FF - Fourier Filter

PURPOSE

Applies filters to 2-D or 3-D Fourier transforms.

SEE ALSO

FF PLOT	[Fourier Filter - design filter]
FP	[Fourier interpolation ||]
FQ	[Filter - Quick Fourier, High, low pass, etc. ||]
FT	[Fourier Transform ||]

USAGE

.OPERATION: FF

.INPUT FILE: FIN001
[Enter name of input file containing Fourier transform.]

.OUTPUT FILE: FOU001
[Enter name of output file containing filtered Fourier file.]

1: low-pass, 2: high-pass,3: Gaussian low-pass, 4: Gausian high-pass 5: Fermi low-pass, 6: Fermi high-pass,7: Butterworth low-pass, 8: Butterworth high-pass, 9: Remez, 10: B Factor

.FILTER TYPE (1-10): 3
[Enter filter option code.

(1) Low-pass truncation. Filter is "top-hat" function that truncates Fourier transform at spatial frequency RAD.

(2) High-pass truncation. Filter is inverse "top-hat" function that passes Fourier transform beyond spatial frequency radius RAD.

(3) Gaussian low-pass. Filter is Gaussian function EXP(-f**2/(2.*RAD**2)), where F is the frequency.

(4) Gaussian high-pass. Filter is complement of Gaussian function: 1.0-EXP(-F**2/(2.*RAD**2)).

(5) Fermi low-pass. Filter is 1/(1+EXP[(F-RAD)/T]) which negotiates between "top-hat" and Gaussian characteristics, depending on the value of T (see below).

(6) Fermi high-pass. As low-pass, but T is replaced by -T.

(7) Butterworth low-pass. Frequencies below pass band are preserved, frequencies above stop band are removed, with a smooth transition in between (pass band < stop band).

Option "7" - Butterworth low-pass. Filter is: 1/(SQRT(1+F/RAD)**(2*ORDER))
In the actual implementation of the filter the parameters, RAD and ORDER are calculated from the pass-band frequency and cut-off frequency specified by the user. Frequencies above pass band are preserved, frequencies below stop band are removed, with a smooth transition in between (pass band > stop band). See NOTE 3 below. See NOTE 3 below.

(8) Butterworth high-pass. Filter is: 1-(1/(SQRT(1+F/RAD)**(2*ORDER)))

(9) Multiplication by the Remez filter designed with the help of 'FF PLOT' command.

(10) B Factor: normalizes amplitudes by a 'B' temperature factor.

For filters [(1) through (5)], the filter radius will be solicited:

.FILTER RADIUS: 0.12
[Enter filter function radius in frequency units. They are of the range 0.0<=f<=0.5. See 'FF PLOT' for discussion of frequency units.]

For the Fermi filters [(5) and (6)], the temperature parameter T will be solicited:

.TEMPERATURE (0=CUTOFF): 0.3
[Roughly within this reciprocal distance (in terms of frequency units), the filter falls off.]

For the Butterworth filters [(7) and (8)], the pass band and stop band frequencies are solicited:

.PASS-BAND FREQUENCY & STOP-BAND FREQUENCY: .2,.4
[Both are in frequency units; 0.0 <= f < =0.5 ]

For the Remez filter (9), the filter file will be solicited:

.FILTER FILE: REM000
[This file can be created by 'FF PLOT', or it can be any Fourier file used as a filter.]

For the B Factor filter (10), the following parameters are solicited:

.ENTER B FACTOR: 0.5

.ENTER D CONSTANT: 1

.ENTER FQ CUTOFF: 0.4
[Where the new amplitude is : AMP = AMP*D(EXP(Bs**2))]

NOTES

1. 'FQ' has similar functions.

2. The FILTER RADIUS can be given either in absolute units or pixel units. If answer is >1.0 it is treated as given in pixel units.

3. In the Butterworth filter the ORDER determines the filter fall off and RAD corresponds to the cut-off radius. In the program RAD and ORDER are calculated from the parameters specified by the user using following equations:
   RAD = fp/((eps)**(2/ORDER))
   ORDER = [2*log(eps/sqrt(a**2-1))]/[log(fp/fs)]
   where fp and fs are the pass-band and stop-band frequencies and parameters "eps" and "a" are given by 0.882 and 10.624 resp. Note that fp and fs deviate from 1.0 and 0.0 by about 0.2 and 0.09 respectively (for a low-pass filter.) For reference see 'FF PLOT'.

SUBROUTINES

FFILTS

CALLER

FOUR1