# Integrating Sensing, Computing, and Communication in 6G Wireless Networks:
Design and Optimization

Qiao Qi, Xiaoming Chen, Ata Khalili, Caijun Zhong, Zhaoyang Zhang, and Derrick
Wing Kwan Ng Qiao Qi, Xiaoming Chen, Caijun Zhong, and Zhaoyang Zhang are with
the College of Information Science and Electronic Engineering, Zhejiang
University, Hangzhou 310027, China. Ata Khalili is with the Institute for
Digital Communications, Friedrich-Alexander-University Erlangen-Nurnberg,
Erlangen 91054, Germany. Derrick Wing Kwan Ng is with the School of Electrical
Engineering and Telecommunications, the University of New South Wales, NSW
2052, Australia.

###### Abstract

The roll-out of various emerging wireless services has triggered the need for
the sixth-generation (6G) wireless networks to provide functions of target
sensing, intelligent computing and information communication over the same
radio spectrum. In this paper, we provide a unified framework integrating
sensing, computing, and communication to optimize limited system resource for
6G wireless networks. In particular, two typical joint beamforming design
algorithms are derived based on multi-objective optimization problems (MOOP)
with the goals of the weighted overall performance maximization and the total
transmit power minimization, respectively. Extensive simulation results
validate the effectiveness of the proposed algorithms. Moreover, the impacts
of key system parameters are revealed to provide useful insights for the
design of integrated sensing, computing, and communication (ISCC).

###### Index Terms:

Integrated sensing, computing, and communication, 6G wireless networks,
beamforming design.

##  I Introduction

The demands of various emerging wireless services, e.g., unmanned vehicle,
holographic communication, and extended reality, require wireless networks to
provide integrated functions of target sensing, intelligent computing and
information communication in a timely manner [1]. In current fifth-generation
(5G) wireless networks, these functions are separated and irreciprocal for
simplicity. Generally speaking, 5G wireless networks can provide enhanced
mobile broadband communications [2], cloud and edge computing [3] but with
limited environment sensing capability [4]. As a result, these functions in 5G
wireless networks cannot satisfy the requirements of emerging wireless
services. In this context, it is desired to study the integrated sensing,
computing, and communication (ISCC) to improve the utilization of limited
system resources for the sixth-generation (6G) wireless networks [5].

In fact, integrated sensing and communication (ISAC) has received considerable
attentions in both academic and industry [6]-[8]. At the preliminary research
stage, as a special case of ISAC, spectrum sharing between radar sensing and
wireless communication has been investigated to improve the spectral
efficiency. In particular, the co-channel interference between radar and
communication systems was carefully mitigated via coordination of the two
separated systems [9]. However, the performance gain of orthogonal spectrum
sharing-based radar sensing and wireless communication is limited. As a
remedy, radar-centric communication integration was proposed by conveying
information over radar signals concurrently via the same spectrum. For
example, the amplitude and the phase of radar signals were controlled
modulating some information bits [10]. Since radar signals can be changed to a
limited extent, the information transmission rate of radar-centric
communication integration is severely unsatisfactory. In contrary, sensing-
aided communication was also proposed to enhance the performance of wireless
communication by using the sensed information. For instance, the authors in
[11] designed an effective beam selection scheme for millimeter-wave
communications based on the sensed environment information, e.g., the shape,
the position, and the materials of surrounding building/car/tree by using a
camera. Yet, the functions of sensing in sensing-aided communication are
constrained. In this context, ISAC based on the integrated wireless
architecture is regarded as a promising approach to enable 6G wireless
networks [12]. By optimizing the wireless transceivers, it is possible to
improve both the sensing and communication performance simultaneously. In
particular, since the base station (BS) will be equipped with a large-scale
antenna array for 6G wireless networks, the superposition coded sensing and
communication signal can be effectively separated by using spatial beamforming
[13]. Inspired by this, the authors in [14] studied a joint communication and
sensing 6G cellular system and discussed the optimal waveform design for
communication as well as sensing by exploiting ultra-high terahertz spectrum
and ultra-massive antenna array.

Meanwhile, integrated computing and communication has attracted significant
interests due to the demands of huge-volume data processing [15, 16].
Originally, cloud computing was commonly adopted by exploring the powerful
computing capability of cloud servers. For example, the authors in [17]
jointly optimized the precoder of communication signals and the computational
resources of cloud servers to minimize the energy consumption at the
terminals. Yet, with the increasing of data volume, cloud computing leads to
prohibitive communication load and latency. To address this issue, edge
computing is applied to wireless networks [18, 19]. By processing the data at
the edge of wireless networks, e.g., the BS, it is likely to increase the
efficiency of computing and communication. For instance, the transmit power of
communication signals and computational resources of edge servers in a multi-
cell wireless network were jointly optimized to maximize the weighted sum of
reductions in time and cost during task completion in [20]. Recently,
computing is implemented via cooperation among the BS and the targeted
devices, namely over-the-air computation, to further increase the computing
efficiency and to decrease the communication load [21]. Specifically, by
taking advantage of the superposition nature of wireless multiple access
channels, numerous computing functions can be realized at the BS by
coordinating the transmit signals of the end devices. In [22], the authors
jointly optimized transmit and receive beamforming to combat the interference
between computing and communication signals. As the processing capability of
the end devices increases, over-the-air computation and federate learning are
gradually combined to enable intelligent computing, namely over-the-air
federate learning (AirFL) [23]. In practice, AirFL trains local computing
models at the end devices with their raw data and aggregates a global
computing model at the BS by exploiting the over-the-air computation. In [24],
the authors derived the tradeoff between communication and computing, and
designed a broadband aggregation algorithm for analog AirFL to reduce the
communication latency.

Moreover, integrated sensing and computing has also become the research focus
with the widespread applications of sensing services. In [25], a novel
distributed framework was developed by combing edge computing and distributed
deep learning for urban environment sensing, which can effectively improve the
computing efficiency and the sensing accuracy. The authors in [26] developed a
multiple-input multiple-output (MIMO) over-the-air computation technique for
realizing multi-modal sensing with high-mobility. The equalization scheme
based on channel feedback was designed to minimize the computation error. In
addition, a real-time big data analytical framework for remote satellite
sensing service was put forward to enhance the computation efficiency in the
context of massive data [27].

Commonly, previous works attempt to integrate two functions in an effective
way. In order to support emerging intelligent applications and services in 6G
wireless networks, it is necessary to integrate sensing, computing and
communication simultaneously. In this paper, we aim to provide feasible and
effective integration schemes by exploiting available resources of 6G wireless
networks. The contributions of this paper are three-fold:

  1. 1.

We put forward a general design framework of ISCC for 6G wireless networks.
Massive MIMO techniques enable the networks to perform multiple-target
sensing, multiple-dimension computing, and multiple-stream communication
simultaneously over the same wireless resources.

  2. 2.

We design two typical joint beamforming algorithms for ISCC from the
perspectives of maximizing the overall performance subject to transmit power
constraints and minimizing the total transmit power consumption subject to
performance requirements, respectively.

  3. 3.

We reveal the impacts of key system parameters on the overall performance of
ISCC, and provide useful guideline for practical system design.

The rest of this paper outlined as follows: Section II briefly introduces a 6G
wireless network integrating sensing, computing and communication. Section III
presents two joint beamforming design algorithms of the weighted overall
performance maximization and the total transmit power minimization,
respectively. Section IV provides extensive simulations to verify the
effectiveness of the proposed algorithms and Section V concludes the paper.

_Notations_ : We use $\Omega_{N}=\\{1,2,\ldots,N\\}$ to denote a set for given
natural number $N$, bold upper (lower) letters to denote matrices (column
vectors), $\|\cdot\|$ to denote $L_{2}$-norm, $|\cdot|$ to denote the absolute
value of a scalar or the size of a dataset, and $\mathbb{E}\\{\cdot\\}$ to
denote expectation. ${{\mathbb{R}}^{m\times n}}$ and ${{\mathbb{C}}^{m\times
n}}$ represent the sets of $m$-by-$n$ dimensional real matrix and complex
matrix, respectively. $(\cdot)^{H}$, $\text{tr}(\cdot)$, $\text{Rank}(\cdot)$
indicates the conjugate transpose, the trace, and the rank of a matrix,
respectively.

##  II System Model

Figure 1: System model for 6G wireless network integrating sensing, computing
and communication.

We consider a 6G wireless network, c.f. Fig. 1, consisting of a BS equipped
with $N$ antennas, and $K$ multi-function sensors equipped with $M$ antennas
each, where $KM\leq N$. The range of interest (ROI), including target sensing
area and interference area, is partitioned into multiple cubes with equal
size, each of which denotes a pixel point111Note that the size of the pixel
represents the accuracy of sensing. In fact, pixels are not necessarily cubic,
and different side lengths can be set individually to change the accuracy of
different dimensions. According to the requirement of sensing accuracy, the
size and the shape of the pixels can be determined based on the wavelength of
the signal, the number of BS antennas, the overall size of the ROI and the
wireless environment. [28]. Without loss of generality, there exist $O$
clutters222In general, the clutters can be considered as scatters over
wireless channels. Hence, the number of clutters and channel state information
can be acquired by channel estimation [29]. within interference range around
the targets as interference for sensing, such a 6G wireless network has three
fundamental tasks, namely target sensing, model computing, and information
communication, which are collaboratively completed by multiple intelligent
sensors and a multi-antenna BS. Specifically, based on the acquired
environmental information333The environmental information can be roughly known
by environmental detection. For example, channel conditions can be obtained by
channel estimation [30] and the location of the ROI can be obtained by beam
sweeping [31]., each intelligent sensor first performs spatial directional
beamforming to control the transmit directions of the superposition of its
sensing, computing, and communication signals, such that the signals reach the
BS adaptively to the channel conditions. As for target sensing, the BS
receives the reflected signals containing the environmental information of the
target area and then designs the corresponding sensing receivers to estimate
the reflection coefficients of targets444Since all the pixel points of the ROI
have the same size, shape and alignment, the reflection coefficients are the
only recognition features in this work. For example, the BS can perform target
classification [32] or target imaging [33] by accurately obtaining their
reflection coefficients.. As for model computing, by exploiting the computing
ability of intelligent sensors that train local models with their datasets,
the BS designs the corresponding computing receivers to obtain the desired
computing signals to facilitate the global model via AirFL. As for information
communication, the BS designs the corresponding communication receivers to
decode the communication signals of each sensor for data transmission. With
the above framework, we can integrate sensing, computing and communication by
a unified transceiver.

Without loss of generality, we assume that the $k$-th sensor constructs the
sensing signals
$\mathbf{s}_{k}^{\text{sens}}={{\left[s_{k,1}^{\text{sens}},\ldots,s_{k,i}^{\text{sens}},\ldots,s_{k,I}^{\text{sens}}\right]}^{T}}\in{\mathbb{R}^{I\times
1}}$ to sense $I$ targets, namely $I$ pixel points in the target area,
generates local model parameters
$\mathbf{s}_{k}^{\text{comp}}={{\left[s_{k,1}^{\text{comp}},\ldots,s_{k,l}^{\text{comp}},\ldots,s_{k,L}^{\text{comp}}\right]}^{T}}\in{\mathbb{R}^{L\times\text{1}}}$
as computing signals to perform AirFL, and logs data
$\mathbf{s}_{k}^{\text{comm}}={{\left[s_{k,1}^{\text{comm}},\ldots,s_{k,j}^{\text{comm}},\ldots,s_{k,J}^{\text{comm}}\right]}^{T}}\in{\mathbb{R}^{J\times
1}}$ as communication signals to transfer the information in each time slot.
Note that the functions of theses three types of signals are different, where
${s}_{k,i}^{\text{sens}}$ is the sensing signal for the $i$-th target from the
$k$-th sensor, which does not carry any data, but is just used to estimate the
reflection coefficients of the targets, ${s}_{k,l}^{\text{comp}}$ is the
computing signal for the pre-processed $l$-th local model parameter at the
$k$-th sensor, which generates from AI model trained by its local dataset, and
${s}_{k,j}^{\text{comm}}$ is the communication signal for the $j$-th measured
data at the $k$-th sensor, which needs to be decoded exactly at the BS. For
ease of analysis, it is assumed that the sensing signals, the computing
signals, and the communication signals are mutually independent and all obey
Gaussian distributed with unit norm, i.e.,
$\mathbb{E}\left\\{s_{k,i}^{\text{sens}}{{\left(s_{k,i}^{\text{sens}}\right)}^{H}}\right\\}=\mathbb{E}\left\\{s_{k,l}^{\text{comp}}{{\left(s_{k,l}^{\text{comp}}\right)}^{H}}\right\\}=\mathbb{E}\left\\{s_{k,j}^{\text{comm}}{{\left(s_{k,j}^{\text{comm}}\right)}^{H}}\right\\}=1,\forall
k\in\Omega_{K},i\in\Omega_{I},l\in\Omega_{L},j\in\Omega_{J}$. Then, the $k$-th
sensor constructs a superposition coded transmit signal as

${{\mathbf{x}}_{k}}=\underbrace{\sum\limits_{i=1}^{I}{{{\mathbf{a}}_{k,i}}s_{k,i}^{\text{sens}}}}_{\text{Sensing}}+\underbrace{\sum\limits_{l=1}^{L}{{{\mathbf{b}}_{k,l}}s_{k,l}^{\text{comp}}}}_{\text{Computing}}+\underbrace{\sum\limits_{j=1}^{J}{{{\mathbf{c}}_{k,j}}s_{k,j}^{\text{comm}}}}_{\text{Communication}},$
(1)

where
${{\mathbf{a}}_{k,i}}\in{\mathbb{C}^{M\times\text{1}}},{{\mathbf{b}}_{k,l}}\in{\mathbb{C}^{M\times\text{1}}},{{\mathbf{c}}_{k,j}}\in{\mathbb{C}^{M\times\text{1}}}$
are the transmit beamforming vectors for corresponding sensing signal,
computing signal, and communication signal, respectively. Consequently, the
received signal at the BS is given by

$\displaystyle\mathbf{y}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\underbrace{\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{I}{{{r}_{i}}{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}s_{k,i}^{\text{sens}}}}}_{\text{Sensing}}+\underbrace{\sum\limits_{k=1}^{K}{\sum\limits_{o=1}^{O}{{{r}_{o}}{{\mathbf{F}}_{k,o}}\sum\limits_{m=1}^{I}{{{\mathbf{a}}_{k,m}}s_{k,m}^{\text{sens}}}}}}_{\text{Interference
from clutters}}$ (2) $\displaystyle+$
$\displaystyle\\!\\!\underbrace{\sum\limits_{k=1}^{K}{\sum\limits_{l=1}^{L}{{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,l}}s_{k,l}^{\text{comp}}}}}_{\text{Computing}}+\underbrace{\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}s_{k,j}^{\text{comm}}}}}_{\text{Communication}}+\underbrace{\mathbf{n}}_{\text{Noise}},$

where $\mathbf{n}$ is the additive white Gaussian noise (AWGN) vector,
$r_{i}\in\mathbb{C},\forall i\in\Omega_{I},$ and $r_{o}\in\mathbb{C},\forall
o\in\Omega_{O},$ are the reflection coefficients of the $i$-th target and the
$o$-th clutter, respectively.
${\mathbf{G}_{k,i}}=\mathbf{g}_{k,i}^{{}^{\prime}}\mathbf{g}_{k,i}^{H}\in\mathbb{C}^{N\times
M}$ represents the cascaded channel in which $\mathbf{g}_{k,i}$ and
$\mathbf{g}_{k,i}^{{}^{\prime}}$ are the channel gains from the $k$-th sensor
to the $i$-th target and from the $i$-th target to the BS, respectively.
${\mathbf{F}_{k,o}}=\mathbf{f}_{k,o}^{{}^{\prime}}\mathbf{f}_{k,o}^{H}\in\mathbb{C}^{N\times
M}$ denotes the cascaded channel, where $\mathbf{f}_{k,o}$ and
$\mathbf{f}_{k,o}^{{}^{\prime}}$ are the channel gains from the $k$-th sensor
to the $o$-th clutter and from the $o$-th clutter to the BS, respectively.
Furthermore, $\mathbf{H}_{k}\in\mathbb{C}^{N\times M}$ is the channel gain
from the $k$-th sensor to the BS. Here, we assume that ${\mathbf{G}_{k,i}}$,
${\mathbf{F}_{k,o}}$, and ${{\mathbf{H}}_{k}}$ remain constant within a time
slot, but fade over time slots independently. In the following, we will
discuss the performance metrics of the three fundamental tasks one-by-one.

###  II-A Sensing for Target Observation

First, we deal with the sensing signal. To improve the accuracy of target
sensing, all sensors collaborate to sense the targets. Thus, a linear unbiased
estimator is deployed at the BS to estimate the reflection coefficients of the
targets. To obtain an accurate reflection coefficient $r_{i},i\in\Omega_{I},$
for target sensing, the BS carries out receive beamforming to enhance the
desired signal and suppress the co-channel interference. Hence, the estimated
reflection coefficient ${{{\hat{r}}}_{i}}$ of the $i$-th target can be
expressed as

$\displaystyle\\!\\!{{{\hat{r}}}_{i}}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\mathbf{v}_{i}^{H}\mathbf{y}$ (3) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\mathbf{v}_{i}^{H}{{r}_{i}}\sum\limits_{k=1}^{K}{{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}s_{k,i}^{\text{sens}}}+\mathbf{v}_{i}^{H}\sum\limits_{k=1}^{K}{\sum\limits_{m=1,m\neq
i}^{I}{{{r}_{m}}{{\mathbf{G}}_{k,m}}{{\mathbf{a}}_{k,m}}s_{k,m}^{\text{sens}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\mathbf{v}_{i}^{H}\sum\limits_{k=1}^{K}{\sum\limits_{o=1}^{O}{{{r}_{o}}{{\mathbf{F}}_{k,o}}\sum\limits_{m=1}^{I}{{{\mathbf{a}}_{k,m}}s_{k,m}^{\text{sens}}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\mathbf{v}_{i}^{H}\sum\limits_{k=1}^{K}{{{\mathbf{H}}_{k}}\left(\sum\limits_{l=1}^{L}{{{\mathbf{b}}_{k,l}}s_{k,l}^{\text{comp}}}+\sum\limits_{j=1}^{J}{{{\mathbf{c}}_{k,j}}s_{k,j}^{\text{comm}}}\right)}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\mathbf{v}_{i}^{H}\mathbf{n},\forall i,$

where $\mathbf{v}_{i}\in\mathbb{C}^{N\times 1}$ is the sensing receive
beamforming vector adopted at the BS for the $i$-th target. Following relevant
parameter estimation works, we adopt the mean squared error (MSE) as the
performance metric of target sensing. It is seen from Eq. (3) that the
$\hat{r}_{i}$ is not only a function of $s_{k,i}^{\text{sens}}$, but also the
other sensing signals from different sensors for different targets, the
computing signals, and communication signals. In this context, we can minimize
the MSE between the estimated reflection coefficient ${{{\hat{r}}}_{i}}$ and
the actual reflection coefficient $r_{i}$ to suppress the interference from
other signals and achieve a high-accuracy target sensing. Thus, the MSE is
given by

$\displaystyle\text{MSE}_{i}^{\text{sens}}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\left({{{\hat{r}}}_{i}}-{{r}_{i}}\right){{\left({{{\hat{r}}}_{i}}-{{r}_{i}}\right)}^{H}}\right\\}$
(4) $\displaystyle=$
$\displaystyle{{\left|\sum\limits_{k=1}^{K}{\mathbf{v}_{i}^{H}{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}}-1\right|}^{2}}R_{\text{i}}^{2}$
$\displaystyle+$ $\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{m=1,m\neq
i}^{I}{R_{\text{m}}^{2}{{\left|\mathbf{v}_{i}^{H}{{\mathbf{G}}_{k,m}}{{\mathbf{a}}_{k,m}}\right|}^{2}}}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{o=1}^{O}{\sum\limits_{m=1}^{I}{R_{o}^{2}{{\left|\mathbf{v}_{i}^{H}{{\mathbf{F}}_{k,o}}{{\mathbf{a}}_{k,m}}\right|}^{2}}}}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\left(\sum\limits_{l=1}^{L}{{{\left|\mathbf{v}_{i}^{H}{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,l}}\right|}^{2}}}+\sum\limits_{j=1}^{J}{{{\left|\mathbf{v}_{i}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}\right|}^{2}}}\right)}$
$\displaystyle+$
$\displaystyle\sigma_{n}^{2}{{\left\|\mathbf{v}_{i}^{H}\right\|}^{2}},\forall
i,$

where $\sigma_{n}^{2}$ is the noise power, and ${{R}_{m}}$ is the root mean
squared (RMS) value of the reflection coefficient ${{r}_{m}},\forall
m\in\\{\Omega_{I}\cup\Omega_{O}\\}$ according to the a-priori probability of
occurrence for each type [32]. To present more intuitively the impact of
reflection coefficient estimation, we provide an application result on target
imaging [33]. As is shown in Fig. 2, the higher the sensing accuracy (the
smaller MSE between the estimated reflection coefficient and the actual
reflection coefficient), the more accurate the target imaging.

Figure 2: Target Imaging results under different sensing accuracy.

###  II-B Computing for Model Aggregation

Secondly, we handle the computing signal. In a time slot, a shared learning
process for AirFL with a global model
$\mathbf{s}^{\text{comp}}\in\mathbb{R}^{L\times 1}$ is trained cooperatively
by multiple intelligent sensors [23, 24]. Each intelligent sensor has its own
local dataset $\mathcal{D}_{k},\forall k\in\Omega_{K}$, and the sum of local
datasets is $\mathcal{D}$, where
$\left|\mathcal{D}\right|=\sum\limits_{k=1}^{K}{\left|{{\mathcal{D}}_{k}}\right|}$.
Thus, the desired global model at the BS by computing local models is given by

$s_{l}^{\text{comp}}=\frac{1}{K}\sum\limits_{k=1}^{K}{s_{k,l}^{\text{comp}}},\forall
l\in\Omega_{L}$ (5)

where $s_{k,l}^{\text{comp}}$ is the $l$-th computing signal contributed by
the pre-processed local model parameter at the $k$-th sensor, and each local
model parameter is multiplied by a weighting factor
${{\xi}_{k}}=\frac{\left|{{\mathcal{D}}_{k}}\right|}{\left|\mathcal{D}\right|}$
before transmission. To reduce the computation error in the presence of
interference and channel fading, we need to implement receive beamforming at
the BS. As a result, the received signal for AirFL at the BS can be expressed
as

$\displaystyle\hat{s}_{l}^{\text{comp}}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{\mathbf{z}_{l}^{H}}{K}\mathbf{y}$ (6)
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{\mathbf{z}_{l}^{H}}{K}\sum\limits_{k=1}^{K}{{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,l}}s_{k,l}^{\text{comp}}+\frac{\mathbf{z}_{l}^{H}}{K}\sum\limits_{k=1}^{K}{\sum\limits_{i=1,i\neq
l}^{L}{{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,i}}s_{k,i}^{\text{comp}}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{\mathbf{z}_{l}^{H}}{K}\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{I}{{{r}_{i}}{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}s_{k,i}^{\text{sens}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{\mathbf{z}_{l}^{H}}{K}\sum\limits_{k=1}^{K}{\sum\limits_{o=1}^{O}{{{r}_{o}}{{\mathbf{F}}_{k,o}}\sum\limits_{m=1}^{I}{{{\mathbf{a}}_{k,m}}s_{k,m}^{\text{sens}}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{\mathbf{z}_{l}^{H}}{K}\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}s_{k,j}^{\text{comm}}}}+\frac{\mathbf{z}_{l}^{H}}{K}\mathbf{n},\forall
l,$

where $\mathbf{z}_{l}\in\mathbb{C}^{N\times 1}$ is the computing receive
beamforming vector at the BS for the $l$-th model parameter. In general, the
distortion of the model aggregation is measured by the MSE between
${s}_{l}^{\text{comp}}$ and ${\hat{s}_{l}^{\text{comp}}}$, which can be
expressed as

$\displaystyle\\!\\!\\!\text{MSE}_{l}^{\text{comp}}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\mathbb{E}\left\\{\left(\hat{s}_{l}^{\text{comp}}-s_{l}^{\text{comp}}\right){{\left(\hat{s}_{l}^{\text{comp}}-s_{l}^{\text{comp}}\right)}^{H}}\right\\}$
(7) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{1}{{{K}^{2}}}\sum\limits_{k=1}^{K}{{{\left|\mathbf{z}_{l}^{H}{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,l}}-1\right|}^{2}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{1}{{{K}^{2}}}\sum\limits_{m=1,m\neq
l}^{L}{\sum\limits_{k=1}^{K}{{{\left|\mathbf{z}_{l}^{H}{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,m}}\right|}^{2}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{1}{{{K}^{2}}}\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{I}{R_{i}^{2}{{\left|\mathbf{z}_{l}^{H}{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}\right|}^{2}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{1}{{{K}^{2}}}\sum\limits_{k=1}^{K}{\sum\limits_{o=1}^{O}{\sum\limits_{m=1}^{I}{R_{o}^{2}{{\left|\mathbf{z}_{l}^{H}{{\mathbf{F}}_{k,o}}{{\mathbf{a}}_{k,m}}\right|}^{2}}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{1}{{{K}^{2}}}\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{{{\left|\mathbf{z}_{l}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}\right|}^{2}}}}+\frac{\sigma_{n}^{2}}{{{K}^{2}}}{{\left\|{{\mathbf{z}}_{l}}\right\|}^{2}},\forall
l.$

Herein, we also provide a classical application case to vividly present the
function of computing, i.e. image recognition based on AirFL[34]. It is seen
from Table I that the higher computing accuracy (the smaller computing MSE
between the aggregated computing signal and the desired global model), the
more accurate the image recognition.

TABLE I: Handwritten-digit identification results under different computing accuracy on IID MNIST dataset trained by CNN network Data image |  |  |  |  |  |  |
---|---|---|---|---|---|---|---
Computing accuracy: High (small MSE) | 0 | 5 | 1 | 9 | 4 | 6 | 4
Computing accuracy: Medium (medium MSE) |  6($\times$) | 5 | 1 | 9 | 4 | 6 | 4
Computing accuracy: Low (large MSE) | 0 |  8($\times$) | 1 |  4($\times$) |  8($\times$) |  4($\times$) | 4
Data image |  |  |  |  |  |  |
Computing accuracy: High (small MSE) | 7 | 1 | 8 | 5 | 2 | 9 | 3
Computing accuracy: Medium (medium MSE) | 7 | 1 | 8 | 5 | 2 |  4($\times$) | 3
Computing accuracy: Low (large MSE) |  4($\times$) | 1 | 8 |  8($\times$) | 2 |  4($\times$) |  8($\times$)

###  II-C Communication for Information Transmission

Finally, we discuss the communication signal. To promote the quality of the
communication signal, the BS performs corresponding receive beamforming to
mitigate the interference. Hence, the received signal at the BS for the
communication signal $s_{k,j}^{\text{comm}},\forall
k\in\Omega_{K},j\in\Omega_{J},$ is given by

$\displaystyle y_{k,j}^{\text{comm}}$ $\displaystyle=$
$\displaystyle\mathbf{u}_{k,j}^{H}\mathbf{y}$ (8) $\displaystyle=$
$\displaystyle\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}s_{k,j}^{\text{comm}}$
$\displaystyle+$ $\displaystyle\sum\limits_{i=1,i\neq
k}^{K}{\sum\limits_{n=1,n\neq
j}^{J}{\mathbf{u}_{k,j}^{H}{{\mathbf{c}}_{i,n}}s_{i,n}^{\text{comm}}}}$
$\displaystyle+$
$\displaystyle\sum\limits_{i=1}^{K}{\sum\limits_{l=1}^{L}{\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}{{\mathbf{b}}_{i,l}}s_{i,l}^{\text{comp}}}}$
$\displaystyle+$
$\displaystyle\mathbf{u}_{k,j}^{H}\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{I}{{{r}_{i}}{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}s_{k,i}^{\text{sens}}}}$
$\displaystyle+$
$\displaystyle\mathbf{u}_{k,j}^{H}\sum\limits_{k=1}^{K}{\sum\limits_{o=1}^{O}{{{r}_{o}}{{\mathbf{F}}_{k,o}}\sum\limits_{m=1}^{I}{{{\mathbf{a}}_{k,m}}s_{k,m}^{\text{sens}}}}}+\mathbf{u}_{k,j}^{H}\mathbf{n},$

where $\mathbf{u}_{k,j}\in\mathbb{C}^{N\times 1}$ is the communication receive
beamforming vector of the $j$-th communication signal from the $k$-th sensor.
Since the received SINR determines the quality of the communication signal, it
is usually regarded as the performance metric for communication. The signal-
to-interference-plus-noise ratio (SINR) related to $s_{k,j}^{\text{comm}}$ can
be expressed as

${{\Gamma}_{k,j}}=\frac{{{\left|\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}\right|}^{2}}}{\sum\limits_{i=1,i\neq
k}^{K}{\sum\limits_{n=1,n\neq
j}^{J}{{{\left|\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}{{\mathbf{c}}_{i,n}}\right|}^{2}}}}+{{X}_{k,j}}+\sigma_{n}^{2}{{\left\|\mathbf{u}_{k,j}\right\|}^{2}}},$
(9)

where

$\displaystyle{{X}_{k,j}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum\limits_{i=1}^{K}{\sum\limits_{l=1}^{L}{{{\left|\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}{{\mathbf{b}}_{i,l}}\right|}^{2}}}}+\sum\limits_{i=1}^{K}{\sum\limits_{m=1}^{I}{R_{m}^{2}{{\left|\mathbf{u}_{k,j}^{H}{{\mathbf{G}}_{i,m}}{{\mathbf{a}}_{i,m}}\right|}^{2}}}}$
(10) $\displaystyle+$
$\displaystyle\\!\\!\\!\sum\limits_{i=1}^{K}{\sum\limits_{o=1}^{O}{\sum\limits_{m=1}^{I}{R_{o}^{2}{{\left|\mathbf{u}_{k,j}^{H}{{\mathbf{F}}_{i,o}}{{\mathbf{a}}_{i,m}}\right|}^{2}}}}}.$

It is observed from (4), (7), and (9) that the performance of sensing,
computing, and communication are jointly determined by the transmit
beamforming vectors $\mathbf{a}_{k,i}$, $\mathbf{b}_{k,l}$, and
$\mathbf{c}_{k,j}$ at the sensors, and receive beamforming vectors
$\mathbf{v}_{i}$, $\mathbf{z}_{l}$, and $\mathbf{u}_{k,j}$ at the BS. Although
the three performance metrics are all desirable to the system, they are
competitive for the system resources. Hence, it is desired to establish a
joint beamforming design framework to enhance the overall performance of ISCC
over the limited resources in 6G wireless networks.

_Remark:_ It is worthy pointing out that our proposed integrated system is
inherently different from multi-user communication systems in terms of signal
waveform, transceiver design and function realization. Firstly, multi-user
communication systems only consider one kind of signal for communication
purpose, while in our proposed integrated system, sensing, computing and
communication signals have different types and serve different purposes.
Secondly, signals of the same type from different users interfere with each
other at the BS in multi-user communication systems, and thus transceiver is
designed to reduce the inter-user interference. In contrary, for the proposed
integrated system, signals of different types interfere with each other, but
signals of the same type may help each other. In this case, the designed
transceiver not only needs to decrease the interference among signals of
different types, but also needs to increase the cooperative gain of signals of
the same type in terms of sensing and computing. Finally, although different
performance metrics such as SINR, data rate and MSE, are chosen for
optimization in multi-user communication systems, they are all used to improve
the performance of the communication function. While for the proposed
integrated system, we select the appropriate performance metrics based on the
functions of sensing, computing and communication. By solving the formulated
multi-objective optimization problem (MOOP), the desired performance of three
different functions can be effectively achieved.

##  III Joint Design of Sensing, Computing and Communication

This section aims at jointly designing transmit and receive beamforming
vectors for ISCC in 6G wireless networks. Since 6G wireless networks have
different priorities among sensing, computing and communication for various
application scenarios, we formulate two typical categories of MOOPs to
investigate the trade-offs among them. The first one is a weighted overall
performance maximization (WOPM) subject to the maximum budget of transmit
power. The second one is a total transmit power minimization (TTPM) while
guaranteeing the quality of service (QoS) requirements on sensing, computing,
and communication, respectively.

###  III-A Weighted Overall Performance Maximization Design

Now, we first study three individual single-objective optimization problem
(SOOP) for sensing, computing, and communication in 6G wireless networks,
respectively, which severs as building blocks for the formulation of WOPM
design. The first SOOP aims at minimizing the weighted sum-MSE for sensing
subject to the maximum transmit power budget, which is formulated as _S-1:
Weighted Sum-MSE of Sensing Minimization:_

$\displaystyle\\!\\!\\!\\!\\!\\!\underset{\begin{smallmatrix}{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},\\\
{{\mathbf{c}}_{k,j}},{{\mathbf{v}}_{i}}\end{smallmatrix}}{\mathop{\text{min
}}}\,\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\sum\limits_{i=1}^{I}{\theta_{i}^{\text{sens}}\text{MSE}_{i}^{\text{sens}}}$
(11) s.t.
$\displaystyle\\!\\!\\!\\!\\!\text{C1:}\sum\limits_{i=1}^{I}{{{\left\|{{\mathbf{a}}_{k,i}}\right\|}^{2}}}+\sum\limits_{l=1}^{L}{{{\left\|{{\mathbf{b}}_{k,l}}\right\|}^{2}}}+\sum\limits_{j=1}^{J}{{{\left\|{{\mathbf{c}}_{k,j}}\right\|}^{2}}}\leq{{P}_{\max,k}},$

where constant $\theta_{i}^{\text{sens}}$ is the weighted coefficient related
to $\text{MSE}_{i}^{\text{sens}}$, and $P_{\max,k}$ is the maximum transmit
power budget at the $k$-th sensor. The objective in _S-1_ is to minimize the
weighted summation of MSEs between estimated reflection coefficients and
actual ones for all $I$ targets depicted in (4), and constraint C1 is the
transmit power limitation at the $k$-th sensor. Note that each SOOP only
concerns one performance metric of the proposed integrated sensing, computing
and communication without considering others. For example, it is seen from
_S-1_ that the communication beam ${{\mathbf{c}}_{k,j}}$ and the computing
beam ${{\mathbf{b}}_{k,l}}$ will naturally be zero to minimize the objective,
while the sensing beam ${{\mathbf{a}}_{k,i}}$ will not be zero due to the term
${{\left|\sum\limits_{k=1}^{K}{\mathbf{v}_{i}^{H}{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}}-1\right|}^{2}}R_{\text{i}}^{2}$
in (4). Hence, no additional conditions related to communication and computing
beams are required. The second SOOP is to minimize the weighted sum-MSE for
computing under the limitation of transmit power consumption, which can be
mathematically expressed as

_S-2: Weighted Sum-MSE of Computing Minimization:_

$\displaystyle\underset{\begin{smallmatrix}{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},\\\
{{\mathbf{c}}_{k,j}},{{\mathbf{z}}_{l}}\end{smallmatrix}}{\mathop{\text{min
}}}\,\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum\limits_{l=1}^{L}{\theta_{l}^{\text{comp}}\text{MSE}_{l}^{\text{comp}}}$
s.t. $\displaystyle\\!\\!\\!\\!\\!\text{C1},$ (12)

where constant $\theta_{l}^{\text{comp}}$ is the weighted coefficient relevant
to $\text{MSE}_{l}^{\text{comp}}$, and the objective is minimizing the
weighted summation of the computation distortions for $L$ model parameters,
which was defined in (7). The third SOOP focuses on the weighted sum-rate
maximization for communication subject to the transmit power limitation, which
is formulated as

_S-3’: Weighted Sum-Rate of Communication Maximization:_

$\displaystyle\underset{\begin{smallmatrix}{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},\\\
{{\mathbf{c}}_{k,j}},{{\mathbf{u}}_{k,j}}\end{smallmatrix}}{\mathop{\text{max}}}\,\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\theta_{k,j}^{\text{comm}}{{r}_{k,j}}}}$
s.t. $\displaystyle\\!\\!\\!\\!\\!\text{C1},$ (13)

where ${{r}_{k,j}}={{\log}_{2}}\left(1+{{\Gamma}_{k,j}}\right)$ is the
achievable rate of the $j$-th communication signal from the $k$-th sensor and
$\theta_{k,j}^{\text{comm}}$ is the weighted coefficient associated with
${{r}_{k,j}}$. _S-3’_ is a classical weighted sum-rate maximization problem
which is non-convex due to the complicated objective function involving log
functions of fractions [35]. Moreover, the transmit beamforming vectors
$\\{{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}\\}$ and
communication receive beamforming vector $\mathbf{u}_{k,j}$ are coupled in the
constraints and objective function. To handle this issue as well as facilitate
the design, we transform _S-3’_ into its equivalent log-MSE minimization
problem via the following theorem.

_Theorem 1:_ The relationship between the received SINR $\Gamma_{k,j}$ and the
minimum MSE (MMSE) $e_{k,j}^{\text{comm}}$ for the communication signal
$s_{k,j}^{\text{comm}}$ can be expressed as

$1+{{\Gamma}_{k,j}}=({e_{k,j}^{\text{comm}}})^{-1},\forall k,j.$ (14)

###### Proof:

Please refer to Appendix A. ∎

According to Theorem 1, we can transform the objective function of _S-3’_ to

$\underset{\begin{smallmatrix}{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},\\\
{{\mathbf{c}}_{k,j}},{{\mathbf{u}}_{k,j}}\end{smallmatrix}}{\mathop{\text{min}}}\,\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\theta_{k,j}^{\text{comm}}{{\log}_{2}}\left(e_{k,j}^{\text{comm}}\right)}},$
(15)

which is equivalent to minimizing the weighted sum-MSE of communication
$\text{MSE}_{k,j}^{\text{comm}}$ with the MMSE receiver
${{\mathbf{u}}_{k,j}}=\Xi^{-1}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}$, given
in Appendix A. Thus, (15) can be reformulated as

$\underset{{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}}{\text{min}}\,\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\theta_{k,j}^{\text{comm}}{{\log}_{2}}\left(\text{MSE}_{k,j}^{\text{comm}}\right)}}$
(16)

However, the transformed objective function (16) still remains non-convex
caused by the structure of sum of logarithmic function. To this end, we
introduce a weight variable $\omega_{k,j}$ for the
$\text{MSE}_{k,j}^{\text{comm}}$ [36], and then (16) is transformed as

$\underset{\begin{smallmatrix}{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},\\\
{{\mathbf{c}}_{k,j,}},{{\omega}_{k,j}}\end{smallmatrix}}{\mathop{\min}}\,\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\theta_{k,j}^{\text{comm}}\left[{{\omega}_{k,j}}\text{MSE}_{k,j}^{\text{comm}}-{{\log}_{2}}\left({{\omega}_{k,j}}\right)\right]}}.$
(17)

As for unconstrained optimization problems in terms of the weight variable
$\omega_{k,j}$, by letting the first-derivative of the objective function (17)
equal to zero, we can obtain the optimal solution as follows

$\omega_{k,j}^{*}={{\left(\ln
2\cdot\text{MSE}_{k,j}^{\text{comm}}\right)}^{-1}},$ (18)

which makes (16) and (17) are equivalent. After the above conversion, _S-3’_
can be transformed as

_S-3: Weighted Sum-Modified MSE of Communication Minimization:_

$\displaystyle\underset{\begin{smallmatrix}{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}},\\\
{{\mathbf{u}}_{k,j}},{{\omega}_{k,j}}\end{smallmatrix}}{\mathop{\text{min
}}}\,\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\theta_{k,j}^{\text{comm}}\left[{{\omega}_{k,j}}\text{MSE}_{k,j}^{\text{comm}}-{{\log}_{2}}\left({{\omega}_{k,j}}\right)\right]}}$
s.t. $\displaystyle\\!\\!\\!\\!\text{C1}.$ (19)

To jointly optimize the three performance of sensing, computing, and
communication, we adopted the weighted sum method to formulate the
corresponding MOOP [37, 38]. The WOPM is formulated as follows:

_M-1: Weighted Overall Performance Maximization:_

$\displaystyle\underset{\begin{smallmatrix}{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}},\\\
{{\mathbf{v}}_{i}},{{\mathbf{u}}_{k,j}},{{\mathbf{z}}_{l}},{{\omega}_{k,j}}\end{smallmatrix}}{\mathop{\min}}\,\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!{{\alpha}_{1}}{{\Psi}_{1}}+{{\alpha}_{2}}{{\Psi}_{2}}+{{\alpha}_{3}}{{\Psi}_{3}}$
s.t. $\displaystyle\\!\\!\\!\\!\text{C1},$ (20)

where ${{\Psi}_{p}},p=1,2,3$ is the normalized objective function555Since the
objective value of _S-1_ , _S-2_ and _S-3_ have different ranges, it is
desired to perform the normalization for each objective function in order to
coordinate the performance of sensing, computing, and communication as well as
to facilitate the convergence of objective value for _M-1_. In this paper, we
define $\mathcal{F}_{p},p=1,2,3,$ as the objective function of _S- $p$_ and
${{\Psi}_{p}}={\left({\mathcal{F}_{p}}-\mathcal{F}_{p}^{*}\right)}/{\left|\mathcal{F}_{p}^{*}\right|}\;$,
where $\mathcal{F}_{p}^{*}$ are the corresponding performance limits of
sensing, computing and communication in _M-1_ , which can be obtained by
respectively solving _S- $p$_ by applying the same algorithm for solving
_M-1_. of _S-1_ , _S-2_ and _S-3_ , respectively. $\alpha_{p}\geq 0$ is the
priority of the $p$-th objective function, which represents the preference of
the system operator and is satisfied with
$\alpha_{1}+\alpha_{2}+\alpha_{3}=1$. By varying the value of $\alpha_{p}$,
_M-1_ can yield different solutions. Note that _M-1_ is equivalent to _S-p_
when $\alpha_{p}=1$ and $\alpha_{q}=0,\forall p\neq q$, which means _M-1_ is a
general formulation of _S-1_ , _S-2_ and _S-3_. Hence, we aim at solving _M-1_
with given priorities $\alpha_{p}$ in the following.

Since multiple variables are inter-coupled in the objective function of _M-1_
, i.e., transmit beams
$\\{\mathbf{a}_{k,i},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}\\}$, receive
beams $\\{{{\mathbf{v}}_{i}},\mathbf{u}_{k,j},{{\mathbf{z}}_{l}}\\}$, and
weight variables $\\{\omega_{k,j}\\}$, _M-1_ is non-convex, which makes it
impossible to obtain an optimal solution in polynomial time. In this context,
we turn to find a sub-optimal solution for exploring the trade-off
relationship among sensing, computing, and communication in 6G wireless
networks. By examining _M-1_ , although it is not a joint convex function of
all variables, it is a convex one in terms of transmit beams, receive beams,
and weight variables, respectively. Based on this observation, an alternating
optimization (AO) method is applied to divide _M-1_ into three subproblems,
i.e., optimizing receive beams by fixing transmit beams and weight variables,
optimizing weight variables by fixing transmit and receive beams, and
optimizing transmit beams by fixing receive beams and weight variables. In
particular, enabled with the AO method, solution procedure for _M-1_ will stop
until the objective value converges in the iterations [39]. Let us first
address the subproblem of optimizing receive beams
$\\{{{\mathbf{v}}_{i}},\mathbf{u}_{k,j},{{\mathbf{z}}_{l}}\\}$ while other
variables remain fixed. By applying the Karush-Kuhn-Tucher (KKT) conditions in
_M-1_ , i.e.,

$\displaystyle\begin{matrix}\frac{\partial}{\partial{{\mathbf{v}}_{i}}}\left(\sum\limits_{p=1}^{3}{{{\alpha}_{p}}{{\Psi}_{p}}}\right)=\mathbf{0}\Rightarrow\frac{\partial}{\partial{{\mathbf{v}}_{i}}}\left(\text{MSE}_{i}^{\text{sens}}\right)=\mathbf{0},\\\
\frac{\partial}{\partial{{\mathbf{z}}_{l}}}\left(\sum\limits_{p=1}^{3}{{{\alpha}_{p}}{{\Psi}_{p}}}\right)=\mathbf{0}\Rightarrow\frac{\partial}{\partial{{\mathbf{z}}_{l}}}\left(\text{MSE}_{l}^{\text{comp}}\right)=\mathbf{0},\\\
\frac{\partial}{\partial{{\mathbf{u}}_{k,j}}}\left(\sum\limits_{p=1}^{3}{{{\alpha}_{p}}{{\Psi}_{p}}}\right)=\mathbf{0}\Rightarrow\frac{\partial}{\partial{{\mathbf{u}}_{k,j}}}\left(\text{MSE}_{k,j}^{\text{comm}}\right)=\mathbf{0},\\\
\end{matrix}$ (21)

we can acquire the optimal receive beamforming vectors, also called MMSE
receiver, as below

${{\mathbf{v}}_{i}}={{\left[{{\bm{\Phi}}_{i}}\left(\sum\limits_{k=1}^{K}{\mathbf{a}_{k,i}^{H}\mathbf{G}_{k,i}^{H}}\right)+\bm{\Phi}\right]}^{-\text{1}}}{{\bm{\Phi}}_{i}},$
(22)
${{\mathbf{z}}_{l}}\text{=}{\bm{\Xi}^{-1}}\sum\limits_{k=1}^{K}{{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,l}}},$
(23)

and

${{\mathbf{u}}_{k,j}}={\bm{\Xi}^{-1}}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}},$
(24)

respectively, where $\bm{\Phi}=\sum\limits_{k=1}^{K}{\sum\limits_{n=1,n\neq
i}^{I}{R_{n}^{2}{{\mathbf{G}}_{k,n}}{{\mathbf{a}}_{k,n}}\mathbf{a}_{k,n}^{H}\mathbf{G}_{k,n}^{H}}}+\sum\limits_{k=1}^{K}{\sum\limits_{o=1}^{O}{\sum\limits_{m=1}^{I}{R_{o}^{2}{{\mathbf{F}}_{k,o}}{{\mathbf{a}}_{k,m}}\mathbf{a}_{k,m}^{H}\mathbf{F}_{k,o}^{H}}}}+\sum\limits_{k=1}^{K}{\sum\limits_{l=1}^{L}{{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,l}}\mathbf{b}_{k,l}^{H}\mathbf{H}_{k}^{H}}}+\sum\limits_{k=1}^{K}{\sum\limits_{m=1}^{J}{{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,m}}}}\mathbf{c}_{k,m}^{H}\mathbf{H}_{k}^{H}+\sigma_{n}^{2}{{\mathbf{I}}_{N}}$,
${{\bm{\Phi}}_{i}}=R_{i}^{2}\sum\limits_{k=1}^{K}{{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}}$,
and
$\bm{\Xi}=\bm{\Phi}+\sum\limits_{k=1}^{K}{\sum\limits_{n=1}^{I}{R_{n}^{2}{{\mathbf{G}}_{k,n}}{{\mathbf{a}}_{k,n}}\mathbf{a}_{k,n}^{H}\mathbf{G}_{k,n}^{H}}}$
is defined in Appendix A. Next, for the second subproblem in terms of
optimizing weight variables $\\{\omega_{k,j}\\}$, the optimal solution is
given in (18), i.e., $\omega_{k,j}^{*}={{\left(\ln
2\cdot\text{MSE}_{k,j}^{\text{comm}}\right)}^{-1}}$. Finally, with the MMSE
receivers and weight variables, the last subproblem of optimizing transmit
beams $\\{\mathbf{a}_{k,i},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}\\}$ for
maximizing the weighted overall performance is a standard convex quadratic
constrained quadratic programming (QCQP) problem, which can be solved by an
inter-point method (IPM) [[42], Chapter 11]. At first, we utilize the barrier
method to transform this subproblem into an unconstrained convex optimization
problem by adding the logarithmic barrier function. Then, the Newton’s method
can be applied to obtain the solution. Specifically, the transformed
unconstrained convex optimization problem can be written as

$\underset{{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}}{\mathop{\min}}\,\varepsilon(\alpha_{1}{{\Psi}_{1}}+\alpha_{2}{{\Psi}_{2}}+\alpha_{3}{{\Psi}_{3}})-\sum\limits_{k=1}^{K}{\log\left(-f_{k}^{\text{con}}\right)},$
(25)

where
$f_{k}^{\text{con}}=\sum\limits_{i=1}^{I}{{{\left\|{{\mathbf{a}}_{k,i}}\right\|}^{2}}}+\sum\limits_{l=1}^{L}{{{\left\|{{\mathbf{b}}_{k,l}}\right\|}^{2}}}+\sum\limits_{j=1}^{J}{{{\left\|{{\mathbf{c}}_{k,j}}\right\|}^{2}}}-{{P}_{\max,k}}$
and $\varepsilon>0$ is a barrier parameter that sets the accuracy of the
approximation. Then, we compute the variables’ gradients and Hessian for
Newton’s method as follows

$\displaystyle\begin{matrix}{{\nabla}_{{{\mathbf{a}}_{k,i}}}}=2\varepsilon\left(\mathbf{T}_{k,i}^{a}{{\mathbf{a}}_{k,i}}-R_{\text{i}}^{2}{{{\tilde{\alpha}}}_{1}}\theta_{i}^{\text{sens}}\mathbf{G}_{k,i}^{H}{{\mathbf{v}}_{i}}\right)+\frac{2{{\mathbf{a}}_{k,i}}}{f_{k}^{\text{con}}},\\\
{{\nabla}_{{{\mathbf{b}}_{k,l}}}}=\frac{2\varepsilon}{{{K}^{2}}}\left(\mathbf{T}_{k,l}^{{b}}{{\mathbf{b}}_{k,l}}-{{{\tilde{\alpha}}}_{2}}\theta_{l}^{\text{comp}}\mathbf{H}_{k}^{H}{{\mathbf{z}}_{l}}\right)+\frac{2{{\mathbf{b}}_{k,l}}}{f_{k}^{\text{con}}},\\\
{{\nabla}_{{{\mathbf{c}}_{k,j}}}}=2\varepsilon\left(\mathbf{T}_{k,l}^{{c}}{{\mathbf{c}}_{k,j}}-{{{\tilde{\alpha}}}_{3}}\theta_{k,j}^{\text{comm}}{{\omega}_{k,j}}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}\right)+\frac{2{{\mathbf{c}}_{k,j}}}{f_{k}^{\text{con}}},\\\
\end{matrix}$ (26)

and

$\displaystyle\begin{matrix}\nabla_{{{\mathbf{a}}_{k,i}}}^{2}=2\varepsilon\mathbf{T}_{k,i}^{{a}}+\frac{4}{{{\left(f_{k}^{\text{con}}\right)}^{2}}}{{\mathbf{a}}_{k,i}}\mathbf{a}_{k,l}^{T}+\frac{2}{f_{k}^{\text{con}}}\mathbf{I},\\\
\nabla_{{{\mathbf{b}}_{k,l}}}^{2}=\frac{2\varepsilon}{{{K}^{2}}}\mathbf{T}_{k,l}^{{b}}+\frac{4}{{{\left(f_{k}^{\text{con}}\right)}^{2}}}{{\mathbf{b}}_{k,l}}\mathbf{b}_{k,l}^{T}+\frac{2}{f_{k}^{\text{con}}}\mathbf{I},\\\
\nabla_{{{\mathbf{c}}_{k,j}}}^{2}=2\varepsilon\mathbf{T}_{k,l}^{{c}}+\frac{4}{{{\left(f_{k}^{\text{con}}\right)}^{2}}}{{\mathbf{c}}_{k,j}}\mathbf{c}_{k,j}^{T}+\frac{2}{f_{k}^{\text{con}}}\mathbf{I},\\\
\end{matrix}$ (27)

with

$\displaystyle\\!\\!\\!\\!\\!\\!\mathbf{T}_{k,i}^{{a}}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!{{{\tilde{\alpha}}}_{1}}\theta_{i}^{\text{sens}}\left(R_{\text{i}}^{2}\mathbf{G}_{k,i}^{H}{{\mathbf{v}}_{i}}\mathbf{v}_{i}^{H}{{\mathbf{G}}_{k,i}}+\sum\limits_{o=1}^{O}{R_{o}^{2}}\mathbf{F}_{k,o}^{H}{{\mathbf{v}}_{i}}\mathbf{v}_{i}^{H}{{\mathbf{F}}_{k,o}}\right)$
(28) $\displaystyle+$
$\displaystyle\\!\\!\\!\frac{{{{\tilde{\alpha}}}_{2}}}{{{K}^{2}}}\sum\limits_{l=1}^{L}{\theta_{l}^{\text{comp}}\left(R_{i}^{2}\mathbf{G}_{k,i}^{H}{{\mathbf{z}}_{l}}\mathbf{z}_{l}^{H}{{\mathbf{G}}_{k,i}}+\sum\limits_{o=1}^{O}{R_{o}^{2}}\mathbf{F}_{k,o}^{H}{{\mathbf{z}}_{l}}\mathbf{z}_{l}^{H}{{\mathbf{F}}_{k,o}}\right)}$
$\displaystyle+$
$\displaystyle\\!\\!\\!{{{\tilde{\alpha}}}_{3}}\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\theta_{k,j}^{\text{comm}}{{\omega}_{k,j}}R_{i}^{2}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}}},$
$\displaystyle+$
$\displaystyle\\!\\!\\!{{{\tilde{\alpha}}}_{3}}\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\theta_{k,j}^{\text{comm}}{{\omega}_{k,j}}\sum\limits_{o=1}^{O}{R_{o}^{2}}\mathbf{F}_{k,o}^{H}{{\mathbf{u}}_{k,j}}\mathbf{u}_{k,j}^{H}{{\mathbf{F}}_{k,o}}}},$
$\displaystyle\mathbf{T_{k,l}^{{b}}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{{{\tilde{\alpha}}}_{1}}{{K}^{2}}\sum\limits_{i=1}^{I}{\theta_{i}^{\text{sens}}\mathbf{H}_{k}^{H}{{\mathbf{v}}_{i}}\mathbf{v}_{i}^{H}{{\mathbf{H}}_{k}}}+{{{\tilde{\alpha}}}_{2}}\theta_{l}^{\text{comp}}\mathbf{H}_{k}^{H}{{\mathbf{z}}_{l}}\mathbf{z}_{l}^{H}{{\mathbf{H}}_{k}}$
(29) $\displaystyle+$
$\displaystyle\\!\\!\\!{{\tilde{\alpha}}}_{3}{{K}^{2}}\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\theta_{k,j}^{\text{comm}}{{\omega}_{k,j}}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}}},$

and

$\displaystyle\mathbf{T}_{k,j}^{{c}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{{{\tilde{\alpha}}}_{1}}\sum\limits_{i=1}^{I}{\theta_{i}^{\text{sens}}\mathbf{H}_{k}^{H}{{\mathbf{v}}_{i}}\mathbf{v}_{i}^{H}\mathbf{H}}+\frac{{{{\tilde{\alpha}}}_{2}}}{{{K}^{2}}}\sum\limits_{l=1}^{L}{\theta_{l}^{\text{comp}}\mathbf{H}_{k}^{H}{{\mathbf{z}}_{l}}\mathbf{z}_{l}^{H}{{\mathbf{H}}_{k}}}$
(30) $\displaystyle+$
$\displaystyle\\!\\!\\!{{{\tilde{\alpha}}}_{3}}\theta_{k,j}^{\text{comm}}{{\omega}_{k,j}}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}},$

where $\nabla_{\mathbf{x}}$ and $\nabla_{\mathbf{x}}^{2}$ denote the gradient
and the Hessian of the variable for the objective function (25), respectively.
Moreover,
${{{\tilde{\alpha}}}_{1}}=\frac{{{\alpha}_{1}}}{\left|\mathcal{F}_{1}^{*}\right|}$,
${{{\tilde{\alpha}}}_{1}}=\frac{{{\alpha}_{2}}}{\left|\mathcal{F}_{2}^{*}\right|}$,
and
${{{\tilde{\alpha}}}_{3}}=\frac{{{\alpha}_{3}}}{\left|\mathcal{F}_{3}^{*}\right|}$.
With the gradients and Hessian, the transmit beams can be computed by the
following updating rules:

$\displaystyle\mathbf{a}_{k,i}^{(t+1)}$ $\displaystyle=$
$\displaystyle\mathbf{a}_{k,i}^{(t)}-\eta_{k,i}^{a}{{\left(\nabla_{{{\mathbf{a}}_{k,i}}}^{2}\right)}^{-1}}{{\nabla}_{{{\mathbf{a}}_{k,i}}}}$
(31) $\displaystyle\mathbf{b}_{k,l}^{(t+1)}$ $\displaystyle=$
$\displaystyle\mathbf{b}_{k,l}^{(t)}-\eta_{k,l}^{b}{{\left(\nabla_{{{\mathbf{b}}_{k,l}}}^{2}\right)}^{-1}}{{\nabla}_{{{\mathbf{b}}_{k,l}}}}$
(32) $\displaystyle\mathbf{c}_{k,j}^{(t+1)}$ $\displaystyle=$
$\displaystyle\mathbf{c}_{k,l}^{(t)}-\eta_{k,l}^{c}{{\left(\nabla_{{{\mathbf{c}}_{k,j}}}^{2}\right)}^{-1}}{{\nabla}_{{{\mathbf{c}}_{k,j}}}}$
(33)

where $\eta_{k,i}^{a},\eta_{k,i}^{b}$ and $\eta_{k,i}^{c}$ are the step sizes
of the transmit beams $\mathbf{a}_{k,i},\mathbf{b}_{k,l}$ and
$\mathbf{c}_{k,j}$, respectively. In summary, the joint design of ISCC for 6G
wireless networks based on the WOPM is concluded as Algorithm 1.

Algorithm 1 : WOPM-based Joint Design of ISCC for 6G wireless networks

Input:
$N,M,K,O,I,L,J,\sigma_{n}^{2},{{\theta}_{k,i}^{\text{sens}}},{{\theta}_{k,l}^{\text{comp}}},{{\theta}_{k,j}^{\text{comm}}},P_{\max,k},\alpha_{p}$
Output:
${{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}},{{\mathbf{v}}_{i}},\mathbf{u}_{k,j},{{\mathbf{z}}_{l}}$

1: Initialize Iteration index $t=1$,
$\mathbf{a}_{k,i}^{(0)}=\mathbf{b}_{k,l}^{(0)}=\mathbf{c}_{k,j}^{(0)}=[\sqrt{\frac{P_{\max,k}}{3M}},0,\ldots,0]^{T},\forall
k,i,l,j$;

2: repeat

3: Compute $\mathbf{v}_{i}^{(t)}$ according to (22) with
$\mathbf{a}_{k,i}^{(t-1)},\mathbf{b}_{k,l}^{(t-1)}$ and
$\mathbf{c}_{k,j}^{(t-1)}$;

4: Compute $\mathbf{z}_{l}^{(t)}$ according to (23) with
$\mathbf{a}_{k,i}^{(t-1)},\mathbf{b}_{k,l}^{(t-1)}$ and
$\mathbf{c}_{k,j}^{(t-1)}$;

5: Compute $\mathbf{u}_{k,j}^{(t)}$ according to (24) with
$\mathbf{a}_{k,i}^{(t-1)},\mathbf{b}_{k,l}^{(t-1)}$ and
$\mathbf{c}_{k,j}^{(t-1)}$;

6: Compute $\omega_{k,j}^{(t)}$ according to (18) with $\mathbf{v}_{i}^{(t)}$,
$\mathbf{z}_{l}^{(t)}$, $\mathbf{u}_{k,j}^{(t)}$,
$\mathbf{a}_{k,i}^{(t-1)},\mathbf{b}_{k,l}^{(t-1)}$ and
$\mathbf{c}_{k,j}^{(t-1)}$ ;

7: Initialize Iteration index $m=1$, barrier parameter $\varepsilon=1$,
amplification factor $\nu$=10;

8: repeat

9: Select feasible step sizes $\eta_{k,l}^{a(m)},\eta_{k,l}^{b(m)}$ and
$\eta_{k,l}^{c(m)}$;

10: Compute $\mathbf{a}_{k,i}^{(m)}$ according to (31) with
$\eta_{k,l}^{a(m)}$,$\mathbf{a}_{k,i}^{(m-1)}$,$\mathbf{v}_{k,i}^{(t)},\mathbf{z}_{l}^{(t)},\mathbf{u}_{k,j}^{(t)}$
and $\omega_{k,j}^{(t)}$;

11: Compute $\mathbf{b}_{k,l}^{(m)}$ according to (32) with
$\eta_{k,l}^{b(m)}$,$\mathbf{b}_{k,l}^{(m-1)}$,$\mathbf{v}_{k,i}^{(t)},\mathbf{z}_{l}^{(t)},\mathbf{u}_{k,j}^{(t)}$
and $\omega_{k,j}^{(t)}$;

12: Compute $\mathbf{c}_{k,j}^{(m)}$ according to (33) with
$\eta_{k,l}^{c(m)}$,$\mathbf{c}_{k,j}^{(m-1)}$,$\mathbf{v}_{k,i}^{(t)},\mathbf{z}_{l}^{(t)},\mathbf{u}_{k,j}^{(t)}$
and $\omega_{k,j}^{(t)}$;

13: if Meeting the centrality condition then

14: Update $\varepsilon=\varepsilon*\nu$;

15: end if

16: Update $m=m+1$;

17: until The duality gap converges

18: Update
$\mathbf{a}_{k,i}^{(t)}=\mathbf{a}_{k,i}^{(m)},\mathbf{b}_{k,l}^{(t)}=\mathbf{b}_{k,l}^{(m)}$
and $\mathbf{c}_{k,j}^{(t)}=\mathbf{c}_{k,j}^{(m)}$;

19: Update $t=t+1$;

20: until The objective value converges

###  III-B Total Transmit Power Minimization Design

Now, let us focus on another MOOP with the goal of minimizing total transmit
power consumption but ensuring the QoS requirements on sensing, computing, and
communication, respectively. The design is formulated as follows:

_M-2: Total Transmit Power Minimization:_

$\displaystyle\underset{\begin{smallmatrix}{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}\\\
,{{\mathbf{v}}_{i}},{{\mathbf{u}}_{k,j}},{{\mathbf{z}}_{l}}\end{smallmatrix}}{\mathop{\text{min
}}}\,\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum\limits_{k=1}^{K}{\left(\sum\limits_{i=1}^{I}{{{\left\|{{\mathbf{a}}_{k,i}}\right\|}^{2}}}+\sum\limits_{l=1}^{L}{{{\left\|{{\mathbf{b}}_{k,l}}\right\|}^{2}}}+\sum\limits_{j=1}^{J}{{{\left\|{{\mathbf{c}}_{k,j}}\right\|}^{2}}}\right)}$
s.t. $\displaystyle\\!\\!\\!\\!\text{C2:\ }\text{
MSE}_{i}^{\text{sens}}\leq{{\delta}_{i}},\forall i\in\Omega_{I},$ (34)
$\displaystyle\\!\\!\\!\\!\text{C3: \
}\text{MSE}_{l}^{\text{comp}}\leq{{\chi}_{l}},\forall l\in\Omega_{L},$
$\displaystyle\\!\\!\\!\\!\text{C4: \
}{{\Gamma}_{k,j}}\geq{{\gamma}_{k,j}},\forall k\in\Omega_{K},j\in\Omega_{J},$

where ${{\delta}_{i}}>0$ is the given tolerable maximum sensing distortion for
$i$-th target object, ${{\chi}_{l}}>0$ is the given maximum tolerable
computation error for the $l$-th model parameter, and ${{\gamma}_{k,j}}>0$
denotes the required minimum SINR for the communication signal
$s_{k,j}^{\text{comm}}$. The objective function of _M-2_ is minimizing the
total transmit power of all sensors, and the constraints C2, C3, and C4 are
the QoS requirements for sensing, computing, and communication, respectively.
However, _M-2_ is not convex because of the variables coupling in constraints
C2, C3 and C4, i.e., transmit beams
$\\{{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}\\}$ and
receive beams $\\{{{\mathbf{v}}_{i}},\mathbf{u}_{k,j},{{\mathbf{z}}_{l}}\\}$,
which makes it hard to find the optimal solution in the polynomial time. Thus,
it is also desired to divide _M-2_ into two subproblems and then solve it by
the AO method. Considering the balance between the system performance and the
computational complexity, it is feasible to apply the MMSE receivers in the
subproblem for optimizing receive beams by fixing transmit beams, which are
given in (22), (23), and (24). For another subproblem for optimizing transmit
beams by fixing receive beams, we need to handle the non-convex constraint C4,
which is equivalent to

$\displaystyle\frac{1}{{{\gamma}_{k,j}}}{{\left|\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}\right|}^{2}}\geq$
$\displaystyle{\sum\limits_{i=1,i\neq k}^{K}{\sum\limits_{n=1,n\neq
j}^{J}{{{\left|\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}{{\mathbf{c}}_{i,n}}\right|}^{2}}}}+{{X}_{k,j}}+\sigma_{n}^{2}{{\left\|\mathbf{u}_{k,j}\right\|}^{2}}}.$
(35)

To address the non-convexity, we introduce an auxiliary variable
${{\mathbf{C}}_{k,j}}={{\mathbf{c}}_{k,j}}\mathbf{c}_{k,j}^{H}$ and substitute
it into (III-B), which is given by

$\displaystyle\frac{1}{{{\gamma}_{k,j}}}\text{tr}\left(\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}{{\mathbf{C}}_{k,j}}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}\right)\geq$
(36) $\displaystyle\sum\limits_{i=1,i\neq k}^{K}{\sum\limits_{n=1,n\neq
j}^{J}\text{tr}\left({\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}{{\mathbf{C}}_{i,n}}\mathbf{H}_{i}^{H}{{\mathbf{u}}_{k,j}}}\right)}+{{X}_{k,j}}+\sigma_{n}^{2}{{\left\|\mathbf{u}_{k,j}\right\|}^{2}}.$

Based on this, the subproblem for the transmit beamforming optimization can be
converted to a standard semi-definite programming (SDP) problem as follows

_M-2’: Equivalent Subproblem of Transmit Beamforming Optimization:_

$\displaystyle\underset{{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{C}}_{k,j}}}{\mathop{\min}}\,\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum\limits_{k=1}^{K}{\left(\sum\limits_{i=1}^{I}{{{\left\|{{\mathbf{a}}_{k,i}}\right\|}^{2}}}+\sum\limits_{l=1}^{L}{{{\left\|{{\mathbf{b}}_{k,l}}\right\|}^{2}}}+\sum\limits_{j=1}^{J}{\text{tr}\left({{\mathbf{C}}_{k,j}}\right)}\right)}$
s.t. $\displaystyle\\!\\!\\!\\!\text{C5: \
}\overline{\text{MSE}}_{i}^{\text{sens}}\leq{{\delta}_{i}},\forall
i\in\Omega_{I},$ (37) $\displaystyle\\!\\!\\!\\!\text{C6: \
}\overline{\text{MSE}}_{l}^{\text{comp}}\leq{{\chi}_{l}},\forall
l\in\Omega_{L},$ $\displaystyle\\!\\!\\!\\!\text{C7: \ }(\ref{SINR1}),$
$\displaystyle\\!\\!\\!\\!\text{C8: \
}{{\mathbf{C}}_{k,j}}\succeq\mathbf{0},\forall k\in\Omega_{K},j\in\Omega_{J},$
$\displaystyle\\!\\!\\!\\!\text{C9: \
}{\text{Rank}\left({{\mathbf{C}}_{k,j}}\right)=1},\forall
k\in\Omega_{K},j\in\Omega_{J},$

where
$\overline{\text{MSE}}_{i}^{\text{sens}}=\text{MSE}_{i}^{\text{sens}}-\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{{{\left|\mathbf{v}_{i}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}\right|}^{2}}+\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\text{tr}\left(\mathbf{v}_{i}^{H}{{\mathbf{H}}_{k}}{{\mathbf{C}}_{k,j}}\mathbf{H}_{k}^{H}{{\mathbf{v}}_{i}}\right)}}}}$
and
$\overline{\text{MSE}}_{l}^{\text{comp}}=\text{MSE}_{l}^{\text{comp}}-\frac{1}{K^{2}}\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{{{\left|\mathbf{z}_{l}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}\right|}^{2}}+\frac{1}{K^{2}}\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{\text{tr}\left(\mathbf{z}_{l}^{H}{{\mathbf{H}}_{k}}{{\mathbf{C}}_{k,j}}\mathbf{H}_{k}^{H}{{\mathbf{z}}_{l}}\right)}}}}$.
Note that constraint C9 for the rank-one limitation of ${{\mathbf{C}}_{k,j}}$
is non-convex, which makes _M-2’_ still non-convexity. To this end, we adopt
the semi-definite relaxation (SDR) technique, namely discarding constraint C9.
As a result, _M-2’_ is restated as

_M-2”: Transformed Subproblem of Transmit Beamforming Optimization:_

$\displaystyle\underset{{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{C}}_{k,j}}}{\mathop{\min}}\,\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum\limits_{k=1}^{K}{\left(\sum\limits_{i=1}^{I}{{{\left\|{{\mathbf{a}}_{k,i}}\right\|}^{2}}}+\sum\limits_{l=1}^{L}{{{\left\|{{\mathbf{b}}_{k,l}}\right\|}^{2}}}+\sum\limits_{j=1}^{J}{\text{tr}\left({{\mathbf{C}}_{k,j}}\right)}\right)}$
s.t. $\displaystyle\\!\\!\\!\\!\text{C5}-\text{C8}.$ (38)

It is found that _M-2”_ is a joint convex problem in terms of transmit beams
$\\{{\mathbf{a}_{k,i}},{\mathbf{b}_{k,l}},{\mathbf{C}_{k,j}}\\}$ due to the
convex constraints C5-C8 and convex objective function, and thus it can be
directly solved via CVX [41]. It is worth mentioning that for the rank of the
optimal solution to _M-2”_ , we have following theorem:

_Theorem 2:_ The rank of the optimal solution ${{\mathbf{C}}_{k,j}^{*}}$ to
_M-2”_ always satisfies
$\text{Rank}\left({{\mathbf{C}}_{k,j}^{*}}\right)=1,\forall k,j$.

###### Proof:

Please refer to Appendix B. ∎

According to Theorem 2, we can perform eigenvalue decomposition (EVD) to get
the unique optimal solution ${{\mathbf{c}}_{k,j}^{*}}$ of _M-2_. The EVD
operation can be presented as

${{\mathbf{c}}_{k,j}^{*}}=\sqrt{{{\lambda}_{k,j}^{\max}}({{\mathbf{C}}_{k,j}^{*}})}{{\bm{\xi}}_{k,j}^{\max}},$
(39)

where ${{\lambda}_{k,j}^{\max}}({{\mathbf{C}}_{k,j}^{*}})$ denotes the maximum
eigenvalue of $\mathbf{C}_{k,j}^{*}$ and ${{\bm{\xi}}_{k,j}^{\max}}$
represents the corresponding unit eigenvector. Hence, the joint design of ISCC
for 6G wireless networks based on TTPM is summarized as Algorithm 2.

Algorithm 2 : TTPM-based Joint Design of ISCC for 6G wireless networks

Input: $K,N,M,I,O,L,J,\sigma_{n}^{2},P_{0},\delta_{i},\chi_{l},\gamma_{k,j}$
Output: $\mathbf{a}_{k,i}$, $\mathbf{b}_{k,l}$,
$\mathbf{c}_{k,j}$,$\mathbf{v}_{i}$,$\mathbf{z}_{l}$,$\mathbf{u}_{k,j}$

1: Initialize iteration index $t=1$,
$\mathbf{a}_{k,i}^{(0)}=\mathbf{b}_{k,l}^{(0)}=\mathbf{c}_{k,j}^{(0)}=[\sqrt{\frac{P_{0}}{3M}},0,\ldots,0]^{T},\forall
k,i,l,j$;

2: repeat

3: Compute $\mathbf{v}_{i}^{(t)}$ according to (22) with
$\mathbf{a}_{k,i}^{(t-1)},\mathbf{b}_{k,l}^{(t-1)}$ and
$\mathbf{c}_{k,j}^{(t-1)}$;

4: Compute $\mathbf{z}_{l}^{(t)}$ according to (23) with
$\mathbf{a}_{k,i}^{(t-1)},\mathbf{b}_{k,l}^{(t-1)}$ and
$\mathbf{c}_{k,j}^{(t-1)}$;

5: Compute $\mathbf{u}_{k,j}^{(t)}$ according to (24) with
$\mathbf{a}_{k,i}^{(t-1)},\mathbf{b}_{k,l}^{(t-1)}$ and
$\mathbf{c}_{k,j}^{(t-1)}$;

6: Obtain
$\\{\mathbf{a}^{(t)}_{k,i},\mathbf{b}^{(t)}_{k,l},\mathbf{C}_{k,j}^{(t)}\\}$
by solving _M-2”_ with fixed
$\\{\mathbf{v}_{i}^{(t)},\mathbf{z}_{l}^{(t)},\mathbf{u}_{k,j}^{(t)}\\}$;

7: Obtain $\mathbf{c}_{k,j}^{(t)}$ by EVD on $\mathbf{C}_{k,j}^{(t)}$
according to (39);

8: $t=t+1$;

9: until convergence

###  III-C Convergence and Complexity Analysis of Proposed Algorithms

Herein, let us discuss the property of convergence and complexity for the
proposed two algorithms.

_Convergence Analysis:_ For Algorithm 1, since _M-1_ is convex for each
optimization variable while fixing the others, the optimal solutions of every
subproblem can be found. In fact, the solutions of each iteration are feasible
in the next iteration. As a result, the objective value of _M-1_ is
monotonically non-increasing during the iterations. Furthermore, due to the
transmit power constraint at the sensors, the objective value of _M-1_ is
lower bounded. Hence, based on the monotone bounded convergence (MBC) theorem
[40], Algorithm 1 would converge for a suitable number of iteration. Next, we
analyze the convergence property of Algorithm 2. First, the adopted MMSE
receivers $\\{\mathbf{v}_{i},\mathbf{z}_{l},\mathbf{u}_{k,j}\\}$ is effective
to ensure the QoS requirements, which can help enhance the system performance.
Moreover, since _M-2”_ is convex for transmit beams
$\\{\mathbf{a}_{k,i}\mathbf{b}_{k,l},\mathbf{c}_{k,j}\\}$ with given MMSE
receivers, it is feasible to get the optimal solution via CVX directly, which
ensures the obtained objective value in the $(t+1)$-th iteration is less than
that in the $t$-th iteration. That implies the total transmit power
consumption is monotonically non-increasing in the iterations. Moreover, due
to the QoS requirements in constraint C2, C3, and C4, the total transmit power
consumption is lower bounded. Thus, Algorithm 2 would converge according to
the MBC theorem.

_Complexity Analysis:_ Since Algorithm 1 and Algorithm 2 are both iterative,
the execution steps are the same for each iteration. Thus, we focus on
analyzing the per-iteration complexity of the proposed algorithms. By noticing
the operations of algorithms, it is observed that the dominating computational
complexities arise from the steps 7-17 for Algorithm 1, i.e., finding optimal
$\\{\mathbf{a}^{(t)}_{k,i},\mathbf{b}^{(t)}_{k,l},\mathbf{c}_{k,j}^{(t)}\\}$
of _M-1_ , and the step 6 for Algorithm 2, i.e., finding optimal
$\\{\mathbf{a}^{(t)}_{k,i},\mathbf{b}^{(t)}_{k,l},\mathbf{C}_{k,j}^{(t)}\\}$
of _M-2”_. Since _M-1_ and _M-2”_ only contain second-order cone (SOC) or
linear matrix inequalities (LMI) constraints, they could both be solved by a
standard IPM [42]. As such, the worst-case runtime by IPM can be used to
depict the computational complexities of algorithm [43]. Specifically, _M-1_
has $K$ SOC constraints of dimension $1$, and the decision variable
$n_{1}=\mathcal{O}(K^{2}M^{2})$. For _M-2”_ , it has $KJ$ LMI constraints of
dimension $1$, $KJ$ LMI constraints of dimension $M$, $I+L$ SOC constraint of
dimension $1$, and the decision variable $n_{2}=\mathcal{O}(KM^{3})$. As a
result, for the solution with a given precision $\epsilon>0$, the worst-case
complexities of Algorithm 1 and Algorithm 2 for per iteration are in the order
of $\ln(1/\epsilon)\varsigma_{1,2}$, c.f. Table II.

TABLE II: The worst-case complexities of proposed algorithms for per iteration Algorithms | Complexity is in order of $\ln\left(1/\varepsilon\right)\varsigma_{\\{1,2\\}}$
---|---
Algorithm 1 | $\varsigma_{1}=\sqrt{2K}\left({{n}_{1}}K+n_{1}^{3}\right)$
Algorithm 2 | $\varsigma_{2}=\sqrt{KJ\left(M+1\right)+2(I+L)}\cdot{{n}_{2}}\cdot\left[KJ\left({{M}^{3}}+{{n}_{2}}{{M}^{2}}+{{n}_{2}}+1\right)+I+L+n_{2}^{2}\right]$

##  IV Simulation Results

TABLE III: Simulation Parameters  Parameters | Values
---|---
Number of BS antennas | $N=64$
Sensor | $K=20,M=3$
Streams | $O=1,I=1,L=1,J=1$
Cell radius | $500\text{ m}$
RMS of reflection coefficient | $R_{i}=R_{o}=1$
Weighted coefficient | $\theta_{i}^{\text{sens}}=\theta_{l}^{\text{comp}}=\theta_{k,j}^{\text{sens}}=1$
Maximum transmit SNR at the sensors |  $\mathrm{SNR}=5$ dB
Noise powers |  $\sigma_{n}^{2}=-50$ dBm
QoS requirements |  $\gamma_{0}=0.1$ dB, $\delta_{0}=0.01,\chi_{0}=0.01$
Performance priority | $\alpha_{1}=\alpha_{2}=\alpha_{3}=1/3$

In this section, we conduct extensive simulations to verify the effectiveness
of the proposed algorithms. Without loss of generality, all sensors are
assumed being randomly distributed within the cell radius. The pass loss is
modeled as $\mathrm{PL}_{\mathrm{dB}}=128.1+37.6\log_{10}(d)$ [44], where $d$
(km) is the distance between the transmitter and receiver. Moreover, it is
assumed that all sensors have the same QoS requirements and the same maximum
transmit power budget, namely $\gamma_{k,j}=\gamma_{0}$,
$\delta_{i}=\delta_{0}$, $\chi_{l}=\chi_{0}$, and $P_{\max,k}=P_{0}$. We use
$\mathrm{SNR}=10\log_{10}(P_{0}/\sigma_{n}^{2})$ to denote the transmit
signal-to-noise ratio (SNR) (in dB) at the sensors. Unless otherwise stated,
the simulation parameters are summarized in Table III.

Figure 3: Convergence behavior of Algorithm 1.

First, we give the convergence performance of Algorithm 1 with different SNRs
at the sensors. As is shown in Fig. 3, the sensing error and the computation
error decrease while the communication rate increases monotonically in the
iterations, and they converge to a stable equilibrium point within few
iterations on average. Hence, the implementation cost of Algorithm 1 is
bearable for practical systems.

Figure 4: The system performance gains versus priorities with different SNRs
for Algorithm 1.

Then, we investigate the effect of priorities of sensing $\alpha_{1}$,
computing $\alpha_{2}$, and communication $\alpha_{3}$ of MOOP on the
performance under different transmit SNRs at the sensors. Here, we present two
different cases. As shown in Fig. 4, for case (a), we fix the weighted
coefficient of communication $\alpha_{3}=1/3$. In this case,
$\alpha_{1}+\alpha_{2}=2/3$, and then we change the value of the priority of
computing $\alpha_{2}$ to obtain different solutions. In case (b), we fix
$\alpha_{1}=\alpha_{2}$ while changing the priority of communication
$\alpha_{3}$ to get various results. It is observed that in case (a) with
increasing of $\alpha_{2}$, the three performance metrics have different
varying trends. Specifically, the computation error slowly decreases, the
sensing error gradually increases, while the communication rate first
decreases and then increases. Moreover, it is seen that in case (b) the
performance of sensing and computation gradually reduces while the performance
of communication gradually improves as $\alpha_{3}$ increases. Hence, it makes
sense to select an appropriate set of priorities to balance the performance of
sensing, computing, and communication. Moreover, it is found that there is a
big gap between the SNR at 5 dB and 0 dB for all three performance metrics,
which means increasing the SNR at the sensors can bring more gain for the
overall system performance.

Figure 5: The system performance gains versus the number of sensors with
different numbers of BS antennas for Algorithm 1.

Fig. 5 shows the influence of the number of sensors $K$ with different numbers
of BS antennas $N$ on the performance of Algorithm 1. For a given SNR at the
sensors, Algorithm 1 with more antennas at the BS can get a better overall
performance. This is because a higher spatial multiplexing gain provided by
the extra antennas are exploited to effectively enhance the performance.
Furthermore, it is seen that with the increasing of the number of sensors,
both the sensing error and the computation error diminishes while the
communication rate increases. That implies the increment of the number of
sensors is conducive to acquire a more accurate estimation on the reflection
coefficient for the target object, a more accurate model aggregation for
AirFL, and a higher weighted sum-rate for information communication, which is
quite attractive for 6G wireless networks with massive sensors.

Figure 6: The comparison of performance on MOOP for different algorithms.

In Fig. 6, we show the performance of Algorithm 1 over three baseline
beamforming algorithms. They are a fixed-MMSE (F-MMSE) algorithm with fixed
MMSE receivers only relevant to the channels, an AO-based match filtering
beamforming (MFBF) algorithm with the match filter receivers, an AO-based
zero-forcing beamforming (ZFBF) algorithm with the zero-forcing transmitters,
respectively. It is seen that the F-MMSE algorithm performs the worst among
the other algorithms due to the fixed receiver, which limits the performance.
The MFBF algorithm outperforms the ZFBF algorithm in the performance of
sensing, but performs worse in the aspects of computing and communication.
Moreover, it is found that Algorithm 1 performs the best for all three
performance since it obtains transmit beams and receive beams both in the
optimal way, which demonstrates the superiority of Algorithm 1.

Figure 7: Convergence behavior of the proposed Algorithm 2.

Fig. 7 checks the convergence performance of Algorithm 2 under different
required minimum SINRs of communication. It is observed that the total
transmit power is progressively lowering in the iterations, and then converges
within no more than 10 iterations on average under different required minimum
SINRs of communication, which verifies the feasibility of Algorithm 2.

Figure 8: Total transmit power versus tolerable maximum computation error with
different maximum tolerable sensing error and required minimum SINR of
communication for Algorithm 2.

Fig. 8 investigates the impacts of the maximum tolerable sensing error
$\delta_{0}$, the maximum tolerable computation error $\chi_{0}$, and the
required minimum SINR of communication $\gamma_{0}$ on the total transmit
power consumption. It can be seen that the total transmit power decreases as
the tolerable maximum computation error increases, since a larger $\chi_{0}$
represents a more relax constraint on the computing accuracy resulting in less
power consumption. Moreover, in the whole region of $\chi_{0}$, the case with
$\delta_{0}=0.001$ and $\gamma_{0}=0.1$ dB requires the smallest total
transmit power consumption, while the case with $\delta_{0}=0.01$ and
$\gamma_{0}=0.1$ dB requires the biggest one. That means a larger $\delta_{0}$
is less restriction on sensing accuracy and a larger $\gamma_{0}$ signifies
higher requirement on communication. Hence, it makes sense to loosen
performance requirements appropriately for reducing the total power
consumption at the sensors.

Figure 9: Total transmit power versus the number of clutters with different
number of BS antennas for Algorithm 2.

Next, we study the influences of the numbers of clutters $O$ and BS antennas
$N$ on the performance of Algorithm 2 in Fig. 9. It is found that the total
transmit power significantly increases as the number of clutters increases,
since clutters as interference component is harmful to the three performance
metrics resulting in more power consumption. Moreover, Algorithm 2 with more
BS antennas performs better thanks to exploiting more array gains. Besides,
the performance gain gap between $N=80$ and $N=96$ is less than that between
$N=64$ and $N=80$, which implies the performance enhancement provided by
adding the number of BS antennas is not infinite. Notice that in practical
systems, the BS equipped with more antennas could effectually decrease the
total transmit power consumption, but it also leads to a higher cost on radio
frequency chains. Hence, it makes sense to select a proper number of BS
antennas for balancing the performance and the cost.

Figure 10: Performance comparison of different algorithm for total transmit
power minimization.

Finally, we compare the performance of different algorithms in the sense of
minimizing the total transmit power in Fig. 10. There are four algorithms,
namely a F-MMSE algorithm, an AO-based MFBF algorithms, an AO-based ZFBF
algorithm, and Algorithm 2. For all algorithms, the total transmit power
increases as the required minimum SINR on communication $\gamma_{0}$
increases. This is because for given sensing and computing requirements, a
larger $\gamma_{0}$ means a more stricter requirement on communication leading
to more transmit power consumption. It is evident that the F-MMSE algorithm
presents the most deficient performance compared to the three AO algorithms.
In addition, although the ZFBF algorithm performs better than the MFBF
algorithm in the low-value region of $\gamma_{0}$, however, it performs worse
in the high-value region of $\gamma_{0}$. More importantly, the proposed
Algorithm 2 achieves the best performance in the whole region of $\gamma_{0}$,
especially for high requirement of communication, which verifies the
advantages of Algorithm 2 for 6G wireless networks integrating sensing,
computing, and communication.

##  V Conclusion

This paper provided a comprehensive architecture for enabling 6G wireless
networks to integrate sensing, computing, and communication. Two typical MOOP-
based algorithms were designed from the perspectives of maximizing the
weighted overall performance and minimizing the total transmit power
consumption. Extensive simulations confirmed the excellent performance of the
proposed algorithms. Moreover, according to the preference of applications for
6G wireless networks, it was feasible to acquire desired performance of
sensing, computing, and communication by varying the system priorities.
Besides, it was found that the overall performance can be improved by
increasing the suitable numbers of sensors and BS antennas.

6G is still in the initial conceptual stage, and many challenges faced by ISCC
has not been well addressed. In this paper, we only make a general integration
of the three isolated functions of sensing, computing and communication to
explore their relationship. To support future emerging various applications
with ultra-high performance requirements, it is necessary to deeply integrate
wireless sensing functions such as positioning, detection, and imaging with
wireless transmission functions, and at the same time exploit widely
distributed computing power for auxiliary processing to achieve cross-
integration of sensing, computing and communication.

##  Appendix A Proof of Theorem 1

For the received communication signal $y_{k,j}^{\text{comm}}$ at the BS, the
MSE related to the $j$-th communication signal from the $k$-th sensor is given
by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{MSE}_{k,j}^{\text{comm}}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\mathbb{E}\left\\{\left(y_{k,j}^{\text{comm}}-s_{k,j}^{\text{comm}}\right){{\left(y_{k,j}^{\text{comm}}-s_{k,j}^{\text{comm}}\right)}^{H}}\right\\}$
(40) $\displaystyle+$
$\displaystyle\\!\\!\\!\sum\limits_{i=1}^{K}{\sum\limits_{m=1}^{J}{\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}{{\mathbf{c}}_{i,m}}}}\mathbf{c}_{i,m}^{H}\mathbf{H}_{i}^{H}{{\mathbf{u}}_{k,j}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\sum\limits_{i=1}^{K}{\sum\limits_{l=1}^{L}{\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}{{\mathbf{b}}_{i,l}}\mathbf{b}_{i,l}^{H}\mathbf{H}_{i}^{H}{{\mathbf{u}}_{k,j}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\sum\limits_{i=1}^{K}{\sum\limits_{n=1}^{I}{R_{n}^{2}\mathbf{u}_{k,j}^{H}{{\mathbf{G}}_{i,n}}{{\mathbf{a}}_{i,n}}\mathbf{a}_{i,n}^{H}\mathbf{G}_{i,n}^{H}{{\mathbf{u}}_{k,j}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\sum\limits_{i=1}^{K}{\sum\limits_{o=1}^{O}{\sum\limits_{m=1}^{I}{R_{o}^{2}\mathbf{u}_{k,j}^{H}{{\mathbf{F}}_{i,o}}{{\mathbf{a}}_{i,m}}\mathbf{a}_{i,m}^{H}\mathbf{F}_{i,o}^{H}{{\mathbf{u}}_{k,j}}}}}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\sigma_{n}^{2}\left\|{{\mathbf{u}}_{k,j}}\right\|-\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}-\mathbf{c}_{k,j}^{H}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}+1.$

Extracting the common factor from the above equation, we define
$\bm{\Xi}=\sum\limits_{i=1}^{K}{\sum\limits_{m=1}^{J}{{{\mathbf{H}}_{i}}{{\mathbf{c}}_{i,m}}}}\mathbf{c}_{i,m}^{H}\mathbf{H}_{i}^{H}+\sum\limits_{i=1}^{K}{\sum\limits_{l=1}^{L}{{{\mathbf{H}}_{i}}{{\mathbf{b}}_{i,l}}\mathbf{b}_{i,l}^{H}\mathbf{H}_{i}^{H}}}+\sum\limits_{i=1}^{K}{\sum\limits_{n=1}^{I}{R_{\text{n}}^{2}{{\mathbf{G}}_{i,n}}{{\mathbf{a}}_{i,n}}\mathbf{a}_{i,n}^{H}\mathbf{G}_{i,n}^{H}}+}\sum\limits_{i=1}^{K}{\sum\limits_{o=1}^{O}{\sum\limits_{m=1}^{I}{R_{o}^{2}{{\mathbf{F}}_{i,o}}{{\mathbf{a}}_{i,m}}\mathbf{a}_{i,m}^{H}\mathbf{F}_{i,o}^{H}}}}+\sigma_{n}^{2}\mathbf{I}$,
then (40) can be rewritten as

$\displaystyle\text{MSE}_{k,j}^{\text{comm}}$ $\displaystyle=$
$\displaystyle\mathbf{u}_{k,j}^{H}\Xi{{\mathbf{u}}_{k,j}}-\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}-\mathbf{c}_{k,j}^{H}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}+1$
(41) $\displaystyle=$
$\displaystyle\\!\\!\\!\left(\mathbf{u}_{k,j}^{H}-\mathbf{c}_{k,j}^{H}\mathbf{H}_{k}^{H}{{\Xi}^{-1}}\right)\Xi{{\left(\mathbf{u}_{k,j}^{H}-\mathbf{c}_{k,j}^{H}\mathbf{H}_{k}^{H}{{\Xi}^{-1}}\right)}^{H}}$
$\displaystyle-$
$\displaystyle\\!\\!\\!\mathbf{c}_{k,j}^{H}\mathbf{H}_{k}^{H}{{\Xi}^{-H}}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}+1.$

It can be concluded that when
${{\mathbf{u}}_{k,j}}=\Xi^{-1}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}$, the
$\text{MSE}_{k,j}^{\text{comm}}$ would be minimized. Consequently, the MMSE
associated with the communication signal $s_{k,j}^{\text{comm}}$ is as below

$\displaystyle{{e}_{k,j}^{\text{comm}}}$ $\displaystyle=$ $\displaystyle
1-\mathbf{c}_{k,j}^{H}\mathbf{H}_{k}^{H}{{\Xi}^{-H}}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}$
(42) $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{{{\Xi}^{H}}-\mathbf{c}_{k,j}^{H}\mathbf{H}_{k}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}}{{{\Xi}^{H}}}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{\mathbf{u}_{k,j}^{H}{{\Xi}^{H}}{{\mathbf{u}}_{k,j}}-\mathbf{u}_{k,j}^{H}\mathbf{c}_{k,j}^{H}\mathbf{H}_{k}^{H}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}{{\mathbf{u}}_{k,j}}}{\mathbf{u}_{k,j}^{H}{{\Xi}^{H}}{{\mathbf{u}}_{k,j}}}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\frac{1}{1+{{\Gamma}_{k,j}}}.$

It is seen that the MMSE of communication signal is the inverse of one plus
SINR and thus helps to transform the objective function of _S-3’_. The proof
is completed.

##  Appendix B Proof of Theorem 2

To start with, let us present two lemmas as follows:

_Lemma 1_ : If $\mathbf{Q}\mathbf{P}=\mathbf{0}$, based on Sylvester’s rank
inequality [45], we have $\text{Rank}(\mathbf{Q})+\text{Rank}(\mathbf{P})\leq
n$, where $\mathbf{Q}\in{\mathbb{C}^{t\times n}}$ and
$\mathbf{P}\in{\mathbb{C}^{n\times s}},\forall t,n,s$.

_Lemma 2_ : For the two matrices of the same size $\mathbf{Q}$ and
$\mathbf{P}$, it always holds true that
$\text{Rank}(\mathbf{Q}+\mathbf{P})\leq\text{Rank}(\mathbf{Q})+\text{Rank}(\mathbf{P})$.

###### Proof:

$\displaystyle\text{Rank}\left(\mathbf{Q}+\mathbf{P}\right)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\text{Rank}\left[\begin{matrix}\mathbf{Q}+\mathbf{P}\\\
\mathbf{0}\\\
\end{matrix}\right]\leq\text{Rank}\left[\begin{matrix}\mathbf{Q}+\mathbf{P}\\\
\mathbf{P}\\\ \end{matrix}\right]=\text{Rank}\left[\begin{matrix}\mathbf{Q}\\\
\mathbf{P}\\\ \end{matrix}\right]$ (43) $\displaystyle\leq$
$\displaystyle\\!\\!\\!\text{Rank}\left(\mathbf{Q}\right)+\text{Rank}\left(\mathbf{P}\right).$

∎

Then, we derive the Lagrangian function of _M-2”_ associated with
$\mathbf{C}_{k,j}$, which is given by

$\displaystyle\mathcal{L}\left({{\mathbf{C}}_{k,j}}\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{K}{\left(\sum\limits_{i=1}^{I}{{{\left\|{{\mathbf{a}}_{k,i}}\right\|}^{2}}}+\sum\limits_{l=1}^{L}{{{\left\|{{\mathbf{b}}_{k,l}}\right\|}^{2}}}+\sum\limits_{j=1}^{J}{\text{tr}\left({{\mathbf{C}}_{k,j}}\right)}\right)}$
(44) $\displaystyle+$
$\displaystyle\sum\limits_{i=1}^{I}{{{\lambda}_{i}}}\left[\overline{\text{MSE}}_{i}^{\text{sens}}-{{\delta}_{i}}\right]$
$\displaystyle+$
$\displaystyle\sum\limits_{l=1}^{L}{{{\beta}_{l}}}\left[\overline{\text{MSE}}_{l}^{\text{comp}}-{{\chi}_{l}}\right]+\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{{{\mu}_{k,j}}{{T}_{k,j}}}}$
$\displaystyle-$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{J}{{{\mathbf{\Theta}}_{k,j}}{{\mathbf{C}}_{k,j}}}},$

where ${{T}_{k,j}}=\sum\limits_{i=1,i\neq k}^{K}{\sum\limits_{n=1,n\neq
j}^{J}{\text{tr}\left(\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}{{\mathbf{C}}_{i,n}}\mathbf{H}_{i}^{H}{{\mathbf{u}}_{k,j}}\right)}}+{{X}_{k,j}}+\sigma_{n}^{2}{{\left\|{{\mathbf{u}}_{k,j}}\right\|}^{2}}-\frac{1}{{{\gamma}_{k,j}}}\text{tr}\left(\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}{{\mathbf{C}}_{k,j}}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}\right)$.
$\lambda_{i},\beta_{l},\mu_{k,j}$, and ${{\mathbf{\Theta}_{k,j}}}$ are
Lagrange multipliers of constraint C5, C6, C7 and C8, respectively. To explore
the optimal solution $\mathbf{C}_{k,j}^{*}$ under the Slater’s condition, we
make use of the KKT conditions as follows:

$\displaystyle\sum\limits_{i=1,i\neq k}^{K}{\sum\limits_{n=1,n\neq
j}^{J}{\text{tr}\left(\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{i}}\mathbf{C}_{i,n}^{*}\mathbf{H}_{i}^{H}{{\mathbf{u}}_{k,j}}\right)}}+{{X}_{k,j}}+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sigma_{n}^{2}{{\left\|{{\mathbf{u}}_{k,j}}\right\|}^{2}}-\frac{1}{{{\gamma}_{k,j}}}\text{tr}\left(\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}\mathbf{C}_{k,j}^{*}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}\right)=0,$
(45a) $\mathbf{\Theta}_{k,j}^{*}\mathbf{C}_{k,j}^{*}=\mathbf{0},$ (45b)
$\displaystyle{{\nabla}_{\mathbf{C}_{k,j}^{*}}}\mathcal{L}$ $\displaystyle=$
$\displaystyle{{\mathbf{I}}_{M}}+\sum\limits_{i=1}^{I}{\lambda_{i}^{*}\mathbf{H}_{k}^{H}{{\mathbf{v}}_{i}}\mathbf{v}_{i}^{H}{{\mathbf{H}}_{k}}}+\sum\limits_{l=1}^{L}{\beta_{l}^{*}}\mathbf{H}_{k}^{H}{{\mathbf{z}}_{l}}\mathbf{z}_{l}^{H}{{\mathbf{H}}_{k}}$
(45c) $\displaystyle-$
$\displaystyle\frac{\mu_{k,j}^{*}}{{{\gamma}_{k,j}}}\mathbf{H}_{k}^{H}{{\mathbf{u}}_{k,j}}\mathbf{u}_{k,j}^{H}{{\mathbf{H}}_{k}}-\bm{\Theta}_{k,j}^{*}=\mathbf{0},$
$\lambda_{i}^{*}\geq 0,\beta_{l}^{*}\geq 0,\mu_{k,j}^{*}\geq
0,\bm{\Theta}_{k,j}^{*}\succeq\mathbf{0}.$ (45d)

Since $X_{k,j}+\sigma_{n}^{2}{{\left\|\mathbf{u}_{k,j}^{H}\right\|}^{2}}>0$ in
(45), we have ${{\mathbf{C}}_{k,j}^{*}}\neq\mathbf{0}$, which means

$\text{Rank}(\mathbf{C}_{k,j}^{*})\geq 1.$ (46)

By exploiting the relationship between $\bm{\Theta}_{k,j}^{*}$ and
$\mathbf{C}_{k,j}^{*}$ in (45b), based on Lemma 1, we obtain that

$\text{Rank}(\bm{\Theta}_{k,j}^{*})+\text{Rank}(\mathbf{C}_{k,j}^{*})\leq M.$
(47)

Next, substituting (46) into (47) yields

$\text{Rank}(\bm{\Theta}_{k,j}^{*})\leq M-1.$ (48)

Then, based on Lemma 2, it is observed from (45c) that

$\text{Rank}(\bm{\Upsilon}_{k,j})+\text{Rank}(\bm{\Theta}_{k,j}^{*})\geq\text{Rank}(\mathbf{I}_{M}),$
(49)

where
$\bm{\Upsilon}_{k,j}=\mathbf{H}_{k}^{H}\left(\frac{\mu_{k,j}^{*}}{{{\gamma}_{k,j}}}{{\mathbf{u}}_{k,j}}\mathbf{u}_{k,j}^{H}-\sum\limits_{i=1}^{I}{\lambda_{i}^{*}{{\mathbf{v}}_{i}}\mathbf{v}_{i}^{H}}-\sum\limits_{l=1}^{L}{\beta_{l}^{*}}{{\mathbf{z}}_{l}}\mathbf{z}_{l}^{H}\right){{\mathbf{H}}_{k}}$.
Since $\bm{\Upsilon}_{k,j}\neq\mathbf{0}$ and $\text{Rank}(\mathbf{I}_{M})=M$,
we obtain

$\text{Rank}(\bm{\Theta}_{k,j}^{*})\geq M-1.$ (50)

Based on the relationship between (48) and (50), we can get
$\text{Rank}(\bm{\Theta}_{k,j}^{*})=M-1$. Substituting it into (47) and
combining (46), it is concluded that

$\text{Rank}(\mathbf{C}_{k,j}^{*})=1,$ (51)

which indicates the adopted SDR is tight. The proof is completed.

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