THE WHAT, WHY, AND HOW OF WAVELETS 1.2. The wavelet transform: Analogies and differences with the windowed Fourier transform. The wavelet transform provides a similar time-frequency description, with a few Important differences. The wavelet transform formulas analogous to(1.1.1) d(112) (T f(a, b)=al d∫(t)v (121) and Tm.A(f∫) m/2dt∫() In both cases we assume that pb satisfies p(t) (for reasons explained in Chapters 2 and 3 Formula(12 2)is again obtained from(1.2.1)by restricting a, b to only dis- crete values: a= a0,b= nboao in this case, with m, n ranging over Z, and ao>1, b0>0 fixed. One similarity between the wavelet and windowed Fourier transforms is clear: both(1. 1. 1)and(1.2.1)take the inner products of f with a family of functions indexed by two labels, g,(8)=e'g(s-t)in(1.1.1),and vla (s)=la -1/2v (=)in(1.2 1). The functions aa, b are calledwavelets" the function a is sometimes called "mother wavelet. "(Note that a and g are implicitly assumed to be real, even though this is by no means essential; if they are not, then complex conjugates have to be introduced in(1.1),(1.2.1) )A typical choice for p is v(t)=(1-t )exp(t/ 2), the second derivative of the Gaussian, sometimes called the mexican hat function because it resembles a cross section of a Mexican hat The mexican hat function is well localized in both time and frequency, and satisfies(1.2.3). As a changes, the p 0(3)=lal -1/2p(s/a) a cover different frequency ranges (large values of the scaling parameter a cor- Respond to small frequencies, or large scale a, i small values of lal correspond gh frequencies or very fine scale w, ) Changing the parameter b as well us to move the time localization center: each va,b(s)is localized around 8 It follows that(1.2.1), lke(1.1.1), provides a time-frequency description of F;wthe difference between the wavelet and windowed Fourier transforms lies in h i pes of the analyzing functions g" and ya, b, as shown in Figure 1.2 g '. all consist of the same envelope function g, translated to the location, and"filled in"with higher frequency oscillations. All the 9,,regardle of the value of w, have the same width. In contrast, the ya, have time-width adapted to their frequency: high frequency y, o are very narrow while low frequency ya, b are much broader. As a result the wavelet transform is better able than the windowed Fourier transform to "zoom in"on very short- lived high frequency phenomena, such as transients in signals(or singularities
CHAPTER 1 (b) v(x) ab with a c 1 b>0 Re g yao with a>1 b<0 FIG 12 Typical shapes of (a)windowed Fourer transform functions 9" ,and g(r-t)can be viewed as translated envelopes eg of the same functions, translated and compressed or stretched in functions or integral kernels) This is illustrated by Figure 1 3, which shows windowed Fourier transforms and the wavelet transform of the same signal f defined by fit)=sin(2nt)+ sin(2Tvt)+y 6(E-ti)+8(t- ta) In practice, this signal is not given by this continuous expression, but by samples and adding a 8-function is then approximated by adding a constant to one sample only. In sampled version, we have then f(nr)=sin(2vnn)+ sin(2TV2nT)+a[5n, m1 +5n, nal For the example in Figure 1. 3a, V1=500 Hz, 22= 1 kHz, T= 178,000 sec(i.e we have 8,000 samples per second), a= 1.5, and n2 -n1 = 32(corresponding o 4 milliseconds bet ween the two pulses). The three spectrograms(graphs of
THE WHAT, WHY, AND HOW OF WAVELETS (a) 3500 3500 3000 2500 1500 1000 500 00 500 (c) 4000 3000 2000 500 (d1 110 110 110 i00 0 0 80 70 70 70 60 50 50 40 40 30 wy 10 10 01c002000300040000100020003000400001000200030004000 FIG 1 3.(a) The aagmal f(t).(b) Windowed Fourer transforms of f wth three dfferent window widths. These are so-called spectrograms: only iRwin ()l as plotted( the phase 1s not rendered on the graph), using grey levels(hagh values black. zero white, intermediate grey levels are assigned proportional to log Irwin()l)n the t(abscissa), w(ordinate)plane (c)Wavelet transform of f. To make the comparson uth(b)we have also plotted Wav()I with the same grey level method, and a linear frequency aris(se, the ordinate corresponds to a).(d] Comparson of the frequency resolution between the three spectrograms and the wavelet transform. I would like to thank Oded Ghitza for generating thus figure
CHAPTER 1 the modulus of the windowed Fourier transform) in Figure 1.3b use standard Hamming windows, with widths 12.8, 6.4,, and 3.2 milliseconds, respectively (Time t varies horizontally, frequency w vertically, on these plots; the grey levels ndicate the value of Twin ()l, with black standing for the highest value. )A the window width increases, the resolution of the two pure tones gets better but it becomes harder or even impossible to resolve the two pulses. Figure 1.3c shows the modulus of the wavelet transform of f computed by means of (complex) Morlet wavelet v(t)=Ce-la(elnt-e-Ra/4),with a=4 make comparison with the spectrograms easier, a linear frequency axis has been used here; for wavelet transforms, a logarithmic frequency axis is more usual. One already sees that the two impulses are resolved even better than with the 3. 2 msec Hamming window(right in Figure 1.3b), while the frequency resolu tion for the two pure tones is comparable with that obtained with the 6.4 msec Hamming window(middle in Figure 1.3b). This comparison of frequency resolu tions is illustrated more clearly by Figure 1.3d: here sections of the spectrograms (i. e, plots of I(Twin)(, tI with fixed t)and of the wavelet transform modulus ((Tav)(, b)l with fixed b)are compared. The dynamic range(ratio between the maxima and the " dip"between the two peaks) of the wavelet transform is comparable to that of the 6.4 msec spectrogram. (Note that the fat horizontal "tail"for the wavelet transform in the graphs in Figure 1. 3d is an artifact of the plotting package used which set a rather high cut-off, as compared with th spectrogram plots; anyway, this cut-off is already at -24 dB In fact, our ear uses a wavelet transform when analyzing sound at least in the very first stage The pressure amplitude oscillations are transmitted from the eardrum to the basilar membrane, which extends over the whole length of the cochlea. The cochlea is rolled up as a spiral inside our inner ear; imagine it unrolled to a straight segment, so that the basilar membrane is also stretched out. We can then introduce a coordinate y along this segment. Experiment and numerical simulation show that a pressure wave which is a pure tone, fu(t) same frequency in time, but with an envelope in y, F(t, y)=elt u(y).In a first approximation, which turns out to be pretty good for frequencies w above 500 Hz, the dependence on w of p(y) corresponds to a shift by log w: there exist one function so that u(y) is very close to o(y -log w). For a general excitation function/, /()=V2r dw f(u)e wut, it follows that the response function F(t,y) is given by the corresponding superposition of"elementary response functions, F(t,y) dw f(w)Fu(t,y) ∫(u)e"φy-logu) If we now introduce a change of parameterization, by defining ve-)=(2π)-1/2叭(x),G(a,t)=F,loga)
THE WHAT, WHY, AND HOW OF WAVELETS then it follows that G(a, t)= dt'f(t)v(a(t-t) which(up to normalization) is exactly a wavelet transform. The dilation param- eter comes in, of course, because of the logarithmic shifts in frequency in the o The occurrence of the wavelet transform in the first stage of our own biological g acoustical analysis suggests that wavelet-based methods for acoustical analysis 3 have a better chance than other methods to lead, e. g, to compression schemes 9.undetectable by our ear W:B Different types of wavelet transform here exist many different types of wavelet transform, all starting from the basic formulas(1.2.1),(1.2). In these notes we will distinguish between A. The continuous wavelet transform(1.2.1),and B. The discrete wavelet transform(1.2.2) Within the discrete wavelet transform we distinguish further between B1. Redundant discrete systems( frames)and B2. Orthonormal (and other)bases of wavelets 1.3, 1. The continuous wavelet transform. Here the dilation and trang. lation parameters a, b vary continuously over R( with the constraint a #0). The wavelet transform is given by formula(1.2.1); a function can be reconstructed rom its wavelet transform by means of the "resolution of identity"formula ∫=C 3.1 zk, where y. 6 (=)=lal -/2v(a),and(,)denotes the L2-inner product.The constant Cy depends only on y and is given by d|(E)》21E (132) Mh aseume Cs oo(otherwise(1.3. 1)does not make sense). If al is in L(R) his is the case in all examples of practical interest), then y is continuous, so Cs can be finite only if 1(0)=0, i.e., dr v(e)=0. a proof for(1.3.1) given in Chapter 2.(Note that we have implicitly assumed that a is real fot somplex th, we should use v instead of v in(1.2.1). In some applications, ch complex ao are useful. wR mula(1.3.1)can be viewed in two different ways: (1)as a way of re- constructing f once its wavelet transform Tway f is known, or(2)as a way to