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 Transformin Signal ProcessingABSTRACTThe 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 inherentpeculiarities. 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 FRFTThe 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 παααααπδαπδαπ-+⎧-≠⎪⎪=-=⎨⎪+=±⎪⎩(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, ()()()2cot 21cot t j g t j e x t αα=-;b. a FT (with its argument scaled by csc α),()()ˆcsc p X u G u α= with ()()12jut G u g t e dt π+∞--∞=⎰c. another product by a chirp, ()()2cot 2ˆu j p pX 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 :()()2tan 2,sec ,0u j p p K t u e K t u αα-=- (5)2 Properties of the fractional Fourier transform2.1 Basic propertiesThe 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 Fouriertransform and the spectrogram. Lohmann generalized eq. (7), and obtained the relationship between the FRFT and Radon-Wigner transform:[]()()2x p W u X u αℜ= (8) where αℜ 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 ϕϕϕϕ+∞-∞-+=⎰ (9)Since the FRFT has such relationship with conventional time-frequency distributions, we want to know whether a more general expression exists. Let()()(),,,x xt 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 rotatedversion of the time-frequency distribution of ()x t , (),p X t f ξ. Thus, the FRFTcorresponds 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 principleSince 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 transformBecause 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 operatorsConvolution 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 transformThe 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. Thedifference 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 processingThe FRFT is a generalization of the classical FT, and processes signals in theunified 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 estimationBecause 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 reconstructionA 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 processingThe 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 thelatter, 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 communicationIn 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 ConclusionsThis 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 thetime-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。
