# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 4 Time Domain Analysis for LRITI Systems
### LRTTI Systems  
**Definition 18** (Linear VV Systems). : _A system \(\mathcal{T}\) is said to be linear if for an input consisting of a weighted sum of VV signals, the output is a weighted sum of the system's response to each of those signals_  
\[\mathcal{T}\Bigg{\{}\sum_{k=1}^{N}a_{k}\mathbf{x}_{k}(t)\Bigg{\}} =\sum_{k=1}^{N}a_{k}\mathcal{T}\big{\{}\mathbf{x}_{k}(t)\big{\}}. \tag{37}\]  
**Remark 9**. : _As previously noted, for CV systems, linearity in (1) implies rotational invariance as given in (2). However for VV systems, only real numbers are considered as \"scalars\", and thus rotation invariance is not automatically inherited from (37) as it was from (1). Therefore, to maintain an alignment between the traditional and generalized theories, we define an additional property, termed rotational invariance._  
**Definition 19** (Rotation-Invariant VV Systems). : _A system \(\mathcal{T}\) is said to be rotation-invariant if a rotation in the input signal results in an identical rotation in the output signal_  
\[\mathbf{x}(t) \stackrel{{\text{rotate}}}{{\longmapsto}} \ \ \ \mathrm{e}^{-\mathrm{j}\theta/2}\mathbf{x}(t)\mathrm{e}^{\mathrm{j} \theta/2}\] \[\mathcal{T}\downarrow \ \ \ \ \ \ \ \ \ \  
**Definition 20** (Time-Invariant VV Systems).: _A system \(\mathcal{T}\) is said to be a time-invariant if a time-shift in the input signal results in an identical time-shift in the output signal_  
\[\mathcal{T}\{\mathbf{x}(t-t_{0})\}=\mathbf{y}(t-t_{0}). \tag{39}\]  
**Remark 10**. : _The time-invariance property of the VV system (39) is a direct extension of the time-invariance property of the CV system (3)._  
**Definition 21** (Sifting Property of Unit Impulse for VV Signals). : _Using the polar representation of a vector-valued signal in (30), we write the generalized sifting property of the unit impulse as_  
\[\mathbf{x}(t) = \int_{-\infty}^{\infty}\!\!\!\mathbf{x}(\tau)\delta(t-\tau)\, \mathrm{d}\tau \tag{40a}\] \[= \int_{-\infty}^{\infty}\!\!\!\!\mathbf{x}(\tau)X^{-\frac{1}{2}}( \tau)\mathbf{e}_{1}\delta(t-\tau)X^{\frac{1}{2}}(\tau)\,\mathrm{d}\tau. \tag{40b}\]  
**Remark 11**. : _The expression in (40a) is a straightforward extension of (4), while (40b) provides the expression that enables us to leverage the rotation invariance (RI) property of LRITI systems._  
**Definition 22** (VV Impulse Response). : _We define the impulse response of a VV LRITI system due to an impulse at time \(t=\tau\) in the \(\mathbf{e}_{1}\) direction as_  
\[\mathbf{h}_{\tau}^{\mathbf{e}_{1}}(t)\triangleq\mathcal{T}\big{\{}\mathbf{e} _{1}\delta(t-\tau)\big{\}}. \tag{41}\]  
_For a time-invariant system, the response to an impulse at \(t=\tau\) can be expressed in terms of the response to an impulse at \(t=0\) as_  
\[\mathbf{h}_{\tau}^{\mathbf{e}_{1}}(t)=\mathbf{h}_{0}^{\mathbf{e}_{1}}(t-\tau) =\mathbf{h}(t-\tau). \tag{42}\]  
_For convenience, we usually drop the subscript and superscript._  
**Remark 12**. : _The impulse response of the VV LRITI systems given in (41) and (42) are similar to CV LTI system (5) and (6), but additionally, a direction must be chosen in which to apply the impulse. We choose, for convention, the reference direction to be \(\mathbf{e}_{1}\)._  
**Theorem 6** (The First Fundamental Theorem). : _Any LRITI system \(\mathcal{T}\{\cdot\}\) can be completely characterized by the impulse response defined in (42).