Module 5: IIR and FIR Filter Design Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School of Electrical Engineering & Telecommunications
|
|
- Austen Hines
- 5 years ago
- Views:
Transcription
1 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
2 IIR filtrs Evry rcursiv digital filtr IIR must cotai at last o closd loop. Each closd loop cotais at last o dlay lmt y M a x L y For rcursiv digital filtrs IIR b For o rcursiv digital filtrs FIR b b Gral digital filtr ELEC97
3 3 Exampls b b a a a b b a a a d ordr FIR filtr ros oly d ordr IIR filtr all pol filtr d ordr IIR filtr with pols ad ros
4 Exampl: Fid th DC gai of th followig filtr: /- a - <a< Th DC gai of ca b obtaid by substitutig o Η. a ; a If th DC gai is udsirabl, you may itroduc a costat gai factor of -a, so that bcoms a dc gai db a 4
5 ELEC97 5 Exrcis Slct such that th filtr has uity gai at. b b a a a
6 Exampl: Cosidr a lowpass filtr y a y- b x Dtrmi b so that. ot: < a < Dtrmi th 3 db badwidth c for th ormalisd filtr i part abov. a b a b a b a b ; ; [ ] a a a a a a po powr alf a a a a a c c c 4 cos cos it cos c / Lowpass filtr
7 Exrcis A first ordr digital filtr is dscribd by a b b a Assum a b < a, b < Aswr : Dtrmi, so that th maximum valu of Comput th 3-db badwidth of th filtr c a b cos ELEC97
8 Exampls: Lowpass filtr : a a a Lowpass filtr : a - Th additio of a ro at - furthr attuats th rspos of th filtr at high frqucis ighpass filtr : 3 a a W ca obtai simpl highpass filtr by rflctig th pol-ro locatios of th lowpass filtr about th imagiary axis. -
9 ELEC97 9 Exampls : 5 x x y filtr ighpass : 4 x x y filtr Lowpass cos 5
10 Digital Filtr Dsig Ovrviw Th most commo digital filtrs fall ito four mai catgoris Low Pass igh Pass Bad Pass Bad Stop FIR filtrs ca b dirctly dsigd i th trasform domai IIR filtrs ar most commoly dsigd by slctig a suitabl aalogu filtr trasfr fuctio, s, ad th trasformig it ito a quivalt digital filtr trasfr fuctio,, usig a mappig fuctio
11 Idal Filtr Magitud Rsposs c c Low Pass Filtr c c igh Pass Filtr c c c c Bad Pass Filtr Bad Stop Filtr ELEC97
12 Spcificatio of a Digital filtr Th maority of filtr dsig tchiqus ar basd o th dsigr providig a spcificatio for a dsird frqucy Magitud Rspos Eg. For a low pass filtr, th dsigr must at last spcify four paramtrs to complt th spcificatio Pass bad cut off frqucy c Stop bad frqucy h Pass bad rippl δ Stop bad attuatio δ A similar st of paramtrs ar usd to spcify th othr typs of filtrs Th Phas Rspos is typically ot spcifid, xcpt, prhaps, wh a liar phas rspos is rquird. ELEC97
13 Spcificatio of a Digital filtr Th dsird magitud spcificatio for a low pass filtr ca th b writt as δ δ δ Pass bad Stop bad Th frqucy rag from c to h is ow as th trasitio bad, which is ffctivly a do t car rgio, i trms of th dsird magitud rspos A st of mappig ruls xists to trasform a digital low pass filtr ito a quivalt vrsio of ay of th othr thr typs of digital filtrs c, typically, a quivalt digital low pass filtr is first dsigd ad th trasformd ito a digital high pass filtr h c
14 Spcificatio of a Digital filtr δ -δ Pass Bad Trasitio Bad Stop Bad δ c h ELEC97
15 Filtr Dsig Filtr dsig pacags utiliss th frqucy magitud rspos providd PLUS a idicatio of whthr a IIR, or FIR, filtr is rquird Dsig tchiqus provid th lowst ordr complxity filtr such that its frqucy rspos is withi th providd magitud rspos spcificatio For a giv magitud rspos th ordr of a FIR filtr, which will mt this spcificatio, will b substatially highr tha th ordr of a IIR filtr, implmtig th sam magitud rspos But, FIR is rquird if liar phas is a cssity! ELEC97
16 FIR Filtrs Systm fuctio cotai oly ros o-rcursiv or rcursiv structurs ar both possibl; th bst ow is th o-rcursiv trasvrsal structur. FIR Filtrs ca hav a xactly liar phas rspos. Th implicatio of this is that o phas distortio is itroducd ito th sigal by th filtr. Th ffcts of usig a limitd umbr of bits to implmt filtrs such as roud off ois ad quatiatio rrors ar much lss svr i FIR tha i IIR. FIR rquirs mor cofficits for sharp cut-off filtrs tha IIR. Thus for a giv amplitud rspos spcificatio, mor procssig tim ad storag will b rquird for FIR implmtatio. Complxity is proportioal to th lgth of th impuls rspos. IIR Filtrs Cotai pols ad ros ormally Oly rcursiv structur is possibl; th most widly usd form is th cascad coctio of first-ordr ad scod ordr sctios. Th phas rsposs of IIR filtrs ar oliar, spcially at th bad dgs. Bcaus of quatiatio of th filtr cofficits, a pol ca i pricipl mov from a positio isid th uit circl to a positio outsid th uit circl ad hc caus istability. IIR rquirs fwr cofficits for sharp cut off filtrs tha FIR. o dirct rlatio btw th complxity ad th lgth of th impuls rspos which is ifiit by dfiitio. FIR filtrs hav o aalogu coutrpart. FIR Aalogu filtrs ca b radily trasformd ito dsig procdurs ar ormally itrativ quivalt IIR digital filtrs mtig similar procdurs. Dsig quatios do ot xist. spcificatios. IIR filtrs ca b dsigd usig 6 dsig formula.
17 Exrcis: Th followig trasfr fuctios rprst two diffrt filtrs mtig idtical amplitud frqucy rspos spcificatios. Dtrmi ad commt o th computatioal ad storag rquirmts. a a a Filtr : b b whr a.49889; a ; a.49889; b ; b ; h h h Filtr : h [ ] [ ] h[ ] [ ] h [ ] [ ] h[ 9] h h h [ 3] h[ 8] [ 4] h[ 7] [ 5] h[ 6] 7
18 As: FIR IIR Filtr Filtr umbr of Multiplicatios 5 umbr of additios 4 Storag locatios cofficits ad data ELEC97
19 Filtr Implmtatio Oc a suitabl filtr has b dsigd i.. th diffrc quatio dtrmid, it is cssary to cosidr issus cocrig its implmtatio Softwar or ardwar Fixd or Floatig Poit umbr of bits pr sampl, cofficit, multiplir ad accumulator Paralll or Cascad 9 ELEC97
20 Filtr Dsig Tchiqus IIR filtrs hav a quivalt aalogu filtr. c, th most commo mthod for dsigig IIR filtrs is, To fid a suitabl aalogu filtr ad th to trasform it ito a quivalt digital filtr FIR filtrs do OT hav a quivalt i th aalogu domai. All FIR filtr dsig tchiqus ar itrativ procdurs. ELEC97
21 IIR Filtr Dsig Tchiqus O dirct mthod for dsigig a simpl IIR filtr or idd a simpl FIR is ow as Pol-Zro Placmt If a IIR filtr is to b dsigd to a rasoabl dgr of accuracy i trms of its dsird frqucy rspos, th pol-ro placmt is OT a accptably accurat tchiqu Two commoly usd idirct dsig tchiqus for IIR filtrs ar Impuls Ivariat Trasformatio Biliar Trasformatio ELEC97
22 Trasformatio Tchiqus Both tchiqus ca trasform a suitabl stabl aalogu trasfr fuctio s ito a stabl digital trasfr fuctio, Both tchiqus us a particular mappig for th LaPlac variabl s i s ito th trasform variabl i Th trasformatio usd must maitai th sstial proprtis i.. stability, filtr typ, cut off frqucy tc. of th aalogu filtr s frqucy magitud rspos ELEC97
23 Impuls Ivariat Trasformatio Th impuls ivariat trasformatio dsigs a digital filtr such that its impuls rspos, h, is a sampld vrsio of th cotiuous impuls rspos of th aalogu filtr, ht I othr words, hht whr T: Samplig Priod This, howvr, dos OT imply that th rsultig digital filtr has th sam frqucy rspos as th origial aalogu filtr ovr th frqucy rag of to f samplig /, i.. << ELEC97
24 Aliasig Effcts All aalogu filtrs hav a frqucy magitud rspos which must hav a o-ro valu at ALL frqucis Wh th cotiuous impuls rspos of th aalogu filtr is sampld, th rsultat digital filtr s frqucy rspos is that of th aalogu filtr REPEATED priodically at a itrval qual to th samplig frqucy c, if th valu of th aalogu filtr s magitud rspos is sigificatly larg at AY frqucy, gratr tha half th samplig frqucy, this WILL rsult i sigificat aliasig of th origial aalogu filtr s frqucy rspos ELEC97
25 Aliasig Effcts c, for xampl, this tchiqu CAOT b usd to trasform a high pass aalogu filtr ito a high pass digital filtr Th samplig priod must b carfully slctd to b high ough to sur that this aliasig ffct has a miimal distortio of th origial dsird aalogu filtr s frqucy rspos Th lsso: Similar bhaviour i th TIME domai for aalogu ad digital filtrs, DOES OT sur similar bhaviour i th FREQUECY domai. 5 ELEC97
26 ELEC97 6 Mappig Rul Cosidr a simpl aalogu low pass filtr with a cut off frqucy of ω c radias s - Th impuls rspos of this filtr is a > c a s s ω t t < t a c t h ω
27 Mappig Rul If w sampl this sigal, with a samplig priod T, to produc th impuls rspos of th digital filtr, w gt ω c T h < This squc has th trasform T 7 ω c ELEC97
28 Mappig Rul Thus th mappig rul to gt from aalogu filtr, s, to quivalt digital filtr,, is giv by ω s ω ct c ot that thr is O rul for mappig ros of th aalogu filtr Oly possibl to dsig crtai typ of low pass filtrs usig this trasformatio 8 ELEC97
29 Trasformatio Clarly, from th mappig rul, a pol of th aalogu filtr at s-ω c is mappd ito a pol of th digital filtr at - ω ct Th trasformatio that has b carrid out is st ; This trasformatio maps th lft half sid of th s pla, i strips /T wid, isid th uit circl o th pla Stabl aalogu filtr bcoms a stabl digital filtr Aliasig occurs bcaus sparat pols i th s pla ca map ito th sam pol i th pla 9 ELEC97
30 Trasformatio Mappig ω 3/T s-pla -pla /T /T σ - -3/T Each strip maps oto th itrior of th uit circl 3 ELEC97
31 Exampl Dtrmi th trasfr fuctio of a low pass filtr, usig th impuls ivariat trasformatio, of th followig aalogu filtr s s 4 s Suggst a suitabl valu for th samplig priod T which should b usd ad, hc, dduc th diffrc quatio for th filtr 3 ELEC97
32 ELEC97 3 Exampl 3T T T T T T T Us trasformatio o EAC trm s 4 s 4 s 4 B A -4 4 ad B A B A s s B Bs A As s B s A s s 4 s
33 Exampl For a trm /sa, a suitabl samplig frqucy would b at fiv to t tims th cut off frqucy of th aalogu filtr I this problm, th filtr is formd as two such trms, hc, w should choos th o with th highr cut off frqucy I this problm, ω c radias s - or / c, a suitabl samplig frqucy f s would b 5/, which is a samplig priod of /5 scods 33 ELEC97
34 Exampl Trasfr fuctio bcoms: 4 T T T 3T.9955 With T, Diffrc Equatio : T y[].9955x[ -].88y[ -]-.58y[ - ] 34 ELEC97
35 Exrcis Usig impuls ivariat mthod dsig a digital filtr to approximat th followig ormalid aalogu trasfr fuctio: s s s Assum that th 3dB cut-off frqucy of th digital filtr is 5 ad th samplig frqucy is.8 35 ELEC97
36 ot: Bfor applyig th impuls ivariat mthod, w d to d-ormali th trasfr fuctio. As: 36 s s c s s s s s s ω ω ω
37 ELEC97 37 Exrcis Th followig is a -trasfr fuctio of a filtr dsigd usig th Impuls Ivariat Trasform, fid s. 3 3 T T T T
38 Biliar Trasformatio Biliar trasformatio is a far mor usful tchiqu as it dos ot hav th aliasig problms associatd with th impuls ivariat trasformatio Ths aros du to th fact that th impuls ivariat trasformatio mappd th s pla, i strips rathr tha i its tirty, isid th uit circl Th biliar trasformatio maps th tir lft had sid of th s pla ito th isid of th uit circl, i o pic 38 ELEC97
39 Trasformatio Mappig ω s-pla -pla σ Imag of th lft had s-pla. 39 ELEC97
40 ELEC97 or...!...! T s T s st st st st st st trms ordr th highr out droppig By st st st st st st st st Trasformatio Equatio
41 Frqucy Warpig Whil th biliar trasformatio dos ot suffr ay aliasig problms, it dos this at th xps of warpig th frqucy axis Th tir aalogu frqucy axis from ω qual to ro to ω qual to ifiity is mappd ito th digital filtr s frqucy rag of qual to ro to qual to Thrfor: ω digital digital aalogu aalogu ω ELEC97
42 Warpig Fuctio owvr, this warpig fuctio will mov all othr importat frqucy poits durig th trasformatio.g. cut off frqucis Without taig this warpig ito accout, a aalogu filtr with a cut off frqucy of f c will OT rsult i a digital filtr with a cut-off frqucy at f c /f s Thrfor, it is cssary to pr-warp ay cut-off frqucis, which must b maitaid accuratly, bfor a suitabl aalogu filtr is slctd This is do usig th Warpig fuctio rlatig digital frqucy to aalogu frqucy ELEC97
43 ELEC97 ta ta cos si mappd oto th uit circl is s W d to cosidr how th s domai frqucy axis T T is that T T T T T s d a warpd pr ω ω ω ω ω Warpig Fuctio
44 Warpig Fuctio Thus a o-liar rlatioship xists btw ω ad as show ωt - ELEC97
45 Practical Dsig Biliar trasformatio is th most commoly usd mthod Dsird magitud spctrum is ffctivly prwarpd ito th aalogu domai Suitabl aalogu filtr typ is slctd from wll ow classical aalogu filtr typs, ithr through dsig quatios or, mor commoly, tabls or filtr dsig hadboos: Buttrworth o rippl i pass or stop bad maximally flat frqucy rspos Chbyshv Typ : Rippl i pass bad, Typ : Rippl i stop bad Elliptical Lowst complxity but rippl i both pass ad stop bad
46 Exampl A simpl aalogu low pass filtr is giv by th trasfr fuctio s ωc s ω c This filtr has a 3 db cut off frqucy at ω c Dtrmi, usig th biliar trasformatio, a quivalt digital low pass filtr Th dsird cut off frqucy is ad th samplig frqucy of th systm is 8 46 ELEC97
47 Exampl Firstly, w caot simply us th biliar trasformatio o s ωc s ω This would rsult i a low pass filtr BUT, th cut off frqucy of th filtr would b warpd d which is OT what w wat ωd ta T ω 5987 rad s c ωat - s 6 ta 6 or f 95 d
48 ω Thrfor, th cut-off frqucy must b pr-warpd f f s prwarpd 8 4 Exampl ta 6 ta rad s T 8 Istad of it is ow 55 Thus th dsird aalogu filtr trasfr fuctio will b ωc s s ω s c 55
49 Exampl Thus th digital filtr s trasfr fuctio will bcom: Diffrc Equatio : y[].99x[].99x[ -].44y[ -]
50 Exampl A ormalisd Low Pass Filtr trasfr fuctio is giv blow. Dtrmi th -trasfr fuctio ad diffrc quatio usig Biliar Trasformatio. Assum fc 3 ad Fs 5 s s 5 ELEC97
51 Exampl Stp: Firstly, dtrmi th rquird digital cut-off frqucy c Stp: Th pr-warp ω 3 c ta 7.95rads c T / s Stp3: Dormalis filtr trasfr fuctio s s s s ω s c ω c ω c 5 ELEC97
52 Exampl Cotiud Stp4: Apply Biliar Trasformatio s s T ELEC97
53 Exrcis It is rquird to dsig a digital filtr to approximat th followig aalogu trasfr fuctio s s s usig th Bi-liar trasformatio mthod obtai th trasfr fuctio, s of th digital filtr assumig a 3dB cut-off frqucy of 5 ad a samplig frqucy of.8. As: ELEC97
54 Filtr Dsig: Pol-Zro Placmt Tchiqu Th positio of pols ad ros i th -pla hav a particular ffct o a filtrs frqucy rspos summarisd as follows: Zros placd ar th uit circl rsult i a otch i th frqucy spctrum Zro placd o th uit circl producs complt rctio of a siusoid at that frqucy Pols placd clos to th uit circl giv ris to larg pas at th associat frqucy By stratgically placig pols ad ros w ca arriv at a particular rspos Tchiqu caot b usd to dsig vry accurat filtrs
55 Exampl Stch th Frqucy Rspos of th systm dscribd by th giv pol-ro plot uit circl -pla 55 ELEC97
56 Frqucy Rspos Amplitud fs/4 fs/ 3fs/4 fs / frqucy 56 ELEC97
57 Dsig Exampl Dsig a filtr usig th pol-ro placmt tchiqu which has th followig spcificatio Complt sigal rctio at DC arrow pass bad at 5 Pass badwidth Samplig frqucy of 5 57 ELEC97
58 Dsig Solutio Complt Sigal Rctio at DC implis d for a ro at > Rcall scod ordr rsoat systm This systm ca b usd to gt a basic bad pass filtr Badwidth is giv by f b b r r cos r f f samplig b b 58 ELEC97
59 ] [.8783 ] [.579 ] [ ] [ ] [ y y x x y Diffrc Equatio ro trm is Zro at rcos b f f r b f f r Pol Trm samplig samplig
60 6 Proy s mthod ] [ m m m M i i i h a b Apart from dsigig IIR filtr from aalogu prototyps, it could b dsigd by miimisig a rror critrio i tim domai. For a IIR filtr with a dsird ad actual impuls rsposs of h d [] ad h[] th last squar rror critrio is Furthr th h[] could b writt as ] [ ] [ d h h E
61 Proy s mthod Accordig to Proy s algorithm first th domiator cofficits, a m ar foud by miimisig th rror. If th iput to th systm is δ[] th a stimat of h[] could b foud for >M as [ ] hr a ar foud by miimisig th followig rror hˆ d a h d [ ] Error M h d [ ] hˆ d [ ] Th th umrator cofficits, b i ar foud by matchig th dsird impuls rspos for a lgth. Matlab commad : proy 6
62 Sha s mthod I this mthod th last squar approach is usd to fid both umrator ad domiator cofficits M sparatly. b If th iput to th systm is δ[] th a stimat of h[] could b foud for >M as hˆ d [ ] a h d First stp: ths domiator cofficits, a, ar foud by miimisig th followig last squar rror this stp is similar to Proy s mthod. Error M h d [ ] hˆ [ ] d 6 [ ] a
63 Scod stp: th cosidr a filtr structur as blow whr a ar th cofficits foud prviously. δ[] Sha s mthod aˆ r b will b foud by miimisig th followig last squar rror. Error h d M whr ad h d [ ] hˆ d υ[] [ ] M b ˆ [ ] [ ] bυ[ ] υ[ ] δ[ ] aˆ υ[ ] h d 63
64 Digital to Digital Trasformatios Low pass to low pass si[ p p ' / ] α α si[ p p ' / ] α whr p & p ' ar th origial & w cut-off frqucis Low pass to high pass α α α cos[ cos[ p p ' / ] p ' / ] p cos[ u L / ] α α cos[ u L / ] α cot u L / ta Low pass to bad pass p l whr & u ar th lowr & uppr cut-off frqucis 64
65 Digital to Digital Trasformatios Low pass to bad stop 65 α α
66 FIR filtr dsig mthods Widowig mthod Frqucy samplig mthod Pars McLlla mthod Last squar mthod 66 ELEC97
67 Widowig mthod Th asist way to obtai a FIR filtr is to simply trucat th impuls rspos of a IIR filtr. h [ ] hd [ ] whr h d [] is th impuls rspos of th dsird IIR filtr, ad h[] is th FIR filtr. I gral h[] ca b thought of as big formd by th product of h d [] ad a widow fuctio w[] as follows: h othrwis [ ] h [ ] w[ ] d W d 67 ELEC97
68 Effct of rctagular widow for a low pass filtr dsig / Th covolutio producs a smard vrsio of th idal low pass frqucy rspos d. Th widr th mai lob of W, th mor spradig, whr as th arrowr th mai lob largr th closr coms to d. Thr is a trad off of maig larg ough so that smarig is miimid, yt small ough to allow rasoabl implmtatio.
69 Som widow fuctios 69
70 Dsig Procdur First, th dsird impuls rspos d to b drivd DTFT of th rquird filtr Th th dsird impuls rspos d to b multiplid by a appropriat widow fuctio. 7 ELEC97
71 For a idal low pass filtr with liar phas of slop - β ad cutoff ω c ca b charactrid i th frqucy domai by d β w c w ω c < ω Thus th corrspodig impuls rspos obtaid by DTFT is, h d [ ] si [ ω ] c β β A causal FIR filtr with impuls rspos h[] ca b obtaid by multiplyig by a widow bgiig at th origi ad dig at - as follows si [ ] [ ωc β ] h ω[ ] β For h[] to b a liar phas, β must b slctd so that th rsultig h[] is symmtric 7 β
72 Exrcis: a Dtrmi th impuls rspos of th lowpass filtr whos magitud rspos is giv by 3 d b To obtai a fiit impuls rspos from a rctagular widow of lgth 9 is usd. Comput th cofficits of th FIR filtr with a liar phas charactristic ad with this fiit impuls rspos. As: a 3 h d [ ] si / 3 As: b <
73 73 [ ] si for d h d d < 3 3 [ ] / 3 si 3 h d To b a liar phas filtr it should b symmtric about 4 9
74 [ ] h [ ] [ ] h d ω h h h h h [ ] [ ] [ ] [ 3] [ 4] Basd o th symmtry th cofficits ar shiftd ad flippd:
75 75 [ ] β β β for d h d d si 3 < β 3 3 [ ] β β β β h d 3 / si β Altrativly you ca icorporat th phas wh calculatig th impuls rspos as blow,
76 [ ] h [ ] [ ] h d ω h h h h h [ ] h[ 8] [ ] h[ 7] 3 4 [ ] h[ 6] [ 3] h[ 5] [ 4]. 333 Basd o th symmtry th cofficits ar shiftig ad flippig:
77 Frqucy-Samplig Filtr Aothr typ of FIR filtr dsig which uss th frqucy sampld valus of th dsird filtr rspos. Dsird filtr Frqucy samplig th filtr Th filtr is implmtd usig th frqucy sampld valus. 77 ELEC97
78 ELEC97 78 Frqucy-Samplig Filtr Although it was implid bfor that all FIR filtrs ar o-rcursiv this is ot strictly tru a part of th FIR filtr ca hav rcursiv structur, providd that th ovr all filtr dos ot hav ay pols Cosidr th followig FIR filtr Th corrspodig filtr systm fuctio is othrwis g h [ ]... p p g g
79 g g Comb Filtr c Rsoator R This ca b implmtd i stags as follows: x / g y - - COMB FILTER -. RESOATOR
80 Exampl 8, g 8 8 x / x / /8 g y / Filtr implmtd usig Comb filtr rsoator structur y /
81 Aalysis of c i Frqucy Domai c ; c si c is show o xt slid for 8. c has ulls qually spacd i th rag corrspodig to ros Shap of magitud rspos givs this filtr its am comb filtr 8 ELEC97
82 /4 Comb Filtr Frqucy Rspos 8 ros o th uit circl. c.5..5 /8 thta :pi/64: *pi; 8; mag abs/*si*thta/; plotthta,mag, axis tight;titl'comb Filtr Frqucy Rspos';
83 Pol-Zro Pattr for Comb Filtr c c Im{} Pol-Zro Pattr for Comb Filtr 8 3/ / /4 R{} uit circl This quatio xplicitly shows qually spacd ros aroud uit circl First ro at -3/ -/ -/4 8 pols
84 Stability of g g Comb Filtr c Rsoator R R has a pol o th uit circl at frqucy i.. at Sic pol is ot isid th uit circl filtr is ot stabl Th ro at of comb filtr cacls th rsoator pol at Th combiatio is thus STABLE
85 Ovrall Frqucy Rspos of Comb Filtr plus Rsoator si si si si ; g g g g
86 Ovrall Systm Frqucy Rspos has a maximum valu g at at / / / thta -pi:pi/64: pi; 8; g; A g/*si*thta/; B sithta/; mag absa./b; plotthta,mag, axis tight; titl'ovrall Filtr Frqucy Rspos';
87 Lt us cosidr a scod-ordr rsoator, whos cofficits ar ral-valud ad th pols ar situatd at th followig ro locatios of th comb filtr. 8 pols 8 3 / Im{} / /4 R{} uit circl -3 / - / - /4
88 Th systm fuctio of th scod-ordr rsoator ca b writt i paralll form as g g R T g Th abov procdur ca b gralisd to iclud ach scod-ordr rsoator cacllig a pair of ros of th comb filtr. Ths rsoators ar all coctd i paralll ad this paralll combiatio is coctd i cascad with th comb filtr to produc th total filtr T giv by:
89 ELEC97 Frqucy Samplig Filtr Ralisatio X / - - Y g g 4 4 g g g g g g g ad W ca show that
90 Frqucy Samplig Filtr Ralisatio ] [ ] [ ] [ ] [ it, trasform of Taig th - ] [ ] [ h Lt th dsird filtr rspos as d []. If this rspos is frqucy sampld at poits th th frqucy sampld valus ar d [] for,, -. Th th impuls rspos of th filtr h[] ca b xprssd as,
91 ] [ ] [ ] [ usig gomtric sris, ] [ ] [
92 Exampl: Lt us cosidr implmtig th liar itrpolator with th comb ad rsoator structur. Th impuls rspos of th liar itrpolator is giv by h ½; h ; h ½ ad h othrwis Solutio: [ cos ] Sic th umbr of lmts i th uit sampl rspos is qual to 3, w choos 3. Th comb filtr systm fuctio is th c 3 3 ELEC97
93 Im c 3 3 R c has thr ros locatd at /3 for, ad. Th ro at will b caclld by a ral rsoator, ad ros at ±/3 will b caclld by a pair of complx rsoators.
94 ; ; * ; ; g g g g R R ] cos [ Th gais of th rsoators ar qual to:
95 Comb ad Rsoator Structur X -3 X /3 - / y / 3 y - Comb / -/ Rsoator ELEC97
96 Exrcis: A frqucy-samplig filtr is show blow ad 3 a - - X - - / Y. a - / - a / - a b c Dtrmi a, a ad a - such that this filtr has a ral impuls rspos h, whr 3 ad Draw th frqucy-samplig filtr structur usig dlay lmts, multiplirs ad addrs. Driv ad xprssio for. Giv a filtr that has th sam frqucy rspos, but ralisd as FIR filtr.
97 Solutio: R Im ; 3 3 a a a ; ; R R 3 Part a:
98 } { R R Part c: h 5; h -4 ad h Th structur is similar to th prvious xampl
99 Advatags Vry simpl: Just sampl th dsird rspos. Rcursiv implmtatio of FIR filtrs gratly rducs th umbr of arithmtic opratios spcially multiplicatios Particularly attractiv solutio if w wat to ma a arrow-bad filtr fw o ro valus Frqucy samplig filtrs ormally hav small itgr cofficits. 99 ELEC97
100 Practical Cosidratios I thory w assum that pols ad ros coicid xactly o th uit circl This rquirs th filtr cofficits to b vry accurat % Th bst w ca do is to gt th pols i th viciity of ros This ca lad to a vry rratic local variatio of th frqucy rspos ELEC97
101 Practical cosidratios I practic, both ros ad pols ar dlibratly locatd ust isid th uit circl Th comb filtr systm fuctio is chos as a c whr a is mad slightly lss tha ELEC97
102 Computr-Aidd dsig of quirippl liar phas FIR filtr dsig I this dsig th obctiv is to miimis th wightd rror btw th dsird ad th actual amplitud rsposs. ε. W [ D ] Wh th pa absolut valu of th rror is miimisd miimax or Chbyshv critrio th rsultig FIR filtr is usually calld quirippl FIR filtr. Pa absolut rror is, ε max max ε
103 Par McCllla Mthod Par McCllla solvd this optimiatio for liar phas FIR filtrs ad that mthod is calld Par McCllla mthod. Matlab commad is FIRPM or rm 3 ELEC97
104 ELEC97 Last Squar Tchiqu Filtrs ca b dsigd basd o miimisig th last wightd squard rror. Th squard rror wh P is th wightig fuctio is; 4 ε d P d
105 If th filtr cofficits, h[] is {b } th W watd to fid th cofficits which could miimi th abov rror. So by taig partial drivativs with rspct to b ad quatig thm to ro, 5 ε d b b P d U L d U L * ε d b b P d U L d U L *
106 Basd o th symmtry proprty for ral cofficits: * - 6 b ε ε d b b P b d U L d U L * [ ] [ ] ε d P d P b b d d U L * ] [ ] [ d r b b U L ε [ ] d P d d P r d ] [ ] [
107 This is th simultaous quatios ad i matrix form 7 ] [ ] [ d r b b U L ε ] [. ] [ ] [. [] ] [ ] [.... ] [... [] ] [ ] [... [] [] U d L d L d b b b r U L r U L r L U r r r L U r r r U L L br - d Thus th filtr cofficits, b ca b foud by solvig ths quatios.
108 Summary Spcificatios of a digital filtr IIR filtr dsig tchiqus Trasform mthods from aalogu prototyps Biliar trasform Impuls ivariat trasform Pol ro placmt Proy s mthod Sha s mthod FIR filtr dsig tchiqus Widowig mthod Frqucy samplig mthod Pars McLlla mthod Last squar mthod 8 ELEC97
Part 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 informationDTFT 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationFractional Sampling using the Asynchronous Shah with application to LINEAR PHASE FIR FILTER DESIGN
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],
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 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 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 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 informationω (argument or phase)
Imagiary uit: i ( i Complx umbr: z x+ i y Cartsia coordiats: x (ral part y (imagiary part Complx cougat: z x i y Absolut valu: r z x + y Polar coordiats: r (absolut valu or modulus ω (argumt or phas x
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 informationNEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES
Digst Joural of Naomatrials ad Biostructurs Vol 4, No, March 009, p 67-76 NEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES A IRANMANESH a*, O KHORMALI b, I NAJAFI KHALILSARAEE c, B SOLEIMANI
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 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 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 informationChapter 4 : Laplace Transform
4. Itroductio Laplace trasform is a alterative to solve the differetial equatio by the complex frequecy domai ( s = σ + jω), istead of the usual time domai. The DE ca be easily trasformed ito a algebraic
More informationECE594I Notes set 6: Thermal Noise
C594I ots, M. odwll, copyrightd C594I Nots st 6: Thrmal Nois Mark odwll Uivrsity of Califoria, ata Barbara rodwll@c.ucsb.du 805-893-344, 805-893-36 fax frcs ad Citatios: C594I ots, M. odwll, copyrightd
More informationINTRODUCTION TO SAMPLING DISTRIBUTIONS
http://wiki.stat.ucla.du/socr/id.php/socr_courss_2008_thomso_econ261 INTRODUCTION TO SAMPLING DISTRIBUTIONS By Grac Thomso INTRODUCTION TO SAMPLING DISTRIBUTIONS Itro to Samplig 2 I this chaptr w will
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 informationBipolar Junction Transistors
ipolar Juctio Trasistors ipolar juctio trasistors (JT) ar activ 3-trmial dvics with aras of applicatios: amplifirs, switch tc. high-powr circuits high-spd logic circuits for high-spd computrs. JT structur:
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 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 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 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 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 informationELG3150 Assignment 3
ELG350 Aigmt 3 Aigmt 3: E5.7; P5.6; P5.6; P5.9; AP5.; DP5.4 E5.7 A cotrol ytm for poitioig th had of a floppy dik driv ha th clodloop trafr fuctio 0.33( + 0.8) T ( ) ( + 0.6)( + 4 + 5) Plot th pol ad zro
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 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 informationMILLIKAN OIL DROP EXPERIMENT
11 Oct 18 Millika.1 MILLIKAN OIL DROP EXPERIMENT This xprimt is dsigd to show th quatizatio of lctric charg ad allow dtrmiatio of th lmtary charg,. As i Millika s origial xprimt, oil drops ar sprayd ito
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 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 informationExponential Moving Average Pieter P
Expoetial Movig Average Pieter P Differece equatio The Differece equatio of a expoetial movig average lter is very simple: y[] x[] + (1 )y[ 1] I this equatio, y[] is the curret output, y[ 1] is the previous
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 informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
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 informationRecursive Implementation of Anisotropic Filters
Rcursiv Implmtatio of Aisotropic Filtrs Zu Yu Dpartmt of Computr Scic, Uivrsit of Tas at Austi Abstract Gaussia filtr is widl usd for imag smoothig but it is wll kow that this tp of filtrs blur th imag
More informationNumerov-Cooley Method : 1-D Schr. Eq. Last time: Rydberg, Klein, Rees Method and Long-Range Model G(v), B(v) rotation-vibration constants.
Numrov-Cooly Mthod : 1-D Schr. Eq. Last tim: Rydbrg, Kli, Rs Mthod ad Log-Rag Modl G(v), B(v) rotatio-vibratio costats 9-1 V J (x) pottial rgy curv x = R R Ev,J, v,j, all cocivabl xprimts wp( x, t) = ai
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 informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signals & Systms Prof. Mark Fowlr ot St #21 D-T Signals: Rlation btwn DFT, DTFT, & CTFT 1/16 W can us th DFT to implmnt numrical FT procssing This nabls us to numrically analyz a signal to find
More informationNumbering Systems Basic Building Blocks Scaling and Round-off Noise. Number Representation. Floating vs. Fixed point. DSP Design.
Numbring Systms Basic Building Blocks Scaling and Round-off Nois Numbr Rprsntation Viktor Öwall viktor.owall@it.lth.s Floating vs. Fixd point In floating point a valu is rprsntd by mantissa dtrmining th
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 informationANOVA- Analyisis of Variance
ANOVA- Aalii of Variac CS 700 Comparig altrativ Comparig two altrativ u cofidc itrval Comparig mor tha two altrativ ANOVA Aali of Variac Comparig Mor Tha Two Altrativ Naïv approach Compar cofidc itrval
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationGeneralizing the DTFT. The z Transform. Complex Exponential Excitation. The Transfer Function. Systems Described by Difference Equations
Geeraliig the DTFT The Trasform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1 The forward DTFT is defied by X e jω = x e jω i which = Ω is discrete-time radia frequecy, a real variable.
More information