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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