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,
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(5.14) |
where

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(5.15) |
where
















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

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.
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

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(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
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|
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(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:
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|
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(5.20) | |
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(5.21) |
Thus, the negative-frequency components of



Next Section:
Filtering and Windowing the Ideal Hilbert-Transform Impulse Response
Previous Section:
Bandpass Filter Design Example