# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 6 Example
### Time-Domain Analysis Using Convolution Definition
Additionally, a comparative summary of the notations and definitions used for traditional CV signals and VV signals is provided in Table 3. Another comparative study of the traditional CV thoery and the proposed VV theory is provided in Table 4.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Description** & **Traditional Formulation** & **Proposed Formulation** \\ \hline Signals & \(\mathsf{x}(t),\mathsf{y}(t),\mathsf{h}(t)\in\mathbb{C}\;\forall\;t\) & \(\mathsf{x}(t),\mathsf{y}(t),\mathsf{h}(t)\in\mathbb{L}^{n}\;\forall\;t\) \\ Polar representation & \(\mathsf{x}(t)=\|\mathsf{x}(t)\|\mathsf{e}^{\mathsf{i}\times\mathsf{x}(t)}\) & \(\mathsf{x}(t)=\mathbf{e}_{1}^{2}\mathsf{x}(t)=\mathbf{e}_{1}\mathsf{x}(t) \mathsf{x}(t)=x(t)\mathsf{X}^{-\frac{1}{2}}(t)\mathbf{e}_{1}\mathsf{X}_{1}^{ \frac{1}{2}}(t)\) \\ Impulse response & \(\mathsf{h}_{\mathsf{r}}(t)=\mathcal{T}\big{\{}\delta(t-\tau)\big{\}}\) & \(\mathsf{h}_{\mathsf{r}}^{\mathsf{e}_{1}}(t)=\mathcal{T}\big{\{}\mathbf{e}_{1} \delta(t-\tau)\big{\}}\) \\ Sifting property & \(\mathsf{x}(t)=\int_{-\infty}^{\infty}\mathsf{x}(\tau)\delta(t-\tau)\,\mathrm{d}\tau\) & \(\mathsf{x}(t)=\int_{-\infty}^{\infty}x(\tau)\mathsf{X}^{-\frac{1}{2}}(\tau) \mathbf{e}_{1}\delta(t-\tau)\mathsf{X}^{\frac{1}{2}}(\tau)\,\mathrm{d}\tau\) \\ Convolution & \(\mathsf{y}(t)=\int_{-\infty}^{\infty}\mathsf{x}(\tau)\mathsf{h}(t-\tau)\, \mathrm{d}\tau\) & \(\mathsf{y}(t)=\int_{-\infty}^{\infty}x(\tau)\mathsf{X}^{-\frac{1}{2}}(\tau) \mathbf{h}(t-\tau)\mathsf{X}^{\frac{1}{2}}(\tau)\,\mathrm{d}\tau\) \\ Periodic exponential signal & \(\mathsf{x}(t)=a_{0}\mathsf{e}^{\mathsf{i}\times t}\) & \(\mathsf{x}(t)=c_{0}\mathsf{e}^{-\mathsf{i}  
One significant difference between CV and VV theory is that the Fourier transform (FT) does not enable decomposition in a space with dimensions higher than a plane. This is because in traditional CV theory, frequency-domain analysis relies on the complex plane, which intrinsically supports only two-dimensional signal representations. For vector-valued signals in \(\mathbb{R}^{N}\), with \(N>2\), this limitation necessitates a channel-wise decomposition--treating each component separately using complex planes resulting in \(N\) scalar complex computations. This approach loses the geometric relationships among components and fails to provide a unified spectral interpretation in arbitrary dimensions. For \(N>2\), a vector-valued signal cannot be naturally decomposed within the complex plane, as complex numbers inherently represent only two-dimensional structures. This motivates the need for more general algebraic frameworks--such as geometric algebra--that can represent and manipulate an arbitrary-dimensional VV signals in a unified and coordinate-free manner. Therefore, we propose to resolve the issue by use of TOPs. For an arbitrary dimension, \(N\) TOPs decomposition provides a principled approach in representing VV signals in a set of mutually orthogonal planes. This framework enables a unified and geometrically meaningful analysis of signals across arbitrary dimensions. TOPs provide frequency-domain analysis for a general VV signal in a finite number of planes, \(\mathbf{j}_{k}\). TOPs enables one to compute the FT of each decomposed signal in terms of planar spectra, \(\mathbf{X}_{\mathbf{j}_{k}}(\omega)\), in their respective planes. To generalize the frequency-domain analysis, we introduced two distinct definitions of the frequency response: a first that is valid for a bivector exponential signal in an arbitrary plane and a second that is valid for a general VV signal decomposed into a set of TOPs. While we can define two different frequency responses, the one which does not use TOPs is of limited utility, because of the lack of a decomposition algorithm. Additionally, an advantage of the partitioning of a signal into TOPs lies in the fact that standard CV theory is essentially a \"planar theory\". Thus, by partitioning the analysis into TOPs, we can leverage much of our existing knowledge and expertise.