Date of publication 0000 00, 0000, date of current version 0000 00, 0000.
XXXXXX

This work is supported in part by the Key Projects of Education Department of
Sichuan Province (Grant 15ZA0134), the Special Funds of Industry Development
of Sichuan Province (Grant zyf-2018-056), the Major Special Funds of Science
and Technology of Sichuan Science and Technology Plan Project (Grant No.
19ZDZX0016 /2019YFG0395), the Demonstration Project of Chengdu Major Science
and Technology Application (Grant No. 2020-YF09-00048-SN), the Sichuan Science
and Technology Program (Grant No. 2021JDRC0003), and the Innovation Fund of
Postgraduate, Xihua University (Grant ycjj2020116).

Corresponding author: Chaojin Qing (e-mail: qingchj@uestc.edu.cn).

# ELM-based Frame Synchronization in Nonlinear Distortion Scenario Using
Superimposed Training

CHAOJIN QING1  WANG YU1  SHUHAI TANG1  CHUANGUI RAO1  AND JIAFAN WANG2  School
of Electrical Engineering and Electronic Information, Xihua University,
Chengdu, 610039, China. (e-mail<EMAIL_ADDRESS> Synopsys Inc., 2025 NE
Cornelius Pass Rd, Hillsboro, OR 97124, USA.

###### Abstract

The requirement of high spectrum efficiency puts forward higher requirements
on frame synchronization (FS) in wireless communication systems. Meanwhile, a
large number of nonlinear devices or blocks will inevitably cause nonlinear
distortion. To avoid the occupation of bandwidth resources and overcome the
difficulty of nonlinear distortion, an extreme learning machine (ELM)-based
network is introduced into the superimposed training-based FS with nonlinear
distortion. Firstly, a preprocessing procedure is utilized to reap the
features of synchronization metric (SM). Then, based on the rough features of
SM, an ELM network is constructed to estimate the offset of frame boundary.
The analysis and experiment results show that, compared with existing methods,
the proposed method can improve the error probability of FS and bit error rate
(BER) of symbol detection (SD). In addition, this improvement has its
robustness against the impacts of parameter variations.

###### Index Terms:

frame synchronization, extreme learning machine, nonlinear distortion,
superimposed training

=-15pt

##  I Introduction

Due to the limited bandwidth resources, wireless communication systems pursue
high spectrum efficiency in the past few decades [1]. As we can see, the
spectrum efficiency of the fifth generation (5G) wireless communication system
is many times higher than that of the fourth generation (4G) wireless
communication system [2], [3]. In wireless communication systems, frame
synchronization (FS) is a fundamental and essential task to guarantee the
overall system performance [4], which usually consumes substantial bandwidth
resources to overcome the synchronization challenge [5]. Thus, during the FS
phase, the contradiction between the high bandwidth resources consumption and
the high spectrum efficiency requirement needs to be resolved. Meanwhile, the
wireless communication system has a large number of nonlinear devices or
blocks, e.g., high power amplifier (HPA), digital to analog converter (DAC),
etc., inevitably causing nonlinear distortion [6], [7]. With limited
considerations for nonlinear distortion, the classical methods (e.g.,
correlation-based FS [8]) and the recent solutions (e.g., compressed sensing-
based FS in [9]) are usually difficult to apply in nonlinear distortion
scenarios [10]. Therefore, the FS is facing challenges from not only the
spectrum efficiency but also the nonlinear distortion.

To cope with nonlinear distortion, machine learning (ML), in particular, deep
learning (DL) has shown its prominent ability [11], [12]. In recent years, DL
has been applied in wireless communication, e.g., signal detection [13],
precoding [14], channel state information (CSI) feedback [15], channel
estimation [16], [17], mobile Internet of Things (IoT) [18], etc. However,
these DL-based approaches exist weaknesses such as long-time training, complex
parameter tuning [10], [19] etc. Different from DL-based scheme, as a single
hidden layer feed-forward neural network, extreme learning machine (ELM) can
learn quickly, randomly generate for input weight and hidden bias, require no
gradient back-propagation, and has good generalization performance [10], [20],
[21]. As one of the effective options, using ELM to deal with nonlinear
distortion is a promising solution.

For saving bandwidth resources and thus improving spectrum efficiency, the FS
using superimposed training sequence is an attractive scheme. Without any
occupation of bandwidth resources, this FS superimposes the training sequence
on data symbols, yielding more transmission symbols than that of non-
superposition mode [22] in the same transmission interval. The superimposed
training-based FS has been investigated in past years, e.g., [22]–[25]. These
promising FSs promote us to develop further explorations, especially for the
scenarios with nonlinear distortion.

Inspired by those advantages of ELM networks and superimposed training, we
investigate an ELM-based FS by using superimposed training, which overcomes
the challenges from spectrum efficiency and nonlinear distortion during the FS
phase. In our work, the merits to cope with nonlinear distortion can be reaped
by ELM networks, and then high spectrum efficiency can be achieved by using
superimposed training. The combination of ELM network and superimposed
training further improves the FS performance in the scenarios of nonlinear
distortion, e.g., the error probability of FS. To our best knowledge, for ELM-
based FS, there are limited works to focus on nonlinear distortion, much less
to focus on superimposed training.

###  I-A RELATED WORKS

We respectively present the related works of DL-based FS and ELM-based FS as
follows.

The DL-based FS has been investigated in [26]–[29]. In [26], an artificial
neural network (ANN)-based synchronization method was proposed. For the end-
to-end communication systems, [27] and [28] investigated the FS based on
neural network (NN). In [27], to achieve the task of FS, a deep neural network
(DNN) was employed to auto-encoder, and a convolutional neural network (CNN)
was developed in [28] to compensate impairments introduced by timing offset
and timing error of sampling. As for [29], a CNN-based FS method was proposed
to convert the one dimension (1D) correlator to two dimension (2D) matrix to
find the frame offset. From [26]–[29], the DL technology provides effective
approaches for FS. Nevertheless, these DL-based FSs are still challenged by
many issues, such as long training time, complex parameter tuning, large
memory requirements, etc.

Relative to DL networks, the ELM network features many advantages [10], [20],
[21], e.g., the gradient back-propagation can be avoided, the output weight
can be obtained by solving the least square (LS) question, and a single-hidden
layer is employed for the feed-forward neural network, etc. In [10], an ELM-
based time-division FS was proposed with the consideration of nonlinear
distortion. In [30], the ELM network was employed to compensate the residual
time offset with time-division mode. Although the error probability of FS is
lower than conventional method, the training sequence for FS still occupies
the bandwidth resources, which reduces systems’ spectrum efficiency. To avoid
the occupation of bandwidth resources, an ELM-based FS method using
superimposed training is proposed in this paper to reduce the error
probability of FS.

###  I-B CONTRIBUTIONS

To overcome the challenges of spectrum efficiency and nonlinear distortion
during the FS phase, the ELM-based FS using superimposed training is
investigated in this paper. The main contributions of this paper are
summarized as follows.

  * •

Firstly, an ELM-based FS method by using superimposed training is proposed. In
contrast to the ELM-based time-division FS scheme in [9], not only the
occupation of bandwidth resources is avoided in the proposed FS method, but
also the smaller error probability of FS is achieved with the same energy
cost.

  * •

Secondly, the superimposed training-based FS is investigated in the scenarios
of nonlinear distortion. Our investigation remedies the deficiencies of the
existing superimposed training-based FS, which cannot work well in the
scenarios of nonlinear distortion and is suitable for practical application.

  * •

Thirdly, extensive experiments are given to verify the effectiveness of the
proposed method in this paper. Compared with the classical correlation method
in [8] and the time division method in [10], both the FS’s error probability
and the symbol detection (SD)’s bit error rate (BER) are reduced with the same
energy consumption. In addition, the proposed FS presents its robustness
against the impacts of parameters.

The remainder of this paper is structured as follows: In Section II, we
describe the system model. The ELM-based FS using superimposed training method
is presented in Section III, followed by the experimental results and analysis
are illustrated in Section IV. Finally, Section V concludes our work.

Notations: Bold face upper case and lower case letters denote matrix and
vector respectively. ${\left(\cdot\right)^{T}}$, ${\left(\cdot\right)^{H}}$,
$\mathbf{(\cdot)^{\dagger}}$, denote the transpose, conjugate transpose, and
matrix pseudo-inverse, respectively. ${\left\|\cdot\right\|_{2}}$ is the
Frobenius norm. $\left|{x}\right|$ denotes the absolute value of $x$ and
$\left|{\mathbf{x}}\right|$ denotes the absolute value operation to the entry-
wise of vector $\mathbf{x}$.

##  II SYSTEM MODEL

At the receiver, the received $M\times 1$ complex-valued signal vector,
denoted as $\mathbf{y}$, can be expressed as

${\mathbf{y}}={\mathbf{H}}{\widetilde{\mathbf{x}}_{{\text{ext}}}}+{\mathbf{n}},$
(1)

where $\mathbf{n}\in\mathbb{C}^{M\times 1}$ represents the complex additive
white Gaussian noise (AWGN) vector with zero-mean and $\sigma^{2}$-variance
entries.
${\widetilde{\mathbf{x}}_{{\text{ext}}}}\in{\mathbb{C}^{\left({2N-L+1}\right)\times
1}}$ denotes the extended vector of transmitted signal with nonlinear
distortion, which can be given by

${\widetilde{\mathbf{x}}_{{\text{ext}}}}={\left[{\underbrace{0,\cdots,0}_{\tau},{{\widetilde{\mathbf{x}}}^{T}},\underbrace{0,\cdots,0}_{N-L-\tau+1}}\right]^{T}},$
(2)

where $N$ and $L$ are the size of search window and the number of multi-path,
respectively. The unknown frame boundary offset is denoted as $\tau$, whose
range is $0\leq\tau\leq N-{L}+1$. In (2),
${\mathbf{\widetilde{x}}}={\left[{{{\widetilde{x}}_{1}},{{\widetilde{x}}_{2}},\cdots,{{\widetilde{x}}_{N}}}\right]^{T}}$
denotes the distorted transmitted signal and can be expressed as

${\mathbf{\widetilde{x}}}={f_{\mathrm{dis}}}\left({\mathbf{x}}\right),$ (3)

where ${f_{\mathrm{dis}}}(\cdot)$ represents the influence of nonlinear
distortion. ${\mathbf{x}}={\left[{{x_{1}},{x_{2}},\cdots,{x_{N}}}\right]^{T}}$
is the superimposed transmitting signal without nonlinear distortion, which
can be given by

${\mathbf{x}}=\sqrt{\rho{E}}{\mathbf{s}}+\sqrt{\left({1-\rho}\right){E}}{\mathbf{c}},$
(4)

where $\rho\in\left[{0,1}\right]$ represents the power proportional
coefficient (PPC), ${E}$ denotes the transmitted power of superimposed
transmitting signal. ${\mathbf{s}}\in{\mathbb{C}^{N\times 1}}$ is the training
sequence and ${\mathbf{c}}\in{\mathbb{C}^{N\times 1}}$ is the modulated data
symbol.

The complex matrix ${\mathbf{H}}\in{\mathbb{C}^{M\times(2N-L+1)}}$ given in
(1) is an $M\times(2N-L+1)$ cyclic matrix, which can be defined as

${{\mathbf{H}}=\left[{\begin{array}[]{*{20}{c}}{{h_{1}}}&0&\cdots\\\
\vdots&{{h_{1}}}&\ddots\\\ {{h_{L}}}&\vdots&\ddots\\\ 0&{{h_{L}}}&\ddots\\\
\vdots&0&\ddots\\\ {}\hfil&\vdots&\ddots\end{array}}\right]},$ (5)

where ${\mathbf{h}}={\left[{{h_{1}},{h_{2}},\cdots,{h_{L}}}\right]^{T}}$
denotes the finite CIR vector of $L$ samples memory, and ${h_{l}}$ represents
the complex-valued channel impulse response (CIR) of the $l$th path,
$l=1,2,\cdots,L$.

With the received signal $\mathbf{y}$ given in (1), we employ ELM network to
implement FS using superimposed training, as well as reducing the influence of
nonlinear distortion.

##  III ELM-based FS

The performance of FS is seriously degraded by the influence of nonlinear
distortion. To conquer this difficulty, the ELM network is introduced into FS
using superimposed training due to its prominent ability to cope with
nonlinear distortion [10]. In the following subsections, we first present the
pre-processing of FS in Section III-A. Then, the ELM-based FS method is given
in Section III-B.

###  III-A PRE-PROCESSING FOR FS

In wireless systems, the FS is usually difficult to obtain, especially for
nonlinear distortion scenarios. In particular, the error probability of DL-
based timing synchronization is far higher than that of matched filtering in
[26]. Similar behaviors are also observed in ELM-based FS experiments. Thus, a
pre-processing of FS is employed to coarsely capture the features of SM.
According to [8], by using the cross-correlation based method, the SM vector
${\mathbf{g}}\in{\mathbb{R}^{N\times 1}}$ can be expressed by

${\mathbf{g}}{\text{ = }}{\left|{{{\mathbf{S}}^{H}}{\mathbf{y}}}\right|^{2}}.$
(6)

Here, the $M\times N$ complex matrix ${\mathbf{S}}$ can be written as

${\mathbf{S}}=\left[{\begin{array}[]{*{20}{c}}{{s_{1}}}&0&\cdots\\\
\vdots&{{s_{1}}}&\ddots\\\ {{s_{N}}}&\vdots&\ddots\\\ 0&{{s_{N}}}&\ddots\\\
\vdots&0&\ddots\\\ {}\hfil&\vdots&\ddots\end{array}}\right],$ (7)

where ${s_{i}},i=1,2,\cdots,N$, represents the $i$th entry of training
sequence ${\mathbf{s}}$. Need to be mentioned that, besides the cross-
correlation based SM in (6), other SMs can also be applied in our method with
the similar processing.

In order to standardize the training of ELM network, the ${\mathbf{g}}$ given
in (6) is normalized as

$\overline{\mathbf{g}}={{{\mathbf{g}}\mathord{\left/{\vphantom{{\mathbf{g}}{\left\|{\mathbf{g}}\right\|}}}\right.\kern-1.2pt}{\left\|{\mathbf{g}}\right\|}}_{2}}.$
(8)

With the normalized SM (i.e., $\overline{\mathbf{g}}$), an ELM network is
utilized to conquer nonlinear distortion and improve SMs for superimposed
training-based FS, which is described in the following subsection.

###  III-B ELM-Based FS

The ELM-based network is employed to improve SMs and decrease the influence of
nonlinear distortion. The ELM-based FS includes offline and online procedures,
which are elaborated in TABLE I and TABLE II, respectively. The offline
training procedure is described as follows.

####  III-B1 OFFLINE TRAINING SPECIFICATION

The offline procedure is elaborated in TABLE I. In the following, we first
describe the data collection for ELM-net training.

  * •

Data Collection For ELM-Net training

For ELM-net training, ${N_{t}}$ samples of input signals and offset labels,
denoted by
$\left\\{{\left({{{\overline{\mathbf{g}}}_{i}},{{\mathbf{T}}_{i}}}\right)}\right\\},i=1,2,\cdots,{N_{t}}$
are collected to form a training set. According to the ${N_{t}}$ collected
${\overline{\mathbf{g}}_{i}}$ forms the input matrix
$\overline{\mathbf{g}}\in{\mathbb{R}^{N\times{N_{t}}}}$, which can be written
as

$\overline{\mathbf{g}}=\left[{{{\overline{\mathbf{g}}}_{1}},{{\overline{\mathbf{g}}}_{2}},\cdots,{{\overline{\mathbf{g}}}_{{N_{t}}}}}\right].$
(9)

Similarly, ${N_{t}}$ offset label vectors ${\mathbf{T}}_{i}$ are converted to
construct the target output matrix $\mathbf{T}\in\mathbb{R}^{N\times N_{t}}$,
which can be given by

${\mathbf{T}=\left[\mathbf{T}_{1},\mathbf{T}_{2},\cdots,\mathbf{T}_{N_{t}}\right]},$
(10)

where the label ${{\mathbf{T}}_{i}}$ can be encoded according to one-hot mode,
i.e.,

${{\bf{T}}_{i}}={\left[{\underbrace{0,\cdots,0}_{{\tau_{i}}},1,\underbrace{0,\cdots,0}_{N-{\tau_{i}}-1}}\right]^{T}},$
(11)

where ${\tau_{i}}$ is the $i$th sample’s frame boundary offset.

In this paper, the input weights
$\mathbf{W}\in{\mathbb{R}^{\widetilde{N}\times{N}}}$ and hidden layer biases
$\mathbf{b}\in\mathbb{R}^{\widetilde{N}\times{1}}$ of ELM network (with
$\widetilde{N}$ hidden neuron number) are respectively randomly chosen, which
is similar to the standard process of ELM network [21]. It should be noted
that the initial values of $\mathbf{W}$ and $\mathbf{b}$ have a certain impact
on the FS performance, we mainly investigate the FS method in this paper.
Admittedly, initial values can further improve the FS’s error probability and
SD’s BER. Then, the input weights $\mathbf{W}$ and hidden layer biases
$\mathbf{b}$ are saved in storage space for later use in the offline and
online procedure.

  * •

Networking Training

As shown in TABLE I, during the training procedure, the training data-set
$\left\\{{\left({\overline{\mathbf{g}},\mathbf{T}}\right)}\right\\}$, input
wights $\mathbf{W}$ and hidden layer biases $\mathbf{b}$ are gradually loaded
from storage space. Then, with
$\left\\{{\left({\overline{\mathbf{g}},\mathbf{T}}\right)}\right\\}$,
$\mathbf{W}$ and $\mathbf{b}$, the output matrix of the hidden layer
${\mathbf{H}}\in{\mathbb{R}^{\widetilde{N}\times{N_{t}}}}$ can be given by

${\mathbf{H}}=\sigma\left({{\mathbf{W}}\overline{\mathbf{g}}+{\mathbf{b}}}\right),$
(12)

where $\sigma\left(\cdot\right)$ represents the activation function, e.g.,
sigmoid [31], hyperbolic tangent [32], rectified linear units (ReLU) [33],
etc. The sigmoid is used in this ELM network [21]. The objective of ELM
network is actually to find the suitable output weights
$\mathbf{\Upsilon}\in{\mathbb{R}^{N\times\widetilde{N}}}$ to approximate the
target output matrix ${\mathbf{T}}$, which can be expressed as [20]

${\mathbf{T}}=\mathbf{\Upsilon}{\mathbf{H}},$ (13)

where
$\mathbf{\Upsilon}=\left[{{{\mathbf{\Upsilon}}_{1}},\cdots,{{\mathbf{\Upsilon}}_{k}},\cdots,{{\mathbf{\Upsilon}}_{\widetilde{N}}}}\right]$,
${\mathbf{\Upsilon}}_{k}$ denotes the output weighting vector connecting the
$k$th hidden neuron and the output neurons. The LS solution
$\widehat{\mathbf{\Upsilon}}$ of ${\mathbf{T}}=\mathbf{\Upsilon}{\mathbf{H}}$
with minimum norm of output weights $\mathbf{\Upsilon}$ is given by [20]

$\widehat{\mathbf{\Upsilon}}=\mathop{\min}\limits_{\mathbf{\Upsilon}}\left\|{\mathbf{\Upsilon}{\mathbf{H}}-{\mathbf{T}}}\right\|={\mathbf{T}}{{\mathbf{H}}^{\dagger}},$
(14)

the output weighting matrix $\mathbf{\Upsilon}$ is learned from offline
training of ELM network, which is saved in storage space for online running.

TABLE I: TRAINING PROCEDURE A training set
$\left\\{{\left.{\left({{{{\mathbf{\bar{g}}}}_{i}},{{\mathbf{T}}_{i}}}\right)}\right|i=1,2,\cdots,{N_{t}}}\right\\}$
generated by SMs and one-hot label and these two are corresponding to the
input and desired output of the ELM, respectively.  step 1 : Randomly chose
the input weights $\mathbf{W}\in\mathbb{R}^{\widetilde{N}\times N}$ and hidden
layer biases $\mathbf{b}\in\mathbb{R}^{\widetilde{N}\times 1}$, respectively,
set the hidden neuron number $\widetilde{N}$. step 2 : With
${\overline{\mathbf{g}}}$, $\mathbf{W}$ and $\mathbf{b}$, the output matrix of
hidden layer $\mathbf{H}\in\mathbb{R}^{\widetilde{N}\times N_{t}}$ is
calculated through activation function $\sigma\left(\cdot\right)$ according to
(12). step 3 : Construct the target output matrix $\mathbf{T}$ using offset
label vector $\mathbf{T}_{i}$ according to (10), compute the output weights
$\mathbf{\Upsilon}\in{\mathbb{R}^{{N}\times{\widetilde{N}}}}$ according to
(14).
---

####  III-B2 ONLINE DEPLOYMENT

With the learned ELM-based network parameters, the online running procedure
can be implemented in this subsection, which is shown in TABLE II.

TABLE II: ONLINE PROCEDURE step 1 : From (6) to (8), perform the preprocessing
to obtain metric vector $\overline{\mathbf{q}}$ as the input of the trained
ELM-net. step 2 : With input metric vector $\overline{\mathbf{q}}$, the
learned output weights $\mathbf{\Upsilon}$, the randomly chosen input weights
$\mathbf{W}$ and hidden layer biases $\mathbf{b}$, the ELM network output
$\mathbf{O}$ can be obtain according to (15). step 3 : Find the location of
the maximum square of the absolute value of the element in $\mathbf{O}$,
(i.e., estimation of frame boundary offset $\widehat{\tau}$) according to
(16).
---

With the input metric vector $\overline{\mathbf{q}}$, which can obtained by
employing the preprocessing procedure according to (6)-(8), the learned output
weights $\mathbf{\Upsilon}$, the random input weights $\mathbf{W}$ and hidden
layer biases $\mathbf{b}$, the ELM network output
${{\mathbf{O}}}\in{\mathbb{R}^{N\times 1}}$ can be written as

${\mathbf{O}}=\mathbf{\Upsilon}\cdot\sigma\left({{\mathbf{W}}{{\overline{\mathbf{q}}}}+{\mathbf{b}}}\right),$
(15)

where ${\mathbf{O}}={\left[{{o_{1}},{o_{2}},\cdots,{o_{N}}}\right]^{T}}$, and
the estimation of frame boundary offset can be expressed as

$\displaystyle{\widehat{\tau}}=$ $\displaystyle\mathop{\arg\max}\limits_{1\leq
j\leq{N}}{|{o_{j}}|^{2}}.$ (16)

With the estimation from (16), the FS is completed by acquiring the frame’s
starting point $\widehat{\tau}$, After FS, the SD is performed, and the
detected symbol $\widehat{\mathbf{c}}$ can be represented as

${\widehat{\mathbf{c}}=\frac{{{\widetilde{\mathbf{x}}_{{\text{est}}}}-\sqrt{\rho{E}}{\mathbf{s}}}}{{\sqrt{1-\rho}}}},$
(17)

where ${\widetilde{\mathbf{x}}_{{\text{est}}}}$ denotes the estimation of
superimposed transmitting signal in the scenario of nonlinear distortion, and
can be obtained by

${\widetilde{\mathbf{x}}_{{\text{est}}}}={{\mathbf{H}}^{\dagger}}{\mathbf{y}}.$
(18)

The FS and SD can be achieved according to (6)-(18), which improve SMs and
address the issues about multi-path interfere and nonlinear distortion.

##  IV EXPERIMENTAL ANALYSIS

In this section, numerical results of the proposed ELM-based FS using
superimposed training are given. Firstly, basic parameters and definitions
involved in simulations are given in Section IV-A. Then, in Section IV-B, the
FS’s error probability and SD’s BER of the proposed scheme with nonlinear
distortion is shown to verify the effectiveness of the proposed ELM-based FS,
followed by the robustness of improvement with different parameters is
discussed in Section IV-C. At last, the computational time complexity is
analyzed in Section IV-D.

[t!](topskip=0pt, botskip=0pt, midskip=0pt)[width=3.3 in]Figure1.pdf Error
probability of FS vs. SNR, where $N=512$, $L=8$ and $\operatorname{EVM}=35\%$.

[t!](topskip=0pt, botskip=0pt, midskip=0pt)[width=3.3 in]Figure2.pdf BER vs.
SNR, where $N=512$, $L=8$ and $\operatorname{EVM}=35\%$.

###  IV-A PARAMETER SETTING

In the simulations, the basic parameters are set as $N=512$, $M=2N=1024$,
$\widetilde{N}=10N=5120$ [20], [34], $L=8$, and ${N_{t}}={10^{5}}$. The
Zadoff-Chu sequence [35] is employed as the training sequence $\mathbf{s}$.
For the time-division method in [10], ${N_{s}}=16$ is considered as the length
of training sequence. By referencing [22] and considering the total
performance of FS’s error probability and SD’s BER, $\rho=0.3$ is adopted in
this paper. The modulated data symbol $\mathbf{c}$ is formed according to the
symbol of quadrature-phase-shift-keying (QPSK) modulation. For the channel
model, the multi-path Rayleigh fading channel with an exponentially-decayed
power coefficient $\eta=0.2$ is considered, where each of the following $L-1$
paths is set as zero-valued with a probability of 0.5 beside the first path to
keep the same situation as [9] and [10]. For the sake of fair comparison, we
assume the superimposed FS and the time-division FS consume the same energy
for transmitting symbols.

[t!](topskip=0pt, botskip=0pt, midskip=0pt)[width=7 in]Figure3.pdf Error
probability of FS and BER against the impact of EVM, where
$\operatorname{EVM}=35\%$, $\operatorname{EVM}=40\%$,
$\operatorname{EVM}=45\%$, and $\operatorname{EVM}=50\%$ are considered,
respectively.

Definitions involved are listed as follows. The signal-to-noise ratio (SNR) in
decibel (dB) is defined as [36]

$SNR=10{\log_{10}}\left({\frac{{{E}}}{{{\sigma^{2}}}}}\right).$ (19)

For nonlinear distortion, the HPA effect is taken into account in these
simulations. The nonlinear amplitude $A\left(x\right)$ and phase
$\Phi\left(x\right)$ are obtained by [37]

$A\left(x\right)=\frac{{{\alpha_{a}}x}}{{1+{\beta_{a}}{x^{2}}}},{\rm{}}\Phi\left(x\right)=\frac{{{\alpha_{\phi}}{x^{2}}}}{{1+{\beta_{\phi}}{x^{2}}}},$
(20)

where ${\alpha_{a}}=1.96$, ${\beta_{a}}=0.99$, ${\alpha_{\phi}}=2.53$, and
${\beta_{\phi}}=2.82$ are considered in the experiments according to [37]. To
measure the distortion intensity, the error vector magnitude (EVM) is used in
this paper, which is expressed as [38]

${\operatorname{EVM}\left(\%\right)=\sqrt{\frac{{\sum\limits_{n=1}^{N}{{{\left|{{{\widetilde{x}}_{n}}-{R_{n}}}\right|}^{2}}}}}{{\sum\limits_{n=1}^{N}{{{\left|{{R_{n}}}\right|}^{2}}}}}}},$
(21)

where ${{\widetilde{x}}_{n}}$ is the $n$-th distorted symbol through HPA,
which denotes the HPA workings in saturated region. ${R_{n}}$ denotes the
desired linear outputs of HPA given the same input without amplification
distortion. In this paper, the EVM is set as $\operatorname{EVM}=35\%$ except
for the robustness analysis against EVMs.

For simplicity, “Prop”, “TD_Corr”, “TD_ELM”, and “Sup_Corr” are used to denote
the proposed ELM-based superimposed FS, the correlation-based time-division FS
in [8], the ELM-based time-division FS in [10], and the correlation-based
superimposed FS method in [22], respectively.

###  IV-B FS and SD PERFORMANCE

To validate the effectiveness of the proposed ELM-based FS using superimposed
training, the error probability of FS and BER of SD under different SNRs are
illustrated in Fig. IV and Fig. IV, respectively.

The effectiveness of the error probability of FS is presented in Fig. IV. It
could be observed that the “Prop” reaches the smallest error probability among
different methods. This reflects the “Prop” obtains the best performance of
error probability for FS, and thus can work well in the scenarios of nonlinear
distortion. In addition, some insights of FS with nonlinear distortion can be
achieved in Fig. IV. Firstly, the error probability of “Sup_Corr” is smaller
than that of “TD_Corr” with the same energy consumption, which embodies the
superiority of superimposed FS compared with time division FS. Secondly, for
relatively high SNR (e.g., $SNR>6$dB), the “TD_ELM” has a smaller error
probability of FS than those of “Sup_Corr” and “TD_Corr”. That is, the ELM
network effectively suppresses the nonlinear distortion for time division FS
in [8], and the ELM-based time-division FS can further obtain better
performance of error probability than that of superimposed FS in [22].
Although the ELM-based time-division FS has shown its effectiveness to
suppress the nonlinear distortion, it can only be effective in a relatively
high SNR region, and the merits of superimposed FS have not been developed.
The “Prop” develops the merits of superimposed training and ELM network, and
thus obtains the smallest error probability of FS in all given SNR region.
Therefore, the combination of superimposed training and ELM network in “Prop”
possesses its effectiveness to improve the error probability of FS in the
scenarios of nonlinear distortion.

[t!](topskip=0pt, botskip=0pt, midskip=0pt)[width=7 in]Figure4.pdf Error
probability of FS and BER against the impact of $L$, where $L=4$, $L=6$,
$L=8$, and $L=10$ are considered, respectively.

Since the training sequence $\mathbf{s}$ is superimposed on the modulated data
symbol $\mathbf{c}$, it needs to be verified whether the superimposed
interference (from the superimposed training) degrades the detection
performance of data symbols. In this paper, the BER of SD is used to measure
the detection performance and is plotted in Fig. IV. From Fig. IV, the “Prop”
achieves the smallest BER for almost all given SNRs. Thus, for the same energy
consumption, the superimposed interference does not degrade the BER
performance. On the contrary, the BER of “Prop” effectively benefits from the
proposed FS, especially for the relatively high SNR region (e.g., $SNR>10$dB).
That is, with simple processing of interference cancellation given in (17),
the “Prop” achieves the best BER performance among all given FS methods. In
particular, the advantages of superimposed training and ELM network can be
separately demonstrated from Fig. IV. Without using superimposition
approaches, “TD_ELM” obtains smaller BER than that of “TD_Corr”, which
reflects the effectiveness of ELM network to deal with nonlinear distortion.
We can also observe that the BER of “Sup_Corr” is smaller than that of
“TD_Corr”. That is, superimposed training used in “Sup_Corr” is useful to
improve the BER performance of “TD_Corr”. Thus, by combining the superimposed
training and ELM network, the BER performance is improved.

As a whole, compared with the “TD_Corr”, “TD_ELM”, and “Sup_Corr”, both the
FS’s error probability and the SD’s BER in the scenarios of nonlinear
distortion are improved by “Prop”. Especially, compared with the “TD_Corr” and
“TD_ELM”, the “Prop” can obtain the chance to transmit more data symbols, and
thus can further improve the spectrum efficiency. By the way, with different
modulation conditions (e.g., BPSK and 16QAM), the proposed method still
improves the FS’s error probability and SD’s BER from Fig. IV and Fig. IV.

###  IV-C ANALYSIS OF PARAMETER IMPACT

In this subsection, the robustness of the proposed scheme against parameter
variation is analysed. The impact of EVM is first discussed, followed by the
number of multi-path (i.e., $L$), the transmitted frame-length $N$, and the
PPC $\rho$. It is worth noting that, besides the change of the impact
parameter (i.e, EVM, $L$, $N$, and $\rho$), other basic parameters remain the
same as those given in Section IV-A during the simulations.

####  IV-C1 IMPACT OF EVM

EVM is usually used to measure the distortion intensity. To analyze the
robustness of the proposed method against different distortion intensities,
Fig. IV-A respectively plots the curves of FS’s error probability and SD’s BER
with different EVMs (i.e, $\mathrm{EVM}=35\%$, $\mathrm{EVM}=40\%$,
$\mathrm{EVM}=45\%$, and $\mathrm{EVM}=50\%$).

From Fig. IV-A, compared with those of “TD_Corr”, “TD_ELM”, and “Sup_Corr”,
the “Prop” method achieves the smallest error probability for each given EVM.
That is, relative to the existing methods, the proposed FS scheme still
improves the FS’s error probability against varying EVMs. With the increase of
EVM, the FS’s error probabilities for all curves in Fig. IV-A (i.e.,
“TD_Corr”, “TD_ELM”, “Sup_Corr”, and “Prop”) increase due to the rise of
distortion intensity. However, the FS’s error probability of “Prop” is smaller
than those of “TD_Corr”, “TD_ELM”, and “Sup_Corr”, especially for the high SNR
region. This reflects the proposed scheme can improve the FS’s error
probability against varying EVMs.

For the SD, Fig. IV-A shows the BER of “Prop” is smaller than those of
“TD_Corr” and “Sup_Corr” in almost all given SNR regions. Especially, in a
relatively high SNR region, e.g., $SNR\geq 12$dB, we can observe the BER of
“Prop” is smaller than those of “TD_Corr”, “Sup_Corr”, and “TD_ELM”. Thus, for
different EVMs, the “Prop” achieves similar or better BER performance.

As a result, against the impact of EVM, the “Prop” possesses its robustness
for improving FS’s error probability and SD’s BER.

[t!](topskip=0pt, botskip=0pt, midskip=0pt)[width=7 in]Figure5.pdf Error
probability of FS and BER against the impact of $N$, where $N=256$, $N=512$,
$N=768$, and $N=1024$ are considered, respectively.

####  IV-C2 IMPACT OF $L$

The FS’s error probability and SD’s BER are usually impacted by the number of
multi-path (i.e., $L$). To demonstrate the robustness of the proposed FS
scheme against the impact of $L$, the error probability of FS and BER of SD
curves are given in Fig. IV-B, where $L=4$, $L=6$, $L=8$, and $L=10$ are
considered, respectively.

From Fig. IV-B, relative to the “TD_Corr”, “TD_ELM”, and “Sup_Corr”, the
“Prop” achieves the minimal error probability of FS for each given $L$. This
reflects the “Prop” improves the FS’s error probability of the existing
methods with the variations of $L$. In addition, with the increase of $L$, the
FS’s error probabilities of “Prop”, “TD_Corr”, “TD_ELM”, and “Sup_Corr” rise
with the enlargement of multi-path interference. Even so, the “Prop” still
presents the ability to cope with the nonlinear distortion and multi-path
interference under different values of $L$, and thus obtains the smallest FS’s
error probability. As a whole, against the impact of $L$, the “Prop” can
robustly reduce FS’s error probability.

From the curves of SD’s BER in Fig. IV-B, the “Prop” achieves the smallest BER
for relatively large $L$ (e.g., $L\geq 8$). This reflects the “Prop” possesses
better BER performance compared with the existing methods when $L\geq 8$. For
the case where $4$dB$\leq SNR\leq 12$dB and $L=4$, the BER of “Prop” is
slightly higher than that of “TD_ELM”. This is because the “TD_ELM” consumes
additional bandwidth resources to avoid the superimposed interference of
“Prop”. Moreover, the ELM network is also employed by “TD_ELM”, and thus
effectively suppresses the nonlinear distortion as well. Nevertheless,
compared with “TD_ELM” for the case where $4$dB $\leq SNR\leq$ $8$dB, the
“Prop” has only slightly higher BER, while transmitting more data symbols to
obtain higher spectrum efficiency. Besides, the BER of “Prop” is still smaller
than that of “TD_ELM” when $SNR\geq 12$dB.

To sum up, against the impact of $L$, the “Prop” can effectively reduce the
FS’s error probability and SD’s BER, especially for the cases of relatively
large $L$ and relatively high SNR.

####  IV-C3 IMPACT OF $N$

Usually, the FS’s error probability and SD’s BER performance are influenced by
the frame-length, i.e., $N$. To validate the robustness against the impact of
$N$, the error probability of FS and BER of SD are illustrated in Fig. IV-C1
with different values of $N$ (i.e., $N=256$, $N=512$, $N=768$, and $N=1024$).

From Fig. IV-C1, for the cases where $N=512$, $N=768$, and $N=1024$, the
“Prop” obtains a smaller FS’s error probability than those of “TD_Corr”,
“TD_ELM”, and “Sup_Corr”. This reflects the “Prop” could reduce the error
probability of FS against the varying $N$. When $N=256$ and $SNR\leq 8$dB, the
“Prop” can still obtain the minimum error probability of FS to embodies the
robustness against the impact of $N$. However, for $SNR\geq 10$dB, such
situation could not be held. As can be seen in Fig. IV-C1(a), the FS’s error
probability of “Prop” is higher than that of “TD_ELM”. This is because the
“TD_ELM” can also suppress the nonlinear distortion as that of the “Prop”,
while the relatively short $N$ brings “Prop” the difficulty to combat the
superimposed interference from the data symbol $\mathbf{c}$. Even so, compared
with the “TD_ELM”, the “Prop” saves the bandwidth resources and significantly
reduces FS’s error probability in a relatively low SNR region (e.g., $SNR\leq
8$dB). In particular, relative to the “TD_ Corr” and “Sup_Corr”, the “Prop”
clearly reduces the error probability of FS for all given SNRs. In addition,
with the increase of $N$, the FS’s error probability of “Prop” decreases. This
reflects the elongated $N$ can effectively suppress the superimposed
interference of the “Prop”, due to the elongation of the superimposed training
sequence $\mathbf{s}$.

Compared with “TD_ELM”, “TD_Corr”, and “Sup_Corr”, the “Prop” has similar or
smaller BER, which demonstrates the “Prop” obtains the similar or better SD’s
BER performance with different values of $N$. Relative to “TD_Corr” and
“Sup_Corr”, the “Prop” reduces the BER for each given $N$, especially in
relatively high SNR region (e.g., $SNR\geq 10$dB). Meanwhile, for the case
where $N=256$ and $0$dB$\leq SNR\leq$ $10$dB, the “Prop” obtains similar BER
performance as that of “TD_ELM”. When $SNR\geq 10$dB, the BER of “Prop” is
slightly higher than that of “TD_ELM”. The reasons are given as follows. On
the one hand, the nonlinear distortion is also suppressed by employing the ELM
network in “TD_ELM”. On the other hand, the “TD_ELM” consumes additional
bandwidth to avoid superimposed interference. Moreover, the corresponding FS’s
error probability of “TD_ELM” is smaller than that of “Prop”, which also
deteriorates the SD’s BER performance. Even so, the “Prop” can save the
bandwidth resource and obtains similar BER performance as that of “TD_ELM”
when $N=256$. Especially, for relatively large $N$ (e.g., $N\geq 512$), the
“Prop” can achieve lower BER than that of “TD_ELM” when $SNR\geq 12$dB.

On a whole, with the varying of $N$, the “Prop” can improve the performance of
FS’s error probability and SD’s BER, and this improvement possesses good
robustness.

TABLE III: Time complexity comparison | TD_ELM | Proposed
---|---|---
Training time | about 19.6 minutes | about 20.2 minutes
Online running time | about 11.6 minutes | about 11.7 minutes

###  IV-D COMPLEXITY ANALYSIS

The training time and online running time between the correlation-based
superimposed FS method in [10] (i.e., “TD_ELM”) and the proposed ELM-based
superimposed FS (i.e., “Prop”) are illustrated in TABLE III to compare the
computational time complexity.

For a fair comparison, $10^{5}$ experiments are conducted for “Prop” and
“TD_ELM” on the same server with Intel Xeon(R) E5-2620 CPU 2.1GHz_16 by using
Matlab software, respectively. During the experiments, only running time is
considered, (i.e., the time to generate data sets and through channel is not
included). From TABLE III, in the training phase, the “TD_ELM” consumes about
19.6 minutes, while the “Prop” costs about 20.2 minutes. In the online running
stage, the “TD_ELM” consumes about 11.6 minutes, and the “Prop” costs about
11.7 minutes. We can see that the average training and online running time of
“Prop” is slightly more than that of “TD_ELM” in each experiment, while the
“Prop” improves the spectrum efficiency with similar or better performance of
FS’s error probability and SD’s BER relative to the existing methods.

##  V CONCLUSION

In this work, we integrated superimposed training-based FS and ELM network to
investigate an ELM-based FS scheme using superimposed training in nonlinear
distortion scenarios. Firstly, a preprocessing procedure is employed to
coarsely reap the features of SM. Then, an ELM network is introduced to
conquer the impact of nonlinear distortion and to obtain the estimation of
frame boundary offset. Compared with some existing methods, the proposed
method can improve the error probability of FS and BER of SD, and those
improvements are robust against parameter variation. In future works, we will
investigate the generalization method of ELM-based FS to alleviate the
difference between the data set from simulation and the data set in real
scenarios.

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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 a 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.
---|---
|  Wang Yu received the B. S. degree from the School of Electrical Engineering
and Electronic Information, Xihua University, Chengdu, China, in 2017, where
he is currently pursuing the M. S. degree under the supervision of Prof. Qing.
His research interests include frame synchronization and frequency
synchronization in wireless communications, and deep learning in physical
layer of wireless communications.
---|---
|  Shuhai Tang received the B. S. degree from the School of Electrical
Engineering and Electronic Information, Xihua University, Chengdu, China, in
2020, where he is currently pursuing the M. S. degree under the supervision of
Prof. Qing. His research interests include synchronization and channel
estimation in OFDM system, and machine learning applications in physical layer
of wireless communications.
---|---
|  Chuangui Rao received B. S. degree in Electrical Engineering and
Information from Southwest Petroleum University, China, in 2017. He is
currently pursuing the M. S. degree with the School of Electrical Engineering
and Electronic Information, Xihua University, Chengdu, China, under the
supervision of Prof. Qing. His research interests include frame
synchronization and channel estimation, and deep learning applications 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.
---|---


# Localized-Diffusive and Ballistic-Diffusive Transitions in Kicked
Incommensurate lattice

Hiroaki S. Yamada  Yamada Physics Research Laboratory, Aoyama 5-7-14-205,
Niigata 950-2002, Japan  Kensuke S. Ikeda  College of Science and Engineering,
Ritsumeikan University, Noji-higashi 1-1-1, Kusatsu 525-8577, Japan

###### Abstract

By using the kicked Harper model, the effect of dynamical perturbations to the
localized and ballistic phases in aperiodic lattice systems is investigated.
The transition from the localized phase to diffusive phase via a critical sub-
diffusion $t^{\alpha}$($t$:time) with $0<\alpha<1$ is observed. In addition,
we first confirmed the existence of the transition from the ballistic phase to
the diffusive phase via a critical super-diffusion with $1<\alpha<2$.

###### pacs:

05.45.Mt,71.23.An,72.20.Ee

Introduction.- In 1D tight-binding model with any disorder the dynamical
quantum states are always localized ishii73  But if it is coupled with noisy
source such as irregular thermal lattice vibrations, the localization is
immediately destroyed and is taken the place by normal diffusion haken72 ;
haken73 . Even very simple time-dependent harmonic perturbations consisting
only a few number of periods may convert the localized wave packet into
irreversible diffusion in Anderson model and kicked Anderson model (Anderson
map) yamada18 ; yamada20 ; yamada22 . The characteristic feature of such
coherent perturbations is that it realizes irreversible diffusion from the
localized state via a critical phase transition we call localization-
delocalization transition (LDT).

1D aperiodic lattice systems have the both features of disordered lattice and
periodoic lattice. Indeed it has ballistic states and critical states in
addition to the localized states. Our question arising here is what would be
the effect of the coherent perturbations on the dynamical states of 1D
aperiodic lattice systems, i.e.., the localized states, the critical states
and the ballistic states

To answer the above question, we investigate the effects of the coherent
perturbation by using kicked Harper model (KHM) that facilitate an efficient
study of the dynamics in aperiodic lattice. In fact, the KHM shows the above
three states intrinsic localized, critical, and ballistic states. The first
question is whether the LDT observed for the coherently perturbed random
system also exists in the aperiodic lattice system. If this is the case,
considering the remarkable duality property of the KHM, a transition from the
ballistic to the diffusive state is also expected. The purpose of the present
paper is to confirm above conjectures.

As stated above, there are three types of main dynamical states of the quantum
wave packet, localized, normal diffusion, and ballistic spread, in the KHM
depending upon the potential strength. In that respect, it is the same as the
Harper model without the kicks. The phase diagram of the localized/delocalized
state in KHM has a nested structure and is quite complicated artuso94 ;
prosen01 ; kolovsky03 ; levi04 . As for the LDT, its presence was reported by
some authors for the KHM driven by the doublly periodic kick or the intensity
modulated periodic kicks. It is also feasible to experimentally observe the
dynamical LDT through the diffusion of wave packets in the pulsed 1D
incommensurate optical lattice such as the KHM sarkar17 .

The purpose of this paper is to show realization of the normal diffusion by
simple harmonic perturbations via the above two transitions paths, where the
harmonic perturbation is polychromatic and is composed of $M$ frequencies
mode. We first show the localized state of KHM responds to such perturbation
in quite similar ways to that of kicked disordered lattice systems such as the
Anderson map yamada18 ; yamada20  and quantum standard map casati89 ;
delande08 ; delande13 ; yamada20 : for monochromatic perturbation $M=1$ the
localizaion still remains and it is $M\geq 2$ that the KHM localized state
undergoes the LDT with increase in the perturbation strength $\epsilon$. We
next show the response of the extended (ballistic) state of KHM to the
harmonic perturbation. We report the wave packet dynamics exhibiting the
ballistic spreading changes into the normal diffusion when the perturbation
strength is increased. The transition occurs through a novel anomalous
diffusion as the critical state, which is characterized by a “super” diffusion
exponent $1<\alpha_{c}<2$. Hereinafter, this transition is referred to as a
ballistic-diffusive transition (BDT) of the wave packet spreading.

Model.- We deal with the dynamically perturbed kicked Harper model (KHM),

$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle
J\sum\limits_{n}^{N}(\hat{b}_{n}^{\dagger}\hat{b}_{n+1}+H.C.)$ (1)
$\displaystyle+$ $\displaystyle
2V\left[1+f_{\epsilon}(t)\right]\delta_{1}(t)\sum\limits_{n}^{N}\cos(2\pi
Qn)\hat{b}_{n}^{\dagger}\hat{b}_{n},$

where $\hat{b}_{n}$($\hat{b}_{n}^{\dagger}$) is the creation(annihilation)
operator of the n-th site, and $N$ is the lattice size.
$\delta_{1}(t)=\sum_{m\in{\mathbb{Z}}}\delta(t-m)$. The
$Q(=\frac{\sqrt{5}-1}{2})$ is an irrational number. $V$ and $J(=-1)$ denote
the potential strength and the hopping energy between adjacent sites,
respectively.

The harmonic perturbation $f_{\epsilon}(t)$ is the sum of harmonic functions

$\displaystyle f_{\epsilon}(t)=\epsilon
f(t)=\frac{\epsilon}{\sqrt{M}}\sum_{i}^{M}\cos(\omega_{i}t),$ (2)

where $M$ and $\epsilon$ are the number of frequency components and the
relative strength of the perturbation, respectively. The frequencies
$\\{\omega_{i}\\}(i=1,...,M)$ are taken as mutually incommensurate numbers of
order $O(1)$. Note that the long-time average of the total power of the
perturbation is normalized to $\overline{f_{\epsilon}(t)^{2}}=\epsilon^{2}/2$.

In the spatial continuous limit, $H(t)$ is given by

$\displaystyle H_{c}(t)=2J\cos(\hat{p}/\hbar)+2V\cos(2\pi
Q\hat{q})[1+f_{\epsilon}(t)]\delta_{1}(t),$ (3)

where $\hat{p}$ and $\hat{q}$ denote the momentum and position operators,
respectively. The unperturbed KHM ($\epsilon=0$) is known to take a localized
state ($|V|>>|J|$), critical state ($V=J$), and extended ($|V|<<|J|$) state
depending on $V$.

The initial wave packet $<n|\Psi(t=0)>=\delta_{n,N/2}$ is localized at the
site $N/2$, and we calculate the time evolution of the wavefunction
$|\Psi(t)>$ using Schr’́odinger equation. We monitor the spread of the wave
function in the site space by the mean square displacement (MSD):
$m_{2}(t)=\sum_{n}(n-N/2)^{2}\left<|\phi(n,t)|^{2}\right>$, where
$\phi(n,t)=<n|\Psi(t)>$ is the site representation of the wave function. The
number of steps is $10^{5}\sim 10^{6}$. We mainly use the system size
$N=2^{16}-2^{17}$, and $\hbar=1/8$.

At the critical state of the transition, an anomalous diffusion $m_{2}(t)\sim
t^{\alpha}$ characterized by diffusion index $\alpha$ is expected. To observe
such a behavior directly we introduce time-dependent the instantaneous
diffusion index $\alpha_{ins}(t)$ defined by
$\alpha_{ins}(t):=\frac{d\log\overline{m_{2}(t)}}{d\log t}$, where the locally
time-averaged MSD $\overline{m_{2}(t)}$ is used in order to reduce fluctuation
in a shorter time-scale.

Localization-delocalization transition (LDT).- We investigate the
delocalization in the KHM with $V=5$. In the case of $M=1$ the localization
maintains for the harmonic perturbaion, and for $M\geq 2$ the delocalization
is observed on a finite time scale, which agrees with the case of kicked
Anderson model yamada20 . Here we focus on the case $M=3$. In Fig.1(a), the
MSD are shown for $M=3$. They indicate localization when $\epsilon$ is small,
but the LDT occurs at a certain critical value $\epsilon_{c}$, and $\epsilon$
exceeds $\epsilon_{c}$, the plot of MSD warping upward (upward deviation) can
be seen in the double-logarithmic plots. For $\epsilon>\epsilon_{c}$ the
normal diffusive behavior $m_{2}\sim t^{1}$ appears as $t\to\infty$. Around
$\epsilon=\epsilon_{c}$, the time-dependence of MSD can be approximately
described by the sub-diffusive spreading. As a result, the time-dependent MSD
displays opposite behaviors for $\epsilon<\epsilon_{c}$ and for
$\epsilon>\epsilon_{c}$, which can directly be confirmed by $\alpha_{ins}(t)$
vs $t$ plot. As shown in Fig.1(b), in the case of $V>>1$, we can see that
there exist an $\epsilon_{c}$ above which the plot of $\alpha_{ins}(t)$
increases and below which it decreases. The limiting tendency of
$\alpha_{ins}(t)$ for $t\to\infty$ seems to be $\alpha_{ins}\to 1$ or
$\alpha_{ins}(t)\to 0$ for $\epsilon>\epsilon_{c}$ or for
$\epsilon>\epsilon_{c}$, respectively. At $\epsilon=\epsilon_{c}$,
$\alpha_{ins}(t)$ fluctuates around $\alpha_{ins}(t)\sim 1/2$, which indicate
the anomalous diffusion of $m_{2}\propto t^{\alpha_{c}}$ with $\alpha_{c}\sim
0.5$.

As shown in Fig.1(c), the scaled MSD $\Lambda(t)=m_{2}(t)/t^{1/2}$ for various
$\epsilon$ has trumpet-shaped pattern that suggests the existence of the LDT
yamada20 .

Figure 1: (Color online) (a)The double-logarithmic plots of $m_{2}(t)$ as a
function of time for different values of the perturbation strength $\epsilon$
increasing from $\epsilon=0.0001$(bottom) to $\epsilon=0.002$(top) in the
polychromatically perturbed KHM of $V=5$ with $M=3$.. $\hbar=1/8$. (b)The
instantaneous diffusion index $\alpha_{ins}(t)$ for some $\epsilon$. The
broken line indicates the critical sub-diffusion line
$\alpha_{ins}(t)=\alpha_{c}=0.5$ predicted by the scaling theory. (c) The
scaled MSD $\Lambda(\epsilon,t)=m_{2}(t)/t^{1/2}$ as functions of time for
increasing perturbation strengths. The results for $\epsilon_{c}\simeq 0.0009$
are shown in thick blue lines.

As shown in Fig.2, for $M\geq 2$ the $M-$dependence of the critical strength
$\epsilon_{c}$ indicates the inverse power-law

$\displaystyle\epsilon_{c}\propto\frac{1}{V}\frac{1}{(M-1)}.$ (4)

The same $M-$dependence even for $V=10$ can be obtained, as seen in Fig.2.
This difference in $\epsilon_{c}$ due to $V$ can be interpreted by the
Maryland transform in appendix. According to the Maryland transform, the
effect of $V$ in the diagonal term is saturated in the region $V>V^{*}(\equiv
0.38)$ for $\hbar=1/8$, and the nature of the off-diagonal term depends on
$\epsilon V$, so the critical value $\epsilon_{c}$ for $V=10$ can be
interpreted as a half of the transition point for $V=5$.

We plotted in Fig.2 dependence of the substantial critical perturbation
strength $\epsilon_{c}V$ upon the color number, which leads to

$\displaystyle\epsilon_{c}V\propto\frac{1}{(M-1)}.$ (5)

An interesting fact is that the result means that $\epsilon_{c}V$(more
precisely $\epsilon_{c}V/\hbar$) does no longer dependent upon $V$ for $|V|\gg
1$. It implies the effect of the quasi-periodic perturbation on the
localization effect saturates as $|V|$ increases. This can immediately be
understood by the Maryland transformed scheme of the quasi-energy eigenvalue
equation for the time-periodic Hamiltonian (1), which is shown in Appendix. It
is interesting that the LDT occurs at a very small relative amplitude of
perturbation, ie, $\epsilon_{c}\sim 10^{-3}$, in contrast to the BDT to be
given later.

Figure 2: (Color online) The critical perturbation strength $V\epsilon_{c}$ as
a function of $(M-1)$ for $V=5$ and $V=10$ in the KHM. The black solid line
with slope $-1$ is shown as a reference.

Ballistic-Diffusive transition (BDT).- Next, we study the effect of the
harmonic perturbation on the extended states in the KHM which occurs in the
case $V\ll 1$. In such a regime the initial wave packet spreads ballistically
as $m_{2}(t)\sim t^{2}$ without the harmonic perturbation. In view of such
dynamical behavior, we henceforth refer to the extended state as a ballistic
state. In the case of $M=1$ the ballistic spreading does not change, and we
investigate the case of $M\geq 2$ in accordance with the case of LDT. We take
the case of $V=0.2$ and $M=3$ as the typical case in accordance with the case
where LDT was observed in Fig.1. The time-dependence of MSD for various values
of $\epsilon$ is displayed in Fig.3(a). When $\epsilon$ is small, the growth
of $m_{2}(t)$ slightly deviates from ballistic spreading in the initial time
regime, but it reaches to ballistic increase as $t\to\infty$. However as
$\epsilon$ is taken large enough, it gradually approaches to normal diffusion
obeying $m_{2}(t)\sim t^{1}$ and finally it increases as $m_{2}(t)\sim t$ even
from the initial stage.

Figure 3: (Color online) (a)The double-logarithmic plots of $m_{2}(t)$ as a
function of time for different values of the perturbation strength $\epsilon$
increasing from $\epsilon=0.02$(top) to $\epsilon=0.8$(bottom) in the
polychromatically perturbed KHM of $V=0.2$ with $M=3$. $\hbar=1/8$. (b)The
instantaneous diffusion index $\alpha_{ins}(t)$. The broken line indicates the
critical super-diffusion line $\alpha_{ins}(t)=\alpha_{c}=1.73$. (c) The
scaled MSD $\Lambda(t)=m_{2}(t)/t^{1.73}$ as functions of time for increasing
strengths $\epsilon$. The results for $\epsilon_{b}\simeq 0.10$ are shown in
thick blue lines.

The above observation suggests a transition from the ballistic spreading to a
normal diffusion. We confirm this directly by the $\alpha_{ins}(t)$ vs
$t$-plot shown in Fig.3(b). In all cases $\alpha_{ins}(t)$ increases in the
initial stage. For $\epsilon$ large enough the increase continues to
$t\to\infty$, whereas $\alpha_{ins}(t)$ decreases as $t\to\infty$ for
$\epsilon$ small enough . There exist a certain $\epsilon=\epsilon_{b}$ at
which $\alpha_{ins}(t)$ approaches to a constant value $\alpha_{b}\sim 1.73$,
indicating the presence of the asymptotic super-diffusion $m_{2}\propto
t^{\alpha_{c}}$ as the critical state.

Figure 4 is an estimate of the diffusion coefficient $D$ in the normal
diffusion region ($\epsilon>\epsilon_{b}$). The time-dependence of MSD shows
the normal diffusion for $\epsilon>\epsilon_{b}$ and the diffusion coefficient
decreases by increasing $\epsilon$. On the other hand, as seen in the previous
part, for $\epsilon>\epsilon_{c}$ $m_{2}(t)$ gradually approaches to the
normal diffusion for $t\to\infty$. Here, in the KHM the $\epsilon-$dependence
of the diffusion coefficient in the normal diffusion region in a wide region
of $\epsilon$($>\max\\{\epsilon_{c},\epsilon_{b}\\}$) is observed in Fig.4.

Figure 4: (Color online) Diffusion coefficient $D$ as a function of $\epsilon$
of the localized ($V=5$) and extended ($V=0.2$) states in the perturbed KHM.
$\hbar=1/8$.

In the localized case ($V=5$), the diffusion coefficient increases with an
increase of $\epsilon$. As $\epsilon$ is increased far beyond $\epsilon_{c}$
and the decrease saturates at a certain level of the order of $O(1)$. On the
other hand, in the extended case ($V=0.2$), the diffusion coefficient
decreases monotonically as $\epsilon$ increases and for $\epsilon$ far beyond
$\epsilon_{b}$, $D$ falls down to the same level as the localized cases for
$\epsilon>>1$. This fact implies that for any $V$ there always exist a
critical point $\epsilon_{c}$ beyond which an irreversible diffusion switches
on, whose diffusion coefficient converges to the same level.

Summary and discussion.- We investigated the dynamical property of the
initially localized wave packet in coherently perturbed KHM. In the localized
cases ($V>>1$), the LDT appeared with increasing the perturbation strength
$\epsilon$. The critical value $M_{c}$ which appear the LDT is $M_{c}=2$. It
also describes the localized and delocalized transitions in the
multidimensional Anderson model that corresponds to the 1D Anderson map and
kicked rotors with the quasi-periodic perturbation. If $M+1$ can be identified
with the spatial dimension $d$, the existence of the LDT is a qualitatively
consistent result with those of the LDT in $d-$dimensional Anderson model
anderson58 ; abrahams79 ; lifshiz88 ; abrahams10 . It was also shown that the
ballistic dynamics without the perturbation makes transition to the normal
diffusion with increase in the perturbation strength. It will be interesting
to see if similar results presented in this paper can be obtained for the
original Harper model with localized, critical and delocalized states as well
as the KHM,

As a result, it was shown that whether the unperturbed state is localized or
ballistic, the increase in the strength of the coherent harmonic perturbation
causes a transition to the normal diffusion. The asymptotic diffusive behavior
is expected as a generic feature of decoherence, when noise is not introduced
into a system. The observed facts presented above implies that in order to
induce quantum diffusion, which is apparently an irreversible quantum
dynamics, we need neither spatial nor temporal stochasticity and only spatial
and temporal aperiodicity is sufficient. This work may give some hints leading
to a deeper understanding of dynamical localization and quantum diffusion in
quasi-periodic systems. It also provides insight into the control of localized
and delocalized states by coherent perturbations in the Floquet engineering.

Acknowledgments: This work is partly supported by Japanese people’s tax via
JPSJ KAKENHI 15H03701, and the authors would like to acknowledge them. They
are also very grateful to Dr. T.Tsuji and Koike memorial house for using the
facilities during this study. The author (H.Y.) would like to acknowledge the
hospitality of the Physics Division of the Nippon Dental University at
Niigata, where part of this work was completed.

Appendix: Maryland transform.- We can regard the time-dependent harmonic
perturbation $f_{\epsilon}(t)$ as the dynamical degrees of freedom. To show
this we introduce the classically canonical action-angle operators
$(\hat{J}_{j}=-i\hbar\frac{\partial_{j}}{\partial_{j}\phi_{j}},\phi_{j})$
representing the harmonic perturbation as the linear modes, and we call them
the color modes. Each quantum oscillator has the action eigenstates $|n_{j}>$
with the action eigenvalue $J_{j}=n_{j}\hbar~{}(n_{j}:$integer) and the energy
$n_{j}\hbar\omega_{j}$, where
$\hat{J}_{j}|m_{j}\rangle=m_{j}\hbar|m_{j}\rangle$($m_{j}\in{\mathbb{Z}}$).
Thus the system (1) is regarded as a quantum system of $(M+1)$-degrees of
freedom spanned by the quantum states $|n>\prod_{j=1}^{M}|n_{j}>$. Then the
Hamiltonian $\tilde{H}_{kick}$ that include the color modes becomes

$\displaystyle\tilde{H}_{kick}(\hat{p},\hat{q},\\{\hat{J}_{j}\\},\\{\hat{\phi}_{j}\\})=2\cos(\hat{p}/\hbar)+$
$\displaystyle 2V\cos(2\pi
Q\hat{q})\left[1+\frac{\epsilon}{\sqrt{M}}\sum_{j}^{M}\cos\phi_{j}\right]\delta_{1}(t)+\sum_{j=1}^{M}\omega_{j}\hat{J}_{j},$
(6)

where $\gamma$ and $|u\rangle$ are the quasi-eigenvalue and quasi-eigenstate.
Here, if the eigenstate representation of $\hat{J}_{j}$ is used, we can obtain
the following $(M+1)-$dimensional tight-binding expression by the Maryland
transform fishman82 ; yamada20 :

$\displaystyle D(n,\\{m_{j}\\})u(n,\\{m_{j}\\})+$
$\displaystyle\sum_{n^{\prime},\\{m_{j}^{{}^{\prime}}\\}}\langle
n,\\{m_{j}\\}|\hat{t}_{KHM}|n^{\prime},\\{m_{j}^{{}^{\prime}}\\}\rangle
u(n^{\prime},\\{m_{j}^{{}^{\prime}}\\})=0,$ (7)

where $\\{m_{j}\\}=(m_{1},....,m_{M})$. Here the diagonal term is

$\displaystyle D(n,\\{m_{j}\\})=\tan\left[\frac{2V\cos(2\pi
Qn)+\hbar\sum_{j}^{M}m_{j}\omega_{j}}{2\hbar}-\frac{\gamma}{2}\right],$ (8)

and the $\hat{t}_{KHM}$ of the off-diagonal term is

$\displaystyle\hat{t}_{KHM}=i\frac{e^{-i\frac{\epsilon 2V}{\sqrt{M}}\cos(2\pi
Q\hat{q})(\sum_{j}^{M}\cos\phi_{j})/\hbar}-e^{i2\cos(\hat{p}/\hbar)/\hbar}}{e^{-i\frac{\epsilon
2V}{\sqrt{M}}\cos(2\pi
Q\hat{q})(\sum_{j}^{M}\cos\phi_{j})/\hbar}+e^{i2\cos(\hat{p}/\hbar)/\hbar}}.$
(9)

The effect of $V$ in the on-site-potential $D(n,\\{m_{j}\\})$ is saturated for
$|V/\hbar|>>\pi$, since the aperiodicity is expressed through the tangent
function. The off-diagonal component controlling inter-site-hopping is
governed by a single parameter $\epsilon V/\hbar$. This is consistent with the
fact that when $|V|$ is sufficiently large, it is the combined parameter
$\epsilon V$ (more precisely $\epsilon V/\hbar$) that governs the feature of
the transition.

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