线性规范变换下的Zak变换与不确定性原理研究
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"这篇研究论文探讨了与线性规范变换（Linear Canonical Transform, LCT）相关的Zak变换和不确定性原理，这些结果主要基于LCT与经典傅里叶变换之间的关系。预计这些发现可能在光学和信号处理等领域具有潜在的应用价值。"
在数学和物理学中，线性规范变换是一种广义的傅里叶变换形式，它涵盖了经典傅里叶变换、分数阶傅里叶变换以及菲涅尔变换等。由于其强大的分析能力，LCT在光学和信号处理等多个领域得到了广泛应用。
Zak变换是另一种数学工具，特别是在周期性函数的研究中非常有用。它将一个函数扩展到整个实数轴上，即使原始函数仅在一个有限区间内定义，这使得对周期性和准周期性信号的分析变得更加方便。在LCT域中，Zak变换提供了更丰富的表示和分析手段，对于理解和操作这类信号具有重要意义。
不确定性原理，又称为海森堡-保罗-韦利不等式，是量子力学中的一个基本概念，它表明一个粒子的位置和动量不能同时被精确测量。在经典数学中，这个原理可以表达为傅里叶变换下的位置空间和频域空间的不确定性。在LCT域下，不确定性原理也有所体现，即在应用LCT后，函数在时间和频率域的分布同样存在一种固有的模糊度，这对理解和优化信号处理算法有直接影响。
论文中，作者 Qingyue Zhang 提出了关于LCT域内Zak变换的新成果，并且探讨了在此域中不确定性原理的具体形式。这些研究成果不仅深化了我们对LCT和Zak变换理论的理解，也为实际应用提供了理论基础，比如在光学系统的设计或信号分析与滤波技术中，可能会利用这些新原理来改进现有方法，提高性能。
这篇研究论文为LCT和Zak变换在不确定性原理框架下的理论发展做出了贡献，其结果有可能为相关领域的工程师和科学家提供新的分析工具和设计思路。通过深入研究这些数学工具和物理原理，我们可以期待在未来的光学和信号处理技术中看到更多的创新和进步。
Zak transform and uncertainty principles
associated with the linear canonical transform
ISSN 1751-9675
Received on 21st November 2015
Revised on 2nd April 2016
Accepted on 13th April 2016
doi: 10.1049/iet-spr.2015.0514
www.ietdl.org
Qingyue Zhang
✉
College of Science, Tianjin University of Technology, Tianjin 300384, People’s Republic of China
✉ E-mail: jczhangqingyue@163.com
Abstract: Several new results about Zak transform and uncertain ty principles in the linear canonical transform (LCT)
domains are presented. The results obtained rely mainly on relationship b etween the LCT and the classical Fourier
transform. The findings will likely have potential applications in optics and signal processing.
1 Introduction
The linear canonical transform (LCT) is the genera lised form of the
classical Fourier transfo rm (FT), the fractional Fourier transform (FrFT)
and the Fresnel transform [1]. As a pow erful analysing tool, LCT has
been widely used in many fields such as optics and signal processin g
[1, 2]. In this paper, we discuss the Zak transform and the uncertainty
principle (Heisenberg–Pauli–W eyl inequality) in the LCT domains.
The classical Zak transform is a certain operation which takes as
input a function of one variable and produces as output a function
of two variables. It has been used successfully in quantum physics
field [3], in time-frequency representation of signals [4], and in
sampling theorem [5]. The results about the classical Zak
transform in the LCT domains appear to be missing in the
literature and we want to fill this gap.
The classical Poisson summation formula is a very important
concept in signal processing community. Li et al.[6] investigated
the classical Poisson summation formula in the FrFT domains.
Zhang and Hou studied the classical Poisson summation formula
associated with the LCT [7]. We derive Poisson summation formula
in the LCT domains by a property of Zak transform. Although
Zhang and Hou had given the classical Poisson summation formula
for the LCT in [7], we reduce the requirements on the signal.
In digital signal and image processing and digital communications,
a continuous signal is usually represented and processed by using its
discrete samples. Then a fundamental problem is how to represent a
continuous signal in terms of a discrete sequence. By the famous
classical Shannon sampling theorem [8, Theorem 1.2], for a
band-limited signal of finite energy, it is completely characterised
by its samples. By a property of Zak transform, we obtain a result,
which generalises classical Shannon sampling theorem.
The uncertainty principle is any of a variety of mathematical
inequalities asserting a fundamental limit to the precision with
which certain pairs of physical properties of a particle, known as
complementary variables, such as position x and momentum p,
can be known simultaneously. It was first introduced by the
German physicist Werner Heisenberg in 1927, and it states that the
more precisely the position of some particle is determined, the less
precisely its momentum can be known, and vice versa [9]. Hence,
the uncertainty principle is also known as Heisenberg’s uncertainty
principle in quantum mechanics. The formal inequality relating the
standard deviation of position σ
x
and the standard deviation of
momentum σ
p
was obtained by Kennard [10] later that year and
by Weyl [11] in 1928
s
x
s
p
≥ h
−
.
Here, h
−
is the reduced Planck constant h/2π.
Recently, researchers discussed the classical uncertainty principle
(Heisenberg–Pauli–Weyl inequality) for LCT [2, 12–22]. However,
to the best of our knowledge, there are no results published about the
Amrein–Berthier–Benedicks qualitative uncertainty principle, the
Donoho–Stark uncertainty principle, the Nazarov uncertainty
principle, the Hardy uncertainty principle and the Beurling
uncertainty principle in the LCT domains. In this paper, we study
these five uncertainty principles in the LCT domains.
Before stating our results, we give some definitions. The LCT with
parameter
A =
ab
cd
of a signal f(t)isdefined as in [2, 16, 18, 19, 22, 23]
L
A
f (u) =
R
f (t)K
A
(t, u)dt, b = 0,
d
√
e
i(cd/2)u
2
f (du), b = 0,
where a, b, c, d are real numbers satisfying ad − bc = 1, i.e. detA =1,
and the kernel of LCT is
K
A
(t, u) =
1
i2
p
b
e
(i/2) (a/b)t
2
−(2/b)tu+(d/b)u
2
,(b = 0).
Denote (U
α
f )(x)=e
iαx
2
f(x), (D
a
f )(x)=|a|
1/2
f(ax), where a ≠ 0.
Then for b ≠ 0, we have
L
A
f =
1
ib
1
|b|
−1
U
d /2b
D
1/b
FU
a/2b
f , (1)
where (Ff )(u) = (1/
2
p
√
)
R
f (x)e
−ixu
dx is the FT of f.
Let
A
1
=
a
1
b
1
c
1
d
1
, A
2
=
a
2
b
2
c
2
d
2
,
and det A
1
= det A
2
= 1.
In Section 2, we give an important property of Zak transform in the
LCT domains and the property’s applications. In Section 3, five
uncertainty principles (the Amrein–Berthier–Benedicks qualitative
uncertainty principle, the Donoho–Stark uncertainty principle, the
Nazarov uncertainty principle, the Hardy uncertainty principle and
the Beurling uncertainty principle) in the LCT domains are
IET Signal Processing
Research Article
IET Signal Process., 2016, Vol. 10, Iss. 7, pp. 791–797
791
&
The Institution of Engineering and Technology 2016
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