Cardinal Exponential Splines: Part II—Think Analog,
Michael A. Unser · 2005
Abstract— By interpreting the Green-function reproductionproperty of exponential splines in signal processing terms, we un-cover a fundamental relation that connects the impulse responsesof allpole analog filters to their discrete counterparts. The link isthat the latter are the B-spline coefficients of the former (whichhappen to be exponential splines). Motivated by this observation,we introduce an extended family of cardinal splines—the gen-eralized E-splines—to generalize the concept for all convolutionoperators with rational transfer functions. We construct the cor-responding compactly supported B-spline basis functions, whichare characterized by their poles and zeros, thereby establishingan interesting connection with analog filter design techniques. Weinvestigate the properties of these new B-splines and present thecorresponding signal processing calculus, which allows us to per-form continuous-time operations, such as convolution, differentialoperators, and modulation, by simple application of the discreteversion of these operators in the B-spline domain. In particular,we show how the formalism can be used to obtain exact, discreteimplementations of analog filters. Finally, we apply our resultsto the design of hybrid signal processing systems that rely ondigital filtering to compensate for the nonideal characteristicsof real-world analog-to-digital (A-to-D) and D-to-A conversionsystems.