# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 1 Introduction  
The study of linear time-invariant (LTI) systems plays an important role in various engineering fields [1, 2]. The most commonly studied LTI systems are likely the complex-valued (CV) LTI systems, which enable a broad area of applications in signal analysis including filtering, system analysis, correlation, signal reconstruction, smoothing, compression, and time-frequency analysis, among others [3, 4, 5, 6, 7]. Within LTI systems analysis, convolution and the Fourier transform (FT) play important roles--convolution along with the impulse response helps to characterize LTI systems in the time domain, while the FT with the frequency response helps to characterize LTI systems in the frequency domain. Several application-driven forms of extended or generalized convolution exist in the literature. A scalar convolution is defined on a vector field using the concept of the scalar product of two vectors [8]. A Clifford convolution, which is an extension of the scalar convolution, is studied in [9]. Several extensions of the convolution operation on multivector fields have been explored extensively [10, 11, 12]. A method for pattern matching on vector fields and vector filter masks using Clifford convolution is presented in [13]. Similarly, various application-driven extensions of FT have been proposed. The quaternion Fourier transform (QFT), which is one of the early extensions of the traditional FT,is proposed by [PERSON] [14] for analyzing 2D-LTI systems and further studied by [PERSON] [15]. A hypercomplex extension to the standard complex Fourier transform is studied in [14], describing the quaternion Fourier transform as a double complex algebraic product. In [16], a hypercomplex Fourier transform is implemented by decomposition into two transforms that are isomorphic to the complex Fourier transform and provide a symplectic decomposition of the hypercomplex Fourier transform. Another quaternion Fourier transforms (QFT) is provided to study the analysis of higher-dimensional linear invariant systems in [17]. The spectral analysis of color images is studied using a generalized quaternion approach in [18]. In [19], a non-commutative multivector FT generalization of the QFT is studied. The Clifford FT for color images with a geometric approach using group action is given in [20]. In [21], the authors introduced a new equation within a Clifford algebra, whose solutions serve as kernels for a novel class of generalized Fourier transforms and are uniquely determined by a system of partial differential equations. A general geometric Fourier transform is defined for the multivector field over \(\mathbb{R}^{p,q}\) with values in GA, \(G^{p,q}\)[22]. A generalization of the QFT to a general non-commutative Fourier transformation of functions from spacetime \(\mathbb{R}^{3,1}\) to the Clifford geometric algebra \(Cl_{3,1}\) is studied and used to further establish two multivector Fourier transforms in [19]. A detailed survey of the application of GA for image and signal processing is provided in [23]. The remainder of this paper is organized as follows. In Section 2, we discuss the traditional CV LTI systems theory, which we intend to generalize. In Section 3, we begin by introducing the notations and terminology we use for geometric algebra (GA). Following this, we present the essential tools to express the VV signals, emphasizing the polar representation and decomposition of VV signals. In Section 4, we present our extension of the traditional CV framework for LTI systems to a VV framework for linear rotation-invariant time-invariant (LRITI) systems, as proposed in [24]. This includes the definition of LRITI systems and VV convolution. Additionally, we provide a generalization of the commutative property for VV convolution. In Section 5, we provide our main contribution, that is, a frequency-domain analysis for VV signals that includes a Fourier transform (FT) for VV signals, the extension of the traditional frequency spectrum/response, and a convolution property for the FT.