Abstract
As the fractional Fourier transform has attracted a considerable amount of attention in the area of optics and signal processing, the discretization of the fractional Fourier transform becomes vital for the application of the fractional Fourier transform. Since the discretization of the fractional Fourier transform cannot be obtained by directly sampling in time domain and the fractional Fourier domain, the discretization of the fractional Fourier transform has been investigated recently. A summary of discretizations of the fractional Fourier transform developed in the last nearly two decades is presented in this paper. The discretizations include sampling in the fractional Fourier domain, discrete-time fractional Fourier transform, fractional Fourier series, discrete fractional Fourier transform (including 3 main types: linear combination-type; sampling-type; and eigen decomposition-type), and other discrete fractional signal transform. It is hoped to offer a doorstep for the readers who are interested in the fractional Fourier transform.
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References
Almeida L B. The fractional Fourier transform and time-frequency representations. IEEE Trans Signal Process, 1994, 42: 3084–3091
Ozaktas H M, Zalevsky Z, Kutay M A. The Fractional Fourier Transform with Applications in Optics and Signal Processing. New York: Wiley, 2000. 1–513
Namias V. The fractional order Fourier transform and its application to quantum mechanics. J Inst Math Appl, 1980, 25: 241–265
Tao R, Deng B, Wang Y. Research progress of the fractional Fourier in signal processing. Sci China Ser F-Inf Sci, 2006, 49(1): 1–25
Tao R, Qi L, Wang Y. Theory and Application of the Fractional Fourier Transform (in Chinese). Beijing: Tsinghua University Press, 2004. 23–49
Zayed A I. On the relationship between the Fourier transform and fractional Fourier transform. IEEE Signal Process Lett, 1996, 3: 310–311
Lohmann A W. Image rotation, Wigner rotation, and the fractional Fourier transform. J Opt Soc Amer A, 1993, 10: 2181–2186
Ozaktas H M, Barshan B, Mendlovic D, et al. Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transform. J Opt Soc Amer A, 1994, 11: 547–559
Lohmann A W, Soffer B H. Relationship between the Radon-Wigner and the fractional Fourier transform. J Opt Soc Amer A, 1994, 11:1798–1801
Mustard D A. The fractional Fourier transform and the Wigner distribution. J Aust Math Soc B, 1996, 38: 209–219
Pei S C, Ding J J. Relations between fractional operations and time-frequency distributions, and their applications. IEEE Trans Signal Process, 2001, 49: 1638–1655
Candan C, Kutay M A, Ozaktas H M. The discrete fractional Fourier transform. IEEE Trans Signal Process, 2000, 48: 1329–1337
Ozaktas H M, Aytur O. Fractional Fourier domains. Signal Process, 1995, 46: 119–124
Kraniauskas P, Cariolaro G, Erseghe T. Method for defining a class of fractional operations. IEEE Trans Signal Process, 1998, 46: 2804–2807
Cariolaro G, Erseghe T, Kraniauskas P, et al. A unified framework for the fractional Fourier transform. IEEE Trans Signal Process, 1998, 46: 3206–3219
Almeida L B. Product and convolution theorems for the fractional Fourier transform. IEEE Signal Process Lett, 1997, 4: 15–17
Zayed A I. A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process Lett, 1998, 5: 101–103
Xia X. On bandlimited signals with fractional Fourier transform. IEEE Signal Process Lett, 1996, 3: 72–74
Erseghe T, Kraniauskas P, Cariolaro G. Unified fractional Fourier transform and sampling theorem. IEEE Trans Signal Process, 1999, 47: 3419–3423
Pei S C, Ding J J. Simplified fractional Fourier transforms. J Opt Soc Amer A, 2000, 17: 2355–2367
Erden M F, Kutay M A, Ozaktas H M. Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration. IEEE Trans Signal Process, 1999, 47: 1458–1462
Zalevsky Z, Mendlovic K. Fractional Wiener filter. Appl Opt, 1996, 35: 3930–3936
Kutay M A, Ozaktas H M, Arikan O, et al. Optimal image restoration with the fractional Fourier transform. J Opt Soc Amer A, 1998, 15: 825–833
Erden M F, Ozaktas H M. Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains. J Opt Soc Amer A, 1998, 15: 1647–1657
Ozaktas H M, Arikan O, Kutay M A, et al. Digital computation of the fractional Fourier transform. IEEE Trans Signal Process, 1996, 44: 2141–2150
Ozaktas H M, Mendlovic D. Fractional Fourier optics. J Opt Soc Amer A, 1995, 12: 743–751
McBride A C, Kerr F H. On Namias’ fractional Fourier transforms. IMA J Appl Math, 1987, 39: 159–175
Mendlovic D, Ozaktas H M, Lohmann A W. Fractional correlation. Appl Opt, 1995, 34: 303–309
Candan C, Ozaktas H M. Sampling and series expansion theorems for fractional Fourier and other transforms. Signal Process, 2003, 83: 2455–2457
Zayed A I, Garcia A G. New Sampling formulae for the fractional Fourier transform. Signal Process, 1999, 77: 111–114
Sharma K K, Joshi S D. Fractional Fourier transform of bandlimited periodic signals and its sampling theorem. Opt Commun, 2005, 256: 272–278
Sharma K K, Joshi S D. On scaling properties of fractional Fourier transform and its relation with other transforms. Optics Commun, 2006, 257: 27–38
Torres R, Pellat-Finet P, Torres Y. Sampling theorem for fractional bandlimited signals: a self-contained proof. Application to digital holography. IEEE Signal Process Lett, 2006, 13: 676–679
Zhang W Q, Tao R. Sampling theorems for bandpass signals with fractional Fourier transform(in Chinese). Acta Electron Sin, 2005, 33(7): 1196–1199
Pei S C, Yeh M H, Luo T L. Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform. IEEE Trans Signal Process, 1999, 47: 2883–2888
Barkat B, Yinguo J. A modified fractional Fourier series for the analysis of finite chirp signals & its application. In: IEEE the 7th International Symposium on Signal Processing and Its Application. New York: IEEE Press, 2003, 1: 285–288
Alieva T, Barbe A. Fractional Fourier and Radon-Wigner transforms of periodic signals. Signal Process, 1998, 69: 183–189
Dickinson B W, Steiglitz K. Eigenvectors and functions of the discrete Fourier transform. IEEE Trans Acoust Speech Signal Process, 1982, ASSP-30: 25–31
Santhanam B, McClellan J H. The DRFT—a rotation in time-frequency space. In: Proc IEEE Int Conf Acoustics Speech Signal Process. New York: IEEE Press, 1995. 921–924
Santhanam B, McClellan J H. The discrete rotational Fourier transform. IEEE Trans Signal Process, 1996, 42: 994–998
Cariolaro G, Erseghe T, Kraniauskas P, et al. Multiplicity of fractional Fourier transforms and their relationships. IEEE Trans Signal Process, 2000, 48: 227–241
Zhao X H, Tao R, Deng B. Practical normalization methods in the digital computation of the fractional Fourier transform. In: Proc IEEE Int Conf Signal Process. New York: IEEE Press, 2004, 1: 105–108
Bultheel A, Sulbaran H M. Computation of the fractional Fourier transform. Appl Comput Harmon Anal, 2004, 16: 182–202
Deng X G, Li Y P, Fan D Y, et al. A fast algorithm for fractional Fourier transform. Opt Commun, 1997, 138: 270–274
Pei S C, Ding J J. Closed-form discrete fractional and affine Fourier transforms. IEEE Trans Signal Process, 2000, 48: 1338–1353
McClellan J H, Parks T W. Eigenvalue and eigenvector decomposition of the discrete Fourier transform. IEEE Trans Audio Eletroacoustics, 1972, AU-20: 66–74
Pei S C, Yeh M H. Discrete fractional Fourier transform. Proc IEEE Int Symp Circ Syst, 1996. 536–539
Pei S C, Yeh M H. Improved discrete fractional Fourier transform. Opt Lett, 1997, 22: 1047–1049
Pei S C, Tseng C C, Yeh M H, et al. Discrete fractional Hartley and Fourier transform. IEEE Trans Circ Syst II, 1998, 45: 665–675
Pei S C, Tseng C C. A new discrete fractional Fourier transform based on constrained eigendecomposition of DFT matrix by Largrange multiplier method. In: Proc IEEE Int Conf Acoustics Speech Signal Process. New York: IEEE Press, 1997. 3965–3968
Pei S C, Tseng C C, Yeh M H. A new discrete fractional Fourier transform based on constrained eigendecomposition of DFT matrix by Largrange multiplier method. IEEE Trans Circuits Syst II, 1999, 46: 1240–1245
Pei S C, Yeh M H, Tseng C C. Discrete fractional Fourier transform based on orthogonal projections. IEEE Trans Signal Process, 1999, 47: 1335–1348
Candan C, Kutay M A, Ozaktas H M. The discrete fractional Fourier transform. In: Proc IEEE Int Conf Acoustics Speech Signal Process. New York: IEEE Press, 1999. 1713–1716
Hanna M T, Seif N P A, Ahmed W A E M. Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular value decomposition of its orthogonal projection matrices. IEEE Trans Circ Syst I, 2004, 51: 2245–2254
Hanna M T, Seif N P A, Ahmed W A E M. Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the direct utilization of the orthogonal projection matrices on its eigenspaces. IEEE Trans Signal Process, 2006, 54: 2815–2819
Pei S C, Hsue W L, Ding J J. Discrete fractional Fourier transform based on new nearly tridiagonal commuting matrices. Proc IEEE Int Conf Acoustics Speech and Signal Process, 2005. 385–388
Pei S C, Hsue W L, Ding J J. Discrete fractional Fourier transform based on new nearly tridiagonal commuting matrices. IEEE Trans Signal Process, 2006, 54: 3815–3828
Candan C. On higher order approximations for Hermite-Gaussian functions and discrete fractional Fourier transforms. IEEE Signal Process Lett, 2007, 14: 699–702
Arikan O, Kutay M A, Ozaktas H M, et al. The discrete fractional Fourier transformation. In: Proc IEEE Int Symp Time-Frequency Time-Scale Anal. New York: IEEE Press, 1996. 205–207
Richman M S, Parks T W. Understanding discrete rotations. In: Proc IEEE Int Conf Acoust Speech Signal Process. New York: IEEE Press, 1997, 3: 2057–2060
Zhu B H, Liu S T, Ran Q W. Optical image encryption based on multifractional Fourier transforms. Opt Lett, 2000, 25(16): 1159–1161
Ran Q W, Yeung D S, E. Tsang C C, et al. General multifractional Fourier transform method based on the generalized permutation matrix group. IEEE Trans Signal Process, 2005, 53(1): 83–98
Qi L, Tao R, Zhou S Y, et al. Detection and parameter estimation of multicomponent LFM signal based on the fractional Fourier transform. Sci China Ser F-Inf Sci, 2004, 47(2): 184–198
Tao R, Li B Z, Wang Y. Spectral analysis and reconstruction for periodic nonuniformly sampled signals in fractional Fourier domain. IEEE Trans Signal Process, 2007, 55(7): 3541–3547
Tao R, Deng B, Zhang W Q, et al. Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain. IEEE Trans Signal Process, 2008, 56(1): 158–171
Zhang F, Tao R. Multirate signal processing based on discrete time fractional Fourier transform(in Chinese). Prog Natl Sci, 2008, 18(1): 93–101
Pei S C, Hsue W L. The multiple-parameter discrete fractional Fourier transform. IEEE Signal Process Lett, 2006, 13: 329–332
Pei S C, Yeh M H. A novel method for discrete fractional Fourier transform computation. In: Proc IEEE Int Symp Circ Syst. New York: IEEE Press, 2001. 585–588
Yeh M H, Pei S C. A method for the discrete fractional Fourier transfor computation. IEEE Trans Signal Process, 2003, 51: 889–891
Hanna M T. On the angular decomposition technique computing the discrete fractional Fourier transform. In: Proc IEEE Int Conf Acoust Speech Signal Process. New York: IEEE Press, 2007. 3988–3991
Huang D F, Chen B S. A multi-input-multi-output system approach for the computation of discrete fractional Fourier transform. Signal Process, 2000, 80: 1501–1513
Zhu Y Q, Qi L, Yang S Y, et al. Calculation of discrete fractional Fourier transform based on adaptive LMS algorithm. Proc IEEE Int Conf Signal Process, 2006, 1: 16–20
Hanna M T. A discrete fractional Fourier transform based on orthonormalized McClellan-Parks eigevectors. In: Proc IEEE Int Conf Circ Syst. New York: IEEE Press, 2003. 81–84
Pei S C, Yeh M H. Two dimensional discrete fractional Fourier transform. Signal Process, 1998, 67: 99–108
Narayanan V A, Prabhu K M M. The fractional Fourier transform: theory, implementation and error analysis. Microprocess Microsy, 2003, 27: 511–521
Pei S C, Yeh M H. Discrete fractional Hadamard transform. In: Pro IEEE Int Symp Circ Syst. New York: IEEE Press, 1999. 179–182
Pei S C, Yeh M H. The discrete fractional cosine and sine transforms. IEEE Trans Signal Processing, 2001, 49: 1198–1207
Cariolaro G, Erseghe T, Kraniauskas P. The fractional discrete cosine transform. IEEE Trans Signal Process, 2002, 50: 902–911
Tseng C C. Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices. IEEE Trans Signal Process, 2002, 50: 866–877
Pei S C, Ding J J. Generalized eigenvectors and fractionalization of offset DFTs and DCTs. IEEE Trans Signal Process, 2004, 52: 2032–2046
Vargas-Rubio J G, Santhanam B. On the multiangle centered discrete fractional Fourier transform. IEEE Signal Process Lett, 2005, 12: 273–276
Pei S C, Yeh M H. Discrete fractional Hilbert transform. Proc IEEE Int Symp Signal Process, 1998. 506–509
Liu Z J, Zhao H F, Liu S T. A discrete fractional random transform. Opt Commu, 2005, 255: 357–365
Yeh M H. Angular decompositions for the discrete fractional signal transforms. In: Proc IEEE Int Symp Circ Syst. New York: IEEE Press, 2003. 93–96
Yeh M H. Angular decompositions for the discrete fractional signal transforms. Signal Process, 2005, 85: 537–547
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Supported by the National Natural Science Foundation of China (Grant Nos. 60232010 and 60572094) and the National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 60625104)
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Tao, R., Zhang, F. & Wang, Y. Research progress on discretization of fractional Fourier transform. Sci. China Ser. F-Inf. Sci. 51, 859–880 (2008). https://doi.org/10.1007/s11432-008-0069-2
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DOI: https://doi.org/10.1007/s11432-008-0069-2