Introduction

The purpose of this work is to develop a unifying formulation of continuous/discrete signal processing using exponential splines. In particular, we want to establish a mathematical link between continuous-time convolution operators, which are described by differential equations, and discrete-time filters, which are characterized by difference equations.

Main Contribution

Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we have proposed a complete and self-contained signal-processing formulation of exponential splines defined on a uniform grid. We have specified the corresponding B-spline basis functions and investigated their reproduction properties (Green function and exponential polynomials); we have also characterized their stability (Riesz bounds).

By interpreting the Green-function reproduction property of exponential splines in signal-processing terms, we have uncovered a fundamental relation that connects the impulse responses of all-pole analog filters to their discrete counterparts. The link is that the latter are the B-spline coefficients of the former (which happen to be exponential splines). Motivated by this observation, we have introduced an extended family of cardinal splines—the generalized E-splines—to generalize the concept for all convolution operators with rational transfer functions. We constructed the corresponding compactly supported B-spline basis functions which are characterized by their poles and zeros, thereby establishing an interesting connection with analog-filter design techniques. We have investigated the properties of these new B-splines and presented the corresponding signal-processing calculus, which allows us to perform continuous-time operations such as convolution, differential operators, and modulation, by simple application of the discrete version of these operators in the B-spline domain. In particular, we have shown how the formalism can be used to obtain exact, discrete implementations of analog filters. Finally, we have applied our results to the design of hybrid signal-processing systems that rely on digital filtering to compensate for the non-ideal characteristics of real-world A-to-D and D-to-A conversion systems.