THE WHAT. WHY, AND HOW OF WAVELETS J1 +J 2…'v(24x+1)l…'v(24x) .1.7.Th o∫∫on阳0,24]and-2, o can be" smeared" intervals(0 21+11, [-21+, o], the dfference u a linear combunation of very stretched out Haar functons 21 to 21+1, and writing f 约+++64++1, where ++1x4+≡号6+,++2+10≡/1 6+=号6bv(2-h-1)-是v(2-h-1z+1 (eee Figure 1.7). This can again be repeated, leading to Jo+J+K 十 2 Cm, vm,t m=-J+1 where support(o+J+K)=[-2J1+K, 2 +K], and s-o+J+K J1 fJo+Ju +K oI Jat Jt It fo mediately that Cm. t vm.L Jo+J1+K 2k2.21/206+12++y/2, which can be made arbitrarily small by taking sufficiently large K. As claimed f can therefore be approximated to arbitrary precision by a finite linear combi- nation of Haar waddle The argument we just saw has implicitly used a"multiresolution" approach ve have written successive coarser and coarser approximations to f(the f3
CHAPTER 1 averaging f over larger and larger intervals), and at every step we have written the difference between the approximation with resolution 2-I, and the next coarser level, with resolution 2, as a linear combination of the th, k. In fact, we have introduced a ladder of spaces(V),ez representing the successive resolution levels: in this particular case, V,=I E L2(R);f piecewise constant on the 2k, 2(k+1)lkEZ. These spaces have the following properties (1)…cvcⅥcVcv1cV2c nez,=10, U,EZ V=L-(r); (3)∫∈V艹∫(22)∈V; (4)∫∈W→∫(-n)∈ Ve for all n∈Z Property 3 expresses that all the spaces are scaled versions of one space(the "multiresolution"aspect). In the Haar example we found then that there exists a function l so tha Proj2!= Proj f+∑,的的k (135) The beauty of the multiresolution approach is that whenever a ladder of spaces V, satisfies the four properties above, together with (5)∈ Vo so that the虮n(x)=φx-n) constitute an orthonormal basis for Vo then there exists v so that (1.3.5)holds. (In the Haar example above,we can take p(r)=1 if0<I< l, (e)=0 otherwise. )The v, k consti- tute automatically an orthonormal basis. It turns out that there are many examples of such"multiresolution analysis ladders, "corresponding to many ex amples of orthonormal wavelet bases. There exists an explicit recipe for the construction of yb; sinceφ∈vcV1, and theφ-L,n(z)=√叭2x-n) constitute an orthonormal basis for V-1(by(3)and(5)above), there exist an=ⅴ{φ-n〉 so thatφ(x)=∑nan叭(2x-n). It then suffices to take v(r)=2(1)"a-n+1 (2r -n). The function o is called a scaling function of the multiresolution analysis. The correspondence multiresolution analysis-or thonormal basis of wavelets will be explained in detail in Chapter 5, and further explored in subsequent chapters. This multiresolution approach is also linked with subband filtering, as explained in $5.6( Chapter 5) Figure 1. 8 shows some examples of pairs of functions p corresponding to different multiresolution analyses which we will encounter in later chapters. The Meyer wavelets(Chapters 4 and 5) have compactly supported Fourier transform o and y themselves are infinitely supported; they are shown in Figure 1.8a.The Battle-Lemarie wavelets(Chapter 5)are spline functions(linear in Figure 1.8b, cubic in Figure1.8c), with knot認 at Z for中号Zfrψ.Bothφ and as have infinite support, and exponential decay; their numerical decay is faster than for the Meyer wavelets(for comparison, the horizontal scale is the same in(a)
THE WHAT WHY AND HOW OF WAVELETS 505 (d) 2 ples of orthonormal wavelet bases. For every w in thu figure, the am的h()=2p(一4)五∈ Z, constitutes an orthonormal bas可P?().The figure plots中 aling function and p for diferent construction thich we will encounter in later chapteri. (a)The Meyer wavelets;(b)and(c)Battle-Lemarie wavelets (d)the Haar wavelet;(e) the neat member of the family of compactly supported wavelets, 2t (f another compactly supported wavelet, with leas asymmetry
16 CHAPTER I (b), and (c)of Figure 1. 8). The Haar wavelet, in Figure 1.8d, has been known since 1910. It can be viewed as the smallest degree Battle-Lemarie wavelet (aHaar = BL, o)or also as the first of a family of compactly supported wavelets constructed in Chapter 6, YHaar 10. Figure 1.e plots the next member of the family of compactly supported wavelets N i, 2 and 2 both have support width 3, and are continuous. In this family of Nal(constructed in $6.4),the regularity increases linearly with the support width(Chapter 7). Finally, Figure 1 8f shows another compactly supported wavelet, with support width 11; and less asymmetry(see Chapter 8) Notes 1.There exist other techniques for time-frequency localization than the win- dowed Fourier transform. A well-known example is the Wigner distribu tion.(See, e. g, Boashash(1990)for a good review on the use of the wigner distribution for signal analysis. )The advantage of the wigner distribution is that unlike the windowed Fourier transform or the wavelet transform does not introduce a reference function(such as the window function or the wavelet)against which the signal has to be integrated The disad- vantage is that the signal enters in the wigner distribution in a quadratic rather than linear way, which is the cause of many interference phenom- ena. These may be useful in some applications, especially for, e. g, signals which have a very short time duration(an example is Janse and Kaiser (1983); Boashash(1990)contains references to many more examples); for signals which last for a longer time, they make the wigner distribution less attractive. Flandrin(1989 shows how the absolute values of both the win dowed Fourier transform and the wavelet transform of a function can also be obtained by "smoothing " its wigner distribution in an appropriate way he phase information is lost in this process however, and reconstruction is not possible any more. 2. The restriction bo= 1, corresponding to(1.3.4), is not very serious: if (1.3.4)provides an orthonormal basis, then so do the vm, n(=)=2-m/2 v(2-mr-nbo), with (=)=bol -1/ (bo r),where bo f0 is arbi- trary. The choice ao =2 cannot be modified by scaling, and in fact ao cannot be chosen arbitrarily. The general construction of orthonormal bases we will expose here can be made to work for all rational choices for >1, as shown in Auscher(1989 ), but the choice ao=2 is the simplest ifferent choices for o correspond of course to different pl. Although the constructive method for orthonormal wavelet bases, called multiresolution analysis, can work only if ao is rational, it is an open question whether there exist orthonormal wavelet bases(necessarily not associated with a multiresolution analysis), with good time-frequency localization, and with irrational ao
CHAPTER The Continuous Wavelet Transform The images of L-functions under the continuous wavelet transform constitute a k.H. s) da址 etts. One of: he simplest examples is th即pd叫 bandlimited functions, discussed in $$2.1 and 2.2. In $2.3 we introduce the E.- concept of band and time limiting; of course no nonzero function can be strictly 1a time-limited (i.e, f(t)=0 for t outside [-T, T]and band-limited (f(E)=0 for * I-n, n2), but one can still introduce time-and-bandwimiting operators We present a short review of the beautiful work of Landau, Pollak, and Slepian on this subject. We then switch to the continuous wavelet transform: the resolution ofthe identity in $2.4 (with a proof of(1.3.1)), the corresponding r k H.s. in $25 *ln 2.6 we briefly show how the one-dimensional results of the earlier sections can be extended to higher dimensions. In $2.7 we draw a parallel with the continuous windowed Fourier transform. ln 52.8 we show how a different kind of time-and-band -limiting operator can be built from the continuous windowed Fourier transform or from the wavelet transform. Finally, we comment in $2 9 on the "zoom-in"property of the wavelet transform 2.1 Bandlimited functions and shannon's theorem YA function f in y,(R)is called bandlimited if its Fourier transform Ff has compact support, i. e, f( )=0 for IE1>f. Let us suppose for simplicity, that =T. Then f can be represented by its Fourier series(see Preliminaries) f)=∑cn df e"mf(E) df einEf(E) f(n)