The fractional Fourier transform and its application to energy localization problems

Applying the fractional Fourier transform (FRFT) and the Wigner distribution on a signal in a cascade fashion is equivalent to a rotation of the time and frequency parameters of the Wigner distribution. We presented in ter Morsche and Oonincx, 2002, an integral representation formula that yields a ﬃ ne transformations on the spatial and frequency parameters of the n -dimensional Wigner distribution if it is applied on a signal with the Wigner distribution as for the FRFT. In this paper, we show how this representation formula can be used to solve certain energy localization problems in phase space. Examples of such problems are given by means of some classical results. Although the results on localization problems are classical, the application of generalized Fourier transform enlarges the class of problems that can be solved with traditional techniques.


INTRODUCTION
In this paper, we generalize the concept of the fractional Fourier transform (FRFT) as introduced by Kober [1] and show its application for solving certain energy localization problems in phase space.In the sequential sections, we will deal with the FRFT; however, here we briefly recall the definition and some properties of the Wigner distribution.This time-frequency representation is the most commonly used tool to analyse the FRFT, see, for example, [2].Relations between fractional operators and other time-frequency distributions were studied in a general fashion in [3].As is probably well known, the Wigner distribution for a signal f with finite energy, that is, f ∈ L 2 (R), is given by Throughout this paper, we use the multidimensional mixed Wigner distribution that reads for all n-dimensional functions f and g with finite energy, that is, f , g ∈ L 2 (R n ), and with (•, •) representing the inner product in R n .In the case g = f , we will use the short notation of the Wigner distribution ᐃᐂ[ f ].Here we briefly recall some properties of the mixed Wigner distribution, which are used throughout this paper.The Wigner distribution is invariant under the action of both translationb and frequency modulation ᏹ ω0 , given byb [ f ](x) = f (x − b) and ᏹ ω0 [ f ](x) = e iω0x f (x), for b, ω 0 ∈ R n and f acting on R n .A straightforward calculation shows that ( This means that a translation over (x 0 , ω 0 ) in the Wigner plane, the phase space related to the Wigner distribution, corresponds to the operator In relation to the FRFT, the following property is of importance.A rotation over π/2 in all dimensions of the Wigner plane is achieved by the action of the Fourier transform Ᏺ n on the signal f ∈ L 2 (R n ), that is, For a comprehensive list of other properties of the Wigner distribution, we refer to [4,5].One last property we want to mention here is the property of satisfying the time and frequency marginals, that is, The sequel of this paper focuses on energy conserving (unitary) operators that correspond to classes of affine transformations in the Wigner plane.In Section 2, the FRFT is discussed as an operator that corresponds to rotation action in the Wigner plane.In Section 3, the whole class of affine transformations in the n-dimensional Wigner plane is presented and studied extensively.Also an integral representation for this class is presented.In Section 4, this representation is used in a mathematical framework for analyzing and solving energy localization problems in the Wigner plane.This framework is based on the Weyl correspondence.Finally, some examples of energy localization problems are discussed in Section 5.The framework of the latter section is used for solving two well-known energy localization problems.

FRACTIONAL FOURIER TRANSFORM
The FRFT on L 2 (R) was originally described by Kober [1] and was later introduced for signal processing by Namias [6] as a Fourier transform (Ᏺ) of fractional order, that is, for α ∈ [−π, π].From this formal definition, an integral representation for Ᏺ α has been derived in a heuristic manner.Later this representation has been formalized in [7,8].The integral representation for functions f ∈ L 2 (R) reads for 0 < |α| < π, with C α = e i((π/4) sgn α−α/2) .For α = 0 and α = π, an expression for the FRFT follows directly from (8), namely, For time-frequency analysis, it is of interest to consider the relation of the FRFT with time-frequency operators like the Wigner distribution.In [2], Almeida showed that the FRFT Ᏺ α gives raise to a rotation in the Wigner plane by an angle α, that is, where R α (x, ω) represents the matrix vector product with matrix In particular, we have a rotation by π/2 in the Wigner plane for Ᏺ π/2 , which is a result that coincides with (5).
The action of the FRFT in the Wigner plane leads us in a natural way to the question, which operators on L 2 (R) act like a linear transformation in the Wigner plane?The following section is devoted to this question.However, instead of operators on L 2 (R), we consider operators acting on L 2 (R n ), since finding a solution for the n-dimensional problem also yields a solution for the one-dimensional problem, but it does not follow straightforwardly from the solution of the one-dimensional case.

AFFINE TRANSFORMATIONS IN THE WIGNER PLANE
Inspired by the FRFT and its action in the Wigner plane, we search for linear operators ᐂ on L 2 (R n ) such that there exist a matrix A ∈ R n×n and a vector b ∈ R n for which holds for all f ∈ L 2 (R n ).Since the translation vector b is the result of the unitary operator ᏺ −b (see ( 4)), it suffices to search for linear operators ᐂ on L 2 (R n ) such that there exists a matrix A ∈ R 2n×2n for which Furthermore, we restrict ourselves to matrices A for which det A = ±1.Operators that yield such transformations A in phase space preserve energy which follows straightforwardly from ( 6) and ( 13) by substitution of variables.In a previous paper [9], we dealt with the problem of classifying all unitary operators on L 2 (R n ) that correspond to a matrix A ∈ R 2n×2n in the sense of (13).Moreover, by polarization, this class of unitary operators will also satisfy In [10], it has been shown that a necessary and sufficient condition on the matrix A, such that a unitary operator ᐂ exists, is that A ∈ R 2n×2n is symplectic.This means that given the 2 × 2 block decomposition the following relations should hold: It can also be shown [11] that for symplectic matrices, we have det A = 1.In the sequel of this paper, we use the notation Sp(n) for all real-valued symplectic 2n × 2n symplectic matrices.
Starting with a symplectic matrix A ∈ R 2n×2n , we derived in [9] an integral representation for a unitary operator Ᏺ A on L 2 (R n ) that satisfies (14).This operator is defined as follows.
Definition 1.Let A ∈ Sp(n) with block decomposition (15).Then for A 12 = 0, the linear operator Ᏺ A on L 2 (R n ) is given by Furthermore, if A 12 = 0, then for all f ∈ L 2 (R n ) and with Here s(A 12 ) denotes the product of the nonzero singular values of A 12 , and vol Ker(A12) (A 22 ) denotes the volume of the simplex spanned by A 22 e 1 , . . ., A 22 e n , with e 1 , . . ., e n any orthonormal basis in the null space of A 12 .
In the particular case for which A 12 is nonsingular, we have vol Ker(A12) (A 22 ) = 1 and s(A 12 ) = det(A 12 ).Furthermore, using the substitution u = A 12 t + A 11 x and conditions (16), formula ( 18) is simplified to which corresponds to the metaplectic representation of Sp(n), as given in [11].
The multidimensional FRFT is a special case of ( 20), namely, it follows from (20) by taking if α i = 2kπ, for all i = 1, . . ., n.Moreover, the FRFT can also be seen as a special case of the operator with Γ ∈ R n×n symmetric and ∆ ∈ R n×n with det ∆ = 0.For simplicity, we also assume ∆ to be symmetric.Of course this operator is also a special case of (18).A generalization of the FRFT in this way was already suggested in [12].

LOCALIZATION PROBLEMS AND THE METAPLECTIC REPRESENTATION
In this section, we consider the celebrated problem in signal processing of maximizing energy in both time and frequency, or space and frequency in more dimensions.This problem has already received much attention in the literature, see, for example, [13,14,15,16].We will show how the representation formula ( 18) can be used to solve a whole class of localization problems if only one problem of this class has already been solved.In the problems we consider here, the goal is to find a function for some bounded weight function σ, called the symbol.Consequently, if with Ω ⊂ R 2n , then (23) represents the energy of f in the Wigner plane within the region Ω.
For solving this maximum energy problem, we introduce the localization operator ᏸ(σ) by for all f , g ∈ L 2 (R n ).Note that by introducing this operator ᏸ(σ), the problem comes down to search for such functions f that maximize (ᏸ(σ) f , f ).The association of a symbol σ with the localization operator ᏸ(σ) is called the Weyl correspondence, see, for example, [11,17].In [14], Flandrin showed that ᏸ(σ) is self-adjoint for real-valued σ.Moreover, it was shown in [18] that if σ is real valued and of finite energy, absolutely integrable, or just bounded, then the eigenvectors of ᏸ(σ) can be chosen to form an orthonormal basis for L 2 (R n ), the set of real-valued eigenvalues is countable, and the possible accumulation point is 0. The function f max that maximizes (23) is given by the eigenvector φ 0 of ᏸ(σ) corresponding to the largest eigenvalue λ 0 of ᏸ(σ).
We now assume that for a certain symbol σ ∈ L ∞ (R 2n ), the function that maximizes (23), f max , and its corresponding fraction of energy λ 0 are known.Then the following lemma gives us the solutions for a whole class of localization problems.
Proof.The proof follows straightforwardly from definition (25) and property (14).We have which completes the proof.
For one-dimensional problems, the following corollary applies.
Corollary 1.Let Ω ⊂ R 2 be an arbitrary bounded region in the Wigner plane and let f max ∈ L 2 (R) be the signal that has maximal energy E max in Ω.Then the signal that has maximal energy To illustrate Corollary 1, the previous result is now applied to two well-known energy localization problems.

EXAMPLES
The two examples we discuss in this section are the maximization of energy on ellipsoidal areas in the Wigner plane and on parallelograms in the time-frequency plane that is related to the Rihaczek distribution.Both problems have already been studied in the literature [14,19] using traditional results on the Wigner distribution.Here we present a way of solving these problems using a generalization of the FRFT.For simplicity, we restrict ourselves to the case of onedimensional signals, where the idea of using the fractional transform for solving such problems can also be visualized in a better way.

Energy concentration on ellipsoidals in the Wigner plane
The problem we consider first is the concentration of energy in a circular region in the Wigner plane.So we consider a region and search for functions f ∈ L 2 (R), with normalized energy f L 2 , for which is maximized.For solving this localization problem, we observe that with ᏸ the localization operator ᏸ(σ) as in (25).We observe that 1 CR is a bounded real-valued symbol, and so we have an orthonormal basis of eigenfunctions with the operator ᏸ(1 CR ) and corresponding positive eigenvalues.The function f max , that maximizes E f (R), is then given by the eigenvector φ 0 of ᏸ(1 CR ) corresponding to the largest eigenvalue λ 0 of ᏸ(1 CR ).Moreover, E max (R) is given by λ 0 .
The eigenvectors of ᏸ(1 CR ) are given by the Hermite functions H k , k ∈ N, which is a result by Janssen in [20].Furthermore, it can be shown [19] that the corresponding eigenvalues satisfy where k ∈ N\{0} with L k being the Laguerre polynomial of degree k.It can be shown that The circular region can also be translated over a vector (x 0 , ω 0 ).As a result of (4), the eigenfunctions of ᏸ(σ) are then given by ᏺ (x0,ω0) H k .The eigenvalues remain the same.
Dilating circular regions in either the time or frequency direction will yield ellipsoidal regions that are orientated along one of these axes.The total class of ellipsoidal regions that are obtained from a circle by means of an area preserving affine transformation is given by A(C R ) − b, with A ∈ R 2×2 , det A = ±1, and b ∈ R 2n .We restrict ourselves to the case det A = 1 since a function that maximizes energy in the regions A(C R ) − b, with det A = −1, is the complex conjugate of the function that maximizes energy in the regions MA(C R ) − b, with Furthermore, since symplectic matrices in R 2×2 are matrices with det A = 1, Corollary 1 applies to this situation, which means that the eigenfunctions of ᏸ(1 A(CR)−b ) are given by ᏺ b Ᏺ A H k and that its eigenvalues satisfy the recursive relations for the eignvalues as presented above.Particularly, we solved the following energy localization problem.Let CR be the ellipsoidal region given by with A ∈ R 2×2 and b ∈ R, then ᏺ b Ᏺ A H 0 is the signal that has maximal energy E max (R) = 1 − e −R 2 in this region of the Wigner plane.Figure 1 illustrates the type of regions one can obtain by starting with the circle C 1 and then transforming it by a symplectic matrix A. In this example, we have chosen for the domains (b), (c), and (d), respectively.Note that the maximal amount of energy a signal can have in each of these regions is (e − 1)/e.

Energy concentration on parallelograms in the Rihaczek plane
The second problem we consider is the maximization of a signal f ∈ L 2 (R), normalized to energy equal to 1, within a rectangular plane in phase space, with respect to the Rihaczek distribution This problem can also be related to the problem of maximizing energy with the localization operator ᏸ(σ).To show this, we introduce the mixed Rihaczek distribution [ f , g] by We will show that for all signals f and g with finite energy if for some x 0 , ω 0 ∈ R + and where ϕ is given by ϕ(x, ω) = e −2ixω .We observe that σ ∞ ≤ 1, and so σ is a bounded symbol.
To prove relation (36), we first write (ᏸ(σ) f , g) as the inner product ] .The latter expression can be rewritten as yielding relation (36).We observe that the mixed Wigner distribution reduces to the Rihaczek distribution for g = f .This means that ᏸ(σ) is the localization operator that corresponds to the rectangular region [−x 0 , x 0 ] × [−ω 0 , ω 0 ] in the Rihaczek plane, that is, the time-frequency plane generated by the Rihaczek distribution.
As far as known, no explicit solution exists for the eigenvector/value problem for this ᏸ(σ).However, some information of this ᏸ(σ) can be obtained by looking at ᏸ(σ) * ᏸ(σ), with ᏸ(σ) * the adjoint of ᏸ(σ).Observe that the eigenvalues of ᏸ(σ) * ᏸ(σ) are directly related to the singular values of opL(σ).For studying ᏸ(σ) * ᏸ(σ), we consider a result by Flandrin.In [14], it was shown that when σ is as in (37), then ᏸ(σ) = Ꮾ(ω 0 )ᏼ(x 0 ), with These projections have been studied extensively by Slepian and Pollak [16,21].In particular, they showed that the eigenfunctions of the operator ᏼ(x 0 )Ꮾ(ω 0 )ᏼ(x 0 ) are given by the prolate spheroidal wave functions (PSWF) ψ k , k ∈ N, (see [22]) and their corresponding eigenvalues depend on the product x 0 ω 0 .Moreover, for x 0 ω 0 → ∞ approximately, the first 2x 0 ω 0 /π eigenvalues that correspond to the PSWF attain a value close to unity.For index numbers in a region around 2x 0 ω 0 /π, the eigenvalues plunge to zero and attain values close to zero afterwards.The number of eigenvalues in the region where the eigenvalues decrease from close to one to close to zero is proportional to log x 0 ω 0 .This asymptotical behaviour has been described rigorously in [21].Furthermore, we observe that the singular values of ᏸ(σ) are given by s k = λ k , where λ k denote the eigenvalues of the operator ᏼ(x 0 )Ꮾ(ω 0 )ᏼ(x 0 ).By definition, its asymptotical behavior is similar to the behaviour of the eigenvalues of ᏼ(x 0 )Ꮾ(ω 0 )ᏼ(x 0 ).
The eigenvectors of ᏸ(σ) * ᏸ(σ) are given by the PSWF.However, they do not give rise to explicit expressions for the eigenfunctions of ᏸ(σ).
As for the circular regions in the Wigner plane, we can also apply a linear transformation A ∈ R 2×2 , with det A = 1, and a translation over b ∈ R 2 on the rectangular region in the Rihaczek plane.This leads to parallelograms In a straightforward way, it can be shown that Lemma 1 also holds for the operator ᏸ(σ) * ᏸ(σ) and so also Corollary 1 holds for ᏸ(σ) * ᏸ(σ).For this situation, it means that the singular values of the operator are given by λ k , where λ k satisfies the previous discussed asymptotical behaviour.The eigenfunctions of are given by ᏺ b Ᏺ A ψ k , with ψ k the PSWF.

CONCLUSIONS
In this paper, we have shown how a generalization of the ndimensional FRFT can be used to analyze certain energy localization problems in the 2n-dimensional phase plane.This generalization is a newly derived representation of so-called metaplectic operators.These operators form a natural extension of the notion of the FRFT in the way that taking the Wigner distribution and a metaplectic operator in a cascade fashion corresponds to a symplectic transformation on the spatial and frequency parameters of the Wigner distribution.
The approach of solving localization problems with metaplectic operators (and their representation formula) has been illustrated by two classical examples in the onedimensional case.
The presented integral representation formula is valid for all choices of the corresponding symplectic transformations in the Wigner plane.On the contrary, classical representation formulas [11] are only available for symplectic transformations with 2 × 2 block decompositions where not all four blocks are singular, which is the case if the metaplectic operator is a d-dimensional Fourier transform on L 2 (R n ), with 0 < d < n.Half way through the eighties, his research interest has changed from splines to signal processing and wavelets.On this subject, he has written application oriented papers and a textbook, and guided several industrial projects.Nowadays, Dr. ter Morsche is the director of education of the bachelor and master course on applied and industrial mathematics in Eindhoven.

Call for Papers
Seamless detection and tracking schemes are able to integrate unthresholded (or below target detection threshold) multiple sensor responses over time to detect and track targets in low signal-to-noise ratio (SNR) and high clutter scenarios.These schemes, also called "track-before-detect (TBD)" algorithms are especially suitable for tracking weak targets that would only very rarely cross a standard detection threshold as applied at the sensor level.
Thresholding sensor responses result in a loss of information.Keeping this information allows some TBD approaches to deal with the classical data association problem effectively in high clutter and low SNR situations.For example, in detection scenarios with simultaneous activation/illumination from different signal sources this feature allows the application of triangulation techniques, where in the case of contact tracking approaches essential information about weak targets would often be lost because these targets did not produce signals that cross the normal detection threshold.Extending this example to a multi-sensor network scenario, a TBD algorithm that can use unthresholded (or below threshold) data has the potential to show improved performance compared to an algorithm that relies on thresholded data.In low SNR situations, this can substantially increase performance particularly in the case of a dense multi-target scenario.
Naturally, TBD algorithms consume high computational processing power: An efficient realization and coding of the TBD scheme is mandatory.
Another issue that arises when using the TBD scheme is the quality of the sensor model: Practical experience with thresholded data shows that a coarser modelling of the likelihood function might be sufficient and often leads to robust algorithms.How much have these sensor models to be improved in order to allow the TBD algorithms to exploit the information provided with the unthresholded data?TBD algorithms that are well known to the tracking community are the likelihood ratio detection and tracking (LRDT), maximum likelihood probabilistic data association (MLPDA ), maximum likelihood probabilistic multihypothesis tracking (MLPMHT), Houghtransform based methods and dynamic programming techniques; also related are the probability hypothesis density (PHD), the histogram probabilistic multi-hypothesis tracking (H-PMHT) algorithms, and, of course, various particle filter approaches.Some of these algorithms are capable of tracking extended targets and performing signal estimation in multi-sensor measurements.
The aim of this special issue is to focus on recent developments in this expanding research area.The special issue will focus on one hand on the development and comparison of algorithmic approaches, and on the other hand on their currently ever-widening range of applications such as in active or passive surveillance scenarios (e.g. for object tracking and classification with image and video based sensors, or scenarios involving chemical, electromagnetic and acoustic sensors).Special interest lies in multi-sensor data fusion and/or multi-target tracking applications.
Authors should follow the EURASIP Journal on Advances in Signal Processing manuscript format described at the journal site http://www.hindawi.com/journals/asp/.Prospective authors should submit an electronic copy of their complete manuscript through the EURASIP JASP Manuscript Tracking System at http://www.hindawi.com/mts/,according to the following timetable:

Call for Papers
The ever-growing need for wireless communications which provide high data rates entails a substantial demand for new spectral resources and more flexible and efficient use of existing resources.Several measurement campaigns conducted in the recent years show that frequency spectrum in the range 30 MHz-3 GHz is most of the time unused leading to low average occupancy rates and motivating to allow more flexible spectrum use.Promising solution to exploit spectrum in a flexible way is via cognitive radio and dynamic spectrum sharing systems which use innovative spectrum management and allow different systems to share the same frequency band.Significant potential improvements offered with such approaches and also positive view from regulatory bodies have led to exploding interest in this field recently.However, such paradigm shift introduces many new design challenges that have to be solved in order to enable proper functioning of the spectrum sharing and cognitive radio systems.Recent research efforts include considerations of different physical layer technologies, spectrum sensing, coexistence mechanisms between legacy and secondary users, and shared medium access among many secondary users.
The objective of this special issue is to showcase the most recent developments and research in this field, as well as to enhance its state-of-the-art.Original and tutorial articles are solicited in all aspects of cognitive radio and spectrum sharing including system and network protocol design, enabling technologies, theoretical studies, practical applications, and experimental prototypes.

Topics of Interest:
Topics of interest include, but are not limited to: • Spectrum measurements and current usage Authors should follow the JWCN manuscript format at http://www.hindawi.com/journals/wcn/.Prospective authors should submit an electronic copy of their complete manuscript through the JWCN's Manuscript Tracking System at http://www.hindawi.com/mts/,according to the following timetable:

Figure 2
illustrates the type of regions one can obtain by starting with the rectangular [−1, 1] × [−1, 1] and then transforming it by the symplectic matrices A as indicated in (33).
Patrick J. Oonincx received his M.S. degree (with honors) in mathematics from Eindhoven University in 1995 with a thesis on generalizations of multiresolution analysis.In 2000, he received the Ph.D. degree in mathematics from University of Amsterdam.His thesis on the mathematics of joint time-frequency/scale analysis has also appeared as a textbook.From 2000 to 2002, he worked as a Postdoctoral Researcher on multiresolution image processing at the Research Institute for Mathematics and Computer Science (CWI) in Amsterdam.Currently, he works as an Assistant Professor in mathematics and signal processing at the Royal Netherlands Naval College, Den Helder, the Netherlands.His research interests are wavelet analysis, timefrequency signal representations, multiresolution imaging, and signal processing for underwater acoustics.Hennie G. ter Morsche received his M.S. degree in 1967 from the University of Nijmegen in the field of nonlinear differential equations.From 1968 to 1978, he worked as a university teacher at the Technische Universiteit Eindhoven.Subsequently, he started at this university his Ph.D. research on spline functions.The thesis, entitled "Interpolational and extremal properties of Lspline functions," was completed in 1982.
• Spectrum regulations • Spectrum sensing and awareness techniques • Dynamic spectrum management • Capacity and achievable data rates in cognitive radio • Multiuser spectrum sharing: • Priority resource allocation • Cooperation and competition of users • Auction-based spectrum sharing • Coexistence of spectrum sharing and legacy narrowband systems • Physical layer design of spectrum sharing systems: • OFDM, OQAM, UWB, CDMA, SDR • MIMO component in spectrum sharing • Applications of cognitive radio & spectrum sharing • Standardization of cognitive radio and spectrum sharing: IEEE P1900, IEEE 802.22,ITU-R activities