Rotor-Based Convolution for Vector-Valued Signals

Traditional linear time-invariant (LTI) systems theory considers systems in which the input and output signals are complex-valued (CV) time-series. We develop an analogous theory within the framework of geometric algebra (GA) to consider linear rotation-invariant time-invariant (LRITI) systems in which the input and output signals are vector-valued (VV) time-series. To that end, we identify the importance of the rotation-invariance in the standard formalism, and point out its inheritance in the standard formalism due to the assumption of linearity with CV scalars. We reinterpret the CV product in the traditional convolution definition as a scale-rotation rather than a product. Based on this reinterpretation, we characterize LRITI systems by defining a convolution operation for VV systems which generalizes the convolution operation using rotors in a GA-based representation.