Hilbert vs Fourier transform; properties of HT

Definition

The Hilbert transform $\hat x(t)$ of a real signal $x(t)$ is obtained by shifting the phase of all its positive-frequency components by $-90^\circ$ and negative-frequency components by $+90^\circ$, keeping amplitudes the same.

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

Example: $\cos\omega_0t \to \sin\omega_0t$; $\sin\omega_0t \to -\cos\omega_0t$.

Hilbert transform vs Fourier transform

Properties of Hilbert transform

1. Same amplitude spectrum: $|\hat X(f)| = |X(f)|$, since $|-j\,\text{sgn}(f)| = 1$.
2. Same energy / power: $\int|\hat x(t)|^2dt = \int|x(t)|^2dt$; also same autocorrelation function.
3. Double transform: $\hat{\hat x}(t) = -x(t)$ (two $-90^\circ$ shifts give $-180^\circ$).
4. Orthogonality: $\int_{-\infty}^{\infty}x(t)\hat x(t)\,dt = 0$.
5. Inverse: $x(t) = -\dfrac{1}{\pi t} * \hat x(t)$.
6. Symmetry: HT of an even function is odd, and of an odd function is even.
7. Linearity: HT of $a x_1 + b x_2$ is $a\hat x_1 + b\hat x_2$.
8. Modulation: for a low-pass $m(t)$ band-limited below $f_c$, HT of $m(t)\cos\omega_ct$ is $m(t)\sin\omega_ct$.

These properties make HT the basis of the phase-shift SSB modulator, $s(t) = m(t)\cos\omega_ct \mp \hat m(t)\sin\omega_ct$.