# Beamforming Design for Semantic-Bit Coexisting Communication System

Maojun Zhang, Guangxu Zhu, Richeng Jin, Xiaoming Chen, Qingjiang Shi, Caijun
Zhong, Kaibin Huang M. Zhang, R. Jin, Xiaoming Chen, and C. Zhong are with the
College of Information Science and Electronic Engineering, Zhejiang
University, Hangzhou, China (Email: $\\{$zhmj, richengjin, chen_xiaoming,
caijunzhong$\\}$@zju.edu.cn.). G. Zhu is with Shenzhen Research Institute of
Big Data, Shenzhen, China (Email: gxzhu@sribd.cn). Qingjiang Shi is with the
School of Software Engineering, Tongji University, Shanghai 200092, China, and
also with Shenzhen Research Institute of Big Data, Shenzhen 518172, China
(email: shiqj@tongji.edu.cn). Kaibin Huang is with The University of Hong
Kong, Hong Kong (email: huangkb@eee.hku.hk).

###### Abstract

Semantic communication (SemCom) is emerging as a key technology for future
sixth-generation (6G) systems. Unlike traditional bit-level communication
(BitCom), SemCom directly optimizes performance at the semantic level, leading
to superior communication efficiency. Nevertheless, the task-oriented nature
of SemCom renders it challenging to completely replace BitCom. Consequently,
it is desired to consider a semantic-bit coexisting communication system,
where a base station (BS) serves SemCom users (sem-users) and BitCom users
(bit-users) simultaneously. Such a system faces severe and heterogeneous
inter-user interference. In this context, this paper provides a new semantic-
bit coexisting communication framework and proposes a spatial beamforming
scheme to accommodate both types of users. Specifically, we consider
maximizing the semantic rate for semantic users while ensuring the quality-of-
service (QoS) requirements for bit-users. Due to the intractability of
obtaining the exact closed-form expression of the semantic rate, a data driven
method is first applied to attain an approximated expression via data fitting.
With the resulting complex transcendental function, majorization minimization
(MM) is adopted to convert the original formulated problem into a multiple-
ratio problem, which allows fractional programming (FP) to be used to further
transform the problem into an inhomogeneous quadratically constrained
quadratic programs (QCQP) problem. Solving the problem leads to a semi-closed
form solution with undetermined Lagrangian factors that can be updated by a
fixed point algorithm. Extensive simulation results demonstrate that the
proposed beamforming scheme significantly outperforms conventional beamforming
algorithms such as zero-forcing (ZF), maximum ratio transmission (MRT), and
weighted minimum mean-square error (WMMSE).

###### Index Terms:

Multi-user MIMO, beamforming design, semantic communication, optimization

##  I Introduction

According to Shannon and Weaver [1], communication could be classified into
three levels, namely technical level concerning _how accurately can the
symbols of communication be transmitted_ , semantic level concerning _how
precisely do the transmitted symbols convey the desired meaning_ , and
effectiveness level concerning _how effectively does the received meaning
affect conduct in the desired way_. In the past forty years, researchers
mainly focused on the first level design, driving the mobile communication
system evolved from the first generation (1G) to the fifth generation (5G).
The transmission rate has been significantly improved and the system capacity
is gradually approaching the Shannon limit. However, the rapid growth of
communication demand in modern society shows no signs of stopping.
Specifically, the upcoming sixth generation (6G) is expected to achieve
transmission rates that are ten times faster than those of 5G [2], which will
enable the support of numerous new applications including virtual and
augmented reality (VR/AR), smart factories, intelligent transportation
systems, etc [3]. This thus prompts an active research area that rethinks the
communication systems at the semantic even effectiveness level.

###  I-A Semantic Communication

To build communication systems at the semantic level, semantic communication
(SemCom) that mines the semantic information from the source, is one of the
most popular candidate technologies in 6G. Compared with the traditional bit-
level digital communication (BitCom) framework that adopts separate source and
channel coding (SSCC) for minimizing bit/symbol error rate, SemCom embraces
joint source and channel coding (JSCC) through neural network, which enables
the extraction of semantic information and demonstrates a better transmission
efficiency compared with BitCom [4]. The authors in [5] proposed to use neural
network to achieve JSCC for image recovery, and optimized the system
performance through end-to-end learning under the criteria of mean square
error (MSE). Then, the authors in [4] incorporated transformer and proposed
DeepSC, which is shown to outperform BitCom, especially in the low signal-to-
noise (SNR) regime. Based on these pioneering works [5, 4], SemCom has then
been extensively studied under different data modalities, including image [6,
7], text [8], speech [9], video [10], and multimodal data [11]. Despite the
potential performance gain of SemCom, critical concerns about its practical
deployment remain. For example, most of the SemCom systems employ analog
symbol transmission, while it has been verified that digital transmission is
more reliable and secure, as well as cost-effective in hardware
implementation. This prompts the development of digital SemCom by designing
codebooks [12] and quantization methods [13, 14] for semantic information.
Besides, the current SemCom heavily relies on neural networks, which are prone
to overfitting the training data collected under certain limited scenarios and
thus lack of generalization capability to deal with the challenges brought by
the dynamic wireless environment. Prompted by this, authors in [15] proposed
an attention-based JSCC scheme that uses channel-wise soft attention to scale
features according to SNR conditions, which enables it applicable to scenarios
with a broad range of SNRs through a single model. Then, given the multiple
antenna cases, a channel-adaptive JSCC scheme that exploits the channel state
information (CSI) and SNR through attention mechanism was further proposed in
[16].

###  I-B Motivations

Although SemCom has shown great potential for 6G, there is a critical issue
that requires further investigation: _Can SemCom completely replace BitCom?_
We believe the answer is no. This is because the task-oriented nature of
SemCom implies that it needs to be tailored for each specific task, which
renders it not suitable for generic transmission tasks. As a result, we
envision that future 6G network will see the co-existence of SemCom and
BitCom, yielding the semantic-bit coexisting system that supports both SemCom
users (sem-users) and BitCom users (bit-users). In the coexisting system, due
to the diverse performance objectives, existing transmission schemes for
BitCom can no longer provide satisfactory services for the sem-users, and thus
need to be redesigned. In response to this, we investigate the beamforming
design for the coexisting multi-user multiple-input single-output (MU-MISO)
system, and try to shed lights on how to adapt the current transmission
algorithms in BitCom to the coexisting system.

###  I-C Related works

The study of multiuser SemCom has received a lot of attention in recent years,
which mainly lies in two directions: resource allocation and interference
management. In terms of resource allocation, a semantic-aware channel
assignment mechanism was proposed in [17], and an optimal semantic-oriented
resource block allocation method was put forward in [18] subsequently. The
main idea of these two works is adjusting communication resources for boosting
the transmission of semantic information. On the other hand, multiuser usually
accompanies with interference, which can cause semantic noise that
significantly degrades the performance [19]. To mitigate the interference,
several methods have been proposed. For instance, the authors in [20] proposed
to jointly optimize the codebook and the decoder, as such the user
interference could be minimized. The authors in [21] further proposed to
dynamically fuse the semantic features to a joint latent representation and
adjust the weights of different user semantic features to combat fading
channels. In addition to the interference from other sem-users, the
interference from bit-users needs to be appropriately mitigated as well. Given
this, the coexistence of sem-users and bit-users was considered in the non-
orthogonal multiple access (NOMA) system [22, 23], where bit-users and sem-
users are viewed as primary users and secondary users, respectively. The
interference issue was addressed through successive interference cancellation
(SIC). However, in the case of MU-MISO that enables multi-users through
spatial multiplexing, the coordination of the two types of users with diverse
transmission objectives remains largely unexplored.

Beanforming is a key technique of MU-MISO systems and has been commonly-used
for interference mitigation [24, 25, 26, 27, 28, 29, 30]. Several linear
design methods have been proposed to tackle the beamforming problem in MU-MISO
systems. Zero forcing (ZF) and maximum ratio transmission (MRT) algorithm are
two simple but effective beamforming algorithms. The former minimizes the user
interference, and the latter maximizes the signal gain at the destination
user. Besides, a well-known iterative algorithm is the weighted minimum mean-
square error (WMMSE) algorithm [24], which achieves high performance by first
transforming the original problem into an MMSE problem and then updating the
variables in an alternative manner. Given high complexity of the WMMSE
algorithm and limited performance of the ZF and the MRT algorithm, researchers
resort to deep learning for developing beamforming scheme with both low
complexity and high performance. With the optimal solution structure revealed
in [25], the data driven method that learns the undetermined parameters in the
solution structure was proposed in [26], and further extended in [27].
Besides, the authors in [31] proposed to use deep unfolding of the WMMSE
algorithm for MU-MISO downlink precoding, which constructs the iteration
process in neural networks. Variants of the deep unfolding-based methods have
been investigated in [28]. The aforementioned schemes aim to maximize the data
rate for BitCom. However, recent research has revealed that the semantic rate
in SemCom has a distinct mapping from SNR to performance [17, 22]. As a
result, existing methods may not be suitable for the semantic-bit coexisting
system, and a new beamforming scheme that takes into account the different
transmission objectives is urgently needed.

###  I-D Contribution and Organization

In this paper, we investigate the transmission design for the semantic-bit
coexisting paradigm in the multiple-antenna communication system.
Specifically, we consider sem-users with the task of image transmission and
propose an adaptive JSCC autoencoder for semantic information extraction and
recovery. A data driven method is adopted to approximate the semantic rate of
sem-users, based on which a beanforming problem that optimizes the performance
of sem-users under the QoS constraints of bit-users is formulated and solved.
The contributions of this paper are summarized as follows:

Figure 1: Semantic-bit coexisting communication system framework

  * •

Targeting the task of image transmission, we propose an effective JSCC scheme
that features a dynamic depth of downsampling, which is realized through the
“early exit” mechanism [11] and the proposed module-by-module training scheme.
On this basis, we further conduct semantic rate approximation on the ImageNet
dataset and build the mapping from the depth of downsampling and SNR to
semantic rate through data regression.

  * •

We propose a beamforming design scheme for the semantic-bit coexisting system.
Specifically, we tackle the primary challenge posed by the transcendental
semantic rate function. By employing majorization-minimization (MM) and
introducing a novel surrogate function, the original objective is transformed
into a multiple-ratio form, which which is further converted to an
inhomogeneous quadratically constrained quadratic programs (QCQP) problem by
fractional programming. The semi-closed form solution for the resulting QCQP
problem is derived, and the original problem is solved in an alternative
manner. Additionally, the alternative algorithm has inspired a low-complexity
beamforming method to address the complexity concern.

  * •

Both theoretical analysis and numerical simulations are presented to validate
the effectiveness of the proposed beamforming scheme in semantic-bit
coexisting communication systems.

The rest of this paper is organized as follows. Section II introduces the
semantic-bit coexisting system model. Section III presents the proposed JSCC
design, the approximation of semantic rate, and the problem formulation. The
optimization problem is solved in Section IV. Then, extensive simulation
results are given in Section V, followed by the concluding remarks in Section
VI.

##  II System Model

In this section, we first present the semantic-bit coexisting system and the
transmission protocol, based on which the performance metrics of sem-users and
bit-users are analyzed, respectively.

###  II-A Semantic-bit Coexisting Communication Framework

We consider a single-cell downlink multi-user multiple-input single-output
(MU-MISO) system shown in Fig. 1. The base station (BS) is equipped with
$N_{t}$ transmit antennas, while the users have a single antenna each. The
users are divided into two groups, namely bit-users with BitCom and sem-users
with SemCom. We denote the bit-users set as
$\mathcal{B}=\\{b_{1},b_{2},...,b_{B}\\}$ and the sem-users set as
$\mathcal{T}=\\{t_{1},t_{2},...,t_{T}\\}$, with $B$ and $T$ being the numbers
of bit-users and sem-users, respectively. The transmit signal vector at the
BS, denoted by $\mathbf{x}\in\mathbb{C}^{N_{t}\times 1}$, is given by

$\displaystyle\mathbf{x}=\sum_{i=1}^{B}\mathbf{v}_{b_{i}}{s}_{b_{i}}+\sum_{j=1}^{T}\mathbf{v}_{t_{j}}s_{t_{j}},$
(1)

where $\mathbf{v}_{b_{i}}\in\mathbb{C}^{N_{t}\times 1}$ and
$\mathbf{v}_{t_{j}}\in\mathbb{C}^{N_{t}\times 1}$ denote the beamforming
vector of the $i$-th bit-user and the $j$-th sem-user, respectively.
Furthermore, we assume that $s_{b_{i}}$ and $s_{t_{j}}$ are zero mean and
$\mathbb{E}[s_{b_{i}}s_{b_{i}}^{H}]=\mathbb{E}[s_{t_{j}}s_{t_{j}}^{H}]=1$, and
the symbols desired for different users are independent from each other.

Then, the received signal $y_{b_{k}}\in\mathbb{C}$ at bit-user $b_{k}$ can be
expressed as

$y_{b_{k}}=\mathbf{h}_{b_{k}}^{H}\mathbf{v}_{b_{k}}s_{b_{k}}+\sum_{i=1,i\neq
k}^{B}\mathbf{h}_{b_{k}}^{H}\mathbf{v}_{b_{i}}s_{b_{i}}\\\
+\sum_{j=1}^{T}\mathbf{h}_{b_{k}}^{H}\mathbf{v}_{t_{j}}s_{t_{j}}+{n}_{b_{k}},\forall
b_{k}\in\mathcal{B},$ (2)

where $\mathbf{h}_{b_{k}}\in\mathbb{C}^{N_{t}\times 1}$ denotes the MISO
channel from the BS to user $b_{k}$ and ${n}_{b_{k}}$ represents the additive
noise which is modeled as a circularly symmetric complex Gaussian random
variable following the distribution
$\mathcal{CN}\big{(}\mathbf{0},\sigma_{b_{k}}^{2}\big{)}$, with
$\sigma_{b_{k}}^{2}$ being the average noise power.

Similarly, let $\mathbf{h}_{t_{k}}\in\mathbb{C}^{N_{t}\times 1}$ denote the
channel from the BS to the sem-user $t_{k}$, the received signal $y_{t_{k}}$
at sem-user $t_{k}$ is given by

$y_{t_{k}}=\mathbf{h}_{t_{k}}^{H}\mathbf{v}_{t_{k}}s_{t_{k}}+\sum_{i=1}^{B}\mathbf{h}_{t_{k}}^{H}\mathbf{v}_{b_{i}}s_{b_{i}}\\\
+\sum_{j=1,j\neq
k}^{T}\mathbf{h}_{t_{k}}^{H}\mathbf{v}_{t_{j}}s_{t_{j}}+{n}_{t_{k}},\forall
t_{k}\in\mathcal{T},$ (3)

where ${n}_{t_{k}}$ is the additive white Gaussian noise with distribution
$\mathcal{CN}(0,\sigma_{t_{k}}^{2})$. Notice that $s_{t_{j}}$ denotes the
symbol stream that contains symbols in the latent representation, which is the
output of the JSCC encoder. The symbols within the latent representation are
transmitted sequentially.

Figure 2: Transmission protocol for bit-users and sem-users

###  II-B Bit-level Communication

We adopt a transmission frame structure consisting of $L+L_{p}$ symbol
intervals, as shown in Fig. 2. We assume slow fading channel, which means that
the channel does not change within a frame and independently fades across
different frames. In this vein, the first $L_{p}$ symbols are utilized for
channel estimation and the remaining $L$ symbols for data transmission. With
the estimated channel state information (CSI), the BS is able to conduct
beamforming. It is worth noting that for sem-users, the target of transmission
is to convey the latent representation from transmitter to receiver. Once this
process completes, no further symbols are required. We assume sem-users should
finish transmission of the latent representation in $M$ symbol intervals
($M<L$) and bit-users uses all $L$ symbol intervals for data transmission,
which leads to the semi-non-orthogonal multiple access (semi-NOMA) [32]. In
this context, the total data transmission component of length $L$ symbol
intervals is further divided into two parts, as shown in Fig. 2. At the shared
period of length $M$ symbol intervals, the BS simultaneously serves all users.
Both bit-users and sem-users will be interfered by each other. The exclusive
period of length $L-M$ symbol intervals is dedicated to data transmission for
bit-users, i.e., $s_{t_{k}}=0,\forall t_{k}\in\mathcal{T}$.111Note that while
we primarily focus on the scenario where the number of transmission symbols
used by bit-users exceeds that of sem-users, it should be emphasized that our
framework and proposed method can be readily extended to situations where the
number of transmission symbols used by sem-users is greater than that of bit-
users. This extension can be achieved by allowing $\frac{M}{L}\geq 1$, where
$M$ represents the length of available symbols within one frame in this
particular scenario. As digital transmission is employed at the bit-user, the
achievable bit rate (bits/s/Hz) during the shared period is given by [1]

$\displaystyle R_{{b_{i},1}}=\log_{2}\left(1+{\gamma}_{b_{i},1}\right),$ (4)

where
$\gamma_{b_{i},1}=\frac{|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B},\mathcal{T}\\}/b_{i}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\sigma_{b_{i}}^{2}}$
denotes the signal-to-interference-plus-noise ratio (SINR) of bit-user $b_{i}$
during the shared period.

Then, at the exclusive period, the BS only serves the bit-users, and the
corresponding achievable bit rate is given by

$\displaystyle R_{{b_{i},2}}=\log_{2}\left(1+{\gamma}_{b_{i},2}\right),$ (5)

where
$\gamma_{b_{i},2}=\frac{|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B}\\}/b_{i}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\sigma_{b_{i}}^{2}}$.

As a result, the overall normalized bit rate of the bit-user $b_{i}$ in a
frame is defined as below.

$\displaystyle
R_{{b_{i}}}=\frac{M}{L}\log_{2}\left(1+{\gamma}_{b_{i},1}\right)+\frac{L-M}{L}\log_{2}\left(1+{\gamma}_{b_{i},2}\right).$
(6)

###  II-C Semantic Communication

In semantic communication, the semantic rate no longer focuses on the symbol
error rate, but on the quality of task completion. Fundamentally, the
performance of semantic communication hinges on the effectiveness of the JSCC
model and the wireless noise intensity. In this sense, the semantic rate can
be generally expressed as

$\displaystyle S_{t_{i}}=\epsilon(\boldsymbol{\Theta},\gamma_{t_{i}}),$ (7)

where $\boldsymbol{\Theta}$ denotes the semantic model composed of deep neural
networks (DNNs) that determines $M$ and the specific method for the extraction
of semantic information, and
$\gamma_{t_{i}}=\frac{|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{j}|^{2}+\sigma_{t_{i}}^{2}}$
is the SINR of the sem-user $t_{i}$ at the shared period. In the context of
image transmission scenario, $\epsilon(\boldsymbol{\Theta},\gamma_{t_{i}})$ is
evaluated under the widely-adopted performance metric called structural
similarity index measure (SSIM).

As mentioned, the overall semantic rate is determined by the adopted JSCC
model and the transmission environment. The former represents the semantic
compression and exploitation ability of the semantic communication, while the
later determines the level of noise disturbation. However, semantic
communication highly relies on neural networks for semantic extraction and
recovery, the black-box nature of which hinders the theoretical analysis,
making $\epsilon(\boldsymbol{\Theta},\gamma_{t_{i}})$ unable to be acquired
precisely. A commonly adopted method for tackling this problem is data
regression [22, 32, 17], which obtains the mapping from $\boldsymbol{\Theta}$
and $\gamma_{t_{i}}$ to $S_{t_{i}}$ through sufficient experimental instances
and curve fitting, which will be further elaborated in the subsequent section.

##  III JSCC Design and Semantic Rate Approximation

In this section, we will elaborate on the design of semantic communication in
detail. Considering the task of image transmission,we first present the
proposed design of the JSCC model. Then, we conduct a series of experiments to
evaluate the performance in different system settings. Building on this, we
approximate the semantic rate with data regression. Finally, the problem that
jointly optimizes the beamforming vectors and the downsampling depth is
formulated.

###  III-A JSCC Network Characterization

Figure 3: JSCC network for image transmission with multi exit mechanism

The proposed JSCC network for image transmission is shown in Fig. 3. At the
encoder part, we consider compressing the original image $\mathbf{s}$ through
multiple downsample modules $\mathcal{D}=\\{\mathscr{D}_{n}\\}_{n=1}^{N}$,
each of which comprises a residual block [33], followed by a convolution
layer. The number of filters in all the convolution layers is set to $C$.
After each downsample module, the image size is reduced by half, and the
number of channels is fixed to $C$. For the upsample module
$\mathcal{U}=\\{\mathscr{U}_{n}\\}_{n=1}^{N}$ at the decoder part, the reverse
process is conducted. Without loss of generality, we consider the image having
a square size, that is, $\mathbf{s}\in\mathbb{R}^{D\times D\times 3}$, with
$D$ being the image size, and “3” the number of channels, respectively.

Additionally, in typical 5G and beyond communication systems, due to the rapid
growth in the number of users and data volumn, the BS needs to allocate
communication resources to individual users. As a result, the available
communication resources for users vary considerably in time and space, which
poses new requirements for semantic communication. Specifically, the JSCC
model should be able to dynamically adjust the number of the transmission
symbols. To this end, we propose a multi-exit mechanism, as illustrated in
Fig. 3. With this mechanism, the decoder can exit early rather than pass all
the downsample modules. Consequently, the size of the latent representation
can be adjusted by selecting the number of passed downsample module. Let $K$
be the number of passed downsample module, then the number of required
transmission symbols $M_{K}$ is given by

$\displaystyle
M_{K}=C\big{(}\frac{D}{2^{K+1}}\big{)}^{2}=\frac{CD^{2}}{4^{K+1}}.$ (8)

###  III-B Semantic Rate Approximation

Before deployment, training is required to obtain the JSCC neural network
$\boldsymbol{\Theta}$. With the multi-exit mechanism, it is desired that the
downsample and upsample modules in $\boldsymbol{\Theta}$ can work
independently, and also be incorporated into deeper models (i.e.,
$\boldsymbol{\Theta}$ with a larger $K$). To this end, we propose a module-by-
module training algorithm. As shown in Algorithm 1, the modules are trained
sequentially, with only the weight parameters in the current layer modules
being updated during the $K$-th round of training, while the upper layer
modules (i.e., $\\{\mathscr{D}_{n}\\}_{n=1}^{K-1}$,
$\\{\mathscr{U}_{n}\\}_{n=1}^{K-1}$) are frozen222“Frozen” means that the
parameters of the module will not change anymore.. Let
$\boldsymbol{\Theta}_{K}$ be the JSCC model with a specific downsampling depth
$K$. The semantic rate defined in (7) is given by
$\epsilon(\boldsymbol{\Theta}_{K},\gamma)$, where $\gamma$ denotes the SNR.
For simplicity, we use the notation $\epsilon(K,\gamma)$ for
$\epsilon(\boldsymbol{\Theta}_{K},\gamma)$ in the rest of this paper, since
$\boldsymbol{\Theta}_{K}$ is uniquely determined by $K$ when the downsampling
and upsampling modules are specified.

Algorithm 1 Algorithm for Module-by-module Training

0: all the modules including $\\{\mathscr{D}_{M}\\}_{M=1}^{N}$,
$\\{\mathscr{U}_{M}\\}_{M=1}^{N}$.

1: for $K=1,2,...,N$ do

2:
$\Theta=\\{\\{\mathscr{D}_{M}\\}_{M=1}^{K},\\{\mathscr{U}_{M}\\}_{M=1}^{K}\\}$.

3: while $\Theta$ not converged do

4: Sample a minibatch of data from training dataset

5: Obtain the gradient $\nabla\mathscr{D}_{K}$ and $\nabla\mathscr{U}_{K}$
through loss calculation and backpropagation.

6: Update $\mathscr{D}_{K}$ and $\mathscr{U}_{K}$ using
$\mathscr{D}_{K}=\mathscr{D}_{K}-\eta\nabla\mathscr{D}_{K}$,
$\mathscr{U}_{K}=\mathscr{U}_{K}-\eta\nabla\mathscr{U}_{K}$.

7: end while

8: Frozen $\mathscr{D}_{K}$ and $\mathscr{U}_{K}$.

9: end for

We train $\boldsymbol{\Theta}$ on the ImageNet dataset [34], a large-scale
image dataset containing over 14 million images, which serves as a standard
benchmark for various computer vision tasks. For the channel setting, we
consider the AWGN channel333Note that we assume AWGN channel for simplicity,
such that the JSCC model can be trained on the BS side, and the training
overhead can thus be significantly reduced. Nevertheless, as we discussed in
the previous work [35], the model trained under AWGN cases can be directly
applied to the MISO cases with minor modification. This is because the final
received signal can be transformed into an equivalent AWGN form when recovery
precoding is adopted at the receiver., where the SNR is fixed to 10 dB in the
training process. Moreover, _mean squared error_ (MSE) criteria is adopted as
the loss function, and the Adam optimizer with the initiate learning rate of
$10^{-4}$ is adopted. After sufficient training, we evaluate the performance
$\epsilon(K,\gamma)$ on the validation dataset with different $K$ and SNR
settings, under SSIM. The evaluation results are depicted in Fig. 4. It can be
seen that with a increasing $K$, i.e., more stringent compression of the
original image, the performance floor when ${\rm SNR}\rightarrow\infty$
decreases monotonically. Besides, for each $K$, $\epsilon(K,\gamma)$ follows
an $S$ shape with respect to $\gamma$ in dB, which is also revealed in [22]
under the text transmission task with the DeepSC model [4]. Therefore, similar
to [22], the generalized logistic function could be utilized to well
approximate $\epsilon(K,\gamma)$, as follows.

$\displaystyle\epsilon(K,\gamma)\approx\tilde{\epsilon}(K,\gamma)$
$\displaystyle=a_{K}+\frac{b_{K}}{c_{K}+{\rm exp}(-10d_{K}\log_{10}(\gamma))}$
$\displaystyle=a_{K}+\frac{b_{K}}{c_{K}+\gamma^{-\alpha_{K}}},$ (9)

where $a_{K}$, $b_{K}$, $c_{K}$, $\alpha_{K}$ are parameters determined by
$K$, and are obtained through curve fitting.

Figure 4: Performance evaluation under different $K$ and SNR settings

###  III-C Problem Formulation

As illustrated in Section II-A, there are two types of users with different
performance metrics in the sematic-bit coexisting communication system. In
this paper, we aim to maximize the semantic rate of sem-users while satisfying
the QoS requirements of bit-users, by jointly optimizing the beamforming
vectors and the downsampling depth. The considered optimization problem can be
formulated as

$\displaystyle\mathbf{P1}:\max_{\mathbf{V},K}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{T}\tilde{\epsilon}\left(K,\gamma_{t_{i}}\right),$
(10a) $\displaystyle{\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\frac{M_{K}}{L}\log_{2}\left(1+\gamma_{b_{i},1}\right)+\frac{L-M_{K}}{L}$
$\displaystyle\qquad\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\cdot\log_{2}\left(1+\gamma_{b_{i},2}\right)\geq\beta_{b_{i}},\forall
b_{i}\in\mathcal{B},$ (10b)
$\displaystyle\sum_{i=1}^{B}\left\|\mathbf{v}_{b_{i}}\right\|^{2}+\sum_{j=1}^{T}\left\|\mathbf{v}_{t_{j}}\right\|^{2}\leq
P_{T},$ (10c)

where
$\gamma_{t_{i}}=\frac{|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{j}|^{2}+\sigma_{t_{i}}^{2}}$,
$\gamma_{b_{i},1}=\frac{|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B},\mathcal{T}\\}/b_{i}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\sigma_{b_{i}}^{2}}$,
$\gamma_{b_{i},2}=\frac{|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B}\\}/b_{i}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\sigma_{b_{i}}^{2}}$,
and
$\mathbf{V}=[\mathbf{V}_{B},\mathbf{V}_{T}]\in\mathbb{C}^{N_{t}\times(B+T)}$,
with
$\mathbf{V}_{B}=[\mathbf{v}_{b_{1}},...,\mathbf{v}_{b_{B}}]\in\mathbb{C}^{N_{t}\times
B}$ and
$\mathbf{V}_{T}=[\mathbf{v}_{t_{1}},...,\mathbf{v}_{t_{T}}]\in\mathbb{C}^{N_{t}\times
T}$. $\beta_{b_{i}}$ denotes the requirements of transmission rate in one
frame from bit-user $b_{i}$. $P_{T}$ denotes the transmit power budget of the
BS. $R_{b_{i},1}$ and $R_{b_{i},2}$ denote the transmission rate defined in
(4) and (5), respectively. $\tilde{\epsilon}(K,\gamma_{t_{i}})$ is the
approximated semantic rate given by (9).

######  Remark 1.

_As shown in $\mathbf{P1}$, beamforming design for a semantic-bit coexisting
system faces some new challenges compared to a BitCom system. Firstly, the
semantic rate $\tilde{\epsilon}(K,\gamma_{t_{i}})$ admits a completely
different form (which is neither convex nor concave) from channel capacity
w.r.t. SINR, which renders the existing interference suppression algorithms
ineffective. Additionally, the performance of sem-users also partially depends
on the downsampling depth $K$ that requires careful design. Unfortunately,
there exists a strong coupling between beamforming design and $K$, making the
problem even more challenging to solve. _

##  IV Joint Optimization of Beamforming and $K$-Configuring for Coexisting
System

In this section, we solve the problem $\mathbf{P1}$ for beamforming design and
configuring $K$ in semantic-bit coexisting MU-MISO systems. As discussed in
Remark 1, it is hard to directly solve the joint optimization problem, and we
thus consider solving $\mathbf{P1}$ by optimizing $\mathbf{V}$ and $K$
alternatively.

###  IV-A Beamforming Design

In this subsection, we optimize the beamforming matrix $\mathbf{V}$ in
$\mathbf{P1}$ with a given $K$, and the subproblem is given below.

$\displaystyle\mathbf{P2}:\max_{\mathbf{V}}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{T}a_{K}+\frac{b_{K}}{c_{K}+(\gamma_{t_{i}})^{-\alpha_{K}}},$
(11a) $\displaystyle{\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle(\ref{eq: qos constraints in p1}),(\ref{eq: power constraints in
p1}).$

As shown in $\mathbf{P2}$, the objective function (11a) is non-convex as (11a)
is a transcendental function of $\mathbf{V}$. Moreover, the fractional
expression exists in the QoS constraints. Therefore, $\mathbf{P2}$ is a NP-
hard problem, indicating that the optimal solution is intractable. We thus
resort to a suboptimal solution. To this end, the problem-solving process is
mainly divided into four steps. Firstly, we relax the power constraint by
regulating the noise intensity with ${{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}/{P_{T}}$. Then, we propose a surrogate function
for approximating the objective function. Next, the transforming method
proposed in [36] is adopted to transform the multiple-ratio fractional
programming (FP) problem into a QCQP problem. Finally, the resulting QCQP
problem is solved in a low-complexity manner.

####  IV-A1 Problem Transformation

It can be observed that the beamforming vectors appear in $\mathbf{P}_{2}$ as
the form of SINR in both the objective function (11a) and the QoS constraints
(10b). Without loss of optimality, similar to [28, 37], the power constraint
can be removed by integrating it to the SINR terms, as follows.

$\displaystyle\mathbf{P3}:\max_{\mathbf{V}}\qquad$
$\displaystyle\sum_{i=1}^{T}a_{K}+\frac{b_{K}}{c_{K}+(\Gamma_{t_{i}})^{-\alpha_{K}}},$
(12a) $\displaystyle{\rm s.t.}\qquad$
$\displaystyle\frac{M_{K}}{L}\log_{2}\left(1+\Gamma_{b_{i},1}\right)+\frac{L-M_{K}}{L}$
$\displaystyle\qquad\cdot\log_{2}\left(1+\Gamma_{b_{i},2}\right)\geq\beta_{b_{i}},\forall
b_{i}\in\mathcal{B}.$ (12b)

where the equivalent SINR terms are given by

$\displaystyle\Gamma_{t_{i}}$
$\displaystyle=\frac{|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{t_{i}}^{2}},$ (13)
$\displaystyle\Gamma_{b_{i},1}$
$\displaystyle=\frac{\left|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}\right|^{2}}{\sum_{j\in\left\\{\mathcal{B},\mathcal{T}\right\\}/b_{i}}\left|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}\right|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}},$ (14)
$\displaystyle\Gamma_{b_{i},2}$
$\displaystyle=\frac{\left|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}\right|^{2}}{\sum_{j\in\left\\{\mathcal{B}\right\\}/b_{i}}\left|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}\right|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}}.$ (15)

It can be found that the optimal solution $\mathbf{V}^{*}$ of the problem
$\mathbf{P2}$ should satisfy the power constraint (10c) with equality. Then
with the optimal solution $\mathbf{V}^{**}$ of the problem $\mathbf{P3}$,
$\mathbf{V}^{*}$ can be obtained by $\sqrt{\frac{{P_{T}}}{{\rm
Tr}(\mathbf{V}^{**}(\mathbf{V}^{**})^{H})}}\mathbf{V}^{**}$. Therefore, we can
resort to solving $\mathbf{P3}$ for obtaining $\mathbf{V}^{*}$.

####  IV-A2 Objective Approximation

The goal is to transform the objective function (12a) such that the
corresponding optimization problem can be appropriately solved. We adopt the
idea of MM, that is, using a surrogate function to approximate the objective
function (12a). Firstly, the semantic rate in $\mathbf{P3}$ is given by

$\displaystyle\tilde{\epsilon}_{K}(\Gamma_{t_{i}})=a_{K}+\frac{b_{K}}{c_{K}+\Gamma_{t_{i}}^{-\alpha_{K}}}.$
(16)

(a) $\alpha_{K}\leq 1$ (b) $\alpha>1$

Figure 5: Objective Approximation with different surrogate functions

The exemplary functions with different $K$ settings are depicted in Fig. 5.
Note that, our objective is to maximize the semantic rate w.r.t. the precoding
matrix $\mathbf{V}$ rather than the SINR term $\\{\Gamma_{t_{i}}\\}$.
Therefore, it is necessary to find an appropriate surrogate function that can
accurately capture the shape of $\epsilon_{K}(\Gamma_{t_{i}})$ and also has a
simple form for ease of handling during the optimization of $\mathbf{V}$. To
this end, we propose to approximate $\tilde{\epsilon}_{K}(\cdot)$ with a
surrogate function for the station point $\Gamma_{t_{i}}^{0}$ in each update
round, as follows.

$\displaystyle\zeta_{K}(\mathbf{V},\Gamma_{t_{i}}^{0})=D(K,\Gamma_{t_{i}}^{0})+E(K,\Gamma_{t_{i}}^{0})\frac{|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}}{F(K,\Gamma_{t_{i}}^{0})|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}+G(K,\Gamma_{t_{i}}^{0})(\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{t_{i}}^{2})},$ (17)

$\displaystyle\mathcal{J}(\Gamma_{t_{i}},{\Gamma_{t_{i}}^{0}},\alpha_{K})=\left\\{\begin{aligned}
\frac{1}{(\Gamma_{t_{i}}^{0})^{\alpha_{K}}}+\frac{\alpha_{K}}{(\Gamma_{t_{i}}^{0})^{\alpha_{K}-1}}(\frac{1}{\Gamma_{t_{i}}}-\frac{1}{\Gamma_{t_{i}}^{0}}),\alpha_{K}\leq
1\\\
\frac{1}{\Gamma_{t_{i}}^{0}+\alpha_{K}(\Gamma_{t_{i}}^{0})^{\alpha-1}(\Gamma_{t_{i}}-\Gamma_{t_{i}}^{0}))},\alpha_{K}>1\end{aligned}\right.$
(18)

Let
$\zeta_{K}(\mathbf{V},\Gamma_{t_{i}}^{0})=a_{K}+\frac{b_{K}}{c_{K}+\mathcal{J}(\Gamma_{t_{i}},{\Gamma_{t_{i}}^{0}},\alpha_{K})}$,
we have
$\tilde{\epsilon}_{K}(\Gamma_{t_{i}})\geq\zeta_{K}(\mathbf{V},\Gamma_{t_{i}}^{0})$.
Therefore we use $\zeta_{K}(\mathbf{V},\Gamma_{t_{i}}^{0})$ as the surrogate
function for $\tilde{\epsilon}_{K}(\Gamma_{t_{i}})$. As shown in Fig. 5, the
proposed surrogate function captured the original
$\tilde{\epsilon}_{K}(\cdot)$ well, which leads to the following optimization
problem:

$\displaystyle\mathbf{P4}:\max_{\mathbf{V}}$
$\displaystyle\sum_{i=1}^{T}\zeta_{K}(\mathbf{V},\Gamma_{t_{i}}^{0}),$ (19a)
$\displaystyle{\rm s.t.}\qquad$ $\displaystyle(\ref{eq: qos constraints in
p3}).$

It can be found that $\zeta_{K}(\cdot)$ exhibits a fractional form of
$\Gamma_{t_{i}}$. Therefore, we resort $\zeta_{K}(\cdot)$ as a function of
beamforming vectors, as shown in (17) on the top of this page, where
$D(K,\Gamma_{t_{i}}^{0})=a_{K}$, $\qquad E(K,\Gamma_{t_{i}}^{0})=b_{K}$,
$F(K,\Gamma_{t_{i}}^{0})=c_{K}+(1-\alpha_{K})(\Gamma_{t_{i}}^{0})^{-\alpha_{K}}$,
and $G(K,\Gamma_{t_{i}}^{0})=\alpha_{K}(\Gamma_{t_{i}}^{0})^{1-\alpha_{K}}$
when $\alpha_{K}\leq 1$;
$D(K,\Gamma_{t_{i}}^{0})=a_{K}+\frac{b_{K}(1-\alpha_{K})(\Gamma_{t_{i}}^{0})^{\alpha_{K}}}{c_{K}(1-\alpha_{K})(\Gamma_{t_{i}}^{0})^{\alpha_{K}}+1}$,
$\qquad
E(K,\Gamma_{t_{i}}^{0})=\frac{\alpha_{K}b_{K}(\Gamma_{t_{i}}^{0})^{\alpha_{K}-1}}{c_{K}(1-\alpha_{K})(\Gamma_{t_{i}}^{0})^{\alpha_{K}}+1}$,
$F(K,\Gamma_{t_{i}}^{0})=\alpha_{K}c_{K}(\Gamma_{t_{i}}^{0})^{\alpha_{K}-1}$,
and
$G(K,\Gamma_{t_{i}}^{0})=c_{K}(1-\alpha_{K})(\Gamma_{t_{i}}^{0})^{\alpha_{K}}+1$
when $\alpha_{K}>1$.

Using the approximation method in each iteration, the problem is transformed
into a multiple-ratio FP problem.

####  IV-A3 Fractional Programming

In this part, we solve the multiple-ratio FP problem $\mathbf{P4}$. Note that,
$\mathbf{P4}$ still cannot be directly solved, because the QoS constraints are
non-convex. Besides, in terms of the beamforming vector, both the objective
and the constraints (12b) are in fractional form. To this end, the alternating
optimization method is considered. We first apply the Lagrangian dual
transformation proposed in [36] to (12b), and the problem can be equivalently
written as

$\displaystyle\mathbf{P5}:\max_{\mathbf{V},\mathbf{y},\mathbf{z}}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{T}\zeta_{K}\big{(}\mathbf{V},\Gamma_{t_{i}}^{0}\big{)},$
(20a) $\displaystyle{\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\frac{M_{K}}{L}R_{b_{i},1}^{\prime}\big{(}y_{i},\mathbf{V}\big{)}$
$\displaystyle+\frac{L-M_{K}}{L}R_{b_{i},2}^{\prime}\big{(}z_{i},\mathbf{V}\big{)}\geq\beta_{b_{i}},\forall
b_{i}\in\mathcal{B},$ (20b)

where $\mathbf{y}=[y_{1},...,y_{B}]^{T}\in\mathbb{R}^{B}$,
$\mathbf{z}=[z_{1},...,z_{B}]^{T}\in\mathbb{R}^{B}$ are auxiliary variables
for the SINR terms,

$\displaystyle R_{b_{i},1}^{\prime}$ $\displaystyle=\log_{2}(1+y_{i})-y_{i}$
$\displaystyle\qquad\qquad+{\frac{(1+y_{i})|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}|^{2}}{\sum_{j\in\left\\{\mathcal{B},\mathcal{T}\right\\}}\left|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}\right|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}}},$ (21)
$\displaystyle R_{b_{i},2}^{\prime}$ $\displaystyle=\log_{2}(1+z_{i})-z_{i}$
$\displaystyle\qquad\qquad+\frac{(1+z_{i})\left|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}\right|^{2}}{\sum_{j\in\left\\{\mathcal{B}\right\\}}\left|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}\right|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}}.$ (22)

Note that, for maximizing $R_{b_{i},1}^{\prime}$ and $R_{b_{i},2}^{\prime}$
with a fixed $\mathbf{V}$, the optimal $\mathbf{y}$ and $\mathbf{z}$ equal to
the corresponding SINR term of user $b_{i}$, as follows.

$\displaystyle y_{i}$
$\displaystyle=\frac{|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B,T}\\}/b_{i}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}},\leavevmode\nobreak\
i\in\\{1,...,B\\},$ (23) $\displaystyle z_{i}$
$\displaystyle=\frac{|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}|^{2}}{\sum_{j\in\\{\mathcal{B}\\}/b_{i}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
i\in\\{1,...,B\\}.$ (24)

Moreover, $R^{\prime}_{b_{i},1}=R_{b_{i},1}$ and
$R^{\prime}_{b_{i},2}=R_{b_{i},2}$ holds for optimal $\\{y_{i}\\}_{i=1}^{B}$
and $\\{z_{i}\\}_{i=1}^{B}$, respectively. The same properties holds for
$R_{b_{i},2}^{{}^{\prime}}$ and $\mathbf{z}$ as well. Therefore, with the
optimal $\mathbf{y}$ and $\mathbf{z}$, the problem $\mathbf{P5}$ can be
reduced to

$\displaystyle\mathbf{P6}:\max_{\mathbf{V}}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{T}\zeta_{K}\big{(}\mathbf{V},\Gamma_{t_{i}}^{0}\big{)},$
(25) $\displaystyle{\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle(\ref{eq: yz added QOS constraints}).$

It can be found that in $\mathbf{P6}$, the sum-of-ratio form exists in both
the objective and the QoS constraints. Therefore, we adopt the quadratic
transformation proposed in [36], which yields the following optimization
problem.

$\displaystyle\mathbf{P7}:\max_{\mathbf{V},\mathbf{x},\mathbf{m},\mathbf{n}}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{T}E(K,\gamma_{t_{i}}^{0})\bigg{[}2x_{i}{\rm
Re}\\{\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}\\}$ $\displaystyle-
x_{i}^{2}\bigg{(}F(K,\gamma_{t_{i}}^{0})|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}+G(K,\gamma_{t_{i}}^{0})$
$\displaystyle\cdot\big{(}$
$\displaystyle\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{t_{i}}^{2}\big{)}\bigg{)}\bigg{]},$
(26a) $\displaystyle{\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\frac{M_{K}}{L}R_{b_{i},1}^{\prime\prime}(\mathbf{V},y_{i},m_{i})$
$\displaystyle+$
$\displaystyle\frac{L-M_{K}}{L}R_{b_{i},2}^{\prime\prime}(\mathbf{V},z_{i},n_{i})\geq\beta_{b_{i}},\forall
b_{i}\in\mathcal{B},$ (26b)

where $R_{b_{i},1}^{\prime\prime}(\mathbf{V},y_{i},m_{i})$ and
$R_{b_{i},1}^{\prime\prime}(\mathbf{V},y_{i},m_{i})$ are given by

$\displaystyle R_{b_{i},1}^{\prime\prime}(\mathbf{V},y_{i},m_{i})$
$\displaystyle=\log\left(1+y_{i}\right)-y_{i}+2m_{i}\sqrt{1+y_{i}}{\rm
Re}\\{\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}\\}$ $\displaystyle-
m_{i}^{2}\big{(}\sum_{j\in\\{\mathcal{B,T}\\}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}\big{)},$ (27)
$\displaystyle R_{b_{i},2}^{\prime\prime}(\mathbf{V},z_{i},n_{i})$
$\displaystyle=\log\left(1+z_{i}\right)-z_{i}+2n_{i}\sqrt{1+z_{i}}{\rm
Re}\\{\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}\\}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-n_{i}^{2}\big{(}\sum_{j\in\\{\mathcal{B}\\}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}\big{)}.$ (28)

In $\mathbf{P7}$, three auxiliary vectors $\mathbf{x}=[x_{1},...,x_{T}]$,
$\mathbf{m}=[m_{1},...,m_{B}]$, and $\mathbf{n}=[n_{1},...,n_{B}]$ are
introduced to transform the original problem to a quadratic programming
problem. More specifically, with a given $\mathbf{V}$, the optimal
$\mathbf{x}$, $\mathbf{m}$, and $\mathbf{n}$ for $\mathbf{P7}$ are given as
follows.

$\displaystyle x_{i}$ $\displaystyle={\rm
Re}\\{\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}\\}\bigg{[}F(K,\Gamma_{t_{i}}^{0})|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}+G(K,\Gamma_{t_{i}}^{0})$
$\displaystyle\cdot$
$\displaystyle{\big{(}\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{t_{i}}^{2}\big{)}}\bigg{]}^{-1},\forall
i=\\{1,...,T\\},$ (29) $\displaystyle m_{i}$
$\displaystyle=\frac{\sqrt{1+y_{i}}{\rm
Re}\\{\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}\\}}{\sum_{j\in\\{\mathcal{B,T}\\}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}},\forall
i=\\{1,...,B\\},$ (30) $\displaystyle n_{i}$
$\displaystyle=\frac{\sqrt{1+z_{i}}{\rm
Re}\\{\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{b_{i}}\\}}{\sum_{j\in\\{\mathcal{B}\\}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2}},\forall
i=\\{1,...,B\\}.$ (31)

Then, in terms of fixed $\mathbf{x}$, $\mathbf{m}$, and $\mathbf{n}$, the
problem $\mathbf{P6}$ can be further reduced to

$\displaystyle\mathbf{P8}:\max_{\mathbf{V}}$
$\displaystyle\sum_{i=1}^{T}E(K,\gamma_{t_{i}}^{0})\bigg{[}2x_{t_{i}}{\rm
Re}\\{\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}\\}$ $\displaystyle-
x_{t_{i}}^{2}\bigg{(}F(K,\Gamma_{t_{i}}^{0})|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}+G(K,\Gamma_{t_{i}}^{0})$
$\displaystyle\big{(}\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{t_{i}}^{2}\big{)}\bigg{)}\bigg{]},$
(32a) $\displaystyle{\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle(\ref{eq: qos constraints after fractional programming}).$

It can be observed that $\mathbf{P8}$ is actually an inhomogeneous and
separable QCQP problem, which can be solved by convex optimization toolboxes
like CVX in MATLAB. However, the complexity of CVX is still unbearable since
$\mathbf{P8}$ needs to be solved in each iteration. Therefore, we derive a
semi-closed form solution for $\mathbf{P8}$ and propose a computationally
efficient fixed point algorithm to search for the Lagrangian multipliers.
Formally, the Lagrangian of $\mathbf{P8}$ is given by

$\displaystyle\mathcal{L}(\mathbf{V},\boldsymbol{\lambda})=$
$\displaystyle\sum_{i=1}^{T}E(K,\gamma_{t_{i}}^{0})\bigg{[}2x_{t_{i}}{\rm
Re}\\{\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}\\}$ $\displaystyle-
x_{t_{i}}^{2}\bigg{(}F(K,\Gamma_{t_{i}}^{0})|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{t_{i}}|^{2}+G(K,\Gamma_{t_{i}}^{0})$
$\displaystyle\cdot\big{(}\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}|\mathbf{h}_{t_{i}}^{H}\mathbf{v}_{j}|^{2}+\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{t_{i}}^{2}\big{)}\bigg{)}\bigg{]}$
$\displaystyle+\sum_{i=1}^{B}\lambda_{i}\bigg{[}\frac{M_{K}}{L}R_{b_{i},1}^{\prime\prime}(\mathbf{V},y_{i},m_{i})$
$\displaystyle+\frac{L-M_{K}}{L}R_{b_{i},2}^{\prime\prime}(\mathbf{V},z_{i},n_{i})-\beta_{b_{i}}\bigg{]},$
(33)

where $\boldsymbol{\lambda}=[\lambda_{1},...,\lambda_{B}]$ is the vector
composed of multiple non-negative Lagrange multipliers.

By taking the first-order derivative of $\mathcal{L}$ over the precoding
vectors (i.e., $\\{\mathbf{v}_{t_{i}}\\}_{i=1}^{T}$,
$\\{\mathbf{v}_{b_{i}}\\}_{i=1}^{B}$) and setting it to zero, we have the
following proposition.

######  Proposition 1.

_For the MU-MISO system with channel
$\\{\\{\mathbf{h}_{b_{i}}\\}_{i=1}^{B},\\{\mathbf{h}_{t_{i}}\\}_{i=1}^{T}\\}$,
the optimal solution of the problem $\mathbf{P8}$ is given by_

$\displaystyle\mathbf{v}_{b_{i}}$
$\displaystyle=\underbrace{\lambda_{i}\frac{M_{K}m_{i}\sqrt{1+y_{i}}+(L-M_{K})n_{i}\sqrt{1+z_{i}}}{L}}_{\rho_{b_{i}}}\mathbf{A}^{-1}\mathbf{h}_{b_{i}},$
(34) $\displaystyle{\rm where}\leavevmode\nobreak\
\mathbf{A}=\frac{\sum_{j=1}^{B}\mu_{j}\sigma_{b_{j}}^{2}+\sum_{j=1}^{T}\nu_{j}\sigma_{t_{j}}^{2}}{P_{T}}\mathbf{I}+\sum_{j=1}^{B}\mu_{j}\mathbf{h}_{b_{j}}\mathbf{h}_{b_{j}}^{H}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{j=1}^{T}\nu_{j}\mathbf{h}_{t_{j}}\mathbf{h}_{t_{j}}^{H},$
(35) $\displaystyle\mathbf{v}_{t_{i}}$
$\displaystyle=\underbrace{x_{i}E(K,\Gamma_{t_{i}}^{0})}_{\varsigma_{t_{i}}}\mathbf{B}_{t_{i}}^{-1}\mathbf{h}_{t_{i}},\vspace{-10mm}$
(36) $\displaystyle{\rm where}\leavevmode\nobreak\
\mathbf{B}_{t_{i}}=\frac{\sum_{j=1}^{B}\mu_{j}\sigma_{b_{j}}^{2}+\sum_{j=1}^{T}\nu_{j}\sigma_{t_{j}}^{2}}{P_{T}}\mathbf{I}$
$\displaystyle+x_{i}^{2}E(K,\Gamma_{t_{i}}^{0})F(K,\Gamma_{t_{i}}^{0})\mathbf{h}_{t_{i}}\mathbf{h}_{t_{i}}^{H}+\sum_{j=1}^{B}\frac{\lambda_{j}M_{K}}{L}m_{j}^{2}\mathbf{h}_{b_{j}}\mathbf{h}_{b_{j}}^{H}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{j=1,j\neq
t_{i}}^{T}\nu_{j}\mathbf{h}_{t_{j}}\mathbf{h}_{t_{j}}^{H},$ (37)

_where
$\mu_{j}=\frac{m_{j}^{2}\lambda_{j}M_{K}+n_{j}^{2}\lambda_{j}(L-M_{K})}{L}$,
$\nu_{j}=x_{j}^{2}E(K,\Gamma_{t_{j}}^{0})G(K,\Gamma_{t_{j}}^{0})$. _

######  Remark 2.

_(Optimal Beamforming Structure) As shown in Proposition 1, it can be observed
that the precoding vectors of both bit-users and sem-users are linear
transformations of their corresponding channel vectors, where the weight
coefficients is divided into linear power allocation coefficients (i.e.,
$\rho_{b_{i}}$ and $\varsigma_{t_{i}}$ in (34) and (36)) and priority
coefficients for interference suppression (i.e., $\mu_{j}$, $\nu_{j}$,
$\frac{\lambda_{j}M_{K}}{L}m_{j}^{2}$ in $\mathbf{A}$ and
$\\{\mathbf{B}_{t_{i}}\\}_{i=1}^{T}$). Comparing (34) and (36), we can find
that $\mathbf{v}_{t_{i}}$and $\mathbf{v}_{b_{i}}$ have different weight
coefficients, indicating that sem-users have different resource allocation
strategies compared to traditional digital communication due to their
different objective functions. Furthermore, according to Proposition 1, the
optimal precoding vector of $\mathbf{P8}$ is only determined by
$\\{\lambda_{i}\\}_{i=1}^{B}$, so we can turn to find the optimal
$\\{\lambda_{i}\\}_{i=1}^{B}$ for solving $\mathbf{P8}$, thereby reducing
computational complexity. _

######  Remark 3.

_(Computation Complexity Analysis) For the calculation of
$\\{\mathbf{v}_{b_{i}}\\}_{i=1}^{B}$, it can be observed that the inverse
matrix is shared among bit-users, thus only needs to be calculated once. The
complexity for calculating $\\{\mathbf{v}_{b_{i}}\\}_{i=1}^{B}$ is given by
$\mathcal{O}(BN_{t}^{2}+TN_{t}^{2}+N_{t}^{3})$. For the calculation of
$\\{\mathbf{v}_{t_{i}}\\}_{i=1}^{T}$, according to the Sherman–Morrison
formula, we have
$\mathbf{B}_{t_{i}}=(\mathbf{C}-\omega_{t_{i}}\mathbf{h_{t_{i}}\mathbf{h}_{t_{i}}^{H}})^{-1}=\mathbf{C}^{-1}+\omega_{t_{i}}\frac{\mathbf{C}^{-1}\mathbf{h}_{t_{i}}\mathbf{h}_{t_{i}}^{H}\mathbf{C}^{-1}}{1-\omega_{t_{i}}\mathbf{h}_{t_{i}}^{H}\mathbf{C}^{-1}\mathbf{h}_{t_{i}}}$,
where $\mathbf{C}$ is defined in (38), and
$\omega_{t_{i}}=\nu_{i}-x_{i}^{2}E(K,\Gamma_{t_{i}}^{0})F(K,\Gamma_{t_{i}}^{0})$.
Therefore, the complexity for calculating $\\{\mathbf{v}_{t_{i}}\\}_{i=1}^{T}$
is given by $\mathcal{O}(BN_{t}^{2}+TN_{t}^{2}+N_{t}^{3})$. In a nutshell, the
overall complexity for calculating the beamforming vectors by (34) and (36) is
$\mathcal{O}(BN_{t}^{2}+TN_{t}^{2}+N_{t}^{3})$. _

Algorithm 2 The proposed method for solving $\mathbf{P8}$

0:
$\boldsymbol{\lambda}=[\lambda_{1},...,\lambda_{B}]=0.01\times\mathbf{1},\boldsymbol{\lambda}^{\prime}=\mathbf{0}$.

1: repeat

2: Update $\boldsymbol{\lambda}^{\prime}$ by
$\boldsymbol{\lambda}^{\prime}\leftarrow\boldsymbol{\lambda}$.

3: Update precoding matrix $\mathbf{V}$ by (34), (36), and
$\\{\lambda_{i}\\}_{i=1}^{B}$.

4: Update $\\{\lambda_{i}\\}_{i=1}^{B}$ by (39).

5: until $\sum_{i=1}^{B}|\lambda_{i}-\lambda_{i}^{\prime}|\leq\xi$

5: the precoding matrix $\mathbf{V}$.

Algorithm 3 The proposed MM-FP algorithm for solving $\mathbf{P2}$

0: the precoding matrix $\mathbf{V}$.

1: repeat

2: Update the SINR related terms, i.e.,
$\Gamma_{t_{i},0}\leftarrow\Gamma_{t_{i}}$, $\mathbf{y}$ with (23),
$\mathbf{z}$ with (24).

3: Update the ratio related terms, i.e., $\mathbf{x}$ with (29), $\mathbf{m}$
with (30), $\mathbf{n}$ with (31).

4: Update the beamforming matrixes $\mathbf{V}$ by solving $\mathbf{P8}$ with
Algorithm 2.

5: until the objective value of $\mathbf{P2}$ converges or the iteration
number reaches the maximum.

6: Normalize the precoding matrix, i.e.,
$\mathbf{V}\leftarrow\sqrt{\frac{P_{T}}{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}}\mathbf{V}$.

$\displaystyle\mathbf{C}$
$\displaystyle=\frac{\sum_{j=1}^{B}\mu_{j}\sigma_{b_{j}}^{2}+\sum_{j=1}^{T}\nu_{j}\sigma_{t_{j}^{2}}}{P_{T}}\mathbf{I}+\sum_{j=1}^{B}\frac{\lambda_{j}M_{K}}{L}m_{j}^{2}\mathbf{h}_{b_{j}}\mathbf{h}_{b_{j}}^{H}+\sum_{j=1}^{T}\nu_{j}\mathbf{h}_{t_{j}}\mathbf{h}_{t_{j}}^{H},$
(38) $\displaystyle\lambda_{i}$
$\displaystyle=\frac{{\beta_{b_{i}}-\frac{M_{K}}{L}(\log(1+y_{i})-y_{i}-m_{i}^{2}\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2})-\frac{L-M_{K}}{L}(\log(1+z_{i})-z_{i}-n_{i}^{2}\frac{{\rm
Tr}(\mathbf{V}\mathbf{V}^{H})}{P_{T}}\sigma_{b_{i}}^{2})}}{{2(\frac{M_{K}}{L}m_{i}\sqrt{1+y_{i}}+\frac{L-M_{K}}{L}n_{i}\sqrt{1+z_{i}})^{2}}{\rm
Re}\\{\mathbf{h}_{b_{i}}^{H}\mathbf{A}^{-1}\mathbf{h}_{b_{i}}\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{\frac{M_{K}}{L}m_{i}^{2}\big{(}\sum_{j\in\\{\mathcal{B,T}\\}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}\big{)}+\frac{L-M_{K}}{L}n_{i}^{2}\big{(}\sum_{j\in\\{\mathcal{B}\\}}|\mathbf{h}_{b_{i}}^{H}\mathbf{v}_{j}|^{2}\big{)}}{{2(\frac{M_{K}}{L}m_{i}\sqrt{1+y_{i}}+\frac{L-M_{K}}{L}n_{i}\sqrt{1+z_{i}})^{2}}{\rm
Re}\\{\mathbf{h}_{b_{i}}^{H}\mathbf{A}^{-1}\mathbf{h}_{b_{i}}\\}}.$ (39)

As discussed in Remark 2, to obtain the optimal solution of $\mathbf{P8}$,
only the Lagrange multipliers $\\{\lambda_{i}^{*}\\}_{i=1}^{B}$ needs to be
determined, where $\lambda_{i}^{*}$ denotes the optimal dual variable. In
addition, it can be found that the optimal solution of $\mathbf{P8}$ should
satisfy the QoS constraints with equality. As a result, we can obtain
$\\{\lambda_{i}^{*}\\}_{i=1}^{B}$ by the fixed-point algorithm. The update
rule is presented in (39). The solution process for $\mathbf{P8}$ is concluded
in Algorithm 2.

So far, the problem $\mathbf{P2}$ has been solved in an alternating manner,
which is summarized in Algorithm 3 and termed as majorization-minimization
fractional programming (MM-FP). There are two groups of introduced variables,
namely SINR related terms ($\\{\gamma_{t_{i},0}\\}_{i=1}^{T}$, $\mathbf{y}$,
$\mathbf{z}$) and the fraction related terms ($\mathbf{x}$, $\mathbf{m}$,
$\mathbf{n}$). As shown in (29)-(31), the two groups are updated in sequence,
followed by the update of $\mathbf{V}$.

In addition, we propose a low-complexity beamforming method to address
practicality concerns. Generally, beamforming involves determining the
precoding direction and allocating power. As shown in (34) and (36), the
precoding direction is determined by the channel vectors,
$\\{m_{i}\\}_{I=1}^{B}$, $\\{n_{i}\\}_{I=1}^{B}$, $\\{x_{i}\\}_{I=1}^{B}$, and
$\\{\lambda_{i}\\}_{I=1}^{B}$. Given this, we consider setting a uniform
$\lambda_{i}$ for all bit-user $b_{i}$ for avoiding iteration, and
approximating the value of $\\{m_{i}\\}_{I=1}^{B}$, $\\{n_{i}\\}_{I=1}^{B}$,
$\\{x_{i}\\}_{I=1}^{B}$ by (29), (30), (31), where the precoding vectors are
replaced by the corresponding channel vector. Consequently, the beamforming
direction, denoted by
$\\{\widetilde{v}_{b_{i}}\\}_{I=1}^{B},\\{\widetilde{v}_{t_{i}}\\}_{I=1}^{T}$,
is established. Then we allocate the power by solving the following problem.

$\displaystyle\mathbf{P9}:$
$\displaystyle\max_{\mathbf{p}_{T},\mathbf{p}_{B}}\sum_{i=1}^{T}a_{K}+\frac{b_{K}}{c_{K}+(\frac{p_{t_{i}}|\mathbf{h}_{t_{i}}^{H}\widetilde{\mathbf{v}}_{t_{i}}|^{2}}{(\sum_{j\in\\{\mathcal{B,T}\\}/t_{i}}p_{j}|\mathbf{h}_{t_{i}}^{H}\widetilde{\mathbf{v}}_{j}|^{2}+\sigma_{t_{i}}^{2})})^{-\alpha_{K}}},$
(40a) $\displaystyle{\rm
s.t.}\frac{M_{K}}{L}\log_{2}\left(1+\frac{p_{b_{i}}\left|\mathbf{h}_{b_{i}}^{H}\widetilde{\mathbf{v}}_{b_{i}}\right|^{2}}{\sum_{j\in\left\\{\mathcal{B},\mathcal{T}\right\\}/b_{i}}p_{j}\left|\mathbf{h}_{b_{i}}^{H}\widetilde{\mathbf{v}}_{j}\right|^{2}+\sigma_{b_{i}}^{2}}\right)+\frac{L-M_{K}}{L}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\cdot\log_{2}\left(1+\frac{p_{b_{i}}\left|\mathbf{h}_{b_{i}}^{H}\widetilde{\mathbf{v}}_{b_{i}}\right|^{2}}{\sum_{j\in\left\\{\mathcal{B}\right\\}/b_{i}}p_{j}\left|\mathbf{h}_{b_{i}}^{H}\widetilde{\mathbf{v}}_{j}\right|^{2}+\sigma_{b_{i}}^{2}}\right)\geq\beta_{b_{i}},\forall
b_{i},$ (40b) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{i=1}^{B}p_{b_{i}}+\sum_{j=1}^{T}p_{t_{j}}\leq P_{T}.$ (40c)

where $\mathbf{p}_{T}=[p_{t_{1}},...,p_{t_{T}}]$ and
$\mathbf{p}_{B}=[p_{b_{1}},...,p_{b_{B}}]$ denote the power allocation vectors
of sem-users and bit-users, respectively.

The low-complexity majorization-minimization fractional programming (LP-MM-FP)
algorithm is concluded in Algorithm 4.

Algorithm 4 The proposed LP-MM-FP algorithm for solving $\mathbf{P2}$

1: $\lambda_{i}\leftarrow\lambda_{0}$, $i=1,...,B$.

2: Calculate $\mathbf{m}$, $\mathbf{n}$, $\mathbf{x}$
,$\\{\Gamma_{t_{i}}\\}_{i=1}^{T}$ by channel vectors and MRT precoding vectors
(i.e.,
$\mathbf{v}_{t_{i}}=\frac{\mathbf{h}_{t_{i}}}{\|\mathbf{h}_{t_{i}}\|},\mathbf{v}_{b_{i}}=\frac{\mathbf{h}_{b_{i}}}{\|\mathbf{h}_{t_{i}}\|}$).

3: Calculate the beamforming directions with $\\{\lambda_{i}\\}_{i=1}^{B}$ and
$\\{\Gamma_{t_{i}}\\}_{i=1}^{T}$, i.e.,
$\widetilde{\mathbf{v}}_{t_{i}}=\frac{\mathbf{B}_{t_{i}}^{-1}\mathbf{h}_{t_{i}}}{\|\mathbf{B}_{t_{i}}^{-1}\mathbf{h}_{t_{i}}\|}$,
$\widetilde{\mathbf{v}}_{t_{i}}=\frac{\mathbf{A}^{-1}\mathbf{b}_{t_{i}}}{\|\mathbf{A}^{-1}\mathbf{h}_{b_{i}}\|}$,
where $\mathbf{A}$, $\mathbf{B}_{t_{i}}$ are given by (35) and (37),
respectively.

4: Calculate power allocation vectors $\mathbf{p}_{T}$ and $\mathbf{p}_{B}$ by
solving problem $\mathbf{P9}$ with
$\\{\widetilde{\mathbf{v}}_{t_{i}}\\}_{i=1}^{T}$ and
$\\{\widetilde{\mathbf{v}}_{b_{i}}\\}_{i=1}^{B}$.

5: Calculate the final precoding vectors by
$\mathbf{v}_{t_{i}}=\sqrt{p_{t_{i}}}\widetilde{\mathbf{v}}_{t_{i}}$,
$\mathbf{v}_{b_{i}}=\sqrt{p_{b_{i}}}\widetilde{\mathbf{v}}_{b_{i}}$.

Algorithm 5 The proposed beamforming algorithm for solving $\mathbf{P1}$

0: The sample set $\mathcal{K}=\\{K|K_{\rm min}\leq K\leq K_{\rm
max},K\in\mathbb{Z}\\}$

1: for $K=K_{\min},...,K_{\rm max}$ in parallel do

2: Obtain $\mathbf{V}_{K}$ through Algorihtm 3 with $K$.

3: Calculate the objective value of problem denoted by $\mathbf{P1}$
$\tau(\mathbf{V}_{K},K)$ with $\mathbf{V}_{K}$ and $K$.

4: end for

5: $K_{\rm opt}=\arg\max_{K\in\mathcal{K}}{\tau(\mathbf{V}_{K},K)}$

5: The beamforming matrix $\mathbf{V}_{K_{\rm opt}}$, $K_{\rm opt}$.

###  IV-B Overall Algorithm

In this subsection, we shall present the proposed method for solving the
problem $\mathbf{P1}$. Firstly, the subproblem that optimizes the beamforming
vector with a fixed $K$ has been tackled in a computation-efficient manner.
Subsequently, recognising that the feasible set for $K$ (i.e., the
downsampling depth) is typically constrained within a narrow integer range, we
employ the exhaustive algorithm to identify the optimal $K$.

The overall algorithm is presented in Algorithm 5. It can be found that
executing Algorithm 5 requires at most $K_{\rm max}-K_{\rm min}$ times the
complexity of Algorithm 3, where $K_{\rm max}$ and $K_{\rm min}$ represent the
maximum and minimum feasible values of $K$, respectively. For conducting
Algorithm 3, it requires multiple iterations, each of which is divided into
three steps. The first step includes the update of SINR related terms, with a
complexity of $\mathcal{O}(BN_{t}+TN_{t})$; the second step includes the
updates of the ratio related terms, with a complexity of
$\mathcal{O}(BN_{t}+TN_{t})$; the third step includes the updates of precoding
matrix, with a complexity of
$\mathcal{O}(L_{2}(BN_{t}^{2}+TN_{t}^{2}+N_{t}^{3}))$, where $L_{2}$ denotes
the the number of iteration rounds for the fixed point method. Therefore, the
complexity for conducting Algorithm 3 is
$\mathcal{O}(L_{1}L_{2}BN_{t}^{2}+L_{1}L_{2}TN_{t}^{2}+L_{1}N_{t}^{3})$, where
$L_{1}$ denotes the iteration number of Algorithm 3. In conclusion, the
complexity for conducting Algorithm 5 is given by $\mathcal{O}((K_{\rm
max}-K_{\rm min})L_{1}L_{2}BN_{t}^{2}+(K_{\rm max}-K_{\rm
min})L_{1}L_{2}TN_{t}^{2}+(K_{\rm max}-K_{\rm min})L_{1}N_{t}^{3})$.

##  V Numerical Results

###  V-A Simulation Setup

(a) Low SNR case (${\rm SNR}=0$ dB)

(b) High SNR case (${\rm SNR}=5$ dB)

Figure 6: Performance comparison of transmission systems

System setup. We consider the clustered Saleh-Valenzuela channel model [38],
in which the channel from the BS to a specific user $i$ is given as follows.

$\displaystyle\mathbf{h}_{i}=\frac{1}{\sqrt{L_{p}}}\sum_{l=1}^{L_{p}}\delta_{i,l}\mathbf{a}_{T}(\theta_{i,l}),$
(41)

where $L_{p}$ is the number of paths and $\delta_{i,l}\sim\mathcal{CN}(0,1)$
is the channel attenuation of the $l$-th path. Without loss of generality, we
set $L_{p}$ to $10$. $\theta_{i,l}$ denotes the azimuth angle of departure
(AoD) at the transmitter, and we assume $\theta_{i,l}$ follows a uniform
distribution from $0$ to $2\pi$. The response vector of Uniform Linear Array
(ULA) at the BS side can be expressed as

$\displaystyle\mathbf{a}_{T}(\theta_{i,l})=[1,e^{-j\pi\sin(\theta_{i,l})},...,e^{-j\pi(N_{t}-1)\sin(\theta_{i,l})}].$
(42)

For the MISO system setting, unless specified, the following system parameters
will be used as the default setting in the experiments: $N_{t}=16$, $B=5$,
$T=3$, $K=3$, ${\rm SNR}=0$ dB, $\beta_{i}=1,\forall i$.444It is worth noting
that the proposed method should be also adaptable to other configurations.
Besides, as mentioned in Section III-B, the ImageNet dataset and SSIM are used
as the training dataset and performance metric, respectively. The images are
resized to $128\times 128$. The length of frame $L$ and the number of filters
$C$ are set to $32,768$ and $128$, respectively. The downsampling depth
$K\in\\{K|2\leq K\leq 6,K\in\mathbb{Z}\\}$. We also conduct performance
evaluation on the Kodak image dataset, which comprises of 24 high-quality
images.

Benchmark schemes. We compare the proposed beamforming algorithm, MM-FP and
LP-MM-FP, with three commonly-adopted beamforming schemes, including the zero
focing (ZF) algorithm, maximum ratio transmission (MRT) algorithm, and
weighted minimum mean-square error (WMMSE) algorithm. Note that the
aforementioned algorithms cannot be directly used for solving $\mathbf{P1}$,
as the QoS constraints may not be satisfied. Given this, we first obtain the
beamforming direction through these algorithms, i.e.,
$\widetilde{\mathbf{v}}_{i}={\mathbf{v}_{i}}/{\|\mathbf{v}_{i}\|_{2}},\forall
i\in\mathcal{B}\cup\mathcal{T}$. Then we reallocate the power by solving the
problem $\mathbf{P9}$, and the final beamforming vector is given by
$\mathbf{v}_{i}=\sqrt{p_{i}}\widetilde{\mathbf{v}}_{i},\forall
i\in\mathcal{B}\cup\mathcal{T}$. The resulting benchmark schemes are named ZF-
PC, MRT-PC, and WMMSE-PC respectively.

###  V-B Evaluation of Coexisting System

In this subsection, we evaluate the effectiveness of the semantic-bit
coexisting system. Firstly, we compare this system with the BitCom system. The
semantic-bit coexisting system utilizes JSCC with neural network for image
transmission, while the BitCom system enables image transmission through
standard modular design, i.e., BPG for source coding, Turbo for channel
coding, and 64QAM for modulation. The two different systems result in two sets
of $\\{a_{K},b_{K},c_{K},\alpha_{K}\\}_{K=1}^{L}$, and we conduct Algorithm 3
for beamforming under the two parameter sets. The results are presented in
Fig. 6, where Fig. 6(a) shows the performance in low SNR case (SNR = $0$dB),
and Fig. 6(b) in high SNR case (SNR = $5$dB). In the low SNR case, the BitCom
system fails to work under any of the examined QoS requirement. This is
because BitCom is sensitive to noise. In the high SNR case, with a low QoS
requirement, sem-users in the BitCom system enjoy low interference from bit-
users, and thanks to the channel coding, the receiver is able to perfectly
decode the BPG bit flow and attains good performance when $\beta_{i}\leq 0.5$.
With the increase of QoS requirements, the strong interference causes the
performance of BitCom to degrade quickly. In the meantime, the semantic-bit
coexisting system only experiences a slight performance degradation as the QoS
requirement increases from 0 to 1.5, thus demonstrating its effectiveness.

Figure 7: Validation of Semantic Rate Approximation

Since data driven method is used to approximate the semantic rate, it is
important to compare the real semantic rate with the approximated one. We
first implement the proposed beamforming scheme in Algorithm 3 and then
compare the image recovery quality (i.e., SSIM) and the objective value in
$\mathbf{P1}$. The performance comparison is presented in Fig. 7, where three
different $K$ settings are considered. The final performance is averaged over
$2,000$ test samples. As shown in Fig. 7, the approximation and simulation
curves almost overlap, and the approximating-based method can well capture the
performance growth trend as SNR increases. This validates the effectiveness of
the data driven method in accurately approximating the semantic rate.

(a) ${\rm SNR=0}$ dB

(b) $\beta_{i}=1,\forall i=1,...,B$

Figure 8: Performance comparison of different beamforming schemes

###  V-C Performance of Beamforming Design

In this subsection, we compare the performance of the proposed beamforming
algorithm with three benchmark schemes. Fig. 8 depicts the semantic
performance using different beamforming schemes in the coexisting system. We
evaluate performance across different SNR and QoS settings. Under different
QoS settings, Fig. 8(a) shows that heuristic beamforming schemes, such as ZF-
PC and MRT-PC, perform poorly since they fail to coordinate beamforming
direction and power for the problem $\mathbf{P2}$. The optimization-induced
method WMMSE-PC achieves better performance than ZF-PC and MRT-PC. However,
WMMSE-PC fails to consider the semantic objective in Fig. 4 and (9), which has
a different mapping relationship between SNR and performance. As a result, the
performance of WMMSE-PC degrades significantly when QoS requirements increase,
which implies that tailored design of beamforming for coexisting systems is
required. The proposed beamforming algorithm outperforms the three benchmark
schemes in all QoS settings, achieving the best performance given all the
examined QoS requirements. Moreover, the proposed LP-MM-FP algorithm achieves
near performance with MM-FP algorithm especially in low QoS regime, while with
much lower computational complexity. We also present some test examples in
Fig. 10, where $\beta_{i}$ is set to 0.8. The recovered image from the system
that adopts the ZF-PC or MRT-PC beamforming schemes has an obvious blur, which
is also reflected in SSIM performance. The system with the WMMSE-PC algorithm
has relatively more noise points in the first and third image. The system with
the proposed beamforming scheme recovers the first and second image clearly,
with some blurs in the third image, yet still outperforms the other three
benchmark schemes in terms of SSIM.

Fig. 8(b) illustrates the performance comparison across different SNR
settings. ZF-PC performs poorly in the low SNR regime, although it can
approach the performance upper bound like WMMSE-PC and the proposed method
when ${\rm SNR\geq 6}$ dB. MRT-PC performs relatively well in the low SNR
regime, but the performance quickly degrades compared with other schemes as
SNR increases since it does not consider user interference for beamforming
design. Similarly, the proposed scheme outperforms these benchmark schemes in
all the examined SNR settings, particularly in the low SNR regime,
demonstrating its robustness. We present some test examples in Fig. 11, and
the recovery performance is consistent with the analytical results in Fig.
8(b). The system that adopts the proposed beamforming scheme achieves the best
SSIM performance in all three recovered images.

###  V-D Evaluation of $K$ Configuring Strategies

Figure 9: Performance comparison of different $K$-setting

This subsection evaluates the effectiveness of different methods for the
configuration of $K$ in a typical loaded scenario, i.e.,
$N_{t}=16,\leavevmode\nobreak\ B=14,\leavevmode\nobreak\ T=1$. We compare the
exhaustive search against two benchmark schemes: Random, which randomly
selects a value of $K$ from $1$ to $L$; the best fixed $K$ setting, which we
found to be the minimum value $K=2$ based on numerical experiments.We present
the performance comparison in Fig. 9, where we use $\Delta\beta_{i}$ to denote
the gap between the currently selected QoS value and the maximum achievable
QoS, and a smaller $\Delta\beta_{i}$ indicates a more stringent QoS
requirement. We observe that the performance of all schemes improves as
$\Delta\beta_{i}$ increases, with the Random algorithm performing noticeably
worse than the other three schemes. The fixed $K$ algorithm has already
achives satisfactory performance in the transmission task considered in this
paper, which can be adopted in the scenarios with limited computation
capability. The considered method optimizes $K$ through the exhaustive method
and can achieves the best performance.

Original | ZF-PC | MRT-PC | WMMSE-PC | MM-FP
---|---|---|---|---
|  |  |  |
SSIM | 0.553 | 0.699 | 0.750 | 0.803
|  |  |  |
SSIM | 0.717 | 0.642 | 0.766 | 0.851

Figure 10: Examples of reconstructed images under ${\rm SNR}=0$ dB,
$\beta_{i}=0.8,\forall i=1,...,B$.

##  VI Conclusion

In this paper, we considered a semantic-user and bit-user coexisting system. A
beamforming problem that maximizes the semantic rate under QoS constraints
from bit-users and power constraint was formulated and solved in an low-
complexity manner. Experiments show that the proposed method significantly
improves the existing beamforming methods dedicated for BitCom. Addressing
issues beyond beamforming in the coexisting system remains an interesting
future direction.

Original | ZF-PC | MRT-PC | WMMSE-PC | MM-FP
---|---|---|---|---
|  |  |  |
SSIM | 0.757 | 0.691 | 0.789 | 0.848
|  |  |  |
SSIM | 0.781 | 0.475 | 0.827 | 0.844

Figure 11: Examples of reconstructed images under SNR=3dB,
$\beta_{b_{i}}=1.0,\forall i=1,...,B$.

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