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lidl20,lidl50,lidl80,oidl20,oidl50,oidl80
Figure 2. Magnitude responses of the line-interpolating PWD ![]() ![]() ![]() ![]() ![]() ![]() |
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We compare the line-interpolating and circle-interpolating PWD operators
in the frequency domain.
At different dip angles,
the magnitude responses of
and
are shown in Figure 2:
When dip angle
, the two operators have similar responses
(Figure 2a and
2d);
when
, the line-interpolating PWD become slightly aliased
(Figure 2b),
while the circle-interpolating PWD
is not aliased
(Figure 2e);
as
increases to
, the former is badly aliased
(Figure 2c),
and the latter is still not aliased
(Figure 2f).
In summary, the line-interpolating PWD has different frequency responses for different dip angles. It may become aliased when the slope is large. The circle-interpolating PWD avoids aliasing for both small and large dip angles.
In line-interpolating PWD,
we must design a digital filter to approximate
the linear phase operator (or phase shift operator)
.
The slope has an infinite range
.
In circle-interpolating PWD,
there are two linear phase operators
and
,
related to the respective directions.
Both the slopes
have a finite range
.
Following Fomel (2002), the phase shift operators can be approximated by the following maxflat fractional delay filter (Thiran, 1971):
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wrap
Figure 3. Phase approximating performances of the maxflat fractional delay filter ![]() ![]() |
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In Figure 3, we show the phase approximating performances
of the maxflat fractional delay filters for different slopes. For
small slope
, the approximations are good, but when the slopes
become large, the phases get wrapped. It is obvious
that the phase wrapping comes when and only when
. The larger
the slope
, the more narrow
the linear-phase frequency bands become.
As mentioned above, in line-interpolating PWC,
the slope
is in the infinite interval
.
For steep structures,
where the slope
becomes larger than
,
there may be phase wrapping in the linear phase approximator.
However, in circle-interpolating PWC,
the ranges of
can be easily controlled by the radius
.
If we choose
,
the circle-interpolating can avoid phase wrapping completely for all dip angles.
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