Windowed Fourier transform of two-dimensional quaternionic signals

In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-\ud valued signals, called the quaternionic windowed Fourier transform (QWFT). Using\ud the spectral representation of the quaternionic Fourier transform (QFT), we derive several\ud important properties such as reconstruction formula, reproducing kernel, isometry, and\ud orthogonality relation. Taking the Gaussian function as window function we obtain quaternionic\ud Gabor filters which play the role of coefficient functions when decomposing\ud the signal in the quaternionic Gabor basis. We apply the QWFT properties and the\ud (right-sided) QFT to establish a Heisenberg type uncertainty principle for the QWFT.\ud Finally, we briefly introduce an application of the QWFT to a linear time-varying system

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

Relaxed quaternionic Gabor expansions at critical density

Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion-valued signal f in the Wiener space can be expanded into a unique l2 series on a lattice at critical density 1, provided one more point is added in the middle of a cell. We call that a relaxed Gabor expansion

Some results on the lattice parameters of quaternionic Gabor frames

Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture do not hold true in the quaternionic case