B. Mawardi, E.S.M. Hitzer, R. Ashino, R. Vaillancourt

**Windowed Fourier transform of two-dimensional quaternionic signals**

*Appl. Math. and Computation*,
216, Iss. 8, pp. 2366-2379, 15 June 2010.

**Abstract:**
In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-valued signals, called the quaternionic windowed Fourier transform (QWFT). Using the spectral
representation of the quaternionic Fourier transform (QFT), we derive several important prop-
erties 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.

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