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 $\pi$ 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 $\rho=0$.I find that cut-off frequency f₀ is almost the same as k₀.  

  $$f_0 \approx k_0$$ (9)

  This satisfies the Equation () very well. In Equation (2), if I set k = k₀, the expression of $\frac {\overline{H}H}{\overline{D}D}$ 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 $\rho=0$ and k₀>0.03, $R_0 \propto k_0^{-1}$, the empirical relationship between the zero-frequency response of 100-point and k₀ is  

  $$R_0 \approx \frac{0.017}{k_0}$$ (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 n_a =50, $\rho=0$. The zero-frequency response here is about $\frac{0.017}{k_0}$;the cut-off frequency is about k₀.
  

• Effect of n_a

  Figure 10 shows the effect of n_a on helix low-cut filter when k₀ = 0.3 and $\rho=0$.For small 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 k₀ =0.3 and $\rho=0$, the empirical expression is  

  $$R_0 \approx \frac{2.7}{n_a}$$ (11)

   
  
  lcut-r00-ak3r0
  Figure 10 The zero-frequency response and cut-off frequency of the helix low-cut filter with k₀ = 0.3, $\rho=0$. The zero-frequency response here is about $\frac{2.7}{n_a}$.
  

• Effect of $\rho$

  $\rho$ 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 $\rho=0.5$and $\rho=1$.For larger $\rho$, the numerical anisotropy is stronger, especially when k₀ is small. Compared with Figure 9, the cut-off frequency at small k₀ increases slightly with $\rho$.

   
  
  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.  

  $$f_0 \approx k_0$$ (12)

  For large n_a, f₀ is almost the same as k₀. Both nonzero $\rho$ and small n_a leads to the anisotropy of cut-off frequency. However, there is a difference between them: $\rho$ causes the average cut-off frequency to increase slightly; small n_a intends to keep it.

  The zero-frequency response R₀ is under the direct control of $\rho$ and influenced by n_a and k₀. If I assume that the influences of n_a and k₀ are independent, the empirical expression of R₀ would be  

  $$R_0 \approx \frac {0.82} {n_a k_0}(1-\rho)$$ (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.