Spectral Transforms
Complex analysis provides immense power and elegance in the analysis of linear time invariant systems. Hamilton's hypercomplex, or quaternion, extension to the complex numbers provides a means to analyze systems whose dynamics can be described by a system of partial differential equations. Complex spectral transformations associate linear time-invariant systems of ordinary differential equations with the geometry of a plane. The hypercomplex spectral transformations associate two-dimensional linear time-invariant (2D-LTI) systems of partial differential equations with the geometry of a sphere. These extended spectral transformations provide an exquisite tool for the analysis of 2D-LTI systems.
In my Ph.D. thesis, the calculus of a class of quaternionic functions, which provides the basis for defining hypercomplex extensions to complex spectral transformations, were defined. Many of the useful properties of complex valued functions, such as analytic, regular, and continuously differentiable, are extended by this class of quaternionic functions. A commutative quaternion algebra is also developed to overcome the noncommutative limitation of Hamilton's quaternionic division algebra.
Two hypercomplex spectral transforms were defined; a quaternionic Laplace and Fourier transform. Both provide many useful operational formulae properties: analogous extensions of the standard complex transformations, and a closed algebra. The two dimensional unit step, unit impulse and sinusoidal stimuli have similar transforms to their one-dimensional counterparts. The Quaternion-Laplace transform provides a generalized root-locus analysis technique. Likewise, the Quaternion-Fourier transform provides a generalized gain-phase frequency response analysis technique. This transform shows utility in the algebraic reduction of 2D-LTI systems described by the double convolution of their Green's functions. The Quaternion-Fourier transform of the double convolution results in a commutative quaternion algebraic product. 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.