Algebraic Signal Processing Theory: 1-D Space

In our paper titled ldquoalgebraic signal processing theory: foundation and 1-D Timerdquo appearing in this issue of the
IEEE

Transactions

on

Signal

Processing
, we presented the algebraic signal processing theory, an axiomatic and general framework for linear signal processing. The basic concept in this theory is the signal model defined as the triple (
A
,
M
,Phi), where
A
is a chosen algebra of filters,
M
an associated
A
-module of signals, and Phi is a generalization of the
z
-transform. Each signal model has its own associated set of basic SP concepts, including filtering, spectrum, and Fourier transform. Examples include infinite and finite discrete time where these notions take their well-known forms. In this paper, we use the algebraic theory to develop infinite and finite
space
signal models. These models are based on a
symmetric
space shift operator, which is distinct from the standard time shift. We present the
space
signal processing concepts of filtering or convolution, ldquo
z
-transform,rdquo spectrum, and Fourier transform. For finite length
space
signals, we obtain 16 variants of space models, which have the 16 discrete cosine and sine transforms (DCTs/DSTs) as Fourier transforms. Using this novel derivation, we provide missing signal processing concepts associated with the DCTs/DSTs, establish them as precise analogs to the DFT, get deep insight into their origin, and enable the easy derivation of many of their properties including their fast algorithms.