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,
where . In the frequency domain, we have
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 , i.e., the positive-frequency components of are multiplied by .
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.
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
Thus, the Hilbert transform is a non-causal linear time-invariant filter.
The complex analytic signal
corresponding to the real signal
then given by
To show this last equality (note the lower limit of 0
instead of the
), it is easiest to apply (4.16) in the frequency
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.
Filtering and Windowing the Ideal Hilbert-Transform Impulse Response
Bandpass Filter Design Example