Hilbert transform properties; energy and power of unit step

Hilbert transform and its properties

The Hilbert transform of a signal $x(t)$ is the signal obtained by shifting the phase of every frequency component of $x(t)$ by $-90^\circ$ (positive frequencies) without changing amplitudes.

$$\hat{x}(t) = x(t) * \frac{1}{\pi t} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau$$

In the frequency domain:

$$\hat{X}(f) = -j\,\text{sgn}(f)\,X(f), \quad H(f) = \begin{cases} -j, & f>0 \\ +j, & f<0 \end{cases}$$

So the Hilbert transformer is an ideal $-90^\circ$ phase shifter with unit gain. Example: HT of $\cos\omega_0 t$ is $\sin\omega_0 t$, and HT of $\sin\omega_0 t$ is $-\cos\omega_0 t$.

Properties

1. $x(t)$ and $\hat{x}(t)$ have the same amplitude spectrum $|X(f)|$.
2. They have the same energy (or power) and the same autocorrelation.
3. HT of HT gives the negative: $\hat{\hat{x}}(t) = -x(t)$.
4. $x(t)$ and $\hat{x}(t)$ are orthogonal: $\int_{-\infty}^{\infty} x(t)\hat{x}(t)\,dt = 0$.
5. HT of an even function is odd and HT of an odd function is even.
6. Inverse HT: $x(t) = -\hat{x}(t) * \frac{1}{\pi t}$.

Use: generating SSB signals (phase-shift method), pre-envelope and analytic signal $x_+(t) = x(t) + j\hat{x}(t)$.

Energy and power of unit step

$u(t) = 1$ for $t \ge 0$, and $0$ for $t<0$.

Energy:

$$E = \int_{-\infty}^{\infty}|u(t)|^2dt = \int_{0}^{\infty}1\,dt = \infty$$

Power:

$$\begin{aligned} P &= \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|u(t)|^2dt \\ &= \lim_{T\to\infty}\frac{1}{2T}\int_{0}^{T}1\,dt = \lim_{T\to\infty}\frac{T}{2T} = \frac12 \end{aligned}$$

Answer: $E = \infty$, $P = 0.5$ W (normalised to 1 Ω). Since energy is infinite but power is finite and non-zero, the unit step is a power signal.