Primer on Hilbert Transform Theory

We need a Hilbert-transform filter to compute the imaginary part of the analytic signal given its real part . That is,

(5.14) |

where . In the frequency domain, we have

(5.15) |

where denotes the frequency response of the Hilbert transform . Since by definition we have for , we must have for , so that for negative frequencies (an allpass response with phase-shift degrees). To pass the positive-frequency components unchanged, we would most naturally define for . However, conventionally, the positive-frequency Hilbert-transform frequency response is defined more symmetrically as for , which gives and ,

In view of the foregoing, the frequency response of the ideal Hilbert-transform filter may be defined as follows:

Note that the point at can be defined arbitrarily since the inverse-Fourier transform integral is not affected by a single finite point (being a ``set of measure zero'').

The ideal filter impulse response is obtained by finding the inverse Fourier transform of (4.16). For discrete time, we may take the inverse DTFT of (4.16) to obtain the ideal discrete-time Hilbert-transform impulse response, as pursued in Problem 10. We will work with the usual continuous-time limit in the next section.

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