Beams Modeled As Single-Degree-Of-Freedom Systems
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THE TWO-POLE BESSEL LOWPASS FILTER
By Tom Irvine Email: [email protected] November 12, 2012
______
Introduction The transfer function of a two-pole Bessel filter is
(1)
The transfer function is taken from Reference 1.
The poles are
(2)
(3)
This following approach is taken from Reference 2.
Define a frequency parameter c as
(4)
Note that T is the time segment duration. It is the inverse of the sampling rate. fo is the filter cutoff frequency.
= for a -3 dB gain at the cutoff frequency.
The scale factor is derived in Appendix A.
1 The frequency parameter c can be interpreted as an analog frequency that corresponds to the normalized frequency ω of the desired digital filter.
Apply the frequency parameter to the transfer function.
(5)
(6)
(7)
Z-transform of Bessel Filter The bilinear transform is defined by
(8)
The purpose of this function is to transform an analog filter into the z-domain. The frequency transformation in equation actually follows from the bilinear transformation in equation (8). Further information regarding this transform is given in Appendix B.
Substitute the bilinear transform into the transfer function in equation (7).
(9)
(10)
(11)
(12)
(13)
(14)
2 (15)
The transfer function can be represented as
(16)
Set L=2. (17)
Multiply through by z 2,
(18)
(19)
(20)
(21)
b2 = (22)
(23)
(24)
The resulting digital recursive filtering relationship is
(25)
Example
3 Figure 1. Transfer Function, Two-Pole Bessel Lowpass Filter, fc = 100 Hz
References
1. Domenic Urzillo, MIL-S-901D Engineering Topics, 83rd Shock and Vibration Symposium, New Orleans, November 5, 2012.
2. Stearns and David, Signal Processing Algorithms in Fortran and C, Prentice Hall, Englewood Cliffs, New Jersey, 1993.
4 APPENDIX A
The two-pole Bessel lowpass filter transfer function is
(A-1)
Set s = jω
Then find the angular frequency ω at which the magnitude is 1/2, which is equivalent to -3 dB gain.
(A-2)
(A-3)
(A-4)
(A-5)
(A-6)
(A-7)
(A-8)
5 (A-9)
(A-10)
(A-11)
(A-12)
(A-13)
(A-14)
(A-15)
(A-16)
(A-17)
(A-18)
6 APPENDIX B
The bilinear transform is defined by
(B-1)
Set (B-2)
(B-3)
(B-4)
(B-5)
(B-6)
(B-7)
(B-8)
(B-9)
7
(B-10)
(B-11)
(B-12)
(B-13)
8