# Hilbert transform

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Hilbert_transform.png
The Hilbert transform, in red, of a square wave, in blue

In mathematics, the Hilbert transform [itex]\left\{\mathcal{H}s\right\}(t)[itex] (also written [itex]\hat{s}(t)[itex]) of the real function s(t) is an integral transform defined by

[itex]

\left\{\mathcal{H}s\right\}(t) = \hat{s}(t) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{s(\tau)}{t-\tau}\,d\tau [itex]

considering the integral as a Cauchy principal value (which avoids singularities at t = τ). The inverse Hilbert transform is:

[itex]

\left\{\mathcal{H}^{-1}\hat{s}\right\}(\tau) = -\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\hat{s}(t)}{\tau-t}\,dt [itex]

where again, the integral is a Cauchy principal value integral.

The Hilbert transform can also be written with a convolution operator as:

[itex]

\hat{s}(t) = {1 \over \pi t} * s(t) [itex]

[itex]\hat{s}(t)[itex] can be generated from [itex]s(t)[itex] by multiplying its frequency spectrum by [itex]-i \sgn(\omega)[itex], where sgn is the signum function and i is the imaginary number. This has the effect of shifting all of its negative frequencies by +90° and all positive frequencies by −90°. Note that the Hilbert transform and the original function are both functions of the same variable.

The Hilbert transform has applications in signal processing.

### Discrete Hilbert transform

The ideal discrete Hilbert transform is in the Z-domain

[itex]

H_{HT}(e^{j\omega}) = \begin{cases} -j, & 0 \leq \omega < \pi \\ j, & -\pi \leq \omega < 0. \end{cases} [itex] Clearly, it is a phase-shifting filter, with a -90 degree phase shift in the upper half plane and +90 degree shift in the lower half plane. However, it is, in the time-domain, and unrealisable system and thus the name ideal discrete Hilbert transform. Still, the impulse response [itex]h_{HT}[n][itex] can be obtained by inverse Discrete Fourier transform which yields

[itex]

h_{HT}[n]= \begin{cases} 0, & \mbox{for }n\mbox{ even},\\ \frac2{\pi n} & \mbox{for }n\mbox{ odd} \end{cases} [itex]

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