Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other approaches <s> A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other approaches <s> Abstract We present a technique based on the quadratic phase transform (QPT) for joint phase parameter estimation of multicomponent chirp signals. The QPT has the ability to analyze chirp signals in the same way as Fourier transform analyzes sinusoidal signals. In view of its practical implementation, a discrete version of the QPT, called the discrete quadratic phase transform (DQPT), is introduced. A novel fast computational procedure of the DQPT is developed. Explicit expressions for the arithmetic operation count are derived. Simulation results are presented to illustrate the usefulness of the DQPT. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other approaches <s> The concept of rotations in continuous-time, continuous-frequency is extended to discrete-time, discrete-frequency as it applies to the Wigner distribution. As in the continuous domain, discrete rotations are defined to be elements of the special orthogonal group over the appropriate (discrete) field. Use of this definition ensures that discrete rotations will share many of the same mathematical properties as continuous ones. A formula is given for the number of possible rotations of a prime-length signal, and an example is provided to illustrate what such rotations look like. In addition, by studying a 90 degree rotation, we formulate an algorithm to compute a prime-length discrete Fourier transform (DFT) based on convolutions and multiplications of discrete, periodic chirps. This algorithm provides a further connection between the DFT and the discrete Wigner distribution based on group theory. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other approaches <s> Abstract A fast algorithm for numerical calculation of arbitrary real order of fractional Fourier transforms is presented. Its computation complexity is comparable to that of fast convolution Fourier transform. Furthermore, it allows one to freely choose the sampling resolutions in both x - and u -space under the restriction of the Nyquist sampling theorem. <s> BIB004 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other approaches <s> The discrete Fourier transform (DFT) has found tremendous applications in almost all fields, mainly because it can be used to match the multiple frequencies of a stationary signal with multiple harmonics. In many applications, wideband and nonstationary signals, however, often occur. One of the typical examples of such signals is chirp-type signals that are usually encountered in radar signal processing, such as synthetic aperture radar (SAR) and inverse SAR imaging. Due to the motion of a target, the radar return signals are usually chirps, and their chirp rates include the information about the target, such as the location and the velocity. In this paper, we study discrete chirp-Fourier transform (DCFT), which is analogous to the DFT. Besides the multiple frequency matching similar to the DFT, the DCFT can be used to match the multiple chirp rates in a chirp-type signal with multiple chirp components. We show that when the signal length N is prime, the magnitudes of all the sidelobes of the DCFT of a quadratic chirp signal are 1, whereas the magnitude of the mainlobe of the DCFT is /spl radic/N. With this result, an upper bound for the number of the detectable chirp components using the DCFT is provided in terms of signal length and signal and noise powers. We also show that the N-point DCFT performs optimally when N is a prime. <s> BIB005 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other approaches <s> This paper presents a general class of fast algorithms for computing the polynomial time-frequency transform (PTFT) of length-apb, where a, b, and p are positive integers. The process of derivation shows some interesting properties that are effectively used for minimization of the computational complexity. By assigning values of a, b, and p, various algorithms, for example, radix-o and split-radix-2/(2a), can be easily obtained to provide the flexibility supporting polynomial time-frequency transforms of various sequence lengths. The detailed analysis on the computational complexities needed by these algorithms is also presented in terms of the numbers of additions and multiplications. It is shown that the proposed algorithms significantly reduce the computational complexity for applications that deal with polynomial phase signals. <s> BIB006 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other approaches <s> This paper presents fast algorithms for computing the polynomial time-frequency transform that deals with a real-valued sequence of length-apb, where a, b and p are positive integers. In particular, it shows that the polynomial time-frequency transform has a conjugate symmetric property, similar to that of the discrete Fourier transform, if the input sequence is real-valued. The computational complexities needed by these proposed algorithms are analyzed in terms of the numbers of real additions and real multiplications. When a=3,4, and 8, comparisons show that the computational complexities required by the proposed algorithms are less than 60% of those needed by the fast algorithms for complex-valued sequences. <s> BIB007
The FRFT can also be realized by the quadratic phase transform (QPT) BIB002 BIB005 , which is defined as Compared with (1), the FRFT can be computed by mapping variables m and n of the QPT onto a and u of the FRFT. With this equivalence of implementation, a number of fast algorithms for the QPT can be used to compute the FRFT BIB002  BIB006 BIB007 . Specifically, the fast algorithms proposed in BIB002 make use of the concept of decimation-in-time decomposition along the m and n domain. These algorithms can reduce the numbers of both complex multiplications and additions by a factor 2log 2 N, where N is the number of samples in the time domain. By further exploiting the periodic and symmetric properties of the QPT and with the radix-2 decimation-in-frequency principle, a computationally more efficient algorithm was proposed in  . Recently, the above fast algorithms are also extended to real value sequences BIB007 , which are useful in voice, audio, and image signal analysis. Several other approaches for obtaining the DFRFT were proposed in the literature. Here, we only provide a brief overview of these techniques. For complete details, a reader should refer to these contributions. Using the group theory, authors in BIB003 proposed that the DFRFT can be obtained as the multiplication of DFT and the periodic chirps. This DFRFT satisfies the rotation property on the WD, and the additivity and reversible property. However, this type of DFRFT can be derived only when the fractional order of the DFRFT equals some specified angles. As well, when the number of points is not prime, it will be very complicated to derive. Using Chirp-Z transforms, a fast numerical algorithm for the DRFT was proposed in BIB004 . This method allows free choice of resolutions in both fractional Fourier spaces, simultaneous data-peeping in any region, and easy implementation. Furthermore, the authors argued that their method is easier to implement in comparison with the method proposed in BIB001 , while maintaining the same computational efficiency.