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



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Comparison to the Optimal Chebyshev FIR Bandpass Filter