That is, the Hilbert transform of is given by
Thus, the Hilbert transform is a non-causal linear time-invariant filter.
The complex analytic signal
corresponding to the real signal
then given by
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.
Comparison to the Optimal Chebyshev FIR Bandpass Filter