Relationships between Convolution and Correlation for Fourier Transform and Quaternion Fourier Transform

In this paper we introduce convolution theorem for the Fourier transform (FT) of \ud two complex functions. We show that the correlation theorem for the FT can be \ud derived using properties of convolution. We develop this idea to derive the \ud correlation theorem for the quaternion Fourier transform (QFT) of the two \ud quaternion functions

Convolution and Correlation Based on Discrete Quaternion Fourier Transform

In this paper we present the generalized convolution and correlation for the two-dimensional discrete quaternion Fourier transforn (DQFT). We provide several new properties of the generalization. These results can be considered as the extensions of the correlation and convolution properties of real and complex Fourier transform to the DQFT domai

Two-Dimensional Quaternionic Windowed Fourier Transform

Signal processing is a fast growing area today and the desired effectiveness in utilization\ud of bandwidth and energy makes the progress even faster. Special signal processors have\ud been developed to make it possible to implement the theoretical knowledge in an efficient\ud way. Signal processors are nowadays frequently used in equipment for radio, transportation,\ud medicine, and production, etc.In this paper, by using the adjoint operator of the (right-sided) QFT, we derive the Plancherel\ud theorem for the QFT. We apply it to prove the orthogonality relation and reconstruction\ud formula of the two-dimensional quaternionic windowed Fourier transform (QWFT). Our\ud results can be considered as an extension and continuation of the previous work of Mawardi\ud et al. (2008).We then present several examples to show the differences between the QWFT and\ud the WFT. Finally, we present a generalization of the QWFT to higher dimensions

Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT

Fourier Multiplier Extension Using Quaternionic Windowed Transform

We begin with an introduction of the quaternionic windowed Fourier\ud transform (QWFT) and define Fourier multiplier for the QWFT to\ud investigate some of its important properties

Construction of Quaternion-Valued Wavelets

In this paper, we introduce quaternion-valued wavelets in the context of\ud the duplex matrix-valued function. We then formulate quaternion scaling and wavelet\ud functions using quaternion multiresolution analysis (QMRA).With these formulations,\ud we obtain coefficients of highpass and lowpass filters of QMR

Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework

Quaternion Algebra-Valued Wavelet Transform

In the previous paper [7], we proposed the two-dimensional continuous\ud quaternion wavelet transform (CQWT). In the present paper,\ud using the orthogonality of harmonic exponential functions we give an\ud alternative proof for inner product relation property of the CQWT