Time frequency scale transforms

TIME-FREQUENCY-SCALE TRANSFORMS ZHAN YANJUN (B.Sc.(Hons)), NUS A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2012 Acknowledgements First and foremost, a very big thank you goes out to my supervisor, Associate Professor Goh Say Song, for his constant encouragement and guidance throughout these few years. He has been a friend and a mentor to me, showing me my strengths and weaknesses and helping me to improve myself, not only in terms of character, but also in terms of my mathematical abilities. Taking time off his busy schedule to meet up with his students, he is a dedicated and motivated educator who puts his student’s well-being before his. Thank you to my family and my relatives for your support. Special thanks also goes out to my graduate coursemates, Charlotte, Ah Xiang Ge, Samuel and Yu Jie. Thank you all for the constructive discussions we have had over the semesters and thank you for teaching me and sharing with me your knowledge on certain subjects and disciplines. Without you all, life would not be so fun and exciting. Last, but not least, thank you to all my teacher friends, my researcher friends, my juniors in NUS, my seniors in NUS, and all the lecturers who have taught me over the years. i Contents Acknowledgements i Contents ii Summary iv 1 Preliminaries 1 1.1 Window Functions and Time-Frequency Analysis . . . . . . . . . . 2 1.2 Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Continuous Transforms . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Semi-Discrete Transforms . . . . . . . . . . . . . . . . . . . 7 1.3 Frames for L2 (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Introducing Modulation to Wavelets . . . . . . . . . . . . . . . . . . 14 2 From Continuous to Discrete Time-Frequency-Scale Transforms 19 2.1 Continuous Transforms . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Semi-Discrete Transforms . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Discrete Transforms: Frames . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Reconstruction from Time-Frequency-Scale Information . . . . . . . 31 2.5 2.4.1 Continuous Transforms . . . . . . . . . . . . . . . . . . . . . 31 2.4.2 Semi-Discrete Transforms . . . . . . . . . . . . . . . . . . . 34 2.4.3 Discrete Transforms: Frames . . . . . . . . . . . . . . . . . . 35 Transforms with Unification of Frequency and Scale Information . . 37 ii CONTENTS iii 2.5.1 Continuous Transforms . . . . . . . . . . . . . . . . . . . . . 37 2.5.2 Semi-Discrete Transforms . . . . . . . . . . . . . . . . . . . 41 3 Nonstationary Time-Frequency-Scale Frames 48 3.1 Construction of Nonstationary Frames . . . . . . . . . . . . . . . . 48 3.2 Nonstationary Gabor Frames . . . . . . . . . . . . . . . . . . . . . 59 3.3 Nonstationary Wavelet Frames . . . . . . . . . . . . . . . . . . . . . 65 3.4 Nonstationary Time-Frequency-Scale Frames . . . . . . . . . . . . . 69 Bibliography 78 Summary The study of time-frequency analysis dates as far back as the early 20th century, when Alfred Haar invented the Haar wavelets (see [11]). Although these were not significantly applied to signal processing in particular, this new era of discoveries impacted the engineering and mathematical worlds. In the 1930s and 1940s, timefrequency analysis arrived together with the revolutionary concept of quantum mechanics, thus starting a whole new discipline in signal processing. One of the mainstream tools to assist us in time-frequency analysis is the continuous wavelet transform. Unlike the Fourier transform, the continuous wavelet transform possesses the flexibility to construct a time-frequency representation of a signal that offers desirable time and frequency localization. To recover the original signal, the inverse continuous wavelet transform can be exploited. The continuous wavelet transform has been extensively studied in the literature (see, for instance, [5], [8], [23] and [24]). In Chapter 1, we state, without proof, some results associated with the ideas of the continuous wavelet transform. Together with the preliminary results on window functions and time-frequency windows, these will facilitate an in-depth discussion of the generalization of the wavelet transform that we are concerned with in general. Section 1.4 introduces the notion of modulation to wavelets. We then compare and contrast the changes in the time-frequency windows of the modulated wavelets with their unmodulated counterparts, and realize that the former offer a more flexible frequency window. One of the main objectives of this thesis is to revisit the continuous wavelet iv v transform, but with the addition of a modulation term. We name this new transform the time-frequency-scale transform. The modulation term contributes another parameter which we can adjust to our advantage. Motivated by the elegance of the reconstruction formulas of the continuous wavelet transforms presented in Section 1.2, we successfully extend the corresponding results with respect to the time-frequency-scale transform in Chapter 2. We begin our discussion in Section 2.1 with the most general version of the time-frequency-scale transform with no restriction of the parameters in the time and scale axes. We then restrict the dilation parameter a by considering only a > 0. Moving on in Section 2.2, we look at a special class of wavelets called the a-adic wavelets. Lastly, in Section 2.3, we further discretize the parameters in the time-frequency-scale transform and consider the resulting collection of functions that forms a frame for L2 (R). To complete the picture, we add in Section 2.4, which takes into account the reconstruction of a signal by using all three parameters, namely the dilation, translation and modulation factors, with the help of a weight function σ(γ). A detailed discussion on the continuous version and the various stages of discretization is included. Section 2.5 addresses time-frequency-scale transforms with unification of frequency and scale information. With the inter-dependency of the dilation and modulation parameters, we explore the assumptions required to implement such a scheme. In particular, we are interested in the relation γj = − aα−j + C, where γj and a−j are the modulation and dilation parameters respectively. Chapter 3 is devoted to devising ways in which we can construct families of frames using modulated wavelets for an increased efficiency in the utility of the time-frequency-scale transform. Chapter 2 emphasized mainly on following the changes in the reconstruction formula from a continuous to semi-discrete transition, whereas in Chapter 3, we venture one step forward and talk about frames. For a greater generalization, we consider nonstationary frames, which is supported by the structural setup of frames. In Section 3.1, we derive a general theorem on vi CHAPTER 0. SUMMARY nonstationary time-frequency-scale frames. Instead of just looking at a particular function to generate a family of frames, we look at how a sequence of functions, through a strategic use of this theorem, produces different families of frames with diverse properties. Setting the scale parameter to 1 in Section 3.2 allows us to generate nonstationary Gabor frames. We look at some examples, and, as a special consequence of taking the sequence of functions to be the same function, derive a well-known result in Gabor analysis (see [4]). Section 3.3 then provides the setting for nonstationary wavelet frames by allowing the modulation parameter to take zero value. One of the main highlights is the main idea behind Section 3.4. We experiment with the inclusion of all three parameters, time, scale and frequency, in the construction of our frames. We scrutinize the scenario where we have different modulation terms integrated in our functions, and we aim to achieve certain advantageous properties of the elements of the constructed frame, such as being real-valued and symmetric. To end off this section, we then present some specific examples of the sequences of modulation parameters {γj }j∈Z . Chapter 1 Preliminaries In this chapter, we recall some definitions and state, without proof, some theorems regarding the continuous wavelet transform. Most of these results can be found in the literature specializing in wavelets and frames (see, for instance, [3], [4], [5], [8] and [18]). In particular, the proofs of the results stated in Sections 1.1, 1.2 and 1.3 can be found in [5]. We adopt a systematic approach to present these statements, following closely what happens as we discretize first the dilation parameter and then the translation parameter. In addition, we will review the concepts of dilation, translation and modulation, and focus on introducing modulation to wavelets. A section is also dedicated to frames and some interesting results that are integral to many proofs in the thesis. This will provide the motivation and also the required tools to spur a discussion on the construction of frames in Chapter 3. A combination of Fourier analysis, functional analysis and linear algebra is essential in fully understanding the concepts of wavelets and frames. References on those background topics include [15], [20] and [21]. 1 2 CHAPTER 1. PRELIMINARIES 1.1 Window Functions and Time-Frequency Analysis Throughout this thesis, we will assume that the signal functions we are working with are measurable, and thus will automatically satisfy all the conditions shown in this section. For each p, where 1 ≤ p < ∞, let Lp (R) denote the class of ∫∞ measurable functions f on R such that the Lebesgue integral −∞ |f (t)|p dt is finite. Each Lp (R) space endowed with the norm {∫ ∥f ∥p := ∞ −∞ |f (t)| dt p } p1 is a Banach space, or a complete normed space. For the case where p = 2, we define the inner product of f ∈ L2 (R) and g ∈ L2 (R) by ∫ ⟨f, g⟩ := ∞ f (t)g(t)dt. −∞ With this inner product, the Banach space L2 (R) becomes a Hilbert space, which is a complete inner product space. Now we introduce the Fourier transform, which is one of our main tools throughout the thesis. Let f ∈ L1 (R). Then the Fourier transform of f is defined by ∫ ∞ f (ω) := e−iωt f (t)dt, −∞ ω ∈ R. By a standard density argument (see, for instance, [5]), the Fourier transform is extended from L1 (R) ∩ L2 (R) to L2 (R). Going straight into the concept of wavelet transforms, we start off by introducing what a window function is. Definition 1.1.1. Let ψ ∈ L2 (R) be a nontrivial function. If tψ(t) ∈ L2 (R), then ψ is called a window function. 1.1. WINDOW FUNCTIONS AND TIME-FREQUENCY ANALYSIS 3 1 Proposition 1.1.2. Any window function ψ satisfies |t| 2 ψ(t) ∈ L2 (R) and ψ ∈ L1 (R). Proposition 1.1.2 shows that any window function lies in both L1 (R) and L2 (R). It also enables us to define the center and radius of a window function. Definition 1.1.3. For any window function ψ ∈ L2 (R), we define its center, µ(ψ), and radius, △(ψ), as follows: 1 µ(ψ) := ∥ψ∥22 1 △(ψ) := ∥ψ∥2 {∫ ∞ −∞ ∫ ∞ t|ψ(t)|2 dt, −∞ (t − µ(ψ)) |ψ(t)| dt 2 2 } 21 . In wavelet analysis, the notions of translation and dilation play a central role. More precisely, we consider the following formulation. Definition 1.1.4. For any window function ψ ∈ L2 (R) and a, b ∈ R, a ̸= 0, we define the translation and dilation of the function as ( − 12 ψb;a (t) := |a| ψ t−b a ) , t ∈ R. (1.1) We say that the original function ψ has been translated by b and dilated by a. With these definitions in mind, let us now investigate the relationship between the centers and radii of ψ and those of ψb;a . Proposition 1.1.5. Let ψ ∈ L2 (R) be a window function. If the center and radius of the window function ψ are given by µ(ψ) and △(ψ) respectively, then the function ψb;a , where a, b ∈ R and a ̸= 0, is a window function whose center is b + aµ(ψ) and radius is |a|△(ψ). Proposition 1.1.6. Let ψ ∈ L2 (R) and suppose that ψ is a window function. If the center and radius of the window function ψ are given by µ(ψ) and △(ψ) respectively, then the function ψb;a , where a, b ∈ R and a ̸= 0, is a window function whose center is µ(ψ) a and radius is 1 △(ψ). |a| 4 CHAPTER 1. PRELIMINARIES We note that the time-frequency window of the function ψ is not arbitrarily flexible in the sense that the centers of the window depend on the dilation term and also the window function used. For example, if we encounter a signal with varying frequencies, it is hard to analyze the signal because in order to change the center of the frequency window, we would have to vary the window function used, or even consider using multiple window functions. There are many ways to tackle this problem, and the technique we employ will be emphasized in Section 1.4, where we will introduce a modulation term to the window function in question. In this way, the center of the window function can be adjusted accordingly when the need arises. 1.2 Wavelet Transforms In this section, we discuss what wavelet transforms are, and also review the various reconstruction formulas associated with the wavelet transform values. 1.2.1 Continuous Transforms Before we attempt to understand fully the function of the wavelet transform, let us familiarize ourselves with some basic definitions. It is known that the Fourier transform alone is not sufficient in extracting instantaneous spectral information from a signal. The continuous wavelet transform addresses this deficiency by providing time-scale information of the signal. Definition 1.2.1. A nontrivial function ψ ∈ L2 (R) is called a basic wavelet or mother wavelet if it satisfies Definition 1.1.1 and the admissibility condition: ∫ Cψ := ∞ −∞ |ψ(ω)|2 dω < ∞. |ω| (1.2) We observe that by Definition 1.1.1, Proposition 1.1.2 and Definition 1.2.1, all mother wavelets are in the function space L1 (R) ∩ L2 (R), and they satisfy what 1.2. WAVELET TRANSFORMS 5 is required for them to be window functions. We investigate how this wavelet interacts with the signal in the continuous wavelet transform. Definition 1.2.2. Let ψ ∈ L2 (R) be a mother wavelet. The continuous wavelet transform relative to ψ of f ∈ L2 (R) is defined as ∫ − 12 (Wψ f )(b, a) := |a| ( ∞ f (t)ψ −∞ ) t−b dt, a a, b ∈ R, a ̸= 0. (1.3) We have constructed a family of wavelets in this way, by translations and dilations. We shall see in the later sections how these operations affect the properties of the wavelet. The formula of the continuous wavelet transform can be written in terms of the inner product of f and the function ψb;a defined in (1.1). Proposition 1.2.3. Let ψ ∈ L2 (R) be a mother wavelet, f ∈ L2 (R). Then for a, b ∈ R and a ̸= 0, (Wψ f )(b, a) as defined in (1.3) can be written as (Wψ f )(b, a) = ⟨f, ψb;a ⟩, where ψb;a is defined in (1.1). An important question in practice is whether a signal can be recovered from the values (Wψ f )(b, a), a, b ∈ R, a ̸= 0. The following theorem shows that not only is this possible, but there is an explicit reconstruction formula. Theorem 1.2.4. Let ψ ∈ L2 (R) be a mother wavelet which defines a continuous wavelet transform Wψ . Then ∫ ∞ −∞ ∫ ∞ −∞ [ ] da (Wψ f )(b, a)(Wψ g)(b, a) 2 db = Cψ ⟨f, g⟩ a for all f, g ∈ L2 (R). In addition, for f ∈ L2 (R), 1 ⟨f, g⟩ = Cψ ∫ ∞ −∞ ∫ ∞ −∞ [(Wψ f )(b, a)⟨ψb;a , g⟩] da db a2 (1.4) for all g ∈ L2 (R), where ψb;a is defined by (1.1) and Cψ by (1.2), which means 6 CHAPTER 1. PRELIMINARIES that 1 f (x) = Cψ ∫ ∞ ∫ −∞ ∞ −∞ [(Wψ f )(b, a)]ψb;a (x) da db a2 weakly. To employ the reconstruction formula (1.4), a good choice of the function g would be the family of Gaussian functions at varying scales. Corollary 1.2.5. Consider the family of Gaussian functions gα , α > 0, defined by x2 1 gα (x) := √ e− 4α , 2 πα x ∈ R. (1.5) Then for any x ∈ R at which f is continuous, 1 f (x) = lim Cψ α→0+ ∫ ∞ −∞ ∫ ∞ −∞ [(Wψ f )(b, a)⟨ψb;a , gα (· − x)⟩] da db. a2 In signal analysis, we are only interested in the positive scale. Restricting ourselves to a > 0, we see that Theorem 1.2.4 still applies, but with a little variation. More precisely, we impose an additional condition on the mother wavelet ψ: ∫ ∞ 0 |ψ(ω)|2 dω = |ω| ∫ ∞ 0 |ψ(−ω)|2 1 dω = Cψ < ∞. |ω| 2 (1.6) Theorem 1.2.6. Let ψ ∈ L2 (R) be a mother wavelet which satisfies (1.6) and defines a continuous wavelet transform Wψ . Then ∫ ∞ −∞ ∫ 0 ∞ [ ] da 1 (Wψ f )(b, a)(Wψ g)(b, a) 2 db = Cψ ⟨f, g⟩ a 2 for all f, g ∈ L2 (R). In addition, for f ∈ L2 (R), 2 ⟨f, g⟩ = Cψ ∫ ∞ −∞ ∫ ∞ [(Wψ f )(b, a)⟨ψb;a , g⟩] 0 da db a2 for all g ∈ L2 (R), where ψb;a is defined by (1.1) and Cψ is as defined in (1.6), 1.2. WAVELET TRANSFORMS 7 which means that 2 f (x) = Cψ ∫ ∞ ∫ −∞ ∞ [(Wψ f )(b, a)] ψb;a (x) 0 da db a2 weakly. 1.2.2 Semi-Discrete Transforms In the previous sub-section, we worked with the premise that the frequency ω, and thus the scale a, can take any value in the frequency axis. In this sub-section, we begin to discretize, or partition this frequency axis into disjoint intervals. We consider a certain type of partitions by taking a = a−j 0 , where a0 ≥ 1. For convenience, we will refer to a0 simply as a throughout this thesis. Definition 1.2.7. A function ψ ∈ L2 (R) is called an a-adic wavelet, where a ≥ 1, if it is a mother wavelet and there exist 0 < A ≤ B < ∞ such that A≤ ∞ ∑ |ψ(a−j ω)|2 ≤ B a.e. (1.7) j=−∞ The condition (1.7) is called the stability condition imposed on the mother wavelet ψ. When a = 2, the mother wavelet is called a dyadic wavelet. By taking the dilation term to be a−j for some a ≥ 1 in (1.3), the new wavelet transform, known as the “normalized” continuous wavelet transform, takes the form (Wjψ f )(b) ( ) 1 := a (Wψ f ) b, j . a j 2 (1.8) The next two results provide information on a-adic wavelets ψ ⋄ that can be used in the reconstruction formula for the semi-discrete setting on hand. Theorem 1.2.8. For any a-adic wavelet ψ ∈ L2 (R), by defining the function 8 CHAPTER 1. PRELIMINARIES ψ ⋄ ∈ L2 (R), via its Fourier transform, as ψ ⋄ (ω) := ∑∞ ψ(ω) k=−∞ |ψ(a−k ω)|2 , (1.9) every f ∈ L2 (R) can be written as f (x) = ∞ ∫ ∑ j=−∞ ∞ −∞ (Wjψ f )(b)[aj ψ ⋄ (aj (x − b))]db a.e. Theorem 1.2.9. Let ψ ∈ L2 (R) be an a-adic wavelet. Then the function ψ ⋄ , whose Fourier transform is defined by (1.9), is also an a-adic wavelet with ∞ ∑ 1 1 ≤ |ψ ⋄ (a−j ω)|2 ≤ B j=−∞ A a.e. As ψ ⋄ is instrumental in the reconstruction formula for the semi-discrete wavelet transform based on ψ, it is an a-adic dual of ψ. This notion of dual is made precise below. Definition 1.2.10. A function ψ ∈ L2 (R) is called an a-adic dual of an a-adic wavelet ψ ∈ L2 (R) if every f ∈ L2 (R) can be expressed as f (x) = = ∞ ∫ ∑ ∞ j=−∞ −∞ ∞ ∫ ∞ ∑ j=−∞ −∞ (Wjψ f )(b)[aj ψ(aj (x − b))]db 3j a 2 (Wψ f )(b, a−j )ψ(aj (x − b))]db a.e. We end off this section with a theorem which narrows down which a-adic duals can be used in the recovery of the original function f . Theorem 1.2.11. Let ψ ∈ L2 (R) be an a-adic wavelet and ψ ∈ L2 (R) satisfy ess sup −∞ 0, defined by (1.5) in Corollary 1.2.5. Then for any fixed γ ∈ R and x ∈ R at which f is continuous, 1 f (x) = lim Cψ α→0+ ∫ ∫ ∞ ∞ −∞ −∞ [(Vψ f )(b, a, γ)⟨ψb;a:γ , gα (· − x)⟩] da db. a2 Proof. Take any x ∈ R at which f is continuous. By setting g(t) = gα (t − x) in (2.5) in Theorem 2.1.3, we have 1 ⟨f, gα (· − x)⟩ = Cψ ∫ ∞ −∞ ∫ ∞ −∞ [(Vψ f )(b, a, γ)⟨ψb;a;γ , gα (· − x)⟩] da db. a2 (2.6) 2.1. CONTINUOUS TRANSFORMS 23 Since ∫ lim+ ⟨f, gα (· − x)⟩ = lim+ α→0 α→0 ∞ −∞ f (t)gα (t − x)dt = lim+ (f ∗ gα )(x) = f (x), α→0 the result follows. So far, we have assumed that the parameter a in the continuous time-frequencyscale transform in (2.1) takes all nonzero real values. However in the investigation of real-life signals, we are only interested in positive values of a. Consequently, there is a problem of reconstructing a signal f based on the values of (Vψ f )(b, a, γ) for a > 0. To this end, similar to handling the analogous problem for the continuous wavelet transform in Theorem 1.2.6, we impose the same condition on the mother wavelet, ψ: ∫ ∞ 0 |ψ(ω)|2 dω = |ω| ∫ 0 ∞ |ψ(−ω)|2 1 dω = Cψ < ∞. |ω| 2 Note that the finiteness of the admissibility condition as defined in (1.2) ensures that these integrals are well defined. With the necessary tools on hand, we are ready to readdress the theorem, but concentrating only on the positive scale. Theorem 2.1.5. Let ψ ∈ L2 (R) be a mother wavelet which satisfies (1.6) and defines a continuous time-frequency-scale transform Vψ . Then for any fixed γ ∈ R, ∫ ∞ −∞ ∫ 0 ∞ [ ] da 1 (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) 2 db = Cψ ⟨f, g⟩ a 2 (2.7) for all f, g ∈ L2 (R). In addition, for f ∈ L2 (R), 2 ⟨f, g⟩ = Cψ ∫ ∞ −∞ ∫ ∞ [(Vψ f )(b, a, γ)⟨ψb;a;γ , g⟩] 0 da db a2 (2.8) for all g ∈ L2 (R), where ψb;a;γ is defined by (2.2) and Cψ by (1.6), which means CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 24 that 2 f (x) = Cψ ∫ ∞ −∞ ∫ ∞ [(Vψ f )(b, a, γ)] ψb;a;γ (x) 0 da db a2 weakly. Proof. Recall from (2.3) that for f ∈ L2 (R), (Vψ f )(b, a, γ) = (Wψ (f (·)e−iγ· ))(b, a). So, for all f, g ∈ L2 (R), ∫ ∫ ] da [ (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) 2 db a 0 ∫−∞ ] da ∞ ∫ ∞[ = (Wψ (f (·)e−iγ· ))(b, a)(Wψ (g(·)e−iγ· ))(b, a) 2 db a −∞ 0 1 = Cψ ⟨f (·)e−iγ· , g(·)e−iγ· ⟩ 2 ∞ ∞ by Theorem 1.2.6. The first part of the theorem then follows from the fact that 1 1 Cψ ⟨f (·)e−iγ· , g(·)e−iγ· ⟩ = Cψ ⟨f, g⟩. 2 2 The proof of the second part of the theorem is similar to that of Theorem 2.1.3. 2.2 Semi-Discrete Transforms In this section, we discretize a strategically, similar to the way we described in Chapter 1. We first define what a normalized time-frequency-scale transform is, and then introduce an a-adic wavelet for the purpose of signal reconstruction. By taking the dilation factor to be a−j , j ∈ Z, for some a ≥ 1 in (2.1), the resulting transform, known as the “normalized” time-frequency-scale transform, takes the form (Vjψ f )(b, γ) ( ) 1 := a (Vψ f ) b, j , γ . a j 2 (2.9) It turns out that the recovery of f from the values (Vjψ f )(b, γ), b, γ ∈ R, is provided by the notions of a-adic wavelets and a-adic duals in Definition 1.2.7 and Theorem 1.2.8. 2.2. SEMI-DISCRETE TRANSFORMS 25 Theorem 2.2.1. For any a-adic wavelet ψ ∈ L2 (R), by defining an a-adic wavelet ψ ⋄ ∈ L2 (R), via its Fourier transform, as ψ ⋄ (ω) := ∑∞ ψ(ω) −k 2 k=−∞ |ψ(a ω)| , every f ∈ L2 (R) can be written as f (x) = ∞ ∫ ∑ j=−∞ ∞ (Vjψ f )(b, γ)[aj eiγx ψ ⋄ (aj (x − b))]db −∞ a.e., where γ ∈ R is fixed. Proof. We know from Theorem 1.2.8 that for any a-adic wavelet ψ ∈ L2 (R), by defining an a-adic dual ψ ⋄ ∈ L2 (R) as above, every f ∈ L2 (R) can be written as ∞ ∫ ∑ f (x) = ∞ −∞ j=−∞ (Wjψ f )(b)[aj ψ ⋄ (aj (x − b))]db a.e. Fix γ ∈ R. We replace f with the function f (·)e−iγ· in the above relation and see from (1.8) and (2.3) that −iγx f (x)e = = = ∞ ∫ ∑ ∞ j=−∞ −∞ ∞ ∫ ∞ ∑ j=−∞ −∞ ∞ ∫ ∞ ∑ j=−∞ −∞ (Wjψ (f (·)e−iγ· ))(b)[aj ψ ⋄ (aj (x − b))]db j a 2 (Wψ (f (·)e−iγ· ))(b, a−j )[aj ψ ⋄ (aj (x − b))]db j a 2 (Vψ f )(b, a−j , γ)[aj ψ ⋄ (aj (x − b))]db. Bringing over the exponential term, we then come to the conclusion that f (x) = ∞ ∫ ∑ j=−∞ ∞ −∞ j a 2 (Vψ f )(b, a−j , γ)[aj eiγx ψ ⋄ (aj (x − b))]db a.e., and the result follows from (2.9). CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 26 Note that we have not mentioned anything about the uniqueness of the a-adic dual. As expected from the discussions in Chapter 1, we would think that this is not the only candidate available. What we are presenting here is just one of the many possibilities that can work hand in hand with the original a-adic wavelet. We emphasize that any a-adic dual will lead to a recovery formula. We have seen in Theorem 1.2.11 that as long as the a-adic dual satisfies (1.10), it is suitable to be an a-adic dual of the original mother wavelet. By similar arguments as above, we conclude that every f ∈ L2 (R) can be written as f (x) = ∞ ∫ ∑ j=−∞ ∞ −∞ (Vjψ f )(b, γ)[aj eiγx ψ(aj (x − b))]db almost everywhere. 2.3 Discrete Transforms: Frames Last, but not least, we look at a special a-adic wavelet, which constitutes a frame. This section will differ from the original results in Chapter 1, because the addition of a modulation term introduces certain new aspects of the dual frame. In addition, we have to take care of the frame operators with respect to different families of frames. We start off by highlighting the link between two different frame operators. Proposition 2.3.1. Suppose that for some γ0 ∈ R, {ψbj,k ;a−j ;γ0 }j,k∈Z forms a frame for L2 (R). Then for every γ ∈ R, {ψbj,k ;a−j ;γ }j,k∈Z also forms a frame for L2 (R) with the same frame bounds. Moreover, if Sγ0 and Sγ are the frame operators with respect to {ψbj,k ;a−j ;γ0 }j,k∈Z and {ψbj,k ;a−j ;γ }j,k∈Z respectively, then ∗ Sγ = Eγ−γ0 Sγ0 Eγ−γ 0 where Eµ denotes the modulation operator as defined in Definition 1.3.5 and Eµ∗ its adjoint operator. 2.3. DISCRETE TRANSFORMS: FRAMES 27 Proof. Given γ0 ∈ R, we first work out that ( j 2 iγt ψbj,k ;a−j ;γ (t) = e a ψ = e i(γ−γ0 )t [ t − bj,k a−j iγ0 t e j 2 ) a ψ ( t − bj,k a−j )] = ei(γ−γ0 )t ψbj,k ;a−j ;γ0 (t). Since we know that {ψbj,k ;a−j ;γ0 }j,k∈Z forms a frame for L2 (R), we have the relation below to hold for some 0 < A ≤ B < ∞: A∥f ∥22 ∞ ∞ ∑ ∑ ≤ |⟨f, ψbj,k ;a−j ;γ0 ⟩|2 ≤ B∥f ∥22 , f ∈ L2 (R). j=−∞ k=−∞ Substituting f (t) = f (t)e−i(γ−γ0 )t in the above, we then see that A∥f (·)e−i(γ−γ0 )· ∥22 ∞ ∞ ∑ ∑ ≤ |⟨f (·)e−i(γ−γ0 )· , ψbj,k ;a−j ;γ0 ⟩|2 ≤ B∥f (·)e−i(γ−γ0 )· ∥22 . j=−∞ k=−∞ Using the fact that ∥f (·)e−i(γ−γ0 )· ∥22 ∫ ∞ = −∞ |f (t)e−i(γ−γ0 )t |2 dt = ∥f ∥22 and that −i(γ−γ0 )· ⟨f (·)e ∫ ,ψ bj,k ;a−j ;γ0 ⟩ = ∞ −∞ ∞ f (t)e−i(γ−γ0 )t ψbj,k ;a−j ;γ0 (t)dt ( ∫ = f (t)e −i(γ−γ0 )t a j 2 eiγ0 t ψ −∞ = ⟨f, ψbj,k ;a−j ;γ ⟩, the inequality now becomes A∥f ∥22 ≤ ∞ ∞ ∑ ∑ j=−∞ k=−∞ |⟨f, ψbj,k ;a−j ;γ ⟩|2 ≤ B∥f ∥22 , ) t − bj,k dt a−j CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 28 proving that for any γ ∈ R, {ψbj,k ;a−j ;γ }j,k∈Z also forms a frame for L2 (R) with the same frame bounds A and B. We now go on to prove the second part of the proposition. By the definition of the frame operator, the frame operator with respect to {ψbj,k ;a−j ;γ0 }j,k∈Z is ∞ ∞ ∑ ∑ Sγ0 f = ⟨f, ψbj,k ;a−j ;γ0 ⟩ψbj,k ;a−j ;γ0 . j=−∞ k=−∞ Similarly, the frame operator with respect with {ψbj,k ;a−j ;γ }j,k∈Z is ∞ ∞ ∑ ∑ Sγ f = ⟨f, ψbj,k ;a−j ;γ ⟩ψbj,k ;a−j ;γ . (2.10) j=−∞ k=−∞ Using the same strategy as before and replacing f (t) with f (t)ei(γ−γ0 )t in (2.10), we see that Sγ (f (·)ei(γ−γ0 )· ) ∞ ∞ ∑ ∑ = ⟨f (·)ei(γ−γ0 )· , ψbj,k ;a−j ;γ ⟩ψbj,k ;a−j ;γ j=−∞ k=−∞ ) ) t − b j,k dt ψbj,k ;a−j ;γ (t) = f (t)ei(γ−γ0 )t e−iγt a ψ −j a −∞ j=−∞ k=−∞ (∫ ( ) ) ∞ ∞ ∞ ∑ ∑ j t − bj,k = dt ψbj,k ;a−j ;γ (t) f (t)e−iγ0 t a 2 ψ a−j −∞ j=−∞ k=−∞ ∞ ∞ ∑ ∑ = ∞ ∞ ∑ ∑ (∫ ∞ ⟨f, ψbj,k ;a−j ;γ0 ⟩ψbj,k ;a−j ;γ . j=−∞ k=−∞ j 2 ( 2.3. DISCRETE TRANSFORMS: FRAMES 29 Multiplying throughout by e−i(γ−γ0 )· , we have that e−i(γ−γ0 )· Sγ (f (·)e−i(γ−γ0 )· ) ( ) ∞ ∞ ∑ ∑ · − bj,k −i(γ−γ0 )· iγ· 2j = ⟨f, ψbj,k ;a−j ;γ0 ⟩e e a ψ a−j j=−∞ k=−∞ ( ) ∞ ∞ ∑ ∑ · − bj,k iγ0 · 2j = ⟨f, ψbj,k ;a−j ;γ0 ⟩e a ψ a−j j=−∞ k=−∞ = ∞ ∞ ∑ ∑ ⟨f, ψbj,k ;a−j ;γ0 ⟩ψbj,k ;a−j ;γ0 = Sγ0 f. j=−∞ k=−∞ By the definition of the modulation operator Eµ , we then come to the conclusion that ∗ Eγ−γ S Eγ−γ0 f = Sγ0 f 0 γ and thus ∗ Sγ = Eγ−γ0 Sγ0 Eγ−γ . 0 Note that we can obtain Sγ by pre- and post-multiplying Sγ0 with the unitary ∗ operators Eγ−γ0 and Eγ−γ respectively. 0 b0 }j,k∈Z for L2 (R) Corollary 2.3.2. Let ψ ∈ L2 (R). If ψ generates a frame {ψj,k b0 where ψj,k is defined by (1.12), then for every γ ∈ R, ψ also generates a frame {ψbj,k ;a−j ;γ }j,k∈Z for L2 (R) with the same frame bounds. b0 Proof. This is an immediate consequence of the above proposition. Since ψj,k = ψbj,k ;a−j ;0 where bj,k = k b, aj 0 we simply take γ0 = 0 in the proposition. Now we look at the reconstruction of signals with the help of the frame operators. Theorem 2.3.3. For any fixed γ ∈ R, each f ∈ L2 (R) can be reconstructed from CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 30 its frame coefficients ⟨f, ψbj,k ;a−j ;γ ⟩, j, k ∈ Z, by applying the transformation ( f = (Sγ−1 Sγ )f = Sγ−1 ∞ ∞ ∑ ∑ ) ⟨f, ψbj,k ;a−j ;γ ⟩ψbj,k ;a−j ;γ j=−∞ k=−∞ = ∞ ∑ ∞ ∑ ⟨f, ψbj,k ;a−j ;γ ⟩Sγ−1 ψbj,k ;a−j ;γ . j=−∞ k=−∞ In addition, by setting ψbj,k ;a−j ;γ := Sγ−1 ψbj,k ;a−j ;γ , j, k ∈ Z, this gives ⟨f, g⟩ = ∞ ∞ ⟨ ∑ ∑ ⟩⟨ f, ψbj,k ;a−j ;γ ⟩ ψbj,k ;a−j ;γ , g , f, g ∈ L2 (R). j=−∞ k=−∞ The proof is a straightforward application of the definition of frame operators, and will not be presented here. Next, we investigate the relationship between the two duals ψbj,k ;a−j ;γ := Sγ−1 ψbj,k ;a−j ;γ , j, k ∈ Z, and ψbj,k ;a−j ;γ0 := Sγ−1 ψbj,k ;a−j ;γ0 , 0 j, k ∈ Z. ∗ We know from Proposition 2.3.1 that Sγ = Eγ−γ0 Sγ0 Eγ−γ . We also know from 0 the proof of the same proposition that ψbj,k ;a−j ;γ (t) = ei(γ−γ0 )t ψbj,k ;a−j ;γ0 (t) and so ∗ −j ;γ (t). Further, ψbj,k ;a−j ;γ0 (t) = e−i(γ−γ0 )t ψbj,k ;a−j ;γ (t) = Eγ−γ ψ 0 bj,k ;a )−1 ( ∗ ψbj,k ;a−j ;γ ψbj,k ;a−j ;γ = Sγ−1 ψbj,k ;a−j ;γ = Eγ−γ0 Sγ0 Eγ−γ 0 ∗ −1 −j ;γ = Eγ−γ S −j ;γ = Eγ−γ0 Sγ−1 Eγ−γ ψ 0 γ0 ψbj,k ;a 0 0 0 bj,k ;a = Eγ−γ0 ψbj,k ;a−j ;γ0 . 2.4. RECONSTRUCTION FROM TIME-FREQUENCY-SCALE INFORMATION 31 So we see that the two duals differ from each other by a modulation term, just like the original two frame systems. 2.4 Reconstruction from Time-Frequency-Scale Information In the previous sections, the recovery and inversion formulas are based on only two out of the three parameters, namely the time and scale parameters, while fixing the modulation parameter. In this section, we introduce a “weight function” σ ∈ L1 (R) such that σ(γ) > 0 for every γ ∈ R. This allows us to include the modulation parameter in the recovery process. One of the advantages of introducing such a weight function is to minimize the effects (if any) of any corrupted parameter. For example, if the information obtained from the time parameter is compromised in the extraction process, we increase the weight of the uncorrupted information from the modulation parameter through the weight function σ. In this sense, we are fully using all three parameters in the recovery process, as compared to only using two out of the three parameters. In the paper [25], the author also discussed about the possibility of introducing a weight function in the calculations. 2.4.1 Continuous Transforms In this section, we explore what happens when we adopt the concept of a weight function in the theorems we have established, starting with the continuous transforms. Theorem 2.4.1. Let ψ ∈ L2 (R) be a mother wavelet which defines a continuous time-frequency-scale transform Vψ . Then for any σ ∈ L1 (R) such that σ(γ) > 0, CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 32 γ ∈ R, ∫ ∞ ∫ −∞ ∞ ∫ [ ] da (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) 2 dbσ(γ)dγ = Cψ ∥σ∥1 ⟨f, g⟩ a −∞ −∞ ∞ (2.11) for all f, g ∈ L2 (R). In addition, for f ∈ L2 (R), 1 ⟨f, g⟩ = Cψ ∥σ∥1 ∫ ∫ ∞ −∞ ∞ ∫ −∞ ∞ [(Vψ f )(b, a, γ)⟨ψb;a;γ , g⟩] −∞ da dbσ(γ)dγ a2 (2.12) for all g ∈ L2 (R), where ψb;a;γ is defined by (2.2) and Cψ by (1.2), which means that 1 f (x) = Cψ ∥σ∥1 ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ −∞ [(Vψ f )(b, a, γ)]ψb;a;γ (x) da dbσ(γ)dγ a2 weakly. Proof. By (2.4) in Theorem 2.1.3, for a fixed γ ∈ R, ∫ ∞ −∞ ∫ ∞ [ −∞ ] da (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) 2 db = Cψ ⟨f, g⟩ a for all f, g ∈ L2 (R). Integrating throughout by σ(γ)dγ, we have ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∫ ∞ ] da [ (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) 2 dbσ(γ)dγ = Cψ ⟨f, g⟩ σ(γ)dγ, a −∞ −∞ ∞ which gives (2.11). Similarly, (2.12) follows from (2.5). Corollary 2.4.2. Consider the family of Gaussian functions gα , α > 0, defined by (1.5). Let σ ∈ L1 (R) such that σ(γ) > 0 for all γ ∈ R. Then for any x ∈ R at which f is continuous, 1 f (x) = lim Cψ ∥σ∥1 α→0+ ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ −∞ [(Vψ f )(b, a, γ)⟨ψb;a:γ , gα (· − x)⟩] da dbσ(γ)dγ. a2 Proof. Once again, we adopt the strategy of integrating both sides with respect to σ(γ)dγ to (2.6) in the proof of Corollary 2.1.4. By setting g(t) = gα (t − x) in 2.4. RECONSTRUCTION FROM TIME-FREQUENCY-SCALE INFORMATION 33 (2.12) in Theorem 2.4.1, we have 1 ⟨f, gα (·−x)⟩ = Cψ ∥σ∥1 ∫ ∞ ∫ −∞ ∫ ∞ −∞ ∞ −∞ [(Vψ f )(b, a, γ)⟨ψb;a;γ , gα (· − x)⟩] da dbσ(γ)dγ. a2 Since ∫ lim+ ⟨f, gα (· − x)⟩ = lim+ α→0 α→0 ∞ f (t)gα (t − x)dt = lim+ (f ∗ gα )(x) = f (x), α→0 −∞ we arrive at the conclusion. Theorem 2.4.3. Let ψ ∈ L2 (R) be a mother wavelet which satisfies (1.6) and defines a continuous time-frequency-scale transform Vψ . Let σ ∈ L1 (R) such that σ(γ) > 0, γ ∈ R. Then ∫ ∞ ∫ −∞ ∞ −∞ ∫ 0 ∞ [ ] da 1 (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) 2 dbσ(γ)dγ = Cψ ∥σ∥1 ⟨f, g⟩ (2.13) a 2 for all f, g ∈ L2 (R). In addition, for f ∈ L2 (R), 2 ⟨f, g⟩ = Cψ ∥σ∥1 ∫ ∫ ∞ −∞ ∫ ∞ ∞ [(Vψ f )(b, a, γ)⟨ψb;a;γ , g⟩] −∞ 0 da dbσ(γ)dγ a2 (2.14) for all g ∈ L2 (R), where ψb;a;γ is defined by (2.2) and Cψ by (1.6), which means that 2 f (x) = Cψ ∥σ∥1 ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ [(Vψ f )(b, a, γ)] ψb;a;γ (x) 0 da dbσ(γ)dγ a2 weakly. Proof. The proof is similar to that of Theorem 2.4.1. Here we employ (2.7) and (2.8) in Theorem 2.1.5, which gives (2.13) and (2.14). A possible extension to the theorems presented above is shown below. By considering a probability space Ω ⊆ R with the probability measure P, we have CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 34 the result that ∫ ∫ Ω ∞ −∞ ∫ [ ∞ −∞ (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) ] da a2 db dP(γ) = Cψ ⟨f, g⟩. Furthermore, if we let P be an absolutely continuous measure with respect to the Lebesgue measure, that is, P(γ) = σ(γ) , ∥σ∥1 where σ ∈ L1 (R) and σ(γ) > 0 for all γ ∈ R, we have the result in Theorem 2.4.1. Lastly, we also note that another possibility of extending these theorems is to take P as the counting measure. 2.4.2 Semi-Discrete Transforms Once again, we adopt the same framework as before, looking at how a-adic wavelets with a weight function can help us in the reconstruction of signals. Theorem 2.4.4. For any a-adic wavelet ψ ∈ L2 (R), by defining an a-adic wavelet ψ ⋄ ∈ L2 (R), via its Fourier transform, as ψ ⋄ (ω) := ∑∞ ψ(ω) k=−∞ |ψ(a−k ω)|2 , every f ∈ L2 (R) can be written as ∞ ∫ ∞ ∫ ∞ 1 ∑ f (x) = (Vjψ f )(b, γ)[aj eiγx ψ ⋄ (aj (x − b))]dbσ(γ)dγ ∥σ∥1 j=−∞ −∞ −∞ a.e., where σ ∈ L1 (R) is a fixed function satisfying σ(γ) > 0, γ ∈ R. Proof. We employ Theorem 2.2.1 from the previous section, which states that every f ∈ L2 (R) can be written as f (x) = ∞ ∫ ∑ j=−∞ ∞ −∞ (Vjψ f )(b, γ)[aj eiγx ψ ⋄ (aj (x − b))]db a.e., where γ ∈ R is fixed. We integrate throughout by σ(γ)dγ, and come to the 2.4. RECONSTRUCTION FROM TIME-FREQUENCY-SCALE INFORMATION 35 conclusion that ∞ ∫ ∑ ∥σ∥1 f (x) = j=−∞ ∞ ∫ −∞ ∞ −∞ (Vjψ f )(b, γ)[aj eiγx ψ ⋄ (aj (x − b))]dbσ(γ)dγ, which proves the theorem. 2.4.3 Discrete Transforms: Frames In this section, we are able to extract more results pertaining to the behavior of the frames as affected by the introduction of the weight function. We also discretize the weight function accordingly, since we are now in the realm of discrete calculations. b0 Proposition 2.4.5. Let ψ ∈ L2 (R). Suppose that {ψj,k }j,k∈Z , defined in (1.12), forms a frame with frame bounds A and B, where 0 < A ≤ B < ∞. Let {σn }n∈Z be ∑ a positive sequence in ℓ1 (Z), that is, n∈Z |σn | < ∞. Then for any real sequence {γn }n∈Z , √ { σn ψbj,k ;a−j ;γn }j,k,n∈Z forms a frame with frame bounds A∥{σn }n∈Z ∥ℓ1 and B∥{σn }n∈Z ∥ℓ1 . b0 Proof. By Corollary 2.3.2, we know that if {ψj,k } forms a frame with frame bounds A and B, then for every n ∈ Z, {ψbj,k ;a−j ;γn }j,k∈Z also forms a frame with the same bounds. As a result, we have the inequality A∥f ∥22 ≤ ∞ ∞ ∑ ∑ |⟨f, ψbj,k ;a−j ;γn ⟩|2 ≤ B∥f ∥22 , f ∈ L2 (R). j=−∞ k=−∞ Multiplying the above equation throughout by σn and summing the resultant relation over n, we have A ∥{σn }n∈Z ∥ℓ1 ∥f ∥22 ≤ ∞ ∞ ∞ ∑ ∑ ∑ n=−∞ j=−∞ k=−∞ |⟨f, √ σn ψbj,k ;a−j ;γn ⟩|2 ≤ B ∥{σn }n∈Z ∥ℓ1 ∥f ∥22 CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 36 for all f ∈ L2 (R). Remark 2.4.6. One way to obtain a positive sequence {σn }n∈Z in ℓ1 (Z) is to take a function σ ∈ L1 (R) satisfying σ(γ) > 0, γ ∈ R, and then set σn := σ(γn ) with γn = nγ0 for some γ0 > 0. Then ∑∞ n=−∞ σ(γn ) < ∞ whenever ∫∞ −∞ σ(γ)dγ < ∞, after imposing appropriate conditions on σ. Lastly, we look at dual frames and the reconstruction formula. Theorem 2.4.7. For every γ ∈ R, let {ψbj,k ;a−j ;γ }j,k∈Z be a frame for L2 (R) with frame operator Sγ , and set ψbj,k ;a−j ;γ := Sγ−1 ψbj,k ;a−j ;γ , j, k ∈ Z. Then for a positive sequence {σn }n∈Z in ℓ1 (Z) and any real sequence {γn }n∈Z , ∞ ⟨ ∞ ∞ ⟩ ⟨√ ⟩ ∑ ∑ ∑ √ 1 f, σn ψbj,k ;a−j ;γn ⟨f, g⟩ = σn ψbj,k ;a−j ;γn , g ∥σn ∥ℓ1 n=−∞ k=−∞ j=−∞ for every f, g ∈ L2 (R). Proof. We know from Theorem 2.3.3 that for a fixed n ∈ Z, we can write the inner product of f and g by ⟨f, g⟩ = ∞ ∞ ⟨ ∑ ∑ ⟩⟨ f, ψbj,k ;a−j ;γn ⟩ ψbj,k ;a−j ;γn , g . k=−∞ j=−∞ Multiplying by σn , we then see that ∞ ∞ ⟨ ⟩⟨√ ⟩ ∑ ∑ √ σn ψbj,k ;a−j ;γn , g . σn ⟨f, g⟩ = f, σn ψbj,k ;a−j ;γn k=−∞ j=−∞ 2.5. TRANSFORMS WITH UNIFICATION OF FREQUENCY AND SCALE INFORMATION 37 Lastly, we sum over n ∈ Z and obtain ∥σn ∥ℓ1 ⟨f, g⟩ = ∞ ∞ ∞ ⟨ ⟩⟨√ ⟩ ∑ ∑ ∑ √ f, σn ψbj,k ;a−j ;γn σn ψbj,k ;a−j ;γn , g . n=−∞ k=−∞ j=−∞ 2.5 Transforms with Unification of Frequency and Scale Information So far we have deduced many modified general theorems from well-known results in wavelet analysis. Another facet of the introduction of the modulation term leads to the question of whether one can integrate the modulation parameter into another available parameter. In this chapter, we explore the possibilities of such an idea, and list out all the new stability conditions and assumptions that are required for the implementation of such a scheme. Throughout this chapter, we will fix a relationship between γ and a, and we see that discretization from the continuous scenario will still give stability. Torr´esani mentioned in [25] that one of the more important issues to take note of when considering such a relation is the finiteness of the integral in the admissibility condition. The notion of interdependent parameters will inevitably bring about the question of whether the integral converges with the new workings that arise. We shall thus embark on the task of answering such queries as we move along from the continuous to the discrete framework. 2.5.1 Continuous Transforms We first look at the continuous wavelet transforms, and divide this section into two main portions, namely when the modulation parameter is a function of the scale parameter, and then vice versa. CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 38 We now restate the reconstruction theorem as inspired by Theorem 2.1.3, this time with a new admissibility condition as given below. Theorem 2.5.1. Let α, C ∈ R and ψ ∈ L2 (R) be a window function satisfying ∫ Cψ,α := ∞ −∞ |ψ(y)|2 dy < ∞, |y − α| (2.15) which defines a continuous time-frequency-scale transform Vψ . Then ∫ ∞ −∞ ∫ ∞ −∞ [ ( ) )] da α α db = Cψ,α ⟨f, g⟩ (Vψ f ) b, a, − + C (Vψ g) b, a, − + C a a a2 ( for all f, g ∈ L2 (R). In addition, for f ∈ L2 (R), 1 ⟨f, g⟩ = Cψ,α ∫ ∞ ∫ −∞ ∞ [ −∞ ] da ) α α (Vψ f ) b, a, − + C ⟨ψb;a;− a +C , g⟩ 2 db a a ( for all g ∈ L2 (R), where ψb;a;− αa +C is defined by (2.2), which means that 1 f (x) = Cψ,α ∫ ∞ −∞ ∫ ( ) α da [(Vψ f ) b, a, − + C ]ψb;a;− αa +C (x) 2 db a a −∞ ∞ weakly. Proof. We first let γ = − αa + C. Then ( ) t−b (Vψ f ) (b, a, γ) = f (t)e |a| ψ dt a −∞ ( ) ∫ ∞ t−b +C )t −i(− α − 12 a |a| ψ = f (t)e dt a −∞ ( ) ∫ ∞ t − b 1 −iα t−b −iCt iαb dt. = f (t)e e a |a|− 2 e ( a ) ψ a −∞ ∫ ∞ −iγt − 12 Continuing, we then let φ(t) := e−iαt ψ(t) and thus φ(ω) = ψ(ω + α). Observing 2.5. TRANSFORMS WITH UNIFICATION OF FREQUENCY AND SCALE INFORMATION 39 that |a|− 2 φ 1 ( t−b ) a = |a|− 2 e−iα( 1 ∫ (Vψ f ) (b, a, γ) = t−b a ) ψ ( t−b ), a ( ∞ f (t)e −iCt e iαb a −∞ − 12 |a| φ ) iαb t−b dt = e a Wφ (f (·)e−iC· )(b, a). a (2.16) By a similar argument, (Vψ g) (b, a, γ) = e iαb a Wφ (g(·)e−iC· )(b, a). So, ∫ ∫ ] da [ (Vψ f ) (b, a, γ) (Vψ g) (b, a, γ) 2 db a −∞ −∞ ∫ ∞∫ ∞ iαb da iαb = e a Wφ (f (·)e−iC· )(b, a)e a Wφ (g(·)e−iC· )(b, a) 2 db a −∞ −∞ −iC· −iC· = Cφ ⟨f (·)e , g(·)e ⟩ = Cφ ⟨f, g⟩ ∞ ∞ by Theorem 1.2.4. This is applicable if ∫ Cφ := ∞ −∞ |φ(ω)|2 dω < ∞, ω which is the case since ∫ Cφ = ∞ −∞ |ψ(ω + α)|2 dω = ω ∫ ∞ −∞ |ψ(ω)|2 dω = Cψ,α < ∞. ω−α One may wonder how we arrived at the relation γ = − αa + C. To explain this, we follow closely the proof presented in Chui’s book [5], and come to the step where we have ∫ ∫ [ ] da (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) db 2 a −∞ −∞ ∫ ∞ ∫ ∞ 1 |ψ(ax − aγ)|2 f (x)g(x) = dadx. 2π −∞ |a| −∞ ∞ ∞ (2.17) CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 40 |ψ(ax−aγ(a))|2 . |a| We now consider the term The rest of the proof requires it to be converted into another term which is independent of x. As such, fixing x ∈ R, we let y = ax − aγ(a). Then dy y y − a2 γ ′ (a) = x − aγ ′ (a) − γ(a) = − aγ ′ (a) = . da a a Our aim is to solve for possible candidates of γ that enable the substitution to go through in (2.17). This can be achieved if a2 γ ′ (a) is a constant. So we consider the differential equation a2 γ ′ (a) = a2 dγ =α da for some α ∈ R. Solving this differential equation using the method of separation of variables, we then see that ∫ 1 dγ = α ∫ 1 da. a2 Hence, γ=− α + C, a α, C ∈ R. We now carry on to prove a result similar to Theorem 2.5.1, but with the dependency of a and γ interchanged. Corollary 2.5.2. Let α, C ∈ R and ψ ∈ L2 (R) be a window function satisfying (2.15), which defines a continuous time-frequency-scale transform Vψ . Then ∫ ∞ −∞ ∫ ∞ −∞ [ ( (Vψ f ) b, ) ( )] α α 1 , γ (Vψ g) b, ,γ dγdb = Cψ,α ⟨f, g⟩ C −γ C −γ α for all f, g ∈ L2 (R). In addition, for f ∈ L2 (R), 1 ⟨f, g⟩ = Cψ,α ∫ ∞ −∞ ∫ ∞ −∞ [ ( (Vψ f ) b, ) ] α 1 α , γ ⟨ψb; C−γ ;γ , g⟩ dγdb C −γ α 2.5. TRANSFORMS WITH UNIFICATION OF FREQUENCY AND SCALE INFORMATION 41 α for all g ∈ L2 (R), where ψb; C−γ ;γ is defined by (2.2), which means that 1 f (x) = Cψ,α ∫ ∞ −∞ ∫ ∞ [ −∞ ( (Vψ f ) b, α ,γ C −γ )] 1 α ψb; C−γ ;γ (x) dγdb α weakly. Proof. By using the same substitution γ = − αa + C as in Theorem 2.5.1, we see that a = α C−γ and thus da dγ α . (C−γ)2 = This leads to the fact that da a2 = α1 dγ. Using this derived substitution in the statement of Theorem 2.5.1, we obtain the result readily. We also note here that since we are using the same relation between γ and a, the admissibility condition stays the same. Let us now illustrate how, given α, C ∈ R, we can choose a suitable ψ ∈ L2 (R) which satisfies (2.15) and thus be used in Theorem 2.5.1 and Corollary 2.5.2. Take α = 1, and select any ψ0 ∈ L2 (R) which satisfies (1.2). We then let ψ(y) = ψ0 (y − 1). Then ∫ ∞ −∞ |ψ(y)|2 dy = |y − 1| ∫ ∞ −∞ |ψ0 (y − 1)|2 dy = |y − 1| ∫ ∞ −∞ |ψ0 (y ′ )|2 ′ dy < ∞ |y ′ | by a change of variables. Other values of α can also be chosen and ψ be defined accordingly. 2.5.2 Semi-Discrete Transforms In this section, instead of just discretizing a as in Section 2.2, we discretize both a and γ. More specifically, similar to the relation γ = − αa + C in the previous section, we take γj = − aα−j + C for some α, C ∈ R, and investigate the properties of the time-frequency-scale transform with such a choice of γj . Definition 2.5.3. A function ψ ∈ L2 (R) is called an a-adic wavelet with respect to γj = − aα−j + C, where a ≥ 1, α, C ∈ R, if it is a window function and there CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 42 exist 0 < A ≤ B < ∞ such that ∞ ∑ A≤ ( ) ψ a−j (ω − γj ) 2 = j=−∞ ∞ ∑ |ψ(a−j (ω − C) + α)|2 ≤ B a.e. (2.18) j=−∞ Note that (2.18) is a more powerful version of (1.7). By taking α = C = 0 in (2.18), we have the usual stability condition for a-adic wavelets. In other words, the usual a-adic wavelets are a-adic wavelets with respect to γj = 0. Proposition 2.5.4. Let ψ ∈ L2 (R) be a window function that satisfies the stability condition (2.18). Then ψ satisfies ∫ ∞ A ln a ≤ α |ψ(y)|2 dy, |y − α| ∫ ∞ −α |ψ(−y)|2 dy ≤ B ln a. |y + α| (2.19) Furthermore, if A = B, then ∫ Cψ,α := ∞ −∞ |ψ(y)|2 dy = 2A ln a. |y − α| Proof. We first note that by integrating over [1 + C, a + C] and substituting y = a−j (ω − C) + α, we obtain the following relationship: ∫ a+C 1+C |ψ(a−j (ω − C) + α)|2 dω = ω−C ∫ a−j+1 +α a−j +α |ψ(y)|2 dy. (y − α) Dividing throughout by ω − C and integrating over [1 + C, a + C] in (2.18), we have ∫ a+C 1+C A dω ≤ ω−C ∫ a+C 1+C ∫ a+C ∞ ∑ B |ψ(a−j (ω − C) + α)|2 dω ≤ dω. ω − C ω − C 1+C j=−∞ With the above fact, we then see that A ln a ≤ ∞ ∫ ∑ j=−∞ a−j+1 +α a−j +α |ψ(y)|2 dy = (y − α) ∫ ∞ α |ψ(y)|2 dy ≤ B ln a. (y − α) 2.5. TRANSFORMS WITH UNIFICATION OF FREQUENCY AND SCALE INFORMATION 43 For the other case, we instead integrate over [−a + C, −1 + C] and let −y = a−j (ω − C) + α to obtain the following relationship: ∫ −1+C −a+C |ψ(a−j (ω − C) + α)|2 dω = −ω + C ∫ a−j+1 −α a−j −α |ψ(−y)|2 dy. (y + α) Dividing throughout by −ω + C and integrating over [−a + C, −1 + C] in (2.18), we have ∫ −1+C −a+C ∫ A dω ≤ −ω + C ∫ −1+C ∞ ∑ |ψ(a−j (ω − C) + α)|2 B dω ≤ dω. −ω + C −ω + C −a+C j=−∞ −1+C −a+C As a result, A ln a ≤ ∞ ∫ ∑ j=−∞ a−j+1 −α a−j −α |ψ(−y)|2 dy = (y + α) ∫ ∞ −α |ψ(−y)|2 dy ≤ B ln a. (y + α) To show the last part of the proposition, we observe that if A = B, then ∫ ∞ α |ψ(y)|2 dy = |y − α| ∫ ∞ −α |ψ(−y)|2 dy = A ln a. |y + α| By letting y ′ = −y, ∫ ∞ −α So, ∫ |ψ(−y)|2 dy = |y + α| ∞ −∞ ∫ |ψ(y)|2 dy = |y − α| −∞ α ∫ |ψ(y ′ )|2 − dy ′ = (−y ′ + α) ∞ α |ψ(y)|2 dy + (y − α) ∫ α −∞ ∫ α −∞ |ψ(y ′ )|2 ′ dy = A ln a. (α − y ′ ) |ψ(y ′ )|2 ′ dy = 2A ln a. (α − y ′ ) When α = C = 0, (2.19) in Proposition 2.5.4 is exactly the well-known necessary condition (see, for instance, [5] and [8]) for usual a-adic wavelets. The stability condition (2.18) is essential in the recovery of the signal function f ∈ L2 (R) from its time-frequency-scale transform values (Vψ f )(b, a−j , − aα−j + C). CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 44 Once again, we expect that a dual wavelet comes into play, combining forces with the original a-adic wavelet to give an elegant solution of reconstruction. Theorem 2.5.5. For any a-adic wavelet ψ ∈ L2 (R) with respect to γj = − aα−j +C, where a ≥ 1, α, C ∈ R, by defining the function ψ ⋄ ∈ L2 (R), via its Fourier transform, as ψ ⋄ (ω) := ∑∞ k=−∞ ψ(ω) |ψ(a−k ω + α(1 − a−k ))|2 , (2.20) every f ∈ L2 (R) can be written as f (x) = ∞ ∫ ∑ j=−∞ ∞ −∞ [ (Vjψ f )(b, γj ) ( j iγj x ae ψ ⋄ x−b a−j )] db a.e. where (Vjψ f )(b, γj ) is defined by (2.9). Proof. We first define φ(t) := e−iαt ψ(t) and φ⋄ (t) := e−iαt ψ ⋄ (t), and thus φ(ω) = ψ(ω + α) and φ⋄ (ω) = ψ ⋄ (ω + α). By (2.20), we have that ψ(ω + α) φ⋄ (ω) = ∑∞ k=−∞ |ψ(a−k (ω + α) + α(1 − a−k ))|2 = ∑∞ k=−∞ φ(ω) |ψ(a−k ω + α)|2 . Since φ(a−k ω) = ψ(a−k ω + α), φ(ω) . −k 2 k=−∞ |φ(a ω)| φ⋄ (ω) = ∑∞ (2.21) We have already shown in (2.16) that ) ( iαb j j (Vjψ f ) (b, γj ) = a 2 (Vψ f ) b, a−j , γj = a 2 e a−j Wφ (f (·)e−iC· )(b, a−j ). (2.22) Then ( j aφ ⋄ x−b a−j ) x−b j −iα( a−j ) = ae iαb = e−iCx e a−j ( ⋄ x−b a−j ) ψ )] [ ( α x−b +C )x ⋄ j i(− a−j . ae ψ a−j (2.23) 2.5. TRANSFORMS WITH UNIFICATION OF FREQUENCY AND SCALE INFORMATION 45 Equations (2.22) and (2.23) then give ∞ ∫ ∑ = ∞ j=−∞ −∞ ∞ ∫ ∞ ∑ j=−∞ = e iCx j 2 a e iαb a−j ∞ ∫ ∑ ∞ −∞ j iγj x ae Wφ (f (·)e −∞ j=−∞ ( [ (Vjψ f )(b, γj ) −iC· ψ ⋄ x−b a−j )] db −j iCx − aiαb −j −j j )(b, a )e e ( j 2 a Wφ (f (·)e −iC· ⋄ )(b, a )a φ ( j aφ x−b a−j ⋄ x−b a−j ) db ) db = eiCx f (x)e−iCx = f (x) where the last step is justified by Theorem 1.2.8. Definition 2.5.6. A function ψ ∈ L2 (R) is called an a-adic dual of an a-adic wavelet ψ ∈ L2 (R) with respect to γj = − aα−j + C, where a ≥ 1, α, C ∈ R, if every f ∈ L2 (R) can be expressed as )] x−b ae ψ db f (x) = a−j j=−∞ −∞ [ ( )] ∞ ∫ ∞ ∑ 3j x−b −j iγ x j a 2 (Vψ f )(b, a , γj ) e ψ = db. a−j j=−∞ −∞ ∞ ∫ ∑ ∞ [ (Vjψ f )(b, γj ) ( j iγj x We see that by taking ψ = ψ ⋄ as defined in (2.20), ψ ⋄ is a possible candidate as a dual of ψ. In the proof of Theorem 2.5.5, we considered the function φ(t) = e−iαt ψ(t). The derivation suggests a relationship between the stability condition (2.18) on ψ in terms of a similar condition on φ, which we now make precise. Lemma 2.5.7. For ψ ∈ L2 (R), define φ(t) := e−iαt ψ(t) where α ∈ R. Then ψ is an a-adic wavelet with respect to γj = − aα−j + C, where a ≥ 1, C ∈ R, if and only if φ is an a-adic wavelet. That is, for 0 < A ≤ B < ∞, A≤ ∞ ∑ j=−∞ |ψ(a−j (ω − C) + α)|2 ≤ B a.e. (2.24) CHAPTER 2. FROM CONTINUOUS TO DISCRETE TIME-FREQUENCY-SCALE TRANSFORMS 46 if and only if ∞ ∑ A≤ |φ(a−j ω)|2 ≤ B a.e. (2.25) j=−∞ Proof. Since ψ(t) = eiαt φ(t), (2.24) is equivalent to A≤ ∞ ∑ |φ(a−j (ω − C))|2 ≤ B a.e., j=−∞ which is in turn equivalent to (2.25) since C is just a constant. In addition, the frame bounds A and B remain the same. Theorem 2.5.8. Let ψ ∈ L2 (R) be an a-adic wavelet with respect to γj = − aα−j + C, where a ≥ 1, α, C ∈ R. Then the a-adic dual ψ ⋄ , whose Fourier transform is defined by (2.20), is also an a-adic wavelet with respect to γj = − aα−j + C, with ∞ ∑ 1 1 ≤ |ψ ⋄ (a−j (ω − C) + α)|2 ≤ B j=−∞ A a.e. Proof. By Lemma 2.5.7 and Theorem 1.2.9, since φ is an a-adic wavelet, φ⋄ as defined in (2.21) is also an a-adic wavelet with bounds 1 B and 1 . A This in turn means that ψ ⋄ (t) := eiαt φ⋄ (t) is an a-adic wavelet with respect to γj = − aα−j + C, with the same bounds. In the above we have identified restrictions on both how aj and γj should coexist. It would be unrealistic to randomly select two sequences and hope that they work. To end off this section, we investigate a more general way of checking whether a wavelet is an a-adic dual of another a-adic wavelet. Theorem 2.5.9. Let ψ ∈ L2 (R) be an a-adic wavelet with respect to γj = − aα−j + C, where a ≥ 1, α, C ∈ R. Suppose that ψ ∈ L2 (R) satisfies ess sup −∞ 0 (3.16) 62 CHAPTER 3. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES and sup(γj+1 − γj ) ≤ s for some s < β. (3.17) j∈Z Suppose that 0 < b < 2π |I| = 2π . β−α Then {Eγj Tkb ψ}j,k∈Z forms a frame for L2 (R). Proof. We take ψj (t) := eiγj t ψ(t), j ∈ Z, in our application of Theorem 3.1.1. ∑ First we check that the function j∈Z |ψj (ω)|2 is bounded above. By assumption, since |I| < 2π , b ψ has support in an interval I of length less than 2π , b meaning that ψ(ω − γj ) ̸= 0 for at most ⌊ ⌋ ⌊ ⌋ 2π |I| +1≤ +1 inf j∈Z (γj+1 − γj ) bc values of j ∈ Z, which is independent of ω ∈ R. Since ψ is continuous and nonzero in only an interval, we have that ∥ψ∥2∞ < ∞. As a result, ∑ (⌊ |ψj (ω)| ≤ 2 j∈Z ⌋ ) 2π + 1 ∥ψ∥2∞ < ∞. bc Next we show that (3.16) implies (3.13). To this end, for j ≥ 1, using (3.16), γj − γ0 = (γj − γj−1 ) + (γj−1 − γj−2 ) + · · · + (γ1 − γ0 ) ≥ jc. So lim γj ≥ lim (γ0 + jc) = ∞. j→∞ j→∞ Similarly, for j ≤ −1, γ0 − γj = (γ0 − γ−1 ) + (γ−1 − γ−2 ) + · · · + (γj+1 − γj ) ≥ |j|c, which gives lim γj ≤ lim (γ0 − |j|c) = −∞. j→−∞ j→−∞ We now check the lower bound of ∑ j∈Z |ψj (ω)|2 . Consider the interval J := [0, s], where s is as in (3.17). Then J ⊂ I. Fix ω ∈ R. By (3.13), {[γj , γj+1 ) : 3.2. NONSTATIONARY GABOR FRAMES 63 j ∈ Z} forms a partition of R. So there exists jω ∈ Z such that ω ∈ [γjω , γjω +1 ). Observe that 0 ≤ ω − γjω < γjω +1 − γjω ≤ sup(γj+1 − γj ) ≤ s < β, j∈Z so ω − γjω ∈ [0, γjω +1 − γjω ) ⊂ [0, s] ⊂ J ⊂ I. Therefore, ∑ |ψj (ω)|2 ≥ |ψ(ω − γjω )|2 ≥ inf |ψ(y)|2 , y∈J j∈Z which is bounded below by a positive constant since ψ is strictly positive in the interior of I and hence in J. There are many functions ψ that we can use in Theorem 3.2.2. For example, we can let ψ be a B-spline supported on the interval [α, β], or any C ∞ function which has support in [α, β]. We note that we can easily design a sequence {γj }j∈Z which satisfies the assumptions in Theorem 3.2.2. Taking γj+1 := γj + δj where 0 < c ≤ δj ≤ s < β for j ∈ Z, we see that inf j∈Z (γj+1 − γj ) = inf j∈Z δj ≥ c and supj∈Z (γj+1 − γj ) = supj∈Z δj ≤ s. In Gabor analysis, there is a result (see [3]) of the similar form as Theorem 3.1.1 on sufficient conditions for stationary Gabor frames. Now we shall obtain this result through the generalization provided by Theorem 3.1.1. Corollary 3.2.3. Consider ψ ∈ L2 (R), τ, b > 0, such that ( ) ∑∑ 2πk 1 ψ(ω − jτ )ψ ω − jτ − < ∞. B := sup b ω∈R j∈Z k∈Z b Then {Ekb Tjτ ψ}j,k∈Z forms a Bessel sequence with bound B. If also   ( ) ∑∑ 2πk  1 ∑ ψ(ω − jτ )ψ ω − jτ − |ψ(ω − jτ )|2 − A := inf   > 0, b ω∈R j∈Z b j∈Z k∈Z k̸=0 64 CHAPTER 3. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES then {Ekb Tjτ ψ}j,k∈Z forms a frame for L2 (R) with bounds A and B. Proof. We start off by taking a = 1 in Theorem 3.1.1. Then, we define the family ψj (t) = eijτ t ψ(t) where τ > 0 and j ∈ Z. As a result, ψj (ω) = ψ(ω−jτ ). Workings then show that ψj (t − kb) = eijτ (t−kb) ψ(t − kb) = e−ijτ kb eijτ t ψ(t − kb), and so ∑∑ |⟨f, ψj (· − kb)⟩|2 = j∈Z k∈Z ∑∑ |⟨f, eijτ · ψ(· − kb)⟩|2 . j∈Z k∈Z In addition, ) ( ∑∑ 1 2πk B = sup ψj (ω)ψj ω − b ω∈R j∈Z k∈Z b ( ) ∑∑ 2πk 1 sup ψ(ω − jτ )ψ ω − jτ − 0, a frame in L2 (R) with bounds A and B. Applying Proposition 1.3.6, the corresponding result holds for the collection {Ekb Tjτ ψ}j,k∈Z . By choosing ψ = g in Corollary 3.2.3, we obtain the following theorem on stationary Gabor frames in [3]. 3.3. NONSTATIONARY WAVELET FRAMES 65 Theorem 3.2.4. Consider g ∈ L2 (R), a, b > 0 such that ( ) ∑∑ 1 2πk B := sup g(ω − jτ )g ω − jτ − < ∞. b ω∈R j∈Z k∈Z b Then {Ekb Tjτ g}j,k∈Z forms a Bessel sequence with bound B. If also   ( ) ∑∑ 1 2πk  ∑ A := inf  |g(ω − jτ )|2 − g(ω − jτ )g ω − jτ −  > 0, b ω∈R j∈Z b j∈Z k∈Z k̸=0 then {Ekb Tjτ g}j,k∈Z forms a frame for L2 (R) with bounds A and B. Sufficient conditions for {Ekb Tjτ g}j,k∈Z to form a frame for L2 (R) have been studied since 1988, and one of the pioneers in this field is Daubechies (see [7]). 3.3 Nonstationary Wavelet Frames Now we look at the cases where a > 1. The main difference in this section is that a is no longer 1 and the dilation term appears in the family of frames we are constructing. Theorem 3.3.1. For I := [−β, −α] ∪ [α, β] where 0 < α < β, consider functions φj ∈ L2 (R), j ∈ Z, whose supports are I and satisfy C ≤ |φj (ω)|2 ≤ D, for some C, D > 0. Suppose that 0 < b < L ≥ 1 such that √ L+1 ω ∈ I, 2π |I| β a α, that is, if β > a α and hence a < N αβ . Since √ √ β L β N β > 1, ≤ for 1 ≤ N ≤ L. We only need to consider the case where α α α √ a < L αβ , which tallies with our assumption (3.19). + Next, we see that Sj+ ∩ Sj+L+M = ∅ for M ≥ 1. This can happen if and √ L+M β j j+L+M L+M only if a β < a α, i.e. β < a α and hence a > . Since αβ > 1, α √ √ √ L+M β L+1 β L+1 β ≤ for M ≥ 1. We only need to consider the case where a > , α α α which tallies with the second half of our assumption (3.19). For the negative parts Sj− , similar arguments show that for every j ∈ Z, − Sj− ∩ Sj+N ̸= ∅ for 1 ≤ N ≤ L. This is possible if and only if −aj β < −aj+N α √ − = ∅ for M ≥ 1. which is provided by a < L αβ in (3.19). In addition, Sj− ∩ Sj+L+M √ L+1 β j+L+M j This is equivalent to −a α < −a β which is made possible by a > as α assured (3.19). Since ∪ j∈Z Sj = ∪ a−j ([−β, −α] ∪ [α, β]) = R+ ∪ R− , j∈Z the above properties of Sj , j ∈ Z, show that every nonzero ω ∈ R lies in at most (L + 1) Sj ’s. Fix ω ∈ R \ {0}. Assume that ω lies in the sets Sjω , Sjω +1 , . . . , Sjω +L . Then ∑ j∈Z ∑ jω +L −j |φj (a ω)| = 2 |φj (a−j ω)|2 ≤ (L + 1)D2 j=jω by (3.18). If ω lies in less than (L + 1) Sj ’s, then the estimate will be smaller, and therefore still be bounded by (L + 1)D2 . Since ω ∈ Sjω , it also follows from (3.18) 3.3. NONSTATIONARY WAVELET FRAMES that ∑ 67 |φj (a−j ω)|2 ≥ |ψjω (a−jω ω)|2 ≥ C 2 . j∈Z j Thus, by Theorem 3.1.1, {a 2 φj (aj · −kb)}j,k∈Z forms a frame for L2 (R). Once again, we can construct an example similar in idea to the one shown after the proof of Theorem 3.2.1. This time, however, we note that I = [−β, −α]∪[α, β]. Given α = 1 and β = 2, for every j ∈ Z, let  1 + e−|ω|j2 , φj (ω) :=  0, 1 ≤ |ω| ≤ 2, otherwise. It can then be shown that |φj (ω)|2 is bounded above and below by 1 + 1 e and 1 respectively on [−2, −1] ∪ [1, 2]. Like in Theorem 3.2.1, (3.18) does not allow φj to be continuous. We now consider what happens if we take φj (ω) = ψ(ω), j ∈ Z, and assume ψ to be continuous. Theorem 3.3.2. For I := [−β, −α] ∪ [α, β] where 0 < α < β, consider a continuous function ψ ∈ L2 (R) which has support in I and is strictly positive in the interior of I. Suppose that 0 < b < such that √ L+1 2π |I| = πβ . Assume that there exists an L ≥ 1 β 0. First choose ϵ such that { 0 < ϵ < min β − α β − aα , 2 a+1 } , and define cj := aj (α + ϵ) and dj := aj (β − ϵ) for j ∈ Z. Note that β − aα > 0 √ because a < L αβ ≤ αβ . With this choice of ϵ, we can ensure that the intervals (cj , dj ) make sense and that limj→−∞ cj = 0 and limj→∞ dj = ∞. In addition, there will not be any holes in between the intervals on (0, ∞), since ϵ < β−aα a+1 ensures that cj+1 < dj for every j ∈ Z. Fixing ω > 0, there exists jω ∈ Z such that ( ) ω ∈ (cjω , djω ) = ajω (α + ϵ), ajω (β − ϵ) . This in turn means that a−jω ω ∈ (α + ϵ, β − ϵ) ⊂ [α, β]. So, we have the conclusion that ∑ j∈Z ( ) |ψ(a−j ω)|2 ≥ |ψ a−jω ω |2 ≥ inf y∈[α+ϵ,β−ϵ] |ψ(y)|2 > 0 since ψ is strictly positive in the interior of [α, β]. A similar argument establishes the lower bound for ω < 0, and this completes the proof. 3.4. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES 3.4 69 Nonstationary Time-Frequency-Scale Frames In Section 3.2, we considered Gabor frames, which are frames with time and frequency parameters. In Section 3.3, we looked at wavelet frames, which contain both time and scale parameters. Here, the construction of a more general type of nonstationary frames is discussed, which uses not two, but three parameters, namely the time, frequency and scale parameters. Our goal in this section is to construct nonstationary wavelet frames with various desirable properties. In particular, we want the frame elements to be realvalued, symmetric and smooth, and that their Fourier transforms are compactly supported and continuous. Such a formulation leads to the utilization of functions based on the timefrequency-scale parameters defined as in (1.14). Functions of the form (1.13) are not used here as a similar modification will not result in real-valued frame elements with symmetry. Theorem 3.4.1. Let a > 1. Consider a bounded continuous function φ ∈ L2 (R) whose support is the interval [α, β] with 0 < aα < β, and is strictly positive in the interior of the interval. Define a sequence of nonnegative numbers {γj }j∈Z for which there exist K > 0, 0 < λ < 1 and L ∈ N such that for all j ∈ Z, 0 ≤ γj ≤ K, (3.20) aγj+1 − γj < λ (β − aα) , (3.21) aL+M γj+L+M − γj > β − aL+M α, In addition, assume that 0 < b < π . β+K M ≥ 1. (3.22) Define ψj (t) := eiγj t φ(t) + e−iγj t φ(−t), t ∈ R. j Then {a 2 ψj (aj · −kb)}j,k∈Z forms a frame for L2 (R). Proof. Note that for every j ∈ Z, we have chosen ψj (t) := eiγj t φ(t) + e−iγj t φ(−t), 70 CHAPTER 3. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES where supp φ = [α, β]. Observe that ∫ ψj (ω) = ∞ −iωt e −∞ ∫ ψj (t)dt = ∞ −i(ω−γj )t e −∞ = φ(ω − γj ) + φ(−ω − γj ). ∫ ∞ φ(t)dt + e−i(ω+γj )t φ(−t)dt −∞ Our first aim is to find the support of ψj , j ∈ Z. Fixing j ∈ Z, since supp φ = [α, β], ψj (ω) ̸= 0 if and only if α ≤ ω − γj ≤ β or α ≤ −ω − γj ≤ β, which means that α ≤ ω ≤ β + K or −(β − K) ≤ ω ≤ −α by (3.20). Thus, supp ψj ⊂ (−β − K, −α)∪(α, β +K) for every j ∈ Z. Since we have chosen b < ∑ we are only interested in the function j∈Z |ψj (a−j ω)|2 . π β+K = 2π , β+K−(−β−K) Continuing the proof, we also see that ψj (a−j ω) = φ(a−j ω − γj ) + φ(−a−j ω − γj ). From the fact that supp φ = [α, β], we see that ψj (a−j ω) ̸= 0 if and only if α ≤ a−j ω − γj ≤ β or α ≤ −a−j ω − γj ≤ β. This in turn works out to be aj (α + γj ) ≤ ω ≤ aj (β + γj ) or aj (−β − γj ) ≤ ω ≤ aj (−α − γj ). We then define Sj+ := [aj (α + γj ), aj (β + γj )] and Sj− := [aj (−β − γj ), aj (−α − γj )]. It is easy to verify that these two classes of support will never intersect each other in the frequency line. In this proof, we will only be concentrating on the case where ω > 0, since arguments for the negative frequency case can be reconstructed fully from the arguments for the positive frequency case. Thus we shall look closely at the overlaps of the supports Sj+ . Observe that (3.22) means that for each j ∈ Z, there are at most L overlaps of such sets Sj+ from its right. For this to occur, it suffices to have aj+L+M (α + γj+L+M ) > aj (β + γj ) for all M ≥ 1, which is aL+M γj+L+M − γj > β − aL+M α for all M ≥ 1. This is what we assumed in (3.22). Fix ω > 0. Then the above implies that ω lies in at most L + 1 sets of Sj . We let these sets be Sjω , Sjω +1 , Sjω +2 , . . . , Sjω +L . Recall that we are interested in 3.4. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES 71 the function ∑ |ψj (a−j ω)|2 = j∈Z ∑ |φ(a−j ω − γj ) + φ(−a−j ω − γj )|2 j∈Z = ∑ ∑ jω +L −j |φ(a ω − γj )| = 2 j=jω j∈Z ∑ |φ(a−j ω − γj )|2 jω +L ≤ B 2 = (L + 1)B 2 < ∞ j=jω for some B > 0 since φ is a bounded function. If ω lies in less than (L + 1) Sj ’s, then the bound will be smaller than or equal to (L + 1)B 2 . We now inspect the lower bound of the function ∑ j∈Z |ψj (a−j ω)|2 , ω > 0. We start off by showing that { τ = inf j∈Z β − aα − aγj+1 + γj a+1 } > 0. (3.23) By (3.21), we have aγj+1 − γj < λ(β − aα), and so β − aα − aγj+1 + γj = (1 − λ)(β − aα) + λ(β − aα) − (aγj+1 − γj ) > (1 − λ)(β − aα). This implies that τ as defined in (3.23) is at least (1−λ)(β−aα) , a+1 which is positive because β > aα. Now, choose ϵ > 0 such that { 0 < ϵ < min β−α ,τ 2 } . Using this choice of ϵ, we then let cj := aj (α + γj + ϵ) and dj := aj (β + γj − ϵ). For the set (cj , dj ) to make sense, we need cj < dj . In other words, we need aj (α + γj + ϵ) < aj (β + γj − ϵ) and thus ϵ < β−α . 2 We also note that ( ) β−α 0 ≤ lim cj = lim a (α + γj + ϵ) ≤ lim a α + K + =0 j→−∞ j→−∞ j→−∞ 2 j j 72 CHAPTER 3. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES and ( ) β−α = ∞. lim dj = lim a (β + γj − ϵ) ≥ lim a β − j→∞ j→∞ j→∞ 2 j j Next, we show that there are no holes in the intervals (cj , dj ), j ∈ Z. For this to happen, for all j ∈ Z, cj+1 < dj . As a result, we need to ensure that the inequality aj+1 (α + γj+1 + ϵ) < aj (β + γj − ϵ) and thus ϵ< β − aα − aγj+1 + γj a+1 holds for all j ∈ Z, which is precisely how we have chosen our ϵ. Consequently ∪ j∈Z (cj , dj ) = (0, ∞). Fixing ω > 0, there exists jω ∈ Z such that ω lies in the interval (cjω , djω ) = (ajω (α + γjω + ϵ), ajω (β + γjω − ϵ)). This in turn means that a−jω ω − γjω ∈ (α + ϵ, β − ϵ) ⊂ [α, β]. So, we have the conclusion that ∑ ( ) |ψ(a−j ω)|2 ≥ |φ a−jω ω − γjω |2 ≥ j∈Z inf y∈[α+ϵ,β−ϵ] |φ(y)|2 > 0 since φ is continuous and strictly positive on the interval (α, β). What Theorem 3.4.1 tells us is that if we have a sequence of nonnegative numbers {γj }j∈Z which satisfies the equations (3.20) to (3.22), and all the assumptions stated in the theorem are fulfilled, then we are able to construct frames of the form as prescribed. So, given such a sequence, all we need to do is to check the conditions (3.20) to (3.22) to come to a suitable conclusion. With that in mind, we now show some examples where we define specific sequences of {γj }j∈Z . Proposition 3.4.2. Let a > 1 and 0 < aα < β < aL+1 α for some L ∈ N. Suppose 3.4. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES 73 that {γj }j∈Z is a constant sequence given by γj := c, j ∈ Z, where 0 ≤ c ≤ λF (1) with 0 < λ < 1 and F (x) := β − ax α , ax − 1 x ∈ R. Then the conditions (3.20) to (3.22) in Theorem 3.4.1 are satisfied. Proof. We see that (3.20) is automatically satisfied due to the structure of the sequence {γj }j∈Z . Next, we look at (3.21). Substituting the fact that γj = γj+1 = c, what we need is ac − c < λ(β − aα), which is c < λ(β−aα) a−1 = λF (1). Thus (3.21) is satisfied for all j ∈ Z. Now we consider (3.22). For j ∈ N and M ≥ 1, γj = γj+L+M = c and (3.22) becomes aL+M c − c > β − aL+M α. This gives c > β−aL+M α aL+M −1 = F (L + M ) for all M ≥ 1. By the strictly decreasing property of F , it suffices to check c > F (L + 1). Since β < aL+1 α, we see that F (L + 1) < 0. As c ≥ 0, we then conclude that (3.22) holds for all j ∈ Z. Remark 3.4.3. Observe that taking γj as a constant gives wavelet frames which are similar in structure to those we saw in Section 3.3. Taking this constant to j be 0, we have ψ(t) := φ(t) + φ(−t), and {a 2 ψ(aj · −kb)}j,k∈Z forms a frame for L2 (R). So far, we have seen only constant sequences. But what happens when we consider mixed sequences where the negative indexed terms are 0 and the nonnegative indexed terms take some positive value? There is a “transition” state that we need to take care of when we are considering the equations (3.21) and (3.22). We decompose (3.21) into three parts, namely when j ≥ 0, j ≤ −2 and j = −1. In the first case, since the indices j +1 and j are all nonnegative, we just substitute 74 CHAPTER 3. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES in the explicit formula and check whether (3.21) is fulfilled. For the second case, both the indices have become negative and thus γj+1 = γj = 0. In this case, (3.21) always holds as aα < β. For the last case of j = −1, (3.21) becomes aγ0 < λ(β − aα) as γ−1 = 0. So the assumption γ0 < λ(β − aα) a will ensure (3.21) for j = −1. We now inspect (3.22). Similar to checking (3.21), we partition the index j into three parts, namely when j ≥ 0, −L − 1 ≤ j ≤ −1 and j ≤ −L − 2. For the first case, both the indices j + L + M and j are nonnegative, so we can substitute the explicit formula of γj and check the inequality. In the second case, we fix j = −P for 1 ≤ P ≤ L + 1 and check the inequality aL+M γL+M −P > β − aL+M α (3.24) for M ≥ 1, given that γ−P = 0. Suppose that we assume β < aL+1 α in the application of Theorem 3.4.1. Since β − aL+M α ≤ β − aL+1 α < 0 for all M ≥ 1 and γL+M −P > 0, we combine them to obtain γL+M −P > 0 > β − aL+M α , aL+M which is exactly (3.24). Lastly, we consider the third case. Fixing j = −P for P ≥ L + 2, (3.22) is the same as 0 > β − aL+M α, aL+M γM −(P −L) > β − aL+M α, 1 ≤ M ≤ P − (L + 1), (3.25) M ≥ P − L, (3.26) again due to γj = 0 for j < 0. Since β < aL+1 α, β < aL+M α for M ≥ 1. Observe 3.4. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES 75 that as M − (P − L) ≥ 0 for all M ≥ P − L, we have γM −(P −L) > 0. Hence, γM −(P −L) > 0 > β − aL+M α , aL+M which means that both (3.25) and (3.26) are fulfilled. In short, we have shown that for (3.22), both the cases when −L − 1 ≤ j ≤ −1 and j ≤ −L − 2 are satisfied with the assumption β < aL+1 α. In the next two propositions, we identify two nonconstant sequences of {γj }j∈Z that satisfy the conditions of Theorem 3.1.1. Proposition 3.4.4. Let a > 1 and 0 < aα < β < aL+1 α for some L ∈ N. Define the sequence {γj }j∈Z by γj :=  C, j ≥ 0,  0, j < 0, aj with ( 0 < γ0 = C < λ β −α a ) and 0 < λ < 1. Then the conditions (3.20) to (3.22) in Theorem 3.4.1 are satisfied. Proof. By the derivation of {γj }j∈Z , (3.20) clearly holds. From our preceding discussion, recall that we have three cases to consider for (3.21) and (3.22). When j ≥ 0, we see that ( aγj+1 − γj = a C aj+1 ) − C = 0 < λ (β − aα) . aj When j ≤ −2, (3.21) is immediately satisfied. To check the third case of j = −1, it is enough to ensure that γ0 = C < λ(β − aα) , a which we have already assumed. Thus, (3.21) is satisfied. We have already shown that under the assumption β < aL+1 α, the second and 76 CHAPTER 3. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES third cases of (3.22) are satisfied automatically. It remains to establish the case where j ≥ 0, which is ( a L+M γj+L+M − γj = a ) C L+M aj+L+M − C = 0 > β − aL+M α. j a Since we have assumed that β < aL+1 α ≤ aL+M α, this is also satisfied. Hence the proof is complete. Proposition 3.4.5. Let a > 1 and 0 < aα < β < aL+1 α for some L ∈ N. Define the sequence {γj }j∈Z by γj+1   1 γ + δ, j := a  0, j ≥ 0, j < 0, with ) } { ( β a −α , δ 0 < γ0 < min λ a a−1 and ( ) λ(a − 1) δ ∈ 0, F (1) a where F (x) := β−ax α . ax −1 Then the conditions (3.20) to (3.22) in Theorem 3.4.1 are satisfied. Proof. We claim that for j ≥ 0, if γj < a δ, a−1 then γj+1 < a δ a−1 with our construc- tion of {γj }j∈Z . Indeed, note that γj+1 Since γ0 < a δ, a−1 1 1 = γj + δ < a a ( ) a a δ +δ = δ. a−1 a−1 this shows that γj is bounded above by a δ a−1 for j ≥ 0. Observe that {γj }j≥0 is a strictly increasing sequence. This is because 1 a−1 a−1 γj+1 − γj = γj + δ − γj = − γj + δ > − a a a ( ) a δ + δ = 0. a−1 3.4. NONSTATIONARY TIME-FREQUENCY-SCALE FRAMES 77 We first look at (3.21). For the first case of j ≥ 0, we see that ( aγj+1 − γj = aδ < (a − 1) λ(β − aα) a−1 ) = λ(β − aα). For the second case, when j ≤ −2, we see that aγj+1 − γj = 0 < β − aα from the assumption that β > aα. For the last case of j = −1, (3.21) is provided by the assumption ( γ0 < λ ) β −α . a Since {γj }j≥0 is a strictly increasing sequence, to verify (3.22), it suffices to check the inequality aL+1 γj+L+1 − γj > β − aL+1 α. We have already shown in the discussion before Proposition 3.4.4 that with the assumption β < aL+1 α, we only have to consider what happens when j ≥ 0. Indeed, ( L+1 a γj+L+1 − γj = a L+1 L ∑ 1 aℓ ℓ=0 ) ( δ=a aL+1 − 1 a−1 ) δ > 0 > β − aL+1 α. Hence (3.22) holds for all j ∈ Z. 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DISCRETE TIME- FREQUENCY- SCALE TRANSFORMS 20 the parameters one by one in a strategic and efficient way, until we arrive at the fully discrete case of frames 2.1 Continuous Transforms First, we introduce the time- frequency- scale transform which is essentially the continuous wavelet transform incorporating a modulation term Definition 2.1.1 Let ψ ∈ L2 (R) be a mother wavelet The time- frequency scale transform... the function ψb;a;γ and the role it plays γ in time- frequency- scale transforms While the function ψb;a is less flexible in the time- frequency window, it fits nicely in the construction of frames with desirable properties like being real-valued and symmetric, which we will be discussing in Section 3.4 Chapter 2 From Continuous to Discrete Time- Frequency- Scale Transforms Here we revisit the properties and... Semi-Discrete Transforms In this section, we discretize a strategically, similar to the way we described in Chapter 1 We first define what a normalized time- frequency- scale transform is, and then introduce an a-adic wavelet for the purpose of signal reconstruction By taking the dilation factor to be a−j , j ∈ Z, for some a ≥ 1 in (2.1), the resulting transform, known as the “normalized” time- frequency- scale. .. is as defined in (1.6), 1.2 WAVELET TRANSFORMS 7 which means that 2 f (x) = Cψ ∫ ∞ ∫ −∞ ∞ [(Wψ f )(b, a)] ψb;a (x) 0 da db a2 weakly 1.2.2 Semi-Discrete Transforms In the previous sub-section, we worked with the premise that the frequency ω, and thus the scale a, can take any value in the frequency axis In this sub-section, we begin to discretize, or partition this frequency axis into disjoint intervals... )(b, a, γ) = |a| ∫ ( ∞ f (t)eiγt ψ −∞ ) t−b dt = ⟨f, ψb;a;γ ⟩ a (2.2) 2.1 CONTINUOUS TRANSFORMS 21 Comparing Proposition 2.1.2 with Proposition 1.2.3, the continuous wavelet transform is (Wψ f )(b, a) = ⟨f, ψb;a ⟩ whereas the continuous time- frequency- scale transform is (Vψ f )(b, a, γ) = ⟨f, ψb;a;γ ⟩ In fact, the two transforms are very closely related in the sense that (Vψ f )(b, a, 0) = (Wψ f )(b,... (x) = Cψ ∫ ∞ −∞ ∫ ∞ −∞ [(Vψ f )(b, a, γ)]ψb;a;γ (x) da db a2 weakly Proof As noted in (2.3), for f ∈ L2 (R), (Vψ f )(b, a, γ) = (Wψ (f (·)e−iγ· )) (b, a) CHAPTER 2 FROM CONTINUOUS TO DISCRETE TIME- FREQUENCY- SCALE TRANSFORMS 22 Using this information, it follows from Theorem 1.2.4 that for every f, g ∈ L2 (R), ∫ ∫ [ ] da (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) 2 db a −∞ ∫−∞ ∞ ∫ ∞ ( ) da = Wψ (f (·)e−iγ· ) (b,... − x)⟩ = Cψ ∫ ∞ −∞ ∫ ∞ −∞ [(Vψ f )(b, a, γ)⟨ψb;a;γ , gα (· − x)⟩] da db a2 (2.6) 2.1 CONTINUOUS TRANSFORMS 23 Since ∫ lim+ ⟨f, gα (· − x)⟩ = lim+ α→0 α→0 ∞ −∞ f (t)gα (t − x)dt = lim+ (f ∗ gα )(x) = f (x), α→0 the result follows So far, we have assumed that the parameter a in the continuous time- frequencyscale transform in (2.1) takes all nonzero real values However in the investigation of real-life... these integrals are well defined With the necessary tools on hand, we are ready to readdress the theorem, but concentrating only on the positive scale Theorem 2.1.5 Let ψ ∈ L2 (R) be a mother wavelet which satisfies (1.6) and defines a continuous time- frequency- scale transform Vψ Then for any fixed γ ∈ R, ∫ ∞ −∞ ∫ 0 ∞ [ ] da 1 (Vψ f )(b, a, γ)(Vψ g)(b, a, γ) 2 db = Cψ ⟨f, g⟩ a 2 (2.7) for all f, g ∈ L2... ⟨f, g⟩ = Cψ ∫ ∞ −∞ ∫ ∞ [(Vψ f )(b, a, γ)⟨ψb;a;γ , g⟩] 0 da db a2 (2.8) for all g ∈ L2 (R), where ψb;a;γ is defined by (2.2) and Cψ by (1.6), which means CHAPTER 2 FROM CONTINUOUS TO DISCRETE TIME- FREQUENCY- SCALE TRANSFORMS 24 that 2 f (x) = Cψ ∫ ∞ −∞ ∫ ∞ [(Vψ f )(b, a, γ)] ψb;a;γ (x) 0 da db a2 weakly Proof Recall from (2.3) that for f ∈ L2 (R), (Vψ f )(b, a, γ) = (Wψ (f (·)e−iγ· ))(b, a) So, for all... |a| Adopting the idea of modulation allows us to vary the modulation term γ to suit our needs in time- frequency analysis For example, if we have a signal with very high frequencies that we would like to analyze with a small frequency window (given by a large value of |a|), we can adjust the center of the frequency 18 CHAPTER 1 PRELIMINARIES window by choosing a suitable value of γ The γ term which appears ... Continuous to Discrete Time- Frequency- Scale Transforms 19 2.1 Continuous Transforms 20 2.2 Semi-Discrete Transforms 24 2.3 Discrete Transforms: Frames... wavelet which defines a continuous time- frequency- scale transform Vψ Then for any σ ∈ L1 (R) such that σ(γ) > 0, CHAPTER FROM CONTINUOUS TO DISCRETE TIME- FREQUENCY- SCALE TRANSFORMS 32 γ ∈ R, ∫ ∞ ∫ −∞... signal function f ∈ L2 (R) from its time- frequency- scale transform values (Vψ f )(b, a−j , − aα−j + C) CHAPTER FROM CONTINUOUS TO DISCRETE TIME- FREQUENCY- SCALE TRANSFORMS 44 Once again, we expect