# Low Ambiguity Zone: Theoretical Bounds and Doppler-Resilient Sequence Design
in Integrated Sensing and Communication Systems

Zhifan Ye, Zhengchun Zhou, Pingzhi Fan, Zilong Liu, Xianfu Lei, and Xiaohu
Tang, Manuscript received August 11, 2021; revised December 8, 2021; accepted
January 14, 2022. The work of Zhifan Ye and Zhengchun Zhou was supported by
the NSFC project No.62071397 and No. 62131016, and in part by projects of
central government to guide local scientific and technological development
under Grant No.2021ZYD0001.. The work of Pingzhi Fan was supported by the NSFC
project No.62020106001 and the 111 project No.111-2-14. This article (as an
invited paper) was presented in part at the 2021 IEEE/CIC International
Conference on Communications in China (ICCC), Xiamen, China, in July 2021
[DOI: 10.1109/ICCC52777.2021.9580208]. (Corresponding author: Zhengchun
Zhou.)Z. Ye is with the School of Mathematics, Southwest Jiaotong University,
Chengdu, 611756, China. E-mail<EMAIL_ADDRESS>Zhou is with the Key Lab of
Information Coding and Wireless Comms., and also with the School of
Mathematics, Southwest Jiaotong University, Chengdu, 611756, China. E-mail:
<EMAIL_ADDRESS>Liu is with the School of Computer Science and Electronics
Engineering, University of Essex, Colchester CO4 3SQ, U.K.. E-mail:
<EMAIL_ADDRESS>Fan, X. Lei and X. Tang are with the Key Lab of
Information Coding and Wireless Comms., Southwest Jiaotong University,
Chengdu, 611756, China. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In radar sensing and communications, designing Doppler resilient sequences
(DRSs) with low ambiguity function for delay over the entire signal duration
and Doppler shift over the entire signal bandwidth is an extremely difficult
task. However, in practice, the Doppler frequency range is normally much
smaller than the bandwidth of the transmitted signal, and it is relatively
easy to attain quasi-synchronization for delays far less than the entire
signal duration. Motivated by this observation, we propose a new concept
called low ambiguity zone (LAZ) which is a small area of the corresponding
ambiguity function of interest defined by the certain Doppler frequency and
delay. Such an LAZ will reduce to a zero ambiguity zone (ZAZ) if the maximum
ambiguity values of interest are zero. In this paper, we derive a set of
theoretical bounds on periodic LAZ/ZAZ of unimodular DRSs with and without
spectral constraints, which include the existing bounds on periodic global
ambiguity function as special cases. These bounds may be used as theoretical
design guidelines to measure the optimality of sequences against Doppler
effect. We then introduce four optimal constructions of DRSs with respect to
the derived ambiguity lower bounds based on some algebraic tools such as
characters over finite field and cyclic difference sets.

###### Index Terms:

Radar Sensing and Communications, Ambiguity Function, Low Ambiguity Zone,
Theoretical Bounds, Doppler Resilience, Spectral Constraints, High Mobility
Communications, Optimal Sequences.

##  I Introduction

###  I-A Background

In recent years, integrated sensing and communication (ISAC) systems capable
of simultaneously performing sensing and communication tasks while sharing the
same hardware and bandwidth resources have attracted increasing research
attention [1]. A typical application scenario of ISAC is for connected
autonomous vehicles where multiple sensors and communication devices are
jointly deployed to measure/track/exchange certain key system information such
as speed, vibration, approaching target distance. These ISAC systems enjoy
mutually enhanced radar and communication functions compared to a single radar
or communication system. Specifically, rapid and high-resolution radar sensing
result can be attained by utilizing some important information (e.g.,
geographical locations, carrier frequency and bandwidth, moving speeds and
routes) acquired by the communication module. Also, the communication quality
can be improved, for example, by adjusting the antenna directions for more
accurate beamforming or by better compensating the Doppler estimated from the
radar sensing result.

That said, legacy sensing and wireless communication systems have been studied
as separate research entities. A sensing system often operates in an
environment which is corrupted by noise and may consist of a variety of
clutters. Thus, two primary goals of any sensing system are to attain enhanced
parameter estimation and target resolution. The parameter estimation of a
target includes information related to its size, motion, and location [2]. The
ability of one or more sensors to distinguish different targets that are very
close is characterized by target resolution. On the other hand, wireless
communication systems are required to provide certain minimum quality-of-
services (QoSs) to meet the user demands in various applications [3]. Thanks
to the advancement in digital circuit technology, most of the system functions
of integrated radar sensing and communication applications can be realized
using adaptive and configurable software systems while keeping the same RF
front-end architecture [4]. From the waveform design point of view, there are
four main types of ISAC schemes [1, 5]: 1) coexistence schemes, which utilize
independent waveforms for each functionality (sensing or communication); 2)
communications waveform-based approaches, where communication signals such as
orthogonal frequency-division multiplexing (OFDM) and orthogonal time-
frequency-space (OTFS) are used for sensing; 3) radar waveform-based schemes,
in which digital messages are embedded into radar waveforms; and 4) joint
waveform design approaches, which achieve the joint sensing-communication
system by deploying dedicated dual-function waveforms. This paper is mainly
concerned with the radar waveform-based ISAC scheme in which the waveforms
exhibiting strong resilience to Doppler are needed.

Figure 1: A diagram of an ISAC system.

By generating appropriate waveforms, one may change radar waveforms from pulse
to pulse, and employ the waveform itself as a means of embedding communication
symbols [5, 6, 7], e.g., code shift-keying (CSK) where each waveform
corresponds to a code representing a communication symbol. Implementing such a
CSK scheme requires careful waveform design to ensure that their auto- and
cross- ambiguity function (AF) should be as small as possible, so as to attain
high-resolution radar sensing performance. From the communications
perspective, one needs to deal with the notorious Doppler effect incurred by
high mobility especially in high frequency bands [8]. Thus, it is of strong
interest to study the fundamental limits and systematic constructions of
sequences which are robust to Doppler. Such sequences having low AF values are
called Doppler-resilient sequences (DRSs) [9]. Throughout this work,
sometimes, they may also be referred to as the ISAC waveforms.

###  I-B Motivation

As illustrated in Figure 1, both sensing and communication systems are
integrated in a single hardware platform, with the same waveform and the same
transmitter. Consider a joint sensing-communication platform equipped with a
number of transmit antennas arranged as a uniform linear array. The sensing
receiver employs an array of receive antennas with an arbitrary linear
configuration, while the communication receivers are assumed to be located in
the direction known to the transmitter. The information data rate is mainly
determined by the radar pulse repetition frequency (PRF), whether the system
uses a phased-array or multiple-input, multiple-output (MIMO) configuration,
and the permissible incremental changes in radar waveform structure and
bandwidth. Note that an AF represents the time response of a filter matched to
a signal when the signal is received with a delay and a Doppler shift [10]. A
good AF may be represented by a spike that peaks at the origin and extremely
small values in other delay-Doppler areas. A waveform with such AF property
gives rise to excellent parameter estimation and favourable resolution between
neighboring targets in the region. However, designing DRSs with such globally
low ambiguity function is an extremely difficult task. In many practical
applications [9, 11, 12, 13], the Doppler frequency range can be much smaller
than the bandwidth of the transmitted signal, and it may not be necessary to
consider the whole signal duration. As shown in Figure 1, the three targets
can be distinguished effectively if the AF has very low sidelobes in the
region of interest. Such a small region with low/zero ambiguity function
values is called low/zero ambiguity zone (LAZ/ZAZ) in this paper.

The concept of LAZ/ZAZ, explicitly quantifies the local region of interest,
i.e., the low/zero ambiguity zone (as a design parameter). To derive the
related theoretical bounds with respect to the quantified local zone and
provide theoretical design guidance to such DRSs, it is essential to
understand the the tradeoff between the sequence length, the set size, and the
maximum ambiguity function value. In the literature, theoretical bounds and
optimal constructions of sequences with low/zero correlation properties have
been extensively investigated [14]. By contrast, the theoretical limits on AF
are much more complicated due to the two-dimensional shifts in delay-Doppler
domain. To the best of our knowledge, only Ding _et al._ generalized the
maximum AF lower bound in [15]. Moreover, very few works are known on DRSs
meeting the derived bound in [15]. Although there are some works on sensing
and communications such as [16, 17, 18, 19] which mainly focus on the design,
analysis and optimization of practical ISAC systems, a comprehensive study on
the fundamental limits of ISAC waveforms is missing.

Besides, the design of ISAC waveforms needs to take into account of the
availability and contiguousness of the spectrum. As spectrum may be
underutilized most of the time, it is likely for secondary users to reuse
certain spectrum holes in an opportunistic manner for improved spectrum
utilization efficiency [20]. Specifically, the idle spectrum resources can be
detected by spectrum sensing and then allocated to secondary users with
minimum interference to the primary ones.

###  I-C Contributions and Organization

The main contributions of this paper are summarized as follows.

  * •

To provide more signal design freedom, i.e. more desirable sequences
(waveforms) for integrated sensing and communication systems, a new concept
called LAZ/ZAZ, corresponding to a small area of interest defined by the
maximum Doppler frequency and the maximum delay is proposed. Further,
theoretical bounds on periodic LAZ of unimodular DRSs with respect to the zone
size of LAZ/ZAZ, the sequence length, and the number of sequences are derived.
It is shown that the obtained bounds include the existing bounds on periodic
global ambiguity function as special cases. The theoretical bound on LAZ/ZAZ
shows that one can improve the Doppler resistance at the expense of
appropriate delay resistance, or vice versa.

  * •

To investigate the impact of spectral constraints, i.e., non-contiguous
spectral bands, on sequence design, theoretical bounds on periodic LAZ/ZAZ of
unimodular DRSs with spectral constraints are derived as well. It is also
shown that these bounds include the existing periodic global bounds as special
cases. The results indicate that theoretical bounds on ZAZ are consistent
under spectral constraints or non-spectral constraints.

  * •

To validate the LAZ/ZAZ concepts and the above theoretical bounds, four
systematic constructions of DRSs with and without spectral constraints, which
are optimal with respect to the derived bounds, are presented, based on some
algebraic tools such as characters over finite field and cyclic difference
sets.

The remainder of this paper is organized as follows. A new concept called
low/zero ambiguity zone (LAZ and ZAZ) is introduced, as shown in Section II.
In Section III, we introduce some famous correlation bound of traditional
sequence and some known DRSs. In Section IV, we aim to derive four types of
lower bounds on periodic LAZ/ZAZ of unimodular DRSs with and without spectral
constraints. These bounds are tight in the sense that they can be achieved
with equality by some optimal DRSs, which are presented in Section V and
Section VI. Some examples and potential applications of LAZ/ZAZ DRSs are
provided in Section VII.

##  II Ambiguity function and low/zero ambiguity zone

AF, defined as a two dimensional delay-Doppler correlation function of the
transmitting signals, plays a central role in waveform design and performance
evaluation. The mainlobe width, sidelobe level, and ambiguity peaks of an AF
have direct impact on the range resolution, sidelobe interference, and
ambiguity characteristics.

There are two types of AF: aperiodic AF and periodic AF. In the literature, an
AF delay may range the whole signal duration $T$ (or full period in the case
of periodic AF), and the AF Doppler shift may span the whole signal bandwidth
$1/T$. In general, the ideal aperiodic/periodic AF should exhibit a single
sharp peak at the origin (which is the nominal delay and Doppler for that
matched filter), and small values close to zero elsewhere (thumbtack shape).
However, as the volume underneath an AF square is a constant [10], if the AF
value is lowered in certain area of the delay-Doppler plane, it may rise
somewhere else. To cope with such scenarios and to more efficiently design the
desired signals, this paper proposes a new concept called LAZ, corresponding
to a small area of interest defined by the maximum Doppler frequency and the
maximum delay (instrumented range).

With the aid of massive MIMO, the communication rate of ISAC system can be
improved greatly [5]. Let us assume that the received signal model at receiver
$n$ due to the signal transmitted from transmitter $m$ is

$y_{n,m}(t)=\alpha_{n,m}s_{m}\left(t-\tau_{n,m}\right)e^{j2\pi
f_{n,m}t}+z_{n,m}(t),$ (1)

where $\tau_{n,m},f_{n,m}$ and $\alpha_{n,m}$ represent the time delay,
Doppler shift and reflection coefficients, respectively, corresponding to the
path between the $m$-th transmitter and the $n$-th receiver, and $z_{n,m}(t)$
denotes the noise. After passing through the matched filter bank, the core of
signal processing is the calculation of AF. For a pair of DRSs
$(\mathbf{a},\mathbf{b})$ of length $N$, the discrete aperiodic cross AF is
defined as follows:

$\widetilde{AF}_{\mathbf{a},\mathbf{b}}(\tau,\nu)=\left\\{\begin{array}[]{ll}\sum\limits_{t=0}^{N-1-\tau}a(t)b^{*}(t+\tau)e^{j2\pi\nu
t/N},\\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad 0\leq\tau\leq
N-1;\\\ \sum\limits_{t=-\tau}^{N-1}a(t)b^{*}(t+\tau)e^{j2\pi\nu t/N},\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
1-N\leq\tau<0;\vspace{0.2cm}\\\
0,\quad\quad\quad\quad\quad\quad\quad\quad\quad|\tau|\geq
N.\end{array}\right.$ (2)

Specially, the aperiodic cross AF shall become periodic cross AF when the
summation variable $t=0$ to $N-1$ (modulo $N$), that is,

$AF_{\mathbf{a},\mathbf{b}}(\tau,\nu)=\sum\limits_{t=0}^{N-1}a(t)b^{*}(t+\tau)e^{j2\pi\nu
t/N},$ (3)

where $\tau,\nu$ are called time- and Doppler- shifts, respectively,
$|\tau|,|\nu|\in\mathbb{Z}_{N}$, $j=\sqrt{-1}$ and the summation $t+\tau$ is
modulo $N$. If $\mathbf{a}=\mathbf{b}$, we call it auto-ambiguity function
denoted by $AF_{\mathbf{a}}(\tau,\nu)$. This article will focus on
investigating periodic AF with the aid of certain algebraic tools. The maximum
ambiguity magnitude of DRS family $\mathcal{S}$ is defined as
$\theta_{\max}=\max\\{\theta_{A},\theta_{C}\\},$ where the maximal auto-
ambiguity magnitude

$\theta_{A}=\max\Bigl{\\{}\left|AF_{\mathbf{a}}(\tau,\nu)\right|:\mathbf{a}\in\mathcal{S},(0,0)\neq(|\tau|,|\nu|)\in\mathbb{Z}_{N}\times\mathbb{Z}_{N}\Bigl{\\}},$
(4)

and the maximal cross-ambiguity magnitude

$\theta_{C}=\max\Bigl{\\{}\left|AF_{\mathbf{a},\mathbf{b}}(\tau,\nu)\right|:\mathbf{a}\neq\mathbf{b}\in\mathcal{S},0\leq|\tau|<N,0\leq|\nu|<N\Bigl{\\}}.$
(5)

The ambiguity magnitude of DRS family $\mathcal{S}$ of size $M$ over a region
$\Pi\subseteq(-N,N)\times(-N,N)$ can be defined as

$\begin{split}F_{\Pi}(\mathcal{S})=\max\Bigl{\\{}\left|AF_{\mathbf{a},\mathbf{b}}(\tau,\nu)\right|&:\mathbf{a},\mathbf{b}\in\mathcal{S}~{}\text{and}~{}(\tau,\nu)\in\Pi,\\\
&(0,0)\neq(\tau,\nu)\in\Pi\text{ if
}\mathbf{a}=\mathbf{b}\Bigl{\\}}.\end{split}$ (6)

Figure 2: Illustration of a ZAAZ of one sequence.

Figure 3: Illustration of a ZCAZ between two sequences.

Such a DRS set with maximum ambiguity magnitude
$\theta_{\max}=F_{\Pi}(\mathcal{S})$ over region $\Pi$ is denoted by
$(N,M,\theta_{\max},|\Pi|)$-$\mathcal{S}$, where $|\Pi|$ is the area of $\Pi$.
In particular, we sometimes drop off $\Pi$ and denote the DRS set by
$(N,M,\theta_{\max})$-$\mathcal{S}$ if we consider global ambiguity function
with $\Pi=(-N,N)\times(-N,N)$. It is noted that conventional correlation
function is a special case of the ambiguity function when $\nu=0$. In wireless
communications and radar sensing, low ambiguity is expected at any given delay
and Doppler shift in order to detect and identify targets and achieve
information transmission between devices. However, designing such codes with
low ambiguity function for delay over whole signal duration and Doppler shift
over whole signal bandwidth is an extremely difficult task. Obviously finding
DRSs with LAZ characteristics is a relatively feasible task. In fact, one can
even design DRSs having periodic zero ambiguity zone. Now we give a formal
definition of periodic low/zero ambiguity zone.

###### Definition 1

Let $\mathcal{S}$ be a set consisting of $M$ distinct sequences of period $N$,
for a small nonnegative real number $\theta$, the periodic low ambiguity zone
with maximum ambiguity magnitude $\theta$ for sequence set $\mathcal{S}$ is
defined as

$\begin{split}\Pi_{\theta}&(\mathcal{S})=\Bigl{\\{}\Pi:F_{\Pi}(\mathcal{S})\leq\theta\Bigl{\\}}.\end{split}$
(7)

In practice, the LAZ is required as large as possible. With this goal in mind,
define

$\begin{split}\Pi_{\max}=\Bigl{\\{}\Pi\in\Pi_{\theta}&(\mathcal{S}):|\Pi|=\max\\{|A|:A\in\Pi_{\theta}(\mathcal{S})\\}\Bigl{\\}},\end{split}$
(8)

such a sequence set is denoted by $(N,M,\theta,\Pi_{\max})$ DRS set.
Specially, if $\theta=0$, that corresponds to the periodic zero ambiguity
zone. Similarly, aperiodic LAZ and $(N,M,\widetilde{\theta},\Pi_{\max})$
aperiodic DRS set can be defined.

An ideal ambiguity function may be represented by a spike that peaks at the
origin and takes zero everywhere. Such an ambiguity function provides perfect
resolution between neighboring targets regardless of how close they are to
each other. An ideal ambiguity function does not exist due to the bound to be
derived in this paper. Yet, similar to the existing sequences with zero
correlation zone (ZCZ) [21], it is possible to construct a set of sequences
which possess zero auto-ambiguity zone (ZAAZ) and zero cross-ambiguity zone
(ZCAZ), as illustrated in Fig. 2 and Fig. 3.

##  III A Review of Celebrated Correlation Bounds and DRSs

A conventional correlation is a one-dimensional function of time-shifts. Let
us define the (periodic) cross-correlation function between sequences
$\mathbf{a}=[a(0),a(1),\cdots,a(N-1)]$ and
$\mathbf{b}=[b(0),b(1),\cdots,b(N-1)]$ with period $N$ as

$R_{\mathbf{a},\mathbf{b}}(\tau)=\sum\limits_{t=0}^{N-1}a(t)b^{*}(t+\tau),$
(9)

where $\mathbf{a}$ and $\mathbf{b}$ are complex unit sequences, and $t+\tau$
is calculated over modulo $N$. If $\mathbf{a}=\mathbf{b}$, the cross-
correlation function of $\mathbf{a}$ and $\mathbf{b}$ reduces to the auto-
correlation function of $\mathbf{a}$, denoted by $R_{\mathbf{a}}(\tau)$. Let
$\mathcal{S}$ be a set consisting of $M$ distinct complex unit sequences of
period $N$. Then the maximal auto-correlation magnitude of $\mathcal{S}$ is
defined as

$\lambda_{A}=\max\\{\left|R_{\mathbf{a}}(\tau)\right|:\mathbf{a}\in\mathcal{S},0<|\tau|<N\\}.$
(10)

Similarly, the maximal cross-correlation magnitude of $\mathcal{S}$ is defined
as

$\lambda_{C}=\max\\{\left|R_{\mathbf{a},\mathbf{b}}(\tau)\right|:\mathbf{a}\neq\mathbf{b}\in\mathcal{S},0\leq|\tau|<N\\}.$
(11)

Moreover, the maximal correlation magnitude of $\mathcal{S}$ is defined as

$\lambda_{\max}=\max\\{\lambda_{A},\lambda_{C}\\}.$ (12)

The set $\mathcal{S}$ is called a complex unimodular $(N,M,\lambda_{\max})$
sequence family of period $N$ with the family size $M$ and the maximum
correlation $\lambda_{\max}$. In [22], Welch developed a lower bound on
$\lambda_{\max}$ given by

$\lambda_{\max}\geq N\sqrt{\frac{M-1}{NM-1}}.$ (13)

In 1979, Sarwate [23] established a trade-off of $\lambda_{A}$ and
$\lambda_{C}$, i.e.,

$\frac{N-1}{N(M-1)}\cdot\left(\frac{\lambda_{A}^{2}}{N}\right)+\left(\frac{\lambda_{C}^{2}}{N}\right)\geq
1.$ (14)

These bounds are important guidelines for conventional sequence design in
which Doppler is not a major concern. A number of optimal conventional
sequence sets with respect to these bounds have been reported in the
literature (see [24, 25, 26, 27] and references therein). Based on the Welch
bound, Tang and Fan derived the following set size upper bound of zero
correlation zone sequences which are useful in quasi-synchronous code-division
multiple-access (QS-CDMA) systems: $MZ_{cz}\leq N,$ where

$\begin{split}Z_{cz}=\mbox{max}\Bigl{\\{}T:R_{{\bf a},{\bf
b}}(\tau)=0~{}~{}\mbox{for}~{}&0<|\tau|<T~{}\mbox{or}~{}\tau=0\\\
&\mbox{and}~{}{\bf a}\neq{\bf b},\forall~{}{\bf a},{\bf
b}\in\mathcal{S}\Bigl{\\}}.\end{split}$ (15)

So far, many families of ZCZ sequence sets have been reported in the
literature [28, 29, 30, 31].

It is noted that traditional sequence design generally assumes the
availability of a contiguous spectral band, meaning that the sequence energy
can be allocated to all the carriers of such a spectral band. Such a design
paradigm may not be able to carry on anymore due to the increasingly congested
spectrum. Recently, significant research attention has been paid on sequence
design over non-contiguous spectrum bands. Liu _et al._ developed in [20] a
series of periodic and aperiodic-correlation lower bounds for spectral
constrained sequences (SCSs) by carrying out convex optimization in the
frequency domain. Specifically, consider
$\mathbf{C}_{i}=[C_{i}(0),C_{i}(1),\ldots,C_{i}(N-1)]$ which is the frequency
domain dual of an SCS with energy $N$, i.e.,
$\left|C_{i}(f)\right|^{2}=0,~{}\text{if}~{}f\in\Omega,$ where
$\Omega\subseteq\\{0,1,\ldots,N-1\\}$ is called the spectral constraint set
which consists of all the forbidden carrier indices. For a frequency-domain
DRS dual set
$\mathcal{C}=\\{\mathbf{C}_{0},\mathbf{C}_{1},\cdots,\mathbf{C}_{M-1}\\}$, we
assume $|\Omega|=N-L$, where $L$ denotes the number of all the admissible
carriers over which active power transmission are allowed. The maximum
periodic correlation magnitude of an $(N,M,\lambda_{\max})$ SCS family
$\mathcal{S}$ satisfies the following lower bound111This is essentially (44)
of [20].

$\lambda_{\max}\geq N\sqrt{\frac{N(M-1)+N-L}{L(NM-1)}}.$ (16)

The above bound reduces to the well-known Welch bound when the number of
forbidden carriers is set to zero, i.e., $L=N$. In 2013, Ding _et al._
generalized the Welch bound for DRS sets [15]. Formally, for any
$(N,M,\theta_{\max})$ DRS set $\mathcal{S}$, one has

$\displaystyle\theta_{\max}\geq N\sqrt{\frac{NM-1}{N^{2}M-1}}.$ (17)

The study of DRSs has been attracting increasing research attention in recent
years. Ding et al. proposed a class of sequences which asymptotically meets
the Welch bound for DRSs in [15]. With the aid of additive character and
multiplicative character over finite field, Wang and Gong constructed several
families of polyphase sequences having low ambiguity amplitudes [32, 33, 34].
Schmidt provided a direct proof for the Wang-Gong construction by the Weil
bound of hybrid character sums [35]. Using the theory of finite-unit norm
tight frames, Benedetto and Donatelli computed the ambiguity amplitudes of the
Frank-Zadoff-Chu sequences in [36]. In addition, there are many researches to
shape local or global ambiguity function by optimization algorithm in the
literature ([37, 38, 39] and references therein).

Whilst each of the above DRSs is designed with the availability of a
contiguous spectral band, we are also concerned with DRSs which are subject to
certain spectral hole constraints (SHSs). In modern communication and radar
systems operating in congested white space, an SHS often occurs when certain
non-contiguous carriers are not allowed for active power allocation. A
sequence satisfying an SHS is called a spectrally-constrained sequence (SCS).
In [40], correlation lower bounds and constructions of SCSs are studied in the
context of non-contiguous orthogonal frequency division multiplexing (OFDM)
systems. These bounds are extended and generalized by Liu _et al._ to single-
and multi- channel SCSs in [20]. Moreover, two classes of optimal unimodular
SCSs whose spectral hole positions are cyclic difference sets are proposed.
Recently, a new optimal SCS set with comb-like spectral hole positions is
developed in [41]. Popovic constructed a family of SCSs with flexible
parameters based on modulatable constant-amplitude zero-autocorrelation
(CAZAC) sequences [42]. Despite these extensive research attempts, SCSs which
exhibit Doppler resilience have not been reported, to the best of our
knowledge.

##  IV Bounds on Periodic LAZ/ZAZ of Unimodular DRS sets

###  IV-A Lower Bound of DRSs with LAZ/ZAZ

Unimodular DRSs, in which each sequence is polyphase consisting of complex-
valued elements with absolute value of one, are highly desirable in many
communication systems for maximum power transmission efficiency. Before the
context of the derivations of the main theorem, we present below a lemma to
show some properties of ambiguity function for unimodular DRS family. Proofs
of all lemmas are presented in the appendix.

###### Lemma 1

For any unimodular DRS $\mathbf{u}$, we have

$\left|AF_{\mathbf{u}}(0,\nu)\right|=0,~{}\text{for}~{}\nu\neq 0.$ (18)

Now we present our main theorem and a simple derivation by forming a “fat”
matrix which consists of all the possible time- and Doppler- shifted versions
of sequences. Such a derivation sheds some light for obtaining the other
bounds in this paper.

###### Theorem 1

(Main Theorem) For any $(N,M,\theta_{\max},|\Pi|)$ unimodular DRS set
$\mathcal{S}$, where $\Pi=(-Z_{x},Z_{x})\times(-Z_{y},Z_{y})$, we have

$\theta_{\max}\geq\frac{N}{\sqrt{Z_{y}}}\sqrt{\frac{MZ_{x}Z_{y}/N-1}{MZ_{x}-1}}.$
(19)

Specially, if $\theta_{\max}=0$, it reduces to

$MZ_{x}Z_{y}\leq N.$ (20)

Therefore, the area of ZAZ $\Pi_{max}$ subjects to

$\left|\Pi_{max}\right|\leq\frac{4N}{M}.$ (21)

###### Proof:

Let $\mathcal{S}=\\{\mathbf{u}_{i}:1\leq i\leq M\\}$ with
$\mathbf{u}_{i}=[u_{i}(0),u_{i}(1),$ $\ldots,u_{i}(N-1)]$. Define the
following matrix from the obtained DRSs as
$\mathbf{U}_{(Z_{x},Z_{y})}=\left[\mathbf{U}_{1}^{(Z_{x},Z_{y})},\mathbf{U}_{2}^{(Z_{x},Z_{y})},\ldots,\mathbf{U}_{M}^{(Z_{x},Z_{y})}\right],$
where
$\mathbf{U}_{j}^{(Z_{x},Z_{y})}=\left[\mathbf{U}_{j}^{(Z_{x})}(0),\mathbf{U}_{j}^{(Z_{x})}(1),\ldots,\mathbf{U}_{j}^{(Z_{x})}(Z_{y}-1)\right],$
and

$\begin{array}[]{lc}\mathbf{U}_{j}^{(Z_{x})}(\nu_{i})=\vspace{0.5ex}\\\
\begin{bmatrix}u_{j}(0)&u_{j}(1)e^{\frac{j2\pi\nu_{i}}{N}}&\dots&u_{j}(Z_{x}-1)e^{\frac{j2\pi\nu_{i}(Z_{x}-1)}{N}}\\\
u_{j}(1)e^{\frac{j2\pi\nu_{i}}{N}}&u_{j}(2)e^{\frac{j2\pi
2\nu_{i}}{N}}&\dots&u_{j}(Z_{x})e^{\frac{j2\pi\nu_{i}Z_{x}}{N}}\\\
\vdots&\vdots&\ddots&\vdots\\\
u_{j}(N-1)e^{\frac{j2\pi\nu_{i}(N-1)}{N}}&u_{j}(0)&\dots&u_{j}(Z_{x}-2)e^{\frac{j2\pi\nu_{i}(Z_{x}-2)}{N}}\end{bmatrix}\end{array}$
(22)

Note that the following identity holds:

$\parallel\mathbf{U}_{(Z_{x},Z_{y})}^{H}\mathbf{U}_{(Z_{x},Z_{y})}\parallel_{F}^{2}=\parallel\mathbf{U}_{(Z_{x},Z_{y})}\mathbf{U}_{(Z_{x},Z_{y})}^{H}\parallel_{F}^{2},$
(23)

where $\parallel\cdot\parallel_{F}$ stands for the Frobenius norm, and
$(\cdot)^{H}$ denotes the transpose conjugate operator. The right-hand side
term of the equation (23) is equal to

$\begin{split}&N(MZ_{x}Z_{y})^{2}\\\ &+\sum_{l\neq
m=0}^{N-1}\left|\sum_{j=1}^{M}\sum_{\nu_{i}=0}^{Z_{y}-1}\sum_{\tau=0}^{Z_{x}-1}u_{j}(l+\tau)u^{*}_{j}(m+\tau)e^{\frac{j2\pi\nu_{i}(l-m)}{N}}\right|^{2}.\end{split}$
(24)

Hence,

$\parallel\mathbf{U}_{(Z_{x},Z_{y})}\mathbf{U}_{(Z_{x},Z_{y})}^{H}\parallel_{F}^{2}\geq
N(MZ_{x}Z_{y})^{2}.$ (25)

On the other hand, the left-hand side term of (23) can be expanded as follows:

$\begin{split}&\sum_{j=1}^{M}\sum_{\tau=1-Z_{x}}^{Z_{x}-1}\sum_{\nu=1-Z_{y}}^{Z_{y}-1}\big{(}Z_{x}-|\tau|\big{)}\big{(}Z_{y}-|\nu|\big{)}\left|AF_{\mathbf{u}_{j}}(\tau,\nu)\right|^{2}\\\
&+\sum_{j,k=1\atop j\neq
k}^{M}\sum_{\tau=1-Z_{x}}^{Z_{x}-1}\sum_{\nu=1-Z_{y}}^{Z_{y}-1}\big{(}Z_{x}-|\tau|\big{)}\big{(}Z_{y}-|\nu|\big{)}\left|AF_{\mathbf{u}_{j},\mathbf{u}_{k}}(\tau,\nu)\right|^{2}.\end{split}$
(26)

Based on Lemma 1, we have

$\begin{split}\sum_{j=1}^{M}\sum_{\nu=1-Z_{y}}^{Z_{y}-1}Z_{x}&\big{(}Z_{y}-|\nu|\big{)}\left|AF_{\mathbf{u}_{j}}(0,\nu)\right|^{2}\\\
&=MZ_{x}Z_{y}\left|AF_{\mathbf{u}_{j}}(0,0)\right|^{2}=MZ_{x}Z_{y}N^{2}.\end{split}$
(27)

Therefore, we obtain

$MZ_{x}Z_{y}N^{2}+MZ_{x}Z_{y}(MZ_{x}Z_{y}-Z_{y})\theta_{\max}^{2}\geq
N(MZ_{x}Z_{y})^{2}.$ (28)

The result then follows. ∎

As a special case, some known bounds can be derived from Theorem 1, we give
the following corollary to illustrate.

###### Corollary 1

With the same notations as before.

  * •

When $Z_{x}=Z_{y}=N$, a lower bound of global ambiguity is given as

$\displaystyle\theta_{\max}\geq\sqrt{N}.$ (29)

In addition, we can get the Sarwate bound of DRS set as follows

$\frac{(N-1)\theta_{A}^{2}}{(MN-1)N}+\frac{(M-1)\theta_{C}^{2}}{MN-1}\geq 1.$
(30)

  * •

When $Z_{y}=1$, (4) reduces to the Tang-Fan-Matsufuji bound for ZCZ sequences
in [28].

###### Remark 1

The global ambiguity function lower bound in (17) is also a special case of
Theorem 1, which can be derived from (23), (26), (82) without the action of
Lemma 1. However, (29) in Corollary 1 is strictly tighter than the lower bound
in (17). Such a lower bound has not been reported before, to the best of our
knowledge. In the next section of this paper, we will show that this lower
bound is tight in the sense that one can construct an infinite family of
sequences meeting the equality of (29).

###  IV-B Lower Bound for SCSs with LAZ/ZAZ

We aim to derive a lower bound of SCSs with LAZ/ZAZ in this subsection.
Firstly, we present the ambiguity function in frequency domain based on the
symmetric discrete Fourier transform, i.e.,

$\begin{split}A(f)=\frac{1}{\sqrt{N}}\sum\limits_{t=0}^{N-1}a(t)e^{-j2\pi
ft/N},\\\ a(t)=\frac{1}{\sqrt{N}}\sum\limits_{f=0}^{N-1}A(f)e^{j2\pi
ft/N},\end{split}$ (31)

where $\mathbf{A}$ denotes the frequency domain dual sequence of $\mathbf{a}$
in (3). Similarly, we define $\mathbf{B}$ for the sequence $\mathbf{b}$.
Therefore, we have

$\begin{split}AF_{\mathbf{a},\mathbf{b}}(\tau,\nu)=\sum\limits_{f=0}^{N-1}A(f-\nu)B^{*}(f)e^{-j2\pi
f\tau/N},\end{split}$ (32)

where $|\tau|,|\nu|\in\mathbb{Z}_{N}$. Let us consider the frequency domain
dual sequence $\mathbf{C}_{j}=[C_{j}(0),\ldots,C_{j}(N-1)]$ of a SCS
$\mathbf{c}_{j}$ with $L\leq N$ admissible carriers. That is,

$\left|C_{j}(f)\right|^{2}=\left\\{\begin{array}[]{ll}\frac{N}{L},&f\not\in\Omega;\vspace{1ex}\\\
0,&\text{ otherwise. }\end{array}\right.$ (33)

From now on, by substituting $\mathbf{C}_{j}$ for $\mathbf{u}_{j}$, we will
analyze ambiguity function in the frequency domain with the technique in the
proof of Theorem 1. Unlike the aforementioned cases, the derivation of lower
bound of SCSs with LAZ is more complex. This is because the right-hand side
term of (23) is hard to determine, and we only have

$\begin{split}\parallel\mathbf{U}_{(Z_{x},Z_{y})}\mathbf{U}_{(Z_{x},Z_{y})}^{H}\parallel_{F}^{2}\geq\sum_{f=0}^{N-1}\left(\sum_{j=1}^{M}Z_{x}\sum_{\nu=0}^{Z_{y}-1}|C_{j}(\nu+f)|^{2}\right)^{2}.\end{split}$
(34)

Based on the above analysis, we present a lower bound for SCSs with LAZ as
follows.

###### Theorem 2

For any $(N,M,\theta_{\max},|\Pi|)$ SCS set $\mathcal{S}$, where
$\Pi=(-Z_{x},Z_{x})\times(-Z_{y},Z_{y})$, we have

$\theta^{2}_{\max}\geq\frac{N(MZ_{x}Z_{y}-N)}{MZ_{x}Z_{y}-1}.$ (35)

Specially, if $\theta_{\max}=0$, it reduces to

$MZ_{x}Z_{y}\leq N.$ (36)

Therefore, the area of ZAZ $\Pi_{max}$ also subjects to

$\left|\Pi_{max}\right|\leq\frac{4N}{M}.$ (37)

###### Proof:

Based on Cauchy-Schwarz inequality and (34), we have

$\begin{split}\parallel\mathbf{U}_{(Z_{x},Z_{y})}\mathbf{U}_{(Z_{x},Z_{y})}^{H}&\parallel_{F}^{2}\geq\\\
&\frac{1}{N}\cdot
N\sum_{f=0}^{N-1}\left(\sum_{j=1}^{M}Z_{x}\sum_{\nu=0}^{Z_{y}-1}|C_{j}(\nu+f)|^{2}\right)^{2}\\\
&\geq\frac{1}{N}\left(\sum_{f=0}^{N-1}\sum_{j=1}^{M}Z_{x}\sum_{\nu=0}^{Z_{y}-1}|C_{j}(\nu+f)|^{2}\right)^{2}\\\
&=\frac{1}{N}\left(\sum_{j=1}^{M}Z_{x}\sum_{\nu=0}^{Z_{y}-1}\sum_{f=0}^{N-1}|C_{j}(\nu+f)|^{2}\right)^{2}\\\
&=N(MZ_{x}Z_{y})^{2}.\end{split}$ (38)

It is similar with the proof of Theorem 1, based on (23) and (26), we have

$MZ_{x}Z_{y}N^{2}+MZ_{x}Z_{y}(MZ_{x}Z_{y}-1)\theta_{\max}^{2}\geq
N(MZ_{x}Z_{y})^{2}.$ (39)

Hence,

$\theta^{2}_{max}\geq\frac{N(MZ_{x}Z_{y}-N)}{MZ_{x}Z_{y}-1}.$ (40)

∎

###### Remark 2

As far as we know, [40] is the only known work which considered the lower
bound of SCS with LCZ. Specifically, it may be seen as a special case of our
derived bound in Theorem 2. By setting $Z_{y}=1$, (34) becomes

$\begin{split}\sum_{f=0}^{N-1}\left(\sum_{j=1}^{M}Z_{x}|C_{j}(f)|^{2}\right)^{2}=\frac{N^{2}(MZ_{x})^{2}}{L}.\end{split}$
(41)

Then, by (23) and (26), we have

$MZ_{x}N^{2}+MZ_{x}(MZ_{x}-1)\lambda_{\max}^{2}\geq\frac{N^{2}(MZ_{x})^{2}}{L}.$
(42)

Hence,

$\lambda^{2}_{max}\geq\frac{N^{2}(MZ_{x}-L)}{L(MZ_{x}-1)},$ (43)

which is the lower bound in [40].

When $Z_{x}=Z_{y}=N$, equations (23)-(26) still hold and the right-hand side
term of (23) reduces to $N(MN^{2})^{2}$. In this case, the left-hand side term
of (23) can be estimated by the property of correlation for SCS family in [20]
as follows

$\begin{split}\sum_{\tau=0}^{N-1}\left|AF_{\mathbf{c}_{j},\mathbf{c}_{k}}(\tau,0)\right|^{2}=N\sum_{f=0}^{N-1}\left|C_{j}(f)\right|^{2}\left|C_{k}(f)\right|^{2}=\frac{N^{3}}{L},\end{split}$
(44)

which is independent of $j,k$. Then we have

$\begin{split}N(MN^{2})^{2}\leq&MN^{2}\left(\frac{N^{3}}{L}+N(N-1)\theta_{A}^{2}\right)\\\
&+M(M-1)N^{2}\left(\frac{N^{3}}{L}+N(N-1)\theta_{C}^{2}\right).\end{split}$
(45)

Hence,

$\theta_{\max}^{2}\geq\frac{N^{2}(L-1)}{L(N-1)}.$ (46)

Associative with the lower bound of correlation for SCS family again, we
obtain the global ambiguity lower bound of SCSs as follows.

###### Theorem 3

For any $(N,M,\theta_{\max})$ SCS set $\mathcal{S}$, we have

$\begin{split}\theta_{A}\geq\max\left\\{N\sqrt{\frac{N-L}{L(N-1)}},N\sqrt{\frac{L-1}{L(N-1)}}\right\\}\\\
\end{split}$ (47)

and

$\begin{split}\theta_{C}\geq\max\left\\{\frac{N}{\sqrt{L}},N\sqrt{\frac{L-1}{L(N-1)}}\right\\}.\end{split}$
(48)

###  IV-C Lower bound of DRSs with aperiodic LAZ

Similar to the proof of Theorem 1 for periodic LAZ, we can obtain the
following lower bound of aperiodic ambiguity function for LAZ based on
(23)-(28). The key point is to replace
$[u_{j}(0),u_{j}(1)e^{\frac{j2\pi\nu_{i}}{N}},\cdots,u_{j}(N-1)e^{\frac{j2\pi\nu_{i}(N-1)}{N}}]^{T}$
with
$[u_{j}(0),u_{j}(1)e^{\frac{j2\pi\nu_{i}}{N}},\dots,u_{j}(N-1)e^{\frac{j2\pi\nu_{i}(N-1)}{N}},\mathbf{0}_{1\times\left(Z_{x}-1\right)}]^{T}$
in the derivation.

###### Theorem 4

For any $(N,M,\widetilde{\theta}_{\max},|\Pi|)$ unimodular DRS set
$\mathcal{S}$, where $\Pi=(-Z_{x},Z_{x})\times(-Z_{y},Z_{y})$, we have

$\widetilde{\theta}^{2}_{\max}\geq
N^{2}\frac{MZ_{x}Z_{y}-N-Z_{x}+1}{\left(N+Z_{x}-1\right)\left(MZ_{x}-1\right)Z_{y}}.$
(49)

###### Remark 3

When $Z_{y}=1$, the bound of (49) is exactly the Tang-Fan-Matsufuji bound for
aperiodic correlation function derived in [43].

##  V Two Classes of Optimal Doppler resilient Sequences

###  V-A Characters and exponential sums over finite fields

In this subsection, we present a brief introduction to the characters sum and
Weil bound which are important tools for the constructions of our proposed
DRSs. Throughout this work, we assume that $p$ is a prime and $n$ is a
positive integer. Let $q=p^{n}$ and $\mathbb{F}_{q}$ denote the finite field
with $q$ elements. Let $Tr(\cdot)$ be the absolute trace function from
$\mathbb{F}_{q}$ to $\mathbb{F}_{p}$ which is defined by

$Tr(x)=x+x^{p}+\cdots+x^{p^{n-1}},x\in\mathbb{F}_{q}.$ (50)

An additive character of $\mathbb{F}_{q}$ is a nonzero function $\chi$ from
$\mathbb{F}_{q}$ to the set of complex numbers with absolute value of 1 such
that $\chi(x+y)=\chi(x)\chi(y)$ for any pair $(x,y)\in\mathbb{F}^{2}_{q}$. For
each $a\in\mathbb{F}_{q}$, the function

$\chi_{a}(x)=\omega_{p}^{Tr(ax)},~{}x\in\mathbb{F}_{q},$ (51)

defines an additive character of $\mathbb{F}_{q}$, where $\omega_{p}$ is a
primitive $p$-th complex root of unity. When $a=0$, $\chi_{0}(x)=1$ for all
$x\in\mathbb{F}_{q}$, and is called the trivial additive character of
$\mathbb{F}_{q}$. Let us recall the following:

###### Lemma 2

[44] Let $\chi$ be a nontrivial additive character of $\mathbb{F}_{q}$ and
$f(x)\in\mathbb{F}_{q}[x]$ with $\operatorname{deg}(f)=d\geqslant 1$ and
$\operatorname{gcd}(d,q)=1$. Then

$\left|\sum_{x\in F_{q}}\chi(f(x))\right|\leqslant(d-1)\sqrt{q}.$ (52)

###  V-B Optimal unimodular Doppler-resilient sequences

The concept of cubic sequence was first proposed in 1980 by Alltop to
construct sequence family with good correlation properties. Although each
cubic sequence does not have zero periodic autocorrelation sidelobes, the
cubic sequence family meets the Welch bound in terms of its maximum periodic
cross-correlation function. Formally, for a positive integer $N$, a cubic
sequence family $\mathcal{U}$ is defined as follows:

###### Construction 1

Define $\mathcal{U}=\\{\mathbf{u}_{j}:0\leq j\leq N-1\\},$ where
$\mathbf{u}_{j}=\\{u_{j}(0),u_{j}(1),\ldots,u_{j}(N-1)\\},$ and

$u_{j}(t)=\omega_{N}^{t^{3}+jt}.$ (53)

It is proved that the maximum periodic cross correlation magnitude of a cubic
sequence family nearly achieves the Welch bound for any prime $N\geq 5$ in
[25]. Next, we consider a generic cubic sequence, which is defined as

$u_{a,b,c}(t)=\omega_{N}^{at^{3}+bt^{2}+ct},$ (54)

where $a,b,c\in\mathbb{Z}_{N}$ with $a\neq 0$. We will show that each generic
cubic sequence also possesses low cross ambiguity function, if $N$ is an odd
prime number greater than 4. To proceed, let us present the following Lemma
which is useful for our subsequent proof.

###### Lemma 3

Let $V_{N}(x,y)=\sum\limits_{t=0}^{N-1}\omega_{N}^{xt^{2}+yt}$, where $x,y$
are integers, $N$ is an odd integer and $g=\gcd(x,N)$, then

$|V_{N}(x,y)|^{2}=\left\\{\begin{array}[]{ll}Ng;{}{}&g\mid y,\\\
0;{}{}&\rm{otherwise}.\end{array}\right.$ (55)

###### Theorem 5

Let $N=p$ be an odd prime. The distribution of the cross-ambiguity function
between $\mathbf{u}_{a_{1},b_{1},c_{1}}$ and $\mathbf{u}_{a_{2},b_{2},c_{2}}$
on family $\mathcal{U}$ is given as follows,

$|AF(\tau,\nu)|=\left\\{\begin{array}[]{ll}p;{}{}&{\rm if}~{}a_{1}=a_{2},p\mid
x,p\mid y,\\\ \sqrt{p};{}{}&{\rm if}~{}a_{1}=a_{2},p\nmid x,\\\ 0;{}{}&{\rm
if}~{}a_{1}=a_{2},p\mid x,p\nmid y,\\\ \leq 2\sqrt{p};{}{}&{\rm
if}~{}a_{1}\neq a_{2}.\end{array}\right.$ (56)

where

$\left\\{\begin{array}[]{ll}x=b_{1}-3a_{2}\tau-b_{2},\\\
y=c_{1}-3a_{2}\tau^{2}-2b_{2}\tau-c_{2}+\nu.\end{array}\right.$ (57)

###### Proof:

Based on the definition of ambiguity function, we have

$\left|AF(\tau,\nu)\right|=\left|\sum_{t=0}^{p-1}\omega_{p}^{(a_{1}-a_{2})t^{3}+xt^{2}+yt}\right|,$
(58)

where $x,y$ are defined in $(\ref{xy})$.

If $a_{1}=a_{2}$, this theorem holds from Lemma 3. Otherwise, let
$f(t)=(a_{1}-a_{2})t^{3}+xt^{2}+yt$ and $\chi=\chi_{1}$, based on Lemma 2, we
have

$\left|\sum_{t=0}^{p-1}\omega_{p}^{(a_{1}-a_{2})t^{3}+xt^{2}+yt}\right|\leq
2\sqrt{p},~{}\text{where}~{}a_{1}-a_{2}\neq 0.$ (59)

The result then follows. ∎

###### Theorem 6

The generic cubic sequence is optimal with respect to the ambiguity lower
bound in Corollary 1.

###### Proof:

Let $N=p$, by Theorem 5 in this paper, we know

$|AF_{\mathbf{u}_{a,b,c}}(\tau,\nu)|=\left\\{\begin{array}[]{ll}p;{}{}&\tau=\nu=0,\\\
0;{}{}&\tau=0,\nu\neq 0,\\\ \sqrt{p};{}{}&\rm{otherwise}.\end{array}\right.$
(60)

Hence, the maximum ambiguity magnitude $\theta_{\max}$ of the cubic sequence
set satisfies $\theta_{\max}=\sqrt{N}.$ Then the result follows. ∎

###### Remark 4

To the best of our knowledge, the generic cubic sequence is the first known
optimal unimodular sequence meeting the ambiguity lower bound in equation
(29). However, the lowest cross ambiguity between distinct cubic sequences is
close to $2\sqrt{p}$. Whether there exists a DRS set, with more than one
sequence, reaching the novel bound is still unknown.

###  V-C Optimal unimodular sequences with zero ambiguity zone

We consider certain quadratic phase CAZAC sequences which have found many
applications in radar sensing, communications, coding theory, and signal
processing. Specifically, for any integer $N$, a quadratic phase sequence
$\mathbf{u}:\mathbb{Z}_{N}\rightarrow\mathbb{C}$ is defined by
$u(t)=\omega_{N}^{at^{2}+bt},~{}0\leq t\leq N-1.$ When $N$ is odd,
$\gcd(a,N)=1$ leads to the general Wiener waveform, which is a famous CAZAC
sequence.

Since the quadratic phase sequence can be seen as a special case of the
general cubic sequence if the coefficient of cubic term is zero. The
distribution of $p$-periodic quadratic phase sequence is at most three-valued
based on Theorem 5. For any positive integer $N$, the auto-ambiguity and ZAZ
of $N$-periodic quadratic phase sequence are proposed subsequently.

###### Theorem 7

The auto-ambiguity function of sequence $\mathbf{u}$ with
$u(t)=\omega_{N}^{at^{2}+bt}$ is given as follows,

$|AF_{\mathbf{u}}(\tau,\nu)|=\left\\{\begin{array}[]{ll}N;{}{}&\nu\equiv
2a\tau\bmod N,\\\ 0;{}{}&\rm otherwise.\end{array}\right.$ (61)

###### Proof:

The above identity follows by

$|AF_{\mathbf{u}}(\tau,\nu)|=\sum_{t=0}^{N-1}\omega_{N}^{(\nu-2a\tau)t}.$ (62)

∎

###### Theorem 8

Let $r=\gcd{(2a,N)}$, if $r>1$, the maximum zero auto-ambiguity zone of
quadratic phase sequences is given below:

$\Pi=\left(-\frac{N}{r},\frac{N}{r}\right)\times\left(-r,r\right),$ (63)

which is optimal with respect to the ambiguity lower bound in equation $(5)$ .

###### Proof:

By Theorem 7, we observe the peak points of auto-ambiguity function near the
origin are $(\frac{N}{r},0)$ and $\left(h,r\right)$, where $\frac{2a}{r}\cdot
h\equiv 1\bmod\frac{N}{r}$. Since $r\mid y$ for any peak points $(x,y)$, there
does not exist non-zero ambiguity function value in the region $\Pi$. Note
that the area of $\Pi$ is $N$ and this completes the proof. ∎

Theorem 7 and Theorem 8 indicate that each quadratic phase sequence has a zero
auto-ambiguity zone. Next, we propose a construction of quadratic phase
sequence family with optimal zero ambiguity zone.

###### Construction 2

With the same notations as above, let $b_{i}=i\lfloor\frac{N}{M}\rfloor$,
define a sequence family $\mathcal{S}$ with integer $M$ as
$\mathcal{S}=\\{\mathbf{s}_{i}:0\leq i\leq M-1\\},$ where
${\mathbf{s}_{i}}=[s_{i}(0),s_{i}(1),\cdots,s_{i}(N-1)],$ is given by

$s_{i}(t)=\omega_{N}^{at^{2}+b_{i}t},0\leq t\leq N-1.$ (64)

Then we obtain an $(N,M,0,4\lfloor\frac{N}{M}\rfloor)$ unimodular DRS set
$\mathcal{S}$, which is optimal with respect to the ambiguity lower bound in
equation $(\ref{LAZ-bound})$ if $M\mid N$.

###### Proof:

By Theorem 7, we observe that the zero ambiguity zone of $\mathcal{S}$ is in
the area of

$\Pi=\left(-\frac{N}{r},\frac{N}{r}\right)\times\left(-r,r\right).$ (65)

Similar to Theorem 7, the peak points $(\tau,\nu)$ of cross-ambiguity between
${\mathbf{s}_{i}}$ and ${\mathbf{s}_{j}}$ satisfy $v\equiv 2a\tau-
b_{i}+b_{j}\mod N$. Hence, no peak points exist in the area of
$\left(-\frac{\min_{i\neq j}\\{b_{i}-b_{j}\\}}{r},\frac{\min_{i\neq
j}\\{b_{i}-b_{j}\\}}{r}\right)\times(-r,r)$. If $M\mid N$, the area of $\Pi$
is $\frac{4N}{M}$ and this completes the proof. ∎

##  VI Optimal SCSs with Doppler Resilience

In this section, we construct an optimal SCS with respect to the bound in
Theorem 3 using cyclic difference sets. First, we give the definition of
cyclic difference set as follows.

For any subset
$\mathcal{D}=\\{d_{0},d_{1},\ldots,d_{n-1}\\}\in\mathbb{Z}_{N}$, the
difference function of $\mathcal{D}$ is defined as
$d_{\mathcal{D}}(\varepsilon)=|(\varepsilon+\mathcal{D})\cap\mathcal{D}|,~{}~{}\varepsilon\in\mathbb{Z}_{N}.$
Then $\mathcal{D}$ is said to be an $(N,n,d)$ cyclic difference set if and
only if $d_{\mathcal{D}}(\varepsilon)$ takes on the value $d$ for $N-1$ times
when $\varepsilon$ ranges over the nonzero elements of $\mathbb{Z}_{N}$.

###### Construction 3

Let $\mathcal{D}$ be an $(N,n,1)$ cyclic difference set, a frequency domain
sequence $\mathbf{C}$ with $N-n$ spectrum holes is defined as

$C(f)=\left\\{\begin{array}[]{ll}\sqrt{\frac{N}{n}}(-1)^{f};{}{}&f\in\mathcal{D},\\\
0;{}{}&\rm{otherwise}.\end{array}\right.$ (66)

Then the time domain dual sequence $\mathbf{c}$ is given as

$c(t)=\frac{1}{\sqrt{N}}\sum_{f=0}^{N-1}C(f)e^{j2\pi ft/N}.$ (67)

###### Lemma 4

The distribution of auto-ambiguity function value for the time domain dual
sequence $\mathbf{c}$ is given as follows,

$|AF_{\mathbf{c}}(\tau,\nu)|=\left\\{\begin{array}[]{ll}N;{}{}&\tau=0,\nu=0,\\\
\frac{N\sqrt{n-1}}{n};{}{}&\tau\neq 0,\nu=0,\\\
\frac{N}{n};{}{}&\rm{otherwise}.\end{array}\right.$ (68)

###### Theorem 9

The normalized time domain dual sequence $\mathbf{c}$ in Construction 3 is an
optimal SCS with respect to the bound in Theorem 3.

###### Proof:

Based on the property of cyclic difference set, we have $n(n-1)=d(N-1)=N-1$.
Therefore, the lower bound of Theorem 3 reduces to
$\max\\{\frac{N\sqrt{n-1}}{n},\frac{N}{n}\\}$, and this completes the proof. ∎

Finally, we present a simple construction of SCS family based on orthogonal
sequence family whose maximum ambiguity value achieves the lower bound in
Theorem 2.

###### Construction 4

Let $\mathcal{O}=\\{\mathbf{O}_{i}:i=0,1,\ldots,N-1\\}$ is an orthogonal
unimodular sequence family with period $N$ in the frequency domain. Let $k$ be
an integer, define a sequence family with period $kN$ as
$\\{\mathbf{T}_{i}:i=0,1,\ldots,N-1\\},$ where
${\mathbf{T}_{i}}=[T_{i}(0),T_{i}(1),\cdots,T_{i}(kN-1)]$ is given by

$T_{i}(f)=\left\\{\begin{array}[]{ll}\sqrt{k}O_{i}(\frac{f}{k});&\rm{if}~{}f\equiv
0\bmod k,\\\ 0;&\rm{otherwise}.\end{array}\right.$ (69)

Let $\mathbf{t}_{i}$ be the time domain dual sequence of $\mathbf{T}_{i}$, and
$\mathcal{T}=\\{\mathbf{t}_{i}:i=0,1,\ldots,N-1\\}$. Then we obtain a
$(kN,N,0,4k)$ Doppler-resilient SCS set $\mathcal{T}$ with zero ambiguity
zone, which is optimal with respect to the bound in Theorem 2.

###### Proof:

Based on the definition of ambiguity function, we have

$\begin{split}|AF_{\mathbf{t}_{i},\mathbf{t}_{j}}(&\tau,\nu)|=\left\\{\begin{array}[]{ll}k\sum\limits_{f=0}^{N-1}O_{i}(f-\frac{\nu}{k})O_{j}^{*}(f)\omega_{N}^{f\tau};{}{}&k\mid\nu,\\\
0;{}{}&\rm{otherwise}.\end{array}\right.\end{split}$ (70)

Hence, $Z_{y}\geq k$. Since $\mathcal{O}$ is an orthogonal sequence family, we
have

$|AF_{\mathbf{t}_{i},\mathbf{t}_{j}}(\tau,0)|=0,~{}\text{for}~{}i\neq j,$ (71)

so $Z_{x}\geq 1$. Therefore, $NZ_{x}Z_{y}\geq kN$. By recalling the Theorem 2
completes the proof. ∎

##  VII Examples and potential applications

###  VII-A Examples of optimal DRSs with periodic ZAZ and LAZ

In this section, we present four examples to illustrate our proposed
constructions.

###### Example 1

Let $N=32$, $M=2$ and $0\leq t\leq 31$, define

$u(t)=e^{\frac{j2\pi(2t^{2})}{32}}~{}~{}\rm{and}~{}~{}v(t)=e^{\frac{j2\pi(2t^{2}+16t)}{32}}.$
(72)

The ambiguity function value of $\mathcal{S}=\\{\mathbf{u},\mathbf{v}\\}$ is
given in Fig. 4, where each point refers to an ambiguity peak point and zero
otherwise. The ZAZ of the sequence family is $(-4,4)\times(-4,4)$, which is
optimal with respect to the bound in Theorem 1.

Figure 4:  ZAZ of the sequence family $\mathcal{S}$ in Example 1.

###### Example 2

Let $p=31$ and $0\leq t\leq 30$, define

$u(t)=e^{\frac{j2\pi
t^{3}}{31}}~{}~{}\rm{and}~{}~{}v(t)=e^{\frac{j2\pi(t^{3}+15t)}{31}},$ (73)

then the magnitude of auto-ambiguity function value of sequence $\mathbf{u}$
is the same as $\mathbf{v}$, which is presented in Fig. 5.

Figure 5:  Auto-ambiguity function of sequence $\mathbf{u}$ in Example 2.
Figure 6: LAZ of sequence set $\mathcal{S}$ in Example 2.

Specifically,

$\begin{split}|AF_{\mathbf{u}}(\tau,\nu)|=\left\\{\begin{array}[]{ll}31;{}{}&\tau=\nu=0,\\\
0;{}{}&\tau=0,\nu\neq 0,\\\
\sqrt{31};{}{}&\rm{otherwise},\end{array}\right.\end{split}$ (74)

which is optimal with respect to (29) in Corollary 1. In addition, sequences
$\mathbf{u}$ and $\mathbf{v}$ form a $(31,2,\sqrt{31},(-31,31)\times(-15,15))$
DRS set $\mathcal{S}$, the low ambiguity zone is presented in Fig. 6.

###### Example 3

Let $D=\\{4,5,8,10\\}$ be a $(13,4,1)$ cyclic difference set with $N=13$, then

$C(f)=\left[0,0,0,0,\sqrt{\frac{13}{4}},-\sqrt{\frac{13}{4}},0,0,\sqrt{\frac{13}{4}},0,\sqrt{\frac{13}{4}},0,0\right].$
(75)

The magnitude of auto-ambiguity function value of corresponding time domain
sequence $\mathbf{c}$ is shown in Fig. 7. Specially,

$|AF_{\mathbf{c}}(\tau,\nu)|=\left\\{\begin{array}[]{ll}13;{}{}&\tau=0,\nu=0,\\\
13\sqrt{3}/4;{}{}&\tau\neq 0,\nu=0,\\\
13/4;{}{}&\rm{otherwise},\end{array}\right.$ (76)

which is optimal with respect to the bound in Theorem 3.

Figure 7:  Auto-ambiguity function of sequence $\mathbf{u}$ in Example 3.

###### Example 4

Let $N=8,M=2$, define

$\mathbf{T_{1}}=\sqrt{2}\cdot[1,0,1,0,1,0,1,0]$ (77)

and

$\mathbf{T_{2}}=\sqrt{2}\cdot[1,0,-1,0,1,0,-1,0].$ (78)

The magnitude of auto-ambiguity function value of corresponding time domain
sequence family $\mathcal{T}$ is shown in Fig. 8. The ZAZ is
$(-2,2)\times(-2,2)$ which is optimal with respect to the bound of Theorem 2.

Figure 8:  ZAZ of the sequence family $\mathcal{T}$ in Example 4.

###  VII-B Examples of DRSs with aperiodic LAZ

In general, it is more difficult to study the aperiodic AF than periodic one,
and is very challenging to obtain DRSs with aperiodic AF via algebraic and
combinatorial method. However, one can obtain DRSs with aperiodic LAZ based on
some optimization algorithms as reported in [38, 39, 45]. In the following, to
illustrate the existence of DRSs with aperiodic LAZ, we give an example based
on the heuristic algorithm in [46].

###### Example 5

Let $\mathbf{s}$ be a binary sequence of length 128 given by

$\begin{split}\mathbf{s}=&[-+--++-+-++++++---+--++\\\
&-+---++-+-++++++--++-+-\\\ &\---+-+++--+-----++-+-++\\\
&++--+---+-+++--+-+++++-\\\ &+--+++-+++--+-+++++--+-\\\
&+++-+++++---+],\end{split}$ (79)

where $+$ and $-$ denote $1$ and $-1$ respectively. The magnitude of auto-
ambiguity function value of $\mathbf{s}$ is shown in Fig. 9 and the aperiodic
LAZ is $(-4,4)\times(-4,4)$.

Figure 9: Aperiod LAZ of the sequence $\mathbf{s}$ in Example 5.

###  VII-C Potential applications

There are three types of sensing code signal design, corresponding to three
different ISAC systems [1, 47], i.e.

  1. 1.

Communication-centric design (communications signal-based ), such as OFDM/OTFS
signals, spread-spectrum signals, Protocol-oriented design. The performance
metric for such signal design include correlation function, PAPR, SINR, etc.

  2. 2.

Radar-centric design (radar signal-based), such as CW signals, Pulsed signals,
FMCW, SFCW, Phase-coded radar signals, PTM-GCP, NS-GCP, etc. The performance
metric for radar-centric signal design is mainly AF, other metric include,
mutual information (MI), CRLB, etc.

  3. 3.

Joint design normally involves solving a joint optimization problem to balance
radar and communication performance, relying on prior channel state knowledge.
In joint design, the objective function and the constraints can use various
metric and their combinations, for communications, and for radar sensing.

The DRSs with periodic/aperiodic LAZ/ZAZ characteristics discussed in this
paper can be applied to radar-centric ISAC systems, which can embed data bits
into radar signals by modifying the radar signal to incorporate digital
modulations, or by index modulation (IM) to send data bits via the indices of
certain radar parameters (MIMO, FAR), although the data rate in this ISAC
system is normally limited. For this radar-centric design, it is very
important for the radar sensing signal to have superior ambiguity function. In
many cases, the maximum tolerable Doppler shift is much smaller than the
sensing signal bandwidth; further, it is not necessary to consider the maximum
unambiguous distance range. Thus one may only need to consider a small area of
interests near-origin zone, i.e. the low or zero ambiguity zone (LAZ/ZAZ). In
fact, the proposed new LAZ/ZAZ concepts, the derived theoretical bounds and
the presented DRSs with periodic LAZ/ZAZ are the main contributions of this
paper. The discussed DRSs with periodic/aperiodic LAZ/ZAZ characteristics can
find potential applications in ISAC systems under the frameworks developed in
[5, 7, 12, 13].

The first potential application is to implement the dual functions of radar
and communication by embedding information in our obtained sequences. In [5]
and [7], for example, a distinctive Gold/Kasami sequence is sent to represent
a communication symbol through code keying, whereas the resultant pseudo-
random communication signals are used as radar waveform. It is pointed out in
[5] and [7] that the data rate and the sensing performance are determined by
the size and periodic AF of the employed sequence family, respectively. Under
this framework, the sequence family constructed in Section V can also be used
to implement the dual functions of radar and communication. Specially, we give
a simple comparison of the sequence family $\mathcal{U}$ generated by
Construction 1 with the Gold/Kasami sequence families. For certain sequence
length $N$, the sizes of Gold sequence family, Kasami sequence family and the
proposed family $\mathcal{U}$ are $N+1,\sqrt{N+1}$ and $N-1$,
respectively222$N=2^{2k+1}-1$ for Gold sequence family, $N=2^{2k}-1$ for
Kasami sequence family, and $N$ is a prime for $\mathcal{U}$. Hence, we choose
approximate period for comparison.. Therefore, compared to the Gold sequence
family (resp., Kasami sequence family) with almost the same length, the
proposed sequence family $\mathcal{U}$ can support almost the same (resp.,
larger) date rate when they are employed under the framework developed in [7].
Besides, the proposed sequence family $\mathcal{U}$ enjoys excellent periodic
AF, thus leading to potential improved sensing performance.

A second potential application is for enhanced preamble and sensing waveform
in future vehicular communication-radar systems, under the frameworks
developed in [12, 13]. For communications, such sequences can be used as
preambles to achieve the tasks of synchronization and channel estimation due
to their good correlation properties. On the other hand, they can also be
employed to construct radar waveforms for improving sensing, thanks to their
very low AF sidelobe in a local region. Following this framework, our proposed
sequences can also be used to perform the dual functions of radar and
communications.

##  VIII Conclusion

In this paper, in order to meet very low ambiguity function requirement within
a small area of interest defined by the certain Doppler frequency and delay,
and to design the desired optimal signals for sensing and communication
systems, a new concept called low ambiguity zone, as well as zero ambiguity
zone, was proposed. Based on the new concept, we derived four lower bounds on
periodic LAZ/ZAZ of unimodular DRSs with and without spectral constraints.
These bounds may be used as benchmarks to measure the Doppler resilience of
unimodular sequences. Also, we presented four types of optimal DRSs with
respect to a collection of the proposed bounds. It should be noted that our
ambiguity lower bounds of SCSs are derived with the assumption that all the
sequences share the identical spectral hole constraint $\Omega$. New ambiguity
lower bounds of SCSs may be possible when different spectral hole constraints
are to be considered. In addition, it is also interesting, although much more
difficult, to investigate aperiodic ambiguity lower bounds and the associated
optimal DRSs.

##  IX Appendix

###  IX-A Proof of Lemma 1

If $\mathbf{u}_{j}$ and $\mathbf{u}_{k}$ are polyphase sequences, for
arbitrary time shift $\tau$, we have the ambiguity function satisfying

$\begin{split}&\sum_{\nu=0}^{N-1}\left|AF_{\mathbf{u}_{j},\mathbf{u}_{k}}(\tau,\nu)\right|^{2}\\\
=&\sum_{\nu=0}^{N-1}\sum_{t=0}^{N-1}u_{j}(t)u^{*}_{k}(t+\tau)e^{\frac{j2\pi\nu
t}{N}}\sum_{s=0}^{N-1}u^{*}_{j}(s)u_{k}(s+\tau)e^{-\frac{j2\pi\nu s}{N}}\\\
=&\sum_{t=0}^{N-1}\sum_{s=0}^{N-1}u_{j}(t)u^{*}_{k}(t+\tau)u^{*}_{j}(s)u_{k}(s+\tau)\sum_{\nu=0}^{N-1}e^{\frac{j2\pi\nu(t-s)}{N}}\\\
=&N\sum_{t=0}^{N-1}|u_{j}(t)|^{2}|u_{k}(t+\tau)|^{2}=N^{2}.\end{split}$ (80)

Since $AF_{\mathbf{u}_{j}}(0,0)=N,$ we have

$\left|AF_{\mathbf{u}_{j}}(0,\nu)\right|=0,~{}\text{for}~{}\nu\neq 0.$ (81)

###  IX-B Proof of Corollary 1

If $Z_{x}=Z_{y}=N$, by denoting $\mathbf{U}_{(Z_{x},Z_{y})}$ as $\mathbf{U}$,
we have

$\parallel\mathbf{U}\mathbf{U}^{H}\parallel_{F}^{2}=N(MN^{2})^{2}.$ (82)

Based on the definition of ambiguity magnitude of DRS family and (18), (23),
(26), we have

$MN^{4}+MN^{2}(N^{2}-N)\theta_{A}^{2}+MN^{2}(MN^{2}-N^{2})\theta_{C}^{2}\geq
N(MN^{2})^{2}.$ (83)

Hence, $\theta_{\max}\geq\sqrt{N}.$ The rest of the proof is similar to the
above, so we omit it.

###  IX-C Proof of Lemma 3

By definition

$\begin{split}|V_{N}(x,y)|^{2}&=\sum\limits\limits_{t=0}^{N-1}\omega_{N}^{xt^{2}+yt}\sum\limits_{s=0}^{N-1}\omega_{N}^{-xs^{2}-ys}\\\
&=\sum\limits_{t=0}^{N-1}\sum\limits_{s=0}^{N-1}\omega_{N}^{(t-s)(x(t+s)+y)}\\\
&=\sum\limits_{k=0}^{N-1}\sum\limits_{s=0}^{N-1}\omega_{N}^{k(x(2s+k)+y)}\\\
&=\sum\limits_{k=0}^{N-1}\omega_{N}^{xk^{2}+yk}\sum\limits_{s=0}^{N-1}\omega_{N}^{2kxs}.\end{split}$
(84)

If $N$ is odd, then

$\sum\limits_{s=0}^{N-1}\omega_{N}^{2kxs}=\left\\{\begin{array}[]{ll}N;{}{}&N\mid
kx,\\\ 0;{}{}&\text{otherwise}.\end{array}\right.$ (85)

Hence, let $g=\gcd(x,N)$, we have

$\begin{split}|V_{N}(x,y)|^{2}&=N\sum\limits_{N\mid
kx}\omega_{N}^{xk^{2}+yk}=N\sum\limits_{e=0}^{g-1}\omega_{N}^{\frac{yeN}{g}}\\\
=N\sum\limits_{e=0}^{g-1}\omega_{g}^{ye}&=\left\\{\begin{array}[]{ll}Ng;{}{}&g\mid
y,\\\ 0;{}{}&otherwise.\end{array}\right.\end{split}$ (86)

This completes the proof.

###  IX-D Proof of Lemma 4

By the definition of the ambiguity function in equation $(\ref{bound1})$ , if
$\nu\neq 0$, we have

$|AF_{\mathbf{c}}(\tau,\nu)|=\left|\sum\limits_{f\in\mathcal{D}\cap\mathcal{D}+\nu}C^{*}(f)C(f-\nu)e^{-j2\pi
f\tau/N}\right|$ (87)

and

$|AF_{\mathbf{c}}(\tau,0)|=\frac{N}{n}\left|\sum\limits_{f\in\mathcal{D}}e^{-j2\pi
f\tau/N}\right|.$ (88)

Since $\mathcal{D}$ is an $(N,n,1)$ cyclic difference set, we have
$\left|\mathcal{D}\cap\left(\mathcal{D}+v\right)\right|=1$, for $v\neq 0$,
then $|AF_{\mathbf{c}}(\tau,\nu)|=\frac{N}{n}.$ If $v=0$, we have

$|AF_{\mathbf{c}}(\tau,0)|^{2}=\frac{N}{n}\left|\sum\limits_{f,f^{\prime}\in\mathcal{D}}e^{-j2\pi(f-f^{\prime})\tau/N}\right|.$
(89)

Thus, we have $|AF_{\mathbf{c}}(\tau,0)|=\frac{N\sqrt{n-1}}{n}$, for $\tau\neq
0$, based on the properties of the cyclic difference set. Then the result
follows.

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|  Zhifan Ye received the B.S. and M.S. degrees in mathematics from Fujian
Normal University, Fuzhou, China, in 2013 and 2016, respectively. He is
currently pursuing the Ph.D. degree with the School of Mathematics from
Southwest Jiaotong University, Chengdu, China. His research interests include
sequence design, sensing and communication.
---|---
|  Zhengchun Zhou received the B.S. and M.S. degrees in mathematics and the
Ph.D. degree in information security from Southwest Jiaotong University,
Chengdu, China, in 2001, 2004, and 2010, respectively. From 2012 to 2013, he
was a postdoctoral member in the Department of Computer Science and
Engineering, the Hong Kong University of Science and Technology. From 2013 to
2014, he was a research associate in the Department of Computer Science and
Engineering, the Hong Kong University of Science and Technology. Since 2001,
he has been in the Department of Mathematics, Southwest Jiaotong University,
where he is currently a professor. His research interests include sequence
design, Boolean function, coding theory, and compressed sensing. He is an
associated editor of Advances in Mathematics of Communications and IEICE
Transactions on Fundamentals, and was a Guest Editor for special issues of
Cryptography and Communications. Dr. Zhou was the recipient of the National
excellent Doctoral Dissertation award in 2013 (China).
---|---
|  Zilong Liu received his PhD (2014) from School of Electrical and Electronic
Engineering, Nanyang Technological University (NTU), Master Degree (2007) in
the Department of Electronic Engineering from Tsinghua University, and
Bachelor Degree (2004) in the School of Electronics and Information
Engineering from Huazhong University of Science and Technology (HUST). He is a
Lecturer at the School of Computer Science and Electronic Engineering,
University of Essex. From January 2018 to November 2019, he was a Senior
Research Fellow at the Institute for Communication Systems (ICS), Home of the
5G Innovation Centre (5GIC), University of Surrey. Prior to his career in UK,
he spent 9.5 years in the School of Electrical and Electronic Engineering,
Nanyang Technological University (NTU), Singapore, first as a Research
Associate (since July 2008) and then a Research Fellow (since November 2014).
He is generally interested in coding and signal processing for various
communication systems. Details of his research can be found at:
https://sites.google.com/site/zilongliu2357.
---|---
|  Pingzhi Fan (M’93-SM’99-F’15) received his MSc degree in computer science
from the Southwest Jiaotong University, China, in 1987, and PhD degree in
Electronic Engineering from the Hull University, UK, in 1994. He is currently
a distinguished professor and director of the institute of mobile
communications, Southwest Jiaotong University, China, and a visiting professor
of Leeds University, UK (1997-), a guest professor of Shanghai Jiaotong
University (1999-). He is a recipient of the UK ORS Award (1992), the NSFC
Outstanding Young Scientist Award (1998), IEEE VTS Jack Neubauer Memorial
Award (2018), and 2018 IEEE SPS Signal Processing Letters Best Paper Award.
His current research interests include vehicular communications, massive
multiple access and coding techniques, etc. He served as general chair or TPC
chair of a number of international conferences including VTC’ 2016 Spring,
IWSDA’2019, ITW’2018 etc. He is the founding chair of IEEE Chengdu (CD)
Section, IEEE VTS BJ Chapter and IEEE ComSoc CD Chapter. He also served as an
EXCOM member of IEEE Region 10, IET(IEE) Council and IET Asia Pacific Region.
He has published over 300 international journal papers and 8 books (incl.
edited), and is the inventor of 25 granted patents. He is an IEEE VTS
Distinguished Lecturer (2015-2019), and a fellow of IEEE, IET, CIE and CIC.
---|---
|  Xianfu Lei received his Ph.D from Southwest Jiaotong University in 2012. He
has been an Associate Professor with the School of Information Science and
Technology at Southwest Jiaotong University since 2015. From 2012 to 2014, he
worked as a research fellow in the Department of Electrical and Computer
Engineering at Utah State University. Dr Lei’s research interests are 5G/6G
networks, cooperative and energy harvesting networks and physical-layer
security. He has been serving as an Area Editor for IEEE Communications
Letters and an Associate Editor for IEEE Wireless Communications Letters and
IEEE Transactions on Communications. He served as Senior/Associate Editor for
IEEE Communications Letters from 2014-2019. He received the best paper award
in IEEE/CIC ICCC2020, the best paper award in WCSP2018, the WCSP 10-Year
Anniversary Excellent Paper Award, IEEE Communications Letters Exemplary
Editor 2019, and Natural Science Award of China Institute of Communications
(2019).
---|---
|  Xiaohu Tang (Senior Member, IEEE) received the B.S. degree in applied
mathematics from Northwestern Polytechnical University, Xi’an, China, in 1992,
the M.S. degree in applied mathematics from Sichuan University, Chengdu,
China, in 1995, and the Ph.D. degree in electronic engineering from Southwest
Jiaotong University, Chengdu, in 2001. From 2003 to 2004, he was a Research
Associate with the Department of Electrical and Electronic Engineering, The
Hong Kong University of Science and Technology. From 2007 to 2008, he was a
Visiting Professor with the University of Ulm, Germany. Since 2001, he has
been with the School of Information Science and Technology, Southwest Jiaotong
University, where he is currently a Professor. His research interests include
coding theory, network security, distributed storage, and information
processing for big data. Dr. Tang was a recipient of the National Excellent
Doctoral Dissertation Award, China, in 2003, the Humboldt Research Fellowship,
Germany, in 2007, and the Outstanding Young Scientist Award by NSFC, China, in
2013. He served as an Associate Editors for several journals, including IEEE
Transactions on Information Theory and IEICE Transactions on Fundamentals, and
served for a number of technical program committees of conferences.
---|---