×
Stern, Adrian

Sampling of compact signals in offset linear canonical transform domains. (English) Zbl 1122.94023

Signal Image Video Process. 1, No. 4, 359-367 (2007).
Summary: The offset linear canonical transform (OLCT) is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms in engineering such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform (FRST), frequency modulation, time shifting, time scaling, chirping and others. Therefore the OLCT provides a unified framework for studying the behavior of many practical transforms and system responses. In this paper the sampling theorem for OLCT is considered. The sampling theorem for OLCT signals presented here serves as a unification and generalization of previously developed sampling theorems.

Cited in 32 Documents

MSC:

94A20 Sampling theory in information and communication theory
42B99 Harmonic analysis in several variables

Keywords:

regular sampling; offset linear canonical transform; special affine Fourier transform; linear canonical transform; time-frequency representation; uncertainty principle
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Shannon C.E. (1949). Communication in the presence of noise. Proc. IRE 37: 447–457 · doi:10.1109/JRPROC.1949.232969
[2] Pei S.C., Ding J.J. (2003). Eigenfunctions of the offset Fourier, fractional Fourier and linear canonical transforms. J. Opt. Soc. Am. A 20: 522–532 · doi:10.1364/JOSAA.20.000522
[3] Abe S., Sheridan J.T. (1994). Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation. Opt. Lett. 19: 1801–1803 · doi:10.1364/OL.19.001801
[4] Moshinsky M., Quesne C. (1971). Linear canonical transformations and their unitary representations. J. Math. Phys. 12: 1772–1783 · Zbl 0247.20051 · doi:10.1063/1.1665805
[5] Pei S.C., Ding J.J. (2002). Eigenfunctions of linear canonical transform. IEEE Trans. Acoust. Speech. Sig. Process. 50: 11–26 · Zbl 1369.94259
[6] Wolf K.R. (1979). Integral Transforms in Science and Engineering, Chap. 9,10. Plenum Press, New York · Zbl 0409.44001
[7] Almeida L.B. (1994). The fractional Fourier transform and time frequency representation. IEEE Trans. Sig. Process. 42: 3084–3091 · doi:10.1109/78.330368
[8] Ozaktas H.M., Kutay M.A., Zalevsky Z. (2000). The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York
[9] Papoulis A. (1994). Pulse compression, fiber communication and diffraction: a unified approach. J. Opt. Soc. Am. A 11: 3–13 · doi:10.1364/JOSAA.11.000003
[10] Pei S.C., Ding J.J. (2004). Generalized eigenvectors and fractionalization of offset DFTs and DCTs. IEEE Trans. Sig. Proc. 52(7): 2032–2046 · Zbl 1369.42025 · doi:10.1109/TSP.2004.828904
[11] Pei S.C., Yeh M.H., Luo T.L. (1999). Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform. IEEE Trans. Sig. Proc. 47(10): 2883–2888 · Zbl 1009.94535 · doi:10.1109/78.790671
[12] Xia X.G. (1996). On bandlimited signals with fractional Fourier transform. IEEE Sig. Proc. Lett. 3(3): 72–74 · doi:10.1109/97.481159
[13] Zayed A.I. (1996). On the relationship between the Fourier and fractional Fourier transforms. IEEE Sig. Proc. Lett. 3(12): 310–311 · doi:10.1109/97.544785
[14] Candan C., Ozaktas H.M. (2003). Sampling and series expansion theorems for fractional Fourier and other transforms. Sig. Process. 83(11): 2455–2457 · Zbl 1145.94337 · doi:10.1016/S0165-1684(03)00196-8
[15] Gori F. (1981). Fresnel transform and sampling theorem. Opt. Com. 39: 293–297 · doi:10.1016/0030-4018(81)90094-8
[16] Stern A. (2006). Sampling of linear canonical transformed signals. Sig. Process. 87: 1421–1425 · Zbl 1172.94344 · doi:10.1016/j.sigpro.2005.07.031
[17] Kramer H.P. (1959). A generalized sampling theorem. J. Math. Phys. 38: 68–72 · Zbl 0196.31702
[18] Wigner E.P. (1932). On the quantum correlation for thermodynamic equilibrium. Phys. Rev. 40: 749–759 · JFM 58.0948.07 · doi:10.1103/PhysRev.40.749
[19] Cohen L. (1989). Time–frequency distributions–a review. Proc. IEEE 77: 941–981 · doi:10.1109/5.30749
[20] Pei S.C., Ding J.J. (2001). Relations between fractional operations and time–frequency distributions and their applications. IEEE Trans. Sig. Proc. 49: 1638–1655 · Zbl 1369.94258 · doi:10.1109/78.934134
[21] Whittaker M. (1929). The Fourier theory of the cardinal functions. Proc. Math. Soc. Edinburgh 1: 169–176 · JFM 54.0308.02 · doi:10.1017/S0013091500013511
[22] Grochenig K. (2001). Foundation of time–frequency analysis. Birkhauser, Boston
[23] Mallat, S.: A wavelet tour of signal processing, 2nd edn. Academic, San Diego, pp. 31–32 (1999) · Zbl 0998.94510
[24] Ozaktas H.M., Aytur O. (1995). Fractional Fourier domains. Sig. Process. 46: 119–124 · Zbl 0875.94037 · doi:10.1016/0165-1684(95)00076-P
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.