Log-Time Sampling of Signals: Zeta Transform
ABSTRACT
We introduce a general framework for the log-time sampling of continuous-time signals. We define the zeta transform based on the log-time sampling scheme, where the signal x(t)is sampled at time instants tn=Tlogn,n=1,2,....The zeta transform of the log-time sampled signals can be described by a linear combination of Riemann zeta function, which firmly joins the log-time sampling process to the number theory. The instantaneous sampling frequency of the log-sampled signal equals fn=n/T,n=1,2,..., i.e. it increases linearly with the sampling number. We describe the properties of the log-sampled signals and discuss several applications in nonuniform sampling schemes.
We introduce a general framework for the log-time sampling of continuous-time signals. We define the zeta transform based on the log-time sampling scheme, where the signal x(t)is sampled at time instants tn=Tlogn,n=1,2,....The zeta transform of the log-time sampled signals can be described by a linear combination of Riemann zeta function, which firmly joins the log-time sampling process to the number theory. The instantaneous sampling frequency of the log-sampled signal equals fn=n/T,n=1,2,..., i.e. it increases linearly with the sampling number. We describe the properties of the log-sampled signals and discuss several applications in nonuniform sampling schemes.
Cite this paper
nullH. Olkkonen and J. Olkkonen, "Log-Time Sampling of Signals: Zeta Transform," Open Journal of Discrete Mathematics, Vol. 1 No. 2, 2011, pp. 62-65. doi: 10.4236/ojdm.2011.12008.
nullH. Olkkonen and J. Olkkonen, "Log-Time Sampling of Signals: Zeta Transform," Open Journal of Discrete Mathematics, Vol. 1 No. 2, 2011, pp. 62-65. doi: 10.4236/ojdm.2011.12008.
References
[1] J. M. Borwein, D. M. Bradley and R. E. Crandall, “Computational Strategies for the Riemann zeta Function,” Journal of Computational and Applied Mathematics, Vol. 121, No.1-2, 2000, pp. 247-296. doi:10.1016/S0377-0427(00)00336-8 [2] M. Vetterli, P. Marziliano and T. Blu, “Sampling Signals with Finite Rate of Innovation,” IEEE Transactions on Signal Processing, Vol. 50, No. 6, 2002, pp.1417-1428. doi:10.1109/TSP.2002.1003065 [3] I. Maravic and M. Vetterli, “Sampling and Reconstruction of Signals with Finite Rate of Innovation in the Presence of Noise,” IEEE Transactions on Signal Processing, Vol. 53, No. 8, 2005, pp. 2788-2805. doi:10.1109/TSP.2005.850321 [4] J. T. Olkkonen and H. Olkkonen, “Reconstruction of Wireless UWB Pulses by Exponential Sampling Filter,” Wireless Sensor Network, Vol. 2, 2010, pp. 462-466. doi:10.4236/wsn.2010.26057 [5] H. Olkkonen and J. T. Olkkonen, “Measurement and Reconstruction of Transient Signals by Parallel Exponential Filters,” IEEE Transactions on. Circuits and Systems II, Vol. 57, No. 6, 2010, pp. 426-429. doi:10.1109/TCSII.2010.2048375
[1] J. M. Borwein, D. M. Bradley and R. E. Crandall, “Computational Strategies for the Riemann zeta Function,” Journal of Computational and Applied Mathematics, Vol. 121, No.1-2, 2000, pp. 247-296. doi:10.1016/S0377-0427(00)00336-8 [2] M. Vetterli, P. Marziliano and T. Blu, “Sampling Signals with Finite Rate of Innovation,” IEEE Transactions on Signal Processing, Vol. 50, No. 6, 2002, pp.1417-1428. doi:10.1109/TSP.2002.1003065 [3] I. Maravic and M. Vetterli, “Sampling and Reconstruction of Signals with Finite Rate of Innovation in the Presence of Noise,” IEEE Transactions on Signal Processing, Vol. 53, No. 8, 2005, pp. 2788-2805. doi:10.1109/TSP.2005.850321 [4] J. T. Olkkonen and H. Olkkonen, “Reconstruction of Wireless UWB Pulses by Exponential Sampling Filter,” Wireless Sensor Network, Vol. 2, 2010, pp. 462-466. doi:10.4236/wsn.2010.26057 [5] H. Olkkonen and J. T. Olkkonen, “Measurement and Reconstruction of Transient Signals by Parallel Exponential Filters,” IEEE Transactions on. Circuits and Systems II, Vol. 57, No. 6, 2010, pp. 426-429. doi:10.1109/TCSII.2010.2048375