RaneNote BESSEL FILTER CROSSOVER A Beel Filter Croover, and It Relation to Other Croover Beel Function Phae Shift Group Delay Beel, 3dB Down Introduction One of the way that a croover may be contructed from a Beel low-pa filter employ the tandard lowpa to high-pa tranformation. Variou frequency normalization can be choen for bet magnitude and polar repone, although the linear phae approximation in the paband of the low-pa i not maintained at higher frequencie. The reulting croover i compared to the Butterworth and Linkwitz-Riley type in term of the magnitude, phae, and time domain repone. Ray Miller Rane Corporation RaneNote 47 998,, 6 Rane Corporation
A Brief Review of Croover There are many choice for croover today, due epecially to the flexibility of digital ignal proceing. We now have added incentive to examine unconventional croover type. Each type ha it own tradeoff between contraint of flatne, cutoff lope, polar repone, and phae repone. See [] and [] for more complete coverage of croover contraint and type. Much of the content of thi paper i cloely related to previou work by Liphitz and Vanderkooy in [3]. Our enitivity to frequency repone flatne make thi one of the highet prioritie. It i often ued a a tarting point when chooing a croover type. Cutoff lope of at leat db per octave are uually choen becaue of limitation in the frequency range that driver can faithfully reproduce. Even thi i le than optimal for mot driver. Polar repone i the combined magnitude veru litening angle from noncoincident driver [4]. The ideal cae i a large lobe in the polar repone directly in front of the driver, and happen when low-pa and high-pa output are in-phae. The phae repone of a croover i one of it mot ubtle apect, and o i often ignored. A purely linear phae hift, which i equivalent to a time delay, i otherwie inaudible, a i a mall non-linear phae hift. Still, there i evidence that phae coloration i audible in certain circumtance [5], and certainly ome people are more enitive to it than other. A firt-order croover i unique, in that it um with a flat magnitude repone and zero reultant phae hift, although the low-pa and high-pa output are in phae quadrature (9 ), and the driver mut perform over a huge frequency range. The phae quadrature that i characteritic of odd-order croover reult in a moderate hift in the polar repone lobe. In pite of thi, third-order Butterworth ha been popular for it flat ound preure and power repone, and 8 db per octave cutoff lope. Second-order croover have hitorically been choen for their implicity, and a uable db per octave cutoff. Fourth-order Linkwitz-Riley preent an attractive option, with flat ummed repone, 4 db per octave cutoff, and output which are alway in phae with each other, producing optimal polar repone. Steeper cutoff lope are known to require higher order with greater phae hift, which for the linear phae cae i equivalent to more time delay. A number of other novel and ueful deign exit which hould be conidered when chooing a croover. Generating the high-pa output by ubtracting the low-pa output from an appropriately time-delayed verion of the input reult in a linear phae croover, with tradeoff in cutoff lope, polar repone, and flatne []. Overlapping the deign frequencie and equalizing the repone can reult in a linear phae croover [3], with a tradeoff in polar repone. A croover with perfect polar repone can be deigned with a compromie in phae repone or cutoff lope [6]. What i a Beel Croover? The Beel filter wa not originally deigned for ue in a croover, and require minor modification to make it work properly. The purpoe of the Beel filter i to achieve approximately linear phae, linear phae being equivalent to a time delay. Thi i the bet phae repone from an audible tandpoint, auming you don t want to correct an exiting phae hift. Beel are hitorically low-pa or all-pa. A croover however require a eparate high-pa, and thi need to be derived from the low-pa. There are different way to derive a high-pa from a low-pa, but here we dicu a natural and traditional one that maximize the cutoff lope in the high-pa. Deriving thi high-pa Beel, we find that it no longer ha linear phae. Other derivation of the high-pa can improve the combined phae repone, but with tradeoff. Two other iue cloely related to each other are the attenuation at the deign frequency and the ummed repone. The traditional Beel deign i not ideal here. We can eaily change thi by hifting the low or high-pa up or down in frequency. Thi way, we can adjut the low-pa v. high-pa repone overlap, and at the ame time achieve a phae difference between the low-pa and high-pa that i nearly contant over all frequencie. In the fourth order cae thi i 36, or eentially in-phae. In fact, the econd and fourth order cae are comparable to a Linkwitz-Riley with lightly more rounded cutoff!
4 6 8... Figure. Fourth-Order Beel Magnitude Beel Low-Pa and High-Pa Filter The focu of thi paper i on croover derived uing traditional method, which begin with an all-pole lowpa filter with tranfer function (Laplace Tranform) of the form /p(),where p() i a polynomial whoe root are the pole. The Beel filter ue a p() which i a Beel polynomial, but the filter i more properly called a Thomon filter, after one of it developer [7]. Still le known i the fact that it wa actually reported everal year earlier by Kiyau [8]. Beel low-pa filter have maximally flat group delay about Hz [9], o the phae repone i approximately linear in the paband, while at higher frequencie the linearity degrade, and the group delay drop to zero (ee Fig. and ). Thi nonlinearity ha minimal impact becaue it occur primarily when the output level i low. In fact, the phae repone i o cloe to a time delay that Beel low-pa and all-pa filter may be ued olely to produce a time delay, a decribed in []. The high-pa output tranfer function may be generated in different way, one of which i to replace every intance of in the low-pa with /. Thi flip the magnitude repone about the deign frequency to yield the high-pa. Characteritic of the low-pa with repect to Hz are, in the high-pa, with repect to infinite frequency intead. A number of other high-pa derivation are poible, but they reult in compromied cutoff lope or polar repone (ee []). Thee are beyond the cope of thi paper. Thi popular method reult in the general tranfer function (); () i a fourth-order Beel example..8.6.4... Figure. Fourth-Order Beel Group Delay Note the revered coefficient order of the high-pa a compared to the low-pa, once it converted to a polynomial in, and an added n th -order zero at the origin. Thi zero ha a counterpart in the low-pa, an implicit n th -order zero at infinity! The nature of the repone of the high-pa follow from equation (3) below, where i evaluated on the imaginary axi to yield the frequency repone. =j ω, p jω = p jω h, ω h = ω (3)
The magnitude repone of the low-pa and the high-pa are mirror image of each other on a logfrequency cale; the negative ign ha no effect on thi. The phae of the low-pa typically drop near the cutoff frequency from an aymptote of zero a the frequency i increaed, and aymptotically approache a negative value. However, in addition to being mirror image on a log-frequency cale, the phae of the highpa i the negative of the low-pa, which follow from the negative ign in (3). So the phae rie from zero at high frequency, and approache a poitive value aymtotically a the frequency i decreaed. Thi reult in offet curve with imilar hape. Any aymmetry of the -haped phae curve i mirrored between the low-pa and high-pa. See Figure 5 for a econd-order example, where the phae curve alo ha inherent ymmetry. One pecial cae i where the denominator polynomial p() ha ymmetric coefficient, where the n th coefficient i equal to the contant term; the (n-) t coefficient i equal to the linear term, etc. Thi i the cae for Butterworth and therefore the Linkwitz-Riley type [3]. A fourth-order Linkwitz-Riley i given a an example in equation (4). (4).. 4... 3 4 When thi i the cae, coefficient reveral ha no effect on p(), and the high-pa differ from the low-pa only in the numerator term n. Thi numerator can eaily be hown to produce a contant phae hift of 9, 8, 7, or 36 (36 i in-phae in the frequency domain), with repect to the low-pa, when frequency repone i evaluated on the imaginary axi. For the econd-order cae =(jω) = -ω and the minu ign indicate a polarity reveral (or 8 phae hift at all frequencie). Normalization Filter tranfer function are normalized by convention for ω o =, (f = Hz). π and are then deigned for a particular frequency by replacing every intance of in the tranfer function by, ω ω o o =. π. f o c Thi ha the effect of hifting the magnitude and phae repone right or left when viewed on a log-frequency cale. Of coure, it doen t affect the hape of thee repone curve, ince when the tranfer function are evaluated: c. c... c. n n f ωo = f c = jω ωo = f( jy), y = ω ωo where y i a contant multiple of the variable frequency. The group delay, being the negative derivative of the phae with repect to angular frequency, i alo caled up or down. Thi proce can alo be ued to adjut the overlap between the low-pa and high-pa filter, o a to modify the ummed repone. After thi i done, the filter are till normalized a before, and may be deigned for a particular frequency. Adjuting the overlap will be done here with a normalization contant u, which will be applied equally but oppoitely to both the low-pa and high-pa. In the low-pa, i replaced by (/u), and in the high-pa, i replaced by (u). The low-pa repone i hifted right (u > ) or left (u < ) when viewed on a log frequency cale, and the highpa repone i hifted in the oppoite direction. Thee overlap normalization may be baed on the magnitude repone of either output at the deign frequency, choen for the flattet ummed repone, for a particular phae hift, or any other criterion. Normalization influence the ymmetry of p(), but perfect ymmetry i not achievable in general. Thi mean that it will not alway be poible to make the low-pa and high-pa phae repone differ exactly by a contant multiple of 9 for ome normalization. The ituation can be clarified by normalization for c n =, a done by Liphitz and Vanderkooy in [] and [5], where c = for unity gain at Hz. Thi form reveal any inherent aymmetry. Equation (6) how the general lowpa, while (7) i the fourth-order Beel denominator. Note that it become nearly ymmetric, and relatively imilar to the Linkwitz-Riley in (4). n c n k. k... k. n n (5) (6) 9.. 3. 5 4 3.. 4.396. 3.39. 3 4 (7)
Phae-Matched Beel The textbook low-pa Beel i often deigned for an approximate time delay of t o = ω o rather than for the common -3 db or -6 db level at the deign frequency ued for croover. Thi deign i ued a a reference to which other normalization are compared. The low-pa and high-pa have quite a lot of overlap, with very little attenuation at the deign frequency, a hown in Figure 3, for a econd-order Beel with one output inverted. 5 5 The ummed magnitude repone of the Beel normalized by the 45 i fairly flat, within db for the econd-order and fourth-order. We may adjut the overlap lightly for flattet magnitude repone intead, at the expene of the polar repone. Figure 4-6 how the reult of four normalization for the econd-order filter. The 3 db and phae-match normalization are illutrated in Figure 5 and 6. Note that for the econd-order phae-match deign, low-pa and high-pa group delay are exactly the ame. 3 5 5... Time Delay.73-3 db Mag..58 Phae Match.5 Flattet ω k Figure 4. Comparion of Second-Order Beel Sum 3. LP HP Difference Figure 3. Second-Order Beel Croover Beel polynomial of degree three or higher are not inherently ymmetric, but may be normalized to be nearly ymmetric by requiring a phae hift at the deign frequency of 45 per order, negative for the low-pa, poitive for the high-pa. Thi reult in a fairly contant relative phae between the low-pa and high-pa at all other frequencie. Equation (8) how an equation for deriving the normalization contant of the fourth-order Beel, where the imaginary part of the denominator (7) i et to zero for 8 phae hift at the deign frequency. = jω p, ω p ω p 3. =, u = ω p =.5 Thi normalization i not new, but wa preented in a lightly different context in [5], with a normalization contant of.9759, which i the quare of the ratio of the phae-match u in equation (8) to the u implied by equation (6) and (7), the fourth root of /5. Since the phae nonlinearity of the high-pa i now in the paband, the croover reulting from the um of the two approache phae linearity only at lower frequencie. Thi doen t preclude it from being a ueful croover. (8).5.5.5...73-3 db Mag..58 Phae Match Figure 5. Comparion of Second-Order Phae...73-3 db Mag LP.73-3 db Mag HP.73-3 db Mag Sum.58 Phae Match Figure 6. Second-Order Group Delay
4 6 8.. Time-delay Deign.48-3 db Mag Deign.4 Flattet.3 Phae Match *4 Figure 8. Summed Fourth-Order Repone * * *3 *4 *5 *.3: phae match *.4: flattet *3.48: -3 db magnitude *4.64: cancellation *5.: time delay deign Figure 7. Summed Fourth-Order Beel Frequency Repone v. Normalization. Normalization value are relative to time delay deign. The fourth-order i illutrated in Figure 7-9, Figure 7 being a 3-D plot of frequency repone veru normalization. Figure 8 how four cae, which are cro-ection of Figure 7. The phae-match cae ha good flatne a well a the bet polar repone. The fourth-order Linkwitz-Riley i very imilar to the Beel normalized by.3. The third-order Beel magnitude ha comparable behavior. In a real application, phae hift and amplitude variation in the driver will require ome adjutment of the overlap for bet performance. The enitivity of the croover repone to normalization hould be conidered []. Comparion of Type Butterworth, Linkwitz-Riley, and Beel croover may be thought of a very eparate type, while in fact they are all particular cae in a continuou pace of poible croover. The eparate and ummed magnitude repone are ditinct but comparable, a can be een by graphing them together (Figure ). The Beel and Linkwitz-Riley are the mot imilar. The Butterworth ha the harpet initial cutoff, and a +3dB um at croover. The Linkwitz-Riley ha moderate rolloff and a flat um. The Beel ha the widet, mot gradual croover region, and a gentle dip in the ummed repone. All repone converge at frequencie far from the deign frequency. 6 4 3 4 5. -3 db Mag Deign LP ω k -3 db Mag Deign HP -3 db Mag Deign Sum Phae Match Sum Phae Match HP (lightly peaked) Figure 9. Fourth-Order Sum Group Delay 6. Butterworth Linkwitz-Riley Beel ω k Figure. Fourth-Order Magnitude
4 3. Butterworth Linkwitz-Riley Beel ω k Figure. Fourth-Order Group Delay The phae repone alo look imilar, but the amount of peaking in the group delay curve varie omewhat, a hown in Figure. There i no peaking in the Beel low-pa, while there i a little in the highpa for order >. The ummed repone ha only a little peaking. The group delay curve i directly related to the behaviour in the time domain, a dicued in []. The mot overhoot and ringing i exhibited by the Butterworth deign, and the leat by the Beel. Often when dicuing croover, the low-pa tep repone i conidered by itelf, while the high-pa and ummed tep repone i uually far from ideal, except in the cae of the linear phae croover; thi ha been known for ome time [], but tep-repone graph of higher-order croover are generally avoided out of good tate! Table give Beel croover denominator normalized for time delay and phae match. Note the near-perfect ymmetry for the (lat three) phae-match cae. Drag Net Beel Croover at -3 db Rane Beel croover i et for phae match between low-pa and high-pa. Thi minimize lobing due to driver eparation, and alo reult in a pretty flat combined repone. Another popular option i to have the magnitude repone 3 db at the deign frequency. If 3 db i deired at the etting, the frequency etting need to be changed by particular factor. You will need to enter eparate value for low-pa and high-pa: multiply the low-pa and high-pa frequencie by the following factor: Croover Type Low-Pa High-Pa Second-Order ( db/octave).7.786 Third Order (8 db/octave).43.78 Fourth Order (4 db/octave).533.65 For example, for a econd-order low-pa and highpa et to Hz, et the low-pa to 7 Hz and the high-pa to 786 Hz. Normalization with repect to time delay deign Low-pa Denominator Polynomial High-pa Denominator Polynomial 3 3 5 5 3 5 5 3 3 =.5774 9 3 5 4 5 3 3 9 3 4.43.48.463.8 3.8.463.48 3.5 =.386 3.4 4.5 3.4 3.5 4.5 3.4. 4.5. 3.4. 3 4 Table - Beel Croover of Second, Third, and Fourth-Order, Normalized Firt for Time Delay Deign, then for Phae Match at Croover
Summary A Beel croover deigned a decribed above i not radically different from other common type, particularly compared to the Linkwitz-Riley. It doe not maintain linear phae repone at higher frequencie, but ha the mot linear phae of the three dicued, along with fairly good magnitude flatne and minimal lobing for the even order. It i one good choice when the driver ued have a wide enough range to upport the wider croover region, and when good tranient behaviour i deired. A verion of thi RaneNote wa preented at the 5th Convention of the Audio Engineering Society, San Francico, CA, 998 Reference [] S.P. Liphitz and J. Vanderkooy, A Family of Linear-Phae Croover Network of High Slope Derived by Time Delay, J. Aud. Eng. Soc, vol 3, pp- (983 Jan/Feb.). [] Robert M. Bullock,III, Loudpeaker-Croover Sytem: An Optimal Choice, J. Audio Eng. Soc, vol. 3, p486 (98 July/Aug. ) [3] S.P. Liphitz and J. Vanderkooy, Ue of Frequency Overlap and Equalization to Produce High-Slope Linear-Phae Loudpeaker Croover Network, J. Audio Eng. Soc, vol. 33, pp4-6 (985 March) [4] S.H. Linkwitz, Active Croover Network For Non-Coincident Driver, J. Audio Eng Soc, vol. 4, pp -8 (976 Jan/Feb.). [5] S.P. Liphitz, M Pocock and J. Vanderkooy On the Audibility of Midrange Phae Ditortion in Audio Sytem, J. Audio Eng. Soc, vol 3, pp 58-595 (98 Sept.) [6] S.P. Liphitz and J. Vanderkooy, In Phae Croover Network Deign, J. Audio Eng. Soc, vol 34, p889 (986 Nov.) [7] W.E. Thomon, Delay Network Having Maximally Flat Frequency Characteritic, Proc IEEE, part 3, vol. 96, Nov. 949, pp. 487-49. [8] Z. Kiyau, On A Deign Method of Delay Network, J. Int. Electr. Commun. Eng., Japan, vol. 6, pp. 598-6, Augut, 943. [9] L. P. Huelman and P. E. Allen, Introduction to the Theory and Deign of Active Filter, McGraw-Hill, New York, 98, p. 89. [] Denni G. Fink, Time Offet and Croover Deign, J. Audio Eng Soc, vol. 8:9, pp6-6 (98 Sept) [] Wielaw R. Wozczyk, Beel Filter a Loudpeaker Croover, Audio Eng. Soc. Preprint 949 (98 Oct.) [] J.R. Ahley, On the Tranient Repone of Ideal Croover Network, J. Audio Eng. Soc, vol, pp4-44 (96 July) Rane Corporation 8 47th Ave. W., Mukilteo WA 9875-598 USA TEL 45-355-6 FAX 45-347-7757 WEB www.rane.com 4-6