Algebraic Foundations of Generalized Signal Processing: A Unified Approach Across Domains

Year : 2025 | Volume : 15 | Issue : 03 | Page : 33 44
    By

    V. Basil Hans,

  1. Professor, Department of Management & Commerce, Srinivas University in Mangalore, Karnataka, India

Abstract

Using the techniques of algebra, notably polynomial algebras and modules, algebraic signal processing (ASP) is a contemporary, abstract framework that generalizes conventional signal processing— including Fourier analysis, filtering, and convolution. The notion is to use algebraic structures to explain signals, systems, and transformations such that ideas may be understood and generalized across many domains, including time, space, graph, or group. A unifying theoretical framework called ASP generalizes classical signal processing utilizing algebraic structures, especially polynomial algebras and modules. ASP models signal as module elements over a selected algebra; filters are shown as algebra elements operating on the signal space. A systematic approach to signal operations like shifting, filtering, and spectral analysis in a wide spectrum of domains—including time, space, and graphs—is made possible by this abstract concept. By use of module decomposition and representation theory, the framework offers a profound understanding of the design of Fourier-like transforms, therefore enabling innovative ideas in areas including graph signal processing, image analysis, and multidimensional data representation. ASP not only generalizes classical methods but also helps to create novel transforms suited to uneven or non-Euclidean structures by capturing the core of signal processing in algebraic terms.

Keywords: Graph signal processing, spectral analysis, Fourier transform, signal modules, polynomial algebras

[This article belongs to Current Trends in Signal Processing ]

How to cite this article:
V. Basil Hans. Algebraic Foundations of Generalized Signal Processing: A Unified Approach Across Domains. Current Trends in Signal Processing. 2025; 15(03):33-44.
How to cite this URL:
V. Basil Hans. Algebraic Foundations of Generalized Signal Processing: A Unified Approach Across Domains. Current Trends in Signal Processing. 2025; 15(03):33-44. Available from: https://journals.stmjournals.com/ctsp/article=2025/view=228400


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Regular Issue Subscription Review Article
Volume 15
Issue 03
Received 10/05/2025
Accepted 24/06/2025
Published 26/09/2025
Publication Time 139 Days
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