THE CONTINUOUS WAVELET TRANSFORM The eigenvalues An depend on T' and of course; an easy scaling argument (substitute r= Tr, y= Ty in the expression for(Qr Pn Qt f(=)shows that the An depend only on the product Tf. For fixed Tf, the behavior of An as n increases is schematically represented in Figure 2. 3. Typically, the An stay close to 1 for small n, plunge to zero near the threshold value 2Tn / and stay close FIG. 2. 3. The eigenvalues An for Or P]Qr for 2TR/T=25 to zero afterwards. More precisely, for any (arbitrarily smal E>0, there exists 8 constant C so that {m;A≥1-e}≤a-clog(m, #{m;1-∈≥A≥e}≤2 Ce log(Tn), (232) which means that the "plunge region"has width proportional to log (Tn). Since limx-oo z-Ilogr=0, the width of the plange region becomes negligibly small when compared to the threshotd value2mg/r,asTΩ→∞. In fact,(2.3.2)ia rigorous version of the fact that a time- and bandlimited region( -T,r]x(f, corresponds to 2Tn/r"degrees of freedom, " i.e, there exist(up to an error, amall compared to Tn)2T1/ independent functions (and not more) that are essentially timelimited to T, r] and bandlimited to -n2, n. Note that 2Tn/ is exactly the area of (T,r x[-n,], divided by 2T. This num- ber is therefore equal to the number of sampling times within -T,T specified by Shannon's theorem for a function with bandwidth S2; this heuristic way of counting the "independent degrees of freedom"was part of the folklore of com munication theory long before it was justified by Landau, Pollak, and Slepian Independently, it was also known to physicists that a region in phase space <(=space-momentum, or time-frequency such as here )with area S corresponds to S/2x "independent states" in the semiclassical limit (i.e, when S is much larger than A; the expression S/2T corresponds to units such that h= 1).We will extend the definition of Nyquist density from its original sampling back ground, and use it for the critical time-frequency density(2x- present in all these examples It is time to return to the wavelet transform. In what follows we will develop the contimuous version of both the wavelet transform and the windowed Fourier
CHAPTER 2 2. 4. The continuous wavelet transform We restrict ourselves, for the time being, to one-dimensional wavelets. We always suppose that E L2(R); the analyzing wavelet should moreover satisfy the admissibility condition already mentioned in $1.3 C=2r/dK()2<∞o (241) The role of this condition will soon become clear.Ifψ∈Ll(k),thenψis continuous and (2.4. 1)can only be satisfied if v(O)=0, or dr v(z)=0.On the other hand, if dr v(z)=0 and we impose a slightly stronger condition than integrability on v, namely dr(1+1z)a l(e)l oo for some a>0,then l(E)IS CIEI, with B= min(a, 1), and(2.4.1)is satisfed. It follows that, for all practical purposes, (2 4: 1)is equivalent to the requirement that dr v(z)=0 (In practice, we will impose far more stringent decay conditions on w than those needed in this argument We generate a doubly- indexed family of wavelets from w by dilating translating, (x)={a-12p(x-6 where a, bE R, afo(we use negative as well as positive a at this point ).The normalization has been chosen so that Hψ·=lψ for all a,b. We will assume that lyll= 1. The continuous wavelet transform with respect to this wavelet family is then (Twyf)(a,b)=(f,ψ dz∫(x)l1/2y Note that o(r"aˇ∫)(a,b)≤‖f‖ A function f can be recovered from its wavelet transform via the resolution of the identity, as follows. PROPOSITION 2.4. 1. For all S,gEL(R) da db (r"∫)(a,b)(Twag)(a,b)=Cψ《f,g) (24.2) -∞J-co da db oo -oo 02(Twa(a, b)(Twag)(a, 6) //4U岖N(lw d')lH2e“y(aE (243)
THE CONTINUOUS WAVELET TRANSFORM The expression between the first pair of brackets can be viewed as(27)1/ times the Fourier transform of Fa()=|a/2∫(￡)ψ(aE); the second has a similar tr interpretation as(2m)/ times the complex conjugate of the Fourier transform A of Ga(5)=la)/ 9(E)v(aE). By the unitarity of the Fourier transform it follows 2439-2/=/岖( /间()5wa 2r/(页/yP (nterchange is allowed by Fubini's theore C+(, g) (make a change of variables s= af in the second integral) It is now clear why we imposed(2. 4.1): if C+ were infinite, then the resolution of the identity(2.4.2)would not hold Formula(2. 4.2 )can be read =c广广命mabp (244) with convergence of the integral "in the weak sense, "i.e, taking the inner prod uct of both sides of(2.4.)with any gE [(R), and commuting the inner product with the integral over a, b in the right-hand side, leads to a true formula. The convergence also holds in the following, slightly stronger sense da db im。-c Tf)(a,b)ψ4=0.(245) A? "- Here the integral stands for the unique element in L(R)that has inner products with g∈L2(R) given by Mg(Tw)(a, b)(vsa, b, 9) since the absolute value of this is bounded by da db IflI Aa ll all=4B A1-^42)l
CHAPTER 2 we can give a sense to the integral in(2. 4.5)by Riesz'lemma. The proof of (2.4.5) is then simple P(Twˇ∫)ab)ψ A1<|a≤A2 b a2(Tw1织 da db sup Igl=l A1 s sup C-1 ∫dd a(rab)(m9a列 > 1/2 do db aI(T9)(a,6 By Proposition 2.4.1, the expression between the second pair of brackets is igIl= 1, and: the expression between the first pair of brackets converges to zero as A1-0, A2, B-00, because the infinite integral converges. This estab- lishes(2. 4.5) Formula(2.4.5), which shows that any f in L2(R)can be arbitrarily well approximated by a superposition of wavelets, may seem paradoxical: after all wavelets have integral zero, so how can any superposition of them(which nec- essarily still has integral zero) then be a good approximation to f if f itself happens to have nonzero integral? The solution to this paradox does not lie (as solutions to paradoxes so often do) in the mathematical sloppiness of the question. We can easily make it all rigorous: if we take fE L'(R)nL2(R),and if y itself is in L'(R), then one easily checks that the da db o2(Tw f)(a, b)wa. A<|a≤A are indeed all in L(R)(with norm bounded by 2C+ IfHza Wp ll a l volleY /1V)a A2/), and that they have integral zero, whereas f itself, the func tion they are approaching as A1-0, A2, B-oo, may well have nonzero integral
THE CONTINUOUS WAVELET TRANSFORM The explanation to this apparent paradox is that the limit(2.4.5)holds in L2- sense, but not in L-gense. As A+0, A2, B00, 的Q2wb da db becomes a very fat, very stretched-out function, which still has the same integral asf itself, but vanishingly small L2-norm.(This is similar to the observation hat the functions m(z)=(2n) -I for 1rl sn,0 otherwise, satisfy 9n= 1 for alln, even though g(x)→0 for all a,and|gn‖L=(2n)-1/2-0forn→;the in do not converge in L(R).) Several variations on(2.4.4)are possible, in which we restrict ourselves to positive a only (as opposed to the use of both positive and negative a in(2. 4.4)) One possibility is to require that p satisfy an admissibility condition slightly more stringent than(2.4.1), namely dE I5I-1 l(E)1=2n/ dE IEI-l(E) (24.6) Equality of these two integrals follows immediately if, e. g, al is a real function, because then v(-f)=MlE). The resolution of the identity then becomes, with this new C 西T""ˇ∫(a,b)ψ (24.7) to be understood in the same weak or slightly stronger senses as(2.4. 4 ).(The proof of (2.4.7)is entirely analogous to that of(2.4.4). Another variation occurs if f is a real function, and if support w C o, oo) In this case, one easily proves th ∫=2c db Re t"r∫(a,b)ψ], (248) with Cw as defined by(2.4.1). (To prove(2.4.8), use that f(=)=(2r)-1/2 2Re jo df e ff(E),because f(E)=f().)Formula(2.4.8)can of course be other's Hilbert transform. Using a complex wavelet, even for the analysis of eal functions, may have its advantages. In Kronland-Martinet, Morlet, and Grossmann(1987), for example, a complex waveletψ with supportψC阳0,∞)i used and the wavelet transform T wav f is represented by graphs of its modulus and its phase If both f and w are so-called " analytical signals, i.e., if support f and apportψc卩,∞), then T"w"f(a,b)=0ifa<0, 8o that(24.4) immediately simplifies to