Postlude on Hilbert Transform Theory

The Hilbert transform of a real, continuous-time signal may be expressed as the convolution of with the Hilbert transform kernel:

$\displaystyle h(t) \isdef \frac{1}{\pi t} \qquad \hbox{(Hilbert transform \lq\lq kernel'')}$

That is, the Hilbert transform of is given by

$\displaystyle y(t) = (h \ast x)(t) \qquad \hbox{(Hilbert transform of $x$)}.$

Thus, the Hilbert transform is a non-causal linear time-invariant filter. From Fourier theory, we have that the frequency response of the Hilbert transform filter is given by

$\displaystyle H(\omega) \isdef \left\{\begin{array}{ll} -j, & \omega>0 \\ [5pt]... ...ga=0 \\ \end{array} \right.. \qquad\hbox{(Hilbert frequency response)} \protect$

(E.5)

The complex analytic signal corresponding to the real signal is then given by

$\begin{eqnarray*} x_a(t) &\isdef & x(t) + j y(t) \qquad \hbox{(Analytic signal c... ...{\pi}\displaystyle\int_0^{\infty} X(\omega)e^{j\omega t} d\omega \end{eqnarray*}$

To show this last equality (note the lower limit of 0 instead of the usual $-\infty$ ), it is easiest to apply (E.5) in the frequency domain:

$\begin{eqnarray*} X_a(\omega) &\isdef & X(\omega) + j Y(\omega)\\ &\isdef & (X_++X_-) + j (-j X_+ + j X_-) \\ [20pt] &=& 2X_+(\omega) \end{eqnarray*}$

Thus, the negative-frequency components of are canceled, while the positive-frequency components are doubled. This occurs because the Hilbert transform is an allpass filter that provides a degree phase shift at all negative frequencies, and a degree phase shift at all positive frequencies, as indicated in (E.5). The use of the Hilbert transform to create an analytic signal from a real signal is one of its main applications. Of course, as the preceding sections make clear, a Hilbert transform in practice is far from ideal because it must be made finite-duration in some way.

Previous: Comparison to use of the hilbert function
Next: Generalized Window Method

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
FIR Digital Filter Design
Postlude on Hilbert Transform Theory