# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 2 Traditional LTI Systems Theory  
The typical theory of LTI systems allows for the analysis of signals, which are represented as time-series with complex values. These CV quantities are indicated using lowercase sans serif fonts, such as \(j,h,x,y\), while the real-valued quantities are denoted using lowercase italic fonts, such as \(\omega,t,\tau,\theta\). Systems refer to mappings of signals and are denoted here using uppercase calligraphic letters, i.e., \(\mathcal{T},\mathcal{F}\). In the remainder of this section, we provide a summary of standard results that we intend to generalize. We omit the proofs; however, most of the standard results are provided in [1, 2, 25, 26, 27, 28]. **Definition 1** (Linear Systems). : _A CV system \(\mathcal{T}\) is said to be linear if for an input consisting of a weighted sum of complex-valued signals, the output is a weighted sum of the system's response to each of those signals_  
\[\mathcal{T}\left\{\sum_{k=1}^{N}a_{k}x_{k}(t)\right\}=\sum_{k=1}^{N}a_{k} \mathcal{T}\left\{x_{k}(t)\right\} \tag{1}\]  
_where \(a_{k}\in\mathbb{C}\)._  
**Remark 1**. : _In CV systems, rotation invariance (RI) naturally arises from the system's linearity and by considering complex values as \"scalars\". Consequently, a complex-valued system \(\mathcal{T}\) is inherently rotation-invariant in the complex plane if it exhibits linearity_  
\[ \begin{array}{ccc}\mathrm{x}(t)&\stackrel{{\mathrm{linearity}}}{{\longmapsto}}&\sum_{k=1}^{N}\mathrm{x}_{k}(t) \mathrm{e}^{\mathrm{i}\theta_{k}}\\ \mathcal{T}\downarrow&&\downarrow\mathcal{T}\\ \mathrm{y}(t)&\stackrel{{\mathrm{linearity}}}{{\longmapsto}}&\sum_{k=1 }^{N}\mathcal{T}\{\mathrm{x}_{k}(t)\}\mathrm{e}^{\mathrm{i}\theta_{k}}.\end{array}  \tag{2}\]  
**Definition 2** (Time-Invariant Systems). : _A system \(\mathcal{T}\) is said to be time-invariant (TI) if a time-shift in the input signal results in an identical time-shift in the output signal_  
\[\mathcal{T}\{\mathrm{x}(t-t_{0})\}=\mathrm{y}(t-t_{0}). \tag{3}\]  
**Definition 3** (Sifting Property). : _The study of LTI systems in the time domain often begins with the representation of a signal in terms of a superposition of scaled shifted unit impulses_  
\[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]  
**Definition 4** (Impulse Response). : _The impulse response of a system, \(\mathcal{T}\{\cdot\}\), is the response of the system to a unit impulse (Dirac delta) applied at \(\tau\)_  
\[\mathsf{h}_{\tau}(t)=\mathcal{T}\big{\{}\delta(t-\tau)\big{\}}. \tag{5}\]  
**Corollary 1**. : _If a system is time-invariant, then the response of that system to an impulse applied at an arbitrary time \(\tau\) is equal to the response of the system to an impulse applied at the time \(\tau=0\), time-shifted by \(\tau\)_  
\[\mathsf{h}_{\tau}(t)=\mathsf{h}_{0}(t-\tau). \tag{6}\]  
**Theorem 1** (The First Fundamental Theorem). : _Any LTI system \(\mathcal{T}\{\cdot\}\) can be completely characterized by a single signal called the system's impulse response. Moreover, the output \(\mathrm{y}(t)\) of an LTI system \(\mathcal{T}\{\cdot\}\) may then be expressed in terms of the convolution integral_  
\[\mathrm{y}(t) = \mathcal{T}\{\mathrm{x}(t)\} \tag{7a}\] \[\mathrm{x}(t)*\mathsf{h}(t) \triangleq \int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\mathsf{h}(t-\tau) \,\mathrm{d}\tau \tag{7b}\]  
and can be seen in the block diagram provided in Figure 1.  
Figure 1: Block diagram illustrating the first fundamental theorem of LTI systems. **Theorem 2** (**Commutative Property of CV Convolution**). : _The CV convolution for LTI systems defined in (7b) is commutative_  
\[\mathsf{x}(t)*\mathsf{h}(t) = \mathsf{h}(t)*\mathsf{x}(t) \tag{8a}\] \[\int\limits_{-\infty}^{\infty}\mathsf{x}(\tau)\mathsf{h}(t-\tau) \,\mathrm{d}\tau = \int\limits_{-\infty}^{\infty}\mathsf{h}(\tau)\mathsf{x}(t-\tau) \,\mathrm{d}\tau. \tag{8b}\]  
**Remark 2**.