Connections between signal processing and complex analysis
Keywords:
complex analysis, signal processing, analytic functions, residue theory, fourier analysis, riesz basesAbstract
Complex analysis plays a fundamental role in modern signal processing, offering a robust mathematical framework for representing and manipulating signals. Its applications span from sampling and interpolation to fourier and time-frequency analysis. In particular, tools such as riesz bases, frames, and analytic signals leverage complex variables to enhance signal reconstruction, compression, and noise reduction. The shannon-whittaker sampling theorem, deeply rooted in complex analysis, remains a cornerstone of digital signal theory, providing the foundation for signal digitization and reconstruction. Recent advancements have extended this interplay into domains like graph signal processing, neurophysiological data modeling, and complex-valued neural networks, indicating the growing influence of complex analysis in data-intensive applications (makowski et al., 2021; adali et al., 2021). Moreover, conformal mappings and residue theory are increasingly used to optimize filter design and understand pole-zero behavior in transfer functions. This paper explores the theoretical and practical connections between signal processing and complex analysis, highlighting core concepts and recent developments that underscore their synergy.
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