Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> In recent years, the fractional Fourier transform has been the focus of many research papers. In this letter, we show that the fractional Fourier transform is nothing more than a variation of the standard Fourier transform and, as such, many of its properties, such as its inversion formula and sampling theorems, can be deduced from those of the Fourier transform by a simple change of variable. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> We study bandlimited signals with fractional Fourier transform (FRFT). We show that if a nonzero signal f is bandlimited with FRFT F/sub /spl alpha// for a certain real /spl alpha/, then it is not bandlimited with FRFT F/sub /spl beta// for any /spl beta/ with /spl beta//spl ne//spl plusmn//spl alpha/+n/spl pi/ for any integer n. This is a generalization of the fact that a nonzero signal can not be both timelimited and bandlimited. We also provide sampling theorems for bandlimited signals with FRFT that are similar to the Shannon sampling theorem. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> 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. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT’s) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties. <s> BIB004 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> In this note we obtain two new sampling formulae for reconstructing signals that are band limited or time limited in the fractional Fourier transform sense. In both cases, we use samples from both the signal and its Hilbert transform, but each taken at half the Nyquist rate. <s> BIB005 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> The fractional Fourier transform (FRT) is an extension of the ordinary Fourier transform (FT). Applying the language of the unified FT, we develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity. The FRT sampling theorem is derived as an extension of its ordinary counterpart. <s> BIB006 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform. <s> BIB007 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform. <s> BIB008 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT. <s> BIB009 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> We present much briefer and more direct and transparent derivations of some sampling and series expansion relations for fractional Fourier and other transforms. In addition to the fractional Fourier transform, the method can also be applied to the Fresnel, Hartley, and scale transform and other relatives of the Fourier transform. <s> BIB010 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> Abstract In this correspondence, some features of the fractional Fourier transform (FRFT) of the bandlimited periodic signals are discussed. Two sampling relations in the context of FRFT domain bandlimited, periodic signals are also presented. More specifically, it is shown that only 2 K + 1 coefficients are sufficient to reconstruct any FRFT domain bandlimited periodic signal, where K is the order of the highest nonzero harmonic component in the particular FRFT domain in which the signal is bandlimited. <s> BIB011 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT through sampling of FRFT <s> The fractional Fourier transform (FRFT) has become a very active area in signal processing community in recent years, with many applications in radar, communication, information security, etc., This study carefully investigates the sampling of a continuous-time band limited signal to obtain its discrete-time version, as well as sampling rate conversion, for the FRFT. Firstly, based on product theorem for the FRFT, the sampling theorems and reconstruction formulas are derived, which explain how to sample a continuous-time signal to obtain its discrete-time version for band limited signals in the fractional Fourier domain. Secondly, the formulas and significance of decimation and interpolation are studied in the fractional Fourier domain. Using the results, the sampling rate conversion theory for the FRFT with a rational fraction as conversion factor is deduced, which illustrates how to sample the discrete-time version without aliasing. The theorems proposed in this study are the generalizations of the conventional versions for the Fourier transform. Finally, the theory introduced in this paper is validated by simulations. <s> BIB012
A straightforward approach for obtaining the DFRFT is to sample the FRFT, since the sampling theorems for the FRFT of bandlimited and time-limited signals follow from those of the Shannon sampling theorem BIB001 BIB002 BIB005 BIB006 BIB010 BIB011 BIB012 . However, the resultant discrete transform may lose many important properties (i.e., unitarity and reversibility). In addition, the DFRFT obtained by direct sampling of the FRFT lacks closed-form properties and is not additive, meaning that its applications are very limited BIB009 . In order to maintain some of the FRFT properties, a type of DFRFT, derived as a special case of the continuous FRFT, was proposed in BIB007 . Specifically, it was assumed that the input function is a periodic, equally spaced impulse train. Since this type of DFRFT is a special case of continuous FRFT, many properties of the FRFT also exist and have the fast algorithm. However, this type of DFRFT cannot be defined for all values of a due to various imposed constraints. At the same time, Ozaktas et al. proposed two innovative approaches for obtaining the DFRFT through sampling of the FRFT BIB003 . Both of the methods are based on the idea that we can manipulate the expression for the FRFT, such that the form can be appropriately sampled. In particular, the first method begins to simply by multiplying the signal, x(t), with a chirp function: followed by a chirp convolution exp½jpcscðaÞðtÀtÞgðtÞ dt ð35Þ and the last chirp multiplication The convolution operation in the above equation can be achieved by sampling g(t), and then performing the convolution using the fast Fourier transform (FFT). Hence, using the FFT for convolution, provides us with samples of Maintains some important properties of the FRFT Could lack the fast computation algorithm. Cannot be written in a closed form F a ðxðtÞÞ. However, one has to keep in mind that the bandwidth and time-bandwidth product of g(t) can be as large as twice that of x(t). Thus, we need to sample g(t) at the rate twice the original sampling rate used for x(t), which means that the samples of x(t) need to be interpolated. Specifically, assuming that x and X a denote column vectors with N elements containing the samples of x(t) and its DFRFT, respectively, then, in a matrix notation, the above procedure can be given as where D and J are matrices representing the decimation and interpolation operation, K is a diagonal matrix that corresponds to chirp multiplication, and H corresponds to the convolution operation. Eq. (37) provides the samples of the a-th transform in terms of the samples of the original signal. This is a desirable property for a definition of the DFRFT matrix BIB003 . Very similar results were obtained in BIB006 as well. Sample computed FRFTs for impulse and step functions are shown in Fig. 4 . The second approach, proposed in BIB003 , implements a very similar principle. However, the authors rewrote the FRFT into the form: Then, the modulated function ½expðjpcotðaÞtÞxðtÞ is represented by Shannon's interpolation formula: allowing us to obtain the samples of the fractional transform in terms of the samples of the original signal as À2cscðaÞ mn which is a finite summation and Dt represents the sampling interval. In a matrix form, the overall procedure can be represented as for jnj and jmj r N and where A a is a constant. However, both presented cases assumed that the WD of x(t) is zero outside an origin-centered circle of diameter equal to the sampling period BIB003 . Therefore, there might be several DFRFT matrices providing the same result within the accuracy of this approximation. Nevertheless, if the signal energy contained within this circle is approaching to the total signal energy, we know that all of these matrices provide more accurate results. In other words, signals can be recovered from their transforms only within some approximation errors BIB004 BIB008 . In order to alleviate some of the problems associated with the DFRFT proposed in BIB003 , a new type of DFRFT, which is unitary, reversible, and flexible, was proposed in BIB009 . In addition, the closed-form analytic expression of this DFRFT can be obtained. where n = À N, À N +1,y,N and m = À M, ÀM +1,y,M, then, the following two formulas of the DFRFT are in order:  The inverse DFRFT is simply a Hermitian forward DFRFT with Àa. This DFRFT is efficient to calculate and implement. Because there are two chirp multiplications and one FFT, the total number of the multiplication operations required is 2P þðP=2Þlog 2 P, where P=2M+1 is the length of the output. The authors also claimed that this DFRFT also has the lowest complexity among all types of DFRFT that still work similarly to the continuous FRFT BIB009 . However, it does not match the continuous FRFT and lacks many of the characteristics of the continuous FRFT. For example, it is difficult to filter out the chirp noise with this type of DFRFT BIB009 .