{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 43 "90 degree phase differe nce IIR allpass pair" }}{PARA 257 "" 0 "" {TEXT -1 35 "July 2003, Olli Niemitalo, o@iki.fi" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The basic building block is:" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "H_sect := (z, a) -> (a^ 2 - z^(-2)) / (1 - a^2 * z^(-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'H_sectGR6$%\"zG%\"aG6\"6$%)operatorG%&arrowGF)*&,&*$)9%\"\"#\"\"\"\" \"\"*&F3F3*$)9$\"\"#F3!\"\"!\"\"F3,&F4F4*&*$F0F3F3*$)F8\"\"#F3F:F;F:F) F)F)" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 86 "Equation 1. Allpass sec tion taking one multiplication when a^2 is a known coefficient." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "E ach such allpass section has poles located on the real axis at " } {TEXT 257 1 "a" }{TEXT -1 6 " and -" }{TEXT 260 1 "a" }{TEXT -1 17 ", \+ and zeros at 1/" }{TEXT 259 1 "a" }{TEXT -1 8 " and -1/" }{TEXT 258 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We construct two allpass filters from such sections:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "H_1 := z -> H_sect(z, 0.692 3878)*H_sect(z, 0.9360654322959)*H_sect(z, 0.9882295226860)*H_sect(z, \+ 0.9987488452737)*z^(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$H_1GR6# %\"zG6\"6$%)operatorG%&arrowGF(*&**-%'H_sectG6$9$$\"(yQ#p!\"(\"\"\"-F/ 6$F1$\".fHKa1O*!#8F5-F/6$F1$\".goA&H#))*F:F5-F/6$F1$\".PFX)[()**F:F5\" \"\"F1!\"\"F(F(F(" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 56 "Equation 2 . First allpass filter, delayed by one sample." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "H_2 := z - > H_sect(z, 0.4021921162426)*H_sect(z, 0.8561710882420)*H_sect(z, 0.97 22909545651)*H_sect(z, 0.9952884791278);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$H_2GR6#%\"zG6\"6$%)operatorG%&arrowGF(**-%'H_sectG6$9$$\".EC; @>-%!#8\"\"\"-F.6$F0$\".?C)3rh&)F3F4-F.6$F0$\".^ca4Hs*F3F4-F.6$F0$\".y 7z%)G&**F3F4F(F(F(" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 123 "Equation 3. Second allpass filter, has 90 (+- 0.7) degrees relative phase to f irst filter over a long range of frequencies." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plot(\{argu ment(H_1(exp(I*freq))), argument(H_2(exp(I*freq)))\}, freq=0..Pi, phas e = -Pi..Pi);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7[r7$\"\"!F( 7$$\"+01$*R@!#7$!+HWmO5!\"*7$$\"+57')zUF,$!+QOIU=F/7$$\"+:=z>kF,$!+Aia BCF/7$$\"+?Csf&)F,$!+-NbfGF/7$$\"+.`'*p5!#6$\"+Jr\"FB$\"+z/Q;BF/7$$F+F B$\"+%o\\%Q>F/7$$\"+En\"zc#FB$\"+:ZPA;F/7$$\"+ZG!f*HFB$\"+S'Q:N\"F/7$$ \"+n*))QU$FB$\"+]:U:6F/7$$F2FB$\"+gmf*>(!#57$$\"+^M$e8&FB$\"+*3L=(RFeo 7$$\"+$p0=*fFB$\"+WfMW7Feo7$$\"+LzxZoFB$!+)[0E7\"Feo7$$\"+w;LP$)FB$!+$ \\1>j%Feo7$$\"+Feo$!+JPU@?F/ 7$$\"+![:zG#Feo$!+&)o#F/7 $$\"+`lwH_Feo$\"+>sEHCF/7$$\"+4)3T*eFeo$\"+92p(>#F/7$$\"+ykYxlFeo$\"+S A=\")>F/7$$\"+&Go$zrFeo$\"+eP<0=F/7$$\"+gr'p&yFeo$\"+@Fe?;F/7$$\"+?$[t `)Feo$\"+bgIZ9F/7$$\"+S;-$>*Feo$\"+^b#**G\"F/7$$\"+HeV)y*Feo$\"+$\\QR: \"F/7$$\"+*3W'\\5F/$\"+DE4&***Feo7$$\"+/9i46F/$\"+3$)HR()Feo7$$\"+\"G* Qz6F/$\"+E&[*GtFeo7$$\"+tc9T7F/$\"+lH4=hFeo7$$\"+BA!*38F/$\"+3qDA[Feo7 $$\"+#[AMP\"F/$\"+/J=8OFeo7$$\"+.EuS9F/$\"+A0eqBFeo7$$\"+!\\jD]\"F/$\" +R`wS7Feo7$$\"+ocCp:F/$\"+-Z@J=F/$!+7I1%z%Feo7 $$\"+wY,(*=F/$!+\")3p^gFeo7$$\"+3Ypg>F/$!+5m(*)H(Feo7$$\"+:.SJ?F/$!+%H \\vs)Feo7$$\"+QD$\\4#F/$!+A^(e+\"F/7$$\"+Ynwi@F/$!+#)z(R:\"F/7$$\"+@YB CAF/$!+(*4Z%H\"F/7$$\"+W_V\"H#F/$!+=\"\\hX\"F/7$$\"+$zlYN#F/$!+F!\\th \"F/7$$\"+O*f2U#F/$!+t'3qz\"F/7$$\"+U!z`[#F/$!+_C$e)>F/7$$\"+d#HIb#F/$ !+fL]+AF/7$$\"+bW==EF/$!+wUoFCF/7$$\"+b]\"[o#F/$!+oam'o#F/7$$\"+)[ayr# F/$!+#HDx#GF/7$$\"+?R*3v#F/$!+]!f\"zHF/7$$\"+CQ2mFF/$!+(\\`F0$F/7$$\"+ FPD\"y#F/$!+65?HJF/7$$\"+;7:$y#F/$!+83(*QJF/7$$\"+/([]y#F/$\"+3`RMJF/7 $$\"+#>Ypy#F/$\"+[h_CJF/7$$\"+!oV))y#F/$\"+=kg9JF/7$$\"+c'QEz#F/$\"+3G h%4$F/7$$\"+JOV'z#F/$\"+4&4W2$F/7$$\"+$eBS!GF/$\"+?INLIF/7$$\"+MNh6GF/ $\"+2NR\"*HF/7$$\"+*H4k%GF/$\"+v22'y#F/7$$\"+k]?\")GF/$\"+2-2bDF/7$$\" +7oK7HF/$\"+r#e-K#F/7$$\"+f&[M%HF/$\"+y/U]?F/7$$\"+L3jwHF/$\"+?=t5F/7$$\"+*fFX7$F/$!+@'*o@BF/7$$\"+K2mEJF/$!+ #z$QUDF/7$$\"+jQzGJF/$!+jMC#z#F/7$$\"+&*p#48$F/$!+0Oq\"3$F/7$$\"+D,1LJ F/$\"+;)fY&GF/7$$\"+dK>NJF/$\"+af!)=CF/7$$\"+)QEt8$F/$\"+Vw(z$=F/7$$\" +>&f%RJF/$\"+EsoL5F/7$$\"+]EfTJF/$\"+y#yB$=!#:-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-F$6$7jqF'7$F*$!+u\"z'>DF/7$F1$\"+$eT)eGF/7$F6$\"+Y$[hH#F/7$F; $\"+Z@#['=F/7$F@$\"+xkr6:F/7$FF$\"+&p9R@\"F/7$FK$\"+'[#fx&*Feo7$FP$\"+ M^YQtFeo7$FU$\"+ib_rNFeo7$FY$\"+f#HZt%FB7$Fhn$!+EH,m@Feo7$F]o$!++7ktWF eo7$Fbo$!+9u#fQ)Feo7$Fgo$!+Qwyj6F/7$F\\p$!+t!G?W\"F/7$Fap$!+\"4hYo\"F/ 7$Ffp$!+&e`K/#F/7$F[q$!+?$=AM#F/7$F`q$!+!eyzf#F/7$Feq$!+=MT@GF/7$$\"+R sNk8Feo$!+GXvNHF/7$$\"+(=:\"[9Feo$!+F+FVIF/7$$\"+vY0p9Feo$!+cA=pIF/7$$ \"+iT***[\"Feo$!+7:t%4$F/7$$\"+\\O$4^\"Feo$!+I(G*>JF/7$$\"+OJ(=`\"Feo$ \"+Q4SQJF/7$$\"+5@vt:Feo$\"+/[n*3$F/7$Fjq$\"+W.?UIF/7$$\"+\")p9$y\"Feo $\"+1=_jGF/7$F_r$\"+500&[FB7$F]y$ !+_E(f?\"Feo7$Fby$!+!HRYu#Feo7$Fgy$!+^-)*ySFeo7$F\\z$!+k^>-cFeo7$Faz$! +)=@%[oFeo7$Ffz$!+R&4qD)Feo7$F[[l$!+(*G6u%*Feo7$F`[l$!+P>Uy5F/7$Fe[l$! +L%*G,7F/7$Fj[l$!+MY2G8F/7$F_\\l$!+a&pOW\"F/7$Fd\\l$!+5\"4zc\"F/7$Fi\\ l$!+^S&pp\"F/7$F^]l$!+'G#o4=F/7$Fc]l$!+A\"pA$>F/7$Fh]l$!+D@Jg?F/7$F]^l $!+ml[(=#F/7$Fb^l$!+ew)GJ#F/7$Fg^l$!+*4\"ebCF/7$F\\_l$!+l9n(e#F/7$Fa_l $!+iMtLFF/7$Ff_l$!+p(z:(GF/7$$\"+L\\$yD#F/$!+Yka\\HF/7$F[`l$!+!\\H'HIF /7$$\"+#)GC2BF/$!+d]5oIF/7$$\"+>00BBF/$!+i$Hr5$F/7$$\"+IC+FBF/$!+ZX(p6 $F/7$$\"+RV&4L#F/$!++j&o7$F/7$$\"+&HIHL#F/$!+V5\"=8$F/7$$\"+\\i!\\L#F/ $!+0^xOJF/7$$\"+/A)oL#F/$\"+bnVTJF/7$$\"+d\"e)QBF/$\"+oQXOJF/7$$\"+w>w YBF/$\"+;mU;JF/7$F``l$\"+PKC'4$F/7$$\"+lGr(Q#F/$\"+6u15IF/7$Fe`l$\"+!o d2#HF/7$Fj`l$\"+9?fNFF/7$F_al$\"+)fRN_#F/7$Fdal$\"+;:$oH#F/7$Fial$\"+) *pON?F/7$Fcbl$\"+G-+P)GP(Feo7$Fifl$\"+o'*)*)p%Feo7$F^gl$\"+)H![q8Feo7$Fcgl$!+\"y%*\\o#Feo 7$Fhgl$!+zw=/]Feo7$F]hl$!+V$yJn(Feo7$Fbhl$!+Jgm\"3\"F/7$Fghl$!+TZsi9F/ 7$F\\il$!+\\BL0F/7$Ffil$!+4Dj3BF/7$F[jl$!+m\\v*p#F/7 $$\"+!o&f4JF/$!+K8a6GF/7$F`jl$!+5WYIHF/7$$\"+V>'Q6$F/$!+`dhdIF/7$Fejl$ \"+&y=))3$F/7$Fjjl$\"+a+0zFF/7$F_[m$\"+4WR-CF/7$Fd[m$\"+kj^y@F/7$Fi[m$ \"+LbUA>F/7$F^\\m$\"+-LuC;F/7$Fc\\m$\"+0]%=F\"F/7$Fh\\m$\"+`oL3%)Feo7$ F]]m$\"+8!3]y#Feo7$Fb]m$!+kY^giFeo7$Fg]m$!+!='eTJF/-F]^m6&F_^mF(F`^mF( -%+AXESLABELSG6$Q%freq6\"Q&phaseF]fn-%%VIEWG6$;F($\"+aEfTJF/;$!+aEfTJF /Fcfn" 2 467 467 467 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20530 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 46 "Figure 1. 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Phase difference \+ of H_2 and H_1, detail near 0 Hz." }}{PARA 268 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "You can use the 90 degree phase difference of the filters as such, or you can construct a complex fil ter and analyze its properties:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "H := z -> 0.5*(H_2(z)+I*H_1(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HGR6#%\"zG6\"6$%)operatorG%&arrowGF(,&-% $H_2G6#9$$\"\"&!\"\"*&%\"IG\"\"\"-%$H_1GF/F6F1F(F(F(" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 74 "Equation 4. 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