The convolution theorem for the continuous wavelet tranform

The convolution theorem for the continuous wavelet tranform Antonio F. P$erez-Rend$on, Rafael Robles Dpto. Applied Mathematics I, University of Seville, Spain Abstract We study the application of the continuous wavelet transform to perform signal 1ltering processes. We 1rst show that the convolu tion and correlation of two wavelet fu nctions satisfy the requ ired admissibility and regu larity conditions. By using these new wavelet functions to analyze both convolutions and correlations, respectively, we derive convolution and correlation theorems for the continuous wavelet transform and show them to be similar to that of other joint spatial/spatial–frequency or time/frequency representations. We then investigate the e5ect of multiplying the continuous wavelet transform of a given signal by a related transfer function and show how to perform spatially variant 1ltering operations in the wavelet domain. Finally, we present nu merical examples showing the u sefu lness of applying the convolu tion theorem for the continuous wavelet transform to perform signal restoration in the presence of additive noise. Keywords: Continu ou s wavelet transform; Convolu tion; Correlation; Space varying 1ltering; Signal restoration 1. Introduction The received signal in a large number of signal processing systems is modeled as a convolution between the input signal and the impulse response of the system with additive noise. Signal restoration (or deconvolution) problems have been mainly studied under the assumption that the input signal and noise are stationary (thou gh the terms stationary and non-stationary have some precise meanings in stochastic processes, in this work we use these terms in a qualitative sense, i.e., a stationary signal is a signal whose properties do not change much with time or through space), and conventional estimation algorithms such as Wiener or Kalman 1lters can be directly applied. However, the problem of restoration of non-stationary signals such as seismic data, acoustic signals, mechanical vibration, wireless communication, radar and sonar signals, etc., has recently attracted the attention of signal processing researchers. Properties of joint time–frequency representations have allowed to 1nd optimal deconvolution 1lters for non-stationary signal restoration [4,14]. The wavelet transform is of interest for the analysis of non-stationary signals because it provides an alternative to classical linear time–frequency representations with better time and frequency localization properties [32]. On the other hand, the role of the wavelet transformation is to remove the non-stationary properties of the involved signals; consequently, the conventional estimation algorithms for stationary signal processing can be employed in each scale of the wavelet domain [5]. For these reasons, several Nomenclature ˜ f(w) Fourier transform of function f(x) Ef(x; w;) generalized joint distribution of function fwith kernel  ˆ f(a; b) wavelet transform of function fwith wavelet L2(R) set of all measurable functions that are square integrable f∗(x) complex conjugate of function f(x) ·pnorm de1ned in Lebesgue space Lp(R) (f⊗g)(x) convolution of functions f(x) and g(x) (f∗g)(x) correlation of functions f(x) and g(x) (f⊗1g)(x; y) convolution in the 1rst variable of functions fand g (f⊗2g)(x) convolution in the second variable of functions fand g (f∗1g)(x; y) correlation in the 1rst variable of functions fand g (f∗2g)(x) correlation in the second variable of functions fand g (f⊗⊗g)(x) generalized convolution of functions f(x) and g(x) M -order moment of wavelet (x) F2{ˆ f }(a; w) Fourier transform of ˆ f (a; b) in the second variable (b↔w) algorithms have been developed in order to perform signal restoration in the wavelet domain [1,5,17,23,40,43], but the lack of a convolution theorem for the continuous or discrete wavelet transform prevents us to 1nd optimal deconvolution 1lters in the wavelet domain. Therefore, we think that the establishing of such convolution theorems, besides their theoretical importance from a mathematical point of view, have interesting applications in signal processing problems such as signal restoration. Filtering for signals and images is at present extensively used in di5erent areas of signal and image processing such as image enhancement, image restoration, signal coding and image analysis [3]. Traditionally, most optical data processing systems only perform space invariant processing operations, mathematically described in terms of convolutions, which are easily implemented through the Fourier transform (FT). The convolution theorem for this one states that the FT of the convolution of to signals is the product of their respective FT, that is g(x)=+∞ −∞ dtf(t)h(x−t)⇒ ˜g(w)= ˜ f(w)˜ h(w):(1) Therefore, the 1ltering operation is generally reduced to multiply the FT of the signal by the transfer function of the 1lter, and to make the inverse Fourier transformation of the product, which provides a 1ltered version of the signal. The FT performs a global analysis of a given signal or image, and it is especially suited for stationary signals. Nevertheless, in many cases of practical interest the signal under investigation is non-stationary and has a di5erent frequency content at di5erent instants or localizations. One problem with the application of the FT to such non-stationary signals is that high-frequency terms appear and they are not localized, but are added everywhere. On the other hand, signals can also be a5ected by non-stationary processes; for instance, optical defocus may introduce a spatially variant blur in the signal. Space or time-variant 1ltering [39] is mathematically described as a generalized convolution or a Fredholm integral equation of the 1rst kind: g(x)=+∞ −∞ dtf(t)h(x; t):(2) The convolution theorem no longer applies for these space or time-varying 1lters and thus the Fourier domain is less useful for their application [22]. Therefore, we come to the conclusion that spatially variant signals and processes can be better characterized by joint time–frequency or spatial/spatialfrequency representations [8,13,28]. Wigner [41] introduced a bilinear distribution as a joint representation of the phase space in Quantum Mechanics. Later, Ville [37] derived the same distribution, nowadays called the Wigner–Ville distribution, in the 1eld of signal processing. Another way to obtain a joint representation is through the complex spectrogram, that can be expressed as a windowed FT, 1rst introduced by Gabor [10]. Both Wigner–Ville and Gabor distributions belong to the Cohen class of bilinear distributions [7], in which each member is obtained by introducing a particular kernel in the generalized distribution de1ned as Ef(x; w;)= R3 ddd(; ) ×f(+=2) f∗(−=2) ×exp{2i(x −w −)}:(3) There exist convolution theorems for the Wigner– Ville distribution [6] as well as for other joint representations belonging to Cohen class [26]. Such theorems state that the joint representation of the convolution of two signals is the convolution in the time or space variable, at every 1xed frequency, of the respective joint representations, that is Eg(x; w;) =+∞ −∞ dtE f(t; w;)Eh(x−t; w;):(4) Joint representations also allow us to perform space variant 1lterings on a given function by multiplying its joint representation by a function T(x; w), which can be regarded as a space-variant transfer function, and taking the inverse transformation. This process has been carried out with the complex spectrogram [34] and the Wigner–Ville distribution [2,11,34]. The continuous wavelet transform (CWT) [12,15] stands for a scale-space representation and is an e5ective way to analyze non-stationary signals and images. Typical applications are the detection and characterization of singularities [25], fractal analysis [21,38], pattern recognition [20], noise reduction [18], image processing [27], geophysics [9], aeromagnetic processing [31] and the analysis of biomedical [42] and meteorological [35] images. The existence of fast algorithms for the calculus of the CWT [24,36] guarantees its use for solving real problems. The CWT is a scale-space representation of a given function which may be regarded as a joint representation by identifying the scale with a frequency ratio [15]. This fact introduces an adaptability in the analysis which joint representations lack. Nevertheless, there is no convolution theorem for the CWT, and this limits its usability in image and signal processing. We have found in the literature some works relative to convolution related 1ltering and CWT, which are summarized below. In [19], given a function g(x) as the convolution of another function f(x) and a 1lter h(x), let ˆg(a; b) and ˆ f(a; b) denote the CWT with the same wavelet of g(x) and f(x), respectively. Then, it is shown that ˆg(a; b)=+∞ −∞ dtˆ f(a; t)h(b−t):(5) This means that the CWT of the convolution between some function and a 1lter is the convolution in the time or space variable, at every 1xed scale, of the CWT of the function and the very 1lter. In [33], given the convolution g(x) as above, let ˆ f(a; b) denote the CWT of the function f(x) with wavelet (x) and let ˆ h(a; b) denote the CWT of the 1lter h(x) with wavelet (x)= (−x). Then, it is shown that g(x)= 1 C+∞ 0 da a2+∞ −∞ dˆ f(a; )ˆ h(a; x −); (6) where the normalization constant C is de1ned in (21). That is, the convolution (instead of its CWT) between some function and a 1lter is the sum over all scales of the convolution in the time or space variable, at every 1xed scale, of their respective CWT, but with di5erent wavelets. A di5erent approach is given in [16], where wavelet-based 1lters, known as scale 1lters, are generated as follows. Let ˆ fdenote the CWT of a given function with wavelet and let ˜ h(w) denote the FT of the 1lter h(x); then, a scale 1lter v(a)is de1ned as ˜ h(w)= a v(a)˜ ∗(aw) (7) and is calculated via linear squares methods. Then, it is shown that g(x)= a v(a)ˆ f(a; x):(8) A similar process has been used to construct space varying scale 1lters [29,30]. Let us remark that neither result can be regarded as a convolution theorem, because this must formulate the convolution equation in the wavelet domain entirely, i.e., it must show a relationship between the CWT of two functions and their convolution. Nevertheless, these works point out the existence of such a convolution theorem similar to that for joint representations as shown in (4) and the possibility to use di5erent wavelets for every involved function. This possibility is deeply investigated in Section 3. This paper is organized as follows. In Section 2we de1ne some known topics about Fourier and continuous wavelet transforms and introduce the notation we will use in following sections. In Section 3we derive the convolution and correlation theorems for the continuous wavelet transform and show the e5ect of multiplying this by some transfer function. In Section 4 we initially show the validity of the theorems proved in previous section and then apply them to a computer simulation of a signal restoration problem; comparisons with Fourier based deconvolution are provided showing that the signal restoration method proposed in this paper yields better restorations. Finally, we summarize the main conclusions of this work in Section 5. 2. Background We devote this section to de1ne some known topics about FT and CWT and introduce the notation we will use in next section. Most results in this section come from [12,15]. Given 1 6p¡∞, the Lebesgue space Lp(Rm) is de1ned as the set of all measurable functions f:Rm→Cfor which fp=Rm d˜x|f(˜x)|p1=p ¡∞:(9) It is well known that Lp(R) is a Banach space with norm · pand L2(R) is a Hilbert space with inner product f; g=Rdxf(x)g∗(x), where (·)∗denotes complex conjugate. The FT of a function f∈L2(R) is de1ned as ˜ f(w)=+∞ −∞ dxf(x) exp{−2ixw}(10) which also belongs to L2(R). The reconstruction formula is given by f(x)=+∞ −∞ dw˜ f(w) exp{2ixw}:(11) If f∈L1(R)∩L2(R), then ˜ f(w) is a bounded function in L2(R), i.e., there exists a positive real number M such that |˜ f(w)|6Mfor all w∈R. If fand hare complex-valued functions de1ned on R, their convolution f⊗his de1ned as a new function: g(x)=(f⊗h)(x)=+∞ −∞ dtf(t)h(x−t) (12) provided that the integral exists. If f∈L2(R) and h∈L1(R), then such an integral exists almost everywhere and f⊗h∈L2(R) with f⊗h2=f2·h1. If fand hare complex-valued functions de1ned on R2, it is possible to convolve them in just one variable as follows: g1(x; y)=(f⊗1h)(x; y) =+∞ −∞ dtf(t; y)h(x−t; y);(13) g2(x; y)=(f⊗2h)(x; y) =+∞ −∞ dtf(x; t)h(x; y −t);(14) provided that both integrals exist. If f∈L2(R2) and h∈L1(R2), then such integrals exist almost everywhere and g1;g 2∈L2(R2). An operator quite related to the convolution is the so-called correlation operator. If fand hare complex-valued functions de1ned on R, their correlation f∗his de1ned as a new function g(x)=(f∗h)(x)=+∞ −∞ dtf(t)h(t+x);(15) provided that the integral exists. If f∈L2(R) and h∈L1(R), then such an integral exists almost everywhere and f∗h∈L2(R) with f∗h2=f2·h1. When f≡h,f∗fis known as the autocorrelation of f, while otherwise f∗his known as the cross-correlation between fand h. If fand hare complex-valued functions de1ned on R2, it is possible to correlate them in just one variable as follows: g1(x; y)=(f∗1h)(x; y) =+∞ −∞ dtf(t; y)h(t+x; y);(16) g2(x; y)=(f∗2h)(x; y) =+∞ −∞ dtf(x; t)h(x; t +y);(17) provided that both integrals exist. If f∈L2(R2) and h∈L1(R2), then such integrals exist almost everywhere and g1;g 2∈L2(R2). Given two functions f∈L2(R) and h∈L2(R2), their generalized convolution is de1ned as the Fredholm integral equation of the 1rst kind: g(x)=(f⊗⊗h)(x)=+∞ −∞ dtf(t)h(x; t) (18) which is also known as a space-variant 1ltering in optics and engineering literature [2,11], as opposed to the space invariant 1ltering given by the convolution de1ned in (12). Given a 1xed function ∈L2(R), called wavelet, consider its translations and dilations de1ned as ab(x)= 1 |a|1=2x−b a(19) with a; b ∈Rand a= 0. These functions are scaled so that their L2(R) norms are independent of a. The CWT of a function f∈L2(R) with wavelet is now de1ned as ˆ f(a; b)=+∞ −∞ dxf(x) ∗ ab(x) =1 |a|1=2+∞ −∞ dxf(x) ∗x−b a:(20) A wavelet function (x) is said to be admissible if 0¡C =+∞ −∞ dw|˜ (w)|2 |w|¡∞:(21) If this is the case, the CWT ˆ f(a; b) is invertible on its range, and an inverse transform is given by the relation f(x)= 1 C+∞ −∞ dadb a2ˆ f(a; b) ab(x);(22) where the integral over ais understood to exclude a=0 (integrate over 0 ¡6|a|¡∞, then take the limit →0). The most important properties of wavelets are admissibility (21) and regularity conditions. As can be seen in (20), the wavelet transform of a one-dimensional function is two-dimensional and for most practical applications this is not a desirable property. Therefore, one imposes some additional conditions on the wavelet functions in order to make the wavelet transform decrease quickly with scale. These are the regularity conditions and they state that the wavelet function should have some smoothness and concentration in both time and frequency domains. Regularity is a quite complex topic which is related to the concept of vanishing moments; the number of vanishing moments required depends heavily on the application. The moments of a wavelet ∈L2(R) are de1ned as M =+∞ −∞ dtt  (t):(23) Let Lbe the Hilbert space of all measurable functions H(a; b) that are square integrable with respect to the weight function |a|−2, and let Fbe the space of all wavelet transforms with respect to a 1xed wavelet function (x). It has been shown [15] that Fis a proper subspace of L; that is, not every function H(a; b)∈Lis the CWT of some function f∈L2(R). If we apply the reconstruction formula (22)tosucha function H(a; b): h(x)= 1 C+∞ −∞ dadb a2H(a; b) ab(x);(24) then the function h(x) belongs to L2(R) and it is the unique function such that for any other function f∈L2(R)itisH(a; b)−ˆ f (a; b)L¿ H(a; b)−ˆ h (a; b)L. The function h(x) is known as the least-squares approximation in L2(R)tothe inverse transform of H(a; b). It is important to note that in (20) and (22) the wavelet functions are not speci1ed. This is a di5erence between the CWT and the FT, or other transforms. The theory of wavelet transforms deals with the general properties of the wavelets and the wavelet transforms only. The choice of the wavelet function depends on the particular application. For space–frequency analysis, we need a wavelet that is optimally localized in terms of both spatial width and frequency bandwidth. For smooth signals, we generally want a wavelet that is itself smooth and therefore has good frequency localization. In contrast, signals that contain discontinuities are better analyzed using wavelets with good spatial localization to accurately map rapid changes in the signal. There are several other useful properties that the wavelet function can possess, as its orthogonality, complexity, width or shape. 3. The convolution theorem In [33] it is pointed out the possibility to use different wavelet functions for the signal and the 1lter. Assuming that we have selected the best wavelets for both, what is then the best choice for the convolution or correlation? It is somewhat natural to consider the convolution or correlation of the respective wavelets provided to satisfy the admissibility condition and have a suPcient number of vanishing moments (at least, the same number as the wavelet used for the original signal). Theorem 1. Let 1∈L2(R)and 2∈L1(R)∩L2(R)be two admissible wavelets with N1and N2vanishing moments,respectively,and consider two new wavelets 3=( 1⊗ 2)and 4=( 1∗ 2). Then,both 3and 4are admissible wavelets too and have N1+N2vanishing moments. Proof. Since 1∈L2(R) and 2∈L1(R), we know that both 3; 4∈L2(R) too. De1ne Ci=+∞ −∞ dw|˜ i(w)|2 |w|(25) for i=1;:::;4. Since 1(x) and 2(x) are admissible wavelets, then according to (21) we know that 0¡C 1;C 2¡∞. Taking into account some properties of the FT, we have ˜3(w)= ˜ 1(w)˜ 2(w);(26) ˜4(w)= ˜ 1(w)˜ ∗ 2(w);(27) from which we can 1nd |˜ 3(w)|=|˜ 4(w)|=|˜ 1(w)||˜ 2(w)|:(28) Therefore, we get 0¡C 3=C4=+∞ −∞ dw|˜ 1(w)|2|˜ 2(w)|2 |w|:(29) Since 2∈L1(R), its FT ˜ 2(w) is a bounded function, so that there exists a positive real number Msuch that |˜ 2(w)|6Mfor all w∈R. Then, we arrive at: 0¡C 3=C46M2+∞ −∞ dw|˜1(w)|2 |w| =M2C1¡∞:(30) Therefore, the new de1ned wavelets 3and 4satisfy the admissibility condition (21). Calculate now the moment of wavelet 3, de1ned as M3 =+∞ −∞ dtt  3(t):(31) By inserting (12) for 3(t) into above equation and changing the order of integration, we 1nd M3 =+∞ −∞ dx 1(x)+∞ −∞ dtt  2(t−x):(32) If we perform the change of variables y=t−xinto the inner integral and expand (x+y)according to Newton binomial expansion, we arrive at M3 =   k=0  k+∞ −∞ dxx k 1(x) ×+∞ −∞ dyy −k 2(y);(33) that is M3 =   k=0  kM1 kM2 −k:(34) An analogous reasoning provides M4 =   k=0 (−1)k kM1 kM2 −k:(35) Let be 6N1+N2.Ifk6N1, then M 1 k=0 because 1 has N1vanishing moments; otherwise, −k6N1+ N2−k6N1+N2−N1=N2, so that M 2 −k=0 because 2has N2vanishing moments. Therefore, all addends in (34) and (35) are zero and M 3 =M 4 = 0 for all 6N1+N2. Since both the convolution and correlation of two admissible wavelets satisfy the required admissibility and regularity conditions, we may use them to analyze our convolved and correlated signals. Once the signal and the 1lter have been analyzed with two di5erent wavelet functions and their convolution has been analyzed with the convolution of the chosen wavelets, what is the relation between the three obtained CWT? As it was pointed out in Section 1, such a relationship is quite similar to the convolution theorem for joint representations shown in (4). Theorem 2 (Wavelet convolution theorem). Let f∈L2(R)and h∈L1(R)∩L2(R)be two admissible wavelets,and let ˆ f fand ˆ h hdenote the CWT of two functions f∈L2(R)and h∈L1(R)∩L2(R)with wavelets fand h,respectively.If g=(f⊗h)and g=( f⊗ h), then ˆg g(a; b)= 1 |a|1=2(ˆ ff⊗2ˆ h h)(a; b):(36) Proof. The CWT of g(x), given by (20), may be rewritten as ˆg g(a; b)= 1 |a|1=2 R3 dxdtdyf(t)h(x−t) × ∗ f(y) ∗ hx−b a−y:(37) Performing the change of variables =x−tand $= b+ay −t,(37) may be rewritten as ˆg g(a; b)= 1 |a|3=2+∞ −∞ d$+∞ −∞ dtf(t) × ∗ ft−(b−$) a ×+∞ −∞ dh() ∗ h−$ a;(38) from which it is easy to 1nd ˆg g(a; b)= 1 |a|1=2+∞ −∞ d$ˆ ff(a; b −$)ˆ h h(a; $) =1 |a|1=2(ˆ ff⊗2ˆ h h)(a; b):(39) The correlation of two signals is an operation quite related to the convolution. In fact, the correlation theorem for the FT is quite similar to the convolution theorem; the only di5erence is the complex conjugate that appears in the FT of the 1lter, that is, ˜g(w)= ˜ f(w)˜ h∗(w). Therefore, it is hoped that the CWT satis1es a correlation theorem similar to the previous convolution theorem. Theorem 3 (Wavelet correlation theorem). Let f∈L2(R)and h∈L1(R)∩L2(R)be two admissible wavelets,and let ˆ f fand ˆ h hdenote the CWT of two functions f∈L2(R)and h∈L1(R)∩L2(R)with wavelets fand h,respectively.If g=(f∗h)and g=( f∗ h), then ˆg g(a; b)= 1 |a|1=2(ˆ ff∗2ˆ h h)(a; −b):(40) Proof. The CWT of g(x), given by (20), may be rewritten as ˆg g(a; b)= 1 |a|1=2 R3 dxdtdyf(t)h(t+x) × ∗ f(y) ∗ hy+x−b a:(41) Performing the change of variables =t+xand $= b+t−ay, Eq. (41) may be rewritten as ˆg g(a; b)= 1 |a|3=2+∞ −∞ d$+∞ −∞ dtf(t) × ∗ ft−($−b) a ×+∞ −∞ dh() ∗ h−$ a;(42) from which it is easy to 1nd ˆg g(a; b)= 1 |a|1=2+∞ −∞ d$ˆ ff(a; $ −b)ˆ h h(a; $) =1 |a|1=2(ˆ ff∗2ˆ h h)(a; −b):(43) According to above theorems, the CWT allows us to perform any linear 1ltering. Although this transform is far too redundant to yield fast algorithms, the main feature of Theorems 2and 3is that the convolution and correlation operators apply at every scale independently of each other, and this fact allows us to deeply understand the e5ect of the 1lter on every scale of the given signal. This property may be very useful in deconvolution related problems, where it will be possible to design a di5erent algorithm to every scale, depending on the properties of the 1lter at such a scale and the condition number of the corresponding inverse problem. Despite of previous considerations, the main drawback of Theorems 2and 3is that the convolution and correlation operators do not disappear in the wavelet domain, and this implies no time reduction in the application of such operators with respect to the spatial domain. This is not the case with the FT. The convolution and correlation theorems for the FT are powerful tools to perform space invariant 1ltering processes because they reduce these processes to simply multiply the FT of the signal by the transfer function associated with the 1lter. It would be desirable to perform some kind of 1ltering by simply multiplying the CWT by a transfer function; for joint representations this product performs a spatially variant 1ltering on the given signal. The similarity between the convolution theorems for joint representations and CWT leads to think about the possibility to perform such space varying 1ltering operations in the wavelet domain. Let ˆ fdenote the CWT of a function f∈L2(R) with an admissible wavelet ∈L2(R) and let H∈L2(R2). De1ne G(a; b)= ˆ f (a; b)H(a; b), which may not be the CWT of any function in L2(R); the least square approximation to such a function is obtained by applying the reconstruction formula (22)to G(a; b). Therefore, g(x)= 1 C+∞ −∞ dadb a2ˆ f(a; b) ×H(a; b) x−b a(44) with C de1ned in (21). By inserting the de1nition of ˆ f (a; b) according to (20) into above equation and changing the order of integration, we 1nd g(x)= 1 C+∞ −∞ dtf(t) ×+∞ −∞ dadb |a|5=2H(a; b) × ∗t−b a x−b a:(45) If we de1ne h(x; t)= 1 C+∞ −∞ dadb |a|5=2H(a; b) × ∗t−b a x−b a(46) provided the above integral to converge, then g(x)= (f⊗⊗h)(x) is the function in L2(R) whose CWT with wavelet is as close as possible to the product ˆ f(a; b)H(a; b). Although we have not found an inverse formula for obtaining the transfer function H(a; b) which corresponds to a given kernel h(x; t), this result raises an interesting question. Is it possible to 1nd useful 1lters such that their Fourier transfer functions cannot be well approximated by rational functions but their action is diagonal in the wavelet domain? We think that this question might be of interest for the signal processing community, so that future work may relay on it. 4. Numerical examples In this section, two numerical examples are given to illustrate (i) the validity of the theorems proved in previous section and (ii) the usability of such theorems in signal restoration problems in the presence of additive noise. In both examples, the test signals have been analyzed with the 1rst and second derivatives of the well–known normal distribution with zero mean and unit variance, given by 1(x)=2 √xexp −x2 2(47) and 2(x)= 2 3√(1 −x2) exp −x2 2;(48) respectively. It can be shown that both wavelets satisfy the admissibility condition (21) with coePcients C1= 44:5466 and C2=29:6977, the wavelet 1(x) has a unique vanishing moment and the wavelet 2(x) has two vanishing moments. In the following examples we have also employed the convolution 3= 1⊗ 2and the correlation 4= 1∗ 2of the previous wavelets, whose analytical expressions are given by 3(x)= 1 2√6x(x2−6) exp −x2 4(49) and 4(x)=− 3(x) because of the symmetry properties of 1(x) and 2(x). Both new wavelet functions also satisfy the admissibility condition (21) with coePcients C3=C4=52:6379 and have three vanishing moments, in accordance with Theorem 1. All four wavelet functions are depicted in Fig. 1. -4 -2 0 2 4 x -0.6 -0.3 0 0.3 0.6 ψ1(x) Wavelet function -4 -2 0 2 4 x -0.3 0 0.3 0.6 ψ2(x) Wavelet function -6 -3 0 3 6 x -0.8 -0.4 0 0.4 0.8 ψ3(x) Wavelet function -6 -3 0 3 6 x -0.8 -0.4 0 0.4 0.8 ψ4(x) Wavelet function (a) (b) (c) (d) Fig. 1. Wavelet functions used to analyze the test signals: (a) 1rst derivative of a gaussian distribution of zero mean and unit variance, 1, (b) second derivative of a gaussian distribution of zero mean and unit variance, 2, (c) convolution of previous wavelets, 3= 1⊗ 2, and (d) correlation of the 1rst two wavelets, 4= 1∗ 2. Example 1. Consider a test signal de1ned as f(x) = exp −(x−1)2 6cos(3(x−1)) (50) whose plot is shown in Fig. 2(a). The validity of the wavelet correlation theorem is established by calculating the autocorrelation of the test signal in the wavelet domain using Theorem 3and comparing the result with the autocorrelation obtained using the correlation theorem for the FT. The generated test signal is analyzed with wavelets 1(x) and 2(x) (all integrals have been numerically evaluated using the midpoint quadrature rule). According to Theorem 3, both CWT are correlated at every 1xed scale in order to obtain the CWT of the autocorrelation of the test signal with wavelet 4(x). Finally, such an autocorrelation is obtained by applying the reconstruction formula (22) to its CWT.