《A Really Friendly Guide to Wavelets》课程教学资源（书籍文献）A Really Friendly Guide to Wavelets.pdf_大学文库

AReallyFriendlyGuidetoWavelets-oc.Valens1999-c.valensamindless.com 1. Introduction It is well known from Fourier theory that a signal can be expressed as the sum of a, possibly infinite, series of sines and cosines. This sum is also referred to as a Fourier expansion. The big disadvantage of a Fourier expansion however is that it has only frequency resolution and no time resolution. This means that although we might be able to determine all the frequencies present in a signal, we do not know when they are present. To overcome this problem in the past decades several solutions have been developed which are more or less able to represent a signal in the time and frequency domain at the same time The idea behind these time-frequency joint representations is to cut the signal of interest into several parts and then analyze the parts separately. It is clear that analyzing a signal this way will give more information about the when and where of different frequency components, but it leads to a fundamental problem as well: how to cut the signal Suppose that we want to know exactly all the frequency components present at a certain moment in time. We cut out only this very short time window using a Dirac pulse, transform it to the frequency domain and. something is very wrong. The problem here is that cutting the signal corresponds to a convolution between the signal and the cutting window. Since convolution in the time domain is identical to multiplication in the frequency domain and since the Fourier transform of a Dirac pulse contains all possible frequencies the frequency components of the signal will be smeared out all over the frequency axis. In fact this situation is the opposite of the standard Fourier transform since we now have time resolution but no frequency resolution whatsoever ty principle, which, in signal processing terms, states that it is impossible to know the exact frequency and the exact time of occurrence of this frequency in a signal. In other words, a signal can simply not be represented as a point in the time-frequency space The uncertainty principle shows that it is very important how one cuts the signal The wavelet transform or wavelet analysis is probably the most recent solution to overcome the shortcomings of the Fourier transform. In wavelet analysis the use of a fully scalable modulated window solves the signal-cutting problem. The window is shifted along the signal and for every position the spectrum is calculated. Then this process is repeated many times with a slightly shorter(or longer) window for every new cycle. In the end the result will be a collection of time-frequency representations of the signal, all with different resolutions. Because of this collection of representations we can speak of a multiresolution analysis. In the case of wavelets we normally do not speak about time-frequency representations but about time-scale representations, scale being in a way the opposite of frequency, ecause the term frequency is reserved for the Fourier transform Since from literature it is not always clear what is meant by small and large scales, I will define it here as follows the large scale is the big picture, while the small scales show the details. Thus, going from large scale to small scale is in this context equal to zooming in. A Dirac pulse is defined as f(o=I at 1=0 and f(o=0 for all other I

A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 4 1. Introduction It is well known from Fourier theory that a signal can be expressed as the sum of a, possibly infinite, series of sines and cosines. This sum is also referred to as a Fourier expansion. The big disadvantage of a Fourier expansion however is that it has only frequency resolution and no time resolution. This means that although we might be able to determine all the frequencies present in a signal, we do not know when they are present. To overcome this problem in the past decades several solutions have been developed which are more or less able to represent a signal in the time and frequency domain at the same time. The idea behind these time-frequency joint representations is to cut the signal of interest into several parts and then analyze the parts separately. It is clear that analyzing a signal this way will give more information about the when and where of different frequency components, but it leads to a fundamental problem as well: how to cut the signal? Suppose that we want to know exactly all the frequency components present at a certain moment in time. We cut out only this very short time window using a Dirac pulse1 , transform it to the frequency domain and … something is very wrong. The problem here is that cutting the signal corresponds to a convolution between the signal and the cutting window. Since convolution in the time domain is identical to multiplication in the frequency domain and since the Fourier transform of a Dirac pulse contains all possible frequencies the frequency components of the signal will be smeared out all over the frequency axis. In fact this situation is the opposite of the standard Fourier transform since we now have time resolution but no frequency resolution whatsoever. The underlying principle of the phenomena just described is Heisenberg’s uncertainty principle, which, in signal processing terms, states that it is impossible to know the exact frequency and the exact time of occurrence of this frequency in a signal. In other words, a signal can simply not be represented as a point in the time-frequency space. The uncertainty principle shows that it is very important how one cuts the signal. The wavelet transform or wavelet analysis is probably the most recent solution to overcome the shortcomings of the Fourier transform. In wavelet analysis the use of a fully scalable modulated window solves the signal-cutting problem. The window is shifted along the signal and for every position the spectrum is calculated. Then this process is repeated many times with a slightly shorter (or longer) window for every new cycle. In the end the result will be a collection of time-frequency representations of the signal, all with different resolutions. Because of this collection of representations we can speak of a multiresolution analysis. In the case of wavelets we normally do not speak about time-frequency representations but about time-scale representations, scale being in a way the opposite of frequency, because the term frequency is reserved for the Fourier transform. Since from literature it is not always clear what is meant by small and large scales, I will define it here as follows: the large scale is the big picture, while the small scales show the details. Thus, going from large scale to small scale is in this context equal to zooming in. 1 A Dirac pulse is defined as f(t) = 1 at t = 0 and f(t) = 0 for all other t

AReallyFriendlyGuidetoWavelets-oc.Valens1999-c.valensamindless.com In the following sections I will present the wavelet transform and develop a scheme that will allow us to implement the wavelet transform in an efficient way on a digital computer. The transform will be so efficient that it does not even use wavelets anymore. (The careful reader might raise an eyebrow here and ask: Surely you can't be serl But before we continue a disclaimer. Since wavelet theory is not a new thing anymore, it has been around now for fifteen years, say, I will not present a full and in-depth theory here. Several good textbooks on wavelet theory are available and many readable papers with a good review of wavelet theory have been published. The list of references at the end of this report contains pointers to texts with more extensive wavelet theory coverage like(in random order) Kai941, TWei94], [She961,[Bur98][Dau921,[Hub96,[Mal89] TVet92] I do however present some mathematical background in order to tell a coherent and clear tale(I hope Having this said, let's go on to the wavelets 2. The continuous wavelet transform The wavelet analysis described in the introduction is known as the continuous wavelet transform or CWT. More formally it is written as: s)=Jf(:(), (1) where* denotes complex conjugation. This equation shows how a function f(n) is decomposed into a set of basis functions ys(n), called the wavelets. The variables s and t are the new dimensions, scale and translation, after the wavelet transform For completeness sake equation(2)gives the inverse wavelet transform. I will not expand on this since we are not going to use it f(1)=(s,τwt(t)dtds (2) The wavelets are generated from a single basic wavelet y(o), the so-called mother wavelet, by scaling and translation Ys(0) In(3)s is the scale factor, T is the translation factor and the factor s is for energy normalization across the different It is important to note that in(1), (2)and (3)the wavelet basis functions are not specified. This is a difference between the wavelet transform and the Fourier transform, or other transforms. The theory of wavelet transforms "1 am serious, and don' t call me Shirley. Leslie Nielsen as Dr Rumack in the film airplane!(1980)

A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 5 In the following sections I will present the wavelet transform and develop a scheme that will allow us to implement the wavelet transform in an efficient way on a digital computer. The transform will be so efficient that it does not even use wavelets anymore. (The careful reader might raise an eyebrow here and ask: “Surely you can’t be serious?” 2 ) But before we continue a disclaimer. Since wavelet theory is not a new thing anymore, it has been around now for fifteen years, say, I will not present a full and in-depth theory here. Several good textbooks on wavelet theory are available and many readable papers with a good review of wavelet theory have been published. The list of references at the end of this report contains pointers to texts with more extensive wavelet theory coverage like (in random order) [Kai94], [Wei94], [She96], [Bur98], [Dau92], [Hub96], [Mal89], [Vet92]. I do however present some mathematical background in order to tell a coherent and clear tale (I hope). Having this said, let’s go on to the wavelets. 2. The continuous wavelet transform The wavelet analysis described in the introduction is known as the continuous wavelet transform or CWT. More formally it is written as: s f t t dt s ( , ) ( ) ( ) * = ψ ,τ γ τ ∫ , (1) where * denotes complex conjugation. This equation shows how a function ƒ(t) is decomposed into a set of basis functions 5s,-(t), called the wavelets. The variables s and - are the new dimensions, scale and translation, after the wavelet transform. For completeness sake equation (2) gives the inverse wavelet transform. I will not expand on this since we are not going to use it: ∫∫ = γ τ ψ τ τ f t s t d ds s ( ) ( , ) ( ) , . (2) The wavelets are generated from a single basic wavelet 5(t), the so-called mother wavelet, by scaling and translation:       − τ ψ τ = ψ s t s t s 1 ( ) , . (3) In (3) s is the scale factor, - is the translation factor and the factor s -1/2 is for energy normalization across the different scales. It is important to note that in (1), (2) and (3) the wavelet basis functions are not specified. This is a difference between the wavelet transform and the Fourier transform, or other transforms. The theory of wavelet transforms 2 “I am serious, and don’t call me Shirley.” Leslie Nielsen as Dr. Rumack in the film Airplane! (1980)

A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 6 deals with the general properties of the wavelets and wavelet transforms only. It defines a framework within one can design wavelets to taste and wishes. 3. Wavelet properties The most important properties of wavelets are the admissibility and the regularity conditions and these are the properties which gave wavelets their name. It can be shown [She96] that square integrable functions 5(t) satisfying the admissibility condition, ω < +∞ ω Ψ ω ∫ d | | | ( ) | 2 , (4) can be used to first analyze and then reconstruct a signal without loss of information. In (4) 4(7) stands for the Fourier transform of 5(t). The admissibility condition implies that the Fourier transform of 5(t) vanishes at the zero frequency, i.e. | ( ) | 0 0 2 Ψ ω = ω= . (5) This means that wavelets must have a band-pass like spectrum. This is a very important observation, which we will use later on to build an efficient wavelet transform. A zero at the zero frequency also means that the average value of the wavelet in the time domain must be zero, ∫ψ(t)dt = 0 , (6) and therefore it must be oscillatory. In other words, 5(t) must be a wave. As can be seen from (1) the wavelet transform of a one-dimensional function is two-dimensional; the wavelet transform of a two-dimensional function is four-dimensional. The time-bandwidth product of the wavelet transform is the square of the input signal 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 decreasing scale s. 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 concept and we will try to explain it a little using the concept of vanishing moments. If we expand the wavelet transform (1) into the Taylor series at t = 0 until order n (let τ = 0 for simplicity) we get [She96]:          + +      γ = ∑ ψ ∫ = ( 1) ! (0) 1 ( ,0) 0 ( ) dt O n s t p t f s s n p p p . (7)

AReallyFriendlyGuidetoWavelets-oc.Valens1999-c.valensamindless.com Here f)stands for the p" derivative of f and an+l)means the rest of the expansion. Now, if we define the moments of the wavelet by M then we can rewrite(7)into the finite development X60|/0tns+fM2+(2(My3 +…+C"oMny+ (9) 2 From the admissibility condition we already have that the oh moment Mo=0 so that the first term in the right-hand side of (9)is zero. If we now manage to make the other moments up to M, zero as well, then the wavelet transform coefficients Ys, t)will decay as fast as sfor a smooth signal f(t). This is known in literature as the vanishing moments or approximation order. If a wavelet has N vanishing moments, then the approximation order of the wavelet transform is also N. The moments do not have to be exactly zero, a small value is often good enough. In fact, experimental research suggests that the number of vanishing moments required depends heavily on the application Summarizing, the admissibility condition gave us the wave, regularity and vanishing moments gave us the fast decay or the let, and put together they give us the wavelet. More about regularity"can be found for instance in [Bur98]and [Dau92 4. Discrete wavelets Now that we know what the wavelet transform is, we would like to make it practical. However, the wavelet transform as described so far still has three properties that make it difficult to use directly in the form of (1). The first is the redundancy of the CWT In (1)the wavelet transform is calculated by continuously shifting a continuously scalable function over a signal and calculating the correlation between the two. It will be clear that these scaled functions will be nowhere near an orthogonal basis and the obtained wavelet coefficients will therefore be highly edundant. For most practical applications we would like to remove this redundancy Even without the redundancy of the CWT we still have an infinite number of wavelets in the wavelet transform and we would like to see this number reduced to a more manageable count. This is the second problem we have The third problem is that for most functions the wavelet transforms have no analytical solutions and they can be calculated only numerically or by an optical analog computer. Fast algorithms are needed to be able to exploit the 3 There exist functions of which all moments vanish. An example is the function e-x.sin(x+) for x 20(Kor96] ems to stem from the definition that a filter is called K-regular if its z-transform has k zeroes at ze ies to the scaling filter(which has not been mentioned yet)and it is possible only if all wavelet moments to K he scaling filter is formed by the coefficients h(h)in equation( The CWt behaves just like an orthogonal transform in the sense that the inverse wavelet transform permits us to reconstruct the signal by an integration of all the projections of the signal onto the wavelet basis. This is called quasi-orthogonality [She96]

A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 7 Here ƒ(p) stands for the p th derivative of ƒ and "(n+1) means the rest of the expansion. Now, if we define the moments of the wavelet by Mp, ∫ M = t ψ t dt p p ( , (8) ) then we can rewrite (7) into the finite development       γ = + + + + + + + ( ) ! (0) ... 2! (0) 1! (0) (0) 1 ( ,0) 1 2 ( ) 3 2 (2) 2 1 (1) 0 n n n n M s O s n f M s f M s f f M s s s . (9) From the admissibility condition we already have that the 0th moment M0 = 0 so that the first term in the right-hand side of (9) is zero. If we now manage to make the other moments up to Mn zero as well, then the wavelet transform coefficients γ(s,-) will decay as fast as s n+2 for a smooth signal ƒ(t). This is known in literature as the vanishing moments3 or approximation order. If a wavelet has N vanishing moments, then the approximation order of the wavelet transform is also N. The moments do not have to be exactly zero, a small value is often good enough. In fact, experimental research suggests that the number of vanishing moments required depends heavily on the application [Cal96]. Summarizing, the admissibility condition gave us the wave, regularity and vanishing moments gave us the fast decay or the let, and put together they give us the wavelet. More about regularity4 can be found for instance in [Bur98] and [Dau92]. 4. Discrete wavelets Now that we know what the wavelet transform is, we would like to make it practical. However, the wavelet transform as described so far still has three properties that make it difficult to use directly in the form of (1). The first is the redundancy of the CWT. In (1) the wavelet transform is calculated by continuously shifting a continuously scalable function over a signal and calculating the correlation between the two. It will be clear that these scaled functions will be nowhere near an orthogonal basis5 and the obtained wavelet coefficients will therefore be highly redundant. For most practical applications we would like to remove this redundancy. Even without the redundancy of the CWT we still have an infinite number of wavelets in the wavelet transform and we would like to see this number reduced to a more manageable count. This is the second problem we have. The third problem is that for most functions the wavelet transforms have no analytical solutions and they can be calculated only numerically or by an optical analog computer. Fast algorithms are needed to be able to exploit the 3 There exist functions of which all moments vanish. An example is the function ) sin( 4 1/ 4 e x x ⋅ − for x ≥ 0 [Kör96]. 4 The term regularity seems to stem from the definition that a filter is called K-regular if its z-transform has K zeroes at z=eiπ . In wavelet theory this applies to the scaling filter (which has not been mentioned yet) and it is possible only if all wavelet moments up to K-1 vanish [Bur98]. The scaling filter is formed by the coefficients h(k) in equation (17). 5 The CWT behaves just like an orthogonal transform in the sense that the inverse wavelet transform permits us to reconstruct the signal by an integration of all the projections of the signal onto the wavelet basis. This is called quasi-orthogonality [She96]

AReallyFriendlyGuidetoWavelets-oc.Valens1999-c.valensamindless.com ower of the wavelet transform and it is in fact the existence of these fast algorithms that have put wavelet transforms where they are today Let us start with the removal of redundancy As mentioned before the CWT maps a one-dimensional signal to a two-dimensional time-scale joint representation that is highly redundant. The time-bandwidth product of the CWT is the square of that of the signal and for most applications, which seek a signal description with as few components as possible, this is not efficient. To overcome this problem discrete wavelets have been introduced. Discrete wavelets are not continuously scalable and translatable but can only be scaled and translated in discrete steps. This is achieved by modifying the wavelet representation (3) to create [Dau921 Vi(o) Although it is called a discrete wavelet, it normally is a(piecewise)continuous function In(10)j and k are integers nd so>I is a fixed dilation step. The translation factor to depends on the dilation step. The effect of discretizing the wavelet is that the time-scale space is now sampled at discrete intervals. We usually choose so=2 so that the sampling of the frequency axis corresponds to dyadic sampling. This is a very natural choice for computers, the human ear and music for instance For the translation factor we usually choose to= I so that we also have dyadic sampling of the time axis ●鲁垂●鲁●·@。●e●·e Figure I Localization of the discrete wavelets in the time-scale space on a dyadic gric When discrete wavelets are used to transform a continuous signal the result will be a series of wavelet coefficients nd it is referred to as the wavelet series decomposition. An important issue in such a decomposition scheme is of course the question of reconstruction. It is all very well to sample the time-scale joint representation on a dyadic grid, but if it will not be possible to reconstruct the signal it will not be of great use. As it turns out, it is indeed possible to construct a signal from its wavelet series decomposition In [Dau92] it is proven that the necessary and sufficient

A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 8 power of the wavelet transform and it is in fact the existence of these fast algorithms that have put wavelet transforms where they are today. Let us start with the removal of redundancy. As mentioned before the CWT maps a one-dimensional signal to a two-dimensional time-scale joint representation that is highly redundant. The time-bandwidth product of the CWT is the square of that of the signal and for most applications, which seek a signal description with as few components as possible, this is not efficient. To overcome this problem discrete wavelets have been introduced. Discrete wavelets are not continuously scalable and translatable but can only be scaled and translated in discrete steps. This is achieved by modifying the wavelet representation (3) to create [Dau92]         − τ ψ = ψ j j j j k s t k s s t 0 0 0 0 , 1 ( . (10) ) Although it is called a discrete wavelet, it normally is a (piecewise) continuous function. In (10) j and k are integers and s0 > 1 is a fixed dilation step. The translation factor -0 depends on the dilation step. The effect of discretizing the wavelet is that the time-scale space is now sampled at discrete intervals. We usually choose s0 = 2 so that the sampling of the frequency axis corresponds to dyadic sampling. This is a very natural choice for computers, the human ear and music for instance. For the translation factor we usually choose -0 = 1 so that we also have dyadic sampling of the time axis. Figure 1 Localization of the discrete wavelets in the time-scale space on a dyadic grid. When discrete wavelets are used to transform a continuous signal the result will be a series of wavelet coefficients, and it is referred to as the wavelet series decomposition. An important issue in such a decomposition scheme is of course the question of reconstruction. It is all very well to sample the time-scale joint representation on a dyadic grid, but if it will not be possible to reconstruct the signal it will not be of great use. As it turns out, it is indeed possible to reconstruct a signal from its wavelet series decomposition. In [Dau92] it is proven that the necessary and sufficient

AReallyFriendlyGuidetoWavelets-oc.Valens1999-c.valensamindless.com condition for stable reconstruction is that the energy of the wavelet coefficients must lie between two positive IfI is the energy of f(), A>0, B< oo and A, B are independent of f(o). When equation(1 1)is satisfied, the of basis functions i. () withj, k e Z is referred to as a frame with frame bounds A and b. when A=B the rame is tight and the discrete wavelets behave exactly like an orthonormal basis. When A* B exact reconstruction is still possible at the expense of a dual frame. In a dual frame discrete wavelet transform the decomposition wavelet is different from the reconstruction wavelet We will now immediately forget the frames and continue with the removal of all redundancy from the wavelet transform. The last step we have to take is making the discrete wavelets orthonormal. This can be done only with discrete wavelets. The discrete wavelets can be made orthogonal to their own dilations and translations by special choices of the mother wavelet which means if j=m and k=n Yik(tymn(tdt= 2 An arbitrary signal can be reconstructed by summing the orthogonal wavelet basis functions, weighted by the wavelet transform coefficients [She] f(1)=∑Yj,k)y;k(1) n(13)shows the inverse wavelet transform for discrete wavelets, which we had not yet seen. Orthogonality is not essential in the representation of signals. The wavelets need not be orthogonal and in some applications the redundancy can help to reduce the sensitivity to noise[ She 96]or improve the shift invariance of the transform [ Bur98]. This is a disadvantage of discrete wavelets: the resulting wavelet transform is no longer shift invariant, which means that the wavelet transforms of a signal and of a time-shifted version of the same signal are not simply shifted versions of each other 5. A band-pass filter With the redundancy removed, we still have two hurdles to take before we have the wavelet transform in a practical form. We continue by trying to reduce the number of wavelets needed in the wavelet transform and save the problem of the difficult analytical solutions for the end

A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 9 condition for stable reconstruction is that the energy of the wavelet coefficients must lie between two positive bounds, i.e. 2 2 , , 2 A f f , B f j k ≤ ∑ ψ j k ≤ , (11) where ƒ 2 is the energy of ƒ(t), A > 0, B < and A, B are independent of ƒ(t). When equation (11) is satisfied, the family of basis functions 5j,k(t) with j, k Ζ is referred to as a frame with frame bounds A and B. When A = B the frame is tight and the discrete wavelets behave exactly like an orthonormal basis. When A ≠ B exact reconstruction is still possible at the expense of a dual frame. In a dual frame discrete wavelet transform the decomposition wavelet is different from the reconstruction wavelet. We will now immediately forget the frames and continue with the removal of all redundancy from the wavelet transform. The last step we have to take is making the discrete wavelets orthonormal. This can be done only with discrete wavelets. The discrete wavelets can be made orthogonal to their own dilations and translations by special choices of the mother wavelet, which means:      = = ψ ψ = ∫ otherwise if j m and k n t t dt j k m n 0 1 ( ) ( ) * , , . (12) An arbitrary signal can be reconstructed by summing the orthogonal wavelet basis functions, weighted by the wavelet transform coefficients [She96]: =∑γ ψ j k j k f t j k t , , ( . (13) ) ( , ) ( ) Equation (13) shows the inverse wavelet transform for discrete wavelets, which we had not yet seen. Orthogonality is not essential in the representation of signals. The wavelets need not be orthogonal and in some applications the redundancy can help to reduce the sensitivity to noise [She96] or improve the shift invariance of the transform [Bur98]. This is a disadvantage of discrete wavelets: the resulting wavelet transform is no longer shift invariant, which means that the wavelet transforms of a signal and of a time-shifted version of the same signal are not simply shifted versions of each other. 5. A band-pass filter With the redundancy removed, we still have two hurdles to take before we have the wavelet transform in a practical form. We continue by trying to reduce the number of wavelets needed in the wavelet transform and save the problem of the difficult analytical solutions for the end

AReallyFriendlyGuidetoWavelets-oc.Valens1999-c.valensamindless.com Even with discrete wavelets we still need an infinite number of scalings and translations to calculate the wavelet transform. The easiest way to tackle this problem is simply not to use an infinite number of discrete wavelets. Of course this poses the question of the quality of the transform. Is it possible to reduce the number of wavelets to analyze a signal and still have a useful result The translations of the wavelets are of course limited by the duration of the signal under investigation so that have an upper boundary for the wavelets. This leaves us with the question of dilation: how many scales do we need to analyze our signal? How do we get a lower bound? It turns out that we can answer this question by looking at the wavelet transform in a different way If we look at (5)we see that the wavelet has a band-pass like spectrum. From Fourier theory we know that compression in time is equivalent to stretching the spectrum and shifting it upwards F(a)=9 (14) This means that a time compression of the wavelet by a factor of 2 will stretch the frequency spectrum of the wavelet by a factor of 2 and also shift all frequency components up by a factor of 2. Using this insight we can cover the finite spectrum of our signal with the spectra of dilated wavelets in the same way as that we covered our signal in the time domain with translated wavelets. To get a good coverage of the signal spectrum the stretched wavelet spectra should touch each other, as if they were standing hand in hand(see figure 2). This can be arranged by correctly designing Figure 2 aling of the mother wavelet in the Summarizing, if one wavelet can be seen as a band-pass filter, then a series of dilated wavelets can be seen as a band-pass filter bank, If we look at the ratio between the center fred of a wavelet spectrum and the width of this spectrum we will see that it is the same for all wavelets. This ratio is normally referred to as the fidelity factor Q herefore of a ce o filter bank 6. Intermezzo: a constraint As an intermezzo we will now take a look at an important constraint on our signal, which has sneaked in during the cover its frequency spectrum and its time duration with wavelets. Usually this constraint is formally stated as

A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 10 Even with discrete wavelets we still need an infinite number of scalings and translations to calculate the wavelet transform. The easiest way to tackle this problem is simply not to use an infinite number of discrete wavelets. Of course this poses the question of the quality of the transform. Is it possible to reduce the number of wavelets to analyze a signal and still have a useful result? The translations of the wavelets are of course limited by the duration of the signal under investigation so that we have an upper boundary for the wavelets. This leaves us with the question of dilation: how many scales do we need to analyze our signal? How do we get a lower bound? It turns out that we can answer this question by looking at the wavelet transform in a different way. If we look at (5) we see that the wavelet has a band-pass like spectrum. From Fourier theory we know that compression in time is equivalent to stretching the spectrum and shifting it upwards:       ω = a F a F f at 1 { . (14) ( )} This means that a time compression of the wavelet by a factor of 2 will stretch the frequency spectrum of the wavelet by a factor of 2 and also shift all frequency components up by a factor of 2. Using this insight we can cover the finite spectrum of our signal with the spectra of dilated wavelets in the same way as that we covered our signal in the time domain with translated wavelets. To get a good coverage of the signal spectrum the stretched wavelet spectra should touch each other, as if they were standing hand in hand (see figure 2). This can be arranged by correctly designing the wavelets. Figure 2 Touching wavelet spectra resulting from scaling of the mother wavelet in the time domain. Summarizing, if one wavelet can be seen as a band-pass filter, then a series of dilated wavelets can be seen as a band-pass filter bank. If we look at the ratio between the center frequency of a wavelet spectrum and the width of this spectrum we will see that it is the same for all wavelets. This ratio is normally referred to as the fidelity factor Q of a filter and in the case of wavelets one speaks therefore of a constant-Q filter bank. 6. Intermezzo: a constraint As an intermezzo we will now take a look at an important constraint on our signal, which has sneaked in during the last section: the signal to analyze must have finite energy. When the signal has infinite energy it will be impossible to cover its frequency spectrum and its time duration with wavelets. Usually this constraint is formally stated as