Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach

The field of quaternions has made a substantial impact on signal processing research, with numerous studies exploring their applications. Building on this foundation, this article extends the study of sampling theory using the quaternion fractional Fourier Transform (QFRFT). We first propose a generalized sampling expansion (GSE) for fractional bandlimited signals via the QFRFT, extending the classical Papoulis expansion. Next, we design fractional quaternion Fourier filters to reconstruct both the signals and their derivatives, based on the GSE and QFRFT properties. We illustrate the practical utility of the QFRFT-based GSE framework with a case study on signal denoising, demonstrating its effectiveness in noise reduction with the Mean Squared Error (MSE), highlighting the improvement in signal restoration.