Hilbert transform in frequency domain

The Hilbert transform (HT) of a real signal $x(t)$ is a new signal $\hat{x}(t)$ in which every frequency component of $x(t)$ is phase shifted by $-90^\circ$ (for positive frequencies) and $+90^\circ$ (for negative frequencies), with amplitude unchanged.

Time-domain definition

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

So HT is the output of a linear time-invariant filter with impulse response $h(t) = \dfrac{1}{\pi t}$.

Frequency-domain expression

The Fourier transform of $\dfrac{1}{\pi t}$ is $-j\,\text{sgn}(f)$. Using the convolution property:

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

where

$$H(f) = -j\,\text{sgn}(f) = \begin{cases} -j = e^{-j\pi/2}, & f > 0 \\ 0, & f = 0 \\ +j = e^{+j\pi/2}, & f < 0 \end{cases}$$

Therefore:

• $|H(f)| = 1$ for all $f \neq 0$: amplitude spectrum is unchanged.
• $\angle H(f) = -90^\circ$ for $f>0$ and $+90^\circ$ for $f<0$.

 |H(f)|                 angle H(f)
   1 ------+------         +90 ------+
           |                         |
 ----------+-------> f   ------------+-------> f
                                     +------ -90

The Hilbert transformer is thus an ideal wideband $90^\circ$ phase shifter.

Example: $x(t) = \cos 2\pi f_0 t$ has $X(f) = \tfrac12[\delta(f-f_0) + \delta(f+f_0)]$.

$$\hat{X}(f) = \tfrac{1}{2}[-j\delta(f-f_0) + j\delta(f+f_0)] \Rightarrow \hat{x}(t) = \sin 2\pi f_0 t$$

Uses: generation of SSB (phase-shift method), analytic signal $x(t) + j\hat{x}(t)$ and complex envelope in bandpass signal analysis.