Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory

In a previous work, we identified the importance of rotation invariance in the standard complex-valued theory of linear time-invariant (LTI) systems and proposed a generalized vector-valued (VV) definition of convolution that reinterprets the complex-valued product of the traditional formalism as a scale rotation within the framework of geometric algebra. Based on this convolution definition, we characterized linear rotation-invariant time-invariant (LRITI) systems by defining and using a VV impulse response—effectively generalizing time-domain analysis for VV signals and LRITI systems. In this work, we provide a compatible frequency-domain analysis for VV signals and LRITI systems. First, VV bivector exponential signals are shown to be eigenfunctions of LRITI systems. A Fourier transform is defined, and we propose two generalized definitions of frequency response: the first valid for bivector exponentials in an arbitrary plane and the second valid for a general signal decomposed into a set of totally orthogonal planes (TOPs). Finally, we establish a convolution property for the Fourier transform with respect to TOPs. Together, these results provide compatible time-domain and frequency-domain analyses, thereby enabling a more comprehensive analysis of VV signals and LRITI systems.