# ELM-based Timing Synchronization for OFDM Systems by Exploiting Computer-
aided
Training Strategy

Mintao Zhang, Shuhai Tang, Chaojin Qing, Na Yang, Xi Cai, and Jiafan Wang The
authors (_Corresponding author: Chaojin Qing_) are with the School of
Electrical Engineering and Electronic Information, Xihua University, Chengdu
610039, China (E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and
jifanw@gmail.com).

###### Abstract

Due to the implementation bottleneck of training data collection in realistic
wireless communications systems, supervised learning-based timing
synchronization (TS) is challenged by the incompleteness of training data. To
tackle this bottleneck, we extend the computer-aided approach, with which the
local device can generate the training data instead of generating learning
labels from the received samples collected in realistic systems, and then
construct an extreme learning machine (ELM)-based TS network in orthogonal
frequency division multiplexing (OFDM) systems. Specifically, by leveraging
the rough information of channel impulse responses (CIRs), i.e., root-mean-
square (r.m.s) delay, we propose the loose constraint-based and flexible
constraint-based training strategies for the learning-label design against the
maximum multi-path delay. The underlying mechanism is to improve the
completeness of multi-path delays that may appear in the realistic wireless
channels and thus increase the statistical efficiency of the designed TS
learner. By this means, the proposed ELM-based TS network can alleviate the
degradation of generalization performance. Numerical results reveal the
robustness and generalization of the proposed scheme against varying
parameters.

###### Index Terms:

Timing synchronization, orthogonal frequency division multiplexing, computer-
aided, extreme learning machine.

##  I Introduction

Orthogonal frequency division multiplexing (OFDM), as a kind of the modern
wireless and mobile communication systems, has been subject to extensive
research efforts not only from the 5G field communication systems [1] but also
from the wireless local area networks (WLANs) [2] and the Internet-of-Things
(IoT) systems [3]. For the OFDM systems, timing synchronization (TS) plays a
pivotal role in overall system performance, which aims to locate starting of
the receiver fast Fourier transform (FFT) window within the inter-symbol-
interference (ISI)-free region[4]. For the TS, extensive studies have been
conducted so far to improve the timing metric for OFDM systems [5, 6, 7, 8, 9,
10]. However, classic TS schemes inevitably encounter nonlinear effects, e.g.,
channel fading, hardware imperfections, etc, posing a great challenge to avoid
the ISI in the TS stage, and affecting the subsequent signal processing, e.g.,
channel estimation [11] and symbol detection [12], etc.

Due to its powerful capability in coping with nonlinear problems [13], machine
learning (ML) is a promising solution to tackle the challenge of nonlinear
effects in the TS stage and also aroused extensive interest in the design of
the physical layer communications [14, 15, 16, 17, 18, 19] over the past few
years. In ML-based applications, substantial researches have proven that the
predictive performance of supervised learning outperforms that of unsupervised
learning because the supervised ones could directly take feedback for the
prediction [20]. As for synchronization, several supervised learning-based
schemes [21, 22, 23, 24, 25] have been investigated. For example in [21], a
deep neural network (DNN)-based estimator was constructed to estimate the
timing offset and carrier frequency offset (CFO). On the basis of DNN-based
auto-encoders, the authors in [22] and [23] separately implemented the timing
synchronization and the sampling time synchronization. With DNN approaches,
the authors in [24] investigated the packet detection. In [25], an extreme
learning machine (ELM) was employed to estimate the residual symbol timing
offset (STO). From [21, 22, 23, 24, 25], previous studies suggested that the
performance improvements for synchronization can be achieved by employing
supervised learning.

Although supervised learning-based TS could provide performance improvements,
the implementation bottleneck, essentially, will be encountered in the
training data collection. For classification tasks such as TS, training data
set is consist of the collected samples and their corresponding learning
labels (e.g., labels by STOs). In the context of realistic wireless
communication systems, collecting training data is time and labor consuming,
especially for the supervised learning-based TS. For the supervised learning-
based TS learner, the implementation bottleneck of training data collection
and its correspondingly possible solution mainly exist in the following
aspects:

  1. 1.

Precisely labeling the ground truth labels of TS is impractical to the
receiver in realistic wireless communication systems. For one thing, due to
unknown attributes of wireless channels, e.g., STOs, CFOs, and Channel Impulse
Response (CIRs), the availability and optimal timing index of the received
samples cannot be judged correctly. For another, conventional digital
receivers distinguish the start point of collected samples according to the
sampling grid. Namely, the received samples on off-grid are bound to be lost,
making it challenging to precisely label the collected samples. As a result,
the correlation between the estimated-based labels and the ground truth labels
has been weakened, leading to the degradation of learning efficiency for the
supervised learning-based TS learner. _To solve this issue, desired training
samples and their corresponding learning labels can be generated by computers,
tackling the challenge of generating the ground truth labels in TS. Superior
to the collected data, the computer-aided generated training data could
perfectly know the generation parameters and methods (e.g., STO, CFO, path
delays, etc.) while the former one cannot._

  2. 2.

The completeness of training data collection is hindered in realistic wireless
communication systems. In the long term, the realistic wireless channel
environments are mainly affected by the climate or weather factors, such as
the alternation of spring, summer, autumn, and winter, the temperature
variation in the morning and evening, and the environments variation in rainy
and sunny days. In the short term, the realistic wireless channel environments
are mainly affected by obstacles in the cellular, such as vehicles and
pedestrians. Thus, the maximum multi-path delay is time-varying, which makes
it impractical to collect complete training data (including training samples
and their corresponding learning labels) from the realistic wireless channels.
_To alleviate this challenge, more complete training samples can be generated
by computers[26], alleviating the completeness problem of the data collection
in a realistic wireless communication. The insufficiency of training data may
reduce the statistical efficiency of the supervised learning-based TS learner,
while computers can generate rich data that inclose the changes of TS with
environments. Namely, by employing a large number of wireless communication
scenarios, the computer-aided approach can generate data more sufficiently
than collect the desired data from realistic wireless communication systems.
Especially, the generated data features are more conducive and faster to be
obtained than those cannot be directly obtained by digital receivers [27],
avoiding the time and labor consumptions._

  3. 3.

Storing and transmitting the collected large-scale training data may cause
substantial disasters. On the one hand, the supervised learning-based TS
learner requires sufficient data to improve the adaptability to the changing
wireless environments. However, large-scale data collection inevitably leads
to great difficulty in storage capacity. On the other hand, large-scale
training data transmission will bring unacceptable bandwidth requirements to
the current wireless communication systems. Given the rapidly increasing
demand for bandwidth resources, limited transmission bandwidth is mainly taken
by the business data of users. Namely, it is unlikely to set aside valuable
bandwidth resources merely for sending massive collected samples for network
learning. _To mitigate the pressures of storing and transmitting the collected
large-scale training data for the TS, a computer-aided approach can deploy the
corresponding algorithm of data generation at the local devices. By providing
the constraints of data features, the desired data can be generated rapidly
and released temporarily, which alleviates the pressure from the requirement
of huge storage space. Besides, it is not necessary to transmit data for
network training, avoiding the large bandwidth requirement of data
transmission for supervised learning-based TS learners._

In addition, the current supervised learning-based TS learners relying on the
DNN models (e.g., [22, 23, 24]) have achieved remarkable success in improving
the TS performance. However, several public problems are still required to be
solved, such as complex parameter tuning, long-time training, and local
optimal solution [28]. Unlike the DNN-based approaches, ELM is one kind of the
single hidden layer feed-forward neural networks (SLFNs) without the
requirement of gradient back-propagation [29], which may result in less time-
consuming for training the TS learner.

Inspired by the advantages of computer-aided approaches and ELM networks, we
propose the computer-aided training strategy for ELM-based TS in this paper,
which aims to tackle the bottleneck facing in the training data collection and
enhance the generalization performance of the designed TS learner. The main
contributions of our work are summarized as follows:

  1. 1.

We develop a computer-aided approach to generate the training data for the
ELM-based TS network in OFDM systems. As a study case of this paper, by
exploring the partial properties of CIRs, some important data features of
realistic wireless channels can be rapidly captured, e.g., the root-mean-
square (r.m.s.) delay. Compared with classic TS schemes for training data
collection, the proposed scheme has the ability to reduce the consumption of
extracting the available and desired training data from the raw received data,
i.e., less time- and labor-consuming. Meanwhile, due to the good
controllability of computer-aided approaches, as many realistic situations as
possible can be simulated with the assistance of computers, such that
increases the completeness of training data sets. In addition, by generating
training data at the local device, the proposed scheme reduces the burden of
storage space and transmission bandwidth in the training data collection.

  2. 2.

We exploit the rough features of CIRs (i.e., the expected maximum multi-path
delay) to design the training strategy for the ELM-based TS network.
Particularly, the loose constraint- and flexible constraint-based training
strategies are derived from exploring the statistical information of multi-
path delays, which aims to increase the generalization performance of TS
learners and the correlation between the generated learning labels with the
ground truth labels. The proposed training strategies inherit the advantage of
the training strategy of [30] (i.e., increasing the possibility of locating
STOs into the ISI-free region) and also well adapt themselves to the volatile
ground truth labels (i.e., offering a strong correlation between the generated
learning labels and the ground truth labels). With this, the designed training
strategy can improve the generalization performance of the designed TS learner
compared with that of [30]. Moreover, sufficient training data can be rapidly
produced with the assistance of the computer-aided approach, which avoids
collecting the training data from realistic systems.

  3. 3.

We form a lightweight TS framework for the OFDM systems by fusing the classic
synchronizer and the conventional ELM, i.e., the ELM-based TS network.
Particularly, the classic TS scheme is performed to directly extract the
relevant features of timing information from the raw received samples. By this
means, ELM can easily learn the transformation between the input features and
the learning labels, even without the deeper hidden layers being used for
fusing the implicit data features.

TABLE I: Notations Notation | Description
---|---
$\mathbb{R}^{M\times N}$ |  $M$-by-$N$ dimensional real matrix space
$\mathbb{C}^{M\times N}$ |  $M$-by-$N$ dimensional complex matrix space
$\mathbb{E}\\{\cdot\\}$ | Expectation operation
$e$ | Constant $e=2.71828\ldots$
$\mathbf{0}_{M\times N}$ |  $M$-by-$N$ dimensional all-zeros matrix
$\mathbf{1}_{M\times N}$ |  $M$-by-$N$ dimensional all-ones matrix
$\mathds{I}_{N}$ |  $N$-by-$N$ dimensional Identity matrix
$[\mathbf{X}]_{a,b}$ | Entry $(a,b)$ of matrix $\mathbf{X}$
$\mathbf{X}_{a:b}$ | Sub-vector of $\mathbf{X}$ with entries $a$ to $b$
$\mathcal{{CN}}\left(\mu,{\sigma}^{2}\right)$ |  Circularly symmetric complex Gaussian (CSCG) distribution
$\mathcal{U}(a,b)$ | Uniform distribution
$\left(\cdot\right)^{\ast}$ | Conjugate operation
$\left(\cdot\right)^{\textit{T}}$ | Transpose operation
$\left(\cdot\right)^{\textit{H}}$ | Hermitian transpose
$\left(\cdot\right)^{{\dagger}}$ | Moore–Penrose generalized inverse operation
$|\cdot|$ | Absolute operation
${\left\|\cdot\right\|_{2}}$ |  $\ell_{2}$-norm operation
$\lceil\cdot\rceil$ | Integer ceiling operation
$\log_{10}(\cdot)$ | Logarithm base 10
$\Pr\left(\cdot\right)$ | Probability of an event

The rest of this paper is organized as follows. For notation clarity, the
meaning of symbols that have been adopted in this paper is listed in Table I.
Section II briefly describes the system model and problem formulation. Section
III provides the computer-aided training strategies for the ELM-based TS
network and the implementation of the proposed ELM-based TS network. Section
IV comprehensively evaluates the TS performance of the proposed computer-aided
training strategy for ELM-based TS network, and is followed by conclusions in
Section V.

##  II System Model

Figure 1: System model.

We consider an OFDM system with $N$ sub-carriers, which is plotted in Fig 1.
According to the inverse fast Fourier transform (IFFT), the frequency-domain
symbol $\mathbf{d}\in\mathbb{C}^{N\times 1}$ is first transformed into the
time domain, and the OFDM symbol $\mathbf{s}\in\mathbb{C}^{N\times 1}$ in the
time domain is expressed as

${\bf{s}}={{{\bf{F}}_{N}}{\bf{d}}},$ (1)

where $[\mathbf{s}]_{k,1}=s_{k},k\in\\{0,1,\cdots,N-1\\}$ and
$\mathbb{E}\\{|s_{k}|^{2}\\}=\sigma^{2}_{d}$ with $\sigma^{2}_{d}$ being data
power. ${\bf{F}}_{N}\in\mathbb{C}^{N\times N}$ stands for the $N$-by-$N$
dimensional IDFT matrix and
$[{\bf{F}}_{N}]_{k,l}=\frac{1}{{\sqrt{N}}}{e^{j\frac{{2\pi kl}}{N}}}$,
${k,l}\in\\{0,1,\cdots,N-1\\}$. The cyclic prefix (CP) addition matrix
$\mathbf{G}_{\mathrm{cp}}\in\mathbb{R}^{N+L_{c}\times N}$ is employed to
denote the operation of adding the CP for each OFDM symbol [31], where $L_{c}$
represents the length of CP. Then, the transmitted signal vector is given by

$\mathbf{x}=\mathbf{G}_{\mathrm{cp}}\mathbf{s},$ (2)

where $\mathbf{x}$ is the $N_{u}\times 1$ transmitted signal vector with
$N_{u}=N+L_{c}$, and

$\mathbf{G}_{\mathrm{cp}}=\left[{\begin{array}[]{*{20}{c}}{{\mathbf{0}_{{L_{c}}\times{N-{L_{c}}}}}}&{{\mathds{I}_{{L_{c}}}}}\\\
\\\ {{\mathds{I}_{N}}}&{{\mathbf{0}_{N\times{L_{c}}}}}\end{array}}\right].$
(3)

By denoting $\mathbf{x}=[x_{0},x_{1},\cdots,x_{N_{u}}]^{T}$, the received
signal $r_{n}$ over the multi-path fading channel is described as

$r_{n}=\sum\limits_{l=1}^{L}{h_{l}x_{n-\tau-\tau_{l}}{e^{j\frac{2\pi}{N}\varepsilon{n}}}}+w_{n},{n=0,1,\cdots,N_{u-1}},$
(4)

where $L$, $h_{l}$, $\tau$, $\tau_{l}$, and $\varepsilon$ stand for the taps
of propagation channel, the channel coefficient of $l$th path, the unknown
STO, the path delay of $l$th path, and the CFO being normalized by sub-carrier
space, respectively. In (4), $w_{n}$ is the additive white noise satisfying
the independent Circularly symmetric complex Gaussian (CSCG) distribution with
zero-mean and variance $\sigma^{2}$. Without loss of generality, all multi-
path delays are assumed to be shorter than the length of CP, i.e.,
$\tau_{l+1}>\tau_{l},l=1,2,\cdots,P$ [32]. In the TS stage, the STO estimation
is conducted, denoted as $\widehat{\tau}$. Particularly, we adopt an ELM
network to combine with the classic synchronizer to enhance the TS for the
OFDM systems.

One goal of our work is to generate more complete and accurate learning labels
for TS, compared with the traditional labeling of learning labels for the
collected training data in realistic wireless communication scenarios. Another
goal is to avoid seriously occupying the bandwidth resource and storage space
caused by the transmission and storage of collected training data. The
computer-aided approach can appoint the generation method of training data at
local devices, without additionally transmitting the collected data over the
wireless channel. In addition, the training data can be timely generated
without storing them and discarded promptly. To this end, the occupations of
bandwidth resource and extra storage space could be avoided, inspiring us to
develop computer-aided solutions.

##  III Computer-Aided Training Strategy for ELM-Based TS Network

In this section, we first characterize the existing strategy from the
perspective of obtaining learning labels, then elaborate the computer-aided
strategy and finally present the designed TS framework for the OFDM systems.
For the sake of clarity, a $K$-length learning label, denoted as
$\mathbf{T}\in\mathbb{C}^{K\times 1}$, can be represented by using a time-
indexed sequence, i.e.,

$\mathbf{T}=[T_{0},T_{1},\cdots,T_{j},\cdots,T_{K-1}]^{T}.$ (5)

###  III-A Insights of Existing Strategy

Deploying ML has become a promising solution to enhance/replace the
conventional signal processing [33], however, ML-based signal processing still
faces significant challenges with determining the estimated labels for
training samples, and the ELM-based TS network is no exception.

####  III-A1 Case I

If the information of CIRs are perfectly known to the receiver, three types of
existed learning-label designs are mentioned and compared in [30], which are
listed as follows:

  * •

One-hot Coding: When the expected maximum timing offset is known to be less
than the length of OFDM symbol (i.e., $N$), the correct start location of
preamble is employed to design the learning label. By utilizing the one-hot
coding, the label value is given by [34]

${T_{j}}=\left\\{\begin{array}[]{l}1,\ j=\tau+{L_{c}}\\\ 0,\
\textrm{others}\end{array}\right.,$ (6)

where $j\in\\{0,1,\cdots,K-1\\},K=N_{u}=N+L_{c}$, due to the existence of CP.
The vector form of this learning label is
$\mathbf{T}=[\bm{0}_{1\times\tau+L_{c}},1,\bm{0}_{1\times K-\tau-
L_{c}-1}]^{T}$. Since the TS of OFDM system only requires that the STO
estimation falls into the ISI-free region, the timing instants within the ISI-
free region can be regarded as the ground truth labels. Without the use of
cyclic suffix, the nonzero label feature points to the edge of the ISI-free
region such that results in a weak correlation between the generated learning
labels and truth ground labels.

  * •

Labeling ISI-free Midpoint: With the use of known timing offsets and maximum
multi-path delay, the midpoint of ISI-free region is exploited to design
learning label [30]. The midpoint of ISI-free region is denoted as
$p_{c}=\lceil(\tau_{P}+L_{c})/2\rceil$, which is used for designing the label
value [30]

${T_{j}}=\left\\{\begin{array}[]{l}1,\ j=\tau+p_{c}\\\ 0,\
\textrm{others}\end{array}\right..$ (7)

and its vector form is
$\mathbf{T}=[\bm{0}_{1\times\tau+p_{c}},{1},\bm{0}_{1\times K-\tau-
p_{c}-1}]^{T}$. By allocating nonzero value corresponding to the midpoint of
ISI-free region, this label increases the correlation between the nonzero
label feature and the ground truth label, such that increases the possibility
of residing the estimated STO to the ISI-free region.

  * •

Labeling ISI-free Region: By completely exploring the prior information of
ISI-free region, the label value is given by [30]

${T_{j}}=\left\\{\begin{array}[]{l}1,\ \tau+{\tau_{P}}\leq
j\leq\tau+{L_{c}}\\\ 0,\ \textrm{others}\end{array}\right.,$ (8)

and its vector form can be expressed as
$\mathbf{T}=[\bm{0}_{1\times\tau+\tau_{P}},\bm{1}_{1\times
L_{c}-\tau_{P}+1},\bm{0}_{1\times K-\tau-L_{c}-1}]^{T}$. By setting nonzero
label values corresponding to the whole ISI-free region, this label provides
extra information to increase the correlation between the nonzero label
features and the timing indexes of ISI-free region, which greatly enhances the
ELM learning. As a result, the ELM-based TS network combined with the label in
(8) has a lower error probability of TS than that combined with the label in
(7).

However, the complete real-time multi-path delays are difficult to be pre-
known at the receiver, so the completeness of learning labels in (7) and (8)
are facing challenge with the varied maximum multi-path delay.

####  III-A2 Case II

If the information of CIRs cannot be perfectly known at the receiver,
employing the classic TS schemes to collect the training data is a simple and
natural option. By taking the TS scheme of [5] as an example, the learning
labels can be obtained according to the timing metric. With the received
samples of $r_{n}$, the timing metric $M_{j}$, at $j$th lag of correlation, is
given by

${M_{j}}=\frac{{{{\left|{\sum\limits_{d=0}^{\frac{N}{2}}{r_{j+d}^{*}{r_{j+\frac{N}{2}+d}}}}\right|}^{2}}}}{{{{\left|{\sum\limits_{d=0}^{\frac{N}{2}}{r_{j+\frac{N}{2}+d}^{*}{r_{j+\frac{N}{2}+d}}}}\right|}^{2}}}},$
(9)

and then $\widehat{\tau}$ is obtained by
$\widehat{\tau}=\mathop{\arg\max}\limits_{0\leq j\leq
K-1}\left\\{{{M_{j}}}\right\\}$. With the use of $\widehat{\tau}$, the
generated learning label can be expressed as

${T_{j}}=\left\\{\begin{array}[]{l}1,\ j=\widehat{\tau}\\\ 0,\
\textrm{others}\end{array}\right..$ (10)

This label is direct, and its vector form is denoted as
$\mathbf{T}_{\textrm{CT}}\in\mathbb{R}^{K\times 1}$ and given by
$\mathbf{T}_{\textrm{CT}}=[\bm{0}_{1\times\widehat{\tau}},{1},\bm{0}_{1\times
K-\widehat{\tau}-1}]^{T}$. We assume that, $N_{t}$-samples of
$\mathbf{r}_{m}\in\mathbb{C}^{K+N\times 1}$ are required for the model
training. In addition, $\widehat{\tau}_{m}$ and
${\mathcal{R}^{(m)}_{\textrm{ISI-free}}}$ are denoted as the estimated STO and
the ISI-free region for $\mathbf{r}_{m}$, wherein
${\mathcal{R}^{(m)}_{\textrm{ISI-free}}}$ is given by
${\mathcal{R}^{(m)}_{\textrm{ISI-
free}}}=\tau^{(m)}+\\{\tau^{(m)}_{P},\tau^{(m)}_{P}+1,\cdots,L_{c}\\}$. Then,
the metric for the accuracy of estimated learning labels can be expressed as

${P_{\textrm{label}}}=\frac{1}{{{N_{t}}}}\sum\limits_{m=1}^{{N_{t}}}{\theta\left({{{\widehat{\tau}}_{m}}}\right)},$
(11)

where

$\theta\left({{{\widehat{\tau}}_{m}}}\right)=\left\\{\begin{array}[]{l}1,\
{{\widehat{\tau}}_{m}}\in{\mathcal{R}^{(m)}_{\textrm{ISI-free}}}\\\ 0,\
{{\widehat{\tau}}_{m}}\notin{\mathcal{R}^{(m)}_{\textrm{ISI-
free}}}\end{array}\right..$ (12)

The above $\theta\left({{{\widehat{\tau}}_{m}}}\right)$ indicates whether the
estimated STO (i.e., $\widehat{\tau}_{m}$ for generating learning label) is
correct or not.

######  Remark 1.

According to (12), the obtained labels are partially flawed due to the TS
errors. That is, when the estimated $\widehat{\tau}$ points to the edge or
outside of the ISI-free region, the correlation between the estimated labels
(i.e., $\widehat{\tau}$) and the timing indexes of ISI-free region is
seriously impaired. The same issues can be found in other classic TS schemes.
To guarantee the efficiency of training data, a large-scale of training
samples are required to be collected for selection. For example, if the
accuracy of estimated learning labels is $P_{\textrm{label}}$, at least
$N_{t}/P_{\textrm{label}}$ training samples are required to obtain $N_{t}$ of
effective learning labels. As a result, by leveraging the classic TS scheme,
the training data collection is unfriendly to limited storage space and
transmission bandwidth resources.

To alleviate the above challenge, we develop computer-aided training
strategies, which is presented in Section III-B in detailed.

###  III-B Computer-Aided Strategy

It could be obtained from (12) that, the accuracy of estimated learning labels
is heavily influenced by the TS errors. This is because the performance of
classic TS schemes is vulnerable to the impacts of CIRs and noise. To avoid
incorrectly labeling the training samples, computers can be utilized to
generate the training data (including the training samples and their
corresponding learning labels) at the local device, since the computer-aided
generated training data is controllable and known to the transceiver. In the
following, we mainly discuss the strategy of improving the completeness with
respect to the varied maximum multi-path delays.

####  III-B1 Loose Constraint

The completeness of training data is mainly impaired by the profile of CIRs.
In the TS stage, the key insight is to avoid the ISI, i.e., locating the
timing indexes in the ISI-free region. To this end, we propose a loose
constraint for the profile of CIRs to improve the completeness of training
data. In other words, the loose constraint is utilized to generate the
training samples and design the learning labels. The key idea of the proposed
loss constraint is motivated by these two factors: 1) there is a time range
for the realistic maximum multi-path delay to dynamically change; 2)
statistical information of multi-path delays in a special geographical
environment is easier to be obtained compared with acquiring the realistic
multi-path delays themselves. The main motivation is that, the r.m.s. delay is
usually measured prior to the establishment of terrestrial mobile
communication systems e.g., the wireless communication standards for the 4G
and 5G communication systems [35, 36], and so that the r.m.s. delay can
constrain the maximum multi-path delay to a specific time range with high
probability.

Here, we denote $\mathcal{L}$ as the loose constraint for the maximum multi-
path delay in the training stage, which is utilized to generate the maximum
multi-path delays via computer-aided data generation. Specifically, to avoid
ISI, $\tau_{P}<\mathcal{L}<L_{c}$ is given, and $\mathcal{L}$ is sufficient to
tolerate the increase of $\tau_{P}$. Although the underlying mechanism of
$\mathcal{L}$ is similar to the CP in the aspect of avoiding ISI, this loose
constraint is only used to generate training data in the training stage, which
does not need extra overhead to change the frame structure. For $N_{t}$
samples of $\\{\mathbf{r}_{m}\\}^{N_{t}}_{m=1}$ being required,
${\mathcal{R}^{(m)}_{\textrm{ISI-
free}}}=\tau^{(m)}+\\{\mathcal{L},\mathcal{L}+1,\cdots,L_{c}\\}$ is denoted as
the target ISI-free region (i.e., target STOs for designing learning label)
for each sample of $\mathbf{r}_{m}$. Correspondingly, the ground truth label
for the realistic communication systems are denoted as
$\mathcal{R}_{\textrm{truth}}=\tau+\\{\tau_{P},\tau_{P}+1,\cdots,L_{c}\\}$.
Then, for the case where $\tau^{(m)}=\tau$, the correlation between
${\mathcal{R}^{(m)}_{\textrm{ISI-free}}}$ and $\mathcal{R}_{\textrm{truth}}$
is described as

$\mathcal{R}^{(m)}_{\textrm{ISI-free}}\subset\mathcal{R}_{\textrm{truth}},\
\forall m\in\\{1,2,\cdots,N_{t}\\}.$ (13)

According to (13), since $\mathcal{L}$ is sufficient to the possible maximum
multi-path delays, $\mathcal{R}^{(m)}_{\textrm{ISI-free}}$ always contains the
correct learning labels that correspond to the ground truth labels. This
improves the completeness of training data. For example, when $L_{c}=32$ and
$\tau_{P}$ possibly ranges from 16 to 23, $\mathcal{L}$ can be set as $26$ for
generating each training sample.

The works in [30] verified that the ELM network with the learning label in (7)
has good generalization against the multi-path delays, and so we mainly focus
on the improvement for the learning label in (8). With the use of
$\mathcal{L}$, the improved learning label is written as

${T_{j}}=\left\\{\begin{array}[]{l}1,\ \tau+{\cal L}\leq j\leq\tau+L_{c}\\\
0,\ \textrm{others}\end{array}\right.,$ (14)

and its vector form is denoted as
$\mathbf{T}_{\textrm{LC}}\in\mathbb{R}^{K\times 1}$ and given by
$\mathbf{T}_{\textrm{LC}}=[\bm{0}_{1\times\tau+{\cal L}},\bm{1}_{1\times
L_{c}-{\cal L}+1},\bm{0}_{1\times K-\tau-L_{c}-1}]^{T}$. Since $\tau_{P}$ is
smaller than $\cal L$, the supervised learning-based TS learner combined with
$\mathbf{T}_{\textrm{LC}}$ can actively avoid the potentially discriminatory
behaviors that degrade the generalization performance.

However, according to (13), the loose constraint-based strategy for training
data generation lacks a sufficient number of training labels that have a
strong correlation with the ground truth labels, since
$\mathcal{R}_{\textrm{truth}}\neq\mathcal{R}^{(m)}_{\textrm{ISI-free}}$ or
$\mathcal{R}_{\textrm{truth}}\not\subset\mathcal{R}^{(m)}_{\textrm{ISI-
free}}$. On the one hand, if
$\mathcal{R}_{\textrm{truth}}\equiv\mathcal{R}^{(m)}_{\textrm{ISI-free}}$, the
training labels will show a strong correlation with the ground truth labels,
such that facilitates the performance of TS learners. On the other hand, if
$\mathcal{R}_{\textrm{truth}}\subset\mathcal{R}^{(m)}_{\textrm{ISI-free}}$,
the training data set can be regarded as the complete set of possible maximum
multi-path delays, such that improves the statistical efficiency of TS
learner. As a result, the quality and efficiency of the supervised learning-
based TS leaner are degraded.

####  III-B2 Flexible Constraint

Flexible constraint has the ability to improve the adaptability to the various
demands in practical applications, such as the flexible constraint length
Viterbi decoder [37]. For the supervised learning-based TS leaner, the key
insight of introducing the flexible constraint into the design of training
strategy is that: increasing the correlation between the training labels and
the ground truth labels is helpful to improve the statistical power of the
supervised learning-based TS leaner. To this end, we propose the flexible
constraint for generating the training data on the basis of loose constraints.

By denoting $\widetilde{\cal L}$ as the flexible constraint for the maximum
multi-path delay in the training stage, the target STOs for designing learning
label is denoted by ${\mathcal{R}^{(m)}_{\textrm{ISI-
free}}}=\tau^{(m)}+\\{\widetilde{\mathcal{L}}^{(m)},\widetilde{\mathcal{L}}^{(m)}+1,\cdots,L_{c}\\}$,
with $\widetilde{\mathcal{L}}^{(m)}$ being distribution as
$\widetilde{\mathcal{L}}^{(m)}\sim\mathcal{U}(L_{c}/2,\mathcal{L})$.
Similarly, we denote
$\mathcal{R}_{\textrm{truth}}=\tau+\\{\tau_{P},\tau_{P}+1,\cdots,L_{c}\\}$ as
the ground truth labels for the realistic communication systems. Then, for the
cases where $\tau_{P}>L_{c}/2$ and $\tau^{(m)}=\tau$, the correlation between
${\mathcal{R}^{(m)}_{\textrm{ISI-free}}}$ and $\mathcal{R}_{\textrm{truth}}$
is expressed as

$\mathcal{R}^{(m)}_{\textrm{ISI-free}}\supseteq\mathcal{R}_{\textrm{truth}},\
\forall m\in\\{1,2,\cdots,N_{t}\\}.$ (15)

With this, if the amount of training samples is sufficient, the given
$\mathcal{R}^{(m)}_{\textrm{ISI-free}}$ can be approximately regarded as the
complete set of possible maximum multi-path delays in some cases. As for the
cases where $\tau_{P}<L_{c}/2$ and $\tau^{(m)}=\tau$, the correlation given in
(15) can be rewritten as $\mathcal{R}^{(m)}_{\textrm{ISI-
free}}=\mathcal{R}_{\textrm{truth}},\ \exists m\in\\{1,2,\cdots,N_{t}\\}.$
Wherein $\tau_{P}<L_{c}/2$ can be regraded as a case of $\mathcal{L}$, i.e.,
$\mathcal{L}=L_{c}/2$. Therefore, the proposed flexible constraint inherits
the advantages of the proposed loose constraint and also improves the
correlation between the training labels and ground truth labels.

By leveraging $\widetilde{\mathcal{L}}$, the improved learning label is given
by

${T_{j}}=\left\\{\begin{array}[]{l}1,\ \tau+\widetilde{\cal L}\leq
j\leq\tau+L_{c}\\\ 0,\ \textrm{others}\end{array}\right..$ (16)

and its vector form is denoted as
$\mathbf{T}_{\textrm{FC}}\in\mathbb{R}^{K\times 1}$ and given by
$\mathbf{T}_{\textrm{FC}}=[\bm{0}_{1\times\tau+{\cal L}},\bm{1}_{1\times
L_{c}-{\cal L}+1},\bm{0}_{1\times K-\tau-L_{c}-1}]^{T}$. Due to the fact that
$\widetilde{\mathcal{L}}\sim\mathcal{U}\left(L_{c}/2,\mathcal{L}\right)$, the
supervised learning-based TS learner can learn more training data that may
appear in practical environments. This improves the correlation between the
training labels and the ground truth labels, and so that improves the
statistical power of TS learners. Moreover, instead of collecting sufficient
data features from realistic systems, a large number of rich and diverse data
features can be rapidly generated by $\widetilde{\mathcal{L}}$ at local
devices. Thus, not only the occupations of storage space and transmission
bandwidth caused by frequently collecting desired data features are
correspondingly avoided, but also more memory traces to the neural network can
be retained against the changing profile of CIRs.

######  Remark 2.

It is impractical to completely collect training data from realistic systems
since the received samples are unavailable before the completion of
synchronization. As for the supervised learning-based TS learner, it is
available for local devices (i.e., employing the Central Processing Unit) to
generate efficient and more complete training data. By this means, once the
computer-aided approach is adopted, the training samples and their
corresponding learning labels can be timely generated and released. Compared
with the training data collected by the classic TS schemes, more complete and
accurate training data can be generated at the local device, and thus greatly
reduces the storage consumption and bandwidth occupation of collecting large-
scale data from the realistic systems. It is worth mentioning that, the more
complete training data features we obtain, the more statistical efficiency the
neural network will own. Thus, a good generalization capability of ELM-based
TS network and low occupations of storage space and transmission bandwidth can
be achieved by the proposed strategies of loose constraint and flexible
constraint.

###  III-C ELM-Based TS Network Architecture

The proposed ELM-based TS network consists of two phases, i.e., a pre-
processing phase of feature extraction and an ELM batch learning phase. In the
pre-processing phase, the received signals are processed via a classic TS
scheme for feature extraction. The timing metrics calculated from the raw
received samples are regarded as extracted features, which are then filled up
for the ELM batch learning phase. Within timing metrics, the extracted
features should be normalized by the Frobenius norm (i.e., $\ell_{2}$-norm).
Following the pre-processing phase, the ELM batch learning phase is carried
out to learn the mapping relationship between the input features and the
designed learning labels as desired. To achieve our desired, the training data
sets are generated with the proposed computer-aided training strategy (i.e.,
the loose constraint- and flexible constraint-based strategies in Section
III-B), which are controllable, distinct, and reliable for the ELM learning.

For the convenience of expression, preliminary preparation of ELM-based TS
network is characterized below. Selecting $N_{h}$-numbers of hidden layer
nodes, we denote
$\bm{\mathcal{D}}_{\textrm{ca}}=\\{(\mathbf{r}_{m},\mathbf{T}_{m})|\mathbf{r}_{m}\in{\mathbb{C}}^{N_{w}\times
1},\mathbf{T}_{m}\in{\mathbb{R}}^{K\times 1},m=1,2,\cdots,N_{t}\\}$ as the
original training data set. Wherein $N_{w}$ denotes the size of the
observation window for received signals, and $K$ is the size of the
observation window for timing metrics. Specifically, the received samples of
$\\{\mathbf{r}\\}^{N_{t}}_{m=1}$ are obtained according to (1)–(4) without
interferences of CFO and noise. The generation of learning labels (i.e.,
$\\{\mathbf{T}\\}^{N_{t}}_{m=1}$) is given in Section III-B. Besides, the
architecture of the adopted ELM network is presented in Table II.

####  III-C1 Offline Training

TABLE II: Network Architecture | Input Layer | Hidden Layer | Output Layer
---|---|---|---
Neuronal Nodes | $K$ | ${N_{h}}$ | $K$
Activation Function | None | Tanh | None
Normalization |  $\ell_{2}$-norm | None | None

Algorithm 1 summarizes the process of offline training for the ELM-based TS
network, and the detailed description is elaborated as follows.

Step 1) Initial Feature Extraction: Initially extracting the features using
received samples $\\{\mathbf{r}_{m}\\}^{N_{t}}_{m=1}$ from the given training
data set $\bm{\mathcal{D}}_{\textrm{ca}}$. By taking the classic synchronizer
(i.e., TS scheme in [5]) as an example, the specifical preamble consists of
two identical parts. In addition, the observed sub-vector of $\mathbf{r}_{m}$
at $j$th lag of correlation, denoted as
$\mathbf{r}_{m,j}=[\mathbf{r}_{m}]_{j:j+\frac{N}{2}}$, is expressed as

$\mathbf{r}_{m,j}={\left[{{r_{m,j}},\cdots,{r_{m,j+d}},\cdots,{r_{m,j+N-1}}}\right]^{T}},$
(17)

where $j\leq N_{w}-N+1$ and $N_{w}=2N_{u}$ with the aim of observing a
complete preamble at the receiver. Then, the timing metric is employed as the
ELM input and given by

${M_{m,j}}=\frac{{{{\left|{{P_{m,j}}}\right|}^{2}}}}{{{\frac{1}{4}{\left|{{R_{m,j}}}\right|}^{2}}}},$
(18)

where $P_{m,j}$ and $R_{m,j}$ stand for the autocorrelation factor [5] and the
normalization factor [38], respectively, which are defined as

$\left\\{\begin{array}[]{l}{P_{m,j}}=\sum\limits_{d=0}^{\frac{N}{2}-1}{r_{m,j+d}^{*}{r_{m,j+d+\frac{N}{2}}}}\\\
{R_{m,j}}=\sum\limits_{d=0}^{N-1}{r_{m,j+d}^{*}{r_{m,j+d}}}\end{array}\right..$
(19)

The iteration formula for $P_{m,j+1}$ is derived in [5] to reduce the time
complexity, i.e.,

${M_{m,j+1}}=\frac{{{{\left|{{P_{m,j}}+r_{m,j+\frac{N}{2}}^{*}{r_{m,j+N}}-r_{m,j}^{*}{r_{m,j+\frac{N}{2}}}}\right|}^{2}}}}{{{\frac{1}{4}{\left|{{R_{m,j}}+r_{m,j+N}^{*}{r_{m,j+N}}-r_{m,j}^{*}{r_{m,j}}}\right|}^{2}}}}.$
(20)

By buffering $K$-samples of $M_{m,j}$, the timing metric vector
$\mathbf{M}_{m}\in\mathbb{R}^{K\times 1}$ is constructed by

$\mathbf{M}_{m}=[M_{m,0},M_{m,1},\cdots,M_{m,K-1}]^{T},\ K=N_{u}.$ (21)

To increase the accuracy of the classification and the learning efficiency,
$\mathbf{M}_{m}$ is normalized by $\ell_{2}$-norm, which is expressed as

$\mathbf{\widetilde{g}}_{m}=\frac{\mathbf{M}_{m}}{{{{\left\|\mathbf{M}_{m}\right\|}_{2}}}},$
(22)

where $\mathbf{\widetilde{g}}_{m}\in\mathbb{R}^{K\times 1}$ denotes the
normalized timing metric vector. For the followed ELM batch learning, the
above (1)–(4) and (17)–(22) are repeated till $N_{t}$-samples of
$\mathbf{\widetilde{g}}_{m}$ are achieved.

Step 2) ELM Batch Learning: Denoting $N_{t}$-samples of
$\\{(\widetilde{\mathbf{g}}_{m},\mathbf{T}_{m})|\widetilde{\mathbf{g}}_{m}\in{\mathbb{R}}^{K\times
1},\mathbf{T}_{m}\in{\mathbb{R}}^{K\times 1},m=1,2,\cdots,N_{t}\\}$ as the
training data for the ELM learning. To initialize the network, the input
weight matrix $\mathbf{W}\in\mathbb{R}^{N_{h}\times K}$ and hidden bias vector
$\mathbf{b}\in\mathbb{R}^{N_{h}\times 1}$ are randomly selected from Gaussian
variables, i.e., $\mathcal{N}(\mu,\sigma^{2})$. In addition, the universal
hyperbolic tangent (tanh) activation function is selected, since the
classification task using tanh activation function has a lower classification
error than other activation functions in general.

For each feed-in $\widetilde{\mathbf{g}}_{m}$, the hidden layer output matrix
$\mathbf{H}_{m}$ is calculated according to

${\mathbf{H}_{m}}={f_{\sigma}}\left({\mathbf{W}{\widetilde{\mathbf{g}}_{m}}+\mathbf{b}}\right),$
(23)

where $f_{\sigma}(\cdot)$ stands for the universal tanh activation function.
Repeating the above (23) till $N_{t}$-samples of $\mathbf{H}_{m}$ are
collected. Next, on the basis of $N_{t}$-samples of
$\left\\{{\left({{\mathbf{H}_{m}},{\mathbf{T}_{m}}}\right)}\right\\}_{m=1}^{{N_{t}}}$
as attributes, the hidden layer output matrix
$\mathbf{{H}}\in\mathbb{R}^{N_{h}\times N_{t}}$ and the label matrix
$\mathbf{{T}}\in\mathbb{R}^{K\times N_{t}}$ are expressed as

$\left\\{\begin{array}[]{l}{\bf{{H}}}=[{{\bf{H}}_{1}},{{\bf{H}}_{2}},\cdots,{{\bf{H}}_{{N_{t}}}}]\\\
{\bf{{{T}}}}=[{{\bf{T}}_{1}},{{\bf{T}}_{2}},\cdots,{{\bf{T}}_{{N_{t}}}}]\end{array}\right..$
(24)

Then, the output weight matrix $\bm{\beta}\in\mathbb{R}^{K\times N_{h}}$ is
given by

$\bm{\beta}={{\mathbf{T}}}{{\mathbf{H}}^{\dagger}}.$ (25)

The above trained parameters (i.e., $\mathbf{W}$, $\mathbf{b}$, and
$\bm{\beta}$) are saved for online deployment.

######  Remark 3.

The initial feature extraction is vitally important to improve the learning
capability of ML models [39]. As for simply models like ELM network, the
cleverly designed feature extractor could actively extract the pertinent
information of STOs from raw data (e.g., coarse STO estimation), which
benefits the ELM to enhance learning effectiveness. To this end, the classic
TS scheme is employed to implement the feature extraction to capture timing
metrics. Also, since the normalization makes the features between different
dimensions have a certain numerical comparison, to normalize the data in (22)
before feeding it to the network is a good practice, which can greatly improve
the accuracy of the classification.

Algorithm 1 Offline training

Input: Data set $(\mathbf{r}_{m},\mathbf{T}_{m}),m=1,2,\cdots,N_{t}$.
Output: Output weight matrix $\bm{\beta}$.

1:Set $N_{h}$-numbers of hidden layer nodes.

2:Randomly assign input weight matrix $\mathbf{W}$ and hidden bias vector
$\mathbf{b}$.

3:Calculate timing metric ${\mathbf{M}}_{m}$ from
$\\{\mathbf{r}_{m}\\}^{N_{t}}_{m=1}$ by (18)–(22).

4:Compute hidden layer output $\mathbf{H}_{m}$ with the use of
$\mathbf{M}_{m}$, $\mathbf{W}$, and $\mathbf{b}$ by (22)–(23).

5:Set $\mathbf{H}=[\mathbf{H}_{0},\mathbf{H}_{1},\cdots,\mathbf{H}_{N_{t}}]$,
$\widetilde{\mathbf{T}}=[\mathbf{T}_{0},\mathbf{T}_{1},\cdots,\mathbf{T}_{N_{t}}]$.

6:Estimate the output weight matrix $\bm{\beta}$ via
$\bm{\beta}=\widetilde{\mathbf{T}}\mathbf{H}^{{\dagger}}$.

7:return $\bm{\beta}$.

####  III-C2 Online Deployment

In this phase, a straightforward process of ELM-based TS network is carried
out for online deployment. First, according to (1)–(4) and (17)–(22), the
timing metric is calculated from received signals to capture the coarse timing
feature, namely, $\widetilde{\mathbf{g}}\in\mathbb{R}^{N_{w}\times 1}$. Next,
the extracted $\widetilde{\mathbf{g}}$ is fed to the trained ELM network to
capture the network output $\mathbf{O}\in\mathbb{R}^{K\times 1}$, i.e.,

$\mathbf{O}=\bm{\beta}{f_{\sigma}}\left({\mathbf{W}\widetilde{\mathbf{g}}+\mathbf{b}}\right).$
(26)

By expressing $\mathbf{O}$ as $\\{O_{j}\\}^{K-1}_{j=0}$, the estimated STO is
given by

$\widehat{\tau}=\mathop{\arg\max}\limits_{0\leq
j\leq{K}-1}\left\\{{{{O_{j}}}}\right\\}.$ (27)

##  IV Simulation Results

In this section, numerical results are provided to evaluate the TS performance
of the proposed computer-aided strategies (i.e., loose constraint- and
flexible constraint-based strategies). Basic parameters and definitions
involved in simulations are summarized in Section IV-A. To indicate the
effectiveness of the proposed strategies for generating training data, Section
IV-B provides the error probability of TS against the impacts of the accuracy
and completeness of learning labels. Then, Section IV-C presents the
generalization analysis against the varying parameter of maximum multi-path
delay, and a series of robustness analyses are provided in Section IV-D. _In
the simulations, the testing channel models have not been utilized for the
offline training phase, which aims to prove the feasibility of our study._

###  IV-A Parameter Setting

In the simulations, we consider basic parameters as that $N=128$, $L_{c}=32$,
$K=N_{u}=160$, $N_{w}=2N_{u}=320$, $N_{h}=8K$, and $N_{t}=10^{5}$. The signal
bandwidth for transmitting the collected data samples is considered as
$\mathcal{W}=100$MHz [40]. By employing the data type float32, a collected
training sample is 4 Bytes. The transmitted preamble is generated according to
the Zadoff-Chu sequence [9] with the power of transmitted preamble being
constant as $\sigma^{2}_{p}$, and $\sigma^{2}_{d}=\sigma^{2}_{p}$ is
considered. A Rayleigh multi-path fading channel is considered, in which the
exponential power delay profile is defined as
$\sigma^{2}_{p}=\frac{1}{\Sigma^{P}_{p=1}e^{-\eta\left(p-1\right)}}e^{-\eta\left(p-1\right)}$,
with $\eta=0.2$. Except for the testing maximum multi-path delay (i.e.,
$\tau_{P}$) that needs to discuss, the default parameter of $\tau_{P}$ is set
as 20 sample delays. Correspondingly, the default loose constraint (i.e.,
$\cal L$) is set as 26 sample delays, and the flexible constraint (i.e.,
$\widetilde{\cal L}$) satisfies that
$\widetilde{\mathcal{L}}\sim\mathcal{U}(16,26)$.

Some definitions in the simulations are given as follows. The signal-to-noise-
ratio (SNR) in decibel (dB) [28] is considered in the following simulations,
i.e.,

$\mathrm{SNR}_{p}=10\log_{10}\frac{\sigma^{2}_{d}}{\sigma^{2}_{n}}.$ (28)

The error TS means that the estimated STO points outside the ISI-free region,
wherein the error probability of TS, denoted as $P_{e}$, is defined as

$P_{e}=\frac{Q_{0}}{Q_{1}},$ (29)

where $Q_{0}$ denotes the number of arriving preambles with error estimated
STOs, and $Q_{1}$ denotes the total number of transmitted preambles.

For the sake of clarity, we denote “DatCol” as the existed strategy for
collecting training data (including the training samples and the estimated
labels), in which the collection of training data is conducted by the classic
TS scheme (i.e., [8] for example), with $\mathrm{SNR}_{p}=20$dB. And “Ref
[30]” is denoted as the strategy of [30] for generating learning labels.
Correspondingly, “Prop_LC” and “Prop_FC” are denoted as the proposed loose
constraint-based strategy and the proposed flexible constraint-based strategy
for generating training data, respectively. In addition, we adopt the TS
scheme of [8], denoted as “Corr”, as the baseline in the simulations. For a
fair comparison, except for the comparative parameters, the given TS methods
in each figure(e.g., Fig. 2 to Fig. 7) separately have the same parameter
settings.

(a) Impact of the accuracy of learning labels.

(b) Impact of the completeness of training data.

Figure 2: The error probability of TS vs. SNRs, where $N=128$, $L_{c}=32$, and $\eta=0.2$. $\mathcal{L}$ and $\widetilde{\mathcal{L}}$ denote the proposed loose constraint-based strategy and the proposed flexible constraint-based strategy for generating the training data, respectively. $\tau_{P,\textrm{train}}$ and $\tau_{P,\textrm{test}}$ stand for the training maximum multi-path delay and the testing maximum multi-path delay, respectively. TABLE III:  Storage Consumption and Bandwidth Occupation vs. Different Strategies. Strategy |  Storage Consumption (MB) |  Bandwidth Occupation (sec)
---|---|---
“DatCol” | $633.5277$ | $50.6822$
“Prop_LC” and “Prop_FC” | $469.7252$ | $0$

###  IV-B Effectiveness Analysis

In Fig. 2(a) and Fig. 2(b), the accuracy and completeness of learning labels
are validated, respectively. Additionally, the storage consumption and
bandwidth occupation of different strategies for obtaining training data are
given in Table III.

From Fig. 2(a), the error probability of “DatCol” is higher than those of the
proposed “Prop_LC” and “Prop_FC”. This is due the fact that the collected
training data set includes the incorrect learning labels according to (12).
Therefore, the existed strategy of training data collection (i.e., “DatCol”)
inevitably deteriorates the performance of ELM-based TS network, increasing
the error probability of TS. By contrast, the proposed strategies of “Prop_LC”
and “Prop_FC” achieve smaller error probabilities of TS than “DatCol”. This
indicates that the proposed strategy for training data generation avoids
incorrectly estimating learning labels, due to the good controllability of
computer-aided data generation.

To present the merits of the proposed TS methods in improving the TS
correctness when changing wireless scenarios, Fig. 2(b) depicts the error
probability of TS by changing the testing maximum multi-path delay. From Fig
2(b), when $\tau_{P,\textrm{test}}>\tau_{P,\textrm{train}}$, the ELM-based TS
network trained with the strategies of “DatCol” and “Ref [30]” reach higher
error probabilities than those of “Prop_LC” and “Prop_FC”, due to the impacts
of the impaired completeness of training data. Especially for “Ref [30]” in
the whole SNR region, its error probabilities significantly increase when
$\tau_{P,\textrm{test}}>\tau_{P,\textrm{train}}$. For example, when
$\tau_{P,\textrm{test}}=24$, the error probability of “DatCol” is higher than
0.25 for each given value of SNR, and the error probabilities of “Ref [30]”
are higher than 0.15 for the relatively high SNRs. As a result, the ELM-based
TS networks trained with the existed strategies of “DatCol” and “Ref [30]”
hinder themselves to the practical application for the impaired completeness
of training data. By contrast, after training the TS network, the error
probabilities of “Prop_LC” and “Prop_FC” are much smaller than those of
“DatCol” and “Ref [30]” when $\tau_{P,\textrm{test}}=24$, for all given SNRs.
This reflects that the proposed strategies of “Prop_LC” and “Prop_FC” can
generate more complete training data. This is because “Prop_LC” and “Prop_FC”
utilize the channel models with randomly generated multi-path delays (i.e.,
$\tau_{P,\textrm{train}}=\mathcal{L}$ and $\mathcal{L}\sim(16,26)$) for model
training, the diverse $\mathcal{L}$ generates more power delay profiles (PDPs)
to increase the diversity of training data for increasing the completeness of
training data. Then, a good completeness of training data can be obtained to
facilitate the performance of ELM-based TS networks, reducing the error
probability of TS. For example, when $\mathrm{SNR}_{p}\geq 10$dB and
$\tau_{P,\textrm{test}}=24$, the error probabilities of “Prop_LC” and
“Prop_FC” are lower than $0.0583$. Therefore, compared with the existed
strategies of “DatCol” and “Ref [30]” for obtaining training data, the
proposed computer-aided strategies for generating training data (i.e.,
“Prop_LC” and “Prop_FC”) can obtain more complete training data set, providing
improvement in the reduction of TS errors against the parameter of
$\tau_{P,\textrm{test}}$.

According to (12), the estimated learning labels are critically dependent on
the error probability of TS, during the training data collection. By assuming
the training data collected under $\mathrm{SNR}_{p}=20$dB as shown in Fig. 2,
the error probability of TS equals about 0.257 so that the accuracy of
estimated learning labels is about $0.743$. To collect $10^{5}$ effective
training samples (i.e., $10^{5}$ correct learning labels), at least
$1.35\times 10^{5}$ received samples are required. Since the collected samples
need to be transmitted by the device of data collection to the Central
Processing Unit, there are storage and bandwidth occupation. From Table III,
we can see that, the storage cost of the proposed strategies (i.e., “Prop_LC”
and “Prop_FC”) are less than that of “DatCol”, since the method of data
generation is known to the transceiver. That is, the training data can be
timely generated and released, leading to less consumption of storage compared
with “DatCol”. Furthermore, there are no incorrect learning labels and the
data transmission (i.e., bandwidth occupation) is avoided. If the transmission
bandwidth is given by $100$MHz, it could be obtained that “DatCol” requires
50.6822 sec for transmitting the collected samples, as shown in Table III. By
contrast, “Prop_LC” and “Prop_FC” avoid bandwidth occupation for transmitting
training samples during the wireless transmission, i.e., the bandwidth
occupation in sec equals to 0. _This is because, the training data is
generated at local receiver, which does not occupy the transmission
bandwidth._ As a result, compared with “DatCol”, the proposed “Prop_LC” and
“Prop_FC” significantly improve the storage consumption and bandwidth
occupation.

In summary, compared with “DatCol”, the error probability of TS reveals that
the proposed strategies of “Prop_LC” and “Prop_FC” can avoid the incorrect
learning labels and improve the completeness of training data. In addition,
compared with “DatCol” in Table III, the proposed strategies of “Prop_LC” and
“Prop_FC” possess less storage consumption and bandwidth occupation, which are
friendly to practical implementation.

###  IV-C Generalization Analysis

Figure 3: The generalization analysis with respect to the maximum multi-path
delay, where $N=128$, $L_{c}=32$, $\eta=0.2$, and different testing values of
$\tau_{P,\textrm{test}}$ (i.e., $\tau_{P,\textrm{test}}=22$ and
$\tau_{P,\textrm{test}}=24$) are considered.

The more complete the training data is, the stronger the statistical
efficiency of the neural network will be. Hence, we utilize the error
probability of TS to validate the generation performance of the designed TS
learner (i.e., the ELM-based TS network with training strategies of “Prop_LC”
and “Prop_FC”). In addition, we do not discuss the generalization performance
of “DatCol”, since the TS performance of “DatCol” is mainly affected by
incorrect learning labels in the training data collection. Except for the
parameter that needs to discuss, other basic parameters remain the same as
those adopted in Section IV-B.

From Fig. 3, for each given value of $\tau_{P,\textrm{test}}$, the error
probability of “Prop_FC” is less than those of “Corr” and “Ref [30]” in the
whole SNR region. This is due to the fact that the proposed strategy of
“Prop_FC” can generate more complete training data, which improves the
generalization performance against the impact of $\tau_{P,\textrm{test}}$. As
for $\tau_{P,\textrm{test}}=22$, “Ref [30]” reaches a lower error probability
of TS than “Prop_LC”. Even so, with $\tau_{P,\textrm{test}}$ increases to
$24$, the error probability of “Ref [30]” is significantly higher than that of
“Prop_LC”, for the relatively high SNRs. As a result, compared with the
existed strategy of “Ref [30]”, the proposed strategies of “Prop_LC” and
“Prop_FC” are able to deal with the varied maximum multi-path delays.

###  IV-D Robustness Analysis

This subsection presents the robustness analysis of the proposed TS scheme
against the changed parameters such as the length of CP (i.e., $L_{c}$), the
size of OFDM symbol (i.e., $N$), and the decay factor (i.e., $\eta$), etc. In
addition, different TS schemes (e.g., [5, 6]) are adopted as the feature
extraction, which aims to reveal the robustness against the feature
extraction. The error probabilities of the proposed TS scheme are plotted from
Fig. 4 to Fig. 7. For the sake of clarity, “Ref [5]” and “Ref [6]” denote that
the TS schemes mentioned in [5] and [6] are utilized for the feature
extraction, respectively. The ELM-based TS network which directly learns the
received samples is denoted as “DS_Learn”, i.e., leveraging the received
samples as the ELM input without the procedure of feature extraction. In the
following simulations, unless otherwise specified, parameters remain the same
as those mentioned in Section IV-B.

Figure 4: The Error probability of TS vs. SNRs, where $N=128$, $L_{c}=32$,
$\eta=0.2$, $\tau_{P}=20$, and different TS schemes are used for feature
extraction.

####  IV-D1 Robustness Against the TS Scheme for Feature Extraction

Fig. 4 illustrates the error probability of TS to present the robustness
against the impact of feature extraction. From Fig. 4, it could be obtained
that, the error probability of “DS_Learn” is higher than that of “Corr”, even
for the relatively high SNRs. This reflects that the ELM-based TS network
cannot work well without the process of feature extraction. For each method of
feature extraction, the error probabilities of “Prop_LC” and “Prop_FC” are
smaller than that of “Corr”. This is due to the fact that the TS schemes as
feature extraction successfully extract the hidden state information about
STOs, and thus improve the correctness of estimating STOs. In addition, the
ELM-based TS networks with different feature extractions exhibit different
performance improvements in reducing the error probability of TS. Several
reasons may be used to explain this result. 1) The TS scheme of [6] is
vulnerable to the impact of channel fading. 2) The TS scheme of [6] provides a
sharper correlation peak compared with that of [5], but the timing index of
the correlation peak is largely independent of the ISI-free region. 3) The TS
scheme in [5] has robust features highlighting the ISI-free region compared
with that in [6]. Therefore, the ELM-based TS network assisted by the TS
scheme of [5] outperforms that assisted by [6], for the relatively high SNRs.
To sum up, compared with “Corr” and “DS_Learn”, the ELM-based TS network with
feature extraction can reach significantly lower error probabilities of TS.

Figure 5: The error probability of TS vs. SNRs, where $N=128$, $\eta=0.2$, and
different values of CP length (i.e., $L_{c}=16$ and $L_{c}=32$) are
considered. Correspondingly, when $L_{c}=16$, $\mathcal{L}$ is set as 13 and
$\widetilde{\mathcal{L}}$ satisfies that
$\widetilde{\mathcal{L}}\sim\mathcal{U}(8,13)$. As for $L_{c}=32$,
$\mathcal{L}$ is set as 26 and $\widetilde{\mathcal{L}}$ satisfies that
$\widetilde{\mathcal{L}}\sim\mathcal{U}(16,26)$.

####  IV-D2 Robustness Against $L_{c}$

To reveal the robustness of the proposed TS scheme against the impact of
$L_{c}$, Fig. 5 illustrates the error probability of TS by changing $L_{c}$.
From Fig. 5, for each given value of $L_{c}$, the error probabilities of
“Prop_LC” and “Prop_FC” are smaller than that of “Corr” in the whole SNR
region. It is worth noting that, by decreasing $L_{c}$, the error
probabilities of “Prop_LC” and “Prop_FC” are enlarged. Especially, “Prop_FC”
reaches a smaller error probability than “Prop_LC”. The reason is that
“Prop_FC” owns a more complete profile of CIRs, which facilitates the
statistical efficiency of the TS network. Although the ISI-free region is
shortened by narrowing the length of CP, the proposed “Prop_LC” and “Prop_FC”
still obtain a robust improvement in reducing the error probability of TS.

Figure 6: The error probability of TS vs. SNRs, where $\tau_{P}=20$,
$L_{c}=32$, $\eta=0.2$, and different values of symbol size (i.e., $N=128$ and
$N=256$) are considered.

####  IV-D3 Robustness Against $N$

Fig. 6 depicts the error probability of TS against the impacts of varying $N$.
From Fig. 6, for each given value of $N$, the error probability of “Corr” is
higher than those of “Prop_LC” and “Prop_FC” in the whole SNR region. Among
the proposed strategies of “Prop_LC” and “Prop_FC”, “Prop_FC” reveals a
remarkable improvement in the reduction of error provability relative to
“Prop_LC”, for each given $N$. This is due to the fact that, “Prop_FC” has the
ability to improve the correlation between the training labels and the ground
truth labels, and thus provides improvements in reducing error probability of
TS. For example, when $\mathrm{SNR}_{p}=12$dB and $N=256$, the error
probability of “Prop_LC” is around $0.084$, while it achieves 0.014 for
“Prop_FC”. Although the error probability of TS increases with the increase of
$N$, the proposed “Prop_LC” and “Prop_FC” still outperform “Corr”. As a
result, the proposed strategies of “Prop_LC” and “Prop_FC” achieve robust
improvements in the reduction of TS errors against the impact of $N$.

Figure 7: The error probability of TS vs. SNRs, where $\tau_{P}=20$,
$L_{c}=32$, $N=128$, and different values of decay factor (i.e., $\eta=0.05$,
$\eta=0.20$, and $\eta=0.35$) are considered.

####  IV-D4 Robustness Against $\eta$

To reflect the robustness of the proposed TS scheme against the impact of
$\eta$, Fig. 7 depicts the error probability of TS where $\eta$ varies from
0.05 to 0.35 with the interval being 0.15. In Fig. 7, the error probability of
“Corr” decreases with the decrease of $\eta$. Because the correlation
platform/peaks obtained by “Corr” have been enhanced by the large value of
$\eta$, and so that timing instants of the maximum peak appear in advance to
the ISI region of CP. For each given value of $\eta$ in Fig. 7, the error
probability of “Corr” is higher than those of “Prop_LC” and “Prop_FC” in the
whole SNR region. For the relatively high SNRs, the error provability of
“Prop_FC” is smaller than that of “Prop_LC”. This indicates the superiority of
“Prop_FC” in reducing the TS errors. Also, it could be observed that, the
error probabilities of “Prop_LC” and “Prop_FC” decrease with the increase of
$\eta$. This is likely due to the enlarged $\eta$, which strengthens the power
of the first propagation path. The stronger the first propagation path is, the
stronger timing features will be, such that improves the learning efficiency
of ELM-based TS network. In addition, although the differences of these error
probabilities are not obvious to the varied $\eta$, ELM-based TS networks with
the proposed strategies of “Prop_LC” and “Prop_FC” still outperform “Corr” in
the aspect of reducing TS errors. In a word, compared with “Corr”, the
proposed strategies of “Prop_LC” and “Prop_FC” can provide the robust and
effective improvement in reducing the error probability of TS.

##  V Conclusion

In this paper, we propose two computer-aided strategies to generate the
training data, and construct an ELM-based TS network for the OFDM systems.
Compared with training data collected from the realistic systems, the desired
training data could be timely generated by our proposed strategies at the
local device, and this avoids the occupations of storage space and
transmission bandwidth during the training data collection. Furthermore, the
proposed computer-aided training strategies increase the completeness of
training data against the maximum multi-path delay, therefore improving
statistical efficiency of the ELM-based TS network. By simulations, numerical
results exhibit that the proposed ELM-based TS scheme is superior in reducing
the error probability of TS and possesses a good generalization performance
against the maximum multi-path delay.

##  VI Acknowledgement

This work is supported in part by the Sichuan Science and Technology Program:
(Grant-No. 2021JDRC0003, 2023YFG0316, 2021YFG0064), the Demonstration Project
of Chengdu Major Science and Technology Application (Grant No. 2020-YF09-
00048-SN),the Special Funds of Industry Development of Sichuan Province (Grant
No. zyf-2018-056), and the Industry University Research Innovation Fund of
China University: (Grant No. 2021ITA10016).

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