6: The Second-Order Soundfield Microphone

Transcription

1 : Th Scond-Ordr Soundfild Microphon Grzon statd [ that a scond-ordr soundfild microphon may b constructd...usin twlv small cardioid or hyprcardioid capsuls mountd to form th facs of a rular dodcahdron havin a small diamtr... th scond-harmonic aspcts of th dirctional pickup can b drivd from ths by tchniqus similar to th Blumlin diffrnc tchniqu. Sinc th dodcahdron is th simplst rular polyhdron with nin or mor facs, so a dodcahdral array is th simplst arranmnt of microphons from which th nin indpndnt sinals comprisin th scond-ordr B-format st can b obtaind..: Gomtry of th Dodcahdron It is dsirabl to maximis th dr of symmtry prsnt in th cofficints of th -B matrix, sinc this simplifis th mathmatical tratmnt (and will also simplify th vntual implmntation. Th symmtry which is apparnt in th cofficints of th -B matrix in th cas of th first-ordr soundfild microphon is rlatd to th symmtry of th ttrahdron; spcifically, to th prsnc of C oprations in th roup of rotational symmtris of th ttrahdron [. Each C axis passs throuh a vrtx and th cntroid of th opposit fac of th ttrahdron. Rotation by about on of ths axs has th ffct of takin cos( θ cos( φ to sin( θ cos( φ (or vic vrsa and tc.; i.., of inducin a cyclic prmutation of th polar pattrns associatd with th first-ordr sphrical harmonic componnt sinals. Th sam rotation intrchans th facs of th ttrahdron in a corrspondin mannr. Such C oprations ar also prsnt in th roup of symmtris of th dodcahdron. To maximis th symmtry in th -B matrix cofficints for th scond-ordr soundfild microphon, th dodcahdral capsul array is orintd such that appropriat C axs (which pass throuh two diamtrically opposd vrtics ar alind with thos of th ttrahdral first-ordr soundfild microphon array. By inspction, two orintations which satisfy this critrion may b idntifid; slction btwn ths is arbitrary. Th chosn orintation is such that th hihst part of th 8

2 dodcahdron is an d runnin front-back, and th front-most part is a horizontal d - s fiur.. Fiur.: Orintation of Dodcahdron (viwd from cntr-front dirction Two facs, havin th front-most d in common, point symmtrically up and down, with no lft / riht componnt in thir orintation; ths facs ar convnintly lablld and. Thir (outward unit surfac normal vctors ar x û z (. and 9

3 x û z (. Th cosin of th anl btwn vctors normal to two adjacnt facs of a rular polyhdron is qual to th cosin of th dihdral anl at ach d of that polyhdron; hnc, th scalar product of û and û is qual to th cosin of th dihdral anl at th ds of a rular dodcahdron. Now, for any rular polyhdron, ϑ d sin ( π ( cos v sin π f (. whr ϑ d is th dihdral anl, f is th numbr of ds around ach fac, and v is th numbr of ds which mt at ach vrtx [. Rarranin ivs ϑ d sin cos sin ( π v ( π f (. For a dodcahdron, and ; hnc v f sin ϑ d cos( π sin( π / / (. Now sinc, for any anl ϕ, cos( ϕ ± sin ( ϕ (.

4 so w may us th trionomtric doubl anl idntity for sins to obtain sin( ϑ d ϑd ϑd sin cos ϑd sin ( ϑd sin (.7 whr th positiv squar root is takn in th substitution from quation (. bcaus th dihdral anl must by dfinition b lss than 8. W can now find th cosin of th dihdral anl: cos( ϑ d sin ( ϑd ( ( ( (.8 whr th positiv squar root is takn sinc (by inspction th anl in qustion is lss than 9. W can now dtrmin th valus of th lmnts of vctors, û and u. Sinc thy ar unit ˆ

5 x z (.9 and sinc thir scalar product is qual to, x z (. Equations (.9 and (. may b solvd to iv x (. and z (. so that th vctors ar (. and

6 (. Th opposit (backward-facin facs, lablld and, ncssarily hav unit surfac normal vctors which ar obtaind by multiplyin û and û by : (. and (. Th rmainin iht facs can similarly b roupd into pairs for which th surfac normal vctors ar qual in on componnt, qual and opposit in anothr, and zro in th third; furthrmor, th vctors associatd with ach pair diffr from thos associatd with th opposit pair only by a factor of. Thrfor, calculations quivalnt to thos abov may b usd to obtain ths vctors. It is convnint to dfin χ χ (.7a (.7b Th unit surfac normal vctors for th facs of th dodcahdron may thn b xprssd in

7 trms of ths two valus as [ χ T [ χ T [ χ T [ χ T [ χ T [ χ T [ χ T [ χ T [ χ T [ χ T [ χ T [ χ T û χ (.8a û û û χ χ χ χ χ χ χ û û û û χ χ χ χ (.8b (.8c (.8d (.8 (.8f (.8 (.8h (.8i (.8j (.8k (.8l Th constants χ and χ satisfy th followin rlationships: ( χ ( χ ( χ ( χ χ χ χ χ χ χ ( χ ( χ (.9a (.9b (.9c (.9d (.9 (.9f (.9

8 .: Drivation of th -B Matrix Th mthod dscribd in Chaptr for th drivation of th -B matrix cofficints in th cas of th first-ordr soundfild microphon is not applicabl whn th scond-ordr soundfild microphon is considrd. Whil w can writ quations similar to quations (. and (. for any of th zroth-ordr or first-ordr sinals, this lavs us in ach cas with twlv unknown matrix cofficints and only four quations. diffrnt approach is thrfor rquird. Lt a sinal H b a nral linar combination of th twlv -format sinals: H v v v v v v v v v v v v (. or, by substitutin for ach of th -format sinals, H G a b [ ju ju [ a bu ˆ [ a bu ˆ ˆ ˆ ju [ a bu ˆ ˆ [ a b ju [ a bu ˆ ˆ [ a b j [ a bu ˆ [ a b j [ a b [ a b j d ˆ j j j (. ju ju [ a bu ˆ [ a bu ˆ ˆ ˆ po W may obtain xprssions in trms of th matrix cofficints,, tc., for th cofficint of ach sphrical harmonic componnt in th Laplac sris xpansion of H by valuatin th intrals ivn in quation (.. For ach B-format sinal, w can thn quat ths xprssions to th dsird valus of th cofficints; th rsultin quations may thn b solvd to find th matrix cofficints. Not that this mthod is applicabl to th scond-ordr componnt sinals; a mthod mployin a coincidnt capsul approximation could not b usd, sinc it is th phas diffrncs btwn th capsuls that allow ths sinals to b obtaind. Furthrmor, this

9 approach automatically includs th dpndnc of th Laplac sris cofficints on, so that it is not ncssary to dtrmin th frquncy rspons functions sparatly from th matrix cofficints. W considr first th zroth-ordr componnt of H : π K π π / π / π π / π / H cos( φ dφ dθ { [ a b ˆ j d [ a b ˆ d [ a b ˆ j [ a b ˆ j d d [ a b ˆ j [ a b ˆ j d d j j [ a b ˆ [ a b ˆ d d j ˆ j ˆ [ a b ˆ ˆ d [ a b ˆ ˆ d u d u d j ˆ j ˆ [ a b ˆ ˆ d [ a b ˆ ˆ d u d u d } cos( φ dφdθ j ˆ d (. whr G K π a b (. For compactnss, w introduc th notation π π / Λ( f θ, φ ( f ( θ, φcos( φ dφ dθ π / (. Multiplyin out th bracktd factors in quation (. and usin th linarity proprty of intration, w obtain

10 K { a Λ j ( ( u j ˆ b Λ j ( ( u j ˆ Λ b Λ j ( ( u j ˆ Λ b Λ j ( ( u j ˆ Λ b Λ j ( ( u j ˆ Λ b Λ j ( ( u j ˆ Λ b Λ j ( ( j Λ b Λ j ( ( j Λ b Λ j ( ( j Λ bλ j ( ( j Λ bλ j ( ( j Λ b Λ j ( ( j Λ b Λ } a a a a a a a a a a a (. Rarranin ivs ak bk { { Λ Λ j j ( Λ( j j Λ( Λ( j j Λ( Λ( j j Λ( Λ( j j Λ( Λ( j j Λ( Λ( } ( u j ˆ ( u j ˆ Λ ( j ( j Λ Λ ( j ( j Λ Λ ( j ( j Λ Λ ( j ( j Λ Λ ( j ( j Λ Λ } (. It is now ncssary to valuat ach of th intrals in this xprssion. W first obsrv that 7

11 d dφ j j sin( φ j j j sin( φ j sin( φ cos( φ d sin( φ dφ (.7 so that (omittin th constant of intration j sin( φ cos( φ dφ j j sin( φ (.8 Now considr th first intral in th abov xprssion: π π / π / j ˆ d cos( φ dφdθ If a transformation of variabls can b found that taks to sin(φ, thn it will b possibl to us th rsult abov to valuat th intral. W know that χ χ (.9 and cos( θ cos( φ sin( θ cos( φ sin( φ (. so that ˆ u d ˆ χ cos( θ cos( φ χ sin( φ (. Th intrand is xprssd in trms of polar coordinat anls ( θ, φ, with which cartsian coordinats ( x, y, z may b associatd. Th transformd intrand will b xprssd in 8

12 trms of ( θ, φ or ( x, y, z. W rquir thn that sin( φ χ cos( θ cos( φ χ sin( φ (. and, sinc sin(φ is by dfinition th cosin of th anl mad with th z axis, this may also b statd as zˆ ˆ u (. Exprssin both in th oriinal coordinat systm, w thrfor hav ẑ (. and ẑ χ χ (. and it can b sn that th rquird coordinat transformation is a rotation about th y axis. standard rotation matrix may thrfor b usd: x y z M x y z (. whr M χ χ χ χ (.7 9

13 It may asily b vrifid that this is consistnt with th rquirmnts xprssd by quations (. and (.: χ χ χ χ χ χ (.8 Hnc w hav cos( θ cos( φ χ cos( θ cos( φ χ sin( θ cos( φ sin( θ cos( φ sin( φ χ cos( θ cos( φ χ sin( φ sin( φ (.9a (.9b (.9c Th intration now bcoms π π / π / j ˆ d cos( φ dφ dθ π π / π / j sin( φ cos( φ dφ dθ (. or Λ j ˆ d sin( ( j φ Λ ( (. whr th notation π π / Λ ( f ( θ, φ π / f ( θ, φ cos( φ dφ dθ (. indicats that w ar workin in th transformd coordinat systm. Evaluation of th transformd intral is straihtforward:

14 π π / π / j sin( φ cos( φ dφ dθ π dθ π jπ π sin( π π j ( π / π / j j j sin( φ j j [ j j sin( φ j cos( φ dφ π / π / (. Each of th lvn similar intrals valuats to th sam rsult. Th transformation matrics mployd ar listd in ppndix. W now considr th scond st of intrals, such as π π / π / j ˆ d cos( φ dφdθ pplyin th sam coordinat transformation yilds π π / π / j u ˆ ˆ d cos( φ dφdθ π π / π / sin( φ j sin( φ cos( φ dφ dθ (. Usin th mthod of intration by parts, w obtain sin( φ j sin( φ j cos( φ dφ j j j j j j j j sin( φ sin( φ j j sin( sin( φ sin( φ j j sin( sin( φ sin( φ j j sin( φ sin( φ j sin( φ k r j sin( φ k r φ φ sin( φ cos( φ dφ cos( φ dφ j sin( φ (.

15 Hnc, π / π / sin( φ j sin( φ cos( φ dφ j sin( φ k r sin( φ j j j k r k r k r j j sin( cos( k r sin( cos( j k r j j ( j j j j j j ( ( π / π / j (. and so π π / π / sin( φ j sin( φ π π / cos( φ dφ dθ dθ sin( φ π / jπ j ( j sin( φ cos( φ dφ (.7 ain, ach of th lvn similar intrals valuats to th sam rsult. Substitutin ths rsults into quation (. ivs ak bk { { G a b [ [ π [ jπ j ( } j ( } [ aj ( jbj ( (.8 Substantially th sam mthod may b usd to calculat th cofficints of th sphrical harmonics of hihr ordr, althouh thr ar som additional complications. To valuat intrals such as

16 π π / π / j ˆ d cos( θ cos( φ cos( φ dφ dθ it is ncssary to xprss cos( θ cos( φ (and in othr cass sin( θ cos( φ or sin(φ in trms of th transformd coordinat systm. This may b accomplishd by obsrvin that if cos( θ cos( φ cos( θ cos( φ sin( θ cos( φ M sin( θ cos( φ (.9 sin( φ sin( φ thn cos( θ cos( φ cos( θ cos( φ sin( θ cos( φ M sin( θ cos( φ (. sin( φ sin( φ Whn computin th scond-ordr cofficints, intrals such as π π / π / j ˆ d cos( θ cos ( φcos( φ dφdθ aris. Factors such as cos(θ cos ( φ can b xpandd, by application of trionomtric idntitis, into a polynomial in trms of cos( θ cos( φ, sin( θ cos( φ and sin(φ, which can thn b transformd usin invrs matrics as dscribd abov. Sinc a sphrical harmonic is (by dfinition a polynomial in th thr dirction cosins, such an xpansion will always b possibl for any sphrical harmonic of any ordr. Th intrals which rmain to b valuatd aftr ths transformations hav bn prformd bcom mor complicatd as th sphrical harmonic ordr is incrasd. Howvr, it so happns that ach such intral can b valuatd by applyin th mthod of intration by parts and usin a prviously found rsult. list of th intrals which aris is ivn in ppndix. Th followin xprssions ar obtaind for th cofficints of th zroth-ordr, first-ordr

17 and scond-ordr componnts of H : G a b G a b B,, [ G a b G a b [ aj jbj ( ( ( χ [ χ [ [ bj jaj ( bj ( ( ( χ [ χ [ [ bj jaj ( bj ( ( ( χ [ χ [ G a b [ bj jaj ( bj ( ( ([ [ [ [ [ [ jbj ( aj ( jbj ( G, a b, G a b [ [ jbj ( aj ( jbj ( ( χ [ [ ( χ [ [ jbj ( aj ( jbj ( G B, a b [ [ jbj ( aj ( jbj ( G B, a b [ [ jbj ( aj ( jbj ( (.a (.b (.c (.d (. (.f (. (.h (.i Sinc ths nin quations ar not sufficint to dtrmin th twlv matrix cofficints, w nxt considr th cofficints of th third-ordr sphrical harmonics. Procdin as bfor, w obtain:

18 B,,,, G a b B B,, G a b 8 ([ χ [ [ χ [ [ bj ( j7aj ( bj ( ([ χ [ G a b 8 G a b ( G χ a b 8 G a b χ [ [ bj ( j7aj ( bj ( ([ χ [ [ χ [ [ bj ( j7aj ( bj ( ([ χ [ [ χ [ [ bj ( j7aj ( bj ( [ [ χ [ [ bj ( j7aj ( bj ( ( [ χ [ [ χ [ [ bj ( j7aj ( bj ( (.a (.b (.c (.d (. (.f (. In total, w now hav fiftn quations (xcludin that involvin B,, which is indpndnt of th matrix cofficints. Howvr, th third-ordr Laplac sris cofficints ar not all linarly indpndnt; it may b obsrvd that, (, ( B,, B, (.a (.b (.c

19 Hnc, w hav obtaind xactly twlv linarly indpndnt quations in th twlv -B matrix cofficints; ths can thrfor now b dtrmind. For ach B-format sinal, w st th cofficint of th appropriat sphrical harmonic to a suitabl valu, and all othr cofficints to zro; w thn solv th rsultin st of quations for th twlv matrix cofficints. Not that, whil this suitabl valu is for th zroth-ordr and first-ordr componnt sinals, it taks diffrnt valus for th scond-ordr sinals bcaus of th scalin factors which appar in th dfinitions of th scond-ordr sphrical harmonics. For xampl, th cofficint B, is associatd with th sphrical harmonic sin(θ cos ( φ; hnc, to obtain th dsird polar rspons, B, must b st qual to. It is dsird that th -B matrix should b frquncy-indpndnt, as it is for th first-ordr soundfild microphon; hnc, th frquncy rspons functions ar omittd at this sta. For convninc w also disrard th G ( a b factor. Th -B matrix drivd thrfor dpnds only on th omtry of th array; th polar rsponss of th individual capsuls will b takn into account whn dsinin th non-coincidnc compnsation filtrs. Th matrix cofficints obtaind ar shown in tabl.. Not that ach of th sinals S, T and V is obtaind from a rctanular arranmnt of capsuls similar to that dscribd in sction..

20 W X Y Z R S T U V χ χ ( ( 8 χ χ ( ( 8 χ χ ( ( 8 χ χ ( ( 8 χ χ χ χ χ χ χ χ χ χ ( ( 8 χ χ ( ( 8 χ χ ( ( 8 χ χ ( ( 8 Tabl.: -B Matrix Cofficints for Scond-Ordr Soundfild Microphon.: Prsnc of Unwantd Sphrical Harmonic Componnts Usin th mthods dscribd in th prvious sction, it is possibl to obtain xprssions in trms of th -B matrix cofficints for hihr ordr sphrical harmonic componnts of H. Sinc ths matrix cofficints ar now known for ach of th B-format sinals, it is possibl to stablish th prsnc in ach of th B-format sinals of unwantd sphrical harmonic componnts. It is convnint to dfin th functions 7

21 И n ( ν, α j j j ( n n n [ n( ν jn ( α j(n ν jn( α ( n ( ν jn ( α [ j( ν ( nj ( α ( n j ( α (n ν j ( α n n djn ( α (n j( ν ν j dα n ( α n (. and to introduc normalisd capsul polar pattrn constants a a a b b b a b a (.a (.b Non of th sinals contains third-ordr sphrical harmonic componnts, sinc ths wr liminatd as part of th -B matrix dsin procdur. Th fourth-ordr Laplac sris cofficints ar G { ( [ [ ( [ } И 7, G, G { [ И ( 9 [ [ ( 9 [ } И, G, G 9 { [ И ( [ [ ( [ } И 7 B, G [ И B, G [ И (.a (.b (.c (.d (. (.f (. 8

22 B, G [ И B, G [ И (.h (.i Th fifth-ordr cofficints ar ivn by B,,,,,, G G G G G G G B, B, G B, {( χ [ ( χ [ } И {( χ [ χ [ } И {( χ [ ( χ [ } И 9 { 9 {( χ [ ( χ [ } И {( χ [ ( χ [ } И { {( χ [ ( χ [ } И χ [ ( χ [ } И ( χ [ ( χ [ } И (.7a (.7b (.7c (.7d (.7 (.7f (.7 (.7h (.7i (.7j 9

23 B, G 9 { ( χ [ ( χ [ } И (.7k Th sixth-ordr Laplac sris cofficints ar ivn by G { ( [ [ ( [ } И, G, G [ И {( [ [ ( [ } И 7, G 9, G 8 { [ И ( [ [ ( [ } И, G 9, G 7 [ И { ( [ [ ( [ } И B, [ И B, [ И 7 B, 9 [ И (.8a (.8b (.8c (.8d (.8 (.8f (.8 (.8h (.8i (.8j

24 B, [ И B, 9 [ И B, 88 [ И (.8k (.8l (.8m From ths nral xprssions, th Laplac sris cofficints for ach of th B-format sinals ar obtaind by substitutin in th known valus for th -B matrix cofficints. It so happns that th majority of th cofficints in th Laplac sris for ach sinal ar zro; only th non-zro cofficints ar listd hr. For W : { W} GИ 8, { W } GИ, { W} GИ 88, W GИ { } (.9a (.9b (.9c (.9d For X : { X } GИ,, { X } GИ, { X } GИ 9 (.a (.b (.c For Y : { Y } GИ B, (.a

25 { Y } GИ B, B,{ Y } GИ 9 (.b (.c For Z : 9 { Z } GИ, Z GИ, { Z} GИ 9 { } (.a (.b (.c For R : { R} GИ 7, { R} GИ, { R} GИ 8 { R} GИ, { R} GИ 8, R GИ, R GИ 7 { } { } (.a (.b (.c (.d (. (.f (. For S : 7 { S } GИ,, S GИ, { S } GИ { } (.a (.b (.c

26 7 { S } GИ, 88, { S } GИ 88 (.d (. For T : 7 { T } GИ B, B,{ T } GИ B,{ T } GИ 7 B,{ T } GИ 88 B,{ T } GИ 88 (.a (.b (.c (.d (. For U : 7 { U} GИ, { U} GИ 7, U GИ 88 7 { U} GИ, { U} GИ, U GИ 7, { U} GИ 8 { } { } (.a (.b (.c (.d (. (.f (.

27 For V : { V } GИ B, 8 B,{ V } GИ B,{ V } GИ 8 B, V GИ B, V GИ 8 { } { } (.7a (.7b (.7c (.7d (.7 From ths rsults it can b sn that W is corruptd by unwantd sphrical harmonics of ordr six; th first-ordr componnt sinals ar corruptd by spurious fifth-ordr sphrical harmonics; and th scond-ordr sinals contain both fourth-ordr and sixth-ordr unwantd sphrical harmonic componnts. In Chaptr, it was notd that th B-format sinals obtaind from th first-ordr soundfild microphon ar contaminatd by spurious scond-ordr or third-ordr sphrical harmonics. Th scond-ordr soundfild microphon thrfor rprsnts a considrabl improvmnt in this rspct. Sinc th hihr ordr sphrical harmonics hav cofficints which dpnd on hihr ordr sphrical Bssl functions, which in turn rmain small up until ratr valus of, so it may b xpctd that th maximum frquncy to which ffctiv coincidnc is maintaind will xcd that ivn by quation (.9. This advanta will, howvr, probably b opposd to som xtnt by th fact that th array radius is likly to larr for a scondordr soundfild microphon..: Non-Coincidnc Compnsation Filtrin Th author has not considrd th dsin of non-coincidnc compnsation filtrs in dtail; th dsin of practical approximations to th thortically idal charactristics is a mattr of practical implmntation rathr than fundamntal thory, and thrfor outsid th scop of this thsis. Nvrthlss, th followin obsrvations can b mad. By substitutin th cofficints ivn in tabl. into quation (., w obtain th

28 followin rsults: { W} G[ a j ( jb j ( { Z} G[ b j ( ja j ( b j ( B B B, GИ( a, X G b j ( { } [ ja j ( b j ( GИ,{ Y } G[ b j ( ja j ( b j ( GИ { R} G[ jb j ( a j ( jb j (,,,, GИ { S } G [ jb j ( a j ( jb j ( G И U G jb j ( a j ( jb j( G И T G jb j( a j ( jb j ( G И V G jb j( a j ( jb j ( G И { } [ { } [ { } [ (.8a (.8b (.8c (.8d (.8 (.8f (.8 (.8h (.8i Th factors of and in th xprssions for,,,, B,, and B, ar du to th factors of and which appar in th dfinitions of th corrspondin sphrical harmonics (s pa 8; thy thrfor do not imply th nd for compnsatory scalin of th sinals. Th impuls rsponss corrspondin to ths frquncy rspons functions may b found by takin th invrs Fourir transforms. Lt τ r c (.9

29 thn τω (.7 and mployin th invrs Fourir transforms of th sphrical Bssl functions dvlopd in Chaptr, w obtain ( t Fˆ G τ ( t Fˆ { G[ a j ( τω jb j ( τω } G τ ( ˆ t F G τ ( b t a τ r ( t { GИ τω } ( b t a τ t r ( t { GИ τω } τ τ ( b t a τ t b τ t a τ r ( t τ (.7a (.7b (.7c whr τ < t < τ rτ ( t othrwis (.7 It may b notd that th frquncy rsponss of th dsird sphrical harmonic componnts of th zroth-ordr and first-ordr sinals hav th sam form as in th cas of th first-ordr soundfild microphon. Filtrs that hav provd to iv accptabl rsults with th first-ordr soundfild microphon miht wll thrfor b qually suitabl for us with th scond-ordr microphon. In th cas of th scond-ordr sinals, th rquird filtrin is fundamntally diffrnt in on rspct. From quation (., it can b sn that th frquncy rspons function for th scond-ordr sphrical harmonic componnts bcoms zro for. This is bcaus w ar approximatin scond-ordr dirctional drivativs by takin th diffrnc btwn th outputs of first-ordr microphon capsuls. n intration with rspct to tim is thrfor ncssary, not to compnsat for th spacin of th capsuls, but as a fundamntal part of th

30 mthod bin utilisd to obtain th sinals. Th filtrin will thrfor srv a dual purpos so far as ths sinals ar concrnd, sinc at hihr frquncis compnsation for th ffcts of th capsul spacin will still b rquird. It must b notd that suitabl filtrs cannot b dsind on th basis of thory alon. Durin th dvlopmnt of th first-ordr soundfild microphon, it was found that althouh filtrs dvlopd from thortical analysis av a substantial improvmnt ovr arrays without filtrin, to obtain optimum prformanc it was ncssary to tak into account xprimntal information [7. Crtainly on xpcts that this will b th cas with th scond-ordr soundfild microphon as wll, sinc thr will invitably b dparturs from idal bhaviour which ar not rprsntd in th thortical tratmnt..: dditional B-Format Sinal Procssin..: Rotation & Elvation Th rotation and lvation controls for th scond-ordr soundfild microphon must clarly hav an idntical ffct on th zroth-ordr and first-ordr sinals as in th cas of th firstordr microphon. By trionomtric manipulation it may b stablishd that th rotation control modifis th scond-ordr componnt sinals as follows: R R S T U V cos( θ S sin( θ θ θ T sin( S cos( T cos( θ U sin(θ V sin( θ U cos(θ V (.7a (.7b (.7c (.7d (.7 Th ffct of th lvation control on th scond-ordr sinals may similarly b stablishd to b: R ( cos(φ R sin(φ S ( cos(φ U 8 (.7a 7

31 S T U V φ φ U sin( R cos( S sin( cos( φ T sin( φ V φ ( cos(φ R sin(φ S ( cos(φ U sin( φ T cos( φ V (.7b (.7c (.7d (.7..: Sid-Fir / End-Fir Switchin & Invrsion Th compnsatory sinal procssin rquird to facilitat nd-fir opration is: W W X Z Y Y Z X R S R U S T V U R U V T (.7a (.7b (.7c (.7d (.7 (.7f (.7 (.7h (.7i s in th cas of th first-ordr soundfild microphon, invrtd opration rquirs only a polarity rvrsal of som of th B-format sinals: Y Z S T V Y Z S T V (.7a (.7b (.7c (.7d (.7..: Dominanc Th author has provd that it is not possibl to xtnd th dominanc transformation to work 8

32 with th scond-ordr B-format sinal st. Suppos that th transformation can b xtndd to accommodat th scond-ordr componnt sinals. Th transformd sinals must b linar combinations of th xistin sinals; hnc, thr must xist cofficints W, X, Y, U and V, prsumably functions of, such that cos( θ W X cos( θ Y sin( θ U cos(θ V sin(θ (.77 Not that it is sufficint to considr only th pantophonic cas, sinc th priphonic cas ssntially rducs to this for φ. Complications du to th non-zro rspons of R for dirctions in th horizontal plan ar avoidd by usin a notional sinal which ncods only amplitud; whthr this notional sinal is in actuality proportional to W or to a combination of W and R is unimportant. Th scalin of W is also nlctd for convninc. W know that cos( θ [ ( cos( θ ( cos( θ sin( θ ( cos( θ ( cos( θ ( cos( θ (.78a (.78b sin( θ (.78c and also that cos( θ can b found by usin th idntity cos(θ cos ( θ sin ( θ (.79 By takin various valus of θ, it is possibl to nrat a st of simultanous quations which can thn b solvd for W, X, tc. i Lt θ. Thn (.8a 9

33 θ (.8b and W W X X W X U U U (.8 ii Lt θ 8. Thn θ 8 (.8a (.8b and W W X X W X U U U (.8 iii Lt θ 9. Thn [ cos( θ sin( θ (.8a (.8b (.8c and

34 [ [ [ [ cos( θ (.8 so U Y W U Y W (.8 iv Lt. 9 θ Thn [ (.87a cos( θ (.87b sin( θ (.87c and cos( θ (.88 so U Y W U Y W (.89 v Lt. θ Thn

35 ( ( [ ( ( [ (.9a ( ( ( ( ( ( ( ( ( ( cos( θ (.9b ( ( ( ( ( sin( θ (.9c and ( ( [ ( ( ( ( ( ( [ ( ( [ ( ( ( ( ( ( [ ( ( ( ( ( cos( θ (.9 so

36 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( V Y X W V Y X W V Y X W (.9 Thus fiv simultanous quations hav bn obtaind, sufficint to dtrmin th fiv cofficints. By appropriat manipulations w obtain ( X (.9a Y (.9b ( W (.9c U (.9d and ( ( ( V (.9 Howvr, if instad quation (.9 is obtaind by sttin, θ thn w hav instad ( ( ( ( ( V Y X W (.9 Th sin rvrsal on Y is of no consqunc, sinc w still obtain a valu of zro for Y as bfor. Howvr, th sin rvrsal on V mans that th solution is now ( ( ( V (.9

37 W hav thus obtaind a contradiction, sinc quations (.9 and (.9 cannot simultanously b satisfid. Hnc, it is not possibl to find cofficints W, X, tc., indpndnt of θ, such that quation (.77 is satisfid, and so it is not possibl to xtnd th dominanc transformation to th scond-ordr cas.