A sampling theorem for non-bandlimited signals using generalized sinc functions. (English) Zbl 1155.94359
Summary: A ladder shaped filter of two real parameters \(a_{1},a_{2}\in ( - 1,1)\) is introduced in this note. The impulse response of the corresponding Linear Time Invariant (LTI) system is a generalized Sinc function of two parameters. Consequently a generalized Shannon-type sampling theorem is established for a class of non-bandlimited signals with special spectrum properties associated with a ladder shaped filter of two parameters. Finally, a mathematical characterization for the class of non-bandlimited signals satisfying the generalized sampling theorem is offered. These signals are restrictions to the real line of certain analytic functions in stripped domains symmetric about the real axis in the complex plane. For these signals, their spectra in higher frequency bands are measured by the spectrum of their base bands.
MSC:
| 94A20 | Sampling theory in information and communication theory |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
Keywords:
ladder shaped filter of two parameters; sampling theorem; non-bandlimited signals; generalized sinc function; analytic extensionReferences:
| [1] | Qian, T., Unit analytic signals and harmonic measures, J. Math. Anal. Appl., 314, 526-536 (2006) · Zbl 1082.94006 |
| [2] | Qian, T.; Chen, Q. H.; Li, L. Q., Analytic unit quadrature signals with nonlinear phase, Physica D, 203, 1-2, 80-87 (2005) · Zbl 1070.94504 |
| [3] | Boashash, B., Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals, Proc. IEEE, 80, 417-430 (1992) |
| [4] | Bedrosian, E., A product theorem for Hilbert transform, Proc. IEEE, 51, 868-869 (1963) |
| [5] | Cohen, L., Time-Frequency Analysis (1995), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ |
| [6] | Huang, N. E., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. London A, 454, 903-995 (1998) · Zbl 0945.62093 |
| [7] | Garnett, J. B., Bounded Analytic Functions (1987), Academic Press |
| [8] | Huang, N. E.; Shen, Z.; Long, S. R., A new view of nonlinear water waves: The Hilbert spectrum, Annu. Rev. Fluid Mech., 31, 417-457 (1999) |
| [9] | Meyer, Y., Ondelettes et Operateurs (1990), Hermann · Zbl 0694.41037 |
| [10] | Nuttall, A. H., On the quadrature approximation to the Hilbert transform of modulated signals, Proc. IEEE (Letters), 1458-1459 (1966) |
| [11] | Picinbono, B., On instantaneous amplitude and phase of signals, IEEE Trans. Signal Process., 45, 3, 552-560 (1997) |
| [12] | Seip, K., (Interpolation and Sampling in Spaces of Analytic Functions. Interpolation and Sampling in Spaces of Analytic Functions, University Lecture Series, vol.33 (2004), American Mathematical Society) · Zbl 1057.30036 |
| [13] | Chen, Q. H.; Li, L. Q.; Qian, T., Two families of unit analytic signals with nonlinear phase, Physica D, 221, 1-12 (2006) · Zbl 1104.94004 |
| [14] | Qian, Tao; Chen, Qiuhui, Characterization of analytic phase signals, Comput. Math. Appl., 51, 1471-1482 (2006) · Zbl 1138.94007 |
| [15] | Chen, Q. H.; Li, L. Q.; Qian, T., Time-frequency aspects of nonlinear Fourier atoms, Appl. Numer. Harmon. Anal., 303-313 (2006) |
| [16] | Chen, Qiuhui; Li, Luoqing; Qian, Tao, Stability of frames generated by nonlinear Fourier atoms, Internat. J. Wavelets Multiresolution Inform. Process., 3, 465-476 (2005) · Zbl 1086.42018 |
| [17] | Qian, T., Characterization of boundary values of functions in Hardy spaces with applications in signal analysis, J. Integral Equations Appl., 17, 2, 159-198 (2005) · Zbl 1086.30035 |
| [18] | Q. Chen, T. Qian, Sampling theorems of non-bandlimited signals based on non-Fourier atoms 2007, manuscript; Q. Chen, T. Qian, Sampling theorems of non-bandlimited signals based on non-Fourier atoms 2007, manuscript |
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.
