Think Analog-Act Digital
Prof. Michael Unser, Biomedical Imaging Group, EPFL

Seminar • 07 January 2005 • CO-017

Abstract
By interpreting the Green-function reproduction property of exponential splines in signal-processing terms, we uncover 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 introduce an extended family of cardinal splines—the generalized E-splines to generalize the concept for all convolution operators with rational transfer functions. We construct 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 investigate the properties of these new B-splines and present 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 show how the formalism can be used to obtain exact, discrete implementations of analog filters. We also apply 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. The seminar will be followed by an aperitif and a slide show on the speaker's recent trip to India in BM 5.202.