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| Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation | |
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Next: Bibliography
Up: Appendix A: Hilbert transform
Previous: Derivation of the FIR
The ideal frequency response of the Hilbert transform is expressed as
|
(15) |
From equations 4 and A-5, we obtain the difference as
. For
|
(16) |
and
, the Taylor series
of
at center c is expressed
|
(17) |
where
,
.
Consequently, the signum function sgn
is expressed
|
(18) |
We substitute sin
for
, based on
sgn
=sgn(sin
) for
, truncate the
series at the first M terms, and obtain the sinusoidal power series of
the signum function as
|
(19) |
The series in A-9 converges for
; that is,
has to be larger than
1/2. On the other hand, the expansion center
in the
-domain is
associated to the frequency center in the
-domain via the
relation
. Therefore,
must be less than or equal to 1. Accordingly,
is constrained by
and the corresponding
is within the
range
. Clearly, the ideal frequency response is well
approximated within the middle frequency band. Multiplying A-9
by
and substituting
for
sin
, the transfer function for the zero phase FIR of the
Hilbert transform is expressed as
|
(20) |
To obtain the causal transfer function,
is
multiplied by
and the resultant transfer function of
the FIR Hilbert transform of the (2
+2)th-order is
|
(21) |
For
=0, the transfer functions of equations A-4
and A-11 are approximated as
|
(22) |
|
(23) |
We compare equations A-12 and A-13, and we conclude
that these two transfer functions in middle frequency band of the
frequency domain differ by the constant coefficient
.
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| Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation | |
|
Next: Bibliography
Up: Appendix A: Hilbert transform
Previous: Derivation of the FIR
2015-05-07