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Convolution and Fourier transform operators are key concepts of signal processing, providing essential tools for analyzing linear time-invariant (LTI) systems in the time-domain and frequency-domain, respectively. These fundamental operators have been instrumental in advancing the field of signal processing and image processing. The traditional LTI systems theory deals with complex-valued (CV) time series signals. The CV signals representation follows linearity with CV scalars exhibiting an inbuilt rotation-invariance (RI) property. To acknowledge the presence of RI in CV signals, the CV product in the conventional convolution definition is interpreted in terms of a scale rotation. It is important to note that replacing the CV scalars with real scalars results in a loss of the rotational invariance property. We introduce a linear rotation-invariant time-invariant (LRITI) system with vector-valued (VV) signals. We develop an analogous theory to characterize LRITI systems using VV signals with a new tool called geometric algebra (GA). We define the RI property for VV systems using GA where only real numbers are considered as scalars. To begin with the proposed GA-based formulation, we generalize the convolution operation for VV systems using rotor representation. In addition, we provide a compatible frequency-domain analysis for VV signals and LRITI systems. First, VV bivector exponential signals are shown to be eigen-functions of LRITI systems. A Fourier transform is defined, and we propose two generalized definitions of frequency response: the first valid for bivector exponential 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.