Exponential Splines for Signal Processing 
Investigator: Michael Unser 

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 continuoustime convolution operators, which are described by differential equations, and discretetime filters, which are characterized by difference equations. 

Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we have proposed a complete and selfcontained signalprocessing formulation of exponential splines defined on a uniform grid. We have specified the corresponding Bspline basis functions and investigated their reproduction properties (Green function and exponential polynomials); we have also characterized their stability (Riesz bounds).
By interpreting the Greenfunction reproduction property of exponential splines in signalprocessing terms, we have uncovered a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts. The link is that the latter are the Bspline 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 Esplines—to generalize the concept for all convolution operators with rational transfer functions. We constructed the corresponding compactly supported Bspline basis functions which are characterized by their poles and zeros, thereby establishing an interesting connection with analogfilter design techniques. We have investigated the properties of these new Bsplines and presented the corresponding signalprocessing calculus, which allows us to perform continuoustime operations such as convolution, differential operators, and modulation, by simple application of the discrete version of these operators in the Bspline 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 signalprocessing systems that rely on digital filtering to compensate for the nonideal characteristics of realworld AtoD and DtoA conversion systems. 

Collaboration: Dr. Thierry Blu 


Funding: Grant 200020101821 from the Swiss National Science Foundation 



