Lowpass Filter

Circularly symmetric lowpass filters can be designed by assuming an idealized frequency response of the form  

$$H(k_x,k_y) = \left\{ \begin{array} {ll} 1 & \mbox{for $\sqr... ...q{k_c}$\space } \\ 0 & \mbox{otherwise} \end{array} \right.$$ (7)

where k_c is the cutoff frequency in the one-dimensional case. The first step is to design the low-pass filter for the one-dimensional case. This can be done using any of the linear filtering techniques which determine the filter in some least squares sense with respect to the desired response. I used a Remez exchange algorithm Parks and Burrus (1987) which is a Fourier synthesis algorithm for real symmetric spectral functions with the weighted $L_\infty$ norm. It solves the problem: Given an ideal real symmetrical spectrum F₀(f), $f=[0,\pi]$ find a(n), n=0,N such that:

$$max [W(f)\left(\sum_{n=0}^{N} {a(n)\cos(nf) - F_0(f)}\right)]$$

is minimum, where W(f) is weighting function.

Figure shows the desired and the fitted spectrum obtained using a 21 term zero-phase filter. The filter in the space domain is a symmetrical filter as shown in Figure . The filter coefficients correspond to h_n in Figure . The McClellan transformation filter is designed using the 5x5 filter shown in the Appendix. This filter corresponds to the G filter shown in Figure . Once the filter coefficients h_n are computed they are incorporated into the recursive structure shown in Figure . The implementation is done in both space and wavenumber domains. The wavenumber domain response of the filter is shown in Figure . A spike in x-y space was used to test the space-domain implementation. The output of the lowpass filter is a smoothed spike as shown in Figure

 

rez
Figure 3 1-D actual and desired spectra for the lowpass filter. The desired spectrum is given by the continuous curve and the fitted spectrum is given by the dashed curve.

 

park Figure 4 Twenty-one coefficient symmetric 1-D filter in the space domain corresponding to Fig 3. These are the filter coefficients hn used in the Chebyshev filter structure shown in Figure 2.

 

fil2D
Figure 5 Circularly symmetric 2-D filter corresponding to filter in Figure 4, designed using the first order McClellan transformation. The response is in the k_x-k_y domain. The two horizontal axes are the normalized wavenumbers k_x and k_y and they go from -0.5 to 0.5 cycles/s.

 

lowpout Figure 6 Output of Lowpass Filter in the x-y domain. The filter implementation is in the x-y domain. The modified McClellan filter in space-domain is used to do the transformation.