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## Lowpass Filter

Circularly symmetric lowpass filters can be designed by assuming an idealized frequency response of the form
 (7)
where kc 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 norm. It solves the problem: Given an ideal real symmetrical spectrum F0(f), find a(n), n=0,N such that:

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 hn 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 hn 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 kx-ky domain. The two horizontal axes are the normalized wavenumbers kx and ky 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.

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Stanford Exploration Project
11/17/1997