# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 2 Traditional LTI Systems Theory
: _Careful inspection of (8b) reveals two important aspects: commutation of the product of the input signal and impulse response \(\mathsf{x}(\cdot)\mathsf{h}(\cdot)\leftrightarrow\mathsf{h}(\cdot)\mathsf{x} (\cdot)\) and the interchanging of the arguments of the input signal and impulse response \((\tau)\leftrightarrow(t-\tau)\)._  
**Definition 5** (**Fourier Transform**). : _The study of LTI systems in the frequency domain often begins with the definition of the Fourier transform. The Fourier transform of any signal \(\mathsf{x}(t)\) is defined by_  
\[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]0  
**Definition 6** (**Complex Exponential Signals**). : _A CT complex exponential signal is any \(CV\) signal which can be expressed as_  
\[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]1  
_where \(a_{0}\in\mathbb{R}\) and \(\omega_{0}\) is the angular frequency._  
**Remark 3**. : _The complex exponentials of different frequencies are orthogonal under the standard inner product defined on the space of square-integrable functions over an interval. In other words, their inner product is zero unless the frequencies are identical over a period._  
**Definition 7** (**Frequency Response**). : _Any LTI system \(\mathcal{T}\{\cdot\}\) can be characterized in terms of a frequency response function as the FT of the impulse response_  
\[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]2  
**Definition 8** (**Eigenfunction Property**). : _The eigenfunction property of LTI systems is usually stated as follows. The response of an LTI system to a complex exponential input, \(a_{0}\mathrm{e}^{\mathrm{j}(\omega_{0}t+\phi_{0})}\), is_  
\[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]3  
_the same complex exponential with a multiplicative (scale-rotation) factor \(\mathsf{H}(\mathrm{j}\omega_{0})\)--the eigenvalue associated with the complex exponential at frequency \(\omega_{0}\). A block diagram representation for the same is provided in \[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]4_  
**Definition 9** (**Frequency Spectrum**). : _Any signal \(\mathsf{x}(t)\) can be characterized in terms a frequency spectrum_  
\[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]5  
\[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]6 Block diagram illustrating the eigenfunction property of LTI systems. **Theorem 3** (**Convolution Property of the FT**). : _The Fourier transform of the convolution of two signals is equivalent to the product of their Fourier transforms_  
\[\mathrm{x}(t)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\tau)\delta(t-\tau) \,\mathrm{d}\tau. \tag{4}\]7  
**Theorem 4** (**The Second Fundamental Theorem**). : _Any LTI system can be completely characterized by the system's frequency response, which is defined as the Fourier transform of the system's impulse response. Moreover, the spectrum of an LTI system's output can be computed using the convolution property of the FT, by multiplying the input spectrum with the frequency response._