## Helix low-cut filter

For the helix low-cut filter, the main features are the cut-off frequency and zero-frequency response. All the frequencies in this paper are scaled values, taking as the Nyquist frequency.
• Effect of k0

Figure 9 is the zero-frequency response and cut-off frequency of a 100-point low-cut filter for different k0 when .I find that cut-off frequency f0 is almost the same as k0.    (9)
This satisfies the Equation () very well. In Equation (2), if I set k = k0, the expression of is 0.5. According to the definition of cut-off frequency, the energy spectrum of the filter should be 0.5 at the cut-off frequency.

The difference of the cut-off frequencies at various azimuths is very small when k0 > 0.1 and can be ignored. For k0 < 0.1, the difference is obvious.

In Figure 9, the zero-frequency response decreases as k0 increases. For and k0>0.03, , the empirical relationship between the zero-frequency response of 100-point and k0 is    (10)
For k0 <0.02, the zero frequency response curve becomes flat, and reaches the limit of 1. This is easily derived from Equation (2). When k0 turns to zero, the difference between numerator H and denominator and D becomes smaller.

lcut-r00-ka50r0
Figure 9
The zero-frequency response and cut-off frequency of the helix low-cut filter with na =50, . The zero-frequency response here is about ;the cut-off frequency is about k0.

• Effect of na

Figure 10 shows the effect of na on helix low-cut filter when k0 = 0.3 and .For small na, the numerical anisotropy is very strong. Although the mean value of the cut-off frequency remains the same, the azimuthal difference becomes larger when na becomes smaller. The zero-frequency response increases as na decreases, and when k0 =0.3 and , the empirical expression is    (11)

lcut-r00-ak3r0
Figure 10
The zero-frequency response and cut-off frequency of the helix low-cut filter with k0 = 0.3, . The zero-frequency response here is about .

• Effect of

directly controls the zero-frequency response and affects the cut-off frequency as well. Figure 11 shows the cut-off frequency of the 100-point helix low-cut filter as the function of k0 when and .For larger , the numerical anisotropy is stronger, especially when k0 is small. Compared with Figure 9, the cut-off frequency at small k0 increases slightly with .

lcut-rf0-kra50
Figure 11
The zero-frequency response and cut-off frequency of helix low-cut filter.

• Composed effects

Based on the proceeding analysis, I can derive the composed effects of the adjustable parameters on the helix low-cut filter.

The cut-off frequency is mainly governed by k0.    (12)
For large na, f0 is almost the same as k0. Both nonzero and small na leads to the anisotropy of cut-off frequency. However, there is a difference between them: causes the average cut-off frequency to increase slightly; small na intends to keep it.

The zero-frequency response R0 is under the direct control of and influenced by na and k0. If I assume that the influences of na and k0 are independent, the empirical expression of R0 would be    (13)

Equations (8), (12) and (13) describe the quantitative effects of the helix derivative / low-cut filter's adjustable parameters. These empirical formulas make it quantitative for us to choose the adjustable parameters of the helix filter in practice.