Fractional Sampling using the Asynchronous Shah with application to LINEAR PHASE FIR FILTER DESIGN
|
|
- Shannon Manning
- 5 years ago
- Views:
Transcription
1 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.
2 . 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.
3 . 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
4 . 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
5 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 [ / ]/ / Figur. Phas of xpadd i by factor of. 5
6 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 -/) t Figur. h[] rsults wh sychroously sampld.. 6
7 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
8 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
9 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
10 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 -/) t Figur 4. h[] rsults wh sychroously sampld..
11 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 / rg IV [ / ]/.5.5 (b) / 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 - ()/ - - /.
12 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 (ordr odd), or li this.5 (ordr v). t th 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
13 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
14 [ ] 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
15 [ ] 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
16 [ ] 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
17 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
18 Rfrcs [O&S] la V. Opphim, Roald W. Schafr, Discrt-im Sigal Procssig, 989, Prtic-all, Ic., Eglwood Cliffs, J 763 US, Itrt: [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, o pr 3
DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1
DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT
More informationz 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z
Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist
More informationDigital Signal Processing, Fall 2006
Digital Sigal Procssig, Fall 6 Lctur 9: Th Discrt Fourir Trasfor Zhg-Hua Ta Dpartt of Elctroic Systs Aalborg Uivrsity, Dar zt@o.aau.d Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Cours at a glac MM Discrt-ti
More informationDiscrete Fourier Transform (DFT)
Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial
More informationDiscrete Fourier Transform. Nuno Vasconcelos UCSD
Discrt Fourir Trasform uo Vascoclos UCSD Liar Shift Ivariat (LSI) systms o of th most importat cocpts i liar systms thory is that of a LSI systm Dfiitio: a systm T that maps [ ito y[ is LSI if ad oly if
More information15/03/1439. Lectures on Signals & systems Engineering
Lcturs o Sigals & syms Egirig Dsigd ad Prd by Dr. Ayma Elshawy Elsfy Dpt. of Syms & Computr Eg. Al-Azhar Uivrsity Email : aymalshawy@yahoo.com A sigal ca b rprd as a liar combiatio of basic sigals. Th
More informationPart B: Transform Methods. Professor E. Ambikairajah UNSW, Australia
Part B: Trasform Mthods Chaptr 3: Discrt-Tim Fourir Trasform (DTFT) 3. Discrt Tim Fourir Trasform (DTFT) 3. Proprtis of DTFT 3.3 Discrt Fourir Trasform (DFT) 3.4 Paddig with Zros ad frqucy Rsolutio 3.5
More informationSystems in Transform Domain Frequency Response Transfer Function Introduction to Filters
LTI Discrt-Tim Systms i Trasform Domai Frqucy Rspos Trasfr Fuctio Itroductio to Filtrs Taia Stathai 811b t.stathai@imprial.ac.u Frqucy Rspos of a LTI Discrt-Tim Systm Th wll ow covolutio sum dscriptio
More informationChapter 4 - The Fourier Series
M. J. Robrts - 8/8/4 Chaptr 4 - Th Fourir Sris Slctd Solutios (I this solutio maual, th symbol,, is usd for priodic covolutio bcaus th prfrrd symbol which appars i th txt is ot i th fot slctio of th word
More informationDiscrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform
Discrt Fourir Trasform Dfiitio - T simplst rlatio btw a lt- squc x dfid for ω ad its DTFT X ( ) is ω obtaid by uiformly sampli X ( ) o t ω-axis btw ω < at ω From t dfiitio of t DTFT w tus av X X( ω ) ω
More informationLectures 9 IIR Systems: First Order System
EE3054 Sigals ad Systms Lcturs 9 IIR Systms: First Ordr Systm Yao Wag Polytchic Uivrsity Som slids icludd ar xtractd from lctur prstatios prpard by McCllla ad Schafr Lics Ifo for SPFirst Slids This work
More information1985 AP Calculus BC: Section I
985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b
More informationChapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris
More informationChapter 3 Fourier Series Representation of Periodic Signals
Chptr Fourir Sris Rprsttio of Priodic Sigls If ritrry sigl x(t or x[] is xprssd s lir comitio of som sic sigls th rspos of LI systm coms th sum of th idividul rsposs of thos sic sigls Such sic sigl must:
More informationA Simple Proof that e is Irrational
Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural
More informationVI. FIR digital filters
www.jtuworld.com www.jtuworld.com Digital Sigal Procssig 6 Dcmbr 24, 29 VI. FIR digital filtrs (No chag i 27 syllabus). 27 Syllabus: Charactristics of FIR digital filtrs, Frqucy rspos, Dsig of FIR digital
More informationChapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
More informationProblem Value Score Earned No/Wrong Rec -3 Total
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING ECE6 Fall Quiz # Writt Eam Novmr, NAME: Solutio Kys GT Usram: LAST FIRST.g., gtiit Rcitatio Sctio: Circl t dat & tim w your Rcitatio
More informationStatistics 3858 : Likelihood Ratio for Exponential Distribution
Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai
More informationAPPENDIX: STATISTICAL TOOLS
I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio.
More informationPURE MATHEMATICS A-LEVEL PAPER 1
-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio
More informationOption 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges.
Optio Chaptr Ercis. Covrgs to Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Divrgs 8 Divrgs Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Covrgs to Covrgs to 8 Proof Covrgs to π l 8 l a b Divrgt π Divrgt
More informationChapter 11.00C Physical Problem for Fast Fourier Transform Civil Engineering
haptr. Physical Problm for Fast Fourir Trasform ivil Egirig Itroductio I this chaptr, applicatios of FFT algorithms [-5] for solvig ral-lif problms such as computig th dyamical (displacmt rspos [6-7] of
More informationFrequency Response & Digital Filters
Frquy Rspos & Digital Filtrs S Wogsa Dpt. of Cotrol Systms ad Istrumtatio Egirig, KUTT Today s goals Frquy rspos aalysis of digital filtrs LTI Digital Filtrs Digital filtr rprstatios ad struturs Idal filtrs
More informationA Review of Complex Arithmetic
/0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd
More informationDFT: Discrete Fourier Transform
: Discrt Fourir Trasform Cogruc (Itgr modulo m) I this sctio, all lttrs stad for itgrs. gcd m, = th gratst commo divisor of ad m Lt d = gcd(,m) All th liar combiatios r s m of ad m ar multils of d. a b
More informationSolution to 1223 The Evil Warden.
Solutio to 1 Th Evil Ward. This is o of thos vry rar PoWs (I caot thik of aothr cas) that o o solvd. About 10 of you submittd th basic approach, which givs a probability of 47%. I was shockd wh I foud
More informationH2 Mathematics Arithmetic & Geometric Series ( )
H Mathmatics Arithmtic & Gomtric Sris (08 09) Basic Mastry Qustios Arithmtic Progrssio ad Sris. Th rth trm of a squc is 4r 7. (i) Stat th first four trms ad th 0th trm. (ii) Show that th squc is a arithmtic
More informationEC1305 SIGNALS & SYSTEMS
EC35 SIGNALS & SYSTES DEPT/ YEAR/ SE: IT/ III/ V PREPARED BY: s. S. TENOZI/ Lcturr/ECE SYLLABUS UNIT I CLASSIFICATION OF SIGNALS AND SYSTES Cotiuous Tim Sigals (CT Sigals Discrt Tim Sigals (DT Sigals Stp
More information(Reference: sections in Silberberg 5 th ed.)
ALE. Atomic Structur Nam HEM K. Marr Tam No. Sctio What is a atom? What is th structur of a atom? Th Modl th structur of a atom (Rfrc: sctios.4 -. i Silbrbrg 5 th d.) Th subatomic articls that chmists
More informationPeriodic Structures. Filter Design by the Image Parameter Method
Prioic Structurs a Filtr sig y th mag Paramtr Mtho ECE53: Microwav Circuit sig Pozar Chaptr 8, Sctios 8. & 8. Josh Ottos /4/ Microwav Filtrs (Chaptr Eight) microwav filtr is a two-port twork us to cotrol
More informationFigure 2-18 Thevenin Equivalent Circuit of a Noisy Resistor
.8 NOISE.8. Th Nyquist Nois Thorm W ow wat to tur our atttio to ois. W will start with th basic dfiitio of ois as usd i radar thory ad th discuss ois figur. Th typ of ois of itrst i radar thory is trmd
More informationModule 5: IIR and FIR Filter Design Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School of Electrical Engineering & Telecommunications
Modul 5: IIR ad FIR Filtr Dsig Prof. Eliathamby Ambiairaah Dr. Tharmaraah Thiruvara School of Elctrical Egirig & Tlcommuicatios Th Uivrsity of w South Wals Australia IIR filtrs Evry rcursiv digital filtr
More informationResponse of LTI Systems to Complex Exponentials
3 Fourir sris coiuous-im Rspos of LI Sysms o Complx Expoials Ouli Cosidr a LI sysm wih h ui impuls rspos Suppos h ipu sigal is a complx xpoial s x s is a complx umbr, xz zis a complx umbr h or h h w will
More informationMathematical Preliminaries for Transforms, Subbands, and Wavelets
Mahmaical Prlimiaris for rasforms, Subbads, ad Wavls C.M. Liu Prcpual Sigal Procssig Lab Collg of Compur Scic Naioal Chiao-ug Uivrsiy hp://www.csi.cu.du.w/~cmliu/courss/comprssio/ Offic: EC538 (03)5731877
More informationAn Introduction to Asymptotic Expansions
A Itroductio to Asmptotic Expasios R. Shaar Subramaia Asmptotic xpasios ar usd i aalsis to dscrib th bhavior of a fuctio i a limitig situatio. Wh a fuctio ( x, dpds o a small paramtr, ad th solutio of
More information10. Joint Moments and Joint Characteristic Functions
0 Joit Momts ad Joit Charactristic Fctios Followig sctio 6 i this sctio w shall itrodc varios paramtrs to compactly rprst th iformatio cotaid i th joit pdf of two rvs Giv two rvs ad ad a fctio g x y dfi
More informationPartition Functions and Ideal Gases
Partitio Fuctios ad Idal Gass PFIG- You v lard about partitio fuctios ad som uss ow w ll xplor tm i mor dpt usig idal moatomic diatomic ad polyatomic gass! for w start rmmbr: Q( N ( N! N Wat ar N ad? W
More informationThe University of Manchester Analogue & Digital Filters 2003 Section D5: More on FIR digital filter design.
Th Uivrsity of achstr Aalogu & Digital Filtrs 3 Sctio D5: or o FIR digital filtr dsig. D5..Bacgroud: As w hav s bfor, a FIR digital filtr of ordr may b implmtd by programmig th sigal-flow graph show blow.
More informationProbability & Statistics,
Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus Poisso Distributio Poisso Distributio: A radom variabl X is said
More informationTime : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120
Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,
More information(looks like a time sequence) i function of ˆ ω (looks like a transform) 2. Interpretations of X ( e ) DFT View OLA implementation
viw of STFT Digital Spch Procssig Lctur Short-Tim Fourir Aalysis Mthods - Filtr Ba Dsig j j ˆ m ˆ. X x[ m] w[ ˆ m] ˆ i fuctio of ˆ loos li a tim squc i fuctio of ˆ loos li a trasform j ˆ X dfid for ˆ 3,,,...;
More informationPart B: Transform Methods. Professor E. Ambikairajah UNSW, Australia
Par B: rasform Mhods Profssor E. Ambikairaah UNSW, Ausralia Chapr : Fourir Rprsaio of Sigal. Fourir Sris. Fourir rasform.3 Ivrs Fourir rasform.4 Propris.4. Frqucy Shif.4. im Shif.4.3 Scalig.4.4 Diffriaio
More informationFourier Series: main points
BIOEN 3 Lcur 6 Fourir rasforms Novmbr 9, Fourir Sris: mai pois Ifii sum of sis, cosis, or boh + a a cos( + b si( All frqucis ar igr mulipls of a fudamal frqucy, o F.S. ca rprs ay priodic fucio ha w ca
More informationWorksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
More informationWashington State University
he 3 Ktics ad Ractor Dsig Sprg, 00 Washgto Stat Uivrsity Dpartmt of hmical Egrg Richard L. Zollars Exam # You will hav o hour (60 muts) to complt this xam which cosists of four (4) problms. You may us
More informationChapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1
Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationChapter 6: DFT/FFT Transforms and Applications 6.1 DFT and its Inverse
6. Chaptr 6: DFT/FFT Trasforms ad Applicatios 6. DFT ad its Ivrs DFT: It is a trasformatio that maps a -poit Discrt-tim DT) sigal ] ito a fuctio of th compl discrt harmoics. That is, giv,,,, ]; L, a -poit
More informationLECTURE 13 Filling the bands. Occupancy of Available Energy Levels
LUR 3 illig th bads Occupacy o Availabl rgy Lvls W hav dtrmid ad a dsity o stats. W also d a way o dtrmiig i a stat is illd or ot at a giv tmpratur. h distributio o th rgis o a larg umbr o particls ad
More information[ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is
Discrt-tim ourir Trsform Rviw or discrt-tim priodic sigl x with priod, th ourir sris rprsttio is x + < > < > x, Rviw or discrt-tim LTI systm with priodic iput sigl, y H ( ) < > < > x H r rfrrd to s th
More informationSIGNALS AND LINEAR SYSTEMS UNIT-1 SIGNALS
SIGNALS AND LINEAR SYSTEMS UNIT- SIGNALS. Dfi a sigal. A sigal is a fuctio of o or mor idpdt variabls which cotais som iformatio. Eg: Radio sigal, TV sigal, Tlpho sigal, tc.. Dfi systm. A systm is a st
More informationFrequency Measurement in Noise
Frqucy Masurmt i ois Porat Sctio 6.5 /4 Frqucy Mas. i ois Problm Wat to o look at th ct o ois o usig th DFT to masur th rqucy o a siusoid. Cosidr sigl complx siusoid cas: j y +, ssum Complx Whit ois Gaussia,
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationCDS 101: Lecture 5.1 Reachability and State Space Feedback
CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray ad Hido Mabuchi 5 Octobr 4 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls
More informationONLINE SUPPLEMENT Optimal Markdown Pricing and Inventory Allocation for Retail Chains with Inventory Dependent Demand
Submittd to Maufacturig & Srvic Opratios Maagmt mauscript MSOM 5-4R2 ONLINE SUPPLEMENT Optimal Markdow Pricig ad Ivtory Allocatio for Rtail Chais with Ivtory Dpdt Dmad Stph A Smith Dpartmt of Opratios
More informationOn the approximation of the constant of Napier
Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of
More informationMONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx
MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of
More informationln x = n e = 20 (nearest integer)
H JC Prlim Solutios 6 a + b y a + b / / dy a b 3/ d dy a b at, d Giv quatio of ormal at is y dy ad y wh. d a b () (,) is o th curv a+ b () y.9958 Qustio Solvig () ad (), w hav a, b. Qustio d.77 d d d.77
More informationIntroduction to Quantum Information Processing. Overview. A classical randomised algorithm. q 3,3 00 0,0. p 0,0. Lecture 10.
Itroductio to Quatum Iformatio Procssig Lctur Michl Mosca Ovrviw! Classical Radomizd vs. Quatum Computig! Dutsch-Jozsa ad Brsti- Vazirai algorithms! Th quatum Fourir trasform ad phas stimatio A classical
More informationThomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple
5/24/5 A PROOF OF THE CONTINUED FRACTION EXPANSION OF / Thomas J Oslr INTRODUCTION This ar givs aothr roof for th rmarkabl siml cotiud fractio = 3 5 / Hr is ay ositiv umbr W us th otatio x= [ a; a, a2,
More informationSession : Plasmas in Equilibrium
Sssio : Plasmas i Equilibrium Ioizatio ad Coductio i a High-prssur Plasma A ormal gas at T < 3000 K is a good lctrical isulator, bcaus thr ar almost o fr lctros i it. For prssurs > 0.1 atm, collisio amog
More informationExercises for lectures 23 Discrete systems
Exrciss for lcturs 3 Discrt systms Michal Šbk Automatické říí 06 30-4-7 Stat-Spac a Iput-Output scriptios Automatické říí - Kybrtika a robotika Mols a trasfrs i CSTbx >> F=[ ; 3 4]; G=[ ;]; H=[ ]; J=0;
More informationDiscrete Fourier Series and Transforms
Lctur 4 Outi: Discrt Fourir Sris ad Trasforms Aoucmts: H 4 postd, du Tus May 8 at 4:3pm. o at Hs as soutios wi b avaiab immdiaty. Midtrm dtais o t pag H 5 wi b postd Fri May, du foowig Fri (as usua) Rviw
More informationELEC9721: Digital Signal Processing Theory and Applications
ELEC97: Digital Sigal Pocssig Thoy ad Applicatios Tutoial ad solutios Not: som of th solutios may hav som typos. Q a Show that oth digital filts giv low hav th sam magitud spos: i [] [ ] m m i i i x c
More informationClass #24 Monday, April 16, φ φ φ
lass #4 Moday, April 6, 08 haptr 3: Partial Diffrtial Equatios (PDE s First of all, this sctio is vry, vry difficult. But it s also supr cool. PDE s thr is mor tha o idpdt variabl. Exampl: φ φ φ φ = 0
More informationTriple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling
Tripl Play: From D Morga to Stirlig To Eulr to Maclauri to Stirlig Augustus D Morga (186-1871) was a sigificat Victoria Mathmaticia who mad cotributios to Mathmatics History, Mathmatical Rcratios, Mathmatical
More informationENGG 1203 Tutorial. Difference Equations. Find the Pole(s) Finding Equations and Poles
ENGG 03 Tutoial Systms ad Cotol 9 Apil Laig Obctivs Z tasfom Complx pols Fdbac cotol systms Ac: MIT OCW 60, 6003 Diffc Equatios Cosid th systm pstd by th followig diffc quatio y[ ] x[ ] (5y[ ] 3y[ ]) wh
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationDerivation of a Predictor of Combination #1 and the MSE for a Predictor of a Position in Two Stage Sampling with Response Error.
Drivatio of a Prdictor of Cobiatio # ad th SE for a Prdictor of a Positio i Two Stag Saplig with Rspos Error troductio Ed Stak W driv th prdictor ad its SE of a prdictor for a rado fuctio corrspodig to
More informationSTIRLING'S 1 FORMULA AND ITS APPLICATION
MAT-KOL (Baja Luka) XXIV ()(08) 57-64 http://wwwimviblorg/dmbl/dmblhtm DOI: 075/МК80057A ISSN 0354-6969 (o) ISSN 986-588 (o) STIRLING'S FORMULA AND ITS APPLICATION Šfkt Arslaagić Sarajvo B&H Abstract:
More informationCDS 101: Lecture 5.1 Reachability and State Space Feedback
CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray 7 Octobr 3 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls Dscrib th dsig o
More informationOrdinary Differential Equations
Ordiary Diffrtial Equatio Aftr radig thi chaptr, you hould b abl to:. dfi a ordiary diffrtial quatio,. diffrtiat btw a ordiary ad partial diffrtial quatio, ad. Solv liar ordiary diffrtial quatio with fid
More information2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005
Mark Schm 67 Ju 5 GENERAL INSTRUCTIONS Marks i th mark schm ar plicitly dsigatd as M, A, B, E or G. M marks ("mthod" ar for a attmpt to us a corrct mthod (ot mrly for statig th mthod. A marks ("accuracy"
More informationUNIT 2: MATHEMATICAL ENVIRONMENT
UNIT : MATHEMATICAL ENVIRONMENT. Itroductio This uit itroducs som basic mathmatical cocpts ad rlats thm to th otatio usd i th cours. Wh ou hav workd through this uit ou should: apprciat that a mathmatical
More informationContinuous-Time Fourier Transform. Transform. Transform. Transform. Transform. Transform. Definition The CTFT of a continuoustime
Ctiuus-Tim Furir Dfiiti Th CTFT f a ctiuustim sigal x a (t is giv by Xa ( jω xa( t jωt Oft rfrrd t as th Furir spctrum r simply th spctrum f th ctiuus-tim sigal dt Ctiuus-Tim Furir Dfiiti Th ivrs CTFT
More informationLecture 2: Discrete-Time Signals & Systems. Reza Mohammadkhani, Digital Signal Processing, 2015 University of Kurdistan eng.uok.ac.
Lctur 2: Discrt-Tim Signals & Systms Rza Mohammadkhani, Digital Signal Procssing, 2015 Univrsity of Kurdistan ng.uok.ac.ir/mohammadkhani 1 Signal Dfinition and Exampls 2 Signal: any physical quantity that
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationSlide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS
Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationScattering Parameters. Scattering Parameters
Motivatio cattrig Paramtrs Difficult to implmt op ad short circuit coditios i high frqucis masurmts du to parasitic s ad Cs Pottial stability problms for activ dvics wh masurd i oopratig coditios Difficult
More information07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationNET/JRF, GATE, IIT JAM, JEST, TIFR
Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Mathmatical Physics JEST-6 Q. Giv th coditio φ, th solutio of th quatio ψ φ φ is giv by k. kφ kφ lφ kφ lφ (a) ψ (b) ψ kφ (c) ψ
More information2.29 Numerical Fluid Mechanics Spring 2015 Lecture 12
REVIEW Lctur 11: Numrical Fluid Mchaics Sprig 2015 Lctur 12 Fiit Diffrcs basd Polyomial approximatios Obtai polyomial (i gral u-qually spacd), th diffrtiat as dd Nwto s itrpolatig polyomial formulas Triagular
More informationAn Introduction to Asymptotic Expansions
A Itroductio to Asmptotic Expasios R. Shaar Subramaia Dpartmt o Chmical ad Biomolcular Egirig Clarso Uivrsit Asmptotic xpasios ar usd i aalsis to dscrib th bhavior o a uctio i a limitig situatio. Wh a
More informationASSERTION AND REASON
ASSERTION AND REASON Som qustios (Assrtio Rso typ) r giv low. Ech qustio cotis Sttmt (Assrtio) d Sttmt (Rso). Ech qustio hs choics (A), (B), (C) d (D) out of which ONLY ONE is corrct. So slct th corrct
More informationHadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms
Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of
More informationCOMPUTING FOLRIER AND LAPLACE TRANSFORMS. Sven-Ake Gustafson. be a real-valued func'cion, defined for nonnegative arguments.
77 COMPUTNG FOLRER AND LAPLACE TRANSFORMS BY MEANS OF PmER SERES EVALU\TON Sv-Ak Gustafso 1. NOTATONS AND ASSUMPTONS Lt f b a ral-valud fuc'cio, dfid for ogativ argumts. W shall discuss som aspcts of th
More informationMATH 681 Notes Combinatorics and Graph Theory I. ( 4) n. This will actually turn out to be marvelously simplifiable: C n = 2 ( 4) n n + 1. ) (n + 1)!
MATH 681 Nots Combiatorics ad Graph Thory I 1 Catala umbrs Prviously, w usd gratig fuctios to discovr th closd form C = ( 1/ +1) ( 4). This will actually tur out to b marvlously simplifiabl: ( ) 1/ C =
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationCOLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II
COLLECTION OF SUPPLEMENTARY PROBLEMS I. CHAPTER 6 --- Trscdtl Fuctios CALCULUS II A. FROM CALCULUS BY J. STEWART:. ( How is th umbr dfid? ( Wht is pproimt vlu for? (c ) Sktch th grph of th turl potil fuctios.
More informationInternational Journal of Advanced and Applied Sciences
Itratioal Joural of Advacd ad Applid Scics x(x) xxxx Pags: xx xx Cotts lists availabl at Scic Gat Itratioal Joural of Advacd ad Applid Scics Joural hompag: http://wwwscic gatcom/ijaashtml Symmtric Fuctios
More informationTime Dependent Solutions: Propagators and Representations
Tim Dpdt Solutios: Propagators ad Rprstatios Michal Fowlr, UVa 1/3/6 Itroductio W v spt most of th cours so far coctratig o th igstats of th amiltoia, stats whos tim dpdc is mrly a chagig phas W did mtio
More informationVtusolution.in FOURIER SERIES. Dr.A.T.Eswara Professor and Head Department of Mathematics P.E.S.College of Engineering Mandya
LECTURE NOTES OF ENGINEERING MATHEMATICS III Su Cod: MAT) Vtusoutio.i COURSE CONTENT ) Numric Aysis ) Fourir Sris ) Fourir Trsforms & Z-trsforms ) Prti Diffrti Equtios 5) Lir Agr 6) Ccuus of Vritios Tt
More informationReview Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2
MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm
More informationThey must have different numbers of electrons orbiting their nuclei. They must have the same number of neutrons in their nuclei.
37 1 How may utros ar i a uclus of th uclid l? 20 37 54 2 crtai lmt has svral isotops. Which statmt about ths isotops is corrct? Thy must hav diffrt umbrs of lctros orbitig thir ucli. Thy must hav th sam
More informationNarayana IIT Academy
INDIA Sc: LT-IIT-SPARK Dat: 9--8 6_P Max.Mars: 86 KEY SHEET PHYSIS A 5 D 6 7 A,B 8 B,D 9 A,B A,,D A,B, A,B B, A,B 5 A 6 D 7 8 A HEMISTRY 9 A B D B B 5 A,B,,D 6 A,,D 7 B,,D 8 A,B,,D 9 A,B, A,B, A,B,,D A,B,
More informationChapter (8) Estimation and Confedence Intervals Examples
Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit
More information