Analysis of an Adaptive Sampler Based on Weber's Law
Weber's law suggests a logarithmic relationship between perceptual stimuli and human perception. The Weber sampler is an adaptive, nonuniform sampling mechanism that exploits Weber's law to sample the signal at a minimum rate without significant perceptual degradation. In this paper, we introduce and analyze a regularized Weber sampler for smooth deterministic signals as well as smooth random processes. While analysis for a fixed Weber constant δ is analytically intractable except for some special cases, we exploit the fact that the Weber constant is usually small. Under suitable assumptions, for deterministic signals, we provide analytical approximations to the number of samples in a unit time interval and the intersample times. For a random signal, we give analytical approximations to the expectation and probability distribution function of the respective quantities. Our class of deterministic as well as random signals is quite wide, and in particular covers bandlimited signals and signals satisfying second order ordinary differential equations. We also present a number of simulations to demonstrate that our approximations are good up to a Weber constant of 0.2, which is the regime of practical interest. Our results provide a source model for the Weber sampler, which can be used in the study of transmission of perceptual stimuli signals over communication links.