The Algebraic Structure in Signal Processing: Time and Space

The assumptions underlying linear signal processing (SP) produce more structure than vector spaces. We capture this structure by describing the space of filters as an algebra and the space of signals as the associated module. We formulate an algebraic approach to SP that is axiomatically based on the concept of a signal model. Signal models for time are visualized as directed graphs. We construct corresponding models for undirected graphs, which we hence call space models, and show that, in particular, the 16 DCTs and DSTs are Fourier transforms for these finite space models. Finally, we discuss the extension of our theory to separable and nonseparable 2-DSP

[1]  José M. F. Moura,et al.  The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms , 2003, SIAM J. Comput..

[2]  José M. F. Moura,et al.  DCT/DST and Gauss-Markov fields: conditions for equivalence , 1998, IEEE Trans. Signal Process..

[3]  A.K. Krishnamurthy,et al.  Multidimensional digital signal processing , 1985, Proceedings of the IEEE.

[4]  Markus Püschel,et al.  Cooley-Tukey FFT like algorithm for the discrete triangle transform , 2004 .

[5]  M. Puschel,et al.  Fourier transform for the spatial quincunx lattice , 2005, IEEE International Conference on Image Processing 2005.

[6]  M. Puschel,et al.  The discrete triangle transform , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[7]  P. Yip,et al.  Discrete Cosine Transform: Algorithms, Advantages, Applications , 1990 .

[8]  J. Dicapua.  Chebyshev Polynomials , 2019, Fibonacci and Lucas Numbers With Applications.