CHAPTER 2 It follows that -左厂 ds"f(E) 左厂峻∑c √2 ∑a/d f(n) sin丌(x-n) (211) where we have interchanged integral and summation in the third step, which Is only a priori justifiable if >lenI oo(e.g, if only finitely many cn are nonzero). By a standard continuity argument, the final result holds for all band- limited f(for every I, the series is absolutely summable because 2n If(n)r2 2丌∑n|n2<∞) Formula(211) tells us that f is completely determined by its"sampled"values f (n). If we lift the restriction S= T and assume support f c[-n, 2, with n arbitrary, then(2. 1.1 )becomes f()=∑∫(n sin(x-n丌) the function is nowdetermined by its samples f(nD),corresponding to a sampit ing density"of 3/r= Support 4 .(We use the notation A) for the “sie"ofa, et ACR, as mie::tl red by the Lebesgue measure; in this case 1, 2 I= 20. ) Thig sanpling density is usually called the Nyquist density. The expansion(2.1.2)goes by the name of Shannon's theorem (1+)) (1+) FIG. 2.1. Graph of ga The"elementary building block ni in(2.1.2)decay very slowly(they are not even absolutely integrable).Oversampling" makes it possible to write f asa superposition of functions with faster decay. Suppose that f is still bandlimited in【-9,ie., support f C[-9), but that∫i& pled at&rae(1+川 faster than the Nyquist rate, with A>0. Thea f can be recovered from the ∫(nr/g2(1+川)) in the following way. Define ga by
THE CONTINUOUS WAVELET TRANSFORM ≤g (E) ≤|≤(1+), |≥(1+)n feee Figure 2.1). Because ga= I on support f, we have f(E)=f(E)9A(5).We can now repeat the same construction as before f()=∑cn nE/j(1+) 2(1+x)(1+x) n(1+A) des()∑cne(Q+) Ω2(1+), Ga(c) 2(1+)91( 2snz?(1+/2)sin(xA/2) A2(1+))x2 R)These Ga have faster decay than ni: note that if A-o, then GA+"nI i ected. One can obtain even faster decay by choosing 9x smoother, but it not pay to put too much effort into making gx very smooth: true, Ga will Arxe very fast decay for asymptotically large t, but the size of A imposes some rictions on the numerical decay of GA. In other words, a coo choice of ga sleds to Ga decaying faster than any inverse polynomial G2(x)≤CN(X)(1+|)(N+), but the Constant CN()can be very large: it is related to the range of values the Nth, drivative of 9x on [0, Q (1+ A)), so that it is roughly proportional What happens if f is"undersampled, i.e., if support f=[-n, n), but only e f(nx/Q(1-A))are known, where A>0? We have 2(1- dE f(e)en/n(1-A) 1(1-4) √ df emme/n(I-A) (1-A) Uf()+f(E+2(1-刈)+(-20(1-)
CHAPTER 2 where we have used that the ennE/a have period 2a, and where we have assumed as 3(otherwise more terms would intervene in the sum in the last integrand) This means that the undersampled f(nn(i-x)behave exactly as if they were the Nyquist-spaced samples of a function of narrower bandwidth the fourier trans- form of which is obtained by" folding over”∫( see Figure2.2). In the“ folded” version of f, some of the high frequency content of f is found back in lower frequency regions; only the lEI s n (1-2x)are unaffected. This phenomenon is called aliasing; for undersampled acoustic signals, for instance it is very clearly audible as a metallic clipping of the sound 2.2. Bandlimited functions as a special case of a reproducing kernel Hilbert space For any a,月,-∞≤a<B≤o, the set of functions ∫∈L2(瞅R); support∫c, constitutes a closed subspace of L(R),i.e, it is a subspace, and all Cauchy sequences composed of elements of the subspace converge to an element of the subspace. By the unitarity of the Fourier transform on L(R), it follows that the set of all bandlimited functions Bn={∫∈L2(R); support C|-92,2} is a closed subspace of L(). By the Paley-Wiener theorem(see Preliminaries) any function f in Bn has an analytic extension to an entire function on C, which we also denote by f, and which is of exponential type More precisely f(叫)≤在=fztm In fact, Bn consists of exactly those L-functions for which there exists an an- alytic extension to an entire function satisfying a bound of this type. We can ) 1(ξ2Q(1λ) -9-9(1λ) 0 (1-λ)g FIG.2.2. The three term j(E), J(E+ 20(1-A)), and f(E-20(1-A))for KEI s n(1-A) nd their rum(thick line)
THE CONTINUOUS WAVELET TRANSFORM therefore consider Ba to be a hilbert space of entire functions For f in Bn we 2丌 de/d∫v)en dy f(y) sin s(a-y 丌(x-y) (221) (The interchange of integrals in the last step is permissible if fE L, i.e, if f is sufficiently smooth. Since, for all r, (T(e-)-Isin n(t-)is in L2(R), the conclusion then extends to all f in Bn by the standard trick explained in Pre- x: luminaries. )Introducing the notation ex(v)=nl= mu), we can rewrite(2 2.1) ∫(x)=(f,ex (222) Note that er e Bn, since ez(E)=(2r) -1/2e-irf for IEl f, e(E)=0for > Formula(2.2.2)is typical for a reproducing kernel Hilbert space(rk H s). In 8 an r.k. H.s. T of functions, the map associating to a function f its value f(z) at a point s is a continuous map(this does not hold in most Hilbert space of functions, in particular not in L(R)itself), so that there necessarily exists ez∈ such that f(x)=(f,ex) for all∫∈M( by riesz' representation lemma; see Preliminaries). One also writes ∫(x)=/dK(x,y)f(y), 2. where k(y)=ex(y)is the reproducing kernel. In the particular case of Bn, there even exist special In=n so that the er, constitute an orthonormal basis N e for Bn, leading to Shannon's formula (2.1. 2). Such special In need not exist in a a general r k. H s. We will meet several examples of other r.k. H s 's in what follows 4 2.3. Band- and timelimiting Functions cannot be both band- and timelimited: if f is bandlimited (with arbitrary finite bandwidth), then f is the restriction to r of an entire analytic function;if∫ were timelimited as well, suppor∫c[T, with T<∞, thenf =0 would follow (nontrivial analytic functions can only have isolated zeros). Nevertheless, many practical situations correspond to an efective band and timelimiting: imagine, for instance, that a signal gets transmitted (e.g. over a telephone line)in such a way that frequencies above n are lost(most realistic transmission means suffer from this kind of bandlimiting); imagine, als that the signal(such as a telephone conversation) has a finite time duration The transmitted signal is then, for all practical purposes, effectively band-and timelimited. How can this be? And how well can a function be represented by
CHAPTER 2 such a time- and bandlimited representation? Many researchers worked on these problems, until they were elegantly solved by the work of H. Landau, H. Pollack, and D. Slepian, in their series of papers Slepian and Pollak(1961)and Landau and Pollak(1961, 1962). An excellent review, with many more details than are given here, is Slepian(1976) The example mentioned above( sigmal with finite time duration transmit- ted over a bandlimiting channel)can be modeled as follows: let Qt, Pn be the orthonormal projection operators in L(R)defined by (qr∫)(x)=f(x) for =<T,(Qr/∫)(x)=0for|叫>T (Pn)(E)=f(E)for KEI<n,. (Pnf(E)=0 for IEI>f Then a signal which is timelimited to [ r, r] satisfies f= Qrf, and transmit- ting it over a channel with bandwidth S giveB as end product Pnf PQrf provided there is no other distortion). The operator P QT represents the total time band limiting process. How well the transmitted Pn Qrf approaches the original∫ is measured by‖PQr川2/f‖2=( QT Ph QT f,f)/f|2 The maximum value of this ratio is the largest eigenvalue of the symmetric operator Qr Ph QT, given explicitly by dy sin n(z-y) ∫(y)if|l<T (Or Pn Qr f)(e) 丌(x-y) (231) 0 if r>T The eigenvalues and eigenfunctions of this operator are now known explicitly because of a fortunate accident Qt P q T commutes w differential (A)(x)= m-2)-a The eigenfunctions of this operator, which had been studied for different reasons long before their connection with band- and timelimiting was discovered, are called the prolate spheroidal wave functions, and many of their properties are known. Because A commutes with Or P Or (and because the eigenvalues of A are all simple), the prolate spheroidal wave functions are aleo the eigenfunctions of @T Pn Qr(with different eigenvalues, of course). More specifically, if we Corresponding eigenvalues an of A increase as n increases, the red o that the denote the prolate spheroidal wave functions by wn, n E N, orde Qr Pa qr an= Anin QT P OT∫=0钟∫⊥ pin for all n 钟∫fi8 upported on{x;|l≥T An decreases as n increases, and lim An =0