This work explores the versatility of complex-valued signals present in data representation in diverse applications. Integrating complex values into signals and processing structures proves powerful in system modeling, demonstrating effectiveness in many scenarios. The inherent mathematical efficiency of complex numbers enhances both elegance and practical utility, simplifying calculations and expanding capabilities. However, the use of complex notation introduces considerations related to the intrinsic geometry of the data. This tutorial unravels the theoretical foundations and practical applications in adaptive filtering of complex-valued signals, emphasizing the importance of understanding the displayed profiles of the studied signals and data. Using complex augmented notation and widely linear processing, superior performance is attainable in specific cases, providing valuable information to readers interested in signal processing.
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