E. Hitzer,
S. J. Sangwine

**The orthogonal planes split of quaternions**

In K. Guerlebeck (ed.),
Electronic Proceedings of
*The 9th International Conference on Clifford Algebras and their Applications
in Mathematical Physics (ICCA9), 15-10 July 2011, Weimar, Germany*,
(2011).

**Abstract:**
The two-sided quaternionic Fourier transformation (QFT) was introduced in [2]
for the analysis of 2D linear time-invariant partial-differential systems. In further theoretical
investigations [3, 4] a special split of quaternions was introduced, then called plus minus split. In the
current paper we analyze this split further, interpret it geometrically as orthogonal planes split
(OPS), and generalize it. The new general form of the OPS split allows us to find new geometric
interpretations for the action of the QFT on the signal. The second major result of this work is
a variety of new forms of the QFT, their geometric interpretation, and for each form OPS split
theorems, which allow fast and efficient numerical implementation with standard FFT software.

**Keywords:** Quaternion signals, orthogonal planes split, quaternion Fourier transformations,
geometric interpretation, fast implementations.

[2] T. A. Ell,
*Quaternionic-Fourier Transform for Analysis of Two-dimensional Linear Time-
Invariant Partial Differential Systems*,
in Proceedings of the 32nd IEEE Conference on
Decision and Control, December 15-17, 2 (1993), 1830-1841.

[3] E. Hitzer,
*Quaternion Fourier Transform on Quaternion Fields and Generalizations*,
Adv. in App. Cliff. Alg., 17 (2007), 497-517.

[4] E. Hitzer,
*Directional Uncertainty Principle for Quaternion Fourier Transforms*,
Adv. in App. Cliff. Alg., 20(2) (2010), 271-284.

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