Short note: Hilbert transform

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

Time domain:

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

Frequency domain: the Hilbert transformer is a filter with

$$H(f)=-j\,\text{sgn}(f),\qquad \hat X(f)=-j\,\text{sgn}(f)\,X(f)$$

so $\lvert H(f)\rvert=1$ and $\angle H(f)=-90°$ for $f>0$, $+90°$ for $f<0$. It is an ideal wideband $90°$ phase shifter.

Examples: $\widehat{\cos\omega_0t}=\sin\omega_0t$, $\widehat{\sin\omega_0t}=-\cos\omega_0t$.

Properties:

1. $x(t)$ and $\hat x(t)$ have the same amplitude spectrum, energy and power.
2. $\hat{\hat x}(t)=-x(t)$.
3. $x(t)$ and $\hat x(t)$ are orthogonal: $\int x(t)\hat x(t)dt=0$.
4. For low-pass $m(t)$ and $f_c$ above its bandwidth, $\widehat{m(t)\cos\omega_ct}=m(t)\sin\omega_ct$.

Applications:

• SSB generation: $s(t)=m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct$ (phase-shift method).
• Pre-envelope (analytic signal): $x_+(t)=x(t)+j\hat x(t)$, whose spectrum is zero for $f<0$.
• Band-pass signal representation in terms of in-phase and quadrature components, and complex envelope analysis.