As we develop the algebraic signal processing theory, we consider various aspects of linear signal processing besides transform algorithms and multidimensional transforms. Published results are collected below.
Jelena Kovacevic and Markus Püschel
Sampling Theorem Associated with the Discrete Cosine Transform
Proc. ICASSP 2006, Vol. 3
One way of deriving the discrete Fourier transform (DFT) is by equispaced sampling of periodic signals or signals on a circle. In this paper, we show that an analogous derivation can be used to obtain the DCT (type 2). To achieve this goal, we replace the circle by a line graph with symmetric boundary conditions, and define signal space, filter space, and filtering operation appropriately. Further, we derive the corresponding sampling theorem including the proper notions of "bandlimited'' and "sinc function'' in this case. The results show that, in a rigorous sense, the DCT is closely related to the DFT, and can be introduced without concepts from statistical signal processing as is the current practice.
Markus Püschel and Jelena Kovacevic
Real, Tight Frames with Maximal Robustness to Erasures
Proc. DCC 2005
Motivated by the use of frames for robust transmission over the Internet, we present a first systematic construction of real tight frames with maximum robustness to erasures. We approach the problem insteps: we first construct maximally robust frames by using polynomial transforms. We then add tightness as additional property with the help of orthogonal polynomials. Finally, we impose the last requirement of equal norm on the frame and construct, to our best knowledge, the first real, tight, equal norm frames that are maximally robust to erasures.