Short note: LTI system

A linear time-invariant (LTI) system is a system that satisfies both:

• Linearity (superposition): $a x_1(t) + b x_2(t) \to a y_1(t) + b y_2(t)$.
• Time invariance: $x(t - t_0) \to y(t - t_0)$.

Key results:

• An LTI system is completely described by its impulse response $h(t)$ (or $h[n]$). The output for any input is the convolution

$$y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)\,d\tau, \qquad y[n] = \sum_k x[k]h[n-k]$$

• In frequency domain, $Y(j\omega) = H(j\omega)X(j\omega)$; complex exponentials are eigenfunctions.
• Causal iff $h(t) = 0$ for $t < 0$. BIBO stable iff $\int |h(t)|\,dt < \infty$.
• Cascade: $h_1 * h_2$ (order can be swapped). Parallel: $h_1 + h_2$.
• Described by linear constant-coefficient differential/difference equations.

Examples: RC circuit, ideal delay, moving-average filter $y[n] = \frac12(x[n] + x[n-1])$.