Hilbert transformation; properties of LTI system

Hilbert transformation

The Hilbert transform $\hat{x}(t)$ of a signal $x(t)$ shifts the phase of all its positive-frequency components by $-90^\circ$ and negative-frequency components by $+90^\circ$, leaving amplitudes unchanged:

$$\hat{x}(t) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}d\tau = x(t) * \frac{1}{\pi t}, \qquad \hat{X}(f) = -j\,\text{sgn}(f)\,X(f)$$

Example: $\cos 2\pi f_ct \to \sin 2\pi f_ct$. It is used for SSB generation and for the analytic signal $x_+(t) = x(t) + j\hat x(t)$.

Properties of LTI systems

An LTI system is completely described by its impulse response $h(t)$ (or transfer function $H(f)$), and $y(t) = x(t) * h(t)$.

1. Linearity (superposition): $a x_1 + b x_2 \to a y_1 + b y_2$. Allows analysing a complex input as a sum of simple ones (e.g. sinusoids).
2. Time invariance: $x(t - t_0) \to y(t - t_0)$; the system behaves the same at all times.
3. Commutative: $x * h = h * x$.
4. Associative (cascade): $(x * h_1) * h_2 = x * (h_1 * h_2)$; two systems in cascade have $h = h_1 * h_2$ and $H = H_1H_2$.
5. Distributive (parallel): $x * (h_1 + h_2) = x * h_1 + x * h_2$; parallel systems add.
6. Causality: causal if $h(t) = 0$ for $t < 0$.
7. Stability (BIBO): stable if $\int|h(t)|dt < \infty$.
8. Eigenfunction property: an input $e^{j2\pi f_0t}$ gives $H(f_0)e^{j2\pi f_0t}$; sinusoids pass through with only gain and phase change, no new frequencies.