Abstract
Multi-channel sampling for band-limited signals is fundamental in the theory of multi-channel parallel A/D environment and multiplexing wireless communication environment. As the fractional Fourier transform has been found wide applications in signal processing fields, it is necessary to consider the multi-channel sampling theorem based on the fractional Fourier transform. In this paper, the multi-channel sampling theorem for the fractional band-limited signal is firstly proposed, which is the generalization of the well-known sampling theorem for the fractional Fourier transform. Since the periodic nonuniformly sampled signal in the fractional Fourier domain has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived multi-channel sampling theorem and the specific space-shifting and phase-shifting properties of the fractional Fourier transform. Moreover, by designing different fractional Fourier filters, we can obtain reconstruction methods for other sampling strategies.
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References
Almeida L B. The fractional Fourier transform and time-frequency representations. IEEE Trans Signal Proc, 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
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 Proc, 2001, 49: 1638–1655
Ozaktas H M, Aytur O. Fractional Fourier domains. Signal Proc, 1995, 46: 119–124
Cariolaro G, Erseghe T, Kraniauskas P, et al. A unified framework for the fractional Fourier transform. IEEE Trans Signal Proc, 1998, 46: 3206–3219
Almeida L B. Product and convolution theorems for the fractional Fourier transform. IEEE Signal Proc Lett, 1997, 4: 15–17
Zayed A I. A convolution and product theorem for the fractional Fourier transform. IEEE Signal Proc Lett, 1998, 5: 101–103
Ozaktas H M, Arikan O, Kutay M A, et al. Digital computation of the fractional Fourier transform. IEEE Trans Signal Proc, 1996, 44: 2141–2150
Mendlovic D, Ozaktas H M, Lohmann A W. Fractional correlation. Appl Opt, 1995, 34: 303–309
Xia X. On bandlimited signals with fractional Fourier transform. IEEE Signal Proc Lett, 1996, 3: 72–74
Zayed A I. On the relationship between the Fourier transform and fractional Fourier transform. IEEE Signal Proc Lett, 1996, 3: 310–311
Zayed A I, Garcia A G. New Sampling formulae for the fractional Fourier transform. Signal Proc, 1999, 77: 111–114
Erseghe T, Kraniauskas P, Cariolaro G. Unified fractional Fourier transform and sampling theorem. IEEE Trans Signal Proc, 1999, 47(12): 3419–3423
Candan C, Ozaktas H M. Sampling and series expansion theorems for fractional Fourier and other transforms. Signal Proc, 2003, 83: 2455–2457
Zhang W Q, Tao R. Sampling theorems for bandpass signals with fractional Fourier transform. Acta Electron Sin (in Chinese), 2005, 33(7): 1196–1199
Torres R, Pellat-Finet P, Torres Y. Sampling theorem for fractional bandlimited signals: a self-contained proof. Application to digital holography. IEEE Signal Proc Lett, 2006, 13: 676–679
Tao R, B. Li Z, Wang Y. Spectral analysis and reconstruction for periodic nonuniformly sampled signals in fractional Fourier domain. IEEE Trans Signal Proc, 2007, 55(7): 3541–3547
Martone M. A multicarrier system based on the fractional Fourier transform for time-frequency selective channels. IEEE Trans Comm, 2001, 49(6): 1011–1020
Chen E Q, Tao R, Zhang W Q, et al. The OFDM system and equalization algorithm based on the fractional Fourier transform. Acta Electron Sin (in Chinese), 2007, 35(3): 409–414
Vaidyanathan P P, Liu V C. Classical sampling theorems in the context of multirate and polyphase digital filter band structures. IEEE Trans Signal Proc, 1988, 36(9): 1480–1495
Figueiras A R, Marino J B, Gomez R G. On generalized sampling expansions for deterministic signals. IEEE Trans Circuits and Systems, 1981, cas-28(2): 153–154
Papoulis A. Generalized sampling expansion. IEEE Trans Circuits and Systems, 1977, cas-24(11): 652–654
Jenq Y C. Digital spectra of nonuniformly sampled signals: Fundamentals and high-speed waveform digitizers. IEEE Trans Instrum Meas, 1988, 37(2): 245–251
Jenq Y C. Perfect reconstruction of digital spectrum from nonuniformly sampled signals. IEEE Trans Instrum Meas, 1997, 46(3): 649–652
Neagoe V E. Inversion of the Van der Monde matrix. IEEE Signal Proc Lett, 1996, 2: 119–120
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Supported partially 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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Zhang, F., Tao, R. & Wang, Y. Multi-channel sampling theorems for band-limited signals with fractional Fourier transform. Sci. China Ser. E-Technol. Sci. 51, 790–802 (2008). https://doi.org/10.1007/s11431-008-0087-8
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DOI: https://doi.org/10.1007/s11431-008-0087-8