Hilbert Transform

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

That is, the Hilbert transform of is given by

(5.18) |

Thus, the Hilbert transform is a non-causal linear time-invariant filter.

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

(5.19) |

To show this last equality (note the lower limit of **0**
instead of the
usual
), it is easiest to apply (4.16) in the frequency
domain:

(5.20) | |||

(5.21) |

Thus, the negative-frequency components of are canceled, while the positive-frequency components are doubled. This occurs because, as discussed above, 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 (4.16). The use of the Hilbert transform to create an analytic signal from a real signal is one of its main applications. However, 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.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University