New Formulas for Irregular Sampling of Two-Bands Signals

Author(s)
Bernard Lacaze

ABSTRACT

Many sampling formulas are available for processes in baseband (-a,a) at the Nyquist rate a/π. However signals of telecommunications have power spectra which occupate two bands or more. We know that PNS (periodic non-uniform sampling) allow an errorless reconstruction at rate smaller than the Nyquist one. For instance PNS2 can be used in the two-bands case (-a,-b)∪(b,a) at the Landau rate (a-b)/π We prove a set of formulas which are available in cases more general than the PNS2. They take into account two sampling sequences which can be periodic or not and with same mean rate or not.

Many sampling formulas are available for processes in baseband (-a,a) at the Nyquist rate a/π. However signals of telecommunications have power spectra which occupate two bands or more. We know that PNS (periodic non-uniform sampling) allow an errorless reconstruction at rate smaller than the Nyquist one. For instance PNS2 can be used in the two-bands case (-a,-b)∪(b,a) at the Landau rate (a-b)/π We prove a set of formulas which are available in cases more general than the PNS2. They take into account two sampling sequences which can be periodic or not and with same mean rate or not.

Cite this paper

nullB. Lacaze, "New Formulas for Irregular Sampling of Two-Bands Signals,"*Journal of Signal and Information Processing*, Vol. 2 No. 4, 2011, pp. 253-256. doi: 10.4236/jsip.2011.24035.

nullB. Lacaze, "New Formulas for Irregular Sampling of Two-Bands Signals,"

References

[1] H. J. Landau, “Sampling, Data Transmission, and the Nyquist Rate,” Proceedings of the IEEE, Vol. 55, No. 10, 1967, pp. 1701-1706. doi:10.1109/PROC.1967.5962

[2] B. Lacaze, “About Bi-Periodic Samplings,” Sampling Theory in Signal and Image Processing, Vol. 8, No. 3, 2009, pp. 287-306.

[3] B. Lacaze, “Equivalent Circuits for the PNS2 Sampling Scheme,” IEEE Circuits and Systems, Vol. 57, No. 11, 2010, pp. 2904-2914. doi:10.1109/TCSI.2010.2050228

[4] A. J. Jerri, “The Shannon Sampling Theorem. Its Various Extensions and Applications. A Tutorial Review,” Proceedings of the IEEE, Vol. 65, No. 11, 1977, pp. 1565-1596. doi:10.1109/PROC.1977.10771

[5] B. Lacaze, “Reconstruction Formula for Irregular Sampling,” Sampling Theory in Signal and Image Processing, Vol. 4, No. 1, 2005, pp. 33-43.

[6] B. Lacaze, “The Ghost Sampling Sequence Method,” Sampling Theory in Signal and Image Processing, Vol. 8, No. 1, 2009, pp. 13-21.

[7] H. Cramer and M. R. Leadbetter, “Stationary and Related Stochastic Processes,” Wiley, New York, 1966.

[8] A. Papoulis, “Signal Analysis,” McGraw Hill, New York, 1977.

[9] J. R. Higgins, “A Sampling Theorem for Irregular Sample Points,” IEEE Transactions on Information Theory, Vol. 22, No. 5, 1976, pp. 621-622. doi:10.1109/TIT.1976.1055596

[10] J. R. Higgins, “Some Gap Sampling Series for Multiband Signals,” Signal Processing, Vol. 12, No. 3, 1987, pp. 313-319. doi:10.1016/0165-1684(87)90100-9

[11] J. L. Yen, “On Nonuniform Sampling of Bandwidth-Limited Signals,” IRE Transactions on Circuit Theory, Vol. 3, No. 4, 1956, pp. 251-257.

[12] B. Lacaze, “About a Multiperiodic Sampling Scheme,” IEEE Signal Processing Letters, Vol. 6, No. 12, 1999, pp. 307-308. doi:10.1109/97.803430

[13] B. Lacaze, “A Theoretical Exposition of Stationary Processes Sampling,” Sampling Theory in Signal and Image Processing, Vol. 4, No. 3, 2005, pp. 201-230.

[14] K. Seip, “An Irregular Sampling Theorem for Functions Bandlimited in a Generalized Sense,” SIAM Journal on Applied Mathematics, Vol. 47, No. 5, 1987, pp. 1112-1116. doi:10.1137/0147073

[15] A. I. Zayed and P. L. Butzer, “Lagrange Interpolation and Sampling Theorems,” In: F. Marvasti, Ed., Nonuniform Sampling: Theory and Practice, Kluwer Academic, 2001, pp. 123-168. doi:10.1007/978-1-4615-1229-5_3

[16] Y.-P. Lin and P. P. Vaidyanathan, “Periodically Nonuniform Sampling of Bandpass Signals,” IEEE Transactions on Circuits and Systems II, Vol. 45, No. 3, 1998, pp. 340-351.

[17] R. Venkataramani, “Perfect Reconstruction Formulas and Bounds on Aliasing Error in Sub-Nyquist Nonuniform Sampling of Multibands Signals,” IEEE Transactions on Information Theory, Vol. 46, No. 6, 2000, pp. 2173-2183.

[1] H. J. Landau, “Sampling, Data Transmission, and the Nyquist Rate,” Proceedings of the IEEE, Vol. 55, No. 10, 1967, pp. 1701-1706. doi:10.1109/PROC.1967.5962

[2] B. Lacaze, “About Bi-Periodic Samplings,” Sampling Theory in Signal and Image Processing, Vol. 8, No. 3, 2009, pp. 287-306.

[3] B. Lacaze, “Equivalent Circuits for the PNS2 Sampling Scheme,” IEEE Circuits and Systems, Vol. 57, No. 11, 2010, pp. 2904-2914. doi:10.1109/TCSI.2010.2050228

[4] A. J. Jerri, “The Shannon Sampling Theorem. Its Various Extensions and Applications. A Tutorial Review,” Proceedings of the IEEE, Vol. 65, No. 11, 1977, pp. 1565-1596. doi:10.1109/PROC.1977.10771

[5] B. Lacaze, “Reconstruction Formula for Irregular Sampling,” Sampling Theory in Signal and Image Processing, Vol. 4, No. 1, 2005, pp. 33-43.

[6] B. Lacaze, “The Ghost Sampling Sequence Method,” Sampling Theory in Signal and Image Processing, Vol. 8, No. 1, 2009, pp. 13-21.

[7] H. Cramer and M. R. Leadbetter, “Stationary and Related Stochastic Processes,” Wiley, New York, 1966.

[8] A. Papoulis, “Signal Analysis,” McGraw Hill, New York, 1977.

[9] J. R. Higgins, “A Sampling Theorem for Irregular Sample Points,” IEEE Transactions on Information Theory, Vol. 22, No. 5, 1976, pp. 621-622. doi:10.1109/TIT.1976.1055596

[10] J. R. Higgins, “Some Gap Sampling Series for Multiband Signals,” Signal Processing, Vol. 12, No. 3, 1987, pp. 313-319. doi:10.1016/0165-1684(87)90100-9

[11] J. L. Yen, “On Nonuniform Sampling of Bandwidth-Limited Signals,” IRE Transactions on Circuit Theory, Vol. 3, No. 4, 1956, pp. 251-257.

[12] B. Lacaze, “About a Multiperiodic Sampling Scheme,” IEEE Signal Processing Letters, Vol. 6, No. 12, 1999, pp. 307-308. doi:10.1109/97.803430

[13] B. Lacaze, “A Theoretical Exposition of Stationary Processes Sampling,” Sampling Theory in Signal and Image Processing, Vol. 4, No. 3, 2005, pp. 201-230.

[14] K. Seip, “An Irregular Sampling Theorem for Functions Bandlimited in a Generalized Sense,” SIAM Journal on Applied Mathematics, Vol. 47, No. 5, 1987, pp. 1112-1116. doi:10.1137/0147073

[15] A. I. Zayed and P. L. Butzer, “Lagrange Interpolation and Sampling Theorems,” In: F. Marvasti, Ed., Nonuniform Sampling: Theory and Practice, Kluwer Academic, 2001, pp. 123-168. doi:10.1007/978-1-4615-1229-5_3

[16] Y.-P. Lin and P. P. Vaidyanathan, “Periodically Nonuniform Sampling of Bandpass Signals,” IEEE Transactions on Circuits and Systems II, Vol. 45, No. 3, 1998, pp. 340-351.

[17] R. Venkataramani, “Perfect Reconstruction Formulas and Bounds on Aliasing Error in Sub-Nyquist Nonuniform Sampling of Multibands Signals,” IEEE Transactions on Information Theory, Vol. 46, No. 6, 2000, pp. 2173-2183.