Algebraic Signal Processing Theory: 1-D Space

Abstract

Abstract-In our paper titled "Algebraic Signal Processing Theory: Foundation and 1-D Time" 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 ( 8), where is a chosen algebra of filters, an associated -module of signals, and 8 is a generalization of the -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, " -transform," 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.

Algebraic Signal Processing Theory: Foundation and 1-D Time

“…Thus, imposing only shift-invariance, we have identified one of the key players in ASP, namely polynomial algebras. Indeed, as we will see, they provide the underlying structure for finite time SP and the DFT (Section IV) and for finite space SP and the DCTs/DSTs [1].…”

“…Definition 1 makes it possible to apply different signal models to the same vector of numbers. For example, application of a DFT or a DCT to compute the spectrum of a finite-length vector implicitly adopts different signal models for this vector (Section V-B and [1]). …”

“…We also understand the necessity for boundary conditions, the impact of different choices, and how they relate to the problem of signal extension. We use this methodology to derive 1-D and 2-D space signal models in [1] and [3].…”

“…For other finite discrete time models, the Fourier transform becomes the real DFT (RDFT) or the discrete Hartley transform (DHT). In [1], we consider shift-invariant signals indexed by 1-D lattices; we call them 1-D space signals. In their signal models, the polynomial in is a Chebychev polynomial and their Fourier transforms are the 16 variants of the discrete trigonometric transforms (DTTs), e.g., the discrete cosine transform (DCT) and the discrete sine transform (DST).…”

Multidimensional wave field signal theory: Mathematical foundations

“…This would be particularly powerful as it enables the application of a vast array of results from a signal processing point of view and in particular, recent novel results from algebraic signal processing [11][12][13][14][15][16] will find ground-breaking applications.…”