# Deep Learning for 1-Bit Compressed Sensing-based Superimposed CSI Feedback

Chaojin Qing1*, Qing Ye1, Bin Cai1, Wenhui Liu1, Jiafan Wang2*,

1 School of Electrical Engineering and Electronic Information, Xihua
University, Chengdu, China
2 Synopsys Inc., 2025 NE Cornelius Pass Rd, Hillsboro, OR 97124, USA
*<EMAIL_ADDRESS>(C. Qing); jifanw@gmail.com(J. Wang) This work is supported in part by the Sichuan Science and Technology Program (Grant No. 2021JDRC0003), the Major Special Funds of Science and Technology of Sichuan Science and Technology Plan Project (Grant No. 19ZDZX0016 /2019YFG0395), the Demonstration Project of Chengdu Major Science and Technology Application (Grant No. 2020-YF09- 00048-SN), the Key Scientific Research Fund of Xihua University (Grant No. Z1120941), and the Special Funds of Industry Development of Sichuan Province (Grant No. zyf-2018-056).

###### Abstract

In frequency-division duplexing (FDD) massive multiple-input multiple-output
(MIMO) systems, 1-bit compressed sensing (CS)-based superimposed channel state
information (CSI) feedback has shown many advantages, while still faces many
challenges, such as low accuracy of the downlink CSI recovery and large
processing delays. To overcome these drawbacks, this paper proposes a deep
learning (DL) scheme to improve the 1-bit compressed sensing-based
superimposed CSI feedback. On the user side, the downlink CSI is compressed
with the 1-bit CS technique, superimposed on the uplink user data sequences
(UL-US), and then sent back to the base station (BS). At the BS, based on the
model-driven approach and assisted by the superimposition-interference
cancellation technology, a multi-task detection network is first constructed
for detecting both the UL-US and downlink CSI. In particular, this detection
network is jointly trained to detect the UL-US and downlink CSI
simultaneously, capturing a globally optimized network parameter. Then, with
the recovered bits for the downlink CSI, a lightweight reconstruction scheme,
which consists of an initial feature extraction of the downlink CSI with the
simplified traditional method and a single hidden layer network, is utilized
to reconstruct the downlink CSI with low processing delay. Compared with the
1-bit CS-based superimposed CSI feedback scheme, the proposed scheme improves
the recovery accuracy of the UL-US and downlink CSI with lower processing
delay and possesses robustness against parameter variations.

###### Index Terms:

Massive multiple-input multiple-output (MIMO), deep learning (DL), channel
state information (CSI), superimposed CSI feedback, 1-bit compressed sensing
(CS)

##  I Introduction

Massive multiple-input multiple-output (MIMO) has become the key technology of
the fifth generation (5G) wireless communication system, due to its advantages
in system capacity and link robustness [1] [2], etc. As premises of these
advantages, the base station (BS) needs to obtain accurate downlink channel
state information (CSI), and rely on downlink CSI for precoding [3], antenna
selection [4], radio resource allocation [5], and communication interference
management [6], etc. In time division duplex (TDD) mode, the downlink CSI can
be obtained from uplink CSI by exploiting channel reciprocity [7] [8]. For
frequency-division duplex (FDD) mode, it is difficult to develop the channel
reciprocity due to the different frequency bands used by uplink and downlink
[9][10]. Thus, the downlink CSI is usually estimated by users and fed back to
the BS in FDD massive MIMO system [9]. However, due to a large number of
antennas in massive MIMO systems, CSI feedback incurs significant feedback
overhead, resulting in serious uplink bandwidth occupation.

To reduce feedback overhead, lots of compressive sensing (CS)-based CSI
feedback methods have emerged [11, 12, 13, 14]. In recent years, deep learning
(DL)-based CSI feedback methods [15, 16, 17] are proposed to further reduce
feedback overhead. Although the feedback overhead is reduced to some extent,
both CS-based CSI feedback and DL-based CSI feedback still occupy significant
uplink bandwidth resources. To avoid the occupation of uplink bandwidth
resources, the superimposed CSI feedback was proposed in [18], yet causes
mutual interference due to superimposition operation. In [10], [19], and [20],
the 1-bit CS-based, DL-based, and extreme learning machine (ELM)-based
superimposed CSI feedbacks are respectively proposed to reduce this mutual
interference. Inspired by the advantages of superimposed CSI feedback based on
1-bit CS and DL, we propose a DL-based 1-bit superimposed CSI feedback scheme
in this paper.

###  I-A Related Works

In FDD massive MIMO system, the DL-based CSI feedback methods have been
investigated according to the superimposed CSI feedback, e.g., [10], [19], and
[20], and feedback reduction, e.g., [21, 22, 23, 24, 25, 26], etc.

For reducing feedback overhead, the DL-based data-driven CSI feedback can be
divided into two categories. The first category is mainly based on the
combination of CS technique and DL technique, while the other category employs
the DL technique for the quantized data. In the first category, [21] is the
first application of DL for CSI feedback. In [21], the CSI feedback was mainly
based on a convolutional neural network called CsiNet, which achieved superior
performance over various CS-based CSI feedbacks. Yet, the time correlation,
frequency correlation, spatial correlation, feedback delay and feedback
errors, etc., were not considered in CsiNet, and led to limited applications.
To remedy these defects, some improvements have been proposed in [22, 23, 24].
In [22], a CsiNet long short-term memory (CsiNet-LSTM) was proposed by
exploiting the time correlation, which is suitable for practical application
in time-varying channels. The recurrent neural network-based CsiNet in [23]
was developed to capture the temporal and frequency correlations of wireless
channels. Considering the spatial correlation among antennas, the
bidirectional LSTM (Bi-LSTM) and bidirectional convolutional LSTM (Bi-
ConvLSTM) were proposed in [24]. Another category of feedback reduction
proposed for DL-based CSI feedback is mainly based on the quantization
operation, e.g., [25] and [26]. In [25], a bit-level CsiNet+ was proposed,
which made the current CSI feedback network applicable in real communication
systems and minimized the introduced quantitative distortion to improve the
reconstruction quality. By employing the quantization and entropy coding
blocks into a full convolution network, the work of [26] obtained drastic
improvement in CSI reconstruction quality at even extremely low feedback
rates. Although the DL-based CSI feedback in [21, 22, 23, 24, 25, 26] has
achieved significant improvements in feedback reduction compared with the CS-
based approaches, the uplink bandwidth resources were still seriously occupied
due to the massive MIMO scenarios.

To avoid the occupation of uplink bandwidth resources, superimposed CSI
feedback schemes were proposed in [18, 19, 20]. In [18], the downlink CSI was
spread and then superimposed on the uplink user data sequences (UL-US) as
feedback to the BS, while the recoveries of the UL-US and downlink CSI were
deteriorated by superimposition interference. To remedy this defect, a DL-
based superimposed CSI feedback was proposed in [19], and an ELM-based
superimposed CSI feedback with lower computational complexity was proposed in
[20]. Considering the simplicity and cost-effectiveness, a low-consumed CSI
feedback using 1-bit CS has been studied in [27], in which 1-bit operation
means to discard the signal amplitude and only retain its sign information. In
this work, the downlink CSI was quantified by 1-bit CS to achieve low-consumed
feedback, while this work still occupied uplink bandwidth resources. To remedy
this defect, the superimposed CSI feedback and 1-bit CS technique were
combined in [10] and presented many advantages, e.g., the avoidance of uplink-
bandwidth-resource occupation and the reduction of mutual interference, etc.
However, it is facing challenges in recovery accuracy and processing delay
[28], etc.

By integrating the promising advantage of deep learning and inspired by the
superimposed CSI feedback by using 1-bit CS in [10], we propose a DL-based
1-bit superimposed CSI feedback scheme in this paper. First, the downlink CSI
is compressed by the 1-bit CS technique and then superimposed on the UL-US as
feedback to the BS. At the BS, to recover the bit information for both the UL-
US and downlink CSI, a multi-task detection network with transmitted signal
feature extraction is first constructed. Then, with the recovered bits of the
downlink CSI, a lightweight reconstruction network, which consists of an
initial feature extraction of the downlink CSI with simplified traditional
method and a single hidden layer network, is utilized to reconstruct the
downlink CSI with a low processing delay. Specifically, the advantages of
superimposed CSI feedback by using 1-bit CS are inherited, i.e., without any
occupation of uplink bandwidth for CSI feedback, and effective interference
cancellation in [10], and the recovery accuracies for both the UL-US and
downlink CSI are improved.

###  I-B Contributions

In this paper, a DL-based 1-bit superimposed CSI feedback scheme is proposed
to improve the superimposed CSI feedback 1-bit CS approach in [10]. To the
best of our knowledge, there is a little literature focusing on the DL-based
1-bit superimposed CSI feedback method. And there is also no research on the
introduction of deep learning into 1-bit superimposed feedback. The main
contributions of this paper are as follows:

  * •

We propose the DL-based scheme for 1-bit CS-based superimposed CSI feedback.
By using the nonlinear mapping and feature extraction ability of the DL, we
develop a detection network and a reconstruction network to further suppress
nonlinear superimposition interference, and improve the detection and
reconstruction performances. The proposed scheme retains the advantages of
1-bit CS-based superimposed CSI feedback [10], while obtains better recovery
accuracy for both the UL-US and downlink CSI with much lower processing delay.

  * •

We construct a multi-task detection network to recover the bit information for
both the UL-US and downlink CSI, based on the model-driven approach and
assisted by the superimposition-interference cancellation technology. This
detection network is jointly trained to detect the UL-US and downlink CSI
simultaneously, capturing a globally optimized network parameter. We use the
ability that DL solve nonlinear problems to solve the superimposition
separation, which shortens processing delay while improving the detection
performance without any second-order statistical information about channel and
noise.

  * •

We develop a lightweight reconstruction network by using the linear
approximation ability of the traditional superimposed coding aided binary
iterative hard thresholding (SCA-BIHT) algorithm and the advantages of deep
learning to deal with nonlinear problems. In this network, the initial feature
of downlink CSI is extracted by SCA-BIHT algorithm with only a few iterations,
and then a single hidden layer refinement network is constructed to refine the
downlink CSI reconstruction. The reconstruction network not only greatly
reduces the iterations of the traditional SCA-BIHT algorithm to raise
efficiency, but also obtains a better reconstruction performance of the
downlink CSI with a lower processing delay.

The remainder of this paper is structured as follows: In Section II, we
introduce the system model of the 1-bit superimposed CSI feedback. The DL-
based 1-bit superimposed CSI feedback method is presented in Section III and
followed by numerical results in Section IV. Finally, Section V concludes our
work.

Notations: Boldface upper case and lower case letters denote matrix and vector
respectively. ${\left(\cdot\right)^{T}}$ and $\mathbf{(\cdot)^{\dagger}}$
denote transpose and matrix pseudo-inverse respectively. $\mathbf{I}_{P}$ is
the identity matrix of size $P\times P$.
${\mathrm{{\mathbb{BN}}}}\left(\cdot\right)$ denotes the operation of batch
normalization. ${\left\|\cdot\right\|_{2}}$ is the Euclidean norm.
${\rm{sign}}(\cdot)$ denotes an operator of taking symbolic information, e.g.,
the sign function returns +1 for positive numbers and 0 otherwise.
${\rm{Re}}(\cdot)$ and ${\rm{Im}}(\cdot)$ represent real and imaginary part
operations, respectively. $K(\mathbf{x})$ represents computing the best
$k$-term approximation of $\mathbf{x}$ by thresholding. $\odot$ denotes the
operation of Hadamard product for two vectors or matrices.

##  II System Model

The system model is shown in Fig 1. Considering a massive MIMO system that
consists of one BS with $N$ antennas and $U$ single-antenna users, after the
processing of matched-filter, the received signal from user-$u$,
$u=1,2,\ldots,U$, denoted as ${{\mathbf{R}}_{u}}$, is given as

$\displaystyle{{\mathbf{R}}_{u}}={{\mathbf{g}}_{u}}{{\mathbf{x}}_{u}}+{{\mathbf{N}}_{u}},$
(1)

where ${{\mathbf{g}}_{u}}\in{{\mathbb{C}}^{N\times 1}}$ denotes the uplink
channel vector from user-$u$ to the BS,
${{\mathbf{N}}_{u}}\in{{\mathbb{C}}^{N\times P}}$ is the circularly symmetric
complex Gaussian noise (CSCG) of feedback link, $P$ is the length of the UL-
US. To avoid occupying the limited and crowded uplink bandwidth resources
[29][30], $\mathbf{x}_{u}\in{{\mathbb{C}}^{1\times P}}$ adopts superimposition
technology, and denotes the transmitted signal of user-$u$, which is given by
[10]

$\displaystyle{{\mathbf{x}}_{u}}=\sqrt{{\rho{E_{u}}}}{{\mathbf{s}}_{u}}+\sqrt{(1-\rho){E_{u}}}{{\mathbf{d}}_{u}},$
(2)

where ${{\rho}}\in\left[0,1\right]$ is the power proportional coefficient of
the downlink CSI, $E_{u}$ is the transmitted power of user-$u$, and
${{\mathbf{s}}_{u}}\in{{\mathbb{C}}^{1\times P}}$ and
${{\mathbf{d}}_{u}}\in{{\mathbb{C}}^{1\times P}}$ stand for the modulated
superimposition signal and the UL-US, respectively.

Figure 1: System model.

In this paper, the downlink CSI, satisfying
${{\mathbf{h}}_{u}}\in{{\mathbb{C}}^{1\times N}}$, is a sparse vector with
$K$-sparsity [10], i.e., only $K$ non-zero elements in ${{\mathbf{h}}_{u}}$.
According to the 1-bit CS technique [31], ${{\mathbf{h}}_{u}}$ is compressed
by

$\displaystyle\left\\{{\begin{array}[]{*{20}{c}}{{{\mathbf{y}}_{{\mathrm{real,}}u}}=\mathrm{sign}({\mathop{\rm
Re}\nolimits}({{\mathbf{h}}_{u}}){{\mathbf{\Phi}}_{u}})}\\\
{{{\mathbf{y}}_{{\mathrm{imag,}}u}}=\mathrm{sign}({\mathop{\rm
Im}\nolimits}({{\mathbf{h}}_{u}}){{\mathbf{\Phi}}_{u}})}\end{array}}\right.,$
(5)

where ${{\mathbf{\Phi}}_{u}}\in{{\mathbb{R}}^{N\times M}}$ is the measurement
matrix [10], and ${{\mathbf{y}}_{{\mathrm{real,}}u}}\in{{\mathbb{R}}^{1\times
M}}$ and ${{\mathbf{y}}_{{\mathrm{imag,}}u}}\in{{\mathbb{R}}^{1\times M}}$
denote the real and imaginary parts of the compressed CSI, respectively.

For the convenience of digital modulation, the support-set of the downlink CSI
${{\mathbf{h}}_{u}}$, denoted as ${{\mathbf{z}}_{u}}\in{\\{0,1\\}^{1\times
N}}$, is labelled by the bit-form [10], i.e.,

$\displaystyle{z_{u,k}}=\left\\{\begin{array}[]{l}\;1,\;\;{h_{u,k}}\neq 0\\\
\,0,\;\;{h_{u,k}}=0\end{array}\right.\;,\;k=1,2,\ldots N,$ (8)

where ${z_{u,k}}$ and ${h_{u,k}}$ are the k-th element in ${\mathbf{z}_{u}}$
and ${\mathbf{h}_{u}}$, respectively. In order to reconstruct a more accurate
downlink CSI at the BS, ${\bf{z}}_{u}$ needs to be fed back to the BS with
${{\mathbf{y}}_{{\mathrm{real,}}u}}$ and ${{\mathbf{y}}_{{\mathrm{imag,}}u}}$
by using the feedback vector $\mathbf{p}_{u}$. The feedback vector
$\mathbf{p}_{u}$ is formed by merging ${{\mathbf{y}}_{{\mathrm{real,}}u}}$,
${{\mathbf{y}}_{{\mathrm{imag,}}u}}$, ${{\mathbf{z}}_{u}}$ [10], i.e.,

$\displaystyle{{\mathbf{p}}_{u}}={[{{\mathbf{y}}_{{\mathrm{real,}}u}},{{\mathbf{y}}_{{\mathrm{imag,}}u}},{{\mathbf{z}}_{u}}]}.$
(9)

It is worth noting that ${{\mathbf{p}}_{u}}$ can be viewed as a bit stream
with the elements of ${{\mathbf{p}}_{u}}$ only being 0 or 1. With the digital
modulation, we have

$\displaystyle{{\mathbf{w}}_{u}}\buildrel\Delta\over{=}f_{\mathrm{modu}}({{\mathbf{p}}_{u}}),$
(10)

where $f_{\mathrm{modu}}\left(\cdot\right)$ denotes the mapping function of
digital modulation, such as the quadrature phase shift keying (QPSK). In Eq
(10), ${{\mathbf{p}}_{u}}$ is mapped as modulated feedback vector (MFV)
${{\mathbf{w}}_{u}}\in{{\mathbb{C}}^{1\times L}}$, where
${\textit{L}}=\lceil\left({2M+N}\right)/2\rceil$. Without loss of generality,
the UL-US’s length $P$ is larger than $L$ due to main task of user services
[19][20]. Similar to [10] and [20], to superimpose MVF with UL-US, a spread
spectrum method is utilized, which could capture spread spectrum gain to
suppress the interference caused by the superimposition processing. Thus, the
superimposition signal ${{\mathbf{s}}_{u}}$, given in Eq (2), is obtained by
using a spreading matrix to spread the MFV ${{\mathbf{w}}_{u}}$, i.e.,

$\displaystyle{{\mathbf{s}}_{u}}=\frac{1}{\sqrt{L}}{{{{\mathbf{w}}_{u}}{{\mathbf{Q}}_{u}}}},$
(11)

where ${{\mathbf{Q}}_{u}}\in{{\mathbb{R}}^{L\times P}}$ is a spreading matrix,
which satisfies ${{\mathbf{Q}}_{u}}{\mathbf{Q}}_{u}^{T}=P{{\mathbf{I}}_{L}}$,
e.g., the Walsh matrix [32]. By combining Eq (2) and Eq (11), the transmitted
signal of user-$u$ ${{\mathbf{x}}_{u}}$ is rewritten as

$\displaystyle{{\mathbf{x}}_{u}}=\sqrt{\frac{{\rho{E_{u}}}}{L}}{{\mathbf{w}}_{u}}{{\mathbf{Q}}_{u}}+\sqrt{(1-\rho){E_{u}}}{{\mathbf{d}}_{u}}.$
(12)

At the user-$u$, the downlink CSI ${{\mathbf{h}}_{u}}$ is compressed by using
1-bit CS (given in Eq (5)), and thus the transmitted signal
${{\mathbf{x}}_{u}}$ is formed by weighting and superimposing the UL-US
${{\mathbf{d}}_{u}}$ and superimposition signal ${{\mathbf{s}}_{u}}$ according
to Eq (2)–(12). With the received ${\mathbf{R}}_{u}$ at the BS, the detection
network and reconstruction network are designed to detect the UL-US
${{\mathbf{d}}_{u}}$ and superimposition signal ${{\mathbf{s}}_{u}}$, and
recover the downlink CSI ${{\mathbf{h}}_{u}}$, respectively. The detection and
reconstruction networks will be deliberated in Section III.

##  III DL-based Superimposed CSI Feedback Using 1-Bit CS

In this section, according to the superimposed CSI feedback scheme with the
1-bit CS[10], the detection network and reconstruction network are developed
to recover the UL-US and downlink CSI. A transmitted signal feature extraction
is first employed to coarsely extract the feature after equalizing the uplink
wireless channel. Then, with the extracted transmitted signal feature, we
design the detection network and reconstruction network.

###  III-A Transmitted Signal Feature Extraction

From Eq (2)–(12), the transmitted signal ${{\mathbf{x}}_{u}}$ is formed by
superimposing the UL-US ${{\mathbf{d}}_{u}}$ and the modulated superimposition
signal ${{\mathbf{s}}_{u}}$. To recover ${{\mathbf{d}}_{u}}$ and
${{\mathbf{s}}_{u}}$, the transmitted signal ${{\mathbf{x}}_{u}}$ should be
first extracted, and thus the uplink channel ${{\mathbf{g}}_{u}}$ in Eq (1)
needs to be removed by channel equalization. From [10] and [19], the
transmitted signal feature extraction is employed in this paper. That is, the
uplink wireless channel is equalized through zero forcing (ZF) equalization,
so as to extract the transmission signal feature. The feature extraction is
given as

$\displaystyle{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}}={\mathbf{g}}_{u}^{\dagger}{{\mathbf{R}}_{u}}={{\mathbf{x}}_{u}}+{\mathbf{g}}_{u}^{\dagger}{{\mathbf{N}}_{u}},$
(13)

where ${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}}$ denotes the coarse
extracted vector of transmitted signal ${{\mathbf{x}}_{u}}$. It should be
noted that, relative to the use of ZF equalization to extract the transmitted
signal feature, the use of minimum mean square error (MMSE) channel
equalization can obtain better feature extraction performance, while
encounters higher computational complexity. Especially, the MMSE equalization
requires second-order statistics of uplink channel $\mathbf{g}_{u}$ and noise
$\mathbf{N}_{u}$ [10][18], which leads to application difficulties. Therefore,
we use low-complexity ZF equalization to extract the transmitted signal
feature, leaving the feature improvement to the subsequent detection network.

With the extracted transmitted signal feature
${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}}$, we construct the
detection network to detect UL-US ${{\mathbf{d}}_{u}}$ and superimposition
signal ${{\mathbf{s}}_{u}}$. From Eq (11), ${{\mathbf{s}}_{u}}$ is obtained by
spreading the MFV ${{\mathbf{w}}_{u}}$. In addition, the compressed downlink
CSI ${{\mathbf{y}}_{{\mathrm{real,}}u}}$ and
${{\mathbf{y}}_{{\mathrm{imag,}}u}}$ can be recovered from
${{\mathbf{w}}_{u}}$ (given in Eq (5)–(10)).

###  III-B Detection Network

In order to eliminate superimposed interference and obtain better downlink CSI
and UL-US reconstruction accuracy, the detection network is designed by using
unfolding method [33]. That is, the iteration steps in [10] are replaced by
the groups of CSI-Net and Det-Net, including six subnets, i.e., CSI-Net1, Det-
Net1, CSI-Net2, Det-Net2, CSI-Net3, and Det-Net3, in which the UL-US
${{\mathbf{d}}_{u}}$ and MFV ${{\mathbf{w}}_{u}}$ are detected by solving a
multi-task problem.

####  III-B1 Architecture

The architecture of detection network is illustrated in Fig 2. From the
perspective of convenience and ease of implementation, we first use the
easiest single hidden layer neural network architecture to design CSI-Net$i$
and Det-Net$i$ ($i=1,2,3$). After experimental verification, this architecture
is not only easy but also improves performance. The architecture of detection
network is described as follows:

  * •

CSI-Net1, DET-Net1, CSI-Net2, DET-Net2, CSI-Net3, and DET-Net3 are
successively cascaded to form the multi-task network. To reduce mutual
interference, some expert knowledge is inserted between each cascaded subnets,
i.e., the interference cancellation technology [18][19]. In more detail, the
CSI interference reduction (CSI IR) is introduced between the CSI-Net$i$ and
Det-Net$i$ ($i=1,2,3$), while the UL-US interference reduction (UL-US IR) is
inserted between Det-Net$i$ and CSI-Net$(i+1)$ ($i=1,2$).

  * •

The same network structures are employed by the CSI-Net$i$ and Det-Net$i$
($i=1,2,3$). Each subnet consists of an input layer, a hidden layer, and an
output layer with a fully connected mode. For each CSI-Net$i$ (DET-Net$i$)
($i=1,2,3$), the number of neurons in the input layer, hidden layer, and
output layer are $2L$ ($2P$), $4L$ ($4P$), and $2L$ ($2P$), respectively.

  * •

For each subnet, a batch normalization (BN) is employed to normalize its input
sets, converting the subnet input to zero mean and unit variance.

  * •

The activation functions of linear activation, leaky rectified linear unit
(LReLU) [34] and hyperbolic tangent (Tanh) are adopted by the input layer,
hidden layer and output layer of each subnet, respectively.

  * •

The outputs of CSI-Net3 and DET-Net3 are the detected MFV
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}$
(${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}={\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{\left(3\right)}$) and
the detected UL-US ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}$
(${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}={\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{\left(3\right)}$),
respectively.

The network architecture is summarized in Table I.

TABLE I: Architecture of detection network Subnet | CSI-Net$i$ | Det-Net$i$
---|---|---
layer |  Input | Hidden | Output | Input | Hidden | Output
BN |  $\surd$ | $\times$ | $\times$ | $\surd$ | $\times$ | $\times$
Neurons |  2$L$ | 4$L$ | 2$L$ | 2$P$ | 4$P$ | 2$P$
Activation |  None | LReLU | Tanh | None | LReLU | Tanh
Figure 2: Detection network

####  III-B2 Process of detection network

  * •

Data preprocessing

Due to the requirement of real-valued data sets in common DL-based framework,
we transform the coarse extracted complex-valued vectors
${{\mathbf{\mathord{\buildrel\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}\over{x}}}}_{u}}$, UL-US
${{\mathbf{d}}_{u}}$ and MFV ${{\mathbf{w}}_{u}}$ to the real-valued vectors,
i.e.,

$\displaystyle\left\\{{\begin{array}[]{*{20}{c}}{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}^{\mathrm{real}}=\left[{{\rm{Re}}({{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}}_{u}}),{\rm{Im}}({{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}}_{u}})}\right]}\\\
{{\mathbf{d}}_{u}^{\mathrm{real}}=\left[{{\rm{Re}}({{\mathbf{d}}_{u}}),{\rm{Im}}({{\mathbf{d}}_{u}})}\right]}\\\
{{\mathbf{w}}_{u}^{\mathrm{real}}=\left[{{\rm{Re}}({{\mathbf{w}}_{u}}),{\rm{Im}}({{\mathbf{w}}_{u}})}\right]}\end{array}}\right..$
(17)

To match the real-valued operation, the spreading matrix ${{\mathbf{Q}}_{u}}$
is also transformed to real-valued matrix ${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{Q}}}}_{u}}$, which is obtained as

$\displaystyle{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{Q}}}}_{u}}=\left[{\begin{array}[]{*{20}{c}}{{{\mathbf{Q}}_{u}}}&\mathbf{0}\\\
\mathbf{0}&{{{\mathbf{Q}}_{u}}}\end{array}}\right].$ (20)

Then, to train the detection network, ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}^{\mathrm{real}}$ is
employed as the network input, while ${\mathbf{d}}_{u}^{\mathrm{real}}$ and
${\mathbf{w}}_{u}^{\mathrm{real}}$ are used as training labels in the CSI-
Net$i$ and Det-Net$i$ ($i=1,2,3$), respectively. In addition, to facilitate
the unified description of the sub-network input in the detection network, we
use ${\mathbf{\widetilde{w}}}_{u}^{(1)}$ to represent the input
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}^{\mathrm{real}}$ of the
detection network, i.e.,
${\mathbf{\widetilde{w}}}_{u}^{(1)}={\mathbf{\mathord{\buildrel\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}\over{x}}}}_{u}^{real}$.

  * •

Processing procedure

The processing procedure of trained detection network is given in Table II,
and some steps are explained as follows.

TABLE II: Processing procedure Input:
${\mathbf{\widetilde{w}}}_{u}^{(1)}={\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}^{real}$.
---
(1-1): Use CSI-Net1 to detect MFV $\mathbf{w}_{u}$, then capture
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{\left(1\right)}$.
(1-2): Perform CSI IR by using expert knowledge and acquire
${\mathbf{\tilde{d}}}_{u}^{\left(1\right)}$.
(1-3): Detect UL-US by using DET-Net1 to obtain
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{\left(1\right)}$.
(1-4): Use expert knowledge to carry out UL-US IR and get
${\mathbf{\tilde{w}}}_{u}^{\left(1\right)}$.
(2-1): Employ CSI-Net2 to detect MFV and capture
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{\left(2\right)}$.
(2-2): Conduct CSI IR by using expert knowledge and acquire
${\mathbf{\tilde{d}}}_{u}^{\left(2\right)}$.
(2-3): Detect UL-US by using DET-Net2 to obtain
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{\left(2\right)}$.
(2-4): Perform UL-US IR by using expert knowledge and get
${\mathbf{\tilde{w}}}_{u}^{\left(2\right)}$.
(3-1): Use CSI-Net3 to detect MFV, then capture
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{\left(3\right)}$.
(3-2): Use expert knowledge to conduct CSI IR and acquire
${\mathbf{\tilde{d}}}_{u}^{\left(3\right)}$.
(3-3): Detect UL-US by using DET-Net3 to obtain
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{\left(3\right)}$.
Output: ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}={\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{\left(3\right)}$, and
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}={\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{\left(3\right)}$.

Process of CSI-Net$i$: The CSI-Net$i$ ($i=1,2,3$), is used to detect the MFV,
which is expressed as

$\displaystyle{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{(i)}=\sigma_{2}\left({{\mathbf{W}}_{12}^{(i)}\left({\sigma_{1}\left({{\mathbf{W}}_{11}^{(i)}\mathbb{BN}\left({{\mathbf{\widetilde{w}}}_{u}^{(i)}{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{Q}}}}_{u}^{T}}\right){\rm{+}}{\mathbf{b}}_{11}^{(i)}}\right)}\right){\rm{+}}{\mathbf{b}}_{12}^{(i)}}\right),$
(21)

where $\sigma_{1}$ and $\sigma_{2}$ denote the activation functions of LReLU
and Tanh, respectively. In Eq (21), ${\mathbf{W}}_{11}^{(i)}$
(${\mathbf{b}}_{11}^{(i)}$) and ${\mathbf{W}}_{12}^{(i)}$
(${\mathbf{b}}_{12}^{(i)})$ are the weights (biases) of hidden layer and
output layer, respectively. We use CSI-Net$i$ to detect MFV $\mathbf{w}_{u}$
and obtain the network output ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{(i)}$, which is
briefly described in steps (1-1), (2-1), and (3-1) in Table II.

CSI IR: In steps (1-2), (2-2), and (3-2) in Table II, to reduce the
interference from MFV, a spreading is employed by CSI IR, which is expressed
as

$\displaystyle{\mathbf{\widetilde{d}}}_{u}^{(i)}={\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}^{\mathrm{real}}-\sqrt{\frac{{\rho{E_{u}}}}{L}}{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{\left(i\right)}{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{Q}}}}_{u}},$ (22)

where ${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{Q}}}}_{u}}$ is obtained according
to Eq (20). Then, ${\mathbf{\widetilde{d}}}_{u}^{(i)}$ is fed into Det-Net$i$
to detect the UL-US.

Process of Det-Net$i$: The Det-Net$i$ ($i=1,2,3$), is used to detect the UL-
US, which is expressed as

$\displaystyle{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{(i)}=\sigma_{2}\left({{\mathbf{W}}_{22}^{(i)}\left({\sigma_{1}\left({{\mathbf{W}}_{21}^{(i)}\mathbb{BN}\left({{\mathbf{\widetilde{d}}}_{u}^{(i)}}\right){\rm{+}}{\mathbf{b}}_{21}^{(i)}}\right)}\right){\rm{+}}{\mathbf{b}}_{22}^{(i)}}\right),$
(23)

where ${\mathbf{W}}_{21}^{(i)}$ (${\mathbf{b}}_{21}^{(i)}$) and
${\mathbf{W}}_{22}^{(i)}$ (${\mathbf{b}}_{22}^{(i)})$ denote the weights
(biases) of hidden layer and output layer, respectively.

UL-US IR: In steps (1-4) and (2-4) in Table II, to reduce the interference
from the UL-US, the outputs of Det-Net$i$ ($i=1,2,3$) are processed by expert
knowledge, which is expressed as

$\displaystyle{\mathbf{\widetilde{w}}}_{u}^{(i)}={\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}^{\mathrm{real}}-\sqrt{(1-\rho){E_{u}}}{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{(i)}.$ (24)

With the process given in Table II, the UL-US ${{\mathbf{d}}_{u}}$ and MFV
${{\mathbf{w}}_{u}}$ are detected, where the real-valued descriptions of the
detected UL-US ${{\mathbf{d}}_{u}}$ and MFV ${{\mathbf{w}}_{u}}$ are denoted
by ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}$ and
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}$, respectively. Then,
with the detected MFV ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}$, we develop the
reconstruction network to recover the downlink CSI ${{\mathbf{h}}_{u}}$.

###  III-C Reconstruction Network

Figure 3: Reconstruction network TABLE III: Procedure of reconstruction
network Input: ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}$.
---
1): Map the real-valued ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}$ to
$[{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{real,}}u}},{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{imag,}}u}},{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}}]$
2): Use $[{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{real,}}u}},{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{imag,}}u}},{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}}]$ to rough extract the
feature of downlink
CSI and obtain ${{\mathbf{\tilde{h}}}_{u}}$ by the SCA-BIHT algorithm with
$\beta$ times of iteration.
3): Use the refinement network to refine the reconstructed downlink CSI
${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{u}}$.
Output: ${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{u}}$.

A reconstruction network is designed to further improve the reconstruction
accuracy of ${{\mathbf{h}}_{u}}$ on the basis of the reconstruction algorithm,
and to reduce the processing delay caused by multiple iterations of the
reconstruction algorithm. The reconstruction network is given in Fig 3, and
the processing procedure is summarized in Table III. Generally, the
corresponding de-mapping is first employed to restore the compressed downlink
CSI. Then, the reconstruction algorithm given in [10] with reduced complexity
is utilized to perform an initial feature extraction of the downlink CSI.
According to the initial feature extraction, two dense layers are used to
refine the reconstruction of the downlink CSI. These details will be presented
as follows.

####  III-C1 Inverse mapping operation

From Eq (10) and (17), the real-valued ${\mathbf{w}}_{u}^{\mathrm{real}}$ is
formed by digital modulation and the mapping from complex-valued to real-
valued form. Correspondingly, we adopt inverse mapping to recover the complex-
valued and unmodulated forms. An inverse mapping, denoted by
$f_{\mathbb{R}\rightarrow\mathbb{C}}(\cdot)$, is first employed to map the
real-valued ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}$ back to its complex-
valued form. Then, the digital demodulation mapping, denoted as
$f_{{\mathrm{demo}}}(\cdot)$, is used to demodulate this complex-valued
vector. The whole inverse mapping process is expressed as

$\displaystyle[{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{real,}}u}},{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{imag,}}u}},{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}}]\buildrel\Delta\over{=}{f_{{\mathrm{demo}}}}\left(f_{\mathbb{R}\rightarrow\mathbb{C}}\left({\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}\right)\right).$ (25)

Then, the estimation of sparsity $K$ of the downlink CSI, denoted as
${{{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{K}}}}}$, is obtained by
calculating the number of non-zero entries in
${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}}$.

####  III-C2 Initial feature extraction

TABLE IV: Initial feature extraction procedure Input: Compressed downlink CSI
$[{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{real,}}u}},{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{imag,}}u}}]$,
Support-set ${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}}$.
---
1): Initialization: the real part and imaginary part of reconstructed
data are set to zero (i.e.,
${\mathbf{h}}_{\mathrm{real}}^{0}={\mathbf{h}}_{\mathrm{imag}}^{0}={\mathbf{0}}$),
$t=0$, and
maximum number of iterations $\beta$.
2): $t=t+1$.
3):
${\mathbf{h}}_{\mathrm{real}}^{t}=K({\mathbf{h}}_{\mathrm{real}}^{t-1}+({{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{real,}}u}}-\mathrm{sign}({\mathbf{h}}_{\mathrm{real}}^{t-1}{{\mathbf{\Phi}}_{u}})){\mathbf{\Phi}}_{u}^{T})$,
${\mathbf{h}}_{\mathrm{imag}}^{t}=K({\mathbf{h}}_{\mathrm{imag}}^{t-1}+({{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{imag,}}u}}-\mathrm{sign}({\mathbf{h}}_{\mathrm{imag}}^{t-1}{{\mathbf{\Phi}}_{u}})){\mathbf{\Phi}}_{u}^{T})$.
4):
${\mathbf{h}}_{\mathrm{real}}^{t}={\mathbf{h}}_{\mathrm{real}}^{t}\odot{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}},{\mathbf{h}}_{\mathrm{imag}}^{t}={\mathbf{h}}_{\mathrm{imag}}^{t}\odot{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}}$.
5): Go to step 2) if $t<\beta$, else go to next step.
6):
${\mathbf{h}}={\mathbf{h}}_{\mathrm{real}}^{t}+j{\mathbf{h}}_{\mathrm{imag}}^{t}$.
7): Normalization:
${{\mathbf{\tilde{h}}}_{u}}={\mathbf{h}}/{\left\|{\mathbf{h}}\right\|_{2}}$.
Output: ${{\mathbf{\tilde{h}}}_{u}}$.

With ${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{real,}}u}}$,
${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{imag,}}u}}$,
${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}}$ and
${{{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{K}}}}}$, we employ the
reconstruction algorithm, named SCA-BIHT in [19], to conduct an initial
feature extraction of the downlink CSI, while leaving the refinement
reconstruction to the subsequent refinement network. In particular, this
initial feature extraction is executed by SCA-BIHT with only a few iterations
instead of dozens or hundreds of iterations in [10]. Here, $\beta$ times of
iteration are adopted in this paper. The initial feature extraction procedure
is presented in Table IV.

Based on the initial feature extraction, we then input
${{\mathbf{\tilde{h}}}_{u}}$ to a single hidden layer network to refine the
reconstruction accuracy of the downlink CSI $\mathbf{h}_{u}$.

####  III-C3 Refinement network

TABLE V: Architecture of refinement network Layer |  Input | Hidden | Output
---|---|---|---
Batch normalization |  $\surd$ | $\times$ | $\times$
Neuron number |  2$N$ | 4$N$ | 2$N$
Activation function |  None | LReLU | Linear

According to the initial feature extraction, a single hidden layer network is
employed to refine the reconstruction of the downlink CSI, and its network
architecture is summarized in Table V. Similar to CSI-Net$i$ and Det-Net$i$
($i=1,2,3$) of the detection network, the refinement network is also designed
as the easiest single hidden layer neural network architecture.

The the initial feature of downlink CSI ${{\mathbf{\tilde{h}}}_{u}}$ and the
label ${{\mathbf{h}}_{u}}$ are complex-valued, and thus need to be mapped to
real-valued form, i.e.,

$\displaystyle\left\\{\begin{gathered}{\mathbf{\tilde{h}}}_{u}^{{\text{real}}}=\left[{\operatorname{Re}\left({{{{\mathbf{\tilde{h}}}}_{u}}}\right),\operatorname{Im}\left({{{{\mathbf{\tilde{h}}}}_{u}}}\right)}\right]\hfill\\\
{\mathbf{h}}_{u}^{{\text{real}}}=\left[{\operatorname{Re}\left({{{\mathbf{h}}_{u}}}\right),\operatorname{Im}\left({{{\mathbf{h}}_{u}}}\right)}\right]\hfill\\\
\end{gathered}\right..$ (29)

Then, using the refinement network, the refined reconstruction of the downlink
CSI is obtained from the expression

$\displaystyle{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{u}}={{\mathbf{W}}_{32}}\left({\sigma_{1}\left({{{\mathbf{W}}_{31}}\mathbb{BN}\left({\mathbf{\tilde{h}}}_{u}^{{\text{real}}}\right){\rm{+}}{{\mathbf{b}}_{31}}}\right)}\right){\rm{+}}{{\mathbf{b}}_{32}},$
(30)

where ${{\mathbf{W}}_{31}}$ (${{\mathbf{b}}_{31}}$) and ${{\mathbf{W}}_{32}}$
(${{\mathbf{b}}_{32}}$) denote weights (biases) of the hidden layer and output
layer of the refinement network, respectively.

###  III-D Model Training Specification

Since model training is significant for network performance, we give the
training details in this subsection. In the following, we discuss the training
method, data preparation, and loss function, respectively.

#### Training method

In this paper, the detection network and reconstruction network are separately
trained to reduce the complexity of parameter tuning. For detection network,
there are six subnetworks needed to be trained, including the training
parameters ${\mathbf{W}}_{11}^{(i)}$, ${\mathbf{W}}_{12}^{(i)}$,
${\mathbf{W}}_{21}^{(i)}$, ${\mathbf{W}}_{22}^{(i)}$,
${\mathbf{b}}_{11}^{(i)}$, ${\mathbf{b}}_{12}^{(i)}$,
${\mathbf{b}}_{21}^{(i)}$, and ${\mathbf{b}}_{22}^{(i)}$ ($i=1,2,3$). From Fig
2, the detection network is a multi-task network in reality, which generates
the estimated MFV ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}$ and UL-US
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}$, respectively. Thus,
we jointly train the six subnets of detection network to resolve this multi-
task issue. In the reconstruction network, only the refinement network needs
to be trained to optimize its network parameters ${{\mathbf{W}}_{31}}$,
${{\mathbf{W}}_{32}}$, ${{\mathbf{b}}_{31}}$, and ${{\mathbf{b}}_{32}}$. With
the trained detection network and the corresponding initial feature extraction
of reconstruction network, we then train the refinement network solely.

####  III-D1 Data preparation for training

The training set is acquired by a simulation approach, in which a significant
amount of data samples are generated to train two networks, i.e., the
detection network and the refinement network. Specially, these data samples
are generated as follows.

${{\mathbf{h}}_{u}}$ and ${{\mathbf{g}}_{u}}$ are randomly generated on the
basis of the distribution
$\mathcal{CN}\left({0,\left({{1\mathord{\left/{\vphantom{1N}}\right.\kern-1.2pt}N}}\right)}\right)$.
To train the detection network, we first collect the
${{\mathbf{\mathord{\buildrel\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}\over{x}}}}_{u}}$ according to Eq (13)
to form input sets. Then we save the corresponding ${{\mathbf{d}}_{u}}$ and
${{\mathbf{w}}_{u}}$ as target sets, where ${{\mathbf{d}}_{u}}$ is formed by
QPSK modulation with randomly generated Bernoulli binary sequences. All the
complex-valued data sets are converted to real-valued form. For example, the
input and label of the detection network are set as
$\left\\{{\left({{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}}}_{u}^{\mathrm{real}}}\right),\left({{\mathbf{d}}_{u}^{\mathrm{real}},{\mathbf{w}}_{u}^{\mathrm{real}}}\right)}\right\\}$
according to Eq (17). Similarly, the input and label of the refinement network
are set as
$\left\\{{\left({\mathbf{\tilde{h}}}_{u}^{{\mathrm{real}}}\right),\left({{\mathbf{h}}_{u}^{\mathrm{real}}}\right)}\right\\}$
according to Eq (29). In addition, to validate the trained network parameters
during the training phase, a validation set is generated by following the same
generation method of training set, and thus we could capture a set of
optimized network parameters.

####  III-D2 Loss functions

Figure 4: (a) Training loss of the detection network. (b) Validation loss of
the detection network. (c) Training loss of the reconstruction network. (d)
Validation loss of the reconstruction network.

The detection network is trained by optimizing weights and biases of each
subnet, i.e., CSI-Net$i$ and Det-Net$i$, to minimize the loss function
[35][36]. In addition, the $l_{2}$ regularization is employed in the detection
network to avoid gradient explosions [37]. Thus, the loss function for
training the detection network is expressed as

$\displaystyle{\rm{Los}}{{\rm{s}}_{{\rm{\texttt{Det}}}}}{\rm{=los}}{{\rm{s}}_{\rm{1}}}+\alpha_{1}{\left\|{\mathbf{\Theta}_{1}}\right\|_{2}^{2}},$
(31)

where $\alpha_{1}$ is the regularization coefficient and
${\mathbf{\Theta}_{1}}$ denotes the training parameters, i.e., weights and
biases of the detection network. In Eq (31), $\mathrm{loss_{1}}$ represents
the weighted sum of the losses of six subnets, which is given as

$\displaystyle{\rm{loss_{1}=}}\frac{1}{6}\left(\sum\limits_{i{\rm{=}}1}^{3}{\left\|{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{(i)}-{\mathbf{d}}_{u}^{real}}\right\|_{2}^{2}}+\sum\limits_{i{\rm{=}}1}^{3}{\left\|{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{(i)}-{\mathbf{w}}_{u}^{\mathrm{real}}}\right\|_{2}^{2}}\right),$
(32)

where ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{(i)}$ and
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{(i)}$ are the output
of the CSI-Net$i$ and Det-Net$i$, respectively. With this detection network,
we obtain the MFV ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}^{\left(3\right)}$ and
UL-US ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}^{\left(3\right)}$,
i.e., ${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{w}}}}_{u}$ and
${\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}_{u}$.

With the trained detection network, the reconstruction network is trained
according to ${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{real,}}u}}$,
${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{y}}}}_{{\mathrm{imag,}}u}}$, and
${{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{z}}}}_{u}}$, which are detected
by the detection network and expressed in Eq (25). In reconstruction network,
only the refinement network with single hidden layer needs to be optimized,
and thus the loss function is given by

$\displaystyle{\rm{Los}}{{\rm{s}}_{\texttt{Rec}}}=\left\|{{{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}}_{u}}-{\mathbf{h}}_{u}^{\mathrm{real}}}\right\|_{2}^{2}+{\alpha_{2}}{\left\|{{{\mathbf{\Theta}}_{2}}}\right\|_{2}^{2}},$
(33)

where ${{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}}_{u}}$ is the estimated
downlink CSI, $\alpha_{2}$ is the regularization coefficient and
${\mathbf{\Theta}}_{2}$ denotes all training parameters of refinement network.

To reap an effective and feasible regularization coefficient and verify the
generalization performance of detection network and reconstruction network,
Fig 4 compares the convergence behaviors of
${\rm{Los}}{{\rm{s}}_{\texttt{Det}}}$ and
${\rm{Los}}{{\rm{s}}_{\texttt{Rec}}}$ under different regularization
coefficients (i.e., $\alpha_{1/2}=10^{-9},10^{-8},...,10^{-4}$). From Fig 4,
we can observe the convergence values of training loss and validation loss are
almost the same, which indicates the excellent generalization performance of
detection and reconstruction network. In addition, a smaller value of
$\alpha_{1}$ (or $\alpha_{2}$) leads to a smaller convergence value of
training loss or validation loss. Yet according to Eq (33), the value of
${\rm{Los}}{{\rm{s}}_{\texttt{Rec}}}$ is related to $\alpha_{2}$, the
$\alpha_{2}$ that minimizes the ${\rm{Los}}{{\rm{s}}_{\texttt{Rec}}}$ may not
achieve the best reconstruction performance. The optimized $\alpha_{2}$ is
determined by the reconstruction performance of the downlink CSI, which will
be given in the experimental analysis.

By using the trained detection network and reconstruction network, the UL-US
${{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{d}}}}}_{u}}$ and downlink CSI
${{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}}_{u}}$ can be recovered
from the proposed scheme. Compared with the 1-bit CS-based superimposed CSI
feedback scheme in [10], both the recoveries of the UL-US and downlink CSI are
improved by the proposed scheme, while the requirements of second-order
statistics of noise are avoided. Besides, these improvements are robust
against parameter variations, which will be presented in the experimental
analysis.

##  IV Experiment Results

In this section, we give numerical results of the proposed scheme. Definitions
and basic parameters involved in simulations are first given. Then, to verify
the effectiveness of the proposed scheme, we show the bit error rate (BER) of
UL-US and MFV, and the normalized mean squared error (NMSE) of reconstructed
downlink CSI is presented. Finally, we compare the online running time between
the proposed scheme and conventional scheme. The source code is available at
https://github.com/qingchj851/DL-1BitCS-SC-CSI-Feedback2.

###  IV-A Parameter Setting

Definitions involved in simulations are given as follows. The signal-to-noise
ratio (SNR) in decibel (dB) of the signal received at BS from user-$u$ is
defined as [19]

$\mathrm{SNR}=10{\log_{10}}\left({\frac{{{E_{u}}}}{{\sigma_{u}^{2}}}}\right).$
(34)

The NMSE is utilized to evaluate the recovery performance of downlink CSI, and
defined as [19]

$\mathrm{NMSE}={\frac{{\left\|{{{{\mathbf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}}_{u}}-{{\mathbf{h}}_{u}^{\textrm{real}}}}\right\|_{2}^{2}}}{{\left\|{{\mathbf{h}}_{u}^{\textrm{real}}}\right\|_{2}^{2}}}}.$
(35)

In the experiment phase, $P=512$, $N=64$, and the sampling rate $c$ is defined
as $c=M/N$. The measurement matrix is randomly generated and obeys the
Gaussian distribution [38], and it is guaranteed that its row vector and the
column vector of the compressed signal cannot be sparsely represented by each
other. The Walsh matrix generated by the Walsh sequence is employed as the
spreading matrix ${{\mathbf{Q}}_{u}}$ [32]. The UL-US ${\mathbf{d}}_{u}$ is
formed by applying QPSK modulation upon randomly generated Bernoulli binary
sequences. The training input data-sets are generated according to Eq
(1)–(13). Trainings of detection network and reconstruction network are
carried out under the noise-free setting, and this is different from the
training of the DL-based network in [19], where the training SNR is set as
5dB. Testing data-sets are generated by using the same method as the training
data-sets. The sizes of training set, validation set and testing set of
detection network are 60,000, 20,000, and 20,000, respectively. For the
reconstruction network, 45,000, 15,000, and 15,000 samples are respectively
employed for the training, validation, and testing. Both in detection network
and reconstruction network, we use Adam optimizer as the training optimization
algorithm, and the values of epoch and learning rate are set to 50 and 0.001,
respectively. In the simulations, we stop the testing for BER performance when
at least 1000-bit errors are observed [19][20].

Figure 5: BER versus SNR, where $P{\rm{=512}}$, $c=2.0$, and $\rho=0.10$ are
considered.

For the convenience of expression, we utilize “Proposed” and “Ref[10]” to
denote the proposed DL-based 1-bit superimposed CSI feedback and the
traditional 1-bit superimposed feedback (mentioned in [10]), respectively.

###  IV-B BER Performance

In this subsection, the effectiveness and robustness of the detection network
will be verified. To clarify the effectiveness, the comparison of BER’s
performance between “Proposed” and “Ref[10]” is first presented in Fig 5.
Next, to verify the robustness of the detection network, the impacts against
the parameters of $\rho$ and $c$ are given in Fig 6 and Fig 7, respectively.

Figure 6: BER versus SNR, where $P{\rm{=512}}$ and $c=2.0$ are considered.

To verify the effectiveness of the detection network, both the UL-US and MFV’s
BER performances are illustrated due to the UL-US being superimposed with MFV.
Fig 5 depicts the BER curves of the UL-US and MFV in terms of SNR, where
$c=2.0$ and $\rho=0.10$ are considered. From Fig 5, the BERs of UL-US and MFV
obtained by “Proposed” are respectively smaller than those of “Ref[10]” in the
whole given SNR regions. For example, when $\mathrm{SNR}=10$dB, the BER of UL-
US (or MFV) by “Proposed” is around $3.4\times 10^{-3}$ (or $4.5\times
10^{-2}$), while the BER of UL-US (or MFV) of “Ref[10]” is nearly $1.4\times
10^{-2}$ (or $8.5\times 10^{-2}$). That is, compared with “Ref[10]”, both the
UL-US and MFV’s BERs are improved by the proposed detection network.
Especially, these improvements are significant to be observed in the
relatively higher SNR. The possible reason is that the detection network is
trained under noise-free setting.

To verify the robustness of BER performance’s improvement against the impact
of $\rho$, the BER curves with different values of $\rho$, i.e., $\rho=0.05$,
$\rho=0.10$, and $\rho=0.15$, are plotted in Fig 6, where $c=2.0$ is
considered. From Fig 6, for each given $\rho$, the UL-US and MFV’s BERs of the
“Proposed” are respectively smaller than those of the “Ref[10]”. This reflects
that the proposed detection network could improve the BER performance under
different $\rho$ for both UL-US and MFV. As $\rho$ increases from 0.05 to 0.15
for “Proposed”, the BER of UL-US increases while the BER of MFV decreases, and
vice versa. The reason is that the increased (or decreased) $\rho$ aggregates
(or alleviates) the interference of MFV to UL-US, while alleviates (or
aggregates) the interference of UL-US to MFV. On the whole, with the impacts
of different $\rho$, the improvements of UL-US and MFV’s BER performances are
evidently observed. Thus, the proposed detection network guarantees the
improvement of BER performance against the impact of $\rho$.

Figure 7: BER versus SNR, where $P{\rm{=512}}$ and $\rho=0.10$ are considered.

The UL-US and MFV’s BER curves with different values of compression rate $c$
(i.e., $c=2.0$, $c=2.5$, and $c=3.0$) are depicted in Fig 7, and this
validates the improvement of BER performance is robust against the impact of
$c$, where $\rho=0.10$. In Fig 7, for each given $c$, the UL-US and MFV’s BER
performances of the “Proposed” are smaller than those of the “Ref[10]”. This
implies that the proposed detection network could improve UL-US and MFV’s BER
performance of “Ref[10]” for different values of $c$. With the increase of
$c$, for both “Proposed” and “Ref[10]”, the BERs of both UL-US and MFV
increase, and vice versa. The reason is that the spreading gain (i.e., $P/M$)
decreases with the increase of $c$, and thus affects the detection
performances (similar results can be found in [19] and [20]). As a whole,
compared with “Ref[10]”, the BER improvements of UL-US and MFV are evidently
observed for each given $c$. Thus, the proposed detection network shows its
robustness of improving UL-US and MFV’s BER performances against the impact of
$c$.

Figure 8: NMSE versus SNR, where $P{\rm{=512}}$, $c=2.0$, and $\rho=0.10$ are
considered.

To sum up, according to Figs 5 \- 7, the UL-US and MFV’s BER performances of
“Ref[10]” are effectively improved by the proposed detection network, and
these improvements are robust against the impacts of $\rho$ and $c$.

###  IV-C NMSE Performance

With the detected MFV, the downlink CSI can be reconstructed by using the
proposed reconstruction network. To validate the effectiveness of the proposed
reconstruction network, NMSE curves of the downlink CSI recovered from the
proposed reconstruction network and SCA-BIHT [10] are first given in Fig 8.
Then, to demonstrate the robustness of the reconstruction network, the NMSE
performance against the impacts of $\rho$ and $c$ are shown in Figs 9 and 10,
respectively. In addition, we present the influence of regularization
coefficient $\alpha_{2}$ on the NMSE performance in Table VI.

In Fig 8, the NMSE curves of downlink CSI’s recovery are depicted, where
$c=2.0$ and $\rho=0.10$. The “Proposed” employs 8 times of iteration for
initial feature extraction, i.e., $\beta=8$, followed by two dense layers. In
contrast, different iteration values (i.e., $\beta=10$, $\beta=20$,
$\beta=50$, and $\beta=100$) are given for the SCA-BIHT algorithm of
“Ref[10]”. From Fig 8, when $SNR\leq 14$dB, the “Proposed” achieves the
minimum NMSE, even lower than that of “Ref[10]” with $\beta=100$. For example,
when $\mathrm{SNR}=12$dB, the NMSE of “Proposed” is about $8.94\times
10^{-2}$, while that of “Ref[10]” with $\beta=100$ is about $1.43\times
10^{-1}$. That is, with a smaller NMSE, the two dense layers in the
reconstruction network can replace 95 iterations of SCA-BIHT algorithm in the
relatively low SNR region (e.g., $\mathrm{SNR}\leq$14dB), leading to a lower
processing delay. For the case where $\mathrm{SNR}\geq$16dB, the NMSE of
“Proposed” outperforms that of “Ref[10]” with $\beta=10$. Although it shows a
slightly higher NMSE of “Proposed” than “Ref[10]” with $\beta=50$ and $100$,
it compensates the high processing delay of “Ref[10]”. On the whole, the
proposed reconstruction network has a lower processing delay than “Ref[10]”
and shows a better NMSE performance in the relatively low SNR region.
Therefore, the proposed reconstruction network is effective to improve the
NMSE performance of “Ref[10]”.

Figure 9: NMSE versus SNR, where $P{\rm{=512}}$ and $c=2.0$ are considered,
and the $\beta$ of Ref[10] is 10. Figure 10: NMSE versus SNR, where
$P{\rm{=512}}$ and $\rho=0.10$ are considered, and the $\beta$ of Ref[10] is
10.

To verify the robust improvement of NMSE performance against the impact of
$\rho$, the NMSE curves with variant $\rho$ (i.e., $\rho=0.05$, $\rho=0.10$,
and $\rho=0.15$) are plotted in Fig 9. From Fig 9, for each given $\rho$, the
downlink CSI’s NMSE of the “Proposed” is smaller than that of the “Ref[10]”.
With the increase of $\rho$ (increases from 0.05 to 0.15), the NMSE decreases
for both “Ref[10]” and “Proposed”, and vice versa. The reason is that the
downlink CSI can obtain more transmission power with a larger value of $\rho$.
In addition, with the increase of SNR, the curves gradually converge for the
reason that the main influence of NMSE comes from the superimposed
interference in a relatively high SNR region. On the whole, for each given
value of $\rho$ in Fig 9, the NMSE of “Ref[10]” is reduced by the “Proposed”,
especially in the relatively low SNR region (e.g., $\mathrm{SNR}\leq$14dB).
Thus, the proposed reconstruction network possesses its robustness for
improving the NMSE performance against the impact of $\rho$.

Fig 10 plots the NMSE curves of downlink CSI with different values of
compression rate $c$ (i.e., $c=2.0$, $c=2.5$, and $c=3.0$) to validate the
robustness of NMSE performance’s improvement against the impact of $c$. In Fig
10, for each given $c$, the downlink CSI’s NMSE performance of the “Proposed”
is smaller than that of the “Ref[10]”. In addition, for $\mathrm{SNR}\leq
10$dB, the NMSEs of “Proposed” increase as the increase of $c$. The possible
reason is that the higher compression rate results in lower spreading gain
(i.e., $P/M$). In the low SNR region, the main impact of NMSE performance
comes from the noise interference and is limited by the low spread spectrum
gain. Yet, the NMSE’s convergence value of high compression rate is smaller
than that of low compression rate. For example, for the cases where $c=2.0$,
$c=2.5$, and $c=3.0$, the convergence values of “Proposed” NMSE are about
$6.0\times 10^{-2}$, $4.9\times 10^{-2}$, and $4.4\times 10^{-2}$,
respectively. The possible reason is that the higher compression rate brings
more reconstruction information in the high SNR region, where the noise
interference almost disappeared. On the whole, for each given value of $c$ in
Fig 10, the NMSE of “Ref[10]” is reduced by the “Proposed”. Thus, the proposed
reconstruction network possesses its robustness for improving the NMSE
performance against the impact of $c$.

TABLE VI: The effect of regularization coefficient $\alpha_{2}$ on NMSE
performance

SNR (dB) Regularization coefficient |  $\alpha_{2}=10^{-4}$ |  $\mathbf{\alpha_{2}=10^{-5}}$ |  $\alpha_{2}=10^{-6}$ |  $\alpha_{2}=10^{-7}$ |  $\alpha_{2}=10^{-8}$ |  $\alpha_{2}=10^{-9}$
---|---|---|---|---|---|---
0 |  $0.6121$ |  0.6114 |  0.6115 |  0.6116 |  0.6117 |  0.6118
2 |  $0.5675$ |  0.5666 |  0.5668 |  0.5670 |  0.5671 |  0.5671
4 |  $0.5024$ |  0.5015 |  0.5018 |  0.5021 |  0.5022 |  0.5422
6 |  $0.4151$ |  0.4145 |  0.4146 |  0.4148 |  0.4149 |  0.4149
8 |  $0.3063$ |  0.3056 |  0.3058 |  0.3060 |  0.3060 |  0.3061
10 |  $0.1682$ |  0.1673 |  0.1675 |  0.1678 |  0.1678 |  0.1679
12 |  $0.0903$ |  0.0894 |  0.0897 |  0.0900 |  0.0900 |  0.0901
14 |  $0.0683$ |  0.0677 |  0.0678 |  0.0680 |  0.0681 |  0.0681
16 |  $0.0626$ |  0.0618 |  0.0619 |  0.0623 |  0.0623 |  0.0624
18 |  $0.0608$ |  0.0601 |  0.0602 |  0.0605 |  0.0606 |  0.0606

In addition, the influence of regularization coefficient $\alpha_{2}$ on NMSE
performance is given in Table VI, where $c=2.0$, $\rho=0.10$, and different
values of $\alpha_{2}$ (i.e., $\alpha_{2}=10^{-4}$, $\alpha_{2}=10^{-5}$,
$\alpha_{2}=10^{-6}$, $\alpha_{2}=10^{-7}$, $\alpha_{2}=10^{-8}$, and
$\alpha_{2}=10^{-9}$) are considered. From Table VI, the influence of
different values of the regularization coefficient on the NMSE is not very
obvious. Despite all this, among the given values of $\alpha_{2}$, in all SNR
regions, the minimum of NMSE is still observed as $\alpha_{2}=10^{-5}$. Thus,
the NMSE performance in Table VI indicates $\alpha_{2}=10^{-5}$ is a
preferable regularization coefficient.

To sum up, according to Figs 8 \- 10, the downlink CSI’s NMSE performance of
“Ref[10]” is effectively improved by the proposed reconstruction network, and
these improvements are robust against the impacts of $\rho$ and $c$.

Figure 11: Comparison of “Proposed” and “Ref[10]” about online running time by
conducting $10^{5}$ times of experiments, where $P{\rm{=512}}$ and $\rho=0.10$
are considered.

###  IV-D Online Running Time

To illustrate the low processing delay of “Proposed”, i.e., detection network
and reconstruction network, the online running time between “Proposed” and
“Ref[10]” is compared in Fig 11, where $P{\rm{=512}}$, $\rho=0.10$, and
different values of $c$ (i.e., $c=2.0$, $c=2.5$, and $c=3.0$) are considered.
Especially, “Ref [10]” adopts $\beta=10$ and $100$ in the reconstruction
algorithm (i.e., SCA-BIHT algorithm). Here, $\beta=10$ in “Ref[10]” is used to
guarantee the NMSE of the “Proposed” is smaller than that of “Ref [10]”, and
$\beta=100$ in “Ref[10]” is used to present the “Proposed” has a similar NMSE
(in a relatively high SNR region) while significantly lower processing delay
as that of “Ref[10]”. For a fair comparison, $10^{5}$ online-running
experiments are conducted for “Proposed” and “Ref[10]” on the same PC (with
CPU i5-8250U) by using MATLAB software. For each given $c$ in Fig 11, the
online running time of “Proposed” is shorter than that of “Ref[10]”, e.g.,
when $c=2.0$, the online running time of “Proposed” and $\beta=10$
($\beta=100$) in “Ref[10]” are 75.1s and 201.8s (1266.9s), respectively. This
reflects that the proposed 1-bit CS-based superimposed CSI feedback can reduce
the processing delay. It is also noticed that, as $c$ rises from 2.0 to 3.0,
the online running time of both “Proposed” and “Ref[10]” go up. However, the
total increased running time of the “Proposed” is 15.9s, which is far less
than that of “Ref[10]” (e.g. 54.7s for $\beta=10$ and 374.0s for $\beta=100$).
In addition, Fig 11 shows that the online running time of “Ref[10]” is
proportional to the number of iteration. Thus, the NMSE performance might not
be applicable for “Ref[10]” with large iteration number, while the “Proposed”
can avoid this annoyance.

As a whole, compared with “Ref[10]”, the proposed DL-based 1-bit superimposed
CSI feedback significantly reduces the online running time.

##  V Conclusion

The 1-bit CS-based superimposed CSI feedback is still facing many challenges,
such as low recovery accuracy of the UL-US and downlink CSI, and long
processing delay, etc. To remedy these defects, the DL-based 1-bit
superimposed CSI feedback has been investigated in this paper. The constructed
detection network captures optimized network parameters by using joint
training, and thus improves the BER performance of the UL-US. Moreover, the
detection network is also helpful for reconstructing the downlink CSI. With
the detected downlink CSI’s bits from the detection network, the proposed
reconstruction network utilizes the simplified version of SCA-BIHT with a
single hidden layer network, and achieves a significant improvement on NMSE
performance of the downlink CSI recovery. In particular, compared with the
conventional 1-bit CS-based superimposed CSI feedback, the proposed CSI
feedback scheme presents its robustness against parameter variations and
possesses significantly low processing delay.

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