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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.
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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 $ 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):

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


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

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

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

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.
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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 $ 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.

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