Omnidirectional plane-wave destruction

Anti-aliasing ability

lidl20,lidl50,lidl80,oidl20,oidl50,oidl80
Figure 2. Magnitude responses of the line-interpolating PWD $1-Z_1Z_2^p$ (top) and the circle-interpolating PWD $1-Z_1^{p_1}Z_2^{p_2}$ with $r=1$ (bottom): from left to right $\theta =20^\circ ,50^\circ ,80^\circ$ . Here, $w1$ and $w2$ 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 $1-Z_2Z_1^p$ and $1-Z_1^{p_1}Z_2^{p_2}$ are shown in Figure 2: When dip angle $\theta=20^\circ$ , the two operators have similar responses (Figure 2a and 2d); when $\theta=50^\circ$ , the line-interpolating PWD become slightly aliased (Figure 2b), while the circle-interpolating PWD is not aliased (Figure 2e); as $\theta$ increases to $80^\circ$ , 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) $Z_1^p$ . The slope has an infinite range $[-\infty,+\infty]$ . In circle-interpolating PWD, there are two linear phase operators $Z_1^{p_1}$ and $Z_2^{p_2}$ , related to the respective directions. Both the slopes $p_1,p_2$ have a finite range $[-r,r]$ .

Following Fomel (2002), the phase shift operators can be approximated by the following maxflat fractional delay filter (Thiran, 1971):

$$H_1(Z_1)=\frac{B(1/Z_1)}{B(Z_1)}\approx Z_1^p,$$ (3)

where

$$B(Z_1)=\sum_{{k_1}=-N}^N b_{{k_1}}Z_1^{-{k_1}},$$ (4)

$N$ is the filter order and coefficients $b_{{k_1}}$ are polynomial functions of local slopes $p$ (Chen et al., 2013):

$$b_{k_1}(p)= \frac{(2N)!(2N)!}{(4N)!(N+k_1)!(N-k_1)!} \prod_{m=0}^{N-1-k_1}(m-2N+p) \prod_{m=0}^{N-1+k_1}(m-2N-p).$$ (5)

wrap
Figure 3. Phase approximating performances of the maxflat fractional delay filter $H_1(Z_1)$ when $p=0.2,1.2,5.2$ : 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 $p=0.2$ , 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 $p>1$ . The larger the slope $p$ , the more narrow the linear-phase frequency bands become.

As mentioned above, in line-interpolating PWC, the slope $p$ is in the infinite interval $[-\infty,+\infty]$ . For steep structures, where the slope $p$ becomes larger than $1$ , there may be phase wrapping in the linear phase approximator. However, in circle-interpolating PWC, the ranges of $p_1,p_2$ can be easily controlled by the radius $r$ . If we choose $r\leq 1$ , the circle-interpolating can avoid phase wrapping completely for all dip angles.

Omnidirectional plane-wave destruction