# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 5 Frequency Domain Analysis for LRITI Systems
### Example 2 (Illustrating the Orthogonal Bivector Exponential via Rotations)
Substitution of (54) and (55) in (53b) gives  
\[\mathbf{y}(t)=\left[\alpha\mathbf{e}_{1}\mathbf{e}^{\mathbf{j}\omega_{0}t} \mathbf{e}^{\mathbf{j}\omega_{0}t}+\beta\mathbf{e}_{1}\mathbf{e}^{\mathbf{j} \omega_{0}}\right]*\mathbf{h}(t). \tag{56}\]  
Using the FSS property (44) in (56) gives\[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\] \[= \alpha\int\limits_{-\infty}^{\infty}\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\underbrace{\left(\mathrm{e}^ {-\mathrm{j}\omega\frac{\theta}{2}}\mathbf{h}(\tau)\mathrm{e}^{\mathrm{j}\omega \frac{\theta}{2}}\right)}_{\mathbf{h}\mathbf{g}\left(\tau\right)}\mathrm{e}^{ \mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{d}\tau\] \[\quad+\beta\int\limits_{-\infty}^{\infty}\mathrm{e}^{-\mathrm{j} \omega\frac{\theta}{2}}\mathbf{h}(\tau)\mathrm{e}^{\mathrm{j}\omega\frac{ \theta}{2}}\mathrm{d}\tau\] \[= \alpha\int\limits_{-\infty}^{\infty}\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\left[\mathbf{h}_{\mathbf{R}}( \tau)_{\|\mathrm{j}\mathrm{j}_{0}}+\mathbf{h}_{\mathbf{R}}(\tau)_{\bot \mathrm{j}_{0}}\right]\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t- \tau)\right)}\mathrm{d}\tau\] \[\quad+\beta\int\limits_{-\infty}^{\infty}\mathrm{e}^{-\mathrm{j} \omega\frac{\theta}{2}}\mathbf{h}(\tau)\mathrm{e}^{\mathrm{j}\omega\frac{ \theta}{2}}\mathrm{d}\tau\] \[= \alpha\mathrm{e}^{-\mathrm{j}\omega\frac{\omega_{0}}{2}t}\left[ \int\limits_{-\infty}^{\infty}\mathrm{e}^{\mathrm{j}\omega\frac{\omega_{0}}{2} \tau}\mathrm{d}\tau\right]\mathrm{e}^{\mathrm{j}\omega\frac{\omega_{0}}{2}t}\] \[\quad+\alpha\mathrm{e}^{-\mathrm{j}\omega\frac{\omega_{0}}{2}t} \left[\int\limits_{-\infty}^{\infty}\mathbf{h}_{\mathbf{R}}(\tau)_{\bot \mathrm{j}_{0}}\mathrm{d}\tau\right]\mathrm{e}^{\mathrm{j}\omega\frac{\omega_ {0}}{2}t}\] \[\quad+\beta\mathrm{e}^{-\mathrm{j}\omega\frac{\theta}{2}}\left[ \int\limits_{-\infty}^{\infty}\mathbf{h}(\tau)\mathrm{d}\tau\right]\mathrm{e }^{\mathrm{j}\omega\frac{\theta}{2}}\] \[= \alpha\mathrm{e}^{-\mathrm{j}\omega\frac{\omega_{0}}{2}t}\left[ \int\limits_{-\infty}^{\infty}\mathrm{e}^{\mathrm{j}\omega\frac{\omega_{0}}{2} \tau}\mathrm{h}_{\mathbf{R}}(\tau)_{\|\mathrm{j}\mathrm{e}}^{-\mathrm{j} \omega\frac{\omega_{0}}{2}\tau}\mathrm{d}\tau\right]\mathrm{e}^{\mathrm{j} \omega\frac{\omega_{0}}{2}t}\] \[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]0 \[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]1 \[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]2  
where we recognize that the spinor \(S_{\mathrm{j}}(\omega)\) can be written in terms of the Fourier transform of the impulse response as  
\[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]3 \[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]4  
Therefore, we have shown that bivector exponential signals are eigenfunctions of vector-valued convolution and LRITI systems.