- Effect of
*k*_{0}Figure 9 is the zero-frequency response and cut-off frequency of a 100-point low-cut filter for different

*k*when .I find that cut-off frequency_{0}*f*is almost the same as_{0}*k*._{0}(9) *k*=*k*, 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._{0}The difference of the cut-off frequencies at various azimuths is very small when

*k*> 0.1 and can be ignored. For_{0}*k*< 0.1, the difference is obvious._{0}In Figure 9, the zero-frequency response decreases as

*k*increases. For and_{0}*k*>0.03, , the empirical relationship between the zero-frequency response of 100-point and_{0}*k*is_{0}(10) *k*<0.02, the zero frequency response curve becomes flat, and reaches the limit of 1. This is easily derived from Equation (2). When_{0}*k*turns to zero, the difference between numerator_{0}*H*and denominator and*D*becomes smaller.**lcut-r00-ka50r0**The zero-frequency response and cut-off frequency of the helix low-cut filter with

Figure 9*n*_{a}=50, . The zero-frequency response here is about ;the cut-off frequency is about*k*._{0}

- Effect of
*n*_{a}Figure 10 shows the effect of

*n*_{a}on helix low-cut filter when*k*= 0.3 and .For small_{0}*n*_{a}, the numerical anisotropy is very strong. Although the mean value of the cut-off frequency remains the same, the azimuthal difference becomes larger when*n*_{a}becomes smaller. The zero-frequency response increases as*n*_{a}decreases, and when*k*=0.3 and , the empirical expression is_{0}(11) **lcut-r00-ak3r0**The zero-frequency response and cut-off frequency of the helix low-cut filter with

Figure 10*k*= 0.3, . The zero-frequency response here is about ._{0}

- 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

*k*when and .For larger , the numerical anisotropy is stronger, especially when_{0}*k*is small. Compared with Figure 9, the cut-off frequency at small_{0}*k*increases slightly with ._{0}**lcut-rf0-kra50**The zero-frequency response and cut-off frequency of helix low-cut filter.

Figure 11

- 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

*k*._{0}(12) *n*_{a},*f*is almost the same as_{0}*k*. Both nonzero and small_{0}*n*_{a}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*n*_{a}intends to keep it.The zero-frequency response

*R*is under the direct control of and influenced by_{0}*n*_{a}and*k*. If I assume that the influences of_{0}*n*_{a}and*k*are independent, the empirical expression of_{0}*R*would be_{0}(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.

4/20/1999