Linear Time Invariant(LTI) Systems

Def Linear Time Invariant System Linear Time Invariant system is a linear function transformation $\mathcal{L}\mathcal{R}\mathcal{T}$ with invariant time defined by a Fourier pair: impulse response function $g(t)$ and a transfer function $G(\omega )$. In real domain, the output function is given by: $y_{\mathrm{out}}(t)=g(t)\ast y_{\mathrm{in}}(t)={\int }_{-\infty }^{\infty }g(\tau )y_{\mathrm{in}}(t-\tau )d\tau$ Specially, for delta impulse, we have $y_{\mathrm{out}}(t)=g(t)\ast \delta (t)=g(t)$ In Fourier domain, the output function is given by: $Y_{\mathrm{out}}(\omega )=G(\omega )Y_{\mathrm{in}}(\omega )$

Prop For any input function $y_{\mathrm{in}}(t),z_{\mathrm{in}}(t)$ and the corresponding output function $y_{\mathrm{out}}(t),z_{\mathrm{out}}(t)$, we have following properties:

\mathcal{LTI}[y_{\mathrm{in}}(t)+z_{\mathrm{in}}(t)] = y_{\mathrm{out}}(t)+z_{\mathrm{out}}(t) \\ \mathcal{LTI}[y_{\mathrm{in}}(t+\tau)]= y_{\mathrm{out}}(t+\tau) \end{aligned}$$