Hilbert Transform

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

 (5.17)

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.

Next Section:
Matlab, Continued
Previous Section:
Comparison to the Optimal Chebyshev FIR Bandpass Filter