Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2012 Frames for Hilbert spaces and an application to signal processing Kinney Thompson Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Physical Sciences and Mathematics Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/2735 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. Abstract Frames for Hilbert spaces and an application to signal processing Kinney Thompson The goal of this paper will be to study how frame theory is applied within the field of signal processing. A frame is a redundant (i.e. not linearly independent) coordinate system for a vector space that satises a certain Parseval-type norm inequality. Frames provide a means for transmitting data and, when a certain about of loss is anticipated, their redundancy allows for better signal reconstruction. We will start with the basics of frame theory, give examples of frames and an application that illustrates how this redundancy can be exploited to achieve better signal reconstruction. We also include an introduction to the theory of frames in infinite dimensional Hilbert spaces as well as an interesting example. Thesis Director: Dr. Kevin Beanland Frames for Hilbert spaces and an application to signal processing by Kinney Thompson Bachelor of Science Virginia Polytechnic Institute and State University Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mathematics Virginia Commonwealth University 2012 Major Professor Chair, Examining Committee Committee Member Committee Member Committee Member Dean of The Graduate School Dedication This manuscript is dedicated to my family and friends that have supported me as I took this journey with a special dedication to my friend David that left this world far too soon. ii Acknowledgements A special thanks to Dr. Kevin Beanland for all of his help and support with this work. iii Abstract The goal of this paper will be to study how frame theory is applied within the field of signal processing. A frame is a redundant (i.e. not linearly independent) coordinate system for a vector space that satises a certain Parseval-type norm inequality. Frames provide a means for transmitting data and, when a certain about of loss is anticipated, their redundancy allows for better signal reconstruction. We will start with the basics of frame theory, give examples of frames and an application that illustrates how this redundancy can be exploited to achieve better signal reconstruction. We also include an introduction to the theory of frames in infinite dimensional Hilbert spaces as well as an interesting example. iv Contents Dedication ii Acknowledgements iii Abstract iv Chapter 1 Introduction 1 Chapter 2 Fundamentals of Frame Theory 2 Chapter 3 An Infinite Dimensional Example 15 Chapter 4 Application of Frames To Discretized Data 24 Bibliography 29 Appendices 31 Chapter A C++ Source Code 31 v Chapter 1 Introduction This paper consists of three chapters. In chapter 2 we lay the foundation for the theory of frames in finite dimensions. The tools necessary to construct a frame will be provided; starting from a complex scalar and all the way up to a finite dimensional frame and its associated operators. Necessary proofs and definitions will be provided along the way to allow for more clarity. Reference sources will be provided for any information outside the scope of this paper's discussion. In chapter 3 of this paper we will introduce frame theory for an infinite dimensional Hilbert space. The main result in this section is the construction of an infinite dimen- sional frame such that no subset forms a Schauder basis for the space. This result is somewhat technical and it will broken into several pieces to clarify the exposition. Following the previous two chapters one should have a good understanding of the basics of frame theory. In chapter 4 we discuss an application of frame theory to discrete data and the effects of quantization error compounded by the machine's word size limitations and provide conclusions reached. The frames used in the simulation are taken from chapter 2. 1 Chapter 2 Fundamentals of Frame Theory Let us begin by defining what is our fundamental scalar element. Let p C = fx + iy : x; y 2 R and i = −1g p Here R denotes the set of all real numbers. Let jzj = x2 + y2. A complex number can also be expressed in polar coordinate form as z = reiθ where r ≥ 0 and θ is expressed in radians within the range 0 ≤ θ ≤ 2π. The relationship is given by Euler's identity eiθ = cosθ + isinθ: Note that jzj = jreiθj = r. jeiθj = 1 for all values of θ. The complex conjugate for z = x + iy is defined as z = x − iy. Thus one can see that jzj2 = zz. Note that if we look at the polar coordinate form of complex numbers with r = 1 we can naturally define what are called the nth roots of unity. These vectors have some nice properties that will be noticed later within the paper. Definition 1. Let n in N, z in C is a n-th root of unity if zn = 1: There are n such complex numbers. Remark: For each n in N the complex numbers i 2πk fe n : k = 0; :::; n − 1g are the n-th roots of unity. 2 For n in N let n n C = f(ai)i=1 : ai 2 Cg: Definition 2. Let n in N and define a function k · k : Cn ! [0; 1) by v u n n uX 2 k(ai)i=1k = t jaij : i=1 n n Let Hn denote C equipped with this norm function. Let x = (ai)i=1 and y = (bi)i=1 in n n C . Then x + y = (ai + bi)i=1. This is to say, addition between vectors in this space n n is defined coordinate wise. Now for λ in C, λx = λ(ai)i=1 = (λai)i=1. [6, Page 8] Remark: For each n in N the function k · k satisfies the following n (1) k(ai)i=1k = 0 if and only if ai = 0 for all i = 1; :::; n. n n n n (2) k(ai)i=1 + (bi)i=1k ≤ k(ai)i=1k + k(bi)i=1k which can also be written as v v v u n u n u n uX 2 uX 2 uX 2 t jai + bij ≤ t jaij + t jbij : i=1 i=1 i=1 n n (3) kλ(ai)i=1k = jλjk(ai)i=1k: Now we define theinner product function for Hn. Define the following function h·; ·i : Hn × Hn ! C by n n n X h(ai)i=1; (bi)i=1i = aibi i=1 Note that for x in Hn n n 2 X 2 X kxk = jaij = aiai = hx; xi: i=1 i=1 3 The following are all properties of the inner product, h·; ·i. Let x; y; z be elements in Hn and λ in C. (1) hx; yi = hy; xi. (2) hx + y; zi = hx; zi + hy; zi and hx; y + zi = hx; yi + hx; zi. (3) hx; λyi = λhx; yi and hλx, yi = λhx; yi. Definition 3. Let A ⊂ Hn then n X span(A) = f cixi : n 2 N; ci 2 C; xi 2 A for i = 1; : : : ng: i=1 n n Definition 4. A sequence fxkgk=1 is a basis for C if n n Pn n (a) C = spanfxkgk=1 = f i=1 aixi : ai 2 C g n Pn (b) fxkgk=1 is linearly independent. In other words, if k=1 akxk = 0 for some n scalars fakgk=1 then ak = 0 for all k = 1; :::; n: n n n Definition 5. A basis fxkgk=1 is an orthonormal basis for C if fxkgk=1 is a basis, and 8 < 1 k = j hxk; xji = δk;j = : 0 k 6= j: Define ek as the vector with a 1 in the k-th position and 0 in all other places. This n is the canonical orthonormal basis. For each n 2 N,(ek)k=1 is called the unit vector basis of Cn and will be used throughout this paper. Example 1. Another simple example for C3 is e1 = (2; 0; −1) e2 = (0; −1; 0) e3 = (2; 0; 4) One can quickly see that these three vectors are orthonormal and therefore linearly independent. 4 Orthonormal bases have the following reconstruction property. Let x in Hn. Then n X x = hx; xkixk: k=1 n In the case of the canonical orthonormal basis for Hn x = (ak)k=1, hx; eki = ak. The next proposition is a basic fact from linear algebra. We omit the proof. Proposition 1. Let n in N and M be a subspace of Cn. Then there is a sequence d fxkgk=1 in M that forms an orthonormal basis for M where d ≤ n. n Remark: Let fxkgk=1 be orthonormal basis, such that kxkk = 1 for all k = 1; : : : ; n. Then for all x in Hn n X 2 2 jhx; xnij = kxk : k=1 This is called Parseval's Identity. n Proof. Let fxkgk=1 be an orthonormal basis of Hn and x in Hn. kxk2 = jhx; xij n n X X = jh hx; xiixi; hx; xjixjij i=1 j=1 n n X X = hx; xjihxi; hx; xjixji i=1 j=1 (1) n n X X = hx; xii hx; xjihxi; xji i=1 j=1 n X = hx; xiihx; xii i=1 n X 2 = jhx; xiij i=1 Now we introduce the definition of a frame.