Jo Dattorro, Summr 998 Fractioal Samplig usig th sychroous Shah with applicatio to LIER PSE FIR FILER DESIG bstract W ivstigat th fudamtal procss of samplig usig a impuls trai, calld th shah fuctio [Bracwll], that is swd i tim by a costat offst rlativ to absolut tim. W obsrv that wh a origiatig aalog sigal is badlimitd, thr is littl diffrc btw fractioally timshiftig th corrspodig suc, ad sychroously samplig th tim-shiftd aalog sigal. W us ths rsults to xplai a curious paralll i th thortical aalysis of yps II ad IV liar phas FIR (fiit impuls rspos) filtrs that is maifst by a appart 4-priodicity i th "gralizd" [O&S] liar amplitud ad phas.
. h Cotiuous-im Viw of Samplig s(t) is th absolut-tim sychroous shah fuctio [O&S,pg.83,Eu.(3.6)] [Bracwll] s( t) δ( t ) S ( Ω) δ( Ω ) Ω f h asychroous shah (asychroous to tim zro) is similarly writt s ( t) δ( t ) S Ω δ( Ω ) for th samplig priod, ad for <. Samplig usig th asychroous shah rsults i x X s S ( t) x( t) s ( t) x t δ t ( Ω) X( Ω) S ( Ω) X( Ω ) x δ t From th poit of viw of th cotiuous-tim domai, this uatio says that wh w sampl a sigal asychroously to absolut-tim, ach rplicatio i th frucy domai bcoms multiplid by a complx costat / that is dpdt upo th rplicatio umbr ad th asychroy. Wh th th complx costat bcoms ual to for all ad w hav covtioal sychroous samplig. particularly otworthy cas is wh /, for th th complx costat altrats btw ± at vry rplicatio. his lads to 4/ priodicity i th frucy domai.
. h Suc-Domai Viw of Samplig h itrprtatio of asychroous samplig that w hav from Eu.() is sufficit for may purposs. But w ca writ Eu.() uivaltly as follows ad th driv aothr itrprtatio that is ituitivly appalig. X S Ω Ω Ω X Ω ( ) With a chag of variabl, w momtarily mix th discrt ad cotiuous-tim rprstatios X S X ( ) Ω Ω Ω I th cas that X(Ω) i Eu.() is badlimitd to th yuist frucy, th w may say whr X Ω Ω [ X( ) ] if badlimit d ( B) S X ( ) X( ) x( ) δ( t ) I Eu.(B) it is cssary to raliz that Ω dos ot cacl -/ bcaus th argumt of th formr is ot priodic th argumt of th discrt-tim liar phas trm -/ is priodic i. X( ) corrspods to th covtioally sampld discrt-tim domai sigal, ad is always -priodic. [O&S,pg.87,Eu.(3.)] ow if w sparat out th prmatly priodic part from Eu.(), w gt th Fourir trasform of a badlimitd suc X ( ) X ( ) x t δ t x δ t ( 3) ot from Eu.() that X S (Ω) is / -priodic oly wh is ay multipl of. h oly i that S cas dos it follow that X X( ) Ω, his is tru whthr or ot X(Ω) is badlimitd. Ω, for a itgr. [O&S,pg.87,Eu.(3.8)] 3
. Itrprtatio Comparig th cotiuous-tim domai sigal i Eu.() to that i Eu.(3), w s that th oly discrpacy is th absolut tim locatio of th sampls as dtrmid by th shah. h sampl valus ar th sam i both uatios. But th suc x[] is drivd from th sychroous samplig i Eu.(3) as th sampl valus of th tim-shiftd cotiuous-tim sigal x[ ] x( ) (4) Implicit is th covtio that a "suc" is always associatd with sychroous samplig, rgardlss of th actual tim origi of th sampl valus. From th poit of viw of th suc domai, th Fourir trasform is -priodic as idicatd by Eu.(3). I fact, th Fourir trasform of all sucs is -priodic by dfiitio, rgardlss of th udrlyig ad prhaps uow shah sychroicity. his is i sharp cotrast to th cotiuous-tim trasform Eu.() that is ot cssarily /-priodic. W must rmmbr to itrprt th discrt-tim phas as -priodic that is to say, -/ o th lft-had sid of Eu.(3) must b adustd at vry rplicatio as idicatd o th right-had sid. Bcaus w ca dsig a digital filtr havig a frucy rspos arbitrarily clos to - /, th from Eu.(3) w may coclud that wh th origial aalog sigal is badlimitd, thr is littl diffrc btw fractioally tim-shiftig a suc, ad sychroously samplig th corrspodig tim-shiftd aalog sigal. I othr words, from th poit of viw of th suc domai, both of th aformtiod opratios ar idtical. It is thrfor possibl to dsig a digital filtr for th purpos of dlayig ay suc by a fractio of a sampl. What actually bcoms dlayd, i th stady stat, is th badlimitd aalog sigal that uiuly corrspods to that suc. 3 I th cas that X(Ω) wr ot badlimitd, th discrt-tim liar phas trm - / o th lft-had sid of Eu.(3) bcoms ivalid bcaus th phas trm could ot b so simply drivd from th right-had sid. im shiftig ad samplig ar o logr itrchagabl, ad th suc x[] ow corrspods to som othr badlimitd aalog sigal that ca b dtrmid by frucy-domai aliasig of th origial aalog sigal. his -priodic adustmt is th covtioal spcificatio of liar phas i DSP. 3 h fact that thr is associatd with ay suc a uiu badlimitd cotiuous-tim sigal, is a cosuc of Whittar s horm. 4
3. Liar Phas s it turs out, w may us th framwor of th asychroous shah to xplai a curious 4 priodicity phomo that ariss commoly i fudamtal FIR filtr dsig. W bgi with a simpl xampl ad th graliz th rsults. Exampl (z) z - his trasfr fuctio corrspods to a suc h[] δ[] δ[-] hus w ow a priori that its Fourir trasform ( ) - < is a -priodic fuctio of frucy. Each trm of ( ) is also priodic du to th assumptio of liarity. ( ) ca b writt uivaltly as ( ) cos(/) -/ < (4) importat poit hr is to rcogiz that th xpotial trm has b xpadd i th frucy domai by a factor of as i Figur i.., th argumt of - / is 4-priodic as is cos(/). rg IV [ / ]/.5.5-4 - -.5 4 - -.5 - / Figur. Phas of xpadd i by factor of. 5
Eu.(4) is ot xactly i th form of Eu.(3) bcaus ithr cos(/) or th argumt of -/ ar -priodic. hat could b asily rmdid by taig th absolut valu of th cosi trm thus forcig -priodicity ito th argumt of th dlay trm. otwithstadig, ( ) rmais priodic ovrall as ruird by Eu.(3), ad by our itrprtatio of Eu.(3) thr xists a uiu badlimitd aalog sigal that w may associat with ( ). c, th corrspodig badlimitd aalog sigal h(t /) that bcoms sychroously sampld to yild h[] as i Eu.(4), is calculatd blow ad illustratd i Figur. ( ) h( t ) δ( t ) ( t ) h ( t ) cos h si( t ) ( t )( t ) h[ ] h( ) t d h(t -/)..8.6.4. -4-4 t Figur. h[] rsults wh sychroously sampld.. 6
From [O&S,pg.55,Eu.(5.35)] w hav th xprssio for gralizd liar phas i a discrttim sigal or systm ( ) ( ) -α β (5.35) whr ( ) is ral (icludig gativ ral), α is a uitlss costat but ca b cosidrd a ral umbr of sampls dlayd, ad th costat phas shift β has uits of radias. 4 Bcaus ( ) corrspods to a suc, it is cssarily -priodic. ( ) is ot cssarily priodic, howvr. h argumt of -α has th sam priodicity as ( ), ad th priodicity is wh α is a itgr. I Eu.(4) w saw that wh α/, th th priodicity is 4. Bcaus of ths facts, w prfr a otatio that rmids us that th priodicity is rlatd to α ( ) ( α ) -α β ( α ) ( ) (5) Suc symmtry is a sufficit coditio for gralizd liar phas. hr ar oly two cass of symmtry with rgard to h[] that ar of itrst symmtric ad ati-symmtric. symmtric suc (FIR yp I, II) tas tim-symmtric sampls about α of cotiuous fuctios of th form show i Figur 3(a), whil a ati-symmtric suc (FIR yp III, IV) tas timsymmtric sampls about α of fuctios such as that dpictd i Figur 3(b). h (t -α) (a) α t h o (t -α) (b) α t Figur 3. (a) ypical symmtric impuls rspos ctrd about α. β. (b) ypical ati-symmtric impuls rspos ctrd about α. β/. 4 h choic of β is ot arbitrary udr gralizd liar phas. If it wr arbitrary, it would rsult i phas distortio as opposd to "shift". 7
Suc symmtry is a sufficit coditio for liar phas, but ot a cssary coditio. y suc havig a frucy rspos that ca b put ito th gral form of Eu.(5) will b cosidrd liar i phas. Wh α, th Fourir trasform of Figur 3(a) is pur ral (zro phas,β) whras th trasform of Figur 3(b) is pur imagiary (costat phas shift,β/). ssumig that th cotiuous-tim impuls rsposs i Figur 3 ach hav a corrspodig badlimitd frucy rspos, 5 ay valu of α ad ay amout of shah asychroy will rsult i a liar phas suc if w ta a ifiit umbr of sampls abov th yuist rat. [O&S,ch.5.7.] h suc symmtry coditio oly bcoms cssary for liar phas wh w dmad fiit lgth impuls rspos i.., FIR dsig. hat cssity th limits th availabl choics of giv α. s mtiod arly o, th shah asychroy cass ad / ar otworthy. W furthr distiguish ths two cass bcaus thy rprst th oly cass of shah tim-symmtry about absolut tim. s suc symmtry is of itrst, so ar ths two cass of asychroy. But as w alrady lard from Eu.(3) which is th poit of viw of th suc domai, it mas o diffrc whthr w tim-shift a suc, or sychroously sampl th corrspodig tim-shiftd badlimitd aalog sigal. c without loss of grality, w choos to b zro bcaus o cass of suc symmtry will b lost as a cosuc. othlss, w discovr a strog bod btw asychroous samplig, Eu.() ad Eu.(3), ad th gralizd liar phas dscriptio, Eu.(5). h rlatioship may b xprssd, h ( α ) ( ) ( α ) h ( t) δ( t α ) ho ( t) δ( t α ) h ( t α ) δ( t ) ho ( t α ) δ( t ) ( t ) ( ) d α β α β α α β t β β β β ( 6) ( 7) whr α ow assums th arlir rol of. Eu.(6) is ot cssarily -priodic, whil Eu.(7) is that is prcisly th sam rlatioship as Eu.() to Eu.(3). Li i Eu.(), α i Eu.(6) cotrols th priodicity of th gralizd amplitud ( α ). With st to, thr rmai oly two choics of α which produc suc symmtry, hc liar phas α ad α/. h 5 h xplaatio of this ruirmt for badlimitig is ust li th itrprtatio of Eu.(3). 8
pur ralss or imagiariss of ( α ) β is th dtrmid solly by th h(t) wavform symmtry. h dsir for liar phas is that which dmads suc symmtry as wll. c, it is mor difficult to dsig fractioal dlay FIR filtrs for othr valus of dlay, α. Wh th sampl valus h[] ar th ta as i Eu.(4), h [ ] h ( ) yp I: α, β th w hav what is calld th yp I liar phas FIR filtr. [O&S,pg.57] Similarly, th thr othr typs of liar phas FIR filtrs that aris bcaus of suc symmtry ca b foud h[ ] h ( α ) h[ ] h ( ) o h[ ] h ( α ) o yp II: yp IV: α yp III: α, α,, β β β h charactristics of both th suc ad its trasform ar summarizd i abl alog with th valus of th othr rlatd paramtrs. W s from th tabl that α/ will rsult i a 4-priodic gralizd amplitud (ad gralizd liar phas). h sam also holds tru wh α uals ay odd multipl of /. abl. Symmtry ad Sychroicity FIR yp I II III IV h[] sym. sym. ati-sym. ati-sym. ( α ) Sym. Sym. ti-sym. ti-sym. Priodicity 4 4 ( α ) Ral Ral Imag. Imag. α / / β / / 9
Exampl ( ) - / < (8) h magitud ad phas -/ of this frucy rspos ar priodic by dfiitio. h corrspodig badlimitd aalog sigal h (t -/) that bcoms sychroously sampld to yild h[] as i Eu.(4), is calculatd blow ad show i Figur 4. ( ) h( t α ) δ( t ) h( t α ) h ( t ) t d si( ( t ) ) ( t ) si( ( ) ) h[ ] h( ) α, [ O&S, ( 5. 8) c ] ( ) h (t -/).8.6.4. -4-4 -. t Figur 4. h[] rsults wh sychroously sampld..
From our discussio of liar phas, w s that ( ) i Eu.(8) ca b xprssd i th gralizd form of Eu.(5) with β ( ) ( / ) - / < (9) Wh writt i this form, gralizd amplitud ad liar phas, w ow that ( / ) is 4-priodic as is -/ th gralizd liar phas, bcaus α/ i Eu.(6). his priodicity is illustratd i Figur 5. ( ) rmais priodic. ( / ).5 (a) -4-4 / -.5 - rg IV [ / ]/.5.5 (b) -4 - -.5 4 - -.5 - / Figur 5. Gralizd amplitud (a) ad phas (b) for Exampl ar 4 priodic. hat ( / ) must altrat btw ± with a priod of 4 ca b vrifid by shiftig th gralizd liar phas form Eu.(9) whr o assumptio is mad rgardig th priodicity of its idividual trms ( ) ( / ) - / ( () ) ( ()/ ) - ()/ whr - ()/ - - /.
4 pplicatio: LIER PSE FIR FILER DESIG by IDF Cooboo h tchiu of frucy-domai samplig ad IDF is usful wh it is dsird to ma a filtr havig arbitrary frucy rspos. h rsult is a discrt impuls rspos that loos li.5.5 this 5 5 5 3 (ordr odd), or li this.5 (ordr v). t th -.5 -.5 5 5 5 3 ris of ladig o dow a gard path, w prst a mthod of solutio basd upo th cocpts prstd arlir gralizd amplitud ad liar phas. W poit out that whil this may b a ituitiv approach i o ss, thr is a asir approach, basd o th familiar cocpt of magitud ad phas, prstd at th d of this sctio. W cosidr oly th cas that th dsird frucy rspos ( ) is pur ral ad v 6 i.., FIR yps I ad II. W ma th impuls rspos corrspodig to ( ) causal via th dlay whr dsird filtr ordr. If w forgt to icorporat th dlay trm, th w will always gt a impuls rspos that loos li th " v" cas but with a xtra sampl tacd o o d wh is actually odd. hat xtra sampl dstroys th tim-domai symmtry ad th liar phas. (Implicit i this mthod is som udrlyig cotiuous-tim sic() that is big sampld i o of two ways dpdig upo th ordr.) h IDF is h[ ] ( ) d ( ( ) ) d Rcall that wh h[] is fiit lgth, th DF is simply th DF valuatd i frucy by som simpl fuctio of. [ ] ( ) f h w may approximat h[] via th IDF. It is critical that th IFF iput buffr [] b aligd with absolut frucy. ticipatig that, w writ th IDF i th uivalt form: h ( ) [ ] ( ) d ( ) d h scod itgral i () holds th lft half of th first priodic-rplicatio 7 of ( ). 6 FIR filtr yps III ad IV produc phas distortio bcaus of thir costat / phas shift. 7 W assum that ( () ) ( ). c th dlay trm i must b corcd ito priodicity via th subtractio of. o pr 3
Lt th umbr of frucy domai sampls b ual to, th IDF lgth th lgth of our approximatio to h[]. s i th tim domai, w also cosidr two mthods of samplig i th frucy domai, ad w cosidr v or odd udr ach mthod sparatly. For liar phas to occur, th frucy domai sampls must also b symmtrical. But i th discrt frucy domai, th symmtry w ar cocrd with is circular i.., th rsultig pattr of sampls wh viwd alog th uit circl, ot alog th -axis. [Rabir/Gold,pg.3] [ ] d d h DC-ligd frucy domai samplig: Cas, v, odd: yp I liar phas FIR. From uatio (), [ ] [ ] ot that [-] []. [ ] othr usful rprstatio rsults wh w st -. For th w gt, /. his formulatio allows us to thi i trms of th two-sidd spctrum. 3 o pr 3
[ ] d d h DC-ligd frucy domai samplig: (cot.) Cas, odd, v: yp II liar phas FIR. From uatio (), [ ] [ ] [ ] ot that [-] []. h zro is dmadd by th dsird symmtry of th impuls rspos. [Opphim/Schafr,D-SP,pg.65] altrat rprstatio for th two-sidd spctrum is obtaid by sttig - : [ ] /. his formulatio allows us to thi i trms of th two-sidd spctrum. 4 o pr 3
[ ] d d h o-dc-ligd frucy domai samplig: Cas 3, v, odd: yp I liar phas FIR. From uatio (), [ ] [ ] ot that [-] []. { } [ ] othr usful rprstatio rsults wh w st -. For th w gt, ( /)/. his formulatio allows us to thi i trms of th two-sidd spctrum. h valuatio of th IDF rl at th spcifid frucis dmads that h h 5 o pr 3
[ ] d d h o-dc-ligd frucy domai samplig: (cot.) Cas 4, odd, v: yp II liar phas FIR. From uatio (), [ ] [ ] ot that [-] []. { } { } altrat rprstatio for th two-sidd spctrum is obtaid by sttig - : { } { } ( /)/. his formulatio allows us to thi i trms of th two-sidd spctrum. h valuatio of th IDF rl at th spcifid frucis dmads that h h 6 o pr 3
4. Easir Way h forgoig cooboo was prpard from th poit of viw of gralizd amplitud ad liar phas. From our study Phas Rspos (o this sit) w lar that it is i fact asir to solv th problm usig th poit of viw of magitud ad phas for all th cass. Us poits ad th IDF ot a powr-of- typ program. Cas ) is v. Just sampl th frucy domai usig th magitud ad phas rprstatio. h phas is always priodic i. h basbad ad first rplicatio of phas ar rspctivly / ad ( )/. c thir diffrc is a multipl of at th discotiuity. Cas ) is odd. gai sampl th frucy domai usig th magitud ad phas rprstatio. h phas diffrc is a odd multipl of at th phas discotiuity. hat accouts for th ruird sig ivrsio. Rmmbr that for liar phas ad odd, th sampl at z must b zro. Cas 3) is v. h mai diffrc hr is that th frucy sampls miss DC. h phas trm is / (/)/ / (/)/. gai, th phas diffrc at th phas discotiuity is a multipl of. h problm with this mthod is that th IDF buffr, which ruirs absolut frucy aligmt, is loadd with lft shiftd (by / bi) frucy domai DF sampls. So aftr IDF, w must compsat i th tim domai via th modulatio trm /. Cas 4) is odd. h frucy sampl at z is missd, so ot of cocr. h phas diffrc is a odd multipl of at th phas discotiuity, which accouts for th ruird sig ivrsio. im domai compsatio is agai ruird bcaus of th frucy shift ito th IDF buffr. Closd-form solutios ca b foud i [ccllla,pp.5-5]. 7 o pr 3
Rfrcs [O&S] la V. Opphim, Roald W. Schafr, Discrt-im Sigal Procssig, 989, Prtic-all, Ic., Eglwood Cliffs, J 763 US, Itrt: http://www.prhall.com/ [Bracwll] Roald. Bracwll, h Fourir rasform ad Its pplicatios, Scod Editio, Rvisd, 986, cgraw ill [ccllla] ccllla, Burrus, Opphim, t al., Sigal Procssig usig atlab 5, Prtic all, 998 [Rabir/Gold] Rabir, Gold, hory ad pplicatios of Digital Sigal Procssig, Prtic all, 975 8 o pr 3