# 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)
**Definition 26** (Frequency Response): _We define the frequency response as a spinor-valued function (proof is provided in Appendix A.4)_  
\[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]5 \[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]6 \[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]7  
In LTI system analysis, the frequency response is simply the FT of the impulse response (11), but with LRITI system analysis, the frequency response is seen as a spinor-valued function (64) and requires the FT of the scaled and rotated impulse response in an arbitrary plane \(\mathbf{j}\) along with an additional accumulation term. **Corollary 4** (Eigenfunction Property): _The response of an LRITI system to input \(\mathbf{x}(t)=\mathrm{e}^{-\mathrm{i}b\frac{\omega_{0}}{2}t}\mathbf{u}\mathrm{ e}^{\mathrm{i}b\frac{\omega_{0}}{2}t}\) may be expressed in terms of the frequency response as_  
\[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]8  
A block diagram representation of the eigenfunction property is provided in \[\mathbf{y}(t) = \alpha\int\limits_{-\infty}^{\infty}\left[\mathrm{e}^{-\mathrm{j} \omega\left(\frac{\omega_{0}}{2}(t-\tau)\right)}\mathrm{e}^{-\mathrm{j}\omega \frac{\theta}{2}}\right]\mathbf{h}(\tau)\left[\mathrm{e}^{\mathrm{j}\omega\frac {\theta}{2}}\mathrm{e}^{\mathrm{j}\omega\left(\frac{\omega_{0}}{2}(t-\tau) \right)}\right]\mathrm{d}\tau \tag{57a}\]9  
**Remark 19**: _For \(\mathbb{L}^{2}\) space, the eigenfunction property (65) can be expressed as \(\mathbf{y}(t)=\mathbf{u}\,H_{\mathrm{j}0}(\omega_{0};\mathbf{u})\mathrm{e}^{ \mathrm{i}b\omega_{0}t}\), which closely resembles the traditional eigenfunction property (12), except for the presence of the \(\mathbf{u}\) term. This is because \(\mathbf{u}\), is in this case, is associated with the unit element in the complex plane. Regardless, the interpretation remains the same._