EECS 3 Digil Signl Processing Universiy of Cliforni, Berkeley: Fll 007 Gspr November 4, 007 Trnsforms II - Wveles Preliminry version plese repor errors, ypos, nd suggesions for improvemens We follow n pproch similr o [, ]. The emphsis of his documen is on concepul undersnding. No ll semens re mhemiclly enirely correc, bu hey definiely hold for sufficienly nice funcions, such s he ones h re ypiclly used in ime-frequency nd wvele nlysis. The Generl Trnsform. Definiion A useful generl wy of hinking of rnsforms is in he shpe of inner producs wih se of bsis funcions: where denoes he complex conjuge. T x (γ) = x(), φ γ () () = x()φ γ()d, () The ide here is h T denoes wh kind of bsis funcions re being used nd γ is he index of bsis funcion. The bsis funcions re φ γ () for ll vlues of γ. A good wy of hinking bou his is h for fixed γ, he rnsform coefficien T x (γ) is he resul of projecing he originl signl x() ono he bsis elemen φ γ (). An exmple is he Fourier rnsform, where insed of he leer γ, we more ofen use he leer Ω, nd where φ Ω () = e jω. Hence, in line wih he bove generl noion, we could wrie FT x (Ω) = x(), φ Ω () (3) = Of course, we more ofen simply wrie X(Ω) (or X(jΩ)) in plce of FT x (Ω). x()e jω d, (4). Alernive Formulion For our nex sep, we need he (generl) Prsevl/Plncherel formul, which ssers h f()g ()d = F(jΩ)G (jω)dω. (5) π
Ω Ω Ω 0 0 Figure : A concepul picure: We imgine h he bsis elemen φ γ () only lives in he shded box, i.e., h he signl is very smll ouside he inervl 0, nd h is specrum Φ γ (jω) is very smll ouside of he inervl Ω 0 Ω Ω. Using his, we cn rewrie he generl rnsform s T x (γ) = x(), φ γ () (6) = = π x()φ γ ()d (7) = X(jΩ), X(jΩ)Φ γ (jω)d (8) π Φ γ(jω) (9) Hence, we now hve wo good wys of hinking bou rnsforms: For fixed γ, he rnsform coefficien T x (γ) is he resul of projecing he originl signl x() ono he bsis elemen φ γ (), nd equivlenly, of projecing he originl specrum X(jΩ) ono he specrum of he bsis elemen φ γ (), which is π Φ γ(jω). Consider Figure : Merely s hough experimen, le us hink of bsis elemen φ γ () h lives only inside he box illusred in Figure. Then, gre wy of hinking bou he rnsform coefficien T x (γ) is h i ells us how much of he originl signl x() sis inside h box. In lines wih his inuiion, for he Fourier rnsform, he rnsform coefficien T x (Ω) ells us how much of he originl signl x() sis frequency Ω, nd he box shown in Figure is infiniesimlly hin in frequency nd infiniely long in ime. In he nex secion, we will mke precise wh lives mens.
.3 The Heisenberg Box Of A Signl Reconsider he concepul picure given in Figure. Now, we wn o mke his precise. In order o do so, consider ny signl φ(). For simpliciy (nd wihou loss of generliy), we ssume h he signl is normlized such h Noe h by Prsevl, his lso mens h π Φ(jΩ) dω =. φ() d =. (0) We define he following quniies. The middle of he signl φ() is given by m = φ() d. () If you hve ken clss in probbiliy, you will recognize his o be he men vlue of he disribuion φ(). Similrly, we define he middle of he specrum Φ(jΩ) o be m Ω = Ω π Φ(jΩ) dω, () wih similr probbiliy inerpreion. Moreover, we define: σ = σ Ω = ( m ) φ() d, (3) (Ω m Ω ) π Φ(jΩ) dω. (4) Agin, hese cn be undersood s he respecive vrinces of he wo probbiliy disribuions. Wih hese definiions, we cn now drw more precise picure of he ime-frequency box of he signl φ(), s given in Figure. We should lso poin ou h for he Fourier rnsform, he bsis funcions re of he form φ() = e jω0, nd for hose, he bove inegrls do no ll converge, so specil cre is required mhemiclly. However, he righ inuiion is o sy h he Heisenberg box (he erm ppers in [], nd perhps erlier) of he funcion φ() = e jω0 is horizonl line frequency Ω 0..4 The Unceriny Relion So, wh re he possible Heisenberg boxes? Theorem (unceriny relion). For ny funcion φ(), he Heisenberg box mus sisfy σ σ Ω. (5) Th is, Heisenberg boxes cnno be oo smll. Or: rnsforms cnno hve very high ime resoluion nd very high frequency resoluion he sme ime. (Proof: see clss.) 3
Ω σ m Ω σ Ω m Figure : The Heisenberg box of he funcion φ() (i.e., he plce in ime nd frequency where he funcion φ() is relly live). The Shor-ime Fourier Trnsform I hs long been recognized h one of he mos significn drwbcks of he Fourier rnsform is is lck of ime loclizion: An even h is loclized in ime (such s signl disconinuiy) ffecs ll of he frequencies (remember he Gibbs phenomenon). This feure is clerly undesirble for mny engineering sks, including compression nd clssificion. To regin some of he ime loclizion, one could do shor-ime Fourier rnsform, essenilly chopping up he signl ino shor pieces nd king Fourier rnsforms seprely for ech piece. Kind of rivilly, his gives bck some ime loclizion. More generlly, he following form cn be given: STFT x (τ, Ω) = x()g ( τ)e jω d, (6) where he funcion g() is n pproprie window funcion h cus ou piece of he signl x(). Wih he prmeer τ, we cn plce he window wherever we wn. Wih regrd o he generl rnsform, here, insed of he leer γ, we use he pir (τ, Ω), nd φ τ,ω () = g( τ)e jω. (7) Mny differen window funcions g() re being used, bu one of he esies o undersnd is he Gussin window: g() = 4 e σ. (8) πσ Noe h sricly speking, his window is never zero, so i does no relly cu he signl. However, if is lrge, g() is iny, so his is lmos he sme s zero, bu much esier o nlyze. Wih his 4
window, we find he bsis elemens o be φ τ0,ω 0 () = 4 ( τ 0) πσ e σ e jω0. (9) Now, we wn o find explicily he Heisenberg box of his bsis funcion. To his end, we need he Fourier rnsform of he Gussin window, which is known o be G(jΩ) = 4 4πσ e Ω σ, (0) nd hus, using he sndrd ime- nd frequency-shif properies of he Fourier rnsform, Φ τ0,ω 0 (jω) = 4 4πσ e (Ω Ω 0 ) σ e jωτ0. () Now, we cn find he corresponding prmeers of he Heisenberg box s: m = τ 0, () m Ω = Ω 0 (3) σ = σ (4) σω = σ, (5) nd so, we cn drw he corresponding Figure. I is lso ineresing o noe h for he Gussin window, he Heisenberg unceriny relion (Theorem ) is sisfied wih equliy. I cn be shown h he Gussin window is (essenilly) he only funcion h sisfies he unceriny relion wih equliy, see e.g. [, p.3]. In clss, we lso sw he resuling ime-frequency plos for speech signls (see he sepre figures h were hnded ou in clss). 3 Wvele Trnsforms This is n ineresing cse of he generl rnsform where we sr form single funcion someimes clled he moher wvele. ψ(), (6) Then, we build up our dicionry by shifing nd scling he moher wvele, specificlly, ψ m,n () = m/ ψ( m n), (7) where n nd m re rbirry inegers (posiive, negive, or zero). Th is, in plce of he prmeer γ, we will use he pir of inegers (m, n), where m is he scle (he bigger he corser) nd n is he shif. We will ofen denoe he rnsform coefficien s The following key quesions re of obvious ineres: m,n = WT x (m, n) = x(), ψ m,n (). (8) Wh re he condiions such h we cn recover he originl signl x() from he wvele coefficiens m,n? How do we design good moher wveles ψ()? How do we efficienly compue he wvele coefficiens m,n for given signl x()? nd of course mny more... 5
3. The Hr Wvele We will sr wih he Hr wvele: ψ() =, for 0 <,, for <, 0, oherwise. (9) Exercise: Skech ψ 0,0 (), ψ,0 (), ψ,0 (), nd ψ,3 (). Fcs:. The funcions ψ m,n (), ken over ll inegers m nd n, re n orhonorml se. (Esy o verify.). The funcions ψ m,n (), ken over ll inegers m nd n, re in fc n orhonorml bsis for L (R), he spce of ll funcions x() for which x() d is finie. (This is more difficul o prove, see e.g. [, ch.7].) Due o hese fcs, we cn express ny squre-inegrble funcion x() in he following form: x() = m,n ψ m,n (), (30) m= n= which will be clled he Hr expnsion (of more generlly, he wvele expnsion) of he signl x(). Moreover, due o he orhogonliy, we lso know h m,n = x(), ψ m,n () = x()ψ m,n ()d. (3) To undersnd how his wvele works, i is insrucive o consider piecewise consn funcion, s in Figure 3. We here follow he developmen in [, p.]. Specificlly, consider he funcion f (0) () which is piecewise consn over inervls of lengh 0 =, nd ssumes he vlues...,b, b 0, b, b,.... As shown in Figure 3, we cn wrie f (0) () s he sum of wo componens: A sequence of shifed versions of he Hr wvele scle m = (i.e., of he funcions ψ,n ()) nd residul funcion f () (), which is piecewise consn over inervls of lengh =. Specificlly, we cn wrie nd where we use d () () = f () () = n= n= b n b n+ ψ,n (), (3) }{{},n b n + b n+ ϕ,n (), (33) }{{} c n ϕ,n () = {, for n < n +, 0, oherwise. (34) The rel boos now comes from observing h we cn jus coninue long he sme lines nd decompose f () () ino wo prs. 6
f (0) () b 0 b b 3 6 7 d () () 3 6 7 f () () 3 6 7 Figure 3: The Hr wvele work. 7
b n H, n H 0 H, n H 0 H 3, n H 0... Figure 4: The Hr filer bnk. 3. A Filer Bnk Compnion To The Hr Wvele Le us now reconsider Equions (3) nd (33). We cn rewrie hem purely in erms of he coefficiens s,n = b n b n+ (35) c n = b n + b n+, (36) which we esily recognize s filering he sequence b n wih wo differen filers nd hen downsmpling by fcor of wo! In oher words, sring from he coefficiens b n, here is simple filer bnk srucure h compues ll he wvele coefficiens, illusred in Figure 4. For he Hr exmple, he filers re H (z) = ( z) (37) H 0 (z) = ( + z). (38) Noe h hese filers re no quie cusl, bu since hey re FIR, his is no problem ll. 3.3 Tilings Of The Time-frequency Plne An ineresing observion follows by observing h he Hr filer H (z) is ( crude version of) highpss filer, nd H 0 (z) (no less crude version of ) lowpss filer. Neverheless, if we merely go wih his highpss/lowpss picure, nd merge i wih he downsmpling in ime, we cn drw figure like he one given in Figure 5: consider for exmple he wvele coefficien,0. I resuls from highpss-filering, so i perins o he upper hlf of he frequencies. Moreover, i perins only o he ime inervl from 0 o. This is how we found he recngle lbelled,0 in Figure 5, nd his is how you cn find ll he oher recngles in he figure. 8
Ω,,,0,,,3 3,,,0, 3,0 Figure 5: A iling of he ime frequency plne h somewh mirrors wh he Hr wvele is doing. References [] M. Veerli nd J. Kovcevic, Wveles nd subbnd coding. Upper Sddle River, NJ: Prenice Hll, 995. [] S. Mll, A wvele our of signl processing. Sn Diego, CA: Acdemic Press, nd ed., 999. 9