Generalized convolution theorem associated with fractional Fourier transform

Wireless Communications and Mobile Computing

Volumn 14, Issue 13, 2014, Pages 1340-1351

  Shi, Jun ^a,b   Sha, Xuejun ^a   Song, Xiaocheng ^a   Zhang, Naitong ^a  

^a HARBIN INSTITUTE OF TECHNOLOGY   (China)

^b UNIVERSITY OF DELAWARE   (United States)

Author keywords

fractional Fourier domain; fractional Fourier transform; generalized convolution theorem; time shift operator

Indexed keywords

FOURIER ANALYSIS; SIGNAL PROCESSING;

CLOSED-FORM EXPRESSION; CONVOLUTION THEOREMS; FRACTIONAL FOURIER DOMAINS; FRACTIONAL FOURIER TRANSFORMS; MATHEMATICAL TOOLS; TIME-SHIFT OPERATORS;

CONVOLUTION;

EID: 84905973424     PISSN: 15308669     EISSN: 15308677     Source Type: Journal    
DOI: 10.1002/wcm.2254     Document Type: Conference Paper

References (35)

1. Namias V,. The fractional order Fourier transform and its application to quantum mechanics. Journal of the Institute of Mathematics and Its Applications 1980; 25 (3): 241-265.
2. Mendlovic D, Ozaktas HM,. Fractional Fourier transforms and their optical implementation: I. Journal of the Optical Society of America A 1993; 10 (9): 1875-1881.
3. Almeida LB,. The fractional Fourier transform and time-frequency representations. IEEE Transactions on Signal Processing 1994; 42 (11): 3084-3091.
4. Ozaktas HM, Barshan B, Mendlovic D, Onural L,. Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms. Journal of the Optical Society of America A 1994; 11 (2): 547-558.
5. Xia X-G,. On bandlimited signals with fractional Fourier transform. IEEE Signal Processing Letters 1996; 3 (3): 72-74.
6. Erden MF, Kutay MA, Ozaktas HM,. Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration. IEEE Transactions on Signal Processing 1999; 47 (5): 1458-1462.
7. Ozaktas HM, Zalevsky Z, Kutay MA,. The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley: New York, 2000.
8. Martone M,. A multicarrier system based on the fractional Fourier transform for time-frequency-selective channels. IEEE Transactions on Communications 2001; 49 (6): 1011-1020.
9. Zhang J, Wang Z,. ICI analysis for FRFT-OFDM systems to frequency offset in time-frequency selective fading channels. IEEE Communications letters 2010; 14 (10): 888-890.
10. Pei S-C, Ding J-J,. Fractional Fourier transform, Wigner distribution, and filter design for stationary and nonstationary random processes. IEEE Transactions on Signal Processing 2010; 58 (8): 4079-4092.
11. Sejdic̈ E, Djurovic̈ I, Stankovic̈ L,. Fractional Fourier transform as a signal processing tool: an overview of recent developments. Signal Processing 2011; 91: 1351-1369.
12. Almeida LB,. Product and convolution theorems for the fractional Fourier transform. IEEE Signal Processing Letters 1997; 4 (1): 15-17.
13. Sharma KK, Joshi SD,. Papoulis-like generalized sampling expansions in fractional Fourier domains and their application to superresolution. Optics Communications 2007; 278: 52-59.
14. Akay O, Boudreaux-Bartels GF,. Fractional convolution and correlation via operator methods and an application to detection of linear FM signals. IEEE Transactions on Signal Processing 2001; 49 (5): 979-993.
15. Zayed AI,. A convolution and product theorem for the fractional Fourier transform. IEEE Signal Processing Letters 1998; 5 (4): 101-103.
16. Kraniauskas P, Cariolaro G, Erseghe T,. Method for defining a class of fractional operations. IEEE Transactions on Signal Processing 1998; 46 (10): 2804-2807.
17. Torres R, Pellat-Finet P, Torres Y,. Fractional convolution, fractional correlation and their translation invariance properties. Signal Processing 2010; 90: 1976-1984.
18. Shi J, Chi Y, Zhang N,. Multichannel sampling and reconstruction of bandlimited signals in fractional Fourier domain. IEEE Signal Processing Letters 2010; 17 (11): 909-912.
19. Shi J, Zhang N, Liu X,. A novel fractional wavelet transform and its applications. Science China Information Sciences 2012; 55 (6): 1270-1279.
20. Pei S-C, Ding J-J,. Relations between fractional operations and time-frequency distributions, and their applications. IEEE Transactions on Signal Processing 2001; 49 (8): 1638-1655.
21. Zhu B, Liu S,. Optical image encryption based on the generalized fractional convolution operation. Optics Communications 2001; 195: 371-381.
22. Deng B, Tao R, Wang Y,. Convolution theorems for the linear canonical transform and their applications. Science China Information Sciences 2006; 49 (5): 592-603.
23. Wei D, Ran Q, Li Y, Ma J, Tan L,. A convolution and product theorem for the linear canonical transform. IEEE Signal Processing Letters 2009; 16 (10): 853-856.
24. Giang BT, Tuan NM,. Generalized onvolutions for the integral transforms of Fourier type and applications. Fractional Calculus and Applied Analysis 2009; 12 (3): 252-268.
25. Thao NX, Tuan T,. On the generalized convolution for I-transform. Acta Mathematica Vietnamica 2003; 18 (2): 135-145.
26. Britvina LE,. General convolutions of integral transforms and their application to ODE and PDE problems. Mathematical Modeling and Analysis 2006; 11 (1): 23-34.
27. Britvina LE,. Generalized convolutions for the Hankel transform and related integral operators. Mathematische Nachrichten 2007; 280: 962-970.
28. Zayed AI,. On the relationship between the Fourier and the fractional Fourier transforms. IEEE Signal Processing Letters 1996; 3 (12): 310-311.
29. Erseghe T, Kraniauskas P, P C G,. Unified fractional Fourier transform and sampling theorem. IEEE Transactions on Signal Processing 1999; 47 (12): 3419-3423.
30. Kutay MA, Ozaktas HM, Arbox drawings light down and leftan O, Onural L,. Optimal filtering in fractional Fourier domains. IEEE Transactions on Signal Processing 1997; 45 (5): 1129-1143.
31. Papoulis A,. Probability, Random Variables, and Stochastic Processes. McGraw-Hill: New York, 1984.
32. Tao R, Zhang F, Wang Y,. Fractional power spectrum. IEEE Transactions on Signal Processing 2008; 56 (9): 4199-4206.
33. Scharf LL, Thomas JK,. Wiener filters in canonical coordinates for transform coding, filtering, and quantizing. IEEE Transactions on Signal Processing 1998; 46 (3): 647-654.
34. Pei S-C, Ding J-J,. Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing. IEEE Transactions on Signal Processing 2007; 55 (10): 4839-4850.
35. Refregier P, Javidi B,. Optical image encryption based on input plane and Fourier plane random encoding. Optics Letters 1995; 20 (7): 767-769.