部分傅里叶变换在信号处理中的研究发展中英翻译

毕业设计(论文)外文资料翻译

系别：电子信息系

专业：通信工程

班级：B090310

姓名：孙春甫

学号：B09031015

外文出处：知网

附件： 1. 原文； 2. 译文

2013年05月

Research Progress of the Fractional Fourier Transform

in Signal Processing

ABSTRACT

The fractional Fourier transform is a generalization of the classical Fourier transform, which is introduced from the mathematic aspect by Namias at first and has many applications in optics quickly. Whereas its potential appears to have remained largely unknown to the signal processing community until 1990s. The fractional Fourier transform can be viewed as the chirp-basis expansion directly from its definition, but essentially it can be interpreted as a rotation in the time-frequency plane, i.e. the unified time-frequency transform. With the order from 0 increasing to 1, the fractional Fourier transform can show the characteristics of the signal changing from the time domain to the frequency domain. In this research paper, the fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view. Our aim is to provide a course from the definition to the applications of the fractional Fourier transform, especially as a reference and an introduction for researchers and interested readers.

While solving a heat conduction problem in 1807, a French scientist Jean Baptiste Joseph Fourier, suggested the usage of the Fourier theorem. Thereafter, the Fourier transform (FT) has been applied widely in many scientific disciplines, and has played important role in almost all the science and technology domains. However, with the extension of research objects and scope, the FT has been discovered to have shortcomings. Since the FT is a kind of holistic transform, i.e., through which the whole spectrum is obtained, it cannot obtain the local time-frequency character that is essential and pivotal for processing nonstationary signals. So a series of novel signal analysis theories have been put forward to process nonstationary signals, such as: the fractional Fourier transform, the short-time Fourier transform, Wigner-Ville distribution, Gabor transform, wavelet transform, cyclic statistics, AM/FM signal analysis and so on. Hereinto the fractional Fourier transform (FRFT), as a generalization of the classical FT, has caught more and more attention for its inherent

peculiarities. In the last decade, research into the FRFT theory and application was fruitful, resulting in an upsurge in the study of the FRFT.

In 1980, Namias introduced the FRFT as a way to solve certain classes of ordinary and partial differential equations arising in quantum mechanics from classical quadratic Hamiltonians. His results were later refined by McBride and Kerr. They developed an operational calculus to define the FRFT which was the base for the optical version of the FRFT. In 1993, Mendlovic and Ozaktas offered the optical realization of the FRFT to process the optical signal, which was easy to be realized with some optical instruments. So the FRFT has many applications in optics. Although the FRFT may be potentially useful, it appears to have remained largely unknown to the signal processing community for the lack of physical illumination and fast digital computation algorithm until the interpretation as a rotation in the time-frequency plane and the efficient digital computation algorithm of the FRFT emerged in 1993 and 1996 respectively. Thereafter, many relevant research papers have been published. The study of the FRFT did not start too late at home, but still stayed at the immature stage in view of the number and content of the relevant papers. In early 1996, some review papers about the FRFT appeared at home, yet the potential of the FRFT was just explored then. What is more, no review paper of the FRFT from the aspect of signal processing has been published overseas so far. So this paper tries to summarize the research progress of the FRFT in signal processing, and expatiate the theoretic system of the FRFT in the foundation, application-foundation and application fields to provide the reference to relevant researchers.

The organization of this paper is as follows: we first provide the definition of the FRFT and its meaning. The properties and the relation between the FRFT and the conventional time-frequency distribution are depicted in section 2, as well as the uncertainty principle in the FRFT domain. We consider the FRFT domain to be interpreted as the unified time-frequency transform domain. In section 3, we systematically summarize some signal analysis tools based on the FRFT. We summarize the applications of the FRFT in signal processing in section 4. Finally, this paper is concluded in section 5.

1 Definition of the FRFT

The FRFT is defined as：

()[]()()(),p p p X u F x u x t K t u dt +∞-∞==?

, (1)

where ()()()()()22cot 2csc cot 1cot ,,(),2,21j t ut u p j e n K t u t u n t u n παααααπδαπ

δαπ-+?-≠??=-=??+=±??

(2) where /2p απ= indicates the rotation angle of the transformed signal for FRFT, p is the transform order of the FRFT, and the FRFT operator is designated by p F . It is obvious that the FRFT is periodic with period 4. If and only if 41p n =+ ()2/2n αππ=+, then the FRFT is just the same as the FT. Let /

2u u π= and /2t t π= . Then eq. (1) is equivalent to

()[]()()()()()()22cot cot csc 221cot ,2,2,21u t j j jut p p j e x t e dt n X u F x u x u n x u n αααααππαπαπ

+∞--∞?-≠???==??-=+???? (3) eq. (3) shows that the computation of the FRFT corresponds to the following three steps:

a. a product by a chirp, ()()()2

cot 21cot t j g t j e x t αα=-;

b. a FT (with its argument scaled by csc α),

()()?csc p X u G u α= with ()()1

2jut G u g t e dt π+∞

--∞=?

c. another product by a chirp, ()()2

cot 2?u j p p

X u e X u α= It turns up that the FRFT of ()x t exists in the same conditions in which its FT exists; in other words, if ()X ω exits, ()p X u exits too. Using the computation steps above obtained the unified sampling theorem for the FRFT. Based on chirp-periodicity Erseghe et al.[11] generalized the character of the FT (continuous-time, periodic continuous-time, discrete-time, periodic discrete-time) to four corresponding versions of the FRFT, and deduced the unified sampling theorem for the FRFT.

The FRFT can be considered as a decomposition of the signal, for the inverse FRFT is defined as

()()()(),p p p p x t F X t X u K t u du +∞

---∞??==??? (4)

where ()x t is expressed by a class of orthonormal basis function (),p K t u - with weight factors ()p X u . The basis functions are complex exponentials with linear frequency modulation (LFM). For different values of u , they only differ by a time shift and by a phase factor that depends on u :

()()2

tan 2,sec ,0u j p p K t u e K t u αα-=- (5)

2 Properties of the fractional Fourier transform

2.1 Basic properties

The FRFT is a generalization of the FT, so most of the properties of the FT have their corresponding generalization versions of the FRFT. The basic properties of the FRFT are listed in the appendix. An important property, convolution theorem of the FRFT, has not been listed in the appendix, for it is not obtained simply. Interested readers may refer to refs. Another important property will be introduced that the FRFT can be interpreted as a rotation in the time-frequency plane with angle α. The property establishes the direct relationship between the FRFT and the time-frequency distribution, and founds the theory that the FRFT domain can be interpreted as a uniform time-frequency domain, which offers the FRFT the advantage to be used in signal processing. With the Wigner distribution as the example, let R φ denote the operator to rotate a 2-D function clockwise:

[]()(),cos sin sin cos R y t y t t φωφωφφωφ=+-+ (6)

Then the relationship is as follows:

()[](),,x u W t R W t αωω= (7)

where

()*,22jw u p p W t X t X t e d τττωτ+∞

--∞????=+- ? ?????

? ()*,22jw x W t x t x t e d τττωτ+∞

--∞

????=+- ? ?????? express the Wigner distribution of ()p X u , ()x t respectively. Such relations still remain available for the ambiguity function, the modified short-time Fourier

transform and the spectrogram. Lohmann generalized eq. (7), and obtained the relationship between the FRFT and Radon-Wigner transform:

[]()()2

x p W u X u α?= (8) where α? is the operator of the Radon Transform, expressing the integral projection of a 2-D function with angle /2p απ= to axis t. eq. (8) can also be understood as marginal integral after a rotation of the reference frame with angle α, namely:

()()2cos sin ,sin cos x p W u v u v dv X u ????+∞

-∞-+=? (9)

Since the FRFT has such relationship with conventional time-frequency distributions, we want to know whether a more general expression exists. Let

()()(),,,x x

t f t f W d d τθξψτθτθτθ=--?? (10) where (),t f ψ is the transform kernel, (),x W τθ and (),t f ξare the Wigner distribution and the Cohen class of time-frequency distribution of ()x t respectively. Only if the transform kernel (),t f ψ is rotationally symmetric around the origin, then (),p X t f ξ the time-frequency distribution of the FRFT of ()x t is a rotated

version of the time-frequency distribution of ()x t , (),p X t f ξ. Thus, the FRFT

corresponds to rotation of a relatively large class of time-frequency representations.

From the relationship between the FRFT and the time-frequency distributions mentioned above, we see that the FRFT offers an integrative description of the signal from the time domain to the frequency domain. The FRFT can provide more space for time-frequency analysis of signals.

2.2 Uncertainty principle

Since the FRFT domain is a unified time-frequency transform domain, what is the generalization of the conventional uncertain principle in the FRFT domain? Using the conventional uncertain principle and the three decomposition steps of the FRFT mentioned in section 1, we can obtain the uncertain principle between the two FRFT domains with different transform orders.

3 Fractional operator and transform

Because the FRFT is a united time-frequency analysis tool, and can be interpreted as a rotation in the time-frequency plane, we can define some useful fractional operators and transforms based on the FRFT.

3.1 Fractional operators

Convolution and correlation are the two kinds of signal processing operators in common use. The fractional convolution and fractional correlation operator are defined in the time domain and transform domain respectively adapted to signal detection and parameter estimation; adapted to filter design, beam forming and pattern recognition.

In the time-frequency analysis theory, the unitary operator and hermitian operator are two important operators. Unitarity is one of the factors needed to consider in designing a transform operator. And different transform domains usually can be related by some hermitian operators. Thus, it attracts the people’s strong interest to deduce the unitary and hermitian fractional operator. Based on the concept of time-shift operator and frequency-shift operator, which are two basic unitary operators, Akay defined the fractional-shift operator ,T φτ, namely unitary fractional operator, shown in (11).

[]()()2cos sin 2sin ,cos j j t T x t x t e πτφφπτφφττφ-+=- (11)

3.2 Fractional transform

The fractional transforms introduced in this section means some signal analysis tools based on the FRFT, which mainly contains two classes: one is some corresponding generalizations of conventional signal analysis tools based on the FT making use of the fact that the FRFT is the generalization of the FT; the other is some new time-frequency analysis tools based on the time-frequency rotation property of the FRFT. Then we make the summary of the main fractional transforms, and elaborate on their characteristics and advantages respectively.

Some corresponding generalizations. Hilbert transform is an important signal processing tool that has many applications in communication modulation, image edge detection and so on. We can obtain the fractional Hilbert transform by generalizing the transfer function of the Hilbert transform from the frequency domain into the FRFT domain:

[]()()p Hil p p p x t F X H t -??Γ=??? (12)

The essential of the fractional Hilbert transform is still to suppress the negative portion of the ‘spectrum’, similar to the conventional Hilbert transform. The

difference lies in the ‘spectrum’, which is not the FT but the FRFT of a signal. Based on this definition, obtained a discrete version of the fractional Hilbert transform using eigenvector decomposition-type discrete FRFT, and did some digital image edge detection simulations. The design and application of the fractional Hilbert transformer has been further investigated, and several design methods about the FIR, IIR Hilbert transformer are presented, as well as a secure single-sideband (SSB) communication system with the transform order of the FRFT as a secrete key for demodulation.

Sine transform, cosine transform and Hartley transform all belong to the unitary transform, and have already widely been applied in image compression and adaptive filter. Making use of the relationship between them and the FT, we can obtain the fractional sine, cosine, and Hartley transforms. Note: firstly, the fractional sine, cosine, and Hartley transform are all with a period of 2, different from the FRFT with a period of 4; secondly, the fractional sine transform has no even eigenfunctions, and the fractional cosine transform has no odd eigenfunctions. Therefore, it is better to use the fractional cosine transform to process even functions and use the fractional sine transform to process odd functions. Based on the relationship between the FRFT and Radon-Wigner transform shown in (8), it is easy to find that the invert Radon transform of the FRFT may be an available time-frequency analysis tool. According to this clue proposes a new time-frequency analysis method called the tomography time-frequency transform (TTFT), and reduces the cross-terms through the adaptive filter in the FRFT domain.

The adaptive signal expansion is a signal analysis method based on the expanding signal on a group of elementary functions that are energy-limited and fit for analyzing the time-frequency structure. This time-frequency distribution related with adaptive signal expansion is of better time-frequency resolution and free from window effect and cross-term interference.proposes a new signal expansion method based on the FRFT of Gaussian functions as the elementary functions for the reason that the Gaussian functions satisfy the boundary condition of the uncertainty principle. With the application of the FRFT, the selection of elementary function becomes more flexible through changing the transform order of the FRFT, which may result in more precise time-frequency representation of a signal.

4 Applications in signal processing

The FRFT is a generalization of the classical FT, and processes signals in the

unified time-frequency domain. Compared with the FT, the FRFT is more flexible and suitable for processing non stationary signals. What is more, the fast algorithm of the discrete FRFT has been proposed. Thus, the FRFT has found many applications in signal processing.

4.1 Signal detection and parameter estimation

Because the FRFT can be considered as a decomposition of the signal in terms of chirps, it is suitable for the processing of chirp-like signals. Based on the property of the concentration of a chirp energy resulting in a peak in a certain FRFT domain, we can carry out detection and parameter estimation of chirps accurately through searching the peak in the 2-D distribution plane vs. the FRFT domain and the transform order. Using this clue presents a new method for the detection and parameter estimation of multi component linear frequency modulation (LFM) signals. In order to increase the search efficiency and reduce the interference between these components, the Quasi-Newton method and peak mask in cascade are introduced. Error analysis and simulations show that this parameter estimation method is asymptotically unbiased and efficient.

4.2 Phase retrieval and signal reconstruction

A complex signal can be completely reconstructed (except for a constant phase shift) through phase retrieval from the magnitudes of two of its FRFT ()p X u σ+ and ()p X u σ-. The reason for the exception of a constant term is that the fractional power spectrums square of magnitude of the FRFT, of two functions with only the exception of a constant phase are the same. Currently, the iteration-type and Noniteration-type methods are the two main kinds of phase retrieval methods. The Noniteration-type method retrieves the phase through finding the instantaneous frequency in the FRFT domain based on the relationship between the FRFT and time-frequency distributions.

4.3 Applications in image processing

The application of the FRFT in image processing includes digital watermark and image encryption. After the image is processed through 2-D FRFT, the watermark is embedded in the selected transform coefficients in terms of certain rules. Compromises are needed to make to determine the detection threshold and the transform coefficients for embedding the watermark, respectively. For the former, a trade-off is needed between holding robust and avoiding image deformation; for the

latter, between watermark imperceptiveness and probability of false detection (false alarm). In brief, applying the FRFT in image encryption is to execute encryption through multiplying the 2-D FRFT of the original image by a phase key. Decryption is the inverse of encryption, namely, first multiplying by the conjugation of the phase key to erase this key and then recover the original image by the corresponding inverse 2-D FRFT. Encryption based on the FRFT takes better effect than based on the FT or cosine transform due to one extra degree of freedom.

4.4 Applications in radar, sonar, and communication

In addition to beamforming and object recognition, the FRFT has many other applications in radar, sonar, and communication.

With the development of array antenna technology, the array signal processing based on the FRFT has attracted increasing attention. The proposed approach first separates LFM signals in the FRFT domain by using the energy-concentration property of the LFM signal in a certain FRFT domain, and constructs the correlation matrix of the sensor array signals in the FRFT domain. Through estimating the signal and noise subspaces with the eigendecomposition of the correlation matrix, the MUSIC algorithm is used to estimate the DOAs of LFM signals. Simulation results show that the proposed method can give the precise DOA estimation of wide-band LFM signals, and has great performance even when SNR is very low. Whereas this method is for noncoherent LFM signals, the DOA estimation problem still needs further study to settle for coherent LFM signals.

As we all know, the resonance could be excitated when the wavelength of illumination frequency is approximately the same dimensions as the overall length of the object, and can be used to detect and identify the object accurately. Whereas the resonances have a turn-on time, which implies that they evolve only after certain time duration, most previous techniques have used late time signals only.

5 Conclusions

This paper summarizes the research progress of the FRFT in signal processing, and systematically expatiates the theoretic system of the FRFT in the foundation, application-foundation and application fields. The relationship between the FRFT domain and time domain, frequency domain shows clearly that the FRFT is actually a unified time-frequency transform, which reflects the characteristics of a signal in the

time-frequency domain. Unlike usual quadratic time-frequency distributions, it reveals the time-frequency characteristics with a single variable, and does not suffer from cross-terms. Compared with the traditional FT (in fact it is a special condition of the FRFT), the FRFT does better in nonstationary signals processing especially in the chirp-like signals processing. Moreover, one extra degree of freedom (the order p ) may sometimes help to obtain better performance than the usual time-frequency distributions or the FT. And its developed fast algorithms lead to little computation load for good performance. Judging from sections 3 and 5, there are six main applications of the FRFT in signal processing nowadays, which embody the six advantages of the FRFT:

(1) The FRFT is a unified time-frequency transform. With the order from 0 increasing to 1, the FRFT can reveal the characteristic of the signal gradually changing from the time domain to the frequency domain. As a result, the FRFT can provide more space for time-frequency analysis of signals. The direct utilizing mode of the FRFT is the generalization of the applications in the time, frequency domain to the FRFT domain looking for improvement to some extent, e.g. filtering in the FRFT domain.

(2) The FRFT can be considered as a decomposition of a signal in terms of chirps, thus it is fit for processing chirp-like signals which widely exist in radar, communication, sonar and nature.

(3) The pth FRFT can be interpreted as a rotation in the time-frequency plane with angle /2p απ= It is easy to derive the relation between the FRFT and time-frequency transforms, which can be used in instantaneous frequency estimating, phase retrieval or designing new time-frequency transform such as TTFT, signal expansion with the FRFT of Gaussian functions as the elementary functions.

(4) Compared with the FT, one extra degree of freedom exits in the FRFT, which helps to obtain better performance in some applications such as digital watermarking and image encryption.

(5) The FRFT is a linear transform without cross-term interference, and is ascendant in the multicomponent signal processing with additive noise.

(6) The fast algorithms of the FRFT are relatively developed now, which assures that the FRFT is able to be applied in the real-time digital signal processing. And other fractional transform may develop each fast algorithms based on the FRFT, e.g. fractional convolution, fractional correlation, fractional Hartley transform, and so on.

So far many research results about the FRFT have been obtained, but there still remain many theoretical problems to be settled. For example, the sampling theory in the FRFT domain depicted is only about uniform sampling, and yet ununiform sampling is sometimes inevitable in the real sampling case. For another example, the optimal order must be determined in many applications of the FRFT, but there is no effective method at present yet, and the method based on the location of minimum second-order moment of a signal’s FRFT in p axis has its limitation. Therefore, several directions need further study as follows: Improvement of the existing methods, such as determining the optimal order, better fast algorithm, analysis of the window of the STFRFT, further exploration of applications of the FRFT, and so on; combining the FRFT with multi-rate digital signal processing to constitute the system of multi-rate theory in the FRFT domain, which can reinforce the advantage of the FRFT. proposes an approach to increase the efficiency of the discrete FRFT computation based on polyphase and equivalent transform in the multi-rate theory; and generalization of the theory of the FRFT into the theory of the linear canonical transform (LCT). Like the relationship between the FRFT and the FT, the LCT is the generalization of the FRFT. The LCT has three degrees of freedom, so it is more flexible compared with the FRFT and the FT.

部分傅里叶变换在信号处理中的研究发展

摘要

部分傅里叶变换是广义的经典的傅立叶变换，这是纳米亚首先从数学的方面有很多的应用涉及光学快速发展。鉴于它的潜力很大程度上是未知，直到20世纪90年代到信号处理领域才有所发展。可以看出部分傅里叶变换线性调频脉冲是直接从它的定义的基础上展开的，但本质上，它可以被解释为在时间频率平面上的转化即统一的时频变换发展。从0提高到1的顺序，部分傅里叶变换可以显示变化的信号从时域变换到频域的特性发展。部分傅里叶变换在本文中，已经全面和系统地处理从信号处理的角度来看发展。我们的目标是提供自定义的课程部分傅立叶变换的应用，尤其是作为一个研究人员和有兴趣的读者参考和引进发展。

在1807年为了解决热传导问题是，法国科学家基恩·巴蒂斯特·约瑟夫·傅立叶，建议使用的傅里叶定理。随后，傅立叶变换(FT)被广泛应用于许多科学学科，并在几乎所有的科学和技术领域发挥了重要作用。然而，随着研究对象及范围的扩展，已经发现FT的不足。由于FT是整体的变换，即通过它的整个频谱得到一种它不能获得的当前时间是必不可少的和关键的用于处理非平稳信号的频率特性。因此，已提出了一系列新的信号分析理论来处理非平稳信号，如：部分傅立叶变换，短时傅立叶变换，魏格纳分布，Gabor变换，短波变换，循环统计，AM/ FM信号分析等。于是部分傅里叶变换(FRFT)，作为一个概括的经典FT已引起越来越多的关注，其固有的特殊性。在过去的十年，在FRFT的理论和应用领域研究成果丰硕，形成了FRFT研究热潮。

1980年，纳米亚推出在FRFT中利用常微分方程和偏微分方程的一种方法是从经典二次方程Hamilton函数在量子力学中所产生的某些类别的办法来解决。他的研究结果后来被细化而成为麦克布赖德和科尔公式。他们开发了一种运算微积FRFT的来定义，这是基于FRFT在光学的版本。在1993年，Mendlovic和Ozaktas所提供部分傅立叶变换在光学实现用光信号处理，这是很容易在一些光学仪器中实现的，所以FRFT的很多应用都是在光学中。虽然这里把部分傅立叶变换可能会潜在有用的它似乎仍然很大程度上是未知的，但是在信号处理领域缺乏物理照明和快速的数字计算算法在时频平面中的转换，直到诠释和有效的数字计算算法在FRFT基础上分别于1993年和1996年出现。此后，许多相关研究论

文都相继发表。在FRFT 的研究上并不是太晚展开的，但一些仍处于未成熟阶段的相关论文就出现了。在1996年年初，有关FRFT 的一些评论文章出现在部分傅立叶研究过程中，但FRFT 的潜力任再探索中。更重要的是，至今从信号处理方面的FRFT 综述论文在海外都没有出版。因此，本文尝试总结FRFT 在信号处理在研究进展，并阐述了部分傅立叶变换理论体系的基础上，以对应用程序的基础和应用领域的相关研究人员提供参考。

本文的结构如下：我们首先提供部分傅立叶变换的定义及其意义。第2节中在部分傅立叶变换的不确定性原理的属性和FRFT 的和传统的时频分布之间的关系描绘。我们认为被解释为统一的时频变换域部分傅立叶变换域。在第3节中我们系统地总结了一些基于FRFT 的信号分析工具。在第4节中我们总结在FRFT 在信号处理的应用。最后，第5节是文章的结论。

1 FRFT 的定义

FRFT 的定义为：

()[]()()(),p p p X u F x u x t K t u dt +∞

-∞==? (1) 其中

()()()()()22cot 2csc cot 1cot ,,(),2,21j t ut u p j e n K t u t u n t u n παααααπδαπ

δαπ-+?-≠??=-=??+=±??

(2) 其中/2p απ=表示为部分傅立叶变换的信号的旋转角度，p 为这里把部分傅立叶变换的变换顺序，这里把p F 称为部分傅立叶变换的原理指定的点。很明

显，这里部分傅立叶变换看作是是周期性的，周期为4。当且仅当41p n =+ ()2/2n αππ=+时FRFT 和傅里叶变换是一样的。让/

2u u π=和/2t t π=。然后公式(1)等同于

()[]()()()()()()22cot cot csc 221cot ,2,2,21u t j j jut p p j e x t e dt n X u F x u x u n x u n αααααππαπαπ+∞--∞?-≠???==??-=+????(3)

公式(3)表明，这里把部分傅立叶变换计算分为于以下三个步骤：

a. 一个产生的线性调频脉冲，()()()2cot 21cot t j g t j e x t αα=

- b. 一个FT(带其缩放参数的傅里叶变换)，

()()?csc p X u G u α= 和()()12jut G u g t e dt π+∞

--∞

=? c. 另一个产生的线性调频脉冲，()()2

cot 2

?u j p p X u e X u α= 事实证明在部分傅立叶变换的存在，在相同条件下其FT 存在，换句话说，如果部分傅立叶变换的存在时，普通的傅立叶变换也存在。使用上述的计算步骤可以获得统一的采样定理。基于线性调频脉冲广义“傅里叶变换”(连续时间，周期性连续时间，离散时间，周期性离散时间)的特性FRFT 的四个对应的版本，并推导出统一的采样定理在FRFT 。

部分傅立叶变换可以被认为是分解的信号，逆部分阶傅里叶变换被定义为：

()()()(),p p p p x t F X t X u K t u du +∞

---∞??==??? (4)

(4)式中()x t 的标准正交基本函数表示为(),p K t u -其中重要的因素为()p X u 。基本函数复指数函数具有线性调频(LFM)对于不同的u 值，他们相差的一个时间偏移和相位系数取决于：

()()2

tan 2,sec ,0u j p p K t u e K t u αα-=- (5)

2部分傅里叶变换的性质

2.1基本性质

部分傅立叶变换是一个广义的FT ，对FT 的性质有它们的一般化对应的FRFT 的基本性质的重要特性，没有简单地获得部分傅立叶变换的卷积定理还没有被列入文章中。有兴趣的读者可通过参考文献推出另一个重要特性，可以被解释为部分傅立叶变换的时间频率平面中的旋转角度α。建立部分傅立叶变换和时频分布之间的直接关系，并创立部分傅立叶变换域的理论，可以解释为均匀的时频区域，它提供了用于信号处理FRFT 的优势。随着维格纳分布作为例子，让R φ分别表示旋转沿顺时针方向一个二维的函数：

[]()(),cos sin sin cos R y t y t t φωφωφφωφ=+-+ (6)

然后，关系如下：

()[](),,x u W t R W t αωω= (7)

公式中：

()*,22jw u p p W t X t X t e d τττωτ+∞

--∞????=+- ? ?????

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