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