{f_{{k^{\prime}}}}d\cos\left(\theta\right)\sin\left(\varphi\right)}}}\\\
\vdots\\\
\\!\\!{{e^{-\\!j2\pi\\!{f_{1}}\\!\left(\\!{{m^{\prime}}\\!-1}\\!\right)d\\!\cos\left(\theta\right)\\!\sin\left(\\!\varphi\\!\right)}}}\\!\\!\\\
\vdots\\\
\\!\\!\\!\\!\\!\\!{{e^{-\\!j2\pi\\!{f_{{k^{\prime}}}}\\!\left(\\!{{m^{\prime}}\\!-1}\\!\right)d\\!\cos\left(\theta\right)\\!\sin\left(\\!\varphi\\!\right)}}}\\!\\!\\!\\!\\!\\!\end{array}}_{{{\mathbf{A}}_{x^{\prime}}}}\underbrace{\begin{array}[]{*{20}{c}}{{e^{-j2\pi{f_{1}}d\sin\left(\theta\right)\sin\left(\varphi\right)}}}\\\
\\!\\!\\!\\!\vdots\\!\\!\\!\\!\\\
{{e^{-j2\pi\\!{f_{{k^{\prime}}}}d\sin\left(\theta\right)\sin\left(\varphi\right)}}}\\\
\\!\\!\\!\\!\vdots\\!\\!\\!\\!\\\
\\!\\!{{e^{-\\!j2\pi\\!{f_{1}}\\!\left(\\!{{n^{\prime}}\\!-1}\\!\right)d\\!\sin\left(\theta\right)\\!\sin\left(\\!\varphi\\!\right)}}}\\!\\!\\!\\!\\!\\\
\\!\\!\\!\\!\vdots\\!\\!\\!\\!\\\
\\!\\!\\!\\!\\!\\!{{e^{-\\!j2\pi\\!{f_{{k^{\prime}}}}\\!\left(\\!{{n^{\prime}}\\!-1}\\!\right)d\\!\sin\left(\theta\right)\\!\sin\left(\\!\varphi\\!\right)}}}\\!\\!\\!\\!\\!\\!\end{array}}_{{{\mathbf{A}}_{y^{\prime}}}}}\\!\right]^{\text{T}}}\\!,$
(91)

$0<{k^{\prime}}<K$, $0<{m^{\prime}}<M$, and $0<{n^{\prime}}<N$.

Thus, we complete all the processes of the inverse semantic-aware wireless
sensing framework. Specifically, we use Algorithm 2 to sample the task-related
signal spectrums. With the RIS, Algorithm 1 can encode the sensing data, thus
greatly reducing the data volume to be stored or transmitted. We use the self-
supervised decoding Algorithm 3 to recover the original sensing data. Finally,
with the help of Proposition 2, various sensing tasks can be performed. For
example, intrusion detection can be achieved by detecting the change of
estimated 2D AoA, and the human walking trajectory can be tracked by
estimating the 2D AoA and the ToF of the signals.

##  VI Experiments Results

Since the key contribution of this paper is to achieve the inverse semantic-
aware encoding and decoding of the sensing data with the help of RIS, we aim
to answer the following research questions via experiments:

  1. Q1)

Can the proposed self-supervised decoding scheme recover the original signal
spectrums and ensure the accomplishment of sensing tasks?

  2. Q2)

Can the amplitude response matrix of RIS, i.e., codebook, encrypt the sensing
data?

  3. Q3)

Compared with the existing uniform sampling method, can the proposed semantic
hash sampling method help to achieve more accurate completions of the sensing
tasks?

We first present the experimental platform and the parameter setting of our
proposed algorithms, and then answer the above questions through experimental
evaluations.

###  VI-A Experiments Setting

Figure 5: Test scenario and hardware of the receiver.

To collect sensing data from the real-world scenario, we use three access
points (APs) based on the IEEE 802.11ax protocol to build a test platform
[13]. The collected sensing data is used to conduct a comprehensive evaluation
of our proposed algorithms. The specific experimental scenario and hardware
equipment are shown in Fig. 5. Specifically, the test scenario is a conference
room with tables and chairs. Inside the room, one AP acts as a transmitter to
send OFDM wireless signals with the total bandwidth of $160$ MHz and $2048$
sub-carriers. The center frequency of the sub-carriers is $5.805$ GHz. As
shown in Fig. 5 (Part II), the other two APs form an receiver with $L$-shaped
active sensor array via a power splitter to receive signals. Since the
investigation of STAR-RIS hardware is still at a very early stage, we simulate
the amplitude and phase response matrices of the transmissive elements using a
signal processor [23, 24, 33]. During the experiment, the data packet
transmission rate, i.e., transmission frequency, is $100$ Hz, which means
$100$ packets are transmitted per second. The human target walks along the
preset trajectory to complete the data collection.

The experimental platform for running our proposed algorithms is built on a
generic Ubuntu 20.04 system with an AMD Ryzen Threadripper PRO 3975WX 32-Cores
CPU and an NVIDIA RTX A5000 GPU. In the self-supervised decoding Algorithm 3,
two U-net without the skip connections [46] are used as the self-supervised
neural networks, i.e.,
${{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{A}}}}\\!\\!\left({\bm{e}}\right)$ and
${{\bm{\mathcal{T}}}\\!_{{{\bm{\Theta}}_{P}}}}\\!\\!\left({\bm{e}}\right)$.
The input to the network, i.e., ${\bm{e}}$, is a random vector that has the
same size as ${\bm{x}}_{A}$ and ${\bm{x}}_{P}$ to be recovered. During the
decoding of one MetaSpectrum, ${\bm{e}}$ is fixed in each ADMM iteration. In
addition, to avoid the local minimum that the networks stuck in the last
iteration, ${{\bm{\Theta}}_{A}}$ and ${{\bm{\Theta}}_{P}}$ are set to zero
when each ADMM iteration is finished. In other words, both
${{\bm{\Theta}}_{A}}$ and ${{\bm{\Theta}}_{P}}$ are re-trained in each
iteration.

###  VI-B Experiments Performance Analysis

####  VI-B1 Effectiveness and Efficiency of the proposed inverse semantic
decoding method (Q1)

We first set the data compression ratio as $10\%$. As shown in Fig. 1 (Part
I), starting from $3$ seconds, we select one pair of amplitude and phase
spectrums in each $0.1$ second time segment by using Algorithm 2, for RIS-
aided encoding. Using the encoding Algorithm 1 presented in Section IV-A, we
can obtain one amplitude MetaSpectrum and one phase MetaSpectrum as shown in
Fig. 1 (Part III) for every $10$ pairs of signal spectrums. The decoded
results after $15$ iterations of the outer loop are shown in Fig. 1 (Part II).
For both amplitude and phase spectrums, we observe that the difference between
the decoded and the original spectrums is basically negligible. We present a
detailed comparison of the decoded and the original amplitude spectrums in
Fig. 1 (Part IV). This proves the effectiveness of our encoding and decoding
methods.

Figure 6: The variation of the decoded amplitude and phase spectrums at time
$4.5$ s in the experiment, the corresponding semantic hash matrix, the
estimated 2D AoA spectrum by Proposition 2 with the number of outer loop
decoding iterations, where the data compression ratio is $5\%$.

In addition to the visual contrast, we show in Fig. 6 how the proposed
semantic hash matrix changes with the number of outer loop decoding
iterations. We set the data compression ratio as $5\%$. We observe that, as
the number of outer loop iterations increases, both the decoded amplitude and
phase spectrums at time $4.5$ seconds are gradually close to the ground truth
spectrums. Moreover, the Hamming distance between the semantic hash matrices
of the decoded pair of amplitude and phase spectrums and that of the original
signal spectrums is gradually reducing. Specifically, we can see that $12$
iterations can make the Hamming distance only $2$, which takes about $40$
seconds on average. Furthermore, the estimated 2D AoA values using the decoded
spectrum after $12$ iterations are very close to the true values, which
basically has no effect on the practical sensing tasks. This proves the
efficiency of our encoding and decoding methods.

####  VI-B2 Effectiveness of using the amplitude response matrix of the RIS as
the codebook (Q2)

Figure 7: The PSNR values versus the number of outer loop decoding iterations
with or without codebook.

Figure 7 depicts the average peak signal-to-noise ratio (PSNR) values $10$
experiments versus the number of outer loop decoding iterations, with or
without the codebook ${\bf\Phi}_{A}^{\left(i\right)}$. If the codebook is
available, we observe that the PSNR values of both the amplitude and phase
spectrums are increasing as the number of iterations increases, and gradually
reached a plateau after about $10$ iterations. However, if no codebook is
available or the codebook is wrong, the PSNR values decrease as the number of
iterations increases. The reason is that the parameters of two decoding
network, i.e., ${{\bm{\Theta}}_{A}}$ and ${{\bm{\Theta}}_{P}}$, are learned
according to a wrong objective function.

####  VI-B3 Effectiveness of the proposed semantic hash sampling method (Q3)

Figure 8: Comparison between different sampling methods and ground truth in
terms of 2D AoA changes with movement of human.

Based on the sensing data extracted via two different sampling methods, i.e.,
red line for the uniform sampling and blue line for the semantic hash sampling
methods, Fig. 8 displays estimated elevation and azimuth AoA changes over
time. Note that the estimation results under two sampling schemes are obtained
using the decoded amplitude and phase spectrums with the data compression
ratio as $5\%$. First, we observe that the both elevation and azimuth AoA at
every moment can be accurately estimated using the decoded data. This further
validates the effectiveness of our proposed encoding and decoding algorithms
(for Q1). Furthermore, by comparing the blue and red lines, it can be seen
that the proposed semantic hash sampling method is more efficient and
effective than uniform sampling in describing the details of AoA changes, as
shown in the enlarged part in Fig. 8. Because these changes are typically more
informative, this shows the effectiveness of our proposed semantic hash
sampling method. To compare the two schemes numerically, we consider the MSE
between the ground truth and the 2D AoA estimation results after
interpolation. By calculating, we obtain that the estimation error of the
semantic hash sampling is $0.89$, which is $67\%$ lower than that of uniform
sampling scheme whose estimation error is $2.7$.

Figure 9: Comparison of azimuth AoA estimation results that are obtained by
using the original and decoded signals, respectively.

In addition to the walking human, stationary objects such as tables and chairs
in the conference room also reflect wireless signals. Thus, the information
that can be extracted from the signal spectrums at one certain moment is rich.
Taking the azimuth AoA as an example, Fig. 9 shows the comparison of the
azimuth AoA estimation results that are obtained by using the original and
decoded signals, respectively. The data compression ratio is $5\%$. One can
see from Fig. 9 that our encoding and decoding methods preserve semantic
information related to sensing tasks, which can be illustrated from two
aspects. First, the relative magnitude characteristics among different azimuth
AoA estimated from the decoded signals are consistent with the ground truth,
i.e., azimuth AoA estimated from the original signals. For example, the ground
truth shows that the azimuth AoA of the stronger signals are in
$10^{\circ}-40^{\circ}$ and $110^{\circ}-150^{\circ}$, as indicated by the red
and blue boxes, respectively. In addition, the signals with AoA in
$40^{\circ}-110^{\circ}$ are weaker. The above features are almost completely
preserved in the estimation results obtained using the decoded data. Second,
we observe that the AoA estimation results of the first several strongest
signals are almost unchanged before and after the inverse semantic-aware
encoding and decoding, e.g., the signals marked by the red and blue boxes in
Fig. 9, respectively. This indicates that our proposed algorithms can
effectively preserve the phase characteristics (for Q1).

##  VII Conclusion and Future Directions

We have designed an inverse semantic-aware wireless sensing framework. The
amplitude response matrix of the RIS can be effectively used to generate the
codebook as prior knowledge for decoding. We have shown that our proposed RIS-
aided encoding method can achieve effective data compression. When selecting
the signal spectrums to be encoded, our proposed semantic hash sampling method
is significantly better than the widely used uniform sampling method.
Moreover, the self-supervised decoding method can recover signal amplitude and
phase spectrums to achieve various wireless sensing tasks without affecting
the performance. Since the decoding method does not require any pre-training,
it can greatly save network resources. As the demand for sensing data
increases, our proposed framework can contribute to building a resource-
friendly next-generation Internet.

Figure 10: Early-stop decompress results the real video frames in [38] using
the method proposed in [25] and our method, respectively. Six images are
compressed into one image.

There are two potential future research directions.

  * •

Inverse Semantic-aware Transmission of Images. We can consider the inverse
semantic-aware encoding and decoding of images or audio. In the surveillance
service application, a camera shoots a bay to detect boats. The surveillance
videos take up a lot of storage resource. The idea is to compress several
video frames, e.g., six frames as shown in Fig. 10, into one frame. The
original frames can be recovered by using the proposed self-supervised
decoding algorithm.

  * •

Cantor or Szudzik Pairing Compression. In this paper, we encoded the amplitude
and the phase spectrum separately. A possible improvement is to use the
pairing functions, e.g., cantor [52] or szudzik [53] pairing functions, to
combine the two spectrum into one. As shown in Fig. 1, the pairing compression
can be used as an operation after obtaining amplitude and phase spectrums to
further compress the sensing data.

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