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Hamilton's hypercomplex, or quaternion, extension to the complex numbers provides a means to algebraically analyze systems whose dynamics can be described by a system of partial differential equations. The Quaternion-Fourier transformation, defined in this work, associates two dimensional linear time-invariant (2D-LTI) systems of partial differential equations with the geometry of a sphere. This transform provides a generalized gain-phase frequency response analysis technique. It shows full utility in the algebraic reduction of 2D-LTI systems described by the double convolution of their Green's functions. The standard two dimensional complex Fourier transfer function has a phase associated with each frequency axis and does not describe clearly how each axis interacts with the other. The Quaternion-Fourier transfer function gives an exact measure of this interaction by a single phase angle that may be used as a measure of the relative stability of the system. This extended Fourier transformation provides an exquisite tool for the analysis of 2D-LTI systems.>
Todd A. Ell (Mon,) studied this question.