(looks like a time sequence) i function of ˆ ω (looks like a transform) 2. Interpretations of X ( e ) DFT View OLA implementation

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1 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,,,...; ˆ π ˆ m j ˆ. Itrprtatios of X ˆ. ˆ fixd, ˆ variabl; X ˆ ˆ DTFT x[ mw ] [ m] DFT Viw OLA implmtatio ˆ ˆ. variabl, fixd; X ˆ x[ ] w[ ] ˆ j j Liar Filtrig viw filtr ba implmtatio FBS implmtatio viw of STFT viw of STFT 3. Samplig ats i Tim ad Frqucy ˆ j ˆ rcovr x [ ] xactly from X ˆ. tim: W has badwidth of BHrtz B sampls/sc rat F Hammig Widow: B S Hz sampl at L 4F S Hz or vry L/4 sampls L. frqucy: w [ ] is tim limitd to Lsampls d at last L frqucy sampls to avoid tim aliasig i with OLA mthod ca rcovr x xactly usig lowr samplig rats i ithr tim or frqucy,.g., ca sampl vry L sampls ad divid by widow, or ca us fwr tha L frqucy sampls filtr ba chals; but ths mthods ar highly subjct to aliasig rrors with ay modificatios to STFT i ca us widows LPF that ar logr tha L sampls ad still rcovr with < L frqucy chals;.g., idal LPF is ifiit i tim duratio, but with zros spacd sampls apart whr F / is th BW of th idal LPF S 3 4 viw of STFT H H X Y + H - Tr-Dcimatd Filtr Bas h [ ] w[ ] H H h [ ] h [ ] δ [ ] wp [ ] [ ] p [ ] [ r] δ r h [ ] wr [ ] δ [ r] w[ ] + w [ ] +... r d to dsig digital filtrs that match critria for xact rcostructio of x [ ] ad which still wor with modificatios to STFT 5 6

2 Tr-Dcimatd Filtr Bas ca sampl STFT i tim ad frqucy usig lowpass filtr widow which is movd i jumps of <L sampls if L whr: L is th widow lgth, is th widow shift, ad is th umbr of frqucy chals for a giv chal at, th samplig rat of th STFT d oly b twic th badwidth of th widow Fourir trasform dow-sampl STFT stimats by a factor of at th trasmittr up-sampl bac to origial samplig rat at th rcivr fial output formd by covolutio of th up-sampld STFT with a appropriat lowpass filtr, f [] 7 Filtr Ba Chals Fully dcimatd ad itrpolatd filtr ba chals; a aalysis with badpass filtr, dowshiftig frqucy ad dow-samplig; b sythsis with up-samplr followd by lowpass itrpolatio filtr ad frqucy up-shift; c aalysis with frqucy dow-shift followd by lowpass filtr followd by dow-samplig; d sythsis with up-samplig followd by frqucy up-shift followd by badpass filtr. 8 Full Implmtatio of Aalysis-Sythsis Systm Full implmtatio of STFT aalysis/sythsis systm with chal dcimatio by a factor of, ad chal itrpolatio by th sam factor icludig a box for short-tim modificatios. 9 viw of Dow ad Up-Samplig x[] x [] xd [ ] x[ ] j πr/ r X X d i aliasig additio x [ / ], ±, ±,... x [ ] othrwis u Xu X i imagig rmoval Filtr Ba costructio assumig o short tim modificatios X X w rm x m r y PY r j P [ Xr[ f ] [ r] ] r j m m Wr j xm wr mf r P m Wr r rcogizig that wh P, m δ mq q m ca show that th coditio for y x, is r wr + qf r δ q m Filtr Ba costructio frqucy domai quatios, assumig: h[ ] w[ ] ; g[ ] f[ ] ; H W G F j j j j ; ; Y PG π π j j H X X PG H π π j j + X PG H H X + aliasig trms

3 Filtr Ba costructio Filtr Ba costructio coditios for prfct rcostructio ˆ Y X H PG H ad π j j PG H for,,..., Flat Gai Alias Cacllatio 3 4 Dcimatd Aalysis- Sythsis Filtr Ba practical solutio for th cas w f, -bad solutio whr th coditios for xact rcostructio bcom j π π j + 4 P F W P F W ad j π j π j π P F W 4 + P F W aliasig cacls xactly if P P with F W givig W W 4 y ˆ x M j j M π 5 Dcimatd Aalysis- Sythsis Filtr Ba i wh prfct aliasig is limiatd by suitabl choic of filtrs, th ovrall aalysis-sythsis systm has frqucy rspos: j PW [ ] H whr W W F h [ ] w [ ] p[ ] whr w [ ] w[ ] f[ ] p [ ] P [ ] 6 Dcimatd Aalysis- Sythsis Filtr Ba To dsig a dcimatd aalysis/sythsis filtr ba systm, d to dtrmi th followig:. th umbr of chals,, ad th dcimatio/itrpolatio ratio. Usually dtrmid by th dsird frqucy rsolutio. th widow, w[], ad th itrpolatio filtr impuls rspos, f []. Ths ar lowpass filtrs that should hav good frqucy-slctiv proprtis such that th badpass chal rsposs do ot ovrlap ito mor tha o bad o ithr sid of th chal. 3. th complx gai factors, P[],,,,-. Ths costats ar importat for achivig flat ovrall rspos ad alias cacllatio. Maximally Dcimatd Filtr Bas w show hr that it is possibl to obtai xact rcostructio with ad <L. is trmd maximal dcimatio bcaus this is th largst dcimatio factor that ca b usd with a - chal filtr ba ad still achiv xact rcostructio. omial badwidth of th badpass chal sigal is icrasd from ±π/ to ±π dow-samplig by accomplishs frqucy dowshiftig ormally accomplishd by modulatio 7 8 3

4 Maximally Dcimatd Filtr Bas Aalysis-Sythsis Filtr Ba all modulators rmovd 9 Two Chal Filtr Bas Two Chal Filtr Bas i For this two-bad systm, w hav: ; π,, i Applyig th frqucy domai aalysis w gt: Y G H + G H X j j j j j j + G H + G H X π j j j j π π j i ot icludig multiplir / or th gai factors P[], P[] i Aliasig ca b limiatd compltly by choosig th filtrs such that: G H + G H aliasig caclatio π j j j j π i Prfct rcostructio rquirs: G H + G H flat gai coditio j j j j Quadratur Mirror Filtr QMF Bas Quadratur Mirror Filtr QMF Bas i Alias cacllatio achivd if th filtrs ar chos to satisfy th coditios: h h H H [ ] [ ] jπ j j π g [ ] h [ ] G H g h G H [ ] [ ] j j π i ot that thr is oly a sigl lowpass filtr, with impuls rspos h [ ] o which all othr filtrs ar basd; filtr has omial cutoff of π / rad with a arrow trasitio to a stopbad with adquat attuatio to isolat th two bads i Th filtr h[ ] h [ ] is th complmtary highpass filtr jπ i Th filtrs h [ ] ad h[ ] ar calld Quadratur Mirror Filtrs QMF i Substitutig for th filtr rlatioships w gt: H H H H j j j π π π i.., prfct alias cacllatio for ay choic of lowpass filtr j j j Y H H π j j X H X i For prfct rcostructio w rquir that: j π M H H i.., flat magitud with dlay of M sampls, du to th dlay of th causal filtrs of th filtr ba 3 4 4

5 Quadratur Mirror Filtrs i Assum FI lowpass filtr of lgth L sampls such that h[ ] h[ L ], L; this filtr has liar phas with a dlay of L / sampls ad a frqucy rspos of th form: j H A A L/ L must b odd whr is ral. Usig this lowpass filtr i th filtr ba, th coditio for prfct rcostructio bcoms: j j j L j j H π π A A L i If L is odd, w gt: j j j j L H π A + A i liar phas is guaratd, but ot prfct rcostructio 5 Quadratur Mirror Filtrs i For flat gai w d to satisfy th coditio: j π A + A i Solutio is to us high ordr liar phas QMF filtrs i Johsto proposd a algorithm for dsig of th basic lowpass filtr that miimizd th dviatio from uity whil providig a dsird lvl of stopbad attuatio 6 Quadratur Mirror Filtrs Quadratur Mirror Filtrs Lowpass filtr dsigd by widowig Highpass filtr is mirror imag of lowpass filtr π h[ ] w[ ] g[ ] j h[ ] w[ ] g [ ] 7 practical ralizatio of QMF filtrs ot that aliasig is caclld, but th ovrall frqucy rspos is ot prfctly flat 8 Quadratur Mirror Filtrs Issus with Prfct costructio Th motivatio for th subbad approach is to procss diffrt spch proprtis idpdtly i ach bad ad thus big abl to localiz th bad compact badwidth is importat. Also,basd o th origial motivatio of short-tim aalysis, tmporal focus withi a limitd tim duratio is also importat. Filtr ba dsig is thus rsult of a trad-off btw prfct rcostructio ad badwidth coctratio i a joit critrio: J α π s W d + α T d Stop-bad laag π Dviatio from P 9 Wh procssig.g., quatizatio, a o-liar opratio is ivolvd, harmoic distortios ar ivitabl ad prfct rcostructio caot b achivd, as s i th simpl xampl. Thr xists a dsig procdur Smith-Barwll for QMF filtrs that attmpts to mt th dsig costraits as bst as possibl 3 5

6 Smith-Barwll QMF Dsig. Start with a squc of lgth, say, L- L v that satisfis g + g C δ ; that is, v sampls, xcpt at. C, th valu of th squc at, ds to b larg ough to satisfy th positivity coditio s blow.. Factor G z Z{ g } W z W z such that all roots of W z ar isid th uit circl. 3. Positivity coditio: G W W W > 4. Th mirror chal: W W, L W z W z z or quivaltly w w 5. To limiat aliasig: F z W z ad F z W z F z W z or quivaltly f w F z W z or quivaltly f w 3 Smith-Barwll QMF Dsig - Exampl L 8, L 7 Factoriz G z W z W z G z Z{ g } W z W z Lgth of w L, th g has lgth L choos si π / g,, ±, ±,, ± 7 π g [ ] at,. is addd to satisfy th positivity coditio w [ ] W z :lgth L with all roots isid thuit circl w w [ ] f w [ ] f w [ all arrays start with ] 3 QMF Frqucy sposs Exampl Magitud db Phas dgr, x ormalizd frqucy xπ 33 Tr-Structurd Filtr Bas 34 Tr-Structurd Filtr Bas Samplig Pattr for Tr- Structurd Filtr Bas

7 Tr-Structurd Filtr Bas Tr-Structurd QMF Filtrba Quatizd Chal Sigals δ [] [ ] h [ ] g [ ] δ [ ] yˆ [ ] yˆ [ ] Subbad Codig δ [ ] h[ ] g [ ] [] δ [] Quatizatio ois is coctratd i th bad that it is gratd i. Pˆ σ G + σ G y j j j yˆ [ ] 39 advatags of subbad codig ach subbad ca b codd accordig to th prcptual critrio appropriat to that bad quatizatio ois ca b cofid to th bad that producs it low rgy bads ca b codd so as to produc lss ois applicabl to codig i th Kbps rag 4 Practical Exampl of Subbad Codr Subbad Codr Subjctiv Quality subbad codr usig tr-structurd filr ba Crochir,

8 Subbad Codr Wavforms Cocpt of Tim-Frqucy solutio Wh itrstd i th chagig charactristics of th sigal, short tim Fourir trasform or short tim spctral aalysis is usd. j F, t f τ g t τ τ dτ gt is th widow fuctio providig focus i tim. 43 Tmporal sprad : σ t t t c g t dt g t dt Frqucy sprad : σ c G d G d If g t is Gaussia, it achivs th miimum tim -badwidth product σ t σ.5. Cotiuous prolat sphroidal wav fuctio: a badlimitd sigal that hav th highst rgy coctratio i a spcifid tim itrval. To maitai a similar lvl of ucrtaity, th widow fuctio should b short to provid tim rsolutio for high frqucy compots ad log to provid frqucy rsolutio for low frqucy compots. 44 Wavlt-Basd Mthods Aalysis Filtrs Subbad mthod Sythsis Filtrs Bad Q Q - Bad Bad Q Q - Bad Bad Q Q - Bad Aalysis Filtrs Wavlt Trasform Wavlt mthod Q Q - Q Q - Q Q - Sythsis Filtrs Ivrs Trasform What is i th Wavlt Trasform bloc? What is th diffrc? 45 Wavlt Trasform Built o st of xpasio or basis fuctios that ar drivd from a mothr wavlt through scalig ad traslatio: t τ Lt t b a mothr wavlt : a, τ t a a Cotiuous Wavlt Trasform: * * t τ τ * F * w a, τ f t a, τ t dt f t dt f τ a a a a * τ F w a, τ f at t dt strtch or comprss th sigal for aalysis a a a, τ scalogram; F, τ spctrogram F w Lt Ψ F{ t} ad Ψ a, τ F{ a, τ t} dadτ Ivrs trasform: f t Fw a, τ a, τ t C a Ψ Admissibility coditio: C d < Ψ 46 Wavlt ad Fourir Trasform Ψ F t} ad Ψ F{ } F F{ f t} { a, τ a, τ t t τ aψ a a a a, τ t Ψa, τ * t τ τ * F w a, τ f t dt f τ * a a a a * τ W a, F Fw a, τ F f τ * af Ψ a a a W a, is th frqucy domai rprstatio of th wavlt trasform. jτ Wavlt ad Its Spctrum Ψ t / a Ψt σ aσ, < a < a σ,< a / a σ / a σ 47 / a 48 8

9 Scalig ad Traslatio For discrt wavlt trasform, m m a a ad τ a τ a, τ m, τ If a ad τ m/ m m,, t t, m, ar itgrs m/ m t t a a t m, Z ad m, Z dyadic or octav samplig If th st { m, t} is complt, it is calld affi wavlts. * Orthogoality coditio : m, t pq, t dt δ m p δ q m / m F a, τ w a f t a t dt w m, τ f t w m, m, t m Th orthogoality coditio may ot b asy to satisfy for a gral st of wavlts; but it is still possibl to ivrt discrt wavlt trasform, somtims. 49 frqucy STFT ad CWT tim Costat badwidth frqucy tim Costat-Q badwidth 5 Discrt Wavlt Trasforms Othr Simplr Trasforms or Filtr Bas Short-tim aalysis ca also b viwd as data trasformatio, xcutd i a bloc-by-bloc mar, usig for xampl: Discrt Cosi Trasform Discrt Si Trasform Filtr ba basd o DFT/FFT Gral rmars: Discussio so far aims at a filtr ba framwor that allows prfct rcostructio i.., o loss of iformatio i th origial sigal if dsird; But may spch procssig systms do ot rquir prfct rcostructio; rathr, o may wat to focus o xtractio of y paramtrs or faturs from th spch sigal; Thrfor, othr aalysis mthods.g., trasformatios that do ot hav ivrs may still apply. 5 5 DFT ad DCT -poit DFT Discrt Fourir Trasform X x -poit symmtric jπ / implis ~ j j x π / π / X x + r -poit DFT -poit DCT r Possibl xcssiv discotiuity at dgs lats to ral part of -poit DFT wh th sigal is symmtric X x jπ / ad x x X But, b carful about th poit of symmtry!! x cosπ / 53 Discrt Cosi Trasforms Extsivly usd i audio, imag ad vido π c x + x + x cos DCT-I π c x cos + Most commo DCT; quivalt up to a scal factor of to a DFT of 4 ral iputs of v symmtry with th v-idxd lmts st to zro; i.., a artificial shift of half sampl i th origial samplig stup. DCT-II DCT-III π c x + x cos + IDCT v aroud x ad odd aroud x ; c is v aroud ad odd aroud. Still mor variats. 54 9

10 s ij S Discrt Si Trasform [ ] s ij i, j -D cas; limiat o dimsio for -D j + i + π si, i, j,,,, + + lats to imagiary part of ~-poit DFT Discrt Tim Fourir Trasform - visitd Th DTFT discrt tim Fourir trasform of a - poit squc is X x Frqucy is cotiuous. Sampl th DTFT at π/,,,,. Th rsult is th DFT Discrt Fourir Trasform 3-poit squc 8-poit atisymmtric squc Squc corrspods To 8-poit DFT 55 X j / / π π x X If w comput th ivrs DFT, w obtai ~ j j x π / π / X x + r r 56 DFT for Powr Spctral Dsity Estimatio X x P S X S S P, Exampls of o-uiform filtr bas : th chal procssig wightig fuctio Prfct rcostructio may ot b a rquirmt. 57