# Fusion Learning for 1-Bit CS-based Superimposed CSI Feedback with Bi-
Directional
Channel Reciprocity

Chaojin Qing, , Qing Ye, Wenhui Liu, and Jiafan 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).C. Qing, Q. Ye and W. Liu are with School of
Electrical Engineering and Electronic Information, Xihua University, Chengdu,
China (E-mail: qingchj@mail.xhu.edu.cn).Jiafan Wang is with Synopsys Inc.,
2025 NE Cornelius Pass Rd, Hillsboro, OR 97124, USA (E-mail:
jifanw@gmail.com).

###### Abstract

Due to the discarding of downlink channel state information (CSI) amplitude
and the employing of iteration reconstruction algorithms, 1-bit compressed
sensing (CS)-based superimposed CSI feedback is challenged by low recovery
accuracy and large processing delay. To overcome these drawbacks, this letter
proposes a fusion learning scheme by exploiting the bi-directional channel
reciprocity. Specifically, a simplified version of the conventional downlink
CSI reconstruction is utilized to extract the initial feature of downlink CSI,
and a single hidden layer-based amplitude-learning network (AMPL-NET) is
designed to learn the auxiliary feature of the downlink CSI amplitude. Then,
based on the extracted and learned amplitude features, a simple but effective
amplitude-fusion network (AMPF-NET) is developed to perform the amplitude
fusion of downlink CSI and thus improves the reconstruction accuracy for 1-bit
CS-based superimposed CSI feedback while reducing the processing delay.
Simulation results show the effectiveness of the proposed feedback scheme and
the robustness against parameter variations.

###### Index Terms:

Channel state information (CSI), 1-bit compressed sensing (CS), superimposed
CSI feedback, fusion learning, bi-directional reciprocal channel
characteristics.

##  I Introduction

As one of the key technologies of the fifth generation (5G) and beyond
wireless communication systems, massive multiple-input multiple-output (mMIMO)
systems have shown great prospects in providing high spectrum and energy
efficiency [1]. However, to take full advantage of the system, mMIMO
transmitters must rely on sufficiently accurate channel state information
(CSI), which means that the base station (BS) needs to acquire the downlink
CSI accurately and timely [2].

In frequency division duplex (FDD) systems, the downlink CSI usually needs to
be fed back to the BS. However, numerous BS antennas consume huge feedback
overhead[3]. To improve feedback efficiency, different neural network
(NN)-based CSI feedbacks have been aroused in recent years [2, 3, 4]. Yet the
uplink bandwidth resource is still heavily occupied due to the mMIMO
scenarios. This dilemma is overcome by employing superimposed CSI feedback [5,
7, 6], while introducing the superimposed interference between downlink CSI
and uplink user data sequences (UL-US). To alleviate the superimposed
interference, 1-bit compressed sensing (CS)-based superimposed CSI feedback is
proposed in [8] by transforming the estimation problem into a detection
problem. Thus, the superimposed interference is effectively reduced,
especially for the detection of UL-US.

Nevertheless, the reconstruction of downlink CSI in 1-bit CS-based
superimposed CSI feedback is still facing huge challenges. 1) The amplitude
information is discarded, resulting in the reconstruction accuracy of downlink
CSI is unsatisfactory. 2) The commonly used iterative reconstruction for 1-bit
CS causes excessive processing delay. These issues motivate the following
considerations. 1) The discarded amplitude information in 1-bit CS needs to be
retrieved as much as possible without increasing the additional uplink
bandwidth. This promotes us to exploit the multiple modals of amplitude
features. 2) To reduce the processing delay, the structures of the developed
NNs need to be extremely lightweight, and thus the single hidden layer-based
NN is highly desired. Therefore, we propose a fusion learning scheme to take
full advantage of the multimodal amplitude information and lightweight network
architecture.

To the best of our knowledge, the solution of applying the bi-directional
channel reciprocity and multimodal feature-level fusion for 1-bit CS-based
superimposed CSI feedback has not been investigated. The main contributions of
this letter are summarized as follows:

  1. 1.

We exploit the multiple modals of amplitude feature from the same received
signal. From the bi-directional channel reciprocity, an amplitude-learning
network (AMPL-NET) is developed to capture the amplitude correlation between
uplink and downlink CSI in the angle domain. In addition to the reconstructed
amplitude from 1-bit CS, the multimodal amplitudes are extracted without
adding additional hardware devices or reception overhead.

  2. 2.

We develop simplified reconstruction and lightweight NNs to reduce processing
delay. The simplified reconstruction, lightweight AMPL-NET, and amplitude-
fusion network (AMPF-NET) are employed to reduce the processing delay of
original iteration reconstruction. Although both of the retrieved and
reconstructed amplitudes are inaccurate, the lightweight AMPF-NET plays a
critical role in the accuracy improvement. The retrieved amplitude helps to
improve the reconstructed amplitudes and vice versa.

  3. 3.

We develop a paradigm of fusion learning to capture the solutions of both non-
NN and NN-based receivers from a multimodal perspective for 1-bit CS-based
superimposed CSI feedback. Relative to the iteration reconstruction or data-
driven NN, this paradigm presents significant superiority in reducing the
processing delay and computational complexity, and improving the accuracy for
downlink CSI recovery, which facilitates its practical applications.

Figure 1: System model.

Notation: Boldface upper case and lower case letters denote matrix and vector
respectively. ${\left(\cdot\right)^{T}}$ denotes transpose; $\mathbf{I}_{P}$
is the identity matrix of size $P\times P$; ${\left\|\cdot\right\|}$ is the
Euclidean norm; ${{\mathrm{sign}}}(\cdot)$ denotes an operator of taking
symbolic information, e.g., the sign function returns 1 for positive numbers
and 0 otherwise; ${{\mathrm{Re}}}(\cdot)$ and ${{\mathrm{Im}}}(\cdot)$
represent the operation of taking the real and imaginary parts of a complex
value, respectively; $|\cdot|$ denotes the operation of taking the modulus of
a complex value. $E[\cdot]$ represents the expectation operation. $\odot$
represents the Hadamard product. $e$ is a constant ( $e=2.7183...$).

##  II System Model

Considering an FDD mMIMO system that consists of a BS with $N$ antennas and
$L$ single-antenna users, the system model is given in Fig. 1. On the user-$l$
side, $l=1,2,\cdots,L$, the $K$-sparsity downlink CSI
${\mathbf{h}}_{l}\in\mathbb{C}^{1\times N}$ in the angular domain is
compressed according to the 1-bit CS technique [10], i.e.,

$\left\\{{\begin{array}[]{*{20}{c}}{{{\mathbf{p}}_{{\textrm{real,}}l}}={\mathrm{sign}}\left({{\mathrm{Re}}\left({{{\mathbf{h}}_{l}}{{\mathbf{\Phi}}_{l}}}\right)}\right)}\\\
{{{\mathbf{p}}_{{\textrm{imag,}}l}}={\mathrm{sign}}\left({{\mathrm{Im}}\left({{{\mathbf{h}}_{l}}{{\mathbf{\Phi}}_{l}}}\right)}\right)}\end{array}}\right.,$
(1)

with ${{\mathbf{p}}_{{\textrm{real,}}l}}$ and
${{\mathbf{p}}_{{\textrm{imag,}}l}}$ being the real and imaginary parts of the
compressed downlink CSI, respectively. The $\mathbf{\Phi}_{l}$ denotes the
$N\times M$ measurement matrix [8]. For better performance, the support-set
${\mathbf{z}}_{l}\in\\{0,1\\}^{1\times N}$ is employed to mark indexes of zero
elements and non-zero elements of the downlink CSI [8]. According to
$\mathbf{z}_{l}$, ${{\mathbf{p}}_{{\textrm{real,}}l}}$, and
${{\mathbf{p}}_{{\textrm{imag,}}l}}$, the $feedback$ $vector$ (FV) is
constructed as
${{\mathbf{w}}_{l}}=[{{\mathbf{z}}_{l}},{{\mathbf{p}}_{{\textrm{real,}}l}},{{\mathbf{p}}_{{\textrm{imag,}}l}}]$,
where $\mathbf{w}_{l}$ is a $1\times K$ vector with $K=N+M*2$. It is
noteworthy that the $\mathbf{w}_{l}$ is a bit stream format, because the
elements of $\mathbf{w}_{l}$ only contain 0 and 1. With the digital
modulation, the modulated feedback vector $\mathbf{r}_{l}$ with length $T$ is
obtained as

${{\bf{r}}_{l}}\buildrel\Delta\over{=}{f_{{\textrm{modu}}}}({{\bf{w}}_{l}}),$
(2)

where $f_{\textrm{modu}}(\cdot)$ denotes the mapping function of digital
modulation, such as the quadrature phase shift keying (QPSK), and
$T=\left\lceil{K/2}\right\rceil$ for QPSK modulation. Similar to [8], the
spreading spectrum method is employed to reduce superimposed interference.
Then, the superimposition signal $\mathbf{s}_{l}$ is written as
$\mathbf{s}_{l}=\frac{1}{\sqrt{T}}\mathbf{r}_{l}\mathbf{Q}_{l}^{{T}}$, where
$\mathbf{Q}_{l}\in\mathbb{R}^{P\times T}$ denotes the spreading matrix,
satisfying $\mathbf{Q}_{l}^{T}\mathbf{Q}_{l}=P\mathbf{I}_{T}$. With the
superimposition signal and the UL-US signal, the transmitted signal
${\mathbf{x}}_{l}\in\mathbb{C}^{1\times P}$ is given by [8]

$\displaystyle\mathbf{x}_{l}=$ $\displaystyle\sqrt{\rho
E_{l}}\mathbf{s}_{l}+\sqrt{(1-\rho)E_{l}}\mathbf{d}_{l},$ (3)

where $\rho\in[0,1]$ stands for the power proportional coefficient of downlink
CSI, $E_{l}$ represents the transmitted power of user-$l$, and
$\mathbf{d}_{l}\in\mathbb{C}^{1\times P}$ denotes the modulated UL-US signal.
Without loss of generality, due to the main task of uplink services [8], the
length of UL-US is larger than that of the modulated feedback vector, i.e.,
$P>T$. At the BS, after the processing of matched-filter, the received signal
$\mathbf{Y}_{l}$ of the user-$l$, is given by

$\displaystyle\mathbf{Y}_{l}=$
$\displaystyle\mathbf{g}_{l}\mathbf{x}_{l}+\mathbf{N}_{l},$ (4)

where $\mathbf{N}_{l}\in\mathbb{C}^{N\times P}$ represents the circularly
symmetric complex Gaussian (CSCG) noise with zero-mean and variance
${\sigma_{l}^{2}}$ for each feedback link, ${\mathbf{x}}_{l}$ denotes the
transmitted signal of the $l$-th user, and
${{\mathbf{g}}_{l}}={\left[{{g_{l,1}},{g_{l,2}},\ldots,{g_{l,N}}}\right]^{T}}\in{\mathbb{C}^{N\times
1}}$ denotes the uplink channel matrix from the user-$l$ to the BS. With the
received signal $\mathbf{Y}_{l}$, the UL-US detection is utilized to detect
the UL-US ${\bf{\widetilde{d}}}_{l}$ and modulated feedback vector
${\bf{\widetilde{r}}}_{l}$. Then, the simplified downlink CSI reconstruction
is employed to extract the initial amplitude feature
${\bf{\widehat{h}}}_{{\textrm{amp}},l}$ and angle feature
${\bf{\widetilde{h}}}_{{\textrm{ang}},l}$. Meanwhile, the AMPL-NET is designed
to learn the downlink CSI amplitude feature according to bi-directional
channel reciprocity. Subsequently, we develop the AMPF-NET to fuse the
amplitude features based on the simplified downlink CSI reconstruction and
AMPL-NET. Finally, the amplitude fused from AMPF-NET is combined with its
corresponding ${\bf{\widetilde{h}}}_{{\textrm{ang}},l}$ as a fully recovered
downlink CSI ${{\bf{\widetilde{h}}}_{l}}$.

Figure 2: The fusion learning-based feedback scheme.

##  III Fusion Learning-based Feedback Scheme

The fusion learning-based feedback scheme is shown in Fig. 2, in which a
simplified reconstruction method is first employed to reconstruct the downlink
CSI, which is presented in Section III-A. Then, we develop AMPL-NET and AMPF-
NET to learn and fuse the downlink CSI amplitude, respectively. AMPL-NET
learns the downlink CSI amplitude from uplink CSI, and AMPF-NET follows the
idea of multimodal feature-level fusion. Both AMPL-NET and AMPF-NET are
elaborated in Section III-B. In Section III-C, the analysis of computational
complexity is given.

###  III-A Simplified Downlink CSI Reconstruction

After the UL-US detection, the FV $\widetilde{\mathbf{w}}_{l}$ is recovered
from the detected modulated feedback vector ${\bf{\widetilde{r}}}_{l}$ given
in (2). We employ the reconstruction algorithm with
$\widetilde{\mathbf{w}}_{l}$ and $\alpha$ iterations to perform a simplified
version of the downlink CSI reconstruction, i.e.,

$\displaystyle\left[{\begin{array}[]{*{20}{c}}{\bf{\widehat{h}}}^{T}_{{\textrm{amp}},l},{\bf{\widetilde{h}}}^{T}_{{\textrm{ang}},l}\end{array}}\right]^{T}=f_{\textrm{SCA-
BIHT}}(\widetilde{\mathbf{w}}_{l},\alpha),$ (5)

where $f_{\textrm{SCA-BIHT}}(\cdot)$ denotes the SCA-BIHT reconstruction
algorithm [8], and $\alpha\leq 5$ is considered in our work, which is far less
than the iteration times of the iterative method in [8]. Then, the features of
downlink CSI are extracted, i.e., the initial amplitude feature
${\bf{\widehat{h}}}_{{\textrm{amp}},l}$ and angle feature
${\bf{\widetilde{h}}}_{{\textrm{ang}},l}$ of the downlink CSI are obtained for
subsequent recovery.

###  III-B AMPL-NET and AMPF-NET

In order to obtain the downlink CSI amplitude feature which is different from
the conventional feedback perspective, we construct the lightweight and
effective AMPL-NET, which utilizes the bi-directional correlation of CSI
amplitude[4]. Then, a certain downlink CSI amplitude feature, which is called
auxiliary amplitude feature ${{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{{\textrm{amp}},l}}$, is
learned from AMPL-NET to complement the initial amplitude feature
${\bf{\widehat{h}}}_{{\textrm{amp}},l}$. Next, to refine the amplitude of the
downlink CSI, we borrow the idea of multimodal feature-level fusion and design
AMPF-NET, which fuses initial amplitude feature (from the simplified
reconstruction method in [8]) and auxiliary amplitude feature (from AMPL-NET).

####  III-B1 Network Design

According to [11], choosing the number of hidden neurons or layers is still a
challenge in the NN. Based on a large number of experiments, we design the
AMPL-NET and AMPF-NET, both of which are composed of one input layer, one
hidden layer, and one output layer. The network architectures of AMPL-NET and
AMPF-NET are summarized in Table I, and the detailed descriptions are given as
follows.

In AMPL-NET, neuron numbers of the input layer, hidden layer, and output layer
are $N$, $2N$, and $N$, respectively, while in AMPF-NET they are $2N$, $2N$,
and $N$, respectively. A batch normalization is employed to normalize the
input sets in both AMPL-NET and AMPF-NET, forming the network input as zero
mean and unit variance, and the leaky rectified linear unit (LReLU) [12] and
linear activation are employed as activation functions for the hidden layer
and output layer, respectively. For training the AMPL-NET, the uplink channel
${{\mathbf{g}}_{l}}$ in time domain is transformed to the angular domain[13].
By denoting the transformed uplink channel as ${{\mathbf{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\frown}$}}{g}}}_{l}}$, its amplitude is
given by

${{\mathbf{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\frown}$}}{g}}}_{{\textrm{amp}},l}}={\left[{\left|{{{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\frown}$}}{g}}_{l,1}}}\right|,\left|{{{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\frown}$}}{g}}_{l,2}}}\right|,\ldots,\left|{{{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\frown}$}}{g}}_{l,N}}}\right|}\right]^{T}}.$
(6)

Using ${{\mathbf{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\frown}$}}{g}}}_{{\textrm{amp}},l}}$ as
the input of the AMPL-NET, the auxiliary feature
${{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{{\textrm{amp}},l}}$ of the
downlink CSI amplitude is obtained according to

${{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{{\textrm{amp}},l}}=f_{\textrm{AMPL}}({{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{g}}}}_{{\textrm{amp}},l}},\Theta_{\textrm{AMPL}}),$
(7)

where $f_{\textrm{AMPL}}(\cdot)$ and $\Theta_{\textrm{AMPL}}$ denote the
amplitude learning operation and its network parameter, respectively.

In AMPF-NET, the input ${{\mathbf{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\smile}$}}{h}}}_{{\textrm{amp}},l}}\in{\mathbb{R}^{1\times
2N}}$ is spliced by ${\bf{\widehat{h}}}_{{\textrm{amp}},l}$ and
${{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{{\textrm{amp}},l}}$ ,
i.e.,

${{\mathbf{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\smile}$}}{h}}}_{{\textrm{amp}},l}}=[{\bf{\widehat{h}}}_{{\textrm{amp}},l}^{T},{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}^{T}_{{\textrm{amp}},l}}].$
(8)

Then, using the AMPF-NET, the amplitude of the downlink CSI
${\bf{\widetilde{h}}}_{{\textrm{amp}},l}$ is obtained by

${\bf{\widetilde{h}}}_{{\textrm{amp}},l}=f_{\textrm{AMPF}}({{\mathbf{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\smile}$}}{h}}}_{{\textrm{amp}},l}},\Theta_{\textrm{AMPF}}),$
(9)

where $f_{\textrm{AMPF}}(\cdot)$ and $\Theta_{\textrm{AMPF}}$ denote the
amplitude fusion operation and its network parameter, respectively.

####  III-B2 Training and Deployment

The training sets are acquired by simulation, and a significant amount of data
samples are collected to train AMPL-NET and AMPF-NET, respectively.
Specifically, these data samples are generated as follows.

TABLE I: Architecture Of AMPL-NET and AMPF-NET

Layer | Input | Hidden | Output
---|---|---|---
AMPL-NET | AMPF-NET | AMPL-NET | AMPF-NET | AMPL-NET | AMPF-NET
Batch normalization | $\surd$ | $\surd$ | $\times$ | $\times$ | $\times$ | $\times$
Neuron number | $N$ | $2N$ | $2N$ | $2N$ | $N$ | $N$
Activation function | None | None | LReLU | LReLU | Linear | Linear

The bi-directional channels are generated by MATLAB 5G Toolbox, which is
subject to specifications of the Clustered-Delay-Line (CDL) channel model in
3GPP TR 38.901 [14]. Similar to the setting in [13], the frequency-independent
parameters (e.g., the azimuth angle of departure (AoD)) are fixed, while
varying the complex gain of each path between the downlink and uplink
channels. Then, ${{\mathbf{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\frown}$}}{g}}}_{l}}$ and
${{\mathbf{h}}_{l}}$ are obtained by transforming the generated uplink and
downlink channels to the angular domain, respectively[13]. To train the AMPL-
NET, we form input sets according to (6), while we collect amplitude features
${\bf{\widehat{h}}}_{{\textrm{amp}},l}$ and ${{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{{\textrm{amp}},l}}$ of the
downlink CSI to form input sets according to (8) to train the AMPF-NET. Then,
we save the corresponding ${\bf{h}}_{{\textrm{amp}},l}$ from
${{\mathbf{h}}_{l}}$ as target sets of AMPL-NET and AMPF-NET, respectively.
The optimization goal of AMPL-NET is to minimize the mean squared error (MSE)
between ${{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{h}}}}_{{\textrm{amp}},l}}$ and
${{\mathbf{h}}_{{\text{amp}},l}}$, which is derived as

$\begin{gathered}\mathop{\min}\limits_{{\Theta_{{\text{AMPL}}}}}E\left[{{{\left\|{f_{\textrm{AMPL}}({{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{g}}}}_{{\textrm{amp}},l}},\Theta_{\textrm{AMPL}})-{{\mathbf{h}}_{{\text{amp}},l}}}\right\|}^{2}}}\right].\end{gathered}$
(10)

Similarly, the AMPF-NET minimizes the MSE of the fused amplitude, i.e.,
$E\left[{{{\left\|{{{\bf{\widetilde{h}}}_{{\textrm{amp}},l}-{{\mathbf{h}}_{{\text{amp}},l}}}}\right\|}^{2}}}\right]$,
which is further expressed by

$\begin{gathered}\mathop{\min}\limits_{{\Theta_{{\text{AMPF}}}}}E\left[{{{\left\|{f_{\textrm{AMPF}}({{\mathbf{\overset{\lower
5.0pt\hbox{$\smash{\scriptscriptstyle\smile}$}}{h}}}_{{\textrm{amp}},l}},\Theta_{\textrm{AMPF}})-{{\mathbf{h}}_{{\text{amp}},l}}}\right\|}^{2}}}\right].\end{gathered}$
(11)

We perform the training once for both AMPL-NET and AMPF-NET, and save the
trained network parameters for testing.

By using the AMPF-NET, the high precision downlink amplitude
${\bf{\widetilde{h}}}_{{\textrm{amp}},l}$ is obtained. Then, the amplitude
${\bf{\widetilde{h}}}_{{\textrm{amp}},l}$ is combined with its corresponding
angle ${\bf{\widetilde{h}}}_{{\textrm{ang}},l}$ from the simplified downlink
CSI reconstruction, and this leads to the recovered downlink CSI
${{\bf{\widetilde{h}}}_{l}}$, i.e.,
${{\bf{\widetilde{h}}}_{l}}={\bf{\widetilde{h}}}_{{\textrm{amp}},l}\odot{e^{j{\bf{\widetilde{h}}}^{T}_{{\textrm{ang}},l}}}$.
Compared with the 1-bit CS-based superimposed CSI feedback scheme in [8], the
proposed scheme demonstrates a better CSI reconstruction accuracy, and reduces
the online running time and computational complexity by balancing the off-line
training and online running.

###  III-C Complexity Analysis

For description convenience, “Proposed” is used to denote the proposed scheme;
“Ref[8]” represents the conventional 1-bit superimposed feedback in [8];
“Ref[10]” denotes the 1-bit CS feedback method with TDM mode in [10]. The
comparison of computational complexity is given in Table II, where $\beta$
denotes the iteration times required for the downlink CSI reconstruction
schemes in [8] and [10]. SCA-BIHT has the computational complexity of $4MN$
for each iteration [8]. The proposed simplified version of the downlink CSI
reconstruction has fewer iterations than [8] and [10], i.e., $\alpha<\beta$.
In addition, weight number and floating-point operations (FLOPs) are the most
common metrics to describe the NN complexity [13]. From [13], the total NN
weight number of the proposed AMPL-NET and AMPF-NET is $10N^{2}+6N$, and the
total FLOPs number is $20N^{2}-6N$. Thus, the total complexity of the proposed
scheme (including simplified downlink CSI reconstruction, AMPL-NET, and AMPF-
NET) is $4MN\cdot{\alpha}+10N^{2}+6N+20N^{2}-6N=4MN\cdot{\alpha}+30N^{2}$. For
the case where $M=2N$, $\alpha=5$, and $\beta=100$, the computational
complexities of the downlink CSI reconstruction in “Proposed”, “Ref[8]”, and
“Ref[10]” are $70N^{2}$, $800N^{2}$, and $800N^{2}+400N$, respectively.
Therefore, the proposed scheme has lower computational complexity than those
of [8] and [10].

TABLE II: Computational Complexity Of Downlink CSI Reconstruction

Method | Proposed | Ref[8] | Ref[10]
---|---|---|---
Complexity | $4MN\cdot{\alpha}+30N^{2}$ | $4MN\cdot{\beta}$ | $4(M+1)N\cdot{\beta}$
Case ($M=2N$) |  $70N^{2}$ ($\alpha=5$) |  $800N^{2}$ ($\beta=100$) |  $800N^{2}+400N$ ($\beta=100$)

##  IV Experiment results

In this section, we provide numerical results of the proposed scheme.
Definitions and basic parameters involved in simulations are first given in
Section IV-A. Then, to verify the effectiveness of the proposed scheme, the
normalized mean squared error (NMSE) of the reconstructed downlink CSI and the
online running time are given in Section IV-B. Finally, the robustness of the
proposed scheme is verified in Section IV-C. It should be noted that, the
conventional superimposition scheme in [8] needs to verify the bit error rate
(BER) of UL-US. Considering that the same BER can be obtained as that of [8]
due to the same detection scheme, we here only verify the NMSE for the
recovery of downlink CSI.

###  IV-A Parameters Setting

Definitions involved in simulations are given as follows. The equivalent
signal-to-noise ratio (SNR) ratio and NMSE are defined the same way as [8].
During the experiments, $P=512$, $N=64$, and the compression rate $c$ is
defined as $c=M/N$. We adopt the Walsh matrix as the spreading matrix
${{\mathbf{Q}}_{l}}$ [8]. The UL-US ${\mathbf{d}}_{l}$ is formed by using QPSK
modulation. For AMPL-NET and AMPF-NET, training data-sets are generated by (6)
and (8) respectively, and testing data-sets are generated the same as training
data-sets. AMPL-NET and AMPF-NET are trained under the noise-free setting.
Training, validation, and testing sets have $70,000$, $15,000$, and $15,000$
samples, respectively. In this letter, the NMSE performance of the proposed
scheme is compared with those of [8] and [10].

Figure 3: (a) NMSE versus SNR, where $P{\rm{=512}}$, $c=2.0$, and $\rho=0.10$.
(b) Online running time comparison of “Proposed”, “Ref[8]”, and “Ref[10]”for
$10^{5}$ experiments, where $P{\rm{=512}}$, $\rho=0.20$, $\alpha=5$, and
$\beta=100$ is in both “Ref[8]” and “Ref[10]”.

###  IV-B NMSE Performance and Online Running Time

####  IV-B1 NMSE Performance

To validate the effectiveness of the proposed reconstruction scheme, NMSE
curves of the downlink CSI recovered by “Proposed”, “Ref[8]”, and “Ref[10]”
are given in Fig. 3 (a). In Fig. 3 (a), the NMSE curves in terms of the
downlink CSI’s recovery are presented, where $c=2.0$ and $\rho=0.10$ are
considered. The “Proposed” employs $5$ iterations of the simplified downlink
CSI reconstruction for feature extraction, i.e., $\alpha=5$. By contrast,
different iteration times (i.e., $\beta=5$, $\beta=10$, $\beta=20$,
$\beta=50$, and $\beta=100$) are given for downlink CSI reconstructions of
“Ref[8]” and “Ref[10]”. From Fig. 3 (a), the downlink CSI’s NMSE of “Proposed”
is smaller than those of the “Ref[8]” and “Ref[10]” in the whole SNR regions.
For example, when ${\textrm{SNR}}=10$dB, the NMSE of “Proposed” is about
$5.2\times 10^{-2}$, while “Ref[8]” and “Ref[10]” with $\beta=5$ reach about
$2.9\times 10^{-1}$ and $3.2\times 10^{-1}$, respectively. In a word, the
proposed scheme shows a better NMSE performance in all given SNR regions.

####  IV-B2 Online Running Time

To illustrate the improvement of processing time, the online running time of
“Proposed”, “Ref[8]”, and “Ref[10]” is compared in Fig. 3 (b) with $c$ varying
from $2.0$ to $3.0$. Especially, the “Ref[8]” and “Ref[10]” follow the setting
of $\beta=100$ in Fig. 3 (a). For a fair comparison, $10^{5}$ times of online
running experiments are separately conducted for “Proposed”, “Ref[8]”, and
“Ref[10]” on the same personal computer (with CPU i5-8250U) by using MATLAB
software. For each given $c$, the online running time of “Proposed” is shorter
than those of “Ref[8]” and “Ref[10]”. For instance, when $c=3.0$, the online
running time of “Proposed”, “Ref[8]”, and “Ref[10]” are $63.13s$, $1134.26s$,
and $1220.26s$, respectively. This reflects that the proposed scheme reduces
online running time. The reason for the improvement is that the trade-off is
achieved by balancing the off-line training and online running. In addition,
as $c$ varies from $2.0$ to $3.0$, the online running time of “Proposed”,
“Ref[8]”, and “Ref[10]” arises due to the increase of computational
complexity. On the whole, with the low running time and computational
complexity in Table II, the proposed scheme effectively reduces the processing
delay compared with “Ref[8]” and “Ref[10]”.

###  IV-C Robustness Evaluation

To verify the robustness 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. 4 (a). For each given $\rho$, the downlink
CSI’s NMSE of the “Proposed” is smaller than that of the “Ref[8]”. As the
increase of $\rho$ (increases from $0.05$ to $0.15$), the NMSEs decrease for
both “Ref[8]” and “Proposed”, and vice versa. The reason is that the downlink
CSI could 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 the relatively high SNR region. On the whole, for each given
value of $\rho$, the NMSE of “Ref[8]” is reduced by the “Proposed” in all
given SNR regions. Thus, the proposed scheme shows its robustness in improving
the NMSE performance against the impact of $\rho$.

Figure 4: NMSE versus SNR, where $P{\rm{=512}}$, $\alpha=5$, and $\beta=20$,
(a) $c=2.0$, (b) $\rho=0.10$.

To further validate the robustness against the impact of $c$, Fig. 4 (b)
depicts the NMSE performance of the downlink CSI recovery with compression
rate $c$ varying from $2.0$ to $3.0$. In Fig. 4 (b), for each given $c$, the
NMSE of the “Proposed” is smaller than that of the “Ref[8]”. Furthermore, when
${\textrm{SNR}}\leq 14$dB, the NMSE of “Proposed” increases as the increase of
$c$ (i.e., $c$ increases from $2.0$ to $3.0$). The possible reason is that the
higher compression rate results in lower spreading gain (i.e., $P/M$). In the
low SNR region, the NMSE performance is mainly impacted by noise interference
and limited by the low spread spectrum gain. However, the NMSE’s convergence
value of high compression rate is smaller than that of low compression rate.
For example, in the case of $c=2.0$, $c=2.5$, and $c=3.0$, the NMSE’s
convergence values of “Proposed” are about $4.03\times 10^{-2}$, $3.61\times
10^{-2}$, and $3.39\times 10^{-2}$, respectively. The reason for the analysis
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 $c$ , the NMSE of “Ref[8]” is reduced by the
“Proposed”. Thus, the proposed scheme possesses its robustness against the
impact of $c$.

To sum up, from Fig. 3 (a), the downlink CSI’s NMSE performance of “Ref[8]”
and “Ref[10]” is effectively improved by the “Proposed”. Moreover, Fig. 3 (b)
demonstrates that the proposed scheme has a shorter online running time,
thereby obtaining a lower processing delay. In addition, Fig. 4 (a) and Fig. 4
(b) show that the improvement of “Proposed” are robust against the impacts of
$\rho$ and $c$, respectively.

##  V Conclusion

This letter presents a fusion learning scheme for 1-bit CS-based superimposed
CSI feedback for mMIMO wireless communications. With a simplified downlink CSI
reconstruction and a lightweight AMPL-NET, the initial and auxiliary features
of the downlink CSI amplitude are extracted and then refined by a multimodal
feature-level AMPF-NET. Experiments show that, compared with the conventional
1-bit CS-based superimposed CSI feedback, the proposed scheme achieves a
significant improvement on NMSE performance of the downlink CSI recovery.
Besides, it presents a reduction in processing delays and robustness against
parameter variations.

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