The fractional Fourier transform and time-frequency representations
TL;DR: The authors briefly introduce the functional Fourier transform and a number of its properties and present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time- frequencies such as the Wigner distribution, the ambiguity function, the short-time Fouriertransform and the spectrogram.
Abstract: The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. >
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TL;DR: An algorithm for efficient and accurate computation of the fractional Fourier transform for signals with time-bandwidth product N, which computes the fractionsal transform in O(NlogN) time.
Abstract: An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.
1,034 citations
TL;DR: A novel scheme for joint target search and communication channel estimation, which relies on omni-directional pilot signals generated by the HAD structure, is proposed, which is possible to recover the target echoes and mitigate the resulting interference to the UE signals, even when the radar and communication signals share the same signal-to-noise ratio (SNR).
Abstract: Sharing of the frequency bands between radar and communication systems has attracted substantial attention, as it can avoid under-utilization of otherwise permanently allocated spectral resources, thus improving efficiency. Further, there is increasing demand for radar and communication systems that share the hardware platform as well as the frequency band, as this not only decongests the spectrum, but also benefits both sensing and signaling operations via the full cooperation between both functionalities. Nevertheless, the success of spectrum and hardware sharing between radar and communication systems critically depends on high-quality joint radar and communication designs. In the first part of this paper, we overview the research progress in the areas of radar-communication coexistence and dual-functional radar-communication (DFRC) systems, with particular emphasis on application scenarios and technical approaches. In the second part, we propose a novel transceiver architecture and frame structure for a DFRC base station (BS) operating in the millimeter wave (mmWave) band, using the hybrid analog-digital (HAD) beamforming technique. We assume that the BS is serving a multi-antenna user equipment (UE) over a mmWave channel, and at the same time it actively detects targets. The targets also play the role of scatterers for the communication signal. In that framework, we propose a novel scheme for joint target search and communication channel estimation, which relies on omni-directional pilot signals generated by the HAD structure. Given a fully-digital communication precoder and a desired radar transmit beampattern, we propose to design the analog and digital precoders under non-convex constant-modulus (CM) and power constraints, such that the BS can formulate narrow beams towards all the targets, while pre-equalizing the impact of the communication channel. Furthermore, we design a HAD receiver that can simultaneously process signals from the UE and echo waves from the targets. By tracking the angular variation of the targets, we show that it is possible to recover the target echoes and mitigate the resulting interference to the UE signals, even when the radar and communication signals share the same signal-to-noise ratio (SNR). The feasibility and efficiency of the proposed approaches in realizing DFRC are verified via numerical simulations. Finally, the paper concludes with an overview of the open problems in the research field of communication and radar spectrum sharing (CRSS).
846 citations
Cites methods from "The fractional Fourier transform an..."
...Accordingly, the fractional Fourier transform (FrFT) [85], which is built upon orthogonal chirp basis, is used to process the target return....
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TL;DR: Time-frequency domain signal processing using energy concentration as a feature is a very powerful tool and has been utilized in numerous applications and the expectation is that further research and applications of these algorithms will flourish in the near future.
646 citations
TL;DR: This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions, and is exactly unitary, index additive, and reduces to the D FT for unit order.
Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.
604 citations
Cites background or methods from "The fractional Fourier transform an..."
...Another important property not discussed here is the relationship of the fractional Fourier transform to time–frequency representations such as the Wigner distribution [8], [10], [11], [12]....
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...The transform has become popular in the optics and signal processing communities following the works of Ozaktas and Mendlovic [6]–[8], Lohmann [9] and Almeida [12]....
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...Some of the applications explored include optimal filtering in fractional Fourier domains [13]–[16], cost-efficient linear system synthesis and filtering [17]–[21], time-frequency analysis [11], [12], [22], [23], and Fourier optics and optical information processing [24]–[26]....
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...[29] L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,”IEEE Signal Processing Lett., vol. 4, pp. 15–17, Jan. 1997....
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01 Sep 2001
TL;DR: An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.
Abstract: A brief introduction to the fractional Fourier transform and its properties is given. Its relation to phase-space representations (time- or space-frequency representations) and the concept of fractional Fourier domains are discussed. An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.
481 citations
Cites background from "The fractional Fourier transform an..."
...However, interest in the transform really grew with its reinvention/reintroduction by researchers in the fields of optics and signal processing, who noticed its relevance for a variety of application areas [6, 7, 8, 9]....
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References
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01 Jul 1989
TL;DR: A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented with emphasis on the diversity of concepts and motivations that have gone into the formation of the field.
Abstract: A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented. The objective of the field is to describe how the spectral content of a signal changes in time and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic gal is to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency. Although the basic notions have been developing steadily over the last 40 years, there have recently been significant advances. This review is intended to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. >
3,568 citations
TL;DR: A tutorial review of both linear and quadratic representations is given, and examples of the application of these representations to typical problems encountered in time-varying signal processing are provided.
Abstract: A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided. >
1,587 citations
TL;DR: In this article, a generalized operational calculus is developed, paralleling the familiar one for the ordinary transform, which provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians.
Abstract: We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct a table of fractional order Fourier transforms. A generalized operational calculus is developed, paralleling the familiar one for the ordinary transform. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians. The method of solution is first illustrated by its application to the free and to the forced quantum mechanical harmonic oscillator. The corresponding Green's functions are obtained in closed form. The new technique is then extended to three-dimensional problems and applied to the quantum mechanical description of the motion of electrons in a constant magnetic field. The stationary states, energy levels and the evolution of an initial wave packet are obtained by a systematic application of the rules of the generalized operational calculus. Finally, the method is applied to the second order partial differential equation with time-dependent coefficients describing the quantum mechanical dynamics of electrons in a time-varying magnetic field.
1,523 citations
TL;DR: In this article, the degree p = 1 is assigned to the ordinary Fourier transform and the degree P = 1/2 to the fractional transform, where p is the degree of the optical fiber.
Abstract: In this study the degree p = 1 is assigned to the ordinary Fourier transform The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt Commun (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms I approach the subject of fractional Fourier transforms in two other ways First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming Second, I propose two optical setups that are able to perform a fractional Fourier transform
965 citations
TL;DR: In this article, the authors developed a representation for discrete-time signals and systems based on short-time Fourier analysis and showed that a class of linear-filtering problems can be represented as the product of the time-varying frequency response of the filter multiplied by the short time Fourier transform of the input signal.
Abstract: This paper develops a representation for discrete-time signals and systems based on short-time Fourier analysis. The short-time Fourier transform and the time-varying frequency response are reviewed as representations for signals and linear time-varying systems. The problems of representing a signal by its short-time Fourier transform and synthesizing a signal from its transform are considered. A new synthesis equation is introduced that is sufficiently general to describe apparently different synthesis methods reported in the literature. It is shown that a class of linear-filtering problems can be represented as the product of the time-varying frequency response of the filter multiplied by the short-time Fourier transform of the input signal. The representation of a signal by samples of its short-time Fourier transform is applied to the linear filtering problem. This representation is of practical significance because there exists a computationally efficient algorithm for implementing such systems. Finally, the methods of fast convolution age considered as special cases of this representation.
600 citations