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 <i>x</i>(<i>t</i>)is sampled at time instants <i>t_n</i>=<i>T</i>log<i>n</i>,<i>n</i>=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 <i>f_n</i>=<i>n/T</i>,<i>n</i>=1,2,..., <i>i.e</i>. it increases linearly with the sampling number. We describe the properties of the log-sampled signals and discuss several applications in nonuniform sampling schemes.