Windowed Fourier transform of two-dimensional quaternionic signals

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

• Type: Journal
• Year: 2010
• Where: AMC
• Authors: Mawardi Bahri, Eckhard S. M. Hitzer, Ryuichi Ashino, Rémi Vaillancourt
• Keywords: AMC 2010, Heisenberg Type Uncertainty, Quaternionic Fourier Transform, Type Uncertainty Principle