  Electroencephalography (EEG) stands as a crucial tool in neuroscientific
research and clinical diagnostics, providing valuable insights into the
electrical activities of the brain. Traditional EEG signal processing
techniques, predominantly linear and constrained to time-frequency analysis,
often fail to capture the intricate, dynamic nature of brain signals. This
paper introduces a tensor-based multi-dimensional framework for EEG signal
processing and denoising, aimed at overcoming the limitations of current
methods. Utilizing the advanced mathematical construct of tensors, this
framework allows for a more holistic representation and analysis of EEG data,
encompassing multiple dimensions such as time, electrode space, and frequency
bands. We propose innovative algorithms for multi-dimensional Fourier
transforms and adaptive thresholding, specifically tailored to address the
challenges of non-stationary noise and complex signal artifacts in EEG data.
The framework is further enriched with a time-slicing algorithm that
facilitates real-time analysis, crucial for applications like seizure detection
and brain-computer interfacing. Theoretical formulations and simulated
scenarios demonstrate the potential of this framework in significantly
enhancing the accuracy, efficiency, and speed of EEG signal processing. This
approach not only holds promise for advanced EEG analysis but also sets the
stage for future integrations with other neuroimaging modalities, paving the
way for comprehensive and nuanced understanding of brain function.
