Convolution Structure in the Quaternion Simplified Fractional Fourier Transform Domain

Abstract

Quaternions are an extension of the complex number system and hold significant importance in both mathematics and physics. They are widely applied in areas such as computer animation, where they facilitate smooth rotational transformations, and in signal processing and filter design. In image processing, quaternion-based convolution has gained attention as an effective approach for processing multidimensional data. This paper introduces the concept of the fractional quaternion simplified fractional Fourier transform. Furthermore, a new convolution structure associated with this transform is developed and analyzed. Several fundamental properties of the proposed convolution are established, including linearity, shifting, associativity, distributivity, and conjugation. Furthermore, the study highlights how this generalized transform enhances the flexibility and efficiency of quaternion-based signal representation. The results provide a theoretical foundation that can be applied to advanced problems in image processing, communication systems, and applied mathematics where multidimensional data handling is required.