Omnidirectional plane-wave destruction

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## Anti-aliasing ability

lidl20,lidl50,lidl80,oidl20,oidl50,oidl80
Figure 2.
Magnitude responses of the line-interpolating PWD (top) and the circle-interpolating PWD with (bottom): from left to right . Here, and are normalized frequencies (0.5 denotes Nyquist frequency, half the sampling frequency) in vertical and horizontal directions. White color denotes one and dark denotes zero.

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):

 (3)

where

 (4)

is the filter order and coefficients are polynomial functions of local slopes (Chen et al., 2013):

 (5)

wrap
Figure 3.
Phase approximating performances of the maxflat fractional delay filter when : dash lines denote first-order filter and dotted lines denote second-order filter.

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.

 Omnidirectional plane-wave destruction

Next: 2D linear phase approximation Up: From line interpolation to Previous: From line interpolation to

2013-08-09