# Joint Model and Data Driven Receiver Design for Data-Dependent Superimposed
Training Scheme with Imperfect Hardware

Chaojin Qing, Lei Dong, Li Wang, Jiafan Wang, and Chuan Huang C. Qing, L. Dong
and L. Wang are with the School of Electrical Engineering and Electronic
Information, Xihua University, Chengdu, 610039, China (E-mail:
qingchj@mail.xhu.edu.cn). J. Wang is with the Synopsys Inc., 2025 NE Cornelius
Pass Rd, Hillsboro, OR 97124, USA (E-mail: jifanw@gmail.com).C. Huang is with
the School of Science and Engineering, The Chinese University of Hong Kong,
Shenzhen, 518172, China (E-mail: huangchuan@cuhk.edu.cn).

###### Abstract

Data-dependent superimposed training (DDST) scheme has shown the potential to
achieve high bandwidth efficiency, while encounters symbol misidentification
caused by hardware imperfection. To tackle these challenges, a joint model and
data driven receiver scheme is proposed in this paper. Specifically, based on
the conventional linear receiver model, the least squares (LS) estimation and
zero forcing (ZF) equalization are first employed to extract the initial
features for channel estimation and data detection. Then, shallow neural
networks, named CE-Net and SD-Net, are developed to refine the channel
estimation and data detection, where the imperfect hardware is modeled as a
nonlinear function and data is utilized to train these neural networks to
approximate it. Simulation results show that compared with the conventional
minimum mean square error (MMSE) equalization scheme, the proposed one
effectively suppresses the symbol misidentification and achieves similar or
better bit error rate (BER) performance without the second-order statistics
about the channel and noise.

###### Index Terms:

Deep learning (DL), channel estimation, data detection, data-dependent
superimposed training (DDST), nonlinear distortion.

##  I Introduction

Data-dependent superimposed training (DDST) scheme has triggered wide research
interests due to its high bandwidth efficiency [1, 2, 3, 4, 5, 6, 7, 8, 9].
Relative to the conventional superimposed training (ST) scheme [10], the
training sequence in DDST scheme is combined with a judiciously selected data-
dependent sequence. With this modified training sequence, the DDST scheme can
efficiently eliminate the data-induced interference and achieve better
performance for channel estimation and data detection [1, 2, 3, 4, 5, 6, 7, 8,
9]. In essence, the DDST scheme fully cancels the effects of the unknown
transmission data on the performance of the channel estimator, and this leads
to various methods proposed to improve the data detection performance [11].
However, some information-bearing data at certain frequency bins are removed
prior to transmission, preventing linear receiver recovery of transmitted
symbols and resulting in symbol misidentification [2]. Meanwhile, wireless
communication systems with imperfect devices or blocks, e.g., high power
amplifier (HPA), and digital-to-analog converter (DAC), will inevitably cause
nonlinear distortion to the received signals [12], and give rise to great
challenges for existing schemes (e.g., the DDST scheme in [9]). For a mobile
device, the nonlinear distortion may be more serious due to its low-cost
components [13], limited computational resources [14], and low complexity
processing methods [15], etc. The nonlinear distortion destroys the
orthogonality of the training sequence [16] and confuses the modulation
constellation [17], so that the performance of the channel estimation and data
detection degrades seriously at the receiver side, which further aggravates
symbol misidentification.

To guarantee the benefits of DDST schemes, nonlinear distortion has to be
considered during practical applications. The previous study [12] suggests
that machine learning, in particular deep learning (DL), is prominent to solve
nonlinear problems. In recent years, DL-based physical-layer technique shows
its promising prospects in wireless communication system, e.g., signal
detection [18],[19], precoding [20], channel state information (CSI) feedback
[21], [22], and channel estimation [23], [24], etc. For channel estimation and
data detection, however, there are limited works focused on the DL-based DDST
scheme and even less on the nonlinear distortion caused by hardware
imperfection. Since the nonlinear distortion inevitably appears in wireless
communication systems [25], to guarantee the benefit of high bandwidth
efficiency of the DDST scheme, we investigate a joint model and data driven
receiver with the consideration of nonlinear distortion in this paper.

###  I-A Related Works

Relative to the ST scheme, the DDST scheme improved the overall performance of
channel estimation and data detection [1]. But, the partial information of
transmitted data symbols is discarded in the frequency domain, and thus the
data detection is degraded due to the symbol misidentification. In other
words, the channel estimation accuracy of DDST scheme is obtained at the cost
of degradation of data detection performance. To relieve this problem, various
methods have been proposed in [2, 3, 4, 5, 6, 7, 8, 9].

In [2], authors raised a partially DDST (PDDST) scheme to reduce symbol
misidentification by investigating the trade-off between interference
cancellation and frequency integrity. In reality, to improve the data
detection performance, partial performance of channel estimation was
sacrificed in the scheme of PDDST. This scheme was applied to the systems of
orthogonal frequency-division multiplexing in [3] with position shifting of
pilot signals and the restriction of the lowest power sum. In [4], a semi-
blind approach was proposed for channel estimation and data detection, in
which the first-order statistics of received data were developed for an
iteration approach with kernel weighted least squares (LS). With the increased
computational complexity, the semi-blind approach proposed in [4] enhances the
accuracy of channel estimation and therefore improves the data detection
performance. By considering the equi-spaced (square quadrature) amplitude
modulation, [5] investigated the data coding method to remove the error floor
of data detection in DDST scheme. In [5], the discarded symbol information was
retrieved according to the relevancy between encoding and decoding, which
avoids partial symbol misidentification. In [6], to avoid symbol
misidentification, a constellation rotation method was proposed and used to
preserve the partial symbol information discarded by DDST scheme. By using the
signal sub-space technique, [7] theoretically analyzed the phenomenon of
symbol misidentification (also called data identification problem). It was
shown mathematically that the occurrence of symbol misidentification was
related to both the modulation scheme and the pilot pattern. The precoding-
based approach is another family to improve data detection [8] [9]. By
developing the inter-symbols correlation, the precoding-based approach can
effectively retrieve the discarded symbol information. In [8], the Zadoff-Chu
sequence-based precoding matrix was exploited to resolve the symbol
misidentification. In particular, the detailed design criteria for the
precoding matrix were presented in [9], e.g., the precoding matrix should have
the form of the Kronecker product of an identity matrix and a diagonal matrix
with equal-magnitude diagonal elements, which presents good guidance.

Relative to the DDST scheme, except for the semi-blind approach in [4], the
training sequence-based improvements mentioned above require the transmitter
to be modified. In fact, the transmitter of the mobile device faces the
dilemmas of limited battery energy, storage space and computational resources.
This is impractical to modify the transmitter of mobile device by increasing
the processing complexity and storage space. Further, the nonlinear distortion
caused by imperfect hardware is usually time-varying, which makes the
transmitter modification ineffective since the receiver has to deal with the
nonlinear distortion again. In this paper, we investigate a receiver scheme
(i.e., without the modification of transmitter) for base station (BS) or
access point (AP) to improve the symbol misidentification of DDST with the
consideration of nonlinear distortion, which is different from the existing
schemes. To the best of our knowledge, very limited works discuss the impact
of nonlinear distortion on the DDST scheme, the receiver scheme to improve the
DDST, and the application of DL technology to reduce symbol misidentification.
Thus, we investigate a joint model and data driven receiver scheme to improve
the performance of DDST scheme affected by nonlinear distortion.

###  I-B Contributions

Focusing on the nonlinear distortion scenarios, the joint model and data
driven receiver scheme is proposed to improve channel estimation and data
detection in DDST. Our key insight is to view the hardware imperfection and
unsatisfactory recovery of the linear receiver in DDST as a nonlinear problem.
Then, the problem is handled by DL technology which shows its prominent
ability in dealing with nonlinearity. We combine the advantages of linear
receiver and nonlinear solution for tackling the symbol misidentification in
DDST scheme. Specifically, based on the conventional model driven modes, the
LS estimation and zero forcing (ZF) equalization are first employed to
highlight the initial linear features for channel estimation and data
detection. Then, with obtained initial features, we construct the shallow
neural networks, named as CE-Net and SD-Net, to refine the channel estimation
and data detection. Here, continuing with the linear receiver solutions, the
data driven mode is exploited to refine the channel estimation accuracy and
data detection performance by solving a nonlinear problem, thereby improving
the symbol misidentification. Our investigation remedies the deficiencies of
the existing DDST-based scheme which is facing difficulties in application due
to the lack of considerations of hardware imperfection.

The remainder of this paper is structured as follows: In Section II, we
present the system model of DDST scheme with nonlinear distortion. The
proposed receiver scheme is presented in Section III, and the numerical
results are given in Section IV. Finally, Section V concludes our work.

Notations: Bold face upper case and lower case letters denote matrix and
vector, respectively. ${\left(\cdot\right)^{T}}$, ${\left(\cdot\right)^{H}}$,
$\mathbf{(\cdot)^{\dagger}}$, stand for the transpose, conjugate transpose,
and matrix pseudo-inverse, respectively. $\bigotimes$ is the Kronecker
product. ${\mathop{\mathrm{Re}(\cdot)}}$ and ${\mathop{\mathrm{Im}(\cdot)}}$
denote the real and imaginary parts of a complex number, respectively.
$\mathbf{I}_{N}$ represents an $N\times N$ identity matrix.
${\mathrm{diag}(\cdot)}$ is the diagonalization operation of matrix.
${\mathrm{{\mathbb{BN}}}}\left(\cdot\right)$ denotes the operation of batch
normalization. ${\left\|\cdot\right\|_{2}}$ is the Euclidean norm.
${\mathbf{F}_{N}}$ is the normalized $N\times N$ fourier transform matrix [9].
$\max\left({a,b}\right)$ stands for taking the larger of $a$ and $b$.
$\mathbf{x}(M:N)$ represents the sub-vector of $\mathbf{x}$ by extracting
$\mathbf{x}$ from its $M$th entry to $N$th entry.

##  II System Model

As shown in Fig. 1, this paper considers a cyclic prefix (CP)-protected single
carrier system with the DDST scheme. At the transmitter, after the DDST
processing and adding CP, the transmitted signal experiences nonlinear
distortion caused by imperfect hardware. At the receiver, after the CP is
removed, the channel estimation and data detection are then performed. In the
following, we first describe the signal model at the transmitter, and then
present the received signal model.

Figure 1: System model for the DDST scheme with imperfect hardware.

###  II-A Transmit Signal Model

Fig. 2 shows how to construct one data frame with length $N$ at the
transmitter. In the preprocessing phase, a frame of data
$\mathbf{s}\in\mathbb{C}^{N\times 1}$ is first processed to obtain
${\mathbf{{s}}_{\mathrm{tds}}}$ [9], i.e.,

${{\mathbf{{s}}_{\mathrm{tds}}}={\mathbf{\Theta s}}},$ (1)

where $\mathbf{\Theta}\in\mathbb{C}^{N\times N}$ is a DDST preprocessing
matrix and is usually chosen as [1]

$\mathbf{\Theta}={{\mathbf{I}_{N}}-{\mathbf{J}}}.$ (2)

Here, $\mathbf{J}\in\mathbb{C}^{N\times N}$ is subtracted from
$\mathbf{I}_{N}$ to eliminate the interference of data sequence to training
sequence, and is usually defined as [1]

${{\mathbf{J}}=\frac{1}{Q}{{\mathbf{J}}_{Q}}\otimes{{\mathbf{I}}_{P}},}$ (3)

where ${\mathbf{J}_{Q}}\in\mathbb{C}^{Q\times Q}$ is given as

${{{\mathbf{J}}_{Q}}=\left[{\begin{array}[]{*{20}{c}}1&{{e^{\frac{{j2\pi
t\left({Q-1}\right)}}{Q}}}}&{{e^{\frac{{j2\pi
t\left({Q-2}\right)}}{Q}}}}&\ldots&{{e^{\frac{{j2\pi t}}{Q}}}}\\\
{{e^{\frac{{j2\pi t}}{Q}}}}&1&{{e^{\frac{{j2\pi
t\left({Q-1}\right)}}{Q}}}}&\ldots&{{e^{\frac{{j2\pi 2t}}{Q}}}}\\\
{{e^{\frac{{j2\pi 2t}}{Q}}}}&{{e^{\frac{{j2\pi
t}}{Q}}}}&1&\ldots&{{e^{\frac{{j2\pi 3t}}{Q}}}}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ {{e^{\frac{{j2\pi
t\left({Q-1}\right)}}{Q}}}}&{{e^{\frac{{j2\pi
t\left({Q-2}\right)}}{Q}}}}&{{e^{\frac{{j2\pi
t\left({Q-3}\right)}}{Q}}}}&\ldots&1\end{array}}\right]},$ (4)

with $Q$ and $P$ being constants, satisfying ${Q=\lfloor{N}/{P}\rfloor}$, and
$t$ being the shift variable for constellation rotation [6].

Figure 2: Construction of one data frame for DDST.

To estimate the channel, a training sequence with length $N$, denoted as
$\mathbf{c}$, is first constructed and then superimposed onto the processed
data $\mathbf{s}_{\mathrm{tds}}$ to form the transmitted signal. Similar to
[1], [6] and [9], the training sequence $\mathbf{c}$ is constructed by $Q$
repetitive $P$-period random sequences. By superimposing
${\mathbf{{s}_{\mathrm{tds}}}}$ with ${\mathbf{c}}$, the transmitted signal
$\mathbf{x}\in\mathbb{C}^{N\times 1}$ is then given as

${{\mathbf{x}}={\mathbf{{s}_{\mathrm{tds}}}}+{\mathbf{c}}}.$ (5)

Considering the nonlinear distortion caused by imperfect hardware, the real
transmitted signal is given as

${{\mathbf{x}_{\mathrm{dis}}}={f_{\mathrm{dis}}}\left({\mathbf{x}}\right)},$
(6)

where ${\mathbf{x}_{\mathrm{dis}}}$ denotes the distorted version of
$\mathbf{x}$, and ${f_{\mathrm{dis}}}(\cdot)$ represents the nonlinear-
distortion function [26].

###  II-B Received Signal Model

After the CP is removed, the received signal ${\mathbf{y}}$ with length $N$ is
given as

${{\mathbf{y}}={{\mathbf{H}_{c}}}{\mathbf{{x}_{\mathrm{dis}}}}+{\mathbf{v}},}$
(7)

where ${\mathbf{v}}\in\mathbb{C}^{N\times 1}$ denotes the circularly symmetric
complex Gaussian (CSCG) noise with mean zero and variance ${\sigma_{v}^{2}}$,
and ${\mathbf{H}_{c}}\in{{\mathbb{C}}^{N\times N}}$ is a cyclic matrix
reflecting the multi-path channel fading effect [9]. The first column of
${\mathbf{H}_{c}}$ is given as
${{\mathbf{h}}={\left[{{h_{0}}{\rm{,}}\cdots{\rm{,}}{h_{L-1}},0,\ldots,0}\right]^{\rm{T}}}}$
${\in{{\mathbb{C}}^{N\times 1}}}$, with ${L}$ being the number of channel
paths. Then, the received signal $\mathbf{y}$ after removing CP is rewritten
as

${{\mathbf{y}}={{\mathbf{H}_{c}}}{f_{\mathrm{dis}}}\left({\left({{\mathbf{I}_{N}}-{\mathbf{J}}}\right){\mathbf{s}}+{\mathbf{c}}}\right)+{\mathbf{v}}.}$
(8)

##  III Channel Estimation and Data Detection

In practical wireless communication systems, imperfect hardware is usually
employed by mobile devices to reduce the hardware cost, while causing serious
nonlinear distortion for transmitted signal [27], [28]. Nonlinear distortion
destroys the orthogonality between the received and local training sequence
[16] and confuses the modulation constellation [17], which seriously degrade
the performance of channel estimation and data detection.

To tackle this issue, we propose a joint model and data driven scheme to
design the receiver, which is shown in Fig. 3. Based on the conventional
structure of the receiver, the low-complexity and high-practicability LS
estimation and ZF equalization are first employed to capture the initial
features from the model driven channel estimation and data detection. Then, we
embed simple and effective shallow networks to respectively refine the channel
estimation and data detection by using the data driven method [29]. Different
from the conventional receiver, the proposed receiver incorporates linear and
nonlinear processing methods together by exploiting both the model and data
driven approaches. Compared to the conventional receiver using minimum mean
square error (MMSE) channel estimation and MMSE equalization, our receiver can
achieve similar or better detection performance. Besides, the second-order
statistics about the channel and noise could be avoid, which are difficult to
be obtained in practical applications [30].

In the following subsections, we first elaborate the channel estimation in
Section III-A. Then, in Section III-B, the details of data detection are
described. Finally, the online deployment is shown in Section III-C.

Figure 3: Architecture of various DDST receivers.

###  III-A Channel Estimation

The channel estimation for the proposed DDST receiver scheme is shown in Fig.
4, in which the LS estimation is first performed and then followed by a
shallow network named as CE-Net to refine the channel estimation.

####  III-A1 LS Estimation

We employ LS estimation to obtain an initial channel estimation in the
frequency domain, which also serves as the input of the CE-Net. The received
signal ${\mathbf{y}}$ given in (8) and the training sequence ${\mathbf{c}}$
are first transformed into the frequency domain, i.e.,

${\begin{array}[]{*{20}{c}}{{\mathbf{Y}}={{\mathbf{F}}_{N}}{\mathbf{y}}},\\\
{{\mathbf{C}}={{\mathbf{F}}_{N}}{\mathbf{c}}},\end{array}}$ (9)

where $\mathbf{Y}$ and $\mathbf{C}$ are the frequency domain representations
of received signal $\mathbf{y}$ and training sequence $\mathbf{c}$,
respectively. According to the LS algorithm [1], the frequency domain channel
estimator ${{\mathbf{\widehat{H}}_{P}}}\in\mathbb{C}^{P\times 1}$ is given as

${{\widehat{\mathbf{H}}}_{P}}={\left[{\frac{{Y\left(0\right)}}{{C\left(0\right)}},\frac{{Y\left(Q\right)}}{{C\left(Q\right)}},\ldots,\frac{{Y\left({\left({P-1}\right)Q}\right)}}{{C\left({\left({P-1}\right)Q}\right)}}}\right]^{T}},$
(10)

where $Y(k)$ and $C(k)$, $k=0,\ldots,N-1$, are frequency domain values for the
received signal $\mathbf{y}$ and the training sequence $\mathbf{c}$,
respectively. Then, we transform ${{\mathbf{\widehat{H}}_{P}}}$ into the time
domain, and obtain the time domain channel estimator
${\widehat{\mathbf{h}}}\in\mathbb{C}^{P\times 1}$ as

${{{{\widehat{\mathbf{h}}}}}={\mathbf{F}}_{P}^{H}{{\mathbf{\widehat{H}}}_{P}}},$
(11)

and add zero at the end of ${\widehat{\mathbf{h}}}$ to form a vector
$\mathbf{\widetilde{h}}$ with length $N$, i.e.,

${{\mathbf{\widetilde{h}}}={\left[{{\mathbf{{\widehat{\mathbf{h}}}}}^{T},\underbrace{0,\ldots,0}_{N-P}}\right]^{T}}}.$
(12)

The LS estimation $\mathbf{\widehat{H}}_{\mathrm{LS}}$ is obtained by
transforming $\mathbf{\widetilde{h}}$ into the frequency domain, i.e.,

${{{\mathbf{\widehat{H}}}_{\mathrm{LS}}}={{\mathbf{F}}_{N}}{\mathbf{\widetilde{h}}}}.$
(13)

Figure 4: Channel estimation for DDST receiver.

####  III-A2 Refining Estimation

After the LS estimation, we adopt the CE-Net to refine the channel estimation
by alleviating the influence of nonlinear distortion. The main steps are
explained as follows.

Constructing CE-Net: To construct a neural network, the network layers and the
neurons of each layer usually need to be first determined [31]. In CE-Net, the
neurons and layers are first set by referencing the methods in [32], and then
perform a fine parameter tuning according to a large number of experiments.
That is, the architecture of CE-Net is constructed to capture a trade-off
between its structure complexity and the system’s bit error rate (BER)
performance. The CE-Net is presented in Fig. 4, which consists of an input
layer, two hidden layers, and an output layer. Table I summarizes the network
architecture of CE-Net, and its detailed descriptions are given as follows.

TABLE I: Architecture of CE-Net Layer | Input | Hidden 1 | Hidden 2 | Output
---|---|---|---|---
Number of neurons | ${2N}$ | ${2N}$ | ${2N}$ | ${2N}$
Batch normalization | $\surd$ | $\times$ | $\times$ | $\times$
Activation function | None | ReLU | ReLU | Linear

  * •

The neurons of the input layer, hidden layers, and output layer are all set as
${2N}$ to simplify the network structure and accelerate the network training.

  * •

In order to accelerate the convergence of the network training and prevent the
overfitting problem [33], batch normalization (BN) is employed to normalize
the network inputs.

  * •

The activation function rectified linear unit (ReLU) defined as
${f_{a}}\left(x\right)=\max\left({0,x}\right)$, is adopted for the two hidden
layers due to its superiority in alleviating the gradient vanishing problem
[34]. For the output layer, the linear activation function is employed [21].

Channel estimation: We refine the channel estimation by exploiting the CE-Net,
which is given in Fig. 4. Usually, signals in common wireless systems are
complex-valued, and this brings difficulty to the DL-based network framework
which requires a real-valued data set. To address this issue, the complex-
valued ${{\mathbf{\widehat{H}}}_{\mathrm{LS}}}\in{\mathbb{C}}^{N\times 1}$ is
reshaped to real-valued
${{\mathbf{\widetilde{H}}}_{\mathrm{LS}}}\in{\mathbb{R}}^{2N\times 1}$, i.e.,

${{{\bf{\widetilde{H}}}_{\mathrm{LS}}}={{\left[{{\mathop{\rm
Re}\nolimits}\left({{\bf{\widehat{H}}}_{\mathrm{LS}}^{T}}\right),{\mathop{\rm
Im}\nolimits}\left({{\bf{\widehat{H}}}_{\mathrm{LS}}^{T}}\right)}\right]}^{T}}}.$
(14)

Then, the entries of ${{\bf{\widetilde{H}}}_{\mathrm{LS}}}$, i.e.,
$\mathop{\rm Re}\left(\widehat{H}_{\mathrm{LS}}(1)\right)$, $\mathop{\rm
Re}\left(\widehat{H}_{\mathrm{LS}}(2)\right)$, $\cdots$, $\mathop{\rm
Re}\left(\widehat{H}_{\mathrm{LS}}(N)\right)$, $\mathop{\rm
Im}\left(\widehat{H}_{\mathrm{LS}}(1)\right)$, $\cdots$, $\mathop{\rm
Im}\left(\widehat{H}_{\mathrm{LS}}(N)\right)$, compose the inputs of CE-Net.
By using the CE-Net, the refined CSI, denoted as
${{\mathbf{\widetilde{H}}}_{\mathrm{CE}}}\in{\mathbb{R}}^{2N\times 1}$, is
formulated as

${\mathbf{\widetilde{H}}_{\mathrm{CE}}}={f^{\left({\cal
L}\right)}}\left({\mathbf{W}_{\mathrm{CE}}^{\left({\cal
L}\right)}{f^{\left({{\cal
L}-1}\right)}}\left({\cdots{f^{\left(2\right)}}\left({\mathbf{W}_{\mathrm{CE}}^{\left(2\right)}{\mathbf{\widetilde{H}}_{\mathrm{LS}}}+\mathbf{b}_{\mathrm{CE}}^{\left(2\right)}}\right)}\right)+\mathbf{b}_{\mathrm{CE}}^{\left({\cal
L}\right)}}\right),$ (15)

where ${f^{\left(\ell\right)}}$,
$\mathbf{W}_{\mathrm{CE}}^{\left(\ell\right)}$, and
$\mathbf{b}_{\mathrm{CE}}^{\left(\ell\right)}$ are the activation function,
weight matrix and bias vector of the $\ell$th layer
$\forall\ell=2,\cdots,{\cal L}$, respectively. According to (15), we solve the
nonlinear problem to suppress the nonlinear distortion and obtain the refined
CSI without using the second-order statistics about channel and noise.

Model Training: After constructing the CE-Net architecture and determining the
network output, we need to train the CE-Net. The training details, i.e.,
training data collection, loss function description, and parameter
initialization, are described as follows.

  * •

Data collection: For the training of the CE-Net, the training set
$\left\\{{{{\mathbf{\widetilde{H}}}_{\mathrm{LS},}}{\mathbf{\widetilde{H}}_{\mathrm{Label}}}}\right\\}$
is generated as follows.

Considering the frequency-selective fading channel [9], $\mathbf{h}$ defined
in (7) is generated according to the widely adopted channel model COST2100
[35] without loss of generality, where the outdoor semi-urban scenario at the
300MHz band is considered. Then, $\mathbf{h}$ is transformed into the
frequency domain to form label ${\mathbf{{H}}_{\mathrm{Label}}}$, and we map
the complex-valued ${\mathbf{{H}}_{\mathrm{Label}}}$ to real-valued
${\mathbf{\widetilde{H}}_{\mathrm{Label}}}$. After we obtain the receiver
signal $\mathbf{y}$ from (8), the input of the CE-Net
${{\mathbf{\widehat{H}}}_{\mathrm{LS}}}$ is generated according to (9)–(13).
Finally, the complex-valued set of
$\\{{\mathbf{\widehat{H}}}_{\mathrm{LS}}\\}$ is reshaped to the real-valued
set $\\{{\mathbf{\widetilde{H}}}_{\mathrm{LS}}\\}$. We use training set
$\left\\{{{{\mathbf{\widetilde{H}}}_{\mathrm{LS},}}{\mathbf{\widetilde{H}_{\mathrm{Label}}}}}\right\\}$
to train CE-Net. In addition, to validate the trained network parameters
during the training phase, a validation set is also generated by using the
same generation method of training set, and thus we can capture a set of
optimized network parameters.

  * •

Loss function: To train CE-Net, mean squared error (MSE) is employed as the
loss function, which is defined as

${{{L}}_{{\mathrm{CE-
Net}}}=\frac{1}{{{T_{1}}}}\left\|{\mathbf{\widetilde{H}}_{\mathrm{Label}}-{\mathbf{\widetilde{H}}_{\mathrm{CE}}}}\right\|_{2}^{2}+{\alpha_{\mathrm{CE}}}\sum\limits_{\ell=2}^{4}{{{\left\|{\mathbf{W}_{\mathrm{CE}}^{\left(\ell\right)}}\right\|}_{2}^{2}}}},$
(16)

where ${T_{1}}$ denotes the number of training samples, and
$\alpha_{\mathrm{CE}}$ represents the regularization coefficient. Usually, the
regularization item is used to prevent overfitting and improve the
generalization performance [36]. Moreover, CE-Net’s performance (e.g., the CE-
Net’s estimation accuracy) is affected by $\alpha_{\mathrm{CE}}$. Thus, the
regularization coefficient $\alpha_{\mathrm{CE}}$ needs to be optimized, which
will be discussed in Section IV-B.

  * •

Weight and bias initialization: From [37], appropriate initialization can
effectively avoid the gradient exploding or vanishing problem. For expression
convenience, we use
$\left\\{{{{\mathbf{W}}_{\mathrm{CE}}^{\left(\ell\right)}};{{\mathbf{b}}_{\mathrm{CE}}^{\left(\ell\right)}}}\right\\}$,
${\ell}=2,3,4$, to represent the set of training parameters. For CE-Net
training, elements of
$\left\\{{{{\mathbf{W}}_{\mathrm{CE}}^{\left(\ell\right)}};{{\mathbf{b}}_{\mathrm{CE}}^{\left(\ell\right)}}}\right\\}$
are initialized as independent and identically distributed (i.i.d) Glorot
uniform distribution [38].

###  III-B Data Detection

The data detection for the proposed DDST receiver scheme is given in Fig. 5.
To avoid using the second-order statistics of the noise, the ZF equalization
is first employed, followed by a shallow network named as SD-Net to refine the
detection performance of DDST scheme.

Figure 5: Data detection for DDST receiver.

####  III-B1 ZF Equalization

From (5), the training sequence $\mathbf{c}$ is superimposed onto the
transmitted data sequence $\mathbf{s}$. To recover the data sequence, the
training sequence $\mathbf{c}$ has to be removed. Based on the DDST scheme
[9], we remove the training sequence $\mathbf{c}$ according to

${{\mathbf{\widehat{y}}}=\left({{\mathbf{I}_{N}}-{\mathbf{J}}}\right){\mathbf{y}}},$
(17)

where $\mathbf{\widehat{y}}\in{\mathbb{C}}^{N\times 1}$ denotes the recovery
of the received signal $\mathbf{y}$. In this paper, the frequency domain
equalization is utilized to combat the channel effect [9]. Thus,
${\mathbf{\widehat{y}}}$ in (17) is first transformed to the frequency domain
to form $\mathbf{\widehat{Y}}\in{\mathbb{C}}^{N\times 1}$ as

${{\mathbf{\widehat{Y}}}={{\mathbf{F}}_{N}}{\mathbf{\widehat{y}}}}.$ (18)

With the obtained ${\mathbf{\widehat{Y}}}$, the ZF equalization [32] is
employed to highlight the initial features of linear receiver for data
detection. Based on the ${{\mathbf{\widehat{H}}}_{\mathrm{CE}}}$ (i.e., the
refined CSI of CE-Net) and $\mathbf{\widehat{Y}}$, the ZF-based equalization
is given by

${{\mathbf{\widehat{S}}_{\mathrm{ZF}}}={{\mathbf{G}}_{\mathrm{ZF}}}{\mathbf{\widehat{Y}}}},$
(19)

where ${\mathbf{\widehat{S}}_{\mathrm{ZF}}}$ denotes the output of the ZF
equalization, ${{\mathbf{G}}_{\mathrm{ZF}}}\in{\mathbb{C}}^{N\times N}$ is the
ZF equalization matrix (an $N\times N$ diagonal matrix), i.e.,

${{{\mathbf{G}}_{\mathrm{ZF}}}=\left[{\begin{array}[]{*{20}{c}}{\frac{1}{{{\widehat{H}_{\mathrm{CE}}}\left(1\right)}}}&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{\frac{1}{{{\widehat{H}_{\mathrm{CE}}}\left(2\right)}}}&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&\ddots&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{\frac{1}{{{\widehat{H}_{\mathrm{CE}}}\left(N\right)}}}\end{array}}\right]},$
(20)

with ${\widehat{H}_{\mathrm{CE}}(m)},m=1,2,\cdots,N$, being the $m$th entry of
$\mathbf{\widehat{H}}_{\mathrm{CE}}$. In order to facilitate the data
detection in time domain, the time domain output of the ZF equalization,
denoted as ${\mathbf{\widehat{s}}_{\mathrm{ZF}}}$, is obtained according to

${\mathbf{\widehat{s}}_{\mathrm{ZF}}}={\mathbf{F}^{H}_{N}}{\mathbf{\widehat{S}}_{\mathrm{ZF}}}.$
(21)

####  III-B2 Refining Data Detection

After the ZF equalization, the SD-Net is utilized to alleviate the nonlinear
distortion by solving a nonlinear problem. The main steps are explained as
follows.

Constructing SD-Net: Similar to constructing CE-Net, the neurons and layers of
SD-Net are first set by referencing the methods in [32]. Then, we gradually
increase the neurons and layers until the BER can not be reduced significantly
to make a trade off between structure complexity and system’s BER performance.
The SD-Net, with architecture illustrated in Fig. 5, consists of an input
layer, three hidden layers, and an output layer. The network architecture of
SD-Net is summarized in Table II, and the detailed descriptions are given as
follows.

TABLE II: Architecture of SD-Net Layer | Input | Hidden 1 | Hidden 2 | Hidden 3 | Output
---|---|---|---|---|---
Number of neurons | ${2N}$ | ${2N}$ | ${12N}$ | ${6N}$ | ${2N}$
Batch normalization | $\surd$ | $\times$ | $\times$ | $\times$ | $\times$
Activation function | None | ReLU | ReLU | ReLU | Linear

  * •

Unlike the CE-Net, the SD-Net needs more neurons for hidden layers to enhance
the data detection (including the channel equalization) due to the information
discarding in DDST scheme. Based on extensive experiments of network training
and validating, the neurons of input layer, output layer, and the first hidden
layer are all set as ${2N}$, while ${12N}$ and ${6N}$ neurons are employed for
the second hidden layer and the third hidden layer, respectively.

  * •

Similar to the CE-Net, BN is adopted for the input layer of SD-Net, whose
inputs are normalized as zero mean and unit variance, to accelerate
convergence and prevent overfitting [33].

  * •

For the output layer, the linear activation function is adopted, while for the
three hidden layers, the activation function ReLUs are adopted.

Data detection: Based on the complex-valued
${\mathbf{\widehat{s}}_{\mathrm{ZF}}}$ given in (21), the real-valued form
${\mathbf{\widetilde{s}}_{\mathrm{ZF}}}\in\mathbb{R}^{2N\times 1}$ is given by

${{{\bf{\widetilde{s}}}_{\mathrm{ZF}}}={{\left[{{\mathop{\rm
Re}\nolimits}\left({{\bf{\widehat{s}}}_{\mathrm{ZF}}^{T}}\right),{\mathop{\rm
Im}\nolimits}\left({{\bf{\widehat{s}}}_{\mathrm{ZF}}^{T}}\right)}\right]}^{T}}}.$
(22)

The SD-Net is developed to refine the data detection performance by solving a
nonlinear problem. By denoting the output of the SD-Net as
${\mathbf{\widetilde{s}}_{\mathrm{SD}}}$, we have

${\mathbf{\widetilde{s}}_{\mathrm{SD}}}={f^{\left({\cal
I}\right)}}\left({\mathbf{W}_{\mathrm{SD}}^{\left({\cal
I}\right)}{f^{\left({{\cal
I}-1}\right)}}\left({\cdots{f^{\left(2\right)}}\left({\mathbf{W}_{\mathrm{SD}}^{\left(2\right)}{\mathbf{\widetilde{s}}_{\mathrm{ZF}}}+\mathbf{b}_{\mathrm{SD}}^{\left(2\right)}}\right)}\right)+{\mathbf{b}_{\mathrm{SD}}^{\left({\cal
I}\right)}}}\right),$ (23)

where $\mathbf{W}_{\mathrm{SD}}^{\left(i\right)}$ and
$\mathbf{b}_{\mathrm{SD}}^{\left(i\right)}$ are the weight matrix and bias
vector of the $i$th layer $\forall i=2,\cdots,{\cal I}$, respectively. Then,
the real-valued ${\mathbf{\widetilde{s}}_{\mathrm{SD}}}$ is reshaped to the
complex-valued ${\mathbf{\widehat{s}}}$. Compared with the data detection in
DDST with MMSE equalization [1], the refined ${\mathbf{\widehat{s}}}$ can
improve system’s BER performance, and this will be verified in Section IV.

Model training: The training details of the SD-Net, which include the training
data collection, loss function description and network parameters
initialization, are explained as follows.

  * •

Data collection: Similar to the data collection of CE-Net, the basic data
generation method is adopted in SD-Net to acquire the training set
$\\{{{\mathbf{\widetilde{s}}}_{\mathrm{ZF}}},{\mathbf{\widetilde{s}}}\\}$. For
training SD-Net, the trained network parameters of CE-Net, i.e.,
$\left\\{{{{\mathbf{W}}_{\mathrm{CE}}^{(\ell)}};{{\mathbf{b}}_{\mathrm{CE}}^{(\ell)}}}\right\\}$,
$\ell=2,3,4$, are fixed, and we employ the CE-Net to generate the refined CSI
${\mathbf{\widetilde{H}}}_{\mathrm{CE}}$ (or
${\mathbf{\widehat{H}}}_{\mathrm{CE}}$). According to (17)–(21), the detection
symbol ${{\mathbf{\widehat{s}}_{\mathrm{ZF}}}}$ is obtained. Then, we map the
complex-valued ${{\mathbf{\widehat{s}}_{\mathrm{ZF}}}}$ and modulated symbol
${\mathbf{\widetilde{s}}}$ to real-valued formations, and thus form the real-
valued ${\\{{\mathbf{\widetilde{s}}_{\mathrm{ZF}}}\\}}$ and
${\\{\mathbf{\widetilde{s}}\\}}$, respectively. Similar to the generation of
SD-Net’s training set, an SD-Net’s validation set is also generated.

  * •

Loss function: The loss function for SD-Net is given by

${{{L}_{\mathrm{SD-
Net}}}=\frac{1}{{{T_{2}}}}\left\|{\mathbf{\widetilde{s}}-\mathbf{\widetilde{s}}_{\mathrm{SD}}}\right\|_{2}^{2}+{\alpha_{\mathrm{SD}}}\sum\limits_{i=2}^{5}{{{\left\|{\mathbf{W}_{\mathrm{SD}}^{(i)}}\right\|}_{2}^{2}}}},$
(24)

where ${T_{2}}$ is the total number of samples in training set of SD-Net
training, ${\alpha_{\mathrm{SD}}}$ denotes the regularization coefficient of
the SD-Net. Similar to the optimization of ${\alpha_{\mathrm{CE}}}$ (see (16)
for details), the regularization coefficient ${\alpha_{\mathrm{SD}}}$ needs to
be optimized to improve SD-Net’s generalization performance, which is also
discussed in Section IV-B.

  * •

Weight and bias initialization: Similar to CE-Net, the elements of weight and
bias are initialized as i.i.d Glorot uniform distribution for SD-Net. For
expression convenience, we use
$\left\\{{{{\mathbf{W}}_{\mathrm{SD}}^{(i)}};{{\mathbf{b}}_{\mathrm{SD}}^{(i)}}}\right\\}$,
$i=2,3,4,5$, to represent the set of training parameters.

###  III-C Online Running

With the trained network parameters of CE-Net and SD-Net (by off-line
training), the online running procedure is presented in Table III.
Explanations of online running are given as follows.

TABLE III: Online Running Procedure Input: The received signal ${\mathbf{y}}$
and the training sequence ${\mathbf{c}}$
---
1): According to (9)–(13), the LS estimation is performed to obtain
${{\mathbf{\widehat{H}}_{\mathrm{LS}}}}$.
2): Use the CE-Net to acquire the refined channel estimation
${{\mathbf{\widehat{H}}_{\mathrm{CE}}}}$.
3): According to (17)–(21), the ZF equalization is implemented to obtain
${\mathbf{\widehat{s}}_{\mathrm{ZF}}}$.
4): Use the SD-Net to refine the data detection and obtain
${\mathbf{\widehat{s}}}$.
Output: The detected signal ${\mathbf{\widehat{s}}}$.

####  III-C1 Input Requirement

For online running, the received signal ${\mathbf{y}}$ (given in (7)) and the
known training sequence $\mathbf{c}$ are needed. To facilitate the signal
processing in the frequency domain, ${\mathbf{y}}$ and $\mathbf{c}$ are
transformed to the frequency domain according to (9) to form ${\mathbf{Y}}$
and $\mathbf{C}$, respectively.

####  III-C2 Channel Estimation

LS estimation: By using ${\mathbf{Y}}$ and ${\mathbf{C}}$ (given in (9)), the
LS estimation, in step 1) of Table III, is performed according to (9)–(13).
Then, the channel estimation ${{\mathbf{\widehat{H}}_{\mathrm{LS}}}}$ is
obtained, and thus forms the network input of CE-Net (i.e.,
${{\mathbf{\widetilde{H}}_{\mathrm{LS}}}}$) according to (14).

Channel estimation using CE-Net: With network input
${{\mathbf{\widetilde{H}}_{\mathrm{LS}}}}$, the CE-Net refines the channel
estimation, and thus acquires the refined CSI
${{\mathbf{\widetilde{H}}_{\mathrm{CE}}}}$ with real-valued form (given in
(15)). Then, the complex-valued form of refined CSI, i.e.,
${{\mathbf{\widehat{H}}_{\mathrm{CE}}}}$, is obtained according to

${\left\\{{\begin{array}[]{*{20}{l}}{{\mathop{\rm\mathrm{Re}}\nolimits}\left({{{\mathbf{\widehat{H}}}_{\mathrm{CE}}}}\right)={{\mathbf{\widetilde{H}}_{\mathrm{LS}}}}(1:N)}\\\
{{\mathop{\rm\mathrm{Im}}\nolimits}\left({{{\mathbf{\widehat{H}}}_{\mathrm{CE}}}}\right)={{\mathbf{\widetilde{H}}_{\mathrm{LS}}}}(N+1:2N)}\end{array}}\right.}.$
(25)

That is, the real part and imaginary part of
${{\mathbf{\widehat{H}}_{\mathrm{CE}}}}$ are formed by extracting the first
$N$ entries and the last $N$ entries of
${{\mathbf{\widetilde{H}}_{\mathrm{CE}}}}$, respectively. This procedure of
channel estimation by using CE-Net is listed as step 2) of Table III.

####  III-C3 Data Detection

ZF equalization: With the estimated ${{\mathbf{\widehat{H}}_{\mathrm{CE}}}}$,
the ZF equalization in the frequency domain is performed according to
(17)–(19), and thus obtains ${\mathbf{\widehat{S}}_{\mathrm{ZF}}}$. Then,
${\mathbf{\widehat{S}}_{\mathrm{ZF}}}$ is transformed to time-domain to form
${\mathbf{\widehat{s}}_{\mathrm{ZF}}}$ according to (21). This ZF equalization
procedure is given in step 3) of Table III. By utilizing (22), the real-valued
${\mathbf{\widetilde{s}}_{\mathrm{ZF}}}$ is formed based on the complex-valued
${\mathbf{\widehat{s}}_{\mathrm{ZF}}}$ to facilitate the real-valued
requirement of SD-Net.

Data detection using SD-Net: With the network input
${{\mathbf{\widetilde{s}}_{\mathrm{ZF}}}}$, the SD-Net refines the data
detection. Then, the SD-Net outputs the detected symbol
${{\mathbf{\widetilde{s}}_{\mathrm{SD}}}}$ (given in (23)). By extracting the
first $N$ entries and the last $N$ entries of
${{\mathbf{\widetilde{s}}_{\mathrm{SD}}}}$, the real part and imaginary part
of $\mathbf{\widehat{s}}$ are expressed as

${\left\\{{\begin{array}[]{*{20}{l}}{{\mathop{\rm\mathrm{Re}}\nolimits}\left({{{\mathbf{\widehat{s}}}}}\right)={{\mathbf{\widetilde{s}}_{\mathrm{SD}}}}(1:N)}\\\
{{\mathop{\rm\mathrm{Im}}\nolimits}\left({{{\mathbf{\widehat{s}}}}}\right)={{\mathbf{\widetilde{s}}_{\mathrm{SD}}}}(N+1:2N)}\end{array}}\right.}.$
(26)

This data detection by using SD-Net is presented in step 4) of Table III.

From step 1) to 4) in Table III, the refined channel estimation and data
detection can be achieved from the proposed CE-Net and SD-Net with the
received signal ${\mathbf{y}}$ and the known training sequence ${\mathbf{c}}$.
Compared with the conventional DDST with MMSE channel estimation and MMSE
equalization in [1], the proposed scheme can achieve a lower BER at the cost
of off-line training time, which is the usual feature of DL [39]. It is worth
noting that, the performance of the proposed scheme is improved without any
second-order statistic information of wireless channel $\mathbf{h}$ and noise
$\mathbf{v}$.

##  IV Simulation Results

In this section, we present the numerical results for the proposed receiver
scheme. The basic parameters and definitions involved in simulations are first
given in Section IV-A. Then, in Section IV-B, the model training is analyzed
by discussing the regularization coefficient and the training signal-to-noise
ratio (SNR). The effectiveness of the proposed receiver scheme is verified in
Section IV-C. At last, the analysis of parameters robustness are given in
Section IV-D.

###  IV-A Parameter Setting

In the experiments, the following basic parameters are applied unless
otherwise specified. From [1], $N=240$, $P=12$, and $t=0$ are considered. The
channel $\mathbf{h}$ defined in (7) is generated by channel model COST2100
[35] with outdoor semi-urban scenario at the 300MHz band, and the number of
multi-path is set as $L=12$. The transmitted data symbol ${\mathbf{{s}}}$
given in (1) is modulated by QPSK modulation. The training and validation sets
of both CE-Net and SD-Net have 60,000 and 20,000 samples, respectively. Their
batch sizes are all set as 80 samples. We use Adam optimizer as the training
optimization algorithm [40] with parameters ${\beta_{1}}=0.99$ and
${\beta_{2}}=0.999$ [41]. The learning rate is set to 0.0001, and the $L_{2}$
regularization [42] is adopted. The SNR in decibel (dB) is defined as

${\mathrm{SNR}=10{\log_{10}}\left({\frac{{{E_{\mathbf{x}_{\mathrm{dis}}}}}}{{\sigma_{v}^{2}}}}\right)},$
(27)

where ${E_{\mathbf{x}_{\mathrm{dis}}}}$ is the transmitted power of
${\mathbf{x}_{\mathrm{dis}}}$, which is equal to the summation of data-symbol
power ${E_{s}}$ and training-sequence power ${E_{c}}$. In these simulations,
${E_{s}}=0.9{\mathbf{x}_{\mathrm{dis}}}$ and
${E_{c}}=0.1{\mathbf{x}_{\mathrm{dis}}}$. For network training, the mixed SNR
is adopted, i.e., each training sample is generated under a random SNR from
$\mathrm{SNR}=0$dB to $\mathrm{SNR}=45$dB with the interval of 5dB.

For hardware imperfection, the effects of nonlinear distortion caused by HPA
are considered in this paper. From [43], the nonlinear amplitude and phase are
given by

$A\left(x\right)=\frac{{{\alpha_{a}}x}}{{1+{\beta_{a}}{x^{2}}}},~{}{\rm{}}\Phi\left(x\right)=\frac{{{\alpha_{\phi}}{x^{2}}}}{{1+{\beta_{\phi}}{x^{2}}}},$
(28)

where ${\alpha_{a}}=1.96$, ${\beta_{a}}=0.99$, ${\alpha_{\phi}}=2.53$, and
${\beta_{\phi}}=2.82$ are considered in the simulations.

Furthermore, we use error vector magnitude (EVM) to measure distortion
intensity, which is defined as [44]

${{\mathrm{EVM}}=\sqrt{\frac{{\sum\limits_{n\in
N}{{{\left|{{{\widetilde{x}}_{n}}-{R_{n}}}\right|}^{2}}}}}{{\sum\limits_{n\in
N}{{{\left|{{R_{n}}}\right|}^{2}}}}}}\times 100\%},$ (29)

where ${\widetilde{x}_{n}}$ and ${R_{n}}$ are the distorted and ideal
undistorted outputs of HPA, respectively. That is, for the same input,
${\widetilde{x}_{n}}$ and ${R_{n}}$ are the outputs when HPA works in
saturated region and ideal linear region, respectively. Except for the
robustness analysis against EVMs, we set the EVM as ${55\%}$.

The training and testing of proposed method are carried out on a server with
NVIDIA TITAN RTX GPU and Intel Xeon(R) E5-2620 CPU 2.1GHz$\times$16, and the
results of DDST scheme are obtained by using MATLAB simulation on the server
CPU due to the lack of a GPU solution.

For the convenience of expression, the simplified expressions in the
simulations are given as follows.

  * •

“Proposed” is used to denote the proposed receiver scheme, i.e., the CE-Net
and SD-Net are jointly applied.

  * •

“LS_CE + ZF_SD”, “LS_CE + MMSE_SD”, “MMSE_CE + ZF_SD”, and “MMSE_CE + MMSE_SD”
represent “LS channel estimation followed by ZF equalization”, “LS channel
estimation followed by MMSE equalization”, “MMSE channel estimation followed
by ZF equalization”, and “MMSE channel estimation followed by MMSE
equalization” in the conventional DDST scheme, respectively. We utilize “LS_CE
+ ZF_SD” and “MMSE_CE + MMSE_SD” as the baselines of the low computational
complexity and high detection performance of DDST, respectively.

  * •

“CE-Net + ZF_SD”, “CE-Net + MMSE_SD”, “LS_CE + SD-Net”, and “MMSE_CE + SD-Net”
stand for the “proposed CE-Net followed by ZF equalization”, “proposed CE-Net
followed by MMSE equalization”, “LS channel estimation followed by the
proposed SD-Net”, and “MMSE channel estimation followed by the proposed SD-
Net”, respectively.

###  IV-B Regularization Coefficient and Training SNR Optimization

Usually, the model training is influenced by regularization coefficient and
training SNR [45, 46]. To obtain the optimized learning parameters, the
impacts of regularization coefficient and training SNR are analysed by using
the following simulation experiments.

####  IV-B1 Regularization Coefficient

From (16) and (24), the $L_{2}$ regularization is employed to CE-Net and SD-
Net. To capture an optimized regularization coefficient, the training and
validation losses are given in the Fig. 6 and Fig. 7 for CE-Net and SD-Net,
respectively.

In Fig. 6, different values of ${\alpha_{\mathrm{CE}}}$ (i.e.,
${\alpha_{\mathrm{CE}}}={10^{-2}},{10^{-3}},\ldots,{10^{-7}}$) are employed
for CE-Net, where the numbers of the training batches and validation epochs
are set as ${\rm{6}}\times{\rm{1}}{{\rm{0}}^{\rm{4}}}$ and 100, respectively.
From Fig. 6, it could be observed that different regularization coefficients
achieve different convergence values for the training or validation loss. Even
so, the convergence values of the training and validation losses are almost
the same for a given ${\alpha_{\mathrm{CE}}}$, which shows the CE-Net
possesses a good generalization performance. There exists a weak correlation
between the convergence rate and the convergence value. Among the given values
of ${\alpha_{\mathrm{CE}}}$, the fastest convergence rate of training (or
validation) loss is observed when ${\alpha_{\mathrm{CE}}}={10^{-7}}$, while
reaching the maximum convergence value. On the contrary, the slowest
convergence rate of training (or validation) loss is obtained when
${\alpha_{\mathrm{CE}}}={10^{-6}}$, but its convergence value is not the
minimum one. From the plotted curves in Fig. 6, the smallest convergence value
is achieved when ${\alpha_{\mathrm{CE}}}={10^{-5}}$.

Figure 6:  Training and validation losses against the impact of regularization
coefficient of CE-Net. Figure 7:  Training and validation losses against the
impact of regularization coefficient of SD-Net.

In Fig. 7, to harvest an optimized regularization coefficient, different
values of ${\alpha_{\mathrm{SD}}}$ (i.e.,
${\alpha_{\mathrm{SD}}}={10^{-2}},{10^{-3}},\ldots,{10^{-7}}$) are tested in
the SD-Net. The numbers of the training batches and validation epochs are set
as ${\rm{1.8}}\times{\rm{1}}{{\rm{0}}^{\rm{5}}}$ and 300, respectively. From
Fig. 7, for each given regularization coefficient ${\alpha_{\mathrm{SD}}}$,
the convergence values of the training and validation losses are almost the
same, which indicates the good generalization performance of the SD-Net. In
addition, when $\alpha_{\mathrm{SD}}$ is set as $10^{-7}$, the convergence
value of training (or validation) loss is the minimum one while its
convergence rate is neither the fastest nor the slowest (the fastest and
slowest convergence rates are obtained by $\alpha_{\mathrm{SD}}=10^{-2}$ and
$\alpha_{\mathrm{SD}}=10^{-3}$, respectively). This phenomenon shows a weak
correlation between the convergence rate and the convergence value. Even so,
the minimum convergence value of validation loss is still observed with high
probability when $\alpha_{\mathrm{SD}}=10^{-7}$.

In this paper, to obtain a good performance (e.g., the estimation accuracy,
detection rightness, and generalization, etc), we choose the regularization
coefficients according to the minimum validation loss with high probability,
thereby $\alpha_{\mathrm{CE}}=10^{-5}$ and $\alpha_{\mathrm{SD}}=10^{-7}$ are
employed for CE-Net and SD-Net, respectively.

Figure 8:  BER performances against the impact of training SNR, where $N=240$,
$P=12$, $L=12$, and $\mathrm{EVM}=55\%$.

####  IV-B2 Training SNR

Figure 9: BER performance for different methods vs. SNR, where $N=240$,
$P=12$, $L=12$, and $\mathrm{EVM}=55\%$.

As mentioned in [46], the training SNR usually causes DL network’s performance
fluctuation. To analyze the impact of training SNR on model training, the BER
curves with different training SNRs are plotted in Fig. 8.

In Fig. 8, ${{\mathrm{SNR}}=5}$dB (low training SNR), mixed SNR (each training
sample is generated under a random SNR from ${{\mathrm{SNR}}=0}$dB to
${{\mathrm{SNR}}=45}$dB with the interval of 5dB), and ${{\mathrm{SNR}}=45}$dB
(high training SNR) are considered as the training SNRs. For training SNR
being $5$dB, the “Proposed” captures similar BER with “MMSE_CE + MMSE_SD” in
the relatively low testing SNR region (e.g., ${{\mathrm{SNR}}\leq 10}$dB),
while appearing much higher BER than that of “MMSE_CE + MMSE_SD” in the
relatively high testing SNR region (e.g., ${{\mathrm{SNR}}\geq 20}$dB). It
seems that the low training SNR is unsuitable for training the proposed
network worked in the high SNR region. Although the high training SNR can
bring a low BER when the testing SNR is larger than 25dB, it cannot reap a
satisfied BER in the low SNR region (e.g., the testing SNR is lower than
10dB). Hence, the training $\mathrm{SNR}=45$dB is also unsuitable for the
proposed scheme. When mixed SNR is adopted as the training SNR, the BER of
“Proposed” is similar or lower than that of the “MMSE_CE + MMSE_SD” for the
testing SNRs varying from 0dB to 30dB. From Fig. 8, both the low training SNR
and the high training SNR have their drawbacks, especially for the case where
the testing SNR is far from the training SNR. Thus, we employ the mixed SNR as
the training SNR for a trade-off consideration.

###  IV-C Analysis of Simulation Performance

To verify the effectiveness of the proposed receiver scheme, the BER
performance of the proposed receiver is compared against that of the
conventional DDST scheme in [1]. The performance curves of BER are given in
Fig. 9, and partial numerical results are presented in TABLE IV for the
convenience of comparison.

TABLE IV: The BER of Each Method Method SNR | 0dB | 3dB | 6dB | 9dB | 12dB | 15dB | 18dB | 21dB | 24dB | 27dB | 30dB
---|---|---|---|---|---|---|---|---|---|---|---
LS_CE + ZF_SD | 0.470 | 0.455 | 0.433 | 0.401 | 0.359 | 0.306 | 0.2447 | 0.1869 | 0.1375 | 0.1067 | 0.0881
LS_CE + SD_Net | 0.459 | 0.441 | 0.417 | 0.383 | 0.337 | 0.279 | 0.2177 | 0.1647 | 0.1237 | 0.0989 | 0.0846
LS_CE + MMSE_SD | 0.423 | 0.392 | 0.355 | 0.307 | 0.255 | 0.201 | 0.1491 | 0.1122 | 0.0893 | 0.0772 | 0.0715
MMSE_CE + ZF_SD | 0.419 | 0.381 | 0.333 | 0.273 | 0.204 | 0.135 | 0.0810 | 0.0493 | 0.0335 | 0.0254 | 0.0216
MMSE_CE + MMSE_SD | 0.355 | 0.309 | 0.259 | 0.206 | 0.152 | 0.103 | 0.0672 | 0.0442 | 0.0307 | 0.0238 | 0.0206
MMSE_CE + SD_Net | 0.365 | 0.312 | 0.261 | 0.212 | 0.158 | 0.105 | 0.0605 | 0.0359 | 0.0250 | 0.0204 | 0.0185
CE_Net + ZF_SD | 0.408 | 0.372 | 0.324 | 0.259 | 0.186 | 0.113 | 0.0570 | 0.0268 | 0.0136 | 0.0085 | 0.0065
CE_Net + MMSE_SD | 0.343 | 0.302 | 0.255 | 0.199 | 0.140 | 0.085 | 0.0457 | 0.0233 | 0.0127 | 0.0081 | 0.0063
CE_Net + SD_Net | 0.342 | 0.301 | 0.255 | 0.198 | 0.140 | 0.084 | 0.0456 | 0.0221 | 0.0115 | 0.0071 | 0.0049

From Fig. 9 and TABLE IV, the “Proposed” achieves the minimum BER. That is,
the BER performance of “Proposed” outperforms those of “MMSE_CE + SD_Net”,
“CE_Net + ZF_SD”, “CE_Net + MMSE_SD”, and “MMSE_CE + MMSE_SD”, especially in
the relatively high SNR region (e.g., ${{\mathrm{SNR}}\geq 20}$dB). In the
relatively low SNR region (e.g., ${{\mathrm{SNR}}\leq 5}$dB), the “Proposed”
has similar BER performance as those of “MMSE_CE + SD_Net”, “CE_Net + ZF_SD”,
“CE_Net + MMSE_SD”, and “MMSE_CE + MMSE_SD”. The “LS_CE + ZF_SD” has the worst
BER performance due to the lack of channel and noise’s second-order
statistics. As a whole, even compared with “MMSE_CE + MMSE_SD”, the proposed
receiver scheme achieves a lower BER, while avoiding the requirements of
channel and noise’s second-order statistics.

In addition, the effectiveness of the proposed CE-Net and SD-Net is also
verified in Fig. 9. For CE-Net, the BERs of “CE_Net + ZF_SD” and “CE_Net +
MMSE_SD” are not higher than those of “MMSE_CE + ZF_SD” and “MMSE_CE +
MMSE_SD” in all given SNR regions. In particular, in the relatively high SNR
region, the BER of “CE_Net + ZF_SD” or “CE_Net + MMSE_SD” is lower than those
of “MMSE_CE + ZF_SD” and “MMSE_CE + MMSE_SD”. Thus, the proposed CE-Net has
its effectiveness in improving the accuracy of channel estimation. For the SD-
Net, the BER of the “LS_CE + SD_Net” is lower than that of “LS_CE + ZF_SD” in
all SNR regions, and the “MMSE_CE + SD_Net” obtains a lower BER than “MMSE_CE
+ MMSE_SD” when ${{\mathrm{SNR}}>15}$dB while maintaining similar BER for
${{\mathrm{SNR}}\leq 15}$dB. These reflect that the proposed SD-Net improves
the BER performance of DDST without the information of noise’s second-order
statistics.

Figure 10: BER performance against the impact of EVM, where
$\mathrm{EVM}=45\%$, $\mathrm{EVM}=50\%$, $\mathrm{EVM}=55\%$,
$\mathrm{EVM}=60\%$, and $\mathrm{EVM}=65\%$ are considered, respectively.

###  IV-D Analysis of Parameter Impact

In this subsection, the robustness of the proposed receiver scheme against the
impacts of EVM and $L$ is analysed. For the ease of analysis, only one impact
parameter is changed while other basic parameters are freezed during the
simulations.

####  IV-D1 Impact of EVM

Usually, the signals with different amplitudes cause different distortion
degrees when they pass through the same HPA working in the saturated region.
In order to reflect the robustness of the proposed receiver scheme, we analyze
the parameter impact against EVM in Fig. 10.

In Fig. 10, the EVM varies from $45\%$ to $65\%$ with the interval of $5\%$.
For each given EVM, the “Proposed” achieves the minimum BER compared with
“LS_CE + ZF_SD” and “MMSE_CE + MMSE_SD”. This reflects that the “Proposed” can
improve the BER performance of DDST with little impact from EVM. In
particular, for the relatively high SNR region, e.g., ${{\mathrm{SNR}}\geq
20}$dB, the BER of “Proposed” is significantly lower than those of “LS_CE +
ZF_SD” and “MMSE_CE + MMSE_SD”. In addition, the improvement of BER
performance can be observed for each given EVM. Thus, the proposed receiver
scheme can significantly improve the BER performance of DDST scheme, and this
improvement is robust to the EVM variations.

Figure 11: BER performance against the impact of $L$, where $L=4$, $L=6$,
$L=8$, $L=10$, and $L=12$ are considered, respectively.

####  IV-D2 Impact of $L$

The BER performance is usually influenced by the number of multi-path, i.e.,
${L}$. To illuminate the robustness against multi-path impact, the BER
performance is tested with different values of $L$ in Fig. 11, where $L=4$,
$L=6$, $L=8$, $L=10$, and $L=12$ are considered.

From Fig. 11, the “Proposed” achieves the minimum BER for each given ${L}$
compared with “LS_CE + ZF_SD” and “MMSE_CE + MMSE_SD”. This reflects that the
proposed receiver scheme can improve the BER performance of DDST with “LS_CE +
ZF_SD” or “MMSE_CE + MMSE_SD”. Especially in the relatively high SNR region,
e.g., ${{\mathrm{SNR}}\geq 20}$dB, the BER of “Proposed” is significantly
lower than those of “LS_CE + ZF_SD” and “MMSE_CE + MMSE_SD”. This verifies
that the proposed receiver scheme possesses its robustness against the impact
of $L$.

##  V Conclusion

In this paper, a joint model and data driven receiver scheme for DDST is
proposed to alleviate the symbol misidentification caused by hardware
imperfection and unsatisfactory recovery of received symbols. This scheme
exploits the advantages of linear receivers and nonlinear solutions to tackle
nonlinearity. Specifically, the proposed receiver scheme first adopts the
model driven modes, employing LS estimation and ZF equalization to highlight
the initial features of linear receivers. Then, based on the obtained
features, two shallow neural networks, CE-Net and SD-Net are constructed with
data driven mode to refine the channel estimation and data detection,
respectively. Compared with existing DDST schemes with MMSE based channel
estimation and equalization, the CE-Net and SD-Net in the proposed receiver
achieve similar or better data detection performance without the second-order
statistics of the channel and noise. In addition to the effectiveness in
alleviating the hardware imperfection and maintaining the BER performance
against parameter variations, the proposed receiver scheme avoids the
challenge of mobile device modification in DDST systems and promotes the
existing researches move towards practical application.

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|  Chaojin Qing (M’15) received the B.S. degree in communication engineering
from Chengdu University of Information Technology, Chengdu, China, in 2001,
the M.S. and Ph.D. degrees in communications and information systems from the
University of Electronic Science and Technology of China, Chengdu, China, in
2006 and 2011, respectively. From November 2015 to December 2016, he was a
Visiting Scholar with Broadband Communication Research Group (BBCR) of the
University of Waterloo, Waterloo, ON, Canada. From 2001 to 2004, he was a
teacher with the Communications Engineering Teaching and Research Office,
Chengdu University of Information Technology, Chengdu, China. Since 2011, he
has been an Assistant Professor with the School of Electrical Engineering and
Electronic Information, Xihua University, Chengdu, China. He is the author of
more than 50 papers and more than 20 chinese inventions. His research
interests include detection and estimation, massive MIMO systems, and deep
learning in physical layer of wireless communications.
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|  Lei Dong received the B. S. degree from the School of Electrical
Engineering and Electronic Information, Xihua University, Chengdu, China, in
2019, where he is currently pursuing the M. S. degree under the supervision of
Prof. Qing. His research interests include superimposed training based signal
detection and channel estimation, and deep learning in physical layer of
wireless communications.
---|---
|  Li Wang received the B. S. degree from the School of Electrical Engineering
and Electrical Information, Xihua University, Chengdu, China, in 2020, where
she is currently pursuing the M. S. degree under the supervision of Prof.
Qing. Her research interests include channel estimation, signal detection, and
deep learning in physical layer of wireless communications.
---|---
|  Jiafan Wang (S’15) received his B.S. degree and M.S. degree in Electrical
Engineering from University of Electronic Science and Technology of China in
2006 and 2009, respectively. He accomplished the Ph.D. degree in Computer
Engineering at Texas A$\&$M University, College Station, TX, USA in 2017. His
major field of study is smart integrated circuit design, which includes multi-
dimensional non-deterministic gate implementation with systematic optimization
framework, self-training Analog/Digital Mixed-System for circuit feature
calibration, and configurable locking mechanism against Analog IP piracy. He
is currently working in Synopsys Inc. to develop the world’s leading silicon
chip design software in electronic design automation (EDA) industry.
---|---
|  Chuan Huang (S’09–M’13) received the B.S. degree in math and the M.S.
degree in communications engineering from the University of Electronic Science
and Technology of China(UESTC), Chengdu, and the Ph.D. degree in electrical
engineering from Texas A$\&$M University, College Station, TX, USA, in 2012.
From 2012 to 2013, he was with Arizona State University, Tempe, AZ, USA, as a
Post-doctoral Research Fellow, and then promoted to Assistant Research
Professor from 2013 to 2014. He was also a Visiting Scholar with the National
University of Singapore and a Research Associate with Princeton University.
From January 2015 to February 2020, he was a professor at UESTC, Chengdu,
China. He is currently an associate professor with the Chinese University of
Hong Kong, Shenzhen, China. He served as a Symposium Chair of IEEE GLOBECOM
2019 and IEEE ICCC 2019 and 2020. He is now serving as an Editor of IEEE
Transactions on Wireless Communications, IEEE Access, and IEEE Wireless
Communications Letters. His current research interests include wireless
communications and signal processing.
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