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EIGENVALUE ESTIMATES AND SAMPLING
FOR TIME-FREQUENCY LOCALIZATION OPERATORS
BY
SCOTT IZU, B.S., M.S.
A dissertation submitted to the Graduate School
in partial fulfillment of the requirements
for the degree
Doctor of Philosophy
Major Subject: Mathematics
New Mexico State University
Las Cruces New Mexico
July 2009
“Eigenvalue Estimates and Sampling for Time-Frequency Localization Opera-
tors,” a dissertation prepared by Scott Izu in partial fulfillment of the require-
ments for the degree, Doctor of Philosophy, has been approved and accepted by
the following:
Linda LaceyInterim Dean of the Graduate School
Joe LakeyChair of the Examining Committee
July 2009
Committee in charge:
Dr. Joe Lakey, Chair
Dr. Caroline Sweezy
Dr. Tiziana Giorgi
Dr. Chuck Creusere
ii
DEDICATION
I dedicate this work to my wonderful wife Suleica and my father Allen.
iii
ACKNOWLEDGMENTS
I would like to thank my wife, Suleica, for the countless hours of support and
encouragement. She has helped me through many stuggles and been supportive
through it all.
I would like to thank my father, Allen, for the mentoring and guidance through
difficult times. He always had something positive to say and helped me realize
that I was actually growing through each struggle.
I would like to thank my advisor, Joe Lakey. He was willing to change the
traditional roles a student and an advisor play, adapting to my way of communica-
tion and learning. On several occasions, he was willing to debate topics, providing
clarity to my views. He was willing to venture into different research projects, re-
sulting in many valuble experiences and an excellent introduction to the world of
research. He has spent a great deal of time and effort in helping me to succeed
and I appreciate all the work that he has done.
I would like to thank the following for their encouragement and devotion to de-
veloping young minds: Tonia Izu, George Schuttinger, Rupinder Sekhon, Donald
Sarason and Stewart McKechnie.
Finally, I would like to thank the following organizations for their support: the
Ford Foundation, the SAGE Scholars Program, the National Science Foundation
and the Accenture Foundation.
iv
VITA
February 21, 1980 Born in San Jose, California
1998-2000 A.A., De Anza Community CollegeCupertino, California
2001-2003 B.S., University of CaliforniaBerkeley, California
2003-2005 M.S., New Mexico State UniversityLas Cruces, New Mexico
2006-2007 Teaching Assistant, Department of Mathematics,New Mexico State University.
Honors and Awards
Ford Foundation Fellow, 2004-2008.
AGEP Fellow, 2003-2006.
Anna Schrufer Kist Scholarship, 2005.
NSF Summer Research Grant, 2004.
NSF Honorable Mention, 2004.
Highest Honors, 2003, UC Berkeley.
Highest Distinction, 2003, UC Berkeley.
Percy Lionel Davis Award for Excellence in Mathematics, 2003, UC Berkeley.
SAGE Scholar, 2002-2003.
Accenture Foundation Minority Scholarship, 2002-2003.
Accessibility Award, 2003, USDA.
A+ Certification, 2001.
v
PUBLICATIONS
Joseph Lakey and Scott Izu. Time-Frequency Localization and Sampling of
Multiband Signals. Acta Applicandae Mathematicae.
Joseph Lakey, Mike Coombs, Scott Izu and Chris Weaver. On Models for
Coordination of Activity and It’s Disruption. ARO DAAD19-02-1-0211.
FIELD OF STUDY
Major Field: Harmonic Analysis
vi
ABSTRACT
EIGENVALUE ESTIMATES AND SAMPLING
FOR TIME-FREQUENCY LOCALIZATION OPERATORS
BY
SCOTT IZU, B.S., M.S.
Doctor of Philosophy
New Mexico State University
Las Cruces, New Mexico, 2009
Dr. Joe Lakey
In this work, we build on the classical time-frequency analysis tools developed
by Landau, Slepian and Pollack. We focus on prolate spheroidal wave functions
as a tool to analyze time-frequency localized signals. Recent developments in pe-
riodic nonuniform sampling by Venkataramani and Bresler allow us to develop
several sampling formulas and discrete methods for applying time-frequency lo-
calization concepts to multiband signals. Using concepts introduced by Landau,
we explore the behavior of the eigenvalues for time-frequency localization oper-
ators. By extending the work of Khare and George, we are able to give several
vii
alternatives for calculating these eigenvalues. We also include an error analysis
for these calculations.
Our presentation includes a brief history and overview of several concepts
related to time-frequency analysis and sampling. We develop distribution theory
and functional theoretic foundations in order to provide a rigorous basis for our use
of the Poisson summation formula, impulse sampling, synthesis/analysis operators
and Bessel sequences.
Our overview emphasizes possible applications in digital signal processing and
includes several examples. We also include a small code library to demonstrate
how to modularize code based on theory.
viii
CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 THE FOURIER TRANSFORM . . . . . . . . . . . . . . . . . . . 5
2.1 Operator Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Tempered Distributions . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 The Poisson Summation Formula . . . . . . . . . . . . . . . . . . 14
2.4 Periodic Tempered Distributions and Support . . . . . . . . . . . 19
2.5 Fourier Series I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Fourier Series II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Discrete Time Fourier Transform I . . . . . . . . . . . . . . . . . 36
2.8 Discrete Fourier Transform I . . . . . . . . . . . . . . . . . . . . . 38
2.9 Discrete Fourier Transform II . . . . . . . . . . . . . . . . . . . . 47
2.10 Discrete Fourier Transform III . . . . . . . . . . . . . . . . . . . . 55
3 GENERALIZED BASES . . . . . . . . . . . . . . . . . . . . . . . 62
3.1 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Bessel Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Complete Bessel Sequences . . . . . . . . . . . . . . . . . . . . . . 74
3.4 ω-Linearly Independent Bessel Sequences . . . . . . . . . . . . . . 75
ix
3.5 Frame Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.7 Riesz Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.8 Riesz Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.9 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.9.1 Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.9.2 Psuedo Inverse Definition . . . . . . . . . . . . . . . . . . 86
3.9.3 Banach Spaces and Pseudo Inverses . . . . . . . . . . . . . 86
3.9.4 Frame Sequences and Frames . . . . . . . . . . . . . . . . 88
3.9.5 Riesz Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.9.6 Operators associated with Frames . . . . . . . . . . . . . . 89
3.9.7 Duality Principles . . . . . . . . . . . . . . . . . . . . . . . 90
3.10 Wavelet Based Noise Cancellation Algorithm . . . . . . . . . . . . 90
4 SAMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.1 Poisson Summation Formula for L2 . . . . . . . . . . . . . . . . . 98
4.2 Basic Shannon Sampling . . . . . . . . . . . . . . . . . . . . . . . 100
4.3 Shannon Sampling with a Tiling Set . . . . . . . . . . . . . . . . 101
4.4 General Shannon Sampling Formulas . . . . . . . . . . . . . . . . 104
4.5 Periodic Nonuniform Sampling with Only Complete Aliasing . . . 106
4.6 Periodic Nonuniform Sampling . . . . . . . . . . . . . . . . . . . . 110
4.7 More on ΩK-Tiling Sets . . . . . . . . . . . . . . . . . . . . . . . 114
x
5 TIME-FREQUENCY LOCALIZATION OPERATORS . . . . . . 117
5.1 Prolate Spheriodal Wave Functions . . . . . . . . . . . . . . . . . 117
5.2 Related Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3 PSWF Sampling Formula . . . . . . . . . . . . . . . . . . . . . . 124
5.4 Eigenvalues Greater Than 1/2 . . . . . . . . . . . . . . . . . . . . 127
5.5 When No Significant Eigenvalues Exist . . . . . . . . . . . . . . . 133
5.6 Eigenvalue Calculations . . . . . . . . . . . . . . . . . . . . . . . 137
5.7 Matrix Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 140
6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1 Computation Scripts . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.2 Fourier Transform Digital Signal Representations . . . . . . . . . 164
7.3 Fourier Transform Library . . . . . . . . . . . . . . . . . . . . . . 168
7.4 Wavelet Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.5 Eigenvalue Calculations . . . . . . . . . . . . . . . . . . . . . . . 180
7.6 Plots and Configuration . . . . . . . . . . . . . . . . . . . . . . . 182
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
xi
LIST OF TABLES
1 Commutation Properties . . . . . . . . . . . . . . . . . . . . . . . 11
2 Fourier Series I Equations . . . . . . . . . . . . . . . . . . . . . . 23
3 Fourier Series II Equations . . . . . . . . . . . . . . . . . . . . . . 30
4 Fourier Series III Equations . . . . . . . . . . . . . . . . . . . . . 30
5 Discrete Time Fourier Transform I Equations . . . . . . . . . . . 36
6 Discrete Time Fourier Transform II Equations . . . . . . . . . . . 37
7 Discrete Fourier Transform I Equations . . . . . . . . . . . . . . . 40
8 Discrete Fourier Transform II Equations . . . . . . . . . . . . . . 49
9 Discrete Fourier Transform III Equations . . . . . . . . . . . . . . 56
10 Characterizing Bessel Sequences . . . . . . . . . . . . . . . . . . . 84
11 Banach and Hilbert Space Notation . . . . . . . . . . . . . . . . . 87
12 Uniform Sampling Example . . . . . . . . . . . . . . . . . . . . . 113
13 Spectral Splicing . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
14 Related Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 123
15 Eigenvalues for Example 5.20 . . . . . . . . . . . . . . . . . . . . 138
16 First Set of Parameters for Example 5.20 . . . . . . . . . . . . . . 139
17 Second Set of Parameters for Example 5.20 . . . . . . . . . . . . . 139
xii
LIST OF FIGURES
1 Plot of ρ from Example 2.2 . . . . . . . . . . . . . . . . . . . . . . 8
2 Plot of ρ from Example 2.4 . . . . . . . . . . . . . . . . . . . . . . 9
3 Shifts of ψ in Example 2.19 . . . . . . . . . . . . . . . . . . . . . 17
4 Plot of ψ from Example 2.19 . . . . . . . . . . . . . . . . . . . . . 18
5 Plot of ψ from Example 2.19 . . . . . . . . . . . . . . . . . . . . . 18
6 Plot of a from Example 2.25 . . . . . . . . . . . . . . . . . . . . . 24
7 Plot of A from Example 2.25 . . . . . . . . . . . . . . . . . . . . . 24
8 Plot of b from Example 2.25 . . . . . . . . . . . . . . . . . . . . . 25
9 Plot of B from Example 2.25 . . . . . . . . . . . . . . . . . . . . . 25
10 Alternate definition for a in Example 2.25 . . . . . . . . . . . . . 26
11 Alternate definition for A in Example 2.25 . . . . . . . . . . . . . 26
12 Plot of ψ from Example 2.27 . . . . . . . . . . . . . . . . . . . . . 32
13 Plot of ψ from Example 2.27 . . . . . . . . . . . . . . . . . . . . . 32
14 Plot of f from Example 2.27 . . . . . . . . . . . . . . . . . . . . . 33
15 Plot of f from Example 2.27 . . . . . . . . . . . . . . . . . . . . . 33
16 Approximation for f in Example 2.27 . . . . . . . . . . . . . . . . 34
17 Approximation for f in Example 2.27 . . . . . . . . . . . . . . . . 34
18 Plot of g0g1(t) from Example 2.28 . . . . . . . . . . . . . . . . . . 35
19 Plot of g0g1 from Example 2.28 . . . . . . . . . . . . . . . . . . . 35
xiii
20 Plot of b from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 42
21 Plot of B from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 42
22 Plot of c from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 43
23 Plot of C from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 43
24 Plot of d from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 44
25 Plot of D from Example 2.30 . . . . . . . . . . . . . . . . . . . . 44
26 Approximation for c from Example 2.30 . . . . . . . . . . . . . . 45
27 Approximation for C from Example 2.30 . . . . . . . . . . . . . . 45
28 Approximation for c from Example 2.30 . . . . . . . . . . . . . . 46
29 Approximation for C from Example 2.30 . . . . . . . . . . . . . . 46
30 Plot of g0 from Example 2.32 . . . . . . . . . . . . . . . . . . . . 51
31 Plot of g0 from Example 2.32 . . . . . . . . . . . . . . . . . . . . 51
32 Plot of g0g1 from Example 2.32 . . . . . . . . . . . . . . . . . . . 52
33 Plot of g0g1 from Example 2.32 . . . . . . . . . . . . . . . . . . . 52
34 Approximation for g0 from Example 2.32 . . . . . . . . . . . . . . 53
35 Approximation for g0 from Example 2.32 . . . . . . . . . . . . . . 53
36 Approximation for g0 from Example 2.32 . . . . . . . . . . . . . . 54
37 Approximation for g0 from Example 2.32 . . . . . . . . . . . . . . 54
38 Plot of f from Example 2.34 . . . . . . . . . . . . . . . . . . . . . 58
39 Plot of f from Example 2.34 . . . . . . . . . . . . . . . . . . . . . 58
40 Plot of samples of f from Example 2.34 . . . . . . . . . . . . . . . 59
xiv
41 Plot of samples of f from Example 2.34 . . . . . . . . . . . . . . . 59
42 Summary of Fourier Transform Examples . . . . . . . . . . . . . . 60
43 Summary of Fourier Transform Examples . . . . . . . . . . . . . . 61
44 3 Stage Wavelet Transform Analysis Filter Bank . . . . . . . . . . 92
45 3 Stage Wavelet Packet Analysis Filter Bank . . . . . . . . . . . . 95
46 LAPP Lightning Pulse Before Denoising . . . . . . . . . . . . . . 96
47 LAPP Lightning Pulse After Denoising . . . . . . . . . . . . . . . 96
48 Wavelet Packet Matrix . . . . . . . . . . . . . . . . . . . . . . . . 97
49 Wavelet Packet Coefficients . . . . . . . . . . . . . . . . . . . . . 97
50 Support for f from Example 4.3 . . . . . . . . . . . . . . . . . . . 101
51 Support for f from Example 4.5 . . . . . . . . . . . . . . . . . . . 103
52 Support for f from Example 4.8 . . . . . . . . . . . . . . . . . . . 110
53 Support for f from Example 4.10 . . . . . . . . . . . . . . . . . . 113
xv
1 INTRODUCTION
The purpose of this dissertation is to present methodologies and concepts
for dealing with analog signals in a digital setting. There are many applications
for such methodologies including cell phone communication, video cameras and
satellite sensors. Signals are used in our everyday lives and over recent years we
have seen a large transition from analog to digital. Generally speaking, analog
signals have an uncountable domain and range, while digital signals have a finite
domain and range. This advantage along with many others has pushed the desire
to develop digital technologies even though most real signals tend to be analog.
In the late 40’s at Bell National Laboratories, Shannon presented his famous
sampling theorem. The result was that band-limited signals may be completely
represented by countably infinite and evenly spaced samples. Thus, analog band-
limited signals may be viewed as having a countably infinite domain and uncount-
ably infinite range. The Shannon sampling theorem lies at the heart of many dig-
ital technologies used today and is a first step toward bridging the gap between
analog and digital signals. In practice, applications use samples which are finite in
number and quantized in value. This reduces the domain from countably infinite
to finite and range from uncountably infinite to finite. Theoretically, this gives a
further reduction of the space of analog signals which may be represented in the
digital setting. Realistically, this introduces errors into the digital representation.
1
For band-limited functions, Shannon also introduced a sampling formula which
demonstrates how to reconstruct an analog signal from its samples. This process
involves interpolating the samples using sinc functions. Since Shannon, many
others have attempted to generalize his sampling formula to reduce the errors
involved with using samples which are finite in number and quantized in value.
We start our discussion in Chapter 2 with the development of basic time-
frequency concepts surrounding the Fourier Transform. We demonstrate how the
domain of a function may effectively be reduced from uncountably infinite to
countably infinite as one moves from the Fourier Transform to the Fourier Series.
This is the basic concept involved in the Shannon sampling formula. We point
out that the Discrete Time Fourier Transform represents the time-frequency dual
of the Fourier Series. We also demonstrate how the domain of a function may be
further reduced from countably infinite to finite as we move from the Fourier Series
to the Discrete Fourier Transform. This presentation of the Fourier Transform is
similar to that of [13] but there are several differences including the development
of the underlying distribution theory.
Under certain circumstances, the Discrete Time Fourier Transform may be
used to numerically compute the Fourier Transform of a function. This concept
will be used in some of our examples. Under other circumstances, the Discrete
Fourier Transform may be used to numerically compute the Fourier Transform of a
function. That is, under certain circumstances, a finite sequence and its Discrete
2
Fourier Transform may both be interpolated to reconstruct a continuous func-
tion and its Fourier Transform. We give some new methodologies and examples
regarding this concept.
The Shannon sampling formula may be generalized to reduce the error asso-
ciated with using finitely many samples instead of a countably infinite number
of samples. One common method of generalizing the Shannon sampling formula
comes from generalizing the underlying orthonormal basis. In Chapter 3, we intro-
duce several generalizations of an orthonormal basis including Bessel sequences,
frame sequences, frames, Riesz sequences and Riesz bases. Most of the concepts
presented are based on the work of Casazza and Christensen. However, our presen-
tation differs greatly from these authors. At the end of Chapter 3, we will discuss
further insights gained from our presentation and common misconceptions found
in the literature.
In Chapter 4, we present several generalizations of the Shannon sampling for-
mula involving functions band-limited to multiple intervals. Some are based on
applying operators to an orthornomal basis. Others are based on frames and
are typically referred to as periodic nonuniform sampling formulas (see [24]). We
include a formula which bridges the gap between typical Shannon sampling and
periodic nonuniform sampling formulas. In order to bridge this gap, we switch
notation from that which is found in [24] and provide a slight generalization to
their periodic nonuniform sampling formula.
3
All of the sampling formulas presented involve reconstruction of signals with
compact frequency support. Since applications use samples which are finite in
number, it seems that the key to developing good reconstruction formulas lies in
time-frequency localization. Chapter 5 introduces the time-frequency localization
operators. Possible reconstruction formulas involving time-frequency localization
operators have already been developed and the associated error may be expressed
in terms of eigenvalues. New estimates and methods are introduced for calculating
eigenvalues of the time-frequency localization operators.
4
2 THE FOURIER TRANSFORM
In this Chapter, we discuss four settings of the Fourier Transform: on the real
line R, on the unit circle T, on the integers Z and on the modular integers ZN .
The authors of [13] refer to the Fourier Transform in these four settings as the
Fourier Transform (FT), Fourier Series (FS), Discrete Time Fourier Transform
(DTFT) and Discrete Fourier Transform (DFT) respectively.
We start with the definitions and commutation properties of standard oper-
ators used in Fourier Analysis. These operators include the Fourier Transform,
Dilation, Translation, Modulation, Reverse, Time-Limiting and Band-Limiting
operators. We then develop the distribution theory for tempered distributions
which will play an important role as we discuss the Fourier Transform in each
setting. We include the Poisson Summation formula for tempered distributions
which serves as the foundation for many sampling formulas.
We then follow the authors of [13] in an attempt to relate the four settings of
the Fourier Transform. For example, we give the FT representations for FS and
DTFT signals. We also give FT, FS and DTFT representations for different types
of DFT signals. However, there are several differences between our presentation
and that of [13]. First of all, we derive our equations rigorously using distribu-
tion theory. Naturally, many of the equations presented here generalize those of
[13]. Second, we express the time and frequency sample periods explicitly with
5
parameters T and Ω. Finally, we provide additional equations so that the duality
between time and frequency may be fully represented.
One of the main concepts is that impulse sampling FT and FS signals leads
to DTFT and DFT signals, respectively. We will show that periodization in
the time domain is equivalent to impulse sampling in the frequency domain, so
that by duality, periodizing FT and DTFT signals leads to FS and DFT signals,
respectively. While the authors of [13] consider impulse sampling DTFT and DFT
signals, we view this as subsampling and do not present the equations. In a similar
fashion, we ignore periodizing FS and DFT signals.
Our view that any FS signal is the periodization of an FT signal allows us
to obtain the periodization equation for an FT signal and the FT representation
of an FS signal simultaneously. This view also leads to the concept that there
are Fourier Series equations for compactly supported functions which parallel
the Fourier Series equations for periodic functions. These are presented as the
Campbell sampling equations. Similarly, the Discrete Fourier Transform involves
finite discrete sequences which may be viewed as either periodic or finite.
The familiar view of a finite discrete sequence as representing the weights of a
periodic impulse train is discussed in the first DFT Section. A periodic impulse
train satisfies the following four properties: i) it is a tempered distribution and
has a well-defined Fourier Transform, ii) its time weights may be interpolated to
reconstruct the impulse train, iii) its frequency weights may be interpolated to re-
6
construct the Fourier Transform and iv) its time and frequency weights are related
by the Discrete Fourier Transform. In a sense, these properties imply that Fourier
Analysis on periodic impulse trains may be performed on a computer. There is
one major problem. Periodic impulse trains do not model real signals very well.
We end the chapter with a third DFT Section which gives an example of a class
of Schwartz functions which satisfy the following properties: i) the signal and its
Fourier Transform are Schwartz functions, ii) the time samples may be interpo-
lated to reconstruct the signal, iii) the frequency samples may be interpolated to
reconstruct the Fourier Transform and iv) the time and frequency samples are
related by the Discrete Fourier Transform. To the best of our knowledge, this is
the first such example found in the literature.
2.1 Operator Definitions
We start by defining the Fourier Transform and various operators on L2(R).
We then list commutation properties of these operators. See [20] and [22] for more
information on Fourier analysis on the Schwartz space and over groups.
Definition 2.1. The Schwartz space, denoted by S, consists of all infinitely dif-
ferentiable functions which are rapidly decreasing. We say φ is rapidly decreasing
if for every n,m ∈ N there exists a constant C such that
supx∈R|(1 + |x|2)nφ(m)(x)| ≤ C
7
With respect to the Schwartz space, we say φj → φ in S(R) if for every n,m ∈ N,
xnφ(m)j → xnφ(m) uniformly in R.
Example 2.2. The following is an example of a Schwartz function which has
support in [−1, 1]. This function is graphed in Figure 1.
ρ(t) =
e|t|2
|t|2−1 if |t| < 1
0 if |t| ≥ 1
Figure 1: Plot of ρ from Example 2.2
Definition 2.3. The Fourier Transform operator, denoted by F , takes functions
from the time domain to the frequency domain and is defined by the following
equation for any φ ∈ S(R).
Fφ(x) =
∫ ∞−∞
φ(t)e−2πixtdt
8
The Fourier Transform as defined is a continuous bijective linear operator from
S(R) to S(R). Using a density argument, this definition extends uniquely to a
unitary operator of L2(R).
We may use f and f to denote the Fourier Transform and Inverse Fourier
Transform of f , respectively. Sometimes we will want to be clear whether the
discussion involves the time domain or the frequency domain. If this is the case,
we will use t to represent the time variable and ω to represent the frequency
variable. The following two equations represent the Fourier Transform and its
inverse for f ∈ S(R).
f(ω) =∫∞−∞ f(t)e−2πiωtdt
f(t) =∫∞−∞ f(ω)e2πiωtdw
Example 2.4. Let ρ be as defined in Example 2.2. Then, the Fourier Transform
of ρ is a Schwartz function and is shown in Figure 2.
Figure 2: Plot of ρ from Example 2.4
9
Definition 2.5. For real parameters α > 0, β and γ, the Dilation (Dα), Transla-
tion (Tβ), Modulation (Mγ) and Reverse (R) operators are defined as follows.
Dαf(x) =√αf(αx)
Tβf(x) = f(x− β)
Mγf(x) = e−2πiγxf(x)
Rf(x) = f(−x)
Many authors take FD2π to be the definition of the Fourier Transform. That
is, their definition of the Fourier Transform includes a factor of 1√2π
in front of the
integral while the factor of 2π is missing from the exponential in the integral. Using
Definition 2.3, the Reverse operator is equal to the Fourier Transform operator
squared. In addition, we have a simple equation for the Inverse Fourier Transform:
F−1 = FR = RF
Definition 2.6. For sets S and Σ, the Time Limiting (QS) and Band Limiting
(PΣ) operators are defined as follows.
QSf = f1S
PΣ = F−1QΣF
The Time and Band Limiting Operators QS and PΣ are projection opera-
tors (being self-adjoint and idempotent). The ranges of these operators, denoted
10
R(QS) and R(PΣ), represent the space of functions time and band limited to
the sets S and Σ, respectively. Table 1 lists the commutation properties of the
operators defined above. See [25] for further discussion.
Table 1: Commutation Properties
FDα = D 1αF FQS = P−SF
FTβ = MβF DαQS = Q 1αSDα
FMγ = T−γF TβQS = QS+βTβ
FR = RF MγQS = QSMγ
DαTβ = T βαDα RQS = Q−SR
DαMγ = MαγDα FPΣ = QΣF
DαR = RDα DαPΣ = PαΣDα
TβMγ = e2πiβγMγTβ TβPΣ = PΣTβ
TβR = RT−β MγPΣ = PΣ−γMγ
MγR = RM−γ RPΣ = P−ΣR
2.2 Tempered Distributions
We introduce S ′(R), the space of tempered distributions. Then, we extend
the Fourier Transform, translation, modulation, multiplication and convolution to
include S ′(R). See [14] for more information.
11
Definition 2.7. The space of tempered distributions, denoted S ′(R) is the set
of all linear functionals which are continuous over S(R). We say f is continuous
over S(R) if f(φj)→ 0 in C whenever φj → 0 in S(R).
Definition 2.8. The Fourier Transform of a tempered distribution f ∈ S ′(R) is
defined by the following equation for any φ ∈ S(R).
f(φ) = f(φ)
This definition extends the definition of the Fourier Transform over L2(R) to
a unique weakly continuous bijective linear operator from S ′(R) to S ′(R). We
now extend translation, modulation, multiplication and convolution to include
tempered distributions.
Definition 2.9. For real parameters β and γ, the translation and modulation of
a tempered distribution f ∈ S ′(R) are both tempered distributions defined by the
following equations for any φ ∈ S(R).
Tβf(φ) = f(T−βφ)
Mγf(φ) = f(Mγφ)
Definition 2.10. The space OM(R) consists of all infinitely differentiable func-
tions which are slowly increasing. We say φ is slowly increasing if for every m ∈ N
there exists constants n, C such that
supx∈R|φ(m)(x)| ≤ C(1 + |x|2)n
12
Definition 2.11. The product of a tempered distribution f ∈ S ′(R) and a slowly
increasing function g ∈ OM(R) is a tempered distribution defined by the following
equation for any φ ∈ S(R).
(gf)(φ) = f(gφ)
Definition 2.12. The convolution of two tempered distributions f, g ∈ S ′(R)
where g ∈ OM(R) is a tempered distribution defined by the following equation for
any φ ∈ S(R).
g ∗ f(φ) = f(g(T−tφ))
Theorem 2.13. Suppose f, g ∈ S ′(R) where g ∈ OM(R). Then, we have
g ∗ f = gf
Proof. We observe how these distributions act on a test function using the defi-
nitions of multiplication and convolution. Let φ ∈ S(R). Then, we have
gf(φ) = f(gφ)
= f(gφ)
= f(g(Mtφ))
= f(g(T−tφ))
= g ∗ f(φ)
= g ∗ f(φ)
13
There are a few comments to make before leaving this section. First of all,
Theorem 2.13 provides an equivalent way of defining convolution between two
tempered distributions. Second, the commutation property FTβ = MβF still
holds in S ′(R). Finally, we have the following inclusion (see [14]).
S(R) ⊂ OM(R) ⊂ S ′(R)
2.3 The Poisson Summation Formula
We start with the Poisson Summation Formula for Schwartz functions. This is
used to develop the Poisson Summation Formula for distributions whose Fourier
Transform is slowly increasing. We end the section with an application to p-
periodic partitions of unity.
Theorem 2.14. Suppose φ ∈ S(R). Then, for any p > 0, φ satisfies the Poisson
Summation Formula pointwise. That is,
∞∑m=−∞
Tmpφ = 1p
∞∑m=−∞
φ(mp
)M−mp1R
Proof. Since φ is a Schwartz function there exists a constant C such that
|φ(t−mp)| ≤ C1+|t−mp|2
Thus, the partial sums for the summation on the left uniformly converge to a
continous function whose Fourier Series coefficients are φ(mp
). Thus, the summa-
tion on the right converges to the summation on the left in L2([0, p)).
14
Since φ is a Schwartz function, the sequence of coefficients φ(mp
) is in `1.
Thus, the partial sums for the summation on the right uniformly converge to a
continuous function and the Poisson Summation formula holds pointwise.
Theorem 2.15. Suppose g is a tempered distribution such that g ∈ OM(R).
Then, for any p > 0, g satisfies the Poisson Summation formula in the sense of
distributions. That is,
∞∑m=−∞
Tmpg = 1p
∞∑m=−∞
g(mp
)M−mp1R
Proof. We observe how these distributions act on a test function φ ∈ S(R). Using
the commutation properties listed in Table 1, FMmpφ = T−mpφ. The product gφ
is a Schwartz function and satisfies the Poisson Summation Formula as shown in
Theorem 2.14. Evaluating the Poisson Summation Formula at t = 0 yields,
∞∑m=−∞
g(T−mpφ) =∞∑
m=−∞g(Mmpφ)
=∞∑
m=−∞
gφ(mp)
= 1p
∞∑m=−∞
gφ(mp
)
Corollary 2.16. Suppose g is a tempered distribution such that g ∈ OM(R).
Then, for any p > 0, g satisfies the following two equations.
∞∑m=−∞
Tmpg = g ∗∞∑
m=−∞Tmpδ
F( ∞∑m=−∞
Tmpg)
= g(
1p
∞∑m=−∞
Tmpδ)
15
Proof. We observe how these distributions act on a test function φ ∈ S(R) using
the equalities in the proof of Theorem 2.15.
∞∑m=−∞
Tmpg(φ) =∞∑
m=−∞g(T−mpφ)
=∞∑
m=−∞Tmpδ(g(T−tφ))
=(g ∗
∞∑m=−∞
Tmpδ)
(φ)
∞∑m=−∞
Tmpg(φ) = 1p
∞∑m=−∞
gφ(mp
)
=(g 1p
∞∑m=−∞
Tmpδ)
(φ)
Definition 2.17. Suppose g is a tempered distribution such that g ∈ OM(R) and
p > 0. Then, g is a p-partition of unity if it satisfies the following equality in the
sense of tempered distributions.
∞∑m=−∞
Tmpg = 1
Corollary 2.18. Suppose g is a tempered distribution such that g ∈ OM(R) and
p > 0. Then, g is a p-partition of unity if and only if
g(m/p) =
p if m = 0
0 if m 6= 0,m ∈ Z
Proof. From Corollary 2.16, g is a p-partition of unity if and only if
g(
1p
∞∑m=−∞
Tmpδ)
= δ
16
In this section, we have discussed three key concepts related to the Fourier
Series: the Poisson Summation Formula, impulse trains and p-partitions of unity.
We end this section with an example of a p-partition of unity.
Example 2.19. Let ρ be as defined in Example 2.2. Then, the following defines
an example of a Schwartz function which is also a 1-partition of unity and has
support on the interval [−1, 1] (see Figures 3 - 5).
ψ(t) =
0 if t ≤ −1
ρ(t)ρ(t)+ρ(t+1)
if − 1 < t <= 0
ρ(t)ρ(t−1)+ρ(t)
if 0 < t < 1
0 if 1 ≤ t
Figure 3: Shifts of ψ in Example 2.19
Notice that the integer shifts of ψ sum to 1.
17
Figure 4: Plot of ψ from Example 2.19
Figure 5: Plot of ψ from Example 2.19
Notice that ψ(n) = 0 for all n ∈ Z/0. The Schwartz function ψ was estimated
using the following DTFT equation.
ψ(ω) ≈ 1N
∞∑n=−∞
ψ( nN
)e−2πinωN
18
2.4 Periodic Tempered Distributions and Support
We start defining what it means for a tempered distribution to be periodic.
We then provide a theorem stating that a tempered distribution is periodic if and
only if it may be written as the convolution of an impulse train with a distribution
whose Fourier Transform is slowly increasing. We end with a discussion of support
for distributions.
Definition 2.20. A tempered distribution f ∈ S ′(R) is p-periodic if f = Tpf .
Theorem 2.21. Suppose ψ ∈ S(R) is any p-partition of unity and f ∈ S ′(R) is
a p-periodic tempered distribution. Then,
f =∞∑
m=−∞Tpm(ψf)
Proof. We observe how this distribution acts on a test function φ ∈ S(R).
f(φ) = f(( ∞∑
m=−∞Tpmψ
)φ)
=∞∑
m=−∞f((Tpmψ)φ)
=∞∑
m=−∞f(ψT−pmφ)
=∞∑
m=−∞Tpm(ψf)(φ)
Corollary 2.16 and Theorem 2.21 provide two equivalent ways of defining pe-
riodicity of tempered distributions. That is, we may say f is periodic if f may be
19
expressed as the convolution of an impulse train with a distribution whose Fourier
Transform lies in OM. The authors of [14] define periodic for the space of distri-
butions rather than the space of tempered distributions. However, no generality
is lost here, since any periodic distribution extends uniquely to a tempered distri-
bution. That is, if f is a p-periodic distribution and ψ ∈ S(R) is any compactly
supported p-partition of unity, f defines a periodic tempered distribution by the
following equation for all φ ∈ S(R).
f(φ) = f(ψ∞∑
m=−∞
T−mpφ)
Definition 2.22. A tempered distribution f ∈ S ′(R) is said to have support in a
set S if f(φ) = 0 whenever φ ∈ S(R) and φ1S = 0.
Lemma 2.23. Suppose f ∈ S ′(R) has support in S. If g ∈ OM and satisfies the
equation g1S = 1S, then f = gf .
Proof. Suppose φ ∈ S(R). Then, f(φ)− gf(φ) = f(φ− gφ) = 0.
2.5 Fourier Series I
In this section, we follow [13] developing equations for periodic distributions.
We introduce the notation fI to denote a distribution which is being impulse
sampled in the frequency domain. The distribution fI will be defined using a
Schwartz function ψ but will not depend on the choice of ψ. Examples of fI will
be given following the proof and major results are summarized in Table 2.
20
Theorem 2.24. Suppose f is a T -periodic distribution. Let ψ ∈ S(R) be any
T -partition of unity. Let fI be any tempered distribution such that fI ∈ OM(R)
and satisfies the following property for all m ∈ Z.
fI(mT
) = f(MmTψ) (1)
Then, f and fI satisfy the following equations.
f(t) = 1T
∞∑m=−∞
fI(mT
)e2πimtT (2)
f(t) =∞∑
m=−∞
fI(t−mT ) (3)
f(ω) = 1T
∞∑m=−∞
fI(mT
)δ(ω − mT
) (4)
Proof. While we have not assumed that ψ is compactly supported, we will use
a compactly supported T -partition of unity for the proof. Let φ ∈ S(R) be any
compactly supported T -partition of unity so that φf is compactly supported.
Then, from Corollary 2.18 and Theorem 2.15, for m ∈ Z, we have
φf(mT
) = 1T
∞∑l=−∞
φf( lT
)ψ(m−lT
)
= 1T
∞∑l=−∞
φf( lT
)∫∞−∞ ψ(t)e−
2πi(m−l)tT dt
=∫∞−∞
1T
∞∑l=−∞
φf( lT
)e2πiltT Mm
Tψ(t)dt
=∫∞−∞
( ∞∑l=−∞
TlTφf)Mm
Tψ(t)dt
= f(MmTψ)
= fI(mT
)
21
Further, φf is compactly supported so φf ∈ OM(R) (see [14]). Thus, from
Theorems 2.21 and 2.15 we have
f =∞∑
m=−∞TmT (φf)
= 1T
∞∑m=−∞
M−mTφf(m
T)
= 1T
∞∑m=−∞
M−mTfI(
mT
)
=∞∑
m=−∞TmTfI
This is Equation 3. Equation 4 comes from applying Theorem 2.16 to fI .
This proof implies that fI can be chosen independently of ψ since the terms
f(MmTψ) do not depend on the choice of ψ. An example of fI which is compactly
supported and satisfies Equation 1 is fI = φf where φ ∈ S(R) is any compactly
supported T -partition of unity. A more complicated example of fI which is not
compactly supported and satisfies Equation 1 is fI = φ1f + φ2 where φ1 ∈ S(R)
is any compactly supported T -partition of unity and φ2 is any Schwartz function
whose samples are zero. Notice that under these assumptions the periodization
of φ1f + φ2 is equal to the periodization of φ1f .
There are two special cases where fI ∈ OM(R) that we should mention. If
fI is a Schwartz function, then fI ∈ S(R) and Equation 3 holds pointwise. If
fI ∈ L2(R) with compact support in some set S, then Equation 3 holds in L2(S).
One might say the space of periodic tempered distributions is the largest space
where the Fourier Series holds since the Fourier Series will not even converge in
22
the sense of distributions if the coefficients fI(mT
) do not have slow growth. On the
other hand, if these coefficients are in `2, they may be given using the partition
of unity ψ = 1TI0 . That is, Equation 1 becomes the following.
fI(mT
) =∫TI0
f(t)e−2πimtT dt
Table 2: Fourier Series I Equations
Fourier Series Fourier Transform
f(t) = 1T
∞∑m=−∞
fI(mT
)e2πimtT f(t) =
∫∞−∞ f(ω)e2πiωtdω
fI(mT
) = f(MmTψ)
f(t) =∞∑
m=−∞fI(t−mT )
f(ω) = 1T
∞∑m=−∞
fI(mT
)δ(ω − mT
)
We assume that f is a T -periodic distribution, ψ ∈ S(R) is any T -partition of
unity, fI is any tempered distribution such that fI ∈ OM(R) and fI satisfies
Equation 1 of Theorem 2.24.
Example 2.25. Let ρ as in Example 2.2 and let T = 12. Then, the following
defines functions satisfying Equations 2 - 4 (see Figures 6-11).
a(t) =5−√
3i
2ρ(6t+ 2) + 7ρ(6t) +
5 +√
3i
2ρ(6t− 2)
b(t) = 7∞∑
n=−∞
ρ(6t−3n)+5−√
3i
2
∞∑n=−∞
ρ(6t−1−3n)+5 +√
3i
2
∞∑n=−∞
ρ(6t−2−3n)
23
Figure 6: Plot of a from Example 2.25
The Schwartz function a may be periodized to obtain b.
Figure 7: Plot of A from Example 2.25
The Schwartz function A = a may be impulse sampled to obtain B.
24
Figure 8: Plot of b from Example 2.25
The tempered distribution b is periodic.
Figure 9: Plot of B from Example 2.25
The tempered distribution B = b is an impulse train.
25
Figure 10: Alternate definition for a in Example 2.25
The above tempered distribution may be periodized to obtain b.
Figure 11: Alternate definition for A in Example 2.25
The above tempered distribution may be impulse sampled to obtain B.
26
In Example 2.25, we first created a using a generic function based on the Mat-
lab parameters vector, T and Ω (see the Appendix). The idea is that several
other examples may be created using the same function, but varying the param-
eters under the restrictions mentioned within code comment blocks. After a was
created, A, b and B were created using the DTFT, periodizing and impulse sam-
pling algorithms, respectively. In fact, most of the examples created in this section
came from manipulating vector. In addition, one should note how the theory is
transparent from the Matlab code.
2.6 Fourier Series II
In this section, we develop equations for compactly supported distributions.
Theorem 2.26. Let S be a T -tiling set. Suppose f is a distribution whose support
lies in S0 where S0 is a subset of S and dist(S0, Sc) > 0. Let ψ be any Schwartz
function satisfying the following property.
ψ(t) =
1 if t ∈ S0
0 if t−mT ∈ S0 for some m ∈ Z \ 0
Then, f satisfies the following equations.
f(t) = 1T
∞∑m=−∞
f(mT
)e2πimtT ψ(t) (5)
f(ω) = 1T
∞∑m=−∞
f(mT
)ψ(ω − mT
) (6)
f(mT
) = f(MmTψ) for all m ∈ Z (7)
27
Proof. From Definition 2.22 and Lemma 2.23, we know that ψTmTf is identical
to f for m = 0 and identical to 0 for any other m ∈ Z. Furthermore, f ∈ OM(R)
since f is compactly supported (see [14]). Therefore, by Theorem 2.15, we have
f = ψf
=∞∑
m=−∞ψTmTf
= ψ∞∑
m=−∞TmTf
= ψ 1T
∞∑m=−∞
M−mTf(m
T)
Taking the Fourier Transform yields Equation 6 which is known as the Camp-
bell Sampling equation (see [14]). The Campbell Sampling equation turns out to
hold pointwise since ψ ∈ S(R).
Let φ ∈ S(R) represent a compactly supported function which satisfies the
same constraint as ψ. Then, Equation 5 is satisfied if ψ is replaced by φ. Further,
since φ has compact support, we may choose a Schwartz function g such that
g(t) = e−2πimtT whenever φ(t) 6= 0. From Lemma 2.23 and Corollary 2.18,
f(MmTψ) = f(g)
=∫∞−∞
1T
∞∑l=−∞
f( lT
)M− lTφ(t)e
−2πimtT dt
= 1T
∞∑l=−∞
f( lT
)M− lTφ(m
T)
= 1T
∞∑l=−∞
f( lT
)T lTφ(m
T)
= f(mT
)
28
If the coefficients f(mT
) are in `2, the equations above may be given using the
partition of unity ψ = 1S. That is, if f ∈ L2(S) where S is a tiling set, then f
satisfies the following equations.
f(mT
) =∫Sf(t)e−
2πimtT dt
f(t) = 1T
∞∑m=−∞
f(mT
)e2πimtT 1S(t)
f(ω) = 1T
∞∑m=−∞
f(mT
)1S(ω − mT
)
Notice that S need not be relatively compact. If S = TI0, Equation 6 simplifies
to the following known as the Shannon Sampling formula.
f(ω) =∞∑
m=−∞f(m
T)sinc(Tw −m)
The main equations derived in this section are listed in Table 3 and generalize
to a larger class of distributions. For example, suppose ψ1 and ψ2 are Schwartz
functions such that ψ1ψ2 is a T -partition of unity. Then, certain distributions f
and fI satisfy the equations in Table 4. In the case that ψ1 = ψ2, TmTψ1 is an
orthonormal system of translates (see [25]).
We will finish this section with two examples. Both of these examples will
satisfy the equations given in Table 3. However, only the first example will satisfy
the assumptions given in Theorem 2.26. Again, the Matlab code used to generate
these examples is found in the Appendix.
29
Table 3: Fourier Series II Equations
Fourier Series Fourier Transform
f(t) = 1T
∞∑m=−∞
f(mT
)e2πimtT ψ(t)
f(mT
) = f(MmTψ) f(ω) =
∫∞−∞ f(t)e−2πiωtdt
f(ω) = 1T
∞∑m=−∞
f(mT
)ψ(ω − mT
)
We assume that f is compactly supported on some set S0 and ψ is any Schwartz
function which is identical to 1 on S0 and identical to 0 on S0 + mT for any
non-zero integer m.
Table 4: Fourier Series III Equations
Fourier Series Fourier Transform
f(t) = 1T
∞∑m=−∞
fI(mT
)e2πimtT ψ1(t)
f(mT
) = f(MmTψ2) f(ω) =
∫∞−∞ f(t)e−2πiωtdt
f(ω) = 1T
∞∑m=−∞
fI(mT
)ψ1(ω − mT
)
∞∑m=−∞
fI(t−mT ) = (ψ2f) ∗∞∑
m=−∞δ(t−mT )
1T
∞∑m=−∞
fI(mT
)δ(ω − mT
) = (ψ2 ∗ f)∞∑
m=−∞δ(ω − m
T)
These equations generalize those from Table 3.
30
Example 2.27. Let ρ be as defined in Example 2.2 and let T = 12. Define S
to be the T -tiling set [−12,−1
4) ∪ [1
4, 1
2) and S0 = [− 7
16,− 5
16) ∪ [ 5
16, 7
16). Then, the
following assignments give an example of functions satisfying Equations 5 - 7.
Plots of these functions are given in Figures 12-17.
ψ0(t) =
0 if t ≤ −1
ρ(2t+1)3ρ(2t+1)+3ρ(2t+2)
if − 1 < t < −12
13
if − 12≤ t ≤ 1
2
ρ(2t−1)3ρ(2t−2)+3ρ(2t−1)
if 12< t < 1
0 if 1 ≤ t
ψ(t) = 3ψ0(8t+ 3) + 3ψ0(8t− 3)
f(t) = ρ(16t+ 6) + .02δ(8t− 3)
Example 2.28. Let T = 12
and let ψ0 be the T -partition of unity defined in
Example 2.27. Then, the following assignments give an example of functions sat-
isfying the Fourier Series II Equations. Notice g1 does not satisfy the assumptions
of Theorem 2.26. Plots of these functions are given in Figures 18-19.
g0(t) = 4 + e4πit + 2e8πit
g1(t) = ψ0(t)
f(t) = g0g1(t)
31
Figure 12: Plot of ψ from Example 2.27
The Schwartz function ψ is identical to 1 over S0 = [− 716,− 5
16) ∪ [ 5
16, 7
16).
Figure 13: Plot of ψ from Example 2.27
The Schwartz function ψ was estimated using the following DTFT equation.
ψ(ω) ≈ 1N
∞∑n=−∞
ψ( nN
)e−2πinωN
32
Figure 14: Plot of f from Example 2.27
The tempered distribution f is supported in S0 = [− 716,− 5
16) ∪ [ 5
16, 7
16) and is the
sum of Schwartz and impulse functions.
Figure 15: Plot of f from Example 2.27
The tempered distribution f is the sum of Schwartz and periodic functions.
33
Figure 16: Approximation for f in Example 2.27
The approximation f came from interpolating samples of f (see Equation 5).
Figure 17: Approximation for f in Example 2.27
The approximation˜f came from interpolating samples of f (see Equation 6).
34
Figure 18: Plot of g0g1(t) from Example 2.28
The tempered distribution g0g1 satisfies the Campbell Sampling Equation.
Figure 19: Plot of g0g1 from Example 2.28
The tempered distribution g0g1 may be reconstructed from its samples.
35
2.7 Discrete Time Fourier Transform I
In this section, we discuss the Discrete Time Fourier Transform which is the
dual of the Fourier Series. That is, we may repeat our discussion on Fourier Series
but interchange f , n, T and ψ with f , m, Ω and ψ respectively. The result would
be the DTFT equations listed in Tables 5 and 6. Here, we use the notation f I to
denote a distribution which is being impulse sampled in the time domain. Notice
that the organization of the equations within Tables 5 and 6 has been modified.
Table 5: Discrete Time Fourier Transform I Equations
Fourier Transform
f(ω) =∫∞−∞ f(t)e−2πiωtdt
Discrete Time Fourier Transform
f I( nΩ
) = f(M− nΩψ)
f(ω) = 1Ω
∞∑n=−∞
f I( nΩ
)e−2πiωn
Ω
f(t) = 1Ω
∞∑n=−∞
f I( nΩ
)δ(t− nΩ
)
f(ω) =∞∑
n=−∞f I(ω − nΩ)
We assume that f is an Ω-periodic distribution, ψ ∈ S(R) is any Ω-partition
of unity, f I ∈ OM(R) and f I satisfies f I( nΩ
) = f(M− nΩψ). The superscript I
represents the fact that f I is being impulse sampled in time.
36
Table 6: Discrete Time Fourier Transform II Equations
Fourier Transform
f(t) =∫∞−∞ f(ω)e2πiωtdω
Discrete Time Fourier Transform
f( nΩ
) = f(M− nΩψ)
f(ω) = 1Ω
∞∑n=−∞
f( nΩ
)e−2πiωn
Ω ψ(ω)
f(t) = 1Ω
∞∑n=−∞
f( nΩ
)ψ(t− nΩ
)
We assume that f is a tempered distribution, f is compactly supported on some
set Σ0 and ψ is any Schwartz function which is equal to 1 on Σ0 and equal to 0
on Σ0 + nΩ for any non-zero integer n.
Table 5 represents the case where f is Ω-periodic and should be compared with
Table 2. Table 6 represents the case where f is compactly supported and should
be compared with Table 3.
In several of our examples, we have used the DTFT equation to compute
the Fourier Transform of a Schwartz function. This is due to the fact that the
Riemann sum approximations converge uniformly to φ as N approaches ∞. We
have replaced N with Ω to emphasize the fact that we have used a Riemann sum
to approximate the value of the Fourier Transform integral equation.
37
2.8 Discrete Fourier Transform I
In this section, we develop equations for periodic impulse trains. We make
the assumption that TΩ ∈ N which will allow us to take the TΩ point Discrete
Fourier Transform. Again, the I in fI and f I will represent the fact that fI and
f I are being impulse sampled in the frequency and time domains as shown in
Equations 12 and 15. The distributions fI and f I will be defined using Schwartz
functions ψ1 and ψ2 but will not depend on the choice of ψ1 and ψ2. Finally,
examples of fI and f I will be given in the discussion following the proof.
Theorem 2.29. Let TΩ ∈ N. Suppose f is a T -periodic impulse train where the
spacing between impulses is 1Ω
. Let ψ1 ∈ S(R) be any T -partition of unity and
ψ2 be any Ω-partition of unity. Let fI ,fI be any tempered distributions such that
fI , fI ∈ OM(R) and satisfy the following properties for all m,n ∈ Z.
fI(mT
) = f(MmTψ1) (8)
f I( nΩ
) = f(M− nΩψ2) (9)
Then, f , fI and f I satisfy the following equations.
f(t) = 1T
∞∑m=−∞
fI(mT
)e2πimtT (10)
f(t) =∞∑
m=−∞fI(t−mT ) (11)
f(ω) = 1T
∞∑m=−∞
fI(mT
)δ(ω − mT
) (12)
38
f(ω) = 1Ω
∞∑n=−∞
f I( nΩ
)e−2πiωn
Ω (13)
f(ω) =∞∑
n=−∞f I(ω − nΩ) (14)
f(t) = 1Ω
∞∑n=−∞
f I( nΩ
)δ(t− nΩ
) (15)
f I( nΩ
) = 1T
TΩ−1∑m=0
fI(mT
)e2πimnTΩ for all n ∈ Z (16)
fI(mT
) = 1Ω
TΩ−1∑n=0
f I( nΩ
)e−2πimnTΩ for all m ∈ Z (17)
Proof. Using the results of Section 2.5, Equations 10, 11 and 12 hold. Since f is
an impulse train and the impulses of f are spaced by 1Ω
, we have MΩf = f . Thus,
for any n ∈ Z, we have
fI(m+nTΩ
T) = f(Mm+nTΩ
Tψ1)
= MnΩf(MmTψ1)
= fI(mT
)
Using Equations 9 and 12, for any n ∈ Z, we have
f I( nΩ
) = f(M− nΩψ2)
= 1T
∞∑m=−∞
fI(mT
)e2πimnTΩ ψ2(m
T)
= 1T
TΩ−1∑m=0
fI(mT
)e2πimnTΩ
∞∑l=−∞
ψ2(m+lTΩT
)
= 1T
TΩ−1∑m=0
fI(mT
)e2πimnTΩ
Equation 12 implies that f is an Ω-periodic distribution. Using the results of
Section 2.7, Equations 13, 14 and 15 hold. We may repeat the argument above to
show that Equation 17 follows from Equations 8 and 15.
39
The definition of fI and f I do not depend on the choices of ψ1 and ψ2. Fur-
thermore, examples include fI = fφ1 and f I = fφ2 where φ1, φ2 ∈ S(R) are any
compactly supported T and Ω-partitions of unity, respectively.
In this section, we discussed periodic distributions with periodic FS coeffi-
cients. Table 7 lists a summary of the equations from this section.
Table 7: Discrete Fourier Transform I Equations
Fourier Series Fourier Transform
f(t) = 1T
∞∑m=−∞
fI(mT
)e2πimtT f(t) =
∫∞−∞ f(ω)e2πiωtdω
fI(mT
) = f(MmTψ1) f(ω) =
∫∞−∞ f(t)e−2πiωtdt
Discrete Fourier Transform Discrete Time Fourier Transform
f I( nΩ
) = 1T
TΩ−1∑m=0
fI(mT
)e2πimnTΩ f I( n
Ω) = f(M− n
Ωψ2)
fI(mT
) = 1Ω
TΩ−1∑n=0
f I( nΩ
)e−2πimnTΩ f(ω) = 1
Ω
∞∑n=−∞
f I( nΩ
)e−2πiωn
Ω
f(t) =∞∑
m=−∞fI(t−mT )
f(t) = 1Ω
∞∑n=−∞
f I( nΩ
)δ(t− nΩ
)
f(ω) =∞∑
n=−∞f I(ω − nΩ)
f(ω) = 1T
∞∑m=−∞
fI(mT
)δ(ω − mT
)
We assume TΩ ∈ N, f is a T -periodic impulse train with impulses spaced by
1Ω
, ψ1, ψ2 ∈ S(R) are any T -and Ω-partitions of unity, fI ,fI are any tempered
distributions such that fI , fI ∈ OM(R) and fI , f
I satisfy Equations 8 and 9.
40
Example 2.30. Let ρ and ψ be as in Example 2.2 and 2.19. Let T = 12
and
Ω = 6. Define ψ1(t) = ψ( tT
) = ψ(2t) and ψ2(ω) = ψ(ωΩ
) = ψ(ω6). Then, the
following assignments give an example of functions satisfying Equations 8 - 17.
b(t) = 7∞∑
n=−∞ρ(6t− 3n) + 5−
√3i
2
∞∑n=−∞
ρ(6t− 1− 3n) + 5+√
3i2
∞∑n=−∞
ρ(6t− 2− 3n)
c(t) = 76
∞∑n=−∞
δ(t− n2) + 5−
√3i
2
∞∑n=−∞
δ(t− 16− n
2) + 5+
√3i
2
∞∑n=−∞
δ(t− 13− n
2)
D(ω) = 2∞∑
n=−∞ρ( .5ω−3n
.5|n|) + .5
∞∑n=−∞
ρ( .5ω−1−3n.5|n|
) + 1∞∑
n=−∞ρ( .5ω−2−3n
.5|n|)
Plots for these functions are shown in Figures 20 - 29. Notice that we also
have the following equations which give the weights of the delta functions in the
delta trains c and C.
1Ωb( n
Ω) =
76
if n mod 3 = 0
5−√
3i12
if n mod 3 = 1
5+√
3i12
if n mod 3 = 2
, 1TD(m
T) =
4 if m mod 3 = 0
1 if m mod 3 = 1
2 if m mod 3 = 2
The tempered distribution D is an infinite sum of Schwartz functions φn
where the infinite sum of the Schwartz functions φn converges uniformly. For
Figure 24, D was approximated using a finite sum of DTFT approximations for
the Schwartz functions in the summation above.
Notice b may be replaced with g0 from Example 2.28. In addition, since both
are continuous periodic tempered distributions, their Fourier Transform may be
computed by time limiting to one period, taking the Fourier Transform and then
impulse sampling (see Section 2.5).
41
Figure 20: Plot of b from Example 2.30
The tempered distribution b may be impulse sampled to obtain c.
Figure 21: Plot of B from Example 2.30
The tempered distribution B may be periodized to obtain C.
42
Figure 22: Plot of c from Example 2.30
The tempered distribution c is periodic delta train.
Figure 23: Plot of C from Example 2.30
The tempered distribution C = c is a periodic delta train.
43
Figure 24: Plot of d from Example 2.30
The tempered distribution d may be periodized to obtain c.
Figure 25: Plot of D from Example 2.30
The tempered distribution D = d may be impulse sampled to obtain C.
44
Figure 26: Approximation for c from Example 2.30
The approximation c came from interpolating samples of D (see Equation 10).
Figure 27: Approximation for C from Example 2.30
The approximation C came from interpolating samples of b (see Equation 13).
45
Figure 28: Approximation for c from Example 2.30
The approximation c came from periodizing d (see Equation 11).
Figure 29: Approximation for C from Example 2.30
The approximation C came from periodizing B (see Equation 14).
46
2.9 Discrete Fourier Transform II
In this section, we develop several equations for periodic distributions whose
Fourier Series coefficients are supported by a discrete tiling set.
Theorem 2.31. Let TΩ ∈ N and let Σ be any Ω-tiling set. Let ψ1 ∈ S(R) be any
T -partition of unity and let ψ2 be any Schwartz function satisfying the following
property for all m ∈ Z.
ψ2
(mT
)=
1 if m
T∈ Σ
0 if mT6∈ Σ
Suppose f is a T -periodic distribution such that f(MmTψ1) = 0 whenever m ∈ Z
but mT6∈ Σ. Let fI be any tempered distribution such that fI ∈ OM(R) and satisfies
the following property for all m ∈ Z.
fI(mT
) = f(MmTψ1) (18)
Then, f and fI satisfy the following equations.
f(t) = 1T
∑mT∈Σ
fI(mT
)e2πimtT (19)
f(t) =∞∑
m=−∞fI(t−mT ) (20)
f(ω) = 1T
∑mT∈Σ
fI(mT
)δ(ω − mT
) (21)
f( nΩ
) = f(M− nΩψ2) (22)
47
f(ω) = 1Ω
∞∑n=−∞
f( nΩ
)e−2πiωn
Ω ψ2(ω) (23)
f(t) = 1Ω
∞∑n=−∞
f( nΩ
)ψ2(t− nΩ
) (24)
f( nΩ
) = 1T
∑mT∈Σ
fI(mT
)e2πimnTΩ for all n ∈ Z (25)
fI(mT
) = 1Ω
TΩ−1∑n=0
f( nΩ
)e−2πimnTΩ for all m
T∈ Σ (26)
Proof. Using the results of Section 2.5, f satisfies Equations 19, 20 and 21. Thus,
Equation 21 implies that f has compact support in Σ and by Section 2.7, f satisfies
Equations 22, 23 and 24. By evaluating Equation 19, we see that the samples of
f satisfy Equations 25 and 26.
The equations developed in this section are shown in Table 8. Notice that
this section describes a periodic distribution f which has compactly supported FS
coefficients and also descibes a compactly supported distribution f which has pe-
riodic DTFT coefficients. Thus, by duality, this section also describes a compactly
supported distribution f which has periodic FS coefficients.
On the other hand, the previous section described a periodic distribution f
which had periodic FS coefficients. The remaining description, which will be
given in the next section, involves a compactly supported distribution f with
compactly supported FS coefficients. Notice that tiling sets have been used to
describe compactly supported distributions while discrete tiling sets have been
used to describe compactly supported FS coefficients.
48
Table 8: Discrete Fourier Transform II Equations
Fourier Series Fourier Transform
f(t) = 1T
∑mT∈Σ
fI(mT
)e2πimtT f(t) =
∫∞−∞ f(ω)e2πiωtdω
fI(mT
) = f(MmTψ1)
Discrete Fourier Transform Discrete Time Fourier Transform
f( nΩ
) = 1T
∑mT∈Σ
fI(mT
)e2πimnTΩ f( n
Ω) = f(M− n
Ωψ2)
fI(mT
) = 1Ω
TΩ−1∑n=0
f( nΩ
)e−2πimnTΩ f(ω) = 1
Ω
∞∑n=−∞
f( nΩ
)e−2πiωn
Ω ψ2(ω)
f(t) =∞∑
m=−∞fI(t−mT )
f(t) = 1Ω
∞∑n=−∞
f( nΩ
)ψ2(t− nΩ
)
f(ω) = 1T
∑mT∈Σ
fI(mT
)δ(ω − mT
)
We assume TΩ ∈ N, Σ is an Ω-tiling set, f is a T -periodic distribution, f is
compactly supported on Σ0 = mT
: m ∈ Z, mT∈ Σ, ψ1 ∈ S(R) is any T -partition
of unity and ψ2 is any Schwartz function which is equal to 1 on Σ0 and equal to
0 on Σ0 + nΩ for any non-zero integer n. We also assume fI is any tempered
distribution such that fI ∈ OM(R) and fI satifies Equation 18.
49
Example 2.32. Let ρ and ψ0 be as in Example 2.2 and 2.28. Let T = 12
and
Ω = 6. Define ψ1(t) = ψ( tT
) = ψ(2t) and Σ = [−4,−2) ∪ [6, 8) ∪ [10, 12). Then,
the following assignments give an example of functions that satisfy the equations
of Theorem 2.31.
ψ2(ω) = ρ(ω+42
) + ρ(ω−62
) + ρ(ω−102
)
g0(t) = e−8πit + 4e12πit + 2e20πit
g1(t) = ψ0(t)
fI(t) = g0g1(t)
Plots for these functions are shown in Figures 30 - 37. Notice that we also have
the following equations which give the weighted samples of g0 and the weights of
the delta functions in g0.
1Ωg0( n
Ω) =
76
if n mod 3 = 0
5−√
3i12
if n mod 3 = 1
5+√
3i12
if n mod 3 = 2
, 1Tg0g1(m
T) =
4 if m = 3
1 if m = −2
2 if m = 5
The tempered distribution g0 is a T -periodic tempered distribution. Therefore,
its Fourier Transform may be computed by multiplying by a T -partition of unity,
taking the Fourier Transform and then impulse sampling. Since g1 is a T -partition
of unity, we know that g0g1 may be impulse sampled to obtain g0. In addition,
g0g1 is just one example of many slowly increasing functions whose samples satisfy
Equation 18. Here, g0g1 was calculated using the DTFT to approximate the FT.
50
Figure 30: Plot of g0 from Example 2.32
The tempered distribution g0 is periodic with compactly supported FS coefficients.
Figure 31: Plot of g0 from Example 2.32
The tempered distribution g0 has compact support and has DTFT coefficients
which are periodic with period T .
51
Figure 32: Plot of g0g1 from Example 2.32
The tempered distribution g0g1 may be periodized to obtain g0.
Figure 33: Plot of g0g1 from Example 2.32
The tempered distribution g0g1 may be impulse sampled to obtain g0.
52
Figure 34: Approximation for g0 from Example 2.32
The approximation g0 came from interpolating samples of g0 (see Equation 24).
Figure 35: Approximation for g0 from Example 2.32
The approximation ˜g0 came from interpolating samples of g0 (see Equation 23).
53
Figure 36: Approximation for g0 from Example 2.32
The approximation g0 came from interpolating samples of g0g1 (see Equation 19).
Figure 37: Approximation for g0 from Example 2.32
The approximation g0 came from periodizing g0g1 (see Equation 20).
54
2.10 Discrete Fourier Transform III
In this section, we develop several equations for compactly supported distri-
butions whose Fourier Series coefficients are supported by a discrete tiling set.
Theorem 2.33. Let TΩ ∈ N. Let S be any T -tiling set and Σ any Ω-tiling set.
Let ψ ∈ S (R) be any T -partition of unity satisfying the following equation.
ψ(n
Ω
)=
1 if n
Ω∈ S
0 if nΩ6∈ S
Suppose f is a tempered distribution such that f is a linear combination of the
functions in the set TmTψ : m
T∈ Σ. Then, f satisfies the following equations.
f(ω) =∫∞−∞ f(t)e−2πiωtdt (27)
f(t) =∫∞−∞ f(ω)e2πiωtdω (28)
f(ω) = 1T
∑mT∈Σ
f(mT
)ψ(ω − mT
) (29)
f(t) = 1T
∑mT∈Σ
f(mT
)e2πimtT ψ(t) (30)
f( nΩ
) = 1T
∑mT∈Σ
f(mT
)e2πimnTΩ for all n
Ω∈ S (31)
f(mT
) = 1Ω
∑nΩ∈Sf( n
Ω)e−
2πimnTΩ for all m
T∈ Σ (32)
f(t) = 1Ω
∑nΩ∈Sf( n
Ω)[
1T
∑mT∈Σ
e2πimT
(t− nΩ
)ψ(t)]
(33)
f(ω) = 1Ω
∑nΩ∈Sf( n
Ω)[
1T
∑mT∈Σ
e−2πimnTΩ ψ(ω − m
T)]
(34)
55
Proof. Schwartz functions satisfy Equations 27 and 28. By Corollary 2.18, the
coefficients for the linear combination are 1Tf(m
T) and Equation 29 holds. Finally,
Table 1 implies Equation 30 and properties of ψ imply Equations 31 and 32.
Equations 30 and 34 are generalized FS and DTFT equations. Also, ψ may
be replaced by 1S. The equations developed in this section are shown in Table 9.
Table 9: Discrete Fourier Transform III Equations
Fourier Series Fourier Transform
f(t) = 1T
∑mT∈Σ
f(mT
)e2πimtT ψ(t) f(t) =
∫∞−∞ f(ω)e2πiωtdω
f(mT
) =∫∞−∞ f(t)e−
2πimtT dt f(ω) =
∫∞−∞ f(t)e−2πiωtdt
Discrete Fourier Transform Discrete Time Fourier Transform
f( nΩ
) = 1T
∑mT∈Σ
f(mT
)e2πimnTΩ f( n
Ω) =
∫∞−∞ f(ω)e
2πiωnΩ dω
f(mT
) = 1Ω
∑nΩ∈Sf( n
Ω)e−2πimnTΩ f(ω) = 1
Ω
∑nΩ∈Sf( n
Ω)[
1T
∑mT∈Σ
e−2πimnTΩ ψ(ω − m
T)]
f(t) = 1Ω
∑nΩ∈Sf( n
Ω)[
1T
∑mT∈Σ
e2πimT
(t− nΩ
)ψ(t)]
f(ω) = 1T
∑mT∈Σ
f(mT
)ψ(ω − mT
)
We assume that TΩ ∈ N, S is a T -tiling set, Σ is an Ω-tiling set, ψ ∈ S(R) is
a T -partition of unity which is 1 on the set S0 = nΩ
: n ∈ Z, nΩ∈ Σ, ψ is 0
on S0 + mT for any m ∈ Z and f is a tempered distribution such that f is in
spanTmTψ : m
T∈ Σ.
56
Example 2.34. Let ρ and ψ be as in Example 2.2 and 2.19. Let T = 12
and Ω = 6.
Define S = [−13,−1
6) ∪ [0, 1
6) ∪ [1
3, 1
2) and Σ = [−4,−2) ∪ [6, 8) ∪ [10, 12). Then,
the following assignments give an example of functions that satisfy the equations
of Theorem 2.33.
g0(t) = 4 + e4πit + 2e8πit
g2(t) = ψ(6t+ 2) + ψ(6t) + ψ(6t− 2)
f(t) = g0g2(t)
Plots for these functions are shown in Figures 38 - 41. Notice that we also have
the following equations which give the weighted samples of f and the weights of
the delta functions in f .
1Ωf( n
Ω) =
76
if n = 0
5−√
3i12
if n = −2
5+√
3i12
if n = 2
, 1Tf(m
T) =
4 if m = 3
1 if m = −2
2 if m = 5
Figures 42 and 43 summarize most of the examples from this section. These
include a periodic distribution b (FS I), a periodic impulse train c (DFT I), the
product g0g1 of a periodic distribution and a partition of unity g1 (FS II), a
periodic distribution g0 whose FS coefficients are supported by a discrete tiling
set (DFT II) and finally, the product g0g2 of a periodic distribution g0 whose FS
coefficients are supported by a discrete tiling set and a partition of unity g2 whose
samples are supported by a discrete tiling set. (DFT III).
57
Figure 38: Plot of f from Example 2.34
The tempered distribution f is compactly supported and has FS coefficients which
are also compactly supported. Furthermore, the samples of f are related to the
samples of f by Equations 31 and 32.
Figure 39: Plot of f from Example 2.34
58
Figure 40: Plot of samples of f from Example 2.34
The samples of f are supported in a T -tiling set, while the samples of f are
supported in an Ω-tiling set. These samples are related by the DFT.
Figure 41: Plot of samples of f from Example 2.34
59
Figure 42: Summary of Fourier Transform Examples
60
Figure 43: Summary of Fourier Transform Examples
61
3 GENERALIZED BASES
In this chapter we characterize several sequences which generalize orthonormal
bases. We begin with the development of several functional analysis tools. First,
we discuss when a linear operator is bounded. For a bounded linear operator T ,
we also discuss its reduction T and its pseudo inverse T−1. Some of these ideas,
such as the open mapping theorem, may be found in [21] and [23].
Starting with Section 3.2, we assume `2 is the space of square integrable func-
tions and H is a Hilbert space. Under this assumption, there is a 1-1 correspon-
dence between isometries T : `2 → H and orthonormal bases gn ⊂ H. That is,
if T : `2 → H is an isometry, gn = T (δn) defines an orthonormal basis, where δn
is the standard orthonormal basis of `2. Conversely, if gn ⊂ H is an orthonormal
basis, then T (δn) = gn defines an isometry. Recall that a bounded linear operator
can be completely described by its action on an orthonormal basis. In short, we
may say that there is a 1-1 correspondence between isometries T : `2 → H and
orthonormal bases gn ⊂ H through the binding T (δn) = gn.
A similar correspondence holds between other types of bounded linear opera-
tors and the following types of sequences: Bessel sequences, complete Bessel se-
quences, ω-linearly independent Bessel sequences, frame sequences, frames, Riesz
sequences and Riesz bases. Remarkably, for each of these sequences, we can de-
scribe this correspondence with respect to a bounded linear operator T : `2 → H
62
having a combination of the following three properties: T is injective, R(T ) is
closed and T ∗ is injective. This concept was developed independently of the work
in [1], which gives similar characterizations.
We also give characterizations of Bessel sequences, frame sequences, frames,
Riesz sequences and Riesz basis in terms of the four basic operators associated
with Riesz bases: synthesis operator, analysis operator, Gram matrix and frame
operator. Many of the results involving these four operators may be found in the
following works of Casazza and Christensen: [4, 5, 6, 7, 8, 9, 10].
Our presentation differs in significant ways from that in the literature. We have
pulled together pieces from many sources and given several additions. Therefore,
we include the final section of this chapter to compare our work with the literature.
3.1 Functional Analysis
For the following discussion, H1 and H2 will represent Hilbert spaces and
T : H1 → H2 will represent a linear operator. The nullspace and range of T will
be denoted N (T ) and R(T ), respectively.
Theorem 3.1 discusses necessary and sufficient conditions for T to be bounded
(i.e., continuous). We define the reduction T of T by restricting the domain and
codomain of T . Theorem 3.5 discusses necessary and sufficient conditions forR(T )
to be closed. In a sense, this is exactly the case for which the pseudo inverse T−1
exists. Finally, Corollary 3.7 discusses necessary and sufficient conditions for T to
63
be surjective. Throughout the section, we will be mentioning the duality between
T and T ∗. That is, we will often replace T with T ∗ to obtain a result. Some of the
ideas in this section may be extended to the case where H1 and H2 are Banach
spaces. In general, any equivalence which does not involve the composition of the
operators T and T ∗ may be extended to an equivalence for Banach spaces (see
Table 11 and Section 3.9).
Theorem 3.1. Suppose H1 and H2 are Hilbert spaces and T : H1 → H2 is a
linear operator. Then, the following statements are equivalent.
1. There exists B such that ‖Tf‖ ≤ B ‖f‖ for every f ∈ H1
2. There exists B such that ‖T ∗f‖ ≤ B ‖f‖ for every f ∈ H2
3. There exists B such that T ∗T ≤ B2I on H1
4. There exists B such that TT ∗ ≤ B2I on H2
If any of these statements hold, we may use the following value for B.
B2 = ‖T‖2 = ‖T ∗‖2 = ‖T ∗T‖ = ‖TT ∗‖
Proof. We outline the proof. We will show 1 ⇒ 2 and 1 ⇒ 3. The argument for
the opposite directions is similar. Finally, 2⇔ 4 follows from the duality.
64
1⇒ 2. Suppose 1 holds and let f2 ∈ H2. Then,
‖T ∗f2‖ = sup‖f1‖≤1,f1∈H2
|〈T ∗f2, f1〉|
= sup‖f1‖≤1,f1∈H2
|〈T ∗ f2
‖f2‖ , f1〉| ‖f2‖
= sup‖f1‖≤1,f1∈H2
|〈Tf1,f2
‖f2‖〉| ‖f2‖
≤ sup‖f1‖≤1,f1∈H2
‖Tf1‖ ‖f2‖
≤ sup‖f1‖≤1,f1∈H2
B ‖f1‖ ‖f2‖
= B ‖f2‖
1⇒ 3. Suppose 1 holds and let f ∈ H1. Then, T ∗T ≤ B2I on H1 since
〈T ∗Tf, f〉 = 〈Tf, Tf〉
= ||Tf ||2
≤ B2||f ||2
= 〈B2If, f〉
The following Lemmas concern the range of a bounded linear operator.
Lemma 3.2. Suppose H1, H2 are Hilbert spaces, T : H1 → H2 is a bounded
linear operator and h ∈ R(T ). Then, there exists a unique function f1 ∈ H1 such
that T (f1) = h and ‖f1‖ ≤ ‖g‖ for every g such that T(g) = h. Furthermore, the
function f1 ∈ N (T )⊥.
65
Proof. Let h ∈ R(T ). Then, there exists f ∈ H1 such that T (f) = h. Write
f = f1 + f2 where f1 ∈ N (T )⊥ and f2 ∈ N (T ). Then, T (f1) = h, since
T (f1) = T (f1) + T (f2) = T (f1 + f2) = h
Suppose T (g) = h. Then, the following argument shows that ‖f1‖ ≤ ‖g‖ with
equality if and only if g = f1. Since T (g − f1) = 0 and f1 ∈ N (T )⊥, we have
‖g‖2 = ‖f1 + g − f1‖2 = ‖f1‖2 + 2Re〈f1, g − f1〉+ ‖g − f1‖2
= ‖f1‖2 + ‖g − f1‖2
Lemma 3.3. Suppose H1, H2 are Hilbert spaces and T : H1 → H2 is a bounded
linear operator. Then, R(T )⊥ = N (T ∗). In particular, R(T ) is dense in N (T ∗)⊥.
Proof. Observe the following equivalences.
g ∈ R(T )⊥
⇔ 〈T (f), g〉 = 0 for every f ∈ H
⇔ 〈f, T ∗(g)〉 = 0 for every f ∈ H
⇔ g ∈ N (T ∗)
These lemmas inspire the definition of the reduction operator. Here, A⊥ repre-
sents the orthogonal complement of the subspace A. However, for general Banach
spaces, it represents the set of linear functions which annihilate A (see [21]).
66
Definition 3.4. The reduction of T , denoted T , is defined by the following.
T : N (T )⊥ → N (T ∗)⊥, T : f 7→ T (f)
Notice that from Lemmas 3.2 and 3.3, R(T ) is equal to R(T ) and dense in
N (T ∗)⊥. We will use T to denote the reduction of T , which comes from restricting
both the domain and range of T . We will also use the fact that the reduction of
the adjoint of T is the same as the adjoint of the reduction of T .
The following properties are often useful for describing a linear bounded op-
erator T : T is injective, R(T ) is closed and T ∗ is injective. They correspond
to the following properties concerning T : N (T )⊥ = H1, R(T ) = N (T ∗)⊥ and
N (T ∗)⊥ = H2.
Theorem 3.5. Suppose H1, H2 are Hilbert spaces and T : H1 → H2 is a bounded
linear operator. Then, the following statements are equivalent.
1. The reduction of T is a bijective map.
2. The reduction of T ∗ is a bijective map.
3. The reduction of T ∗T is a bijective map.
4. The reduction of TT ∗ is bijective map.
5. R(T ) is closed (R(T ) = N (T ∗)⊥).
6. R(T ∗) is closed (R(T ∗) = N (T )⊥).
67
7. R(T ∗T ) is closed (R(T ∗T ) = N (T )⊥).
8. R(TT ∗) is closed (R(TT ∗) = N (T ∗)⊥).
9. There exists A > 0 such that√A ‖f‖ ≤ ‖Tf‖ for every f ∈ N (T )⊥.
10. There exists A > 0 such that√A ‖f‖ ≤ ‖T ∗f‖ for every f ∈ N (T ∗)⊥.
11. There exists A > 0 such that AI ≤ T ∗T on N (T )⊥.
12. There exists A > 0 such that AI ≤ TT ∗ on N (T ∗)⊥.
If any of these statements hold, we may use the following value for A.
1A
=∥∥∥T−1
∥∥∥2
=∥∥∥(T ∗)−1
∥∥∥2
=∥∥∥(T ∗T )−1
∥∥∥ =∥∥∥(T T ∗)−1
∥∥∥Proof. Again, we will outline the proof. First, we make some opening comments.
Notice that R(T ) and R(T ) are the same and both are dense in N (T ∗)⊥ by
Lemmas 3.2 and 3.3. Thus, R(T ) is closed if and only if R(T ) contains or is equal
to N (T ∗)⊥. Similar statements hold for T ∗, T ∗T and TT ∗.
1 ⇔ 5. The definition of T implies that it is injective. Based on our opening
comments, R(T ) is closed if and only if T is bijective.
1⇔ 9. Suppose 1 holds. By the open mapping theorem T is open. Thus, T−1
is continuous, which is equivalent to being bounded for linear operators. Thus,
there exists an A > 0 such that ‖f‖ =∥∥∥T−1Tf
∥∥∥ ≤ 1√A‖Tf‖ for all f ∈ N (T ∗)⊥.
On the other hand, suppose 9 holds. Let g ∈ N (T ∗)⊥. Based on our opening
comments, there exists a sequence fn ∈ N (T )⊥ such that T (fn) is Cauchy and
68
converges to g. But the assumption implies that fn is Cauchy and thus converges
to some f ∈ N (T )⊥. By continuity, T (f) = g. Since g was arbitrary, R(T )
contains N (T ∗)⊥. Based on our opening comments, R(T ) is equal to N (T ∗)⊥.
5 ⇔ 6. Suppose 5 holds. Then, T−1 exists. Let f ∈ N (T )⊥. Then, for all
g ∈ N (T )⊥, we have 〈f, g〉 = 〈f, T−1T g〉 = 〈T ∗(T−1)∗f, g〉. Since N (T )⊥ is a
Hilbert space, f = T ∗(T−1)∗f . Based on our opening comments, f ∈ R(T ∗).
Since f was arbitrary, R(T ∗) contains N (T )⊥. Again, based on our opening
comments, R(T ∗) is closed. The reverse direction follows from duality.
5⇔ 7. Suppose 5 holds. Let f ∈ R(T ∗). Then, there exists g ∈ N (T ∗)⊥ such
that T ∗(g) = f . By the assumption, there exists h ∈ H1 such that T (h) = g.
Thus, T ∗T (h) = f . Since f was arbitrary, R(T ∗T ) contains R(T ∗). Since R(T ∗)
contains R(T ∗T ), R(T ∗T ) = R(T ∗) which is closed by the implication 5⇔ 6.
On the other hand, suppose 7 holds. Since R(T ∗) contains R(T ∗T ) = N (T )⊥,
based on our opening comments, R(T ∗) is closed. Thus, R(T ) is closed by the
implication 5⇔ 6.
We have shown 1 ⇔ 5 ⇔ 9 and 5 ⇔ 7. The implications 2 ⇔ 6 ⇔ 10 and
6⇔ 8 follow by replacing T and T ∗. The implications 3⇔ 7⇔ 11 and 4⇔ 8⇔
12 may be shown with arguments similar to those found in 1⇔ 5⇔ 9.
Definition 3.6. If any of the equivalences in Theorem 3.5 holds, then we refer to
the operator T−1 as the psuedo inverse of T .
69
Corollary 3.7. Suppose H1 and H2 are Hilbert spaces and T : H1 → H2 is a
bounded linear operator. Then, the following statements are equivalent.
1. T is a surjective map.
2. R(T ∗) is closed and T ∗ is an injective map.
3. TT ∗ is a bijective map.
4. There exists A > 0 such that√A ‖f‖ ≤ ‖T ∗f‖ for every f ∈ H2.
5. There exists A > 0 such that AI ≤ TT ∗ on H2.
If any of these statements hold, we may use the following value for A.
1A
=∥∥∥(T )−1
∥∥∥2
=∥∥∥(T ∗)−1
∥∥∥2
=∥∥∥(T ∗T )−1
∥∥∥ = ‖(TT ∗)−1‖
Proof. Again, we outline the proof.
1 ⇔ 2. Suppose 1 holds. Since R(T ) is closed, Theorem 3.5 implies that
R(T ∗) is closed. Further, R(T ) = H2 is dense in N (T ∗)⊥ so T ∗ is injective. On
the other hand, suppose 2 holds. By Theorem 3.5, R(T ) is closed. Further, R(T )
is equal to its closure N (T ∗)⊥ = H2.
1 ⇔ 3. Suppose 1 holds. Since R(T ∗) ⊂ N (T ), TT ∗ is injective. Theorem
3.5 implies that R(T ∗) is equal to its closure N (T ∗)⊥. Thus, R(TT ∗) is equal to
R(T ) which is the same as R(T ). Thus, TT ∗ is surjective. On the other hand if,
3 holds then T must be surjective.
70
2 ⇔ 4. Suppose 2 holds. Since T ∗ is injective, N (T ∗)⊥ = H2. Since R(T ∗)
is closed, Theorem 3.5 implies 4. On the other hand, suppose 4 holds. Suppose
T ∗(f) = 0. Then,√A ‖f‖ ≤ 0 and f = 0. Since f was arbitrary, T ∗ is injective.
Theorem 3.5 implies that R(T ∗) is closed.
4 ⇔ 5. This implication follows from the fact that A ‖f‖2 = 〈AIf, f〉 and
‖T ∗f‖2 = 〈TT ∗f, f〉 for every A > 0 and f ∈ H2.
3.2 Bessel Sequences
In this section, we characterize Bessel sequences. There is a 1-1 corre-
spondence between bounded linear operators T : `2 → H and Bessel sequences
gn ⊂ H through the binding gn = T (δn). We also characterize Bessel sequences
as the sequences for which T , T ∗, G and S are well defined.
Theorem 3.8. Let `2 denote the space of square summable sequences and H
denote a Hilbert space. If gn ⊂ H, then the following are equivalent.
1. T : `2 → H, T : sn 7→∞∑n=1
sngn is well-defined.
2. T ∗ : H → `2, T ∗ : f 7→ 〈f, gn〉 is well-defined.
3. G : `2 → `2, G : sn 7→ ∞∑n=1
sn〈gn, gm〉 is well-defined (G = T ∗T ).
4. S : H → H, S : f 7→∞∑n=1
〈f, gn〉gn is well-defined (S = TT ∗).
Furthermore, if these operators are well-defined, they are bounded linear oper-
ators satisfying ‖T‖2 = ‖T ∗‖2 = ‖G‖ = ‖S‖.
71
Proof. Before proving the theorem, let us make a few comments. In 1 and 4,
well-defined means that the partial sums for the summations given converge with
respect to the norm for the Hilbert space H. In 2, well-defined means that the
sequence of inner products given is square summable. In 3, well-defined means
that the partial sums for the summations given converge with respect to the inner
product field (typically R or C) for each n ∈ Z and the sequence of terms, given
by evaluating these summations, is square summable.
In order to show partial sums converge, we will show that they are Cauchy
sequences. Once we show an operator is well-defined, we can also assume the
operator is linear and bounded by the Banach Steinhaus Theorem (see [23]). That
is, each of these operators may be written as the pointwise limit of linear and
bounded operators. Therefore, the Banach Steinhaus Theorem implies that they
are also linear and bounded operators.
1⇔ 2. Suppose 1 holds. We will show that T ∗ is well-defined and is the formal
adjoint of T . Let sn ∈ `2, f ∈ H. Then, we have
〈T (sn), f〉 = 〈∞∑n=1
sngn, f〉
=∞∑n=1
sn〈gn, f〉
= 〈sn, 〈f, gn〉〉
= 〈sn, T ∗(f)〉
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On the other hand suppose 2 holds. Let sn ∈ `2. We will show that partial
sums in 1 form a Cauchy sequence and thus converge in H. Let N2, N1 ∈ N and
assume N2 ≥ N1. Then,∥∥∥∥ N2∑n=1
sngn −N1∑n=1
sngn
∥∥∥∥2
=
∥∥∥∥ N2∑n=N1
sngn
∥∥∥∥2
= sup‖f‖=1
∣∣∣⟨ N2∑n=N1
sngn, f⟩∣∣∣2
= sup‖f‖=1
∣∣∣ N2∑n=N1
sn〈gn, f〉∣∣∣2
≤ sup‖f‖=1
( N2∑n=N1
|sn|2)‖T ∗f‖2
≤ ‖T ∗‖2( N2∑n=N1
|sn|2)
≤ ‖T ∗‖2( N2∑n=1
|sn|2 −N1∑n=1
|sn|2)
1⇔ 3. Suppose 1 holds. Then, T ∗ is well-defined and G = T ∗T is well-defined.
On the other hand, suppose 3 holds. Let sn ∈ `2. Again, we will show that
partial sums in 1 form a Cauchy sequence and thus converge in H. Let N2, N1 ∈ N
and assume N2 ≥ N1. Then we have,∥∥∥∥ N2∑n=1
sngn −N1∑n=1
sngn
∥∥∥∥2
=
∥∥∥∥ N2∑n=N1
sngn
∥∥∥∥2
=∣∣∣⟨ N2∑
n=N1
sngn,N2∑
n=N1
sngn
⟩∣∣∣=
∣∣∣ N2∑m=N1
smN2∑
n=N1
sn〈gn, gm〉∣∣∣
≤( N2∑m=N1
|sm|2)1/2( N2∑
m=N1
∣∣∣ N2∑m=N1
sn〈gn, gm〉∣∣∣2)1/2
≤ ||G||( N2∑n=N1
|sn|2)
= ‖G‖( N2∑n=1
|sn|2 −N1∑n=1
|sn|2)
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2⇔ 4. Suppose 2 holds. Then, T is well-defined and S = TT ∗ is well-defined.
On the other hand, suppose 4 holds. Let f ∈ H. Then,⟨ ∞∑n=1
〈f, gn〉gn, f⟩
=∞∑n=1
〈f, gn〉〈gn, f〉
=∞∑n=1
|〈f, gn〉|2
Definition 3.9. If any of the equivalences in Theorem 3.8 hold, then we refer to
the sequence gn ⊂ H as a Bessel sequence.
With respect to Bessel sequences, T is the synthesis or pre-frame operator,
T ∗ is the analysis operator, G is the Gram matrix and S is the frame operator.
The Gram matrix G has entries Gmn = 〈gn, gm〉 which are related to the inner
products involved in the Gram-Schmidt process. In addition, the Gram matrix
G : `2 → `2 is a synthesis operator for the Bessel sequence T ∗gn (recall the 1-1
correspondence between bounded linear operators and Bessel sequences).
3.3 Complete Bessel Sequences
In this section, we characterize Bessel sequences which are complete. There
is a 1-1 correspondence between bounded linear operators T : `2 → H with dense
range and complete Bessel sequences gn ⊂ H through the binding gn = T (δn).
Definition 3.10. Suppose gn is a sequence in a Hilbert space H. Then, gn
is said to be complete if its linear span is dense in H.
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Lemma 3.11. The following are equivalent for a Bessel sequence gn.
1. gn is complete.
2. T ∗ is injective.
Proof. Observe the following equivalences.
gn is complete.
⇔ (spangn)⊥ = 0.
⇔ f ∈ H and f ∈ (spangn)⊥ together imply f = 0.
⇔ f ∈ H and 〈f, gn〉 = 0 for every n ∈ N together imply f = 0.
⇔ f ∈ H and T ∗(f) = 0 together imply f = 0.
⇔ T ∗ is injective.
3.4 ω-Linearly Independent Bessel Sequences
In this section, we characterize Bessel sequences which are ω-linearly indepen-
dent. There is a 1-1 correspondence between bounded linear injective operators
T : `2 → H and ω-linearly independent Bessel sequences gn ⊂ H through the
binding gn = T (δn).
Definition 3.12. Suppose gn is a sequence in a Hilbert space H. Then, gn is
said to be ω-linearly independent if cn = 0 is the only sequence in `2 for which
the partial sumsN∑n=1
cngn converge to 0 in H.
75
Lemma 3.13. The following are equivalent for a Bessel sequence gn.
1. gn is ω-linearly independent.
2. T is injective.
Proof. Observe the following equivalences.
gn is ω-linearly independent.
⇔ cn ∈ `2 andN∑n=1
cngn converges to 0 together imply cn = 0.
⇔ cn ∈ `2 and T (cn) = 0 together imply cn = 0.
⇔ T is injective.
3.5 Frame Sequences
In this section, we characterize frame sequences. There is a 1-1 correspon-
dence between bounded linear operators T : `2 → H with closed range and frame
sequences gn ⊂ H through the binding gn = T (δn). We also characterize frame
sequences using T , T ∗, G and S.
Corollary 3.14. Suppose gn is a Bessel sequence. Let T , T ∗, G and S be the
operators defined in Theorem 3.8. Then, the following statements are equivalent.
1. The reduction of T is a bijective map.
2. The reduction of T ∗ is a bijective map.
76
3. The reduction of G is a bijective map.
4. The reduction of S is bijective map.
5. R(T ) is closed (R(T ) = N (T ∗)⊥).
6. R(T ∗) is closed (R(T ∗) = N (T )⊥).
7. R(G) is closed (R(G) = N (T )⊥).
8. R(S) is closed (R(S) = N (T ∗)⊥).
9. There exists A > 0 such that√A ‖s‖ ≤ ‖T (s)‖ for every s ∈ N (T )⊥.
10. There exists A > 0 such that√A ‖f‖ ≤ ‖T ∗f‖ for every f ∈ N (T ∗)⊥.
11. There exists A > 0 such that AI ≤ G on N (T )⊥.
12. There exists A > 0 such that AI ≤ S on N (T ∗)⊥.
If any of these statements hold, we may use the following value for A.
1A
=∥∥∥(T )−1
∥∥∥2
=∥∥∥(T ∗)−1
∥∥∥2
=∥∥∥(G)−1
∥∥∥ =∥∥∥(S)−1
∥∥∥Proof. This Corollary follows from Theorem 3.5. Notice that G, S are equal to
the compositions T ∗T and T T ∗ respectively.
Definition 3.15. If any of the equivalences in Corollary 3.14 hold, then we refer
to the sequence gn ⊂ H as a frame sequence.
77
3.6 Frames
In this section, we characterize frames. There is a 1-1 correspondence between
bounded linear surjective operators T : `2 → H and frames gn ⊂ H through
the binding gn = T (δn). We also characterize frames using T , T ∗, G and S.
Corollary 3.16. Suppose gn is a Bessel sequence. Let T , T ∗, G and S be the
operators defined in Theorem 3.8. Then, the following statements are equivalent.
1. gn is a complete frame sequence.
2. T is a surjective map.
3. R(T ∗) is closed and T ∗ is an injective map.
4. S is a bijective map.
5. There exists A > 0 such that√A ‖f‖ ≤ ‖T ∗f‖ for every f ∈ H.
6. There exists A > 0 such that AI ≤ S on H.
If any of these statements hold, we may use the following value for A.
1A
=∥∥∥(T )−1
∥∥∥2
=∥∥∥(T ∗)−1
∥∥∥2
=∥∥∥(G)−1
∥∥∥ = ‖S−1‖
Proof. This Corollary follows from Corollary 3.7 and Lemma 3.11. Notice that G
and S are equal to the compositions T ∗T and T T ∗ respectively.
Definition 3.17. If any of the equivalences in Corollary 3.16 hold, then we refer
to the sequence gn ⊂ H as a frame.
78
Notice that if, gn is a frame, then we have the following reconstruction
formulas. For every f ∈ H,
f = SS−1f =∞∑
n=−∞〈f, S−1gn〉gn (35)
f = S−1Sf =∞∑
n=−∞〈f, gn〉S−1gn (36)
Further if gn is a frame, the sequence (S)−1gn is equal to the sequence
S−1gn and is referred to simply as the canonical dual frame. Any sequence hn
which may replace S−1gn in Equations 35 and 36 is known as a dual frame.
Also, notice that gn is a frame if there exists a lower frame bound A > 0
and an upper frame bound B > 0 such that for every f ∈ H,
A ‖f‖2 ≤∞∑n=1
|〈f, gn〉|2 ≤ B ‖f‖2(37)
3.7 Riesz Sequences
In this section, we characterize Riesz Sequences. There is a 1-1 correspondence
between bounded linear injective operators T : `2 → H with closed range and
Riesz sequences gn ⊂ H through the binding gn = T (δn). We also characterize
Riesz sequences using T , T ∗, G and S.
Definition 3.18. Suppose gn, hn are sequences in a Hilbert space H. Then,
gn, hn are said to be biorthogonal if 〈gm, hn〉 = 〈δm, δn〉 for every m,n ∈ Z.
Definition 3.19. We say that gn is minimal if there exists a sequence hn
such that gn, hn are biorthogonal.
79
Corollary 3.20. Suppose gn is a Bessel sequence. Let T , T ∗, G and S be the
operators defined in Theorem 3.8. Then, the following statements are equivalent.
1. gn is an ω-linearly independent frame sequence.
2. R(T ) is closed and T is an injective map.
3. T ∗ is a surjective map.
4. G is a bijective map.
5. There exists A > 0 such that√A ‖cn‖ ≤ ‖T (cn)‖ for every cn ∈ `2.
6. There exists A > 0 such that AI ≤ G on `2.
7. gn is a minimal frame sequence.
8. There exists a Bessel sequence hn such that gn, hn are biorthogonal.
9. S is a bijective map and gn, (S)−1gn are biorthogonal.
If any of these statements hold, we may use the following value for A.
1A
=∥∥∥(T )−1
∥∥∥2
=∥∥∥(T ∗)−1
∥∥∥2
= ‖G−1‖ =∥∥∥(S)−1
∥∥∥Proof. First of all, equivalences 1-6 follow from Theorem 3.7 and Lemma 3.13.
Notice that G and S are equal to the compositions T ∗T and T T ∗ respectively.
2 ⇒ 9. Suppose 2 holds. By Corollary 3.14, S is a bijective map. Now, let
m,n ∈ Z. Since T is injective, δn is the only sequence in `2 such that T (δn) = gn.
Thus, 〈gm, (S)−1gn〉 = 〈(T )−1gm, (T )−1gn〉 = 〈δm, δn〉.
80
9 ⇒ 8. By Theorem 3.8, S−1(gn) is a Bessel sequence if the sequence
N∑n=1
cnS−1(gn) is a Cauchy sequence. However, N2 ≥ N1 implies
∥∥∥∥ N2∑n=1
cnS−1(gn)−
N1∑n=1
cnS−1(gn)
∥∥∥∥ =
∥∥∥∥S−1( N2∑n=N1
cngn
)∥∥∥∥≤
∥∥∥S−1∥∥∥ ‖T‖( N2∑
n=1
|cn|2 −N1∑n=1
|cn|2)1/2
8⇒ 7. Suppose 8 holds. Let Tgn and T ∗hn represent the synthesis and analysis
operators for the Bessel sequences gn and hn, respectively. By Definition 3.19,
gn is minimal. We show gn is a frame sequence by showing that R(Tgn) is
closed (see Definition 3.15 and Corollary 3.14).
Let f ∈ R(Tgn). For each l ∈ N, choose fl ∈ R(Tgn) such that liml→∞
fl = f . Find
a matrix C whose rows are in `2 such that fl =∞∑m=1
Cmlgm for each l ∈ N. The
following argument uses the fact that gn, hn are biorthogonal along with the
fact that Tgn , T ∗hn are continuous to show that f ∈ R(Tgn). Since f was arbitrary,
this implies that the range of Tgn is closed.
f = liml→∞
fl
= liml→∞
∞∑m=1
Cmlgm
= liml→∞
∞∑n=1
〈∞∑m=1
Cmlgm, hn〉gn
= liml→∞
Tgn(T ∗hn(fl))
= Tgn( liml→∞
T ∗hn(fl))
= Tgn(T ∗hn( liml→∞
fl))
= Tgn(T ∗hn(f))
81
7 ⇒ 1 Suppose 7 holds. Let hn be a sequence such that gn, hn are
biorthogonal. Suppose cn ∈ `2 and the partial sumsN∑n=1
cngn converge to zero.
If we can show that cn = 0, then we will know gn is ω-linearly independent
since cn was arbitrary. However, by continuity, for each m ∈ N,
cm = limN→∞
N∑n=1
cn〈gn, hm〉
=⟨
limN→∞
N∑n=1
cngn, hm
⟩= 0
Definition 3.21. If any of the equivalences in Corollary 3.20 hold, then we refer
to the sequence gn ⊂ H as a Riesz sequence.
Notice that if gn is a Riesz sequence, G is the synthesis operator for the
complete Riesz sequence T ∗gn. Also, notice that gn is a Riesz sequence if
there exists a lower Riesz bound A > 0 and an upper Riesz bound B > 0 such
that for every cn ∈ `2,
A∞∑n=1
|cn|2 ≤∥∥∥∥ ∞∑n=1
cngn
∥∥∥∥2
≤ B∞∑n=1
|cn|2 (38)
3.8 Riesz Basis
In this section, we characterize Riesz Basis. There is a 1-1 correspondence
between bounded linear bijective operators T : `2 → H and Riesz bases gn ⊂ H
through the binding gn = T (δn). We also characterize using T , T ∗, G and S.
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Theorem 3.22. Suppose gn is a Bessel sequence. Let T , T ∗, G and S be the
operators defined in Theorem 3.8. Then, the following statements are equivalent.
1. gn is an ω-linearly independent frame.
2. gn is a complete Riesz sequence.
3. T is a bijective map.
4. T ∗ is a bijective map.
5. S is a bijective map and gn, S−1gn are biorthogonal.
Furthermore, ‖T−1‖2= ‖(T ∗)−1‖2
= ‖G−1‖ = ‖S−1‖.
Proof. This Theorem follows from Corollary 3.16 and Corollary 3.20.
Definition 3.23. If any of the equivalences in Corollary 3.22 hold, then we refer
to the sequence gn ⊂ H as a Riesz basis.
If gn is a Bessel sequence, the frame and Riesz upper bounds are equal and
may be given as the norm of either T 2, (T ∗)2, G or S. On the other hand, if
gn is a Riesz basis, the Riesz and frame lower bounds are also equal and may
be given as the norm of either T−2, (T ∗)−2, G−1 or S−1.
We conclude this section with Table 10 which summarizes the characterizations
given through this chapter. Notice that in the last chapter, we used the term
complete Riesz sequence to describe a Riesz basis.
83
Table 10: Characterizing Bessel Sequences
gn is complete gn is a frame sequence gn is ω-linearly independent
R(T ) is dense R(T ) is closed T is injective
T ∗ is injective R(T ∗) is closed R(T ∗) is dense in `2
gn is a frame
T is surjective
S is bijective
gn is a Riesz sequence
T ∗ is surjective
G is bijective
gn is a Riesz basis
This table presents a visual tool for describing the redundant teminology in the
literature surrounding Bessel sequences. The table assumes that gn is a Bessel
sequence and T , T ∗, G and S are the synthesis operator, analysis operator, Gram
matrix and frame operator associated to gn, respectively. Items occupying the
same set of columns are equivalent. As an example, ”gn is complete” is equiva-
lent to ”R(T ) is dense in H.” As another example, ”gn is a Riesz sequence” is
equivalent to the combination of the following properties: ”R(T ∗) is closed” and
”gn is ω-linearly independent.”
84
3.9 Comparison
In this section, we describe additional pieces presented in this chapter. We
also explain how our approach to pseudo inverses differs from that in the litera-
ture. We point out certain examples of characterizations which involve redundant
terminology. We also give another possible characterization for the four frame
operators. Finally, we give some additional insight to certain duality principles
discussed in the literature.
3.9.1 Additions
We have made many small contributions which are not currently found among the
literature in the context of Riesz Bases. Regarding pseudo inverses, we have added
the following equivalences of Theorem 3.5: 2, 3, 4, 7, 8, 11 and 12. Regarding
surjective bounded linear operators, we have added the following equivalences of
Corollary 3.7: 3 and 5. Regarding Bessel sequences, we have added the following
implications of Theorem 3.8: 2⇒ 1, 3⇒ 1 and 4⇒ 1. Regarding complete Bessel
sequences, we have added Lemma 3.10 of Section 3.3. Regarding frame sequences,
we have added the given implication and following equivalences of Corollary 3.14:
3 ⇒ 5, 2, 4, 10, 11 and 12. Regarding frames, we have added the implication
6 ⇒ 5 of Corollary 3.16. Finally, regarding Riesz sequences, we have added the
following equivalences of Theorem 3.20: 1, 5, 7, 8 and 9.
85
3.9.2 Psuedo Inverse Definition
Let us examine the steps taken to defining a pseudo inverse in [9].
1. Assume the bounded linear operator T : H1 → H2 has closed range.
2. Define the reduction T : N (T )⊥ → H2, T : f 7→ T (f).
3. Write T−1 : N (T ∗)⊥ → N (T )⊥.
4. Extend T−1 to the the pseudo inverse T † : H2 → H1.
In step 3 of his approach, the notation T−1 is abused. That is, the domain
and range of T should correspond to the range and domain of T−1, respectively.
Therefore, it makes more sense to rigorously define the reduction T by restricting
both the domain and range of T . Second, the definition of T † is unnecessary.
Rather than using T †, we found it easier to use T−1, which is the inverse of the
reduction operator. One should compare the implication 5 ⇔ 9 of Theorem 3.5
with the proof of Lemma 5.5.4 in [9] which uses the composition TT †T .
3.9.3 Banach Spaces and Pseudo Inverses
In the beginning of Section 3.1, we claimed that the proofs of the section could
be extended to Banach Spaces. Before continuing, see Table 11, which lists some
differences between Banach and Hilbert space notation.
86
Table 11: Banach and Hilbert Space Notation
Banach Space Notation Hilbert Space Notation
Dual X∗1 , X∗2 H∗1 = H1, H∗2 = H2
T T : X1 → X2 T : H1 → H2
T ∗ T ∗ : X∗1 → X∗2 T ∗ : H2 → H1
H1 quotient space(X1/N (T )
)∗= N (T )⊥ H1/N (T ) = N (T )⊥
H∗2 quotient space(⊥N (T ∗)
)∗= X∗2/N (T ∗) N (T ∗)⊥ = H∗2/N (T ∗)
T T : X1/N (T )→ ⊥N (T ∗) T : N (T )⊥ → N (T ∗)⊥
T ∗ T ∗ : X∗2/N (T ∗)→ N (T )⊥ T ∗ : N (T ∗)⊥ → N (T )⊥
This table presents differences in notation for Banach and Hilbert spaces. Hilbert
spaces, by definition, are reflexive which means that the dual of a Hilbert space
H is isometrically isomorphic to H. Other differences involve the quotient space,
pre-annihilator and annihilator. First, assume X is a Banach space. Then, the
quotient space of a subspace A ⊂ X is the collection of equivalence classes on X,
where two vectors are equivalent if their difference lies in A. The pre-annihilator
of a subspace A∗ ⊂ X∗ is the collection of elements in X which are mapped to 0
by the operators in A∗. The annihilator of a subspace A ⊂ X is the collection of
elements in X∗ which map A to 0. Next, assume H is a Hilbert space. Then, the
quotient space, pre-annihilator and annihilator of a subspace A = A∗ ⊂ H are all
isometrically isomorphic.
87
3.9.4 Frame Sequences and Frames
In [10], a frame sequence is defined as a sequence gn ⊂ H which is a frame for
its closed linear span spangn. Parts of Corollary 3.14 are proved in [10] only
after frames are characterized. Based on Table 10, this seems counterintuitive.
In [10], frames are characterized as sequences whose synthesis operator T sat-
isfies the following three properties: T is well-defined, R(T ) is closed and T ∗ is
injective. A separate characterization is given stating that T is well-defined and
surjective. From Section 3.1, we know that these two characterizations are equiv-
alent based on functional analysis. Another characterization is given stating that
T is bounded on N (T )⊥, T satisfies 9 of Corollary 3.14 and gn is complete. In a
similar fashion, we could pick one equivalence each from Theorem 3.8, Corollary
3.14 and Lemma 3.11 to give almost 100 separate characterizations.
Another interesting example can be found in [2] where a sequence is claimed
to be a frame if and only if T ∗ is well-defined, T is surjective and T satisfies 9 of
Corollary 3.14. It is more natural to state T is well-defined and the statement is
redundant because T satisfies 9 of Corollary 3.14 if and only if R(T ) is closed.
3.9.5 Riesz Bases
In [10], Theorem 3.6.6 gives several characterizations of Riesz Bases. The first
states that Riesz Bases are complete Riesz sequences. The second states that
gn is complete and G is a bijective map. Finally, the third states that gn is a
88
complete Bessel sequence and there exists a complete Bessel sequence hn which
is biorthogonal to gn. This is redundant because hn is automatically complete
given the other assumptions. Really, Theorem 3.6.6 should be used to characterize
Riesz sequences since the assumption that gn is complete is repeated. Notice
that we could give over 300 characterizations of Riesz bases by choosing different
equivalences presented in this chapter.
As another example, Theorem 6.1.1 of [10] gives eight equivalences for a frame
gn to be considered a Riesz basis. A frame gn is a Riesz basis if gn is
ω-linearly independent. However, many of the statements given in Theorem 6.1.1
include redundant assumptions which are automatically satisfied by frames.
3.9.6 Operators associated with Frames
We adopt the terminology from [10] for T , T ∗, G and S even though it may
seem awkward that the frame operator exists for Bessel sequences which are not
frames. We also state an additional characterization for each of these operators.
For instance, any bounded linear operator T : `2 → H is the synthesis operator
for some Bessel sequence. Any bounded linear operator T ∗ : H → `2 is the
analysis operator for some Bessel sequence. Any positive bounded linear operator
G : `2 → `2 is the Gram matrix for some complete Bessel sequence. Finally, any
positive bounded linear operator S : H → H over a separable Hilbert space H
with closed range is the frame operator for some Bessel sequence.
89
3.9.7 Duality Principles
In [10], we find the usual definition for Riesz bases, which states that a sequence
gn is a Riesz basis if there exists an orthonormal basis wn and a bounded lin-
ear bijective operator U such that gn = U(wn). However, this statement implies
that U = TgnT∗wn where Tgn and Twn are the synthesis operators for the sequences
gn and wn, respectively. From this form it is clear that well-defined, injec-
tive, closed range and dense range are all properties inherited from Tgn . Similar
definitions could be made for all of the sequences characterized in this chapter.
In [7], R-duals are discussed. Given a Bessel sequence gn and orthonormal
bases vn and wn one defines the R-dual of gn to be the sequence defined
by hn =∞∑m=1
〈vn, gm〉wm for every n ∈ Z. It is then proven explicitly that gn is
complete if and only if hn is ω-linearly independent and vice versa. However,
this definition implies that Thn = TwnT∗gnTvn . From this form it is clear that well-
defined, injective, closed range and dense range are all properties inherited from
the properties of Tgn . Unlike the case above, two properties are switched. That
is, if Tgn is injective, Thn will have dense range and vice versa.
3.10 Wavelet Based Noise Cancellation Algorithm
We now discuss an algorithm which was developed through a joint collabo-
ration between New Mexico State University (NMSU) and Los Alamos National
Laboratories (LANL). The algorithm uses a wavelet frame to denoise lightning
90
signals. We will introduce both wavelets and wavelet matrices (see [12] and [25])
describing their role in our denoising filter. Then, we will demonstrate the algo-
rithm using an actual lightning signal observation. This observation is coherent
taken at very high frequency (VHF) from the FORTE satellite using the Los
Alamos Portable Pulser (LAPP). While more rigorous approaches may be used to
quantify the reduction of the signal to noise ratio (SNR), they are not presented
here since they tend to involve modelling the lightning signal (see [15]).
Let hn and gn be any two sequences in `2 such that gn = (−1)nh1−n for
all n ∈ Z. We will refer to hn as the scaling filter and gn as the wavelet filter.
Definition 3.24. The approximation operator H : `2 → `2 is defined by the
following equation for all sn ∈ `2 and m ∈ Z.
(Hs)m =∑n∈Z
snhn−2m (39)
Definition 3.25. The detail operator G : `2 → `2 is defined by the following
equation for all sn ∈ `2 and m ∈ Z.
(Gs)m =∑n∈Z
sngn−2m (40)
The approximation operator H and detail operator G may be used to create
an analysis filter bank as shown in Figure 44. Since both operators are invertible,
there is also a synthesis filter bank (see [25] for more details). These filter banks
correspond to the analysis and synthesis operators discussed earlier.
91
Figure 44: 3 Stage Wavelet Transform Analysis Filter Bank
Let N ∈ N. Since H maps 2N -periodic sequences to 2N−1 periodic sequences,
H may be represented by a 2N−1 by 2N matrix HN . The entries of HN must be∑l∈Zhn+l2N−2m for m = 1, ..., 2N−1 and n = 1, ..., 2N . This is due to the fact that for
a 2N -periodic sequence sn and m = 1, ..., 2N−1, we have
(Hs)m =∑n∈Z
snhn−2m
=2N∑n=1
∑l∈Zsn+lNhn+l2N−2m
=2N∑n=1
sn∑l∈Zhn+l2N−2m
Similarly, G may be represented by a 2N−1 by 2N matrix GN where the en-
tries of GN are∑l∈Zgn+l2N−2m for m = 1, ..., 2N−1 and n = 1, ..., 2N . The vertical
concatenation of the matrices HN and GN creates a 2N by 2N matrix WN which
defines the Discrete Wavelet Transform. Just as the individual blocks of Figure
44 have matrix representations, the entire analysis filter bank of Figure 44 may
be represented by the following Wavelet matrix W .
92
W =
WN−2 0 0 0
0 I 0 00 0 I 00 0 0 I
[ WN−1 00 I
]WN
The Wavelet matrix W is invertible and its columns are referred to as wavelets.
Furthermore, W may be used to create an analysis operator defined for any se-
quence sn ∈ `2 using the following technique. First, express sn as a linear
combination of 2N new sequences. Then, break up each new sequence into con-
secutive blocks of length 2N , making sure blocks from separate sequences do not
start at the same index. For instance, the ith new sequence may contain blocks
which start at multiples of 2N but are offset by i (..., −2N + i, i, 2N + i, ...).
Finally, use W to transform each block into a set of wavelet coefficients.
The analysis operator described here is linear and bounded. Since this process
is invertible, the corresponding synthesis operator is surjective. Since sn may
be expressed using different linear combinations, the coefficients are not unique,
implying that the synthesis operator is not injective. From the results in this
section, we conclude that the analysis operator described here is based on a wavelet
frame. Each wavelet in the frame comes from zero padding and shifting the
columns of W .
An analysis filter bank makes use of the analysis operator to obtain a set of
coefficients. Similarly, a synthesis filter bank may reconstruct the signal from
these coefficients via the synthesis operator. Observe the following equalities.
93
sj = 12N
2N∑l=1
sj
= 12N
2N∑l=1
0∑n=1−l
sj−n2N∑m=1
W−1l,mWm,l+n + 1
2N
2N−1∑l=1
sj−n2N−l∑n=1
2N∑m=1
W−1l,mWm,l+n
= 12N
0∑n=1−2N
2N∑l=1−n
2N∑m=1
W−1l,mWm,l+nsj−n + 1
2N
2N−1∑n=1
2N−n∑l=1
2N∑m=1
W−1l,mWm,l+nsj−n
=2N−1∑
n=1−2Nrnsj−n
= (s ∗ r)j
where rn =
1
2N
2N∑m=1
2N∑l=1−n
W−1l,mWm,l+n if 1− 2N ≤ n ≤ 0
12N
2N∑m=1
2N−n∑l=1
W−1l,mWm,l+n if 1 ≤ n ≤ 2N − 1
0 otherwise
These equalities represent using the analysis operator to decompose s into
wavelet coefficients and using the synthesis operator to reconstruct s from wavelet
coefficients. These wavelet coefficients may be thought of as frequency coefficients
since the filters h and g may be thought of as lowpass and highpass filters, respec-
tively. However, instead of using the filter r given above, we will replace r with
the filter given below which accentuates wavelet bands where noise is not present
and suppresses wavelet bands where noise is present.
where rn =
12N
2N∑m=1
1
Meanj
2N∑l=1
Wm,lsl+j
2N∑l=1−n
W−1l,mWm,l+n if 1− 2N ≤ n ≤ 0
12N
2N∑m=1
1
Meanj
2N∑l=1
Wm,lsl+j
2N−n∑l=1
W−1l,mWm,l+n if 1 ≤ n ≤ 2N − 1
0 otherwise
94
In Figures 46-49, we demonstrate the effect of our denoising algorithm on a
lightning observation from the FORTE satellite. We use the Daubechies wavelet
filter with 10 vanishing moments (see [11]) and a 9 stage wavelet packet analysis
filter bank (see Figure 45). The numbers in Figure 45 order the wavelet coefficients
from lowest to highest frequency based on a concept called spectral flipping. The
wavelet matrix W associated with Figure 45 is given below.
Figure 45: 3 Stage Wavelet Packet Analysis Filter Bank
W =
WN−2 0 0 0
0 WN−2 0 00 0 WN−2 00 0 0 WN−2
[ WN−1 00 WN−1
]WN
The Matlab code to process our denoising algorithm is given in the Appendix.
One might try using different attenuation coefficients and make use of our algo-
rithm for calculating the spectral flipping indices is given in the appendix.
95
Figure 46: LAPP Lightning Pulse Before Denoising
Figure 47: LAPP Lightning Pulse After Denoising
The lightning occured between samples 5000 and 7000. This occurrence is more
apparent after using the denoising algorithm.
96
Figure 48: Wavelet Packet Matrix
This image matrix depicts the wavelet packet matrix of the denoising algorithm.
Figure 49: Wavelet Packet Coefficients
Each column of this image matrix depicts the wavelet coefficients for a block of
the denoised lightning signal.
97
4 SAMPLING
In this section, we generalize the Shannon Sampling equations for band-limited
functions. We first prove a Poisson Summation formula involving compactly sup-
ported functions in L2(R). We then develop the Shannon sampling formula for
functions in L2(R) based on the DTFT and give two generalizations involving
tiling sets and a general union of intervals. Finally, we develop two periodic
nonuniform sampling formulas based on frames. The first bridges the gap be-
tween the original Shannon formula and periodic nonuniform sampling formulas
found in recent literature. The second gives a slight generalization of the periodic
nonuniform sampling formula given in [24].
For each generalization presented in this section, we compare the sampling
rate to the Landau rate (see [18]), which represents the smallest possible average
sampling rate needed to perfectly reconstruct f from its samples. We also give an
example with respect to a specific set of spectral support.
4.1 Poisson Summation Formula for L2
In this section, we give another form of the Poisson Summation Formula
which will be used in the proofs throughout the chapter. Our approach involves
using the operator notation previously developed as opposed to writing out in-
tegral formulas. This methodology is just a matter of preference, but we prefer
98
this methodology since we may use the commutation and adjoint properties of the
operators involved. In addition, using operator notation gives a short hand nota-
tion for the ideas we are trying to express. We also use the fact that a function
f ∈ L2(R) whose Fourier Transform f is in L1(R) may be evaluated pointwise
using the following operator notation.
f(n) =
∫ ∞−∞
f(ω)e−2πiωtdω = 〈f ,Mn1R〉 (41)
Theorem 4.1. Suppose f ∈ L2(R) has compact support. Let Ω ∈ R. Then, for
any compact interval I, the following equality holds in L2(R).
QI
∞∑n=−∞
TnΩf = 1Ω
∞∑n=−∞
f( nΩ
)M nΩ1I (42)
Proof. Since both∞∑
n=−∞TnΩf and 1
Ω
∞∑n=−∞
f( nΩ
)M nΩ1R are periodic with period Ω, it
is enough to show the above equality holds for I = ΩΣ0. First of all, the function
QΩΣ0
∞∑n=−∞
TnΩf is in L2(ΩΣ0) since only finitely many terms in the summation
are non-zero. Further, the functions D 1ΩMn1Σ0 form an orthonormal basis for
L2(ΩΣ0). Thus, using the properties from Table 1, we have
QΩΣ0
∞∑n=−∞
TΩnf =∞∑
n=−∞〈QΩΣ0
∞∑k=−∞
TΩkf , D 1ΩMn1Σ0〉D 1
ΩMn1Σ0
=∞∑
n=−∞〈DΩf ,Mn1R〉D 1
ΩMn1Σ0
=∞∑
n=−∞D 1
Ωf(n)D 1
ΩMn1Σ0
= 1Ω
∞∑n=−∞
f( nΩ
)M nΩ1ΩΣ0
99
4.2 Basic Shannon Sampling
We now discuss a reconstruction formula related to that given in Section 2.7.
The main difference is that here the equality holds in L2(R) rather than S ′(R).
Another difference is that we are using the Ω-partition of unity 1ΩΣ0 .
Theorem 4.2. Let Σ0 = [−12, 1
2). Let Ω ∈ R. If f is band-limited to Σ = ΩΣ0,
then f satisfies the following equations in L2(R).
f = 1Ω
∞∑n=−∞
f( nΩ
)T nΩR1Σ (43)
f = 1Ω
∞∑n=−∞
f( nΩ
)M nΩ1Σ (44)
Proof. Using Theorem 4.1, we have the following.
f = QΩΣ0 f
= QΩΣ0
∞∑n=−∞
TΩnf
= 1Ω
∞∑n=−∞
f( nΩ
)M nΩ1ΩΣ0
Theorem 4.2 is based on the fact that the functions e−2πinω
Ω 1Σ form an orthog-
onal basis for L2(Σ). It is important to note that if a function is bandlimited
to an interval, Theorem 4.2 may be applied to some modulation of the function.
Equation 43 gives a formula for reconstructing a bandlimited signal after sampling
at the Landau rate Ω, which is equal to the Nyquist rate (supω∈Σ
ω − infω∈Σ
ω) for an
interval. The following theorem extends this result to tiling sets (see [3]).
100
Example 4.3. Suppose f is band-limited to the interval [−5, 5) as indicated by
Figure 50. Then, substituting Ω = 10 and Σ = [−5, 5) into Equation 43 yields
the following reconstruction formula.
f(t) =∞∑
n=−∞f( n
10)sinc(n− 10t)
Figure 50: Support for f from Example 4.3
4.3 Shannon Sampling with a Tiling Set
In many applications, such as Frequency Division Multiple Access (FDMA),
a function actually has frequency support on a union of M intervals. Since there
is an interval containing the M intervals, Theorem 4.2 may be applied to such
functions. However, the sampling rate typically will be much lower than the
Landau rate. For example, if the measure of the M intervals is only 2 percent of
the measure of the smallest interval containing these M intervals (convex hull),
then the Nyquist rate will be 50 times slower than the Landau rate. We now
generalize Theorem 4.2 to a special multiple interval case based on the comments
in Section 2.6 regarding tiling sets (see [3]).
101
Theorem 4.4. Suppose Σ is an Ω-tiling set which is the union of M disjoint
intervals of the form Im = Ω(Om + βm), where the union of the M intervals Om
is the unit interval Σ0 = [−12, 1
2) and βm ∈ Z for m = 0, ...,M − 1. If f is
band-limited to Σ, then f satisfies the following equations in L2(R).
f = 1Ω
∞∑n=−∞
f( nΩ
)M−1∑m=0
T nΩR1Im (45)
f = 1Ω
∞∑n=−∞
f( nΩ
)M−1∑m=0
M nΩ1Im (46)
Proof. Using the properties from Table 1 and Theorem 4.1, we have
f =M−1∑m=0
QΩ(Om+βm)f
=M−1∑m=0
TΩβmQΩOmT−Ωβm f
=M−1∑m=0
TΩβmQΩOm
∞∑n=−∞
TΩnf
= 1Ω
M−1∑m=0
TΩβm
∞∑n=−∞
f( nΩ
)M nΩ1ΩOm
= 1Ω
∞∑n=−∞
f( nΩ
)M−1∑m=0
M nΩ1Ω(Om+βm)
Theorem 4.4 is based on the fact that the functionsM−1∑m=0
e−2πinω
Ω 1Ω(Om+βm) form
an orthogonal basis for L2(Σ). Equation 45 describes reconstruction of a band-
limited function after sampling at the Landau rate Ω.
For an interval I, the formula for 1I will be useful when using Theorem 4.4.
Let |I| and I denote the length and midpoint of the interval I. Then,
1I(t) = |I|e−2πiItsinc(|I|t) (47)
102
Example 4.5. Suppose f is band-limited to the intervals shown in Figure 51.
Then, f has support in a 10-tiling set. Using Ω = 10, M = 2, O0 = [−5, 2.5),
β0 = −2, O1 = [2.5, 5) and β1 = 2, we may apply Theorem 4.4. Therefore, we
may reconstruct f after sampling at the Landau rate of 10 which is 1/5 the rate
required if we used Equation 43. One possible reconstruction formula is given by
the following equation.
f(t) = 110
∞∑n=−∞
f( n10
)M−1∑m=0
1Im( n10− t)
Figure 51: Support for f from Example 4.5
In Theorem 4.4, we may substitute f for T− γΩf given any γ ∈ R. The result
would be the effect of time limiting, modulating, translating and dilating an or-
thonormal basis over the unit interval. Making this substitution into Equation 46
yields the following equation.
f = 1Ω
∞∑n=−∞
f(γ+nΩ
)M−1∑m=0
M γ+nΩ1Im
= 1√Ω
∞∑n=−∞
f(γ+nΩ
)M−1∑m=0
e−2πiβmγD 1ΩTβmMγ+n1Om
103
4.4 General Shannon Sampling Formulas
In many situations, it may be difficult to find the appropriate parameters
necessary to use Theorem 4.4. In order to explore and discuss other possible
generalizations of Theorem 4.2, we will now present three different reconstruction
formulas for the Fourier Transform of a function f which is band-limited to M
disjoint intervals. These formulas are difficult to implement in practice, so we
leave the reconstruction formulas in terms of f . We will discuss advantages and
disadvantages after the theorem is proven.
Theorem 4.6. Suppose Σ has measure Ω and is the union of M disjoint intervals
of the form Im = Ω(Om + βm), where the union of the M intervals Om is the unit
interval Σ0 = [−12, 1
2) and βm ∈ R for m = 0, ...,M − 1. If f is band-limited to Σ,
then f satisfies the following equations in L2(R).
f = 1Ω
∞∑n=−∞
M−1∑m=0
(PImf)( nΩ
)M nΩ1Im (48)
f =∞∑
n=−∞
M−1∑m=0
1|Om|Ω(PImf)( n
|Om|Ω)M n|Om|Ω
1Im (49)
f = 1Ω
∞∑n=−∞
M−1∑m=0
e−2πiβmn(PImf)( nΩ
)M−1∑l=0
e2πiβlnM nΩ1Il
(50)
Proof. We outline the proof. For m = 0, ...,M − 1, the functions MΩβmPImf are
each band-limited to ΩΣ0. Thus, we may substitute f for MΩβmPImf in Equation
44 to obtain Equation 48.
104
Similarly, for m = 0, ...,M − 1, the functions MΩ(Om+βm)PImf are each band-
limited to |Om|ΩΣ0. Thus, we may substitute f for MΩ(Om+βm)PImf and Ω for
|Om|Ω in Equation 44 to obtain Equation 49.
Finally,M−1∑m=0
MΩβmPImf is band-limited to ΩΣ0. Thus, we may substitute f for
M−1∑m=0
MΩβmPImf in Equation 44 to obtain Equation 50.
Equation 48 represents reconstructing each piece PImf separately. The formula
requires M channels each with sampling rate Ω so the total sampling rate is M
times the Landau rate. One advantage of using this formula is that each channel
is sampled at the same points. On the other hand, using this formula requires the
complexity of calculating PImf .
Equation 49 also represents reconstructing each piece PImf separately. The
formula requires M channels each with different sampling rate and the total sam-
pling rate is actually equal to the Landau rate. However, this formula requires
sampling each channel at different points and the complexity of calculating PIm .
Equation 50 explores the concept of using modulation to manipulate the fre-
quency supports of f so that they form a tiling set. The formula requires M
channels each with sampling rate Ω. Unlike the cases above, the interpolation
functions are independent of the channel so the total sampling rate is equal to
the Landau rate. However, just like the cases above, this formula requires the
complexity of calculating PIm .
105
4.5 Periodic Nonuniform Sampling with Only Complete Aliasing
In this section, we introduce periodic nonuniform sampling, which involves
sampling over a periodic set. We view periodic nonuniform sampling as subsam-
pling after sampling at rate ΩK. Therefore, we must assume that no aliasing
occurs with respect to the sampling rate ΩK. On the other hand, aliasing may or
may not occur with respect to the sampling rate Ω.
In Section 4.3, we described how to reconstruct a function from its samples
when the M spectral slices Ω(Om + βm) formed an Ω-tiling set. In that case, no
aliasing occurs with respect to the sampling rate Ω. That is, given any two spectral
slices Im1 , Im2 , there does not exist an n ∈ N such that Im1 + Ωn intersects Im2 .
Our first periodic nonuniform sampling theorem describes the complementary case
in which complete aliasing occurs with respect to the sampling rate Ω. That is,
given any two spectral slices Ij1 , Ij2 there exists an n ∈ N such that Ij1 +Ωn = Ij2 .
See [24] for details on how to generate spectral slices from an arbitrary set.
For the first periodic nonuniform sampling theorem, we will assume that f
is band-limited to an ΩK-tiling set which is the union of J aliases with respect
to the sampling rate Ω. That is, when using the sampling rate Ω, the aliases are
indistinguishable, combining in accordance with the Poisson Summation Formula.
This is comparable to having one equation and J unknowns. In order to distin-
guish between aliases, J different sampling formulas are used. This is comparable
106
to having J equations and J unknowns. Likewise, only certain choices for the J
sampling formulas may be used to distinguish between aliases. We use J sam-
pling formulas based on the DFT. That is, we first choose a J by J matrix W by
extracting a submatrix from the K by K DFT matrix F where F is defined by
Fmn = e−2πimnK for m = 0, ..., K − 1 and n = 0, ..., K − 1. The J rows chosen will
correspond to the first J rows of F while the J columns chosen will depend on the
spacing of the spectral slices. Next, we create each sampling formula so that the
jth sampling formula involves multiplying the aliases by coefficients from the jth
row of W . We will be able to distinguish between aliases using the J sampling
formulas due to the fact that W is invertible (see [24]).
Theorem 4.7. Suppose Σ is an ΩK-tiling set which is the union of J intervals
of the form Ij = Ω(Σ0 + βj), where βj ∈ Z for j = 0, ..., J − 1. Let W represent
the J by J matrix whose entry in the jth row and kth column is [W ]jk = e−2πijβkK .
If f is band-limited to Σ, then f satisfies the following equations in L2(R).
f = 1Ω
∞∑n=−∞
J−1∑j=0
f( j+nKΩK
)J−1∑k=0
[W−1]kj[W ]jkT j+nKΩK
R1Ik (51)
f = 1Ω
∞∑n=−∞
J−1∑j=0
f( j+nKΩK
)J−1∑k=0
[W−1]kj[W ]jkM j+nKΩK
1Ik (52)
Proof. Notice the assumption that Σ is an ΩK-tiling set implies that the βj rep-
resent J of the K integers contained in some K-tiling set. This assumption is
similar to the tiling set assumptions used in Sections 2.8 - 2.10 and is sufficient
for W to be an invertible matrix (see [24]).
107
Before presenting the computations for the proof, let us review some basic
equalities using the operator notation. First of all, TΩβkT−Ωβl is the identity
operator if k = l. In addition, since W is an invertible matrix, we have
J−1∑j=0
[W−1]kj[W ]jl =
1 if k = l
0 otherwise
(53)
Now, assume n ∈ N and j, k ∈ 0, ..., J − 1. From the properties in Table 1,
we have the following equations.
[W ]jkM j+nKΩK
TΩβk = e2πijβkK M j+nK
ΩKTΩβk
= TΩβkM j+nKΩK
(54)
[W ]jkT−ΩβkF = M jΩKT−ΩβkM− j
ΩKF
= M jΩKT−ΩβkFT− j
ΩK
(55)
We now use the assumptions given in the theorem and assume f is band-
limited to Σ. Since Σ is an ΩK-tiling set, QΩΣ0+Ωnf = 0 if n 6∈ β0, ..., βJ−1.
Since T− jΩKf is also band-limited to Σ, we have the following equation.
QΩΣ0M jΩK
J−1∑k=0
T−Ωβk T− jΩKf = M j
ΩK
J−1∑k=0
T−ΩβkQΩΣ0+Ωβk T− jΩKf
= M jΩK
∞∑n=−∞
T−ΩβnQΩΣ0+ΩnT− jΩKf
= QΩΣ0M jΩK
∞∑n=−∞
T−ΩβnT− jΩKf
(56)
108
Note that Equations 53-56 will be applied in a different order than the order
in which they were presented. Applying these equations and using Theorem 4.1
yields the following equalities.
f =J−1∑k=0
QΩ(Σ0+βk)f
=J−1∑k=0
J−1∑l=0
(J−1∑j=0
[W−1]kj[W ]jl)TΩβkT−ΩβlQΩ(Σ0+βl)f
=J−1∑k=0
J−1∑j=0
[W−1]kjTΩβkQΩΣ0
J−1∑l=0
[W ]jlT−Ωβl f
=J−1∑k=0
J−1∑j=0
[W−1]kjTΩβkQΩΣ0M jΩK
J−1∑l=0
T−ΩβlT− jΩKf
=J−1∑k=0
J−1∑j=0
[W−1]kjTΩβkQΩΣ0M jΩK
∞∑n=−∞
T−ΩnT− jΩKf
= 1Ω
J−1∑k=0
J−1∑j=0
[W−1]kjTΩβkQΩΣ0M jΩK
∞∑n=−∞
T− jΩKf( n
Ω)M n
Ω1R
= 1Ω
∞∑n=−∞
J−1∑j=0
T− jΩKf( n
Ω)J−1∑k=0
[W−1]kjTΩβkM j+nKΩK
1ΩΣ0
= 1Ω
∞∑n=−∞
J−1∑j=0
T− jΩKf( n
Ω)J−1∑k=0
[W−1]kj[W ]jkM j+nKΩK
1Ω(Σ0+βk)
Theorem 4.7 is based on the fact that the functions e−2πi(j+nK)ω
ΩK 1Σ form a frame
for L2(Σ) where the functions are indexed by n ∈ Z and j = 0, ..., J − 1. The
functionsJ−1∑k=0
[W−1]kj[W ]jke− 2πi(j+nK)ω
ΩK 1Ik form a dual frame and are also indexed
by n ∈ Z and j = 0, ..., J − 1. Equation 51 gives a formula to reconstruct a band-
limited function after uniform sampling at the Landau rate ΩJ . This should be
compared to Equation 45 which gives a formula to reconstruct a band-limited
function after sampling at the rate ΩK.
109
Example 4.8. Suppose f is band-limited to the intervals shown in Figure 52.
Then, f has support in a 40-tiling set which contains 3 complete aliases with
respect to the sampling rate 10. Using Ω = 10, K = 4, J = 3, β0 = −4, β1 = −2
and β2 = 3, we may apply Theorem 4.7 to obtain following reconstruction formula.
f(t) = 110
∞∑n=−∞
2∑j=0
f( j+4n40
)2∑
k=0
[W−1]kj[W ]jk1Ik(j+4n
40− t)
Figure 52: Support for f from Example 4.8
4.6 Periodic Nonuniform Sampling
We now give a generalization of Theorems 4.4 and 4.7 which involves recon-
struction of a function f from its samples when f is band-limited to an ΩK-tiling
set Σ whose spectral slices Ijm form Ω-tiling sets in one sense and completely alias
in another sense. To be more precise, we assume the spectral slices Ijm satisfy the
following properties (see Table 12).
1. Given j,m1 6= m2, the intersection (Ijm1 +Ωn)∩ Ijm2 is empty for all n ∈ N.
2. Given j1 6= j2,m, there exists an n ∈ N such that (Ij1m + Ωn) = Ij2m.
110
Theorem 4.9. Suppose Σ is an ΩK-tiling set which is the union of J different Ω-
tiling sets Σj. Further suppose that each Ω-tiling set Σj is the union of M disjoint
intervals of the form Ijm = Ω(Om + βjm), where the union of the M intervals
Om is the unit interval Σ0 and βjm ∈ Z for j = 0, ..., J − 1, m = 0, ...,M − 1.
Let Wm represent the J by J matrix whose entry in the jth row and kth column
is [Wm]jk = e−2πijβkm
K . If f is band-limited to Σ, then f satisfies the following
equations in L2(R).
f = 1Ω
∞∑n=−∞
J−1∑j=0
f( j+nKΩK
)M−1∑m=0
J−1∑k=0
[W−1m ]kj[Wm]jkT j+nK
ΩKR1Ikm (57)
f = 1Ω
∞∑n=−∞
J−1∑j=0
f( j+nKΩK
)M−1∑m=0
J−1∑k=0
[W−1m ]kj[Wm]jkM j+nK
ΩK1Ikm (58)
Proof. Using the properties from Table 1 and Theorem 2.15, we have
f =M−1∑m=0
J−1∑k=0
QΩ(Om+βmk)f
=M−1∑m=0
J−1∑k=0
J−1∑l=0
(J−1∑j=0
[W−1m ]kj[Wm]jl)TΩβkmT−ΩβlmQΩ(Om+βlm)f
=M−1∑m=0
J−1∑k=0
J−1∑j=0
[W−1m ]kjTΩβkmQΩOm
J−1∑l=0
[Wm]jlT−Ωβlm f
=M−1∑m=0
J−1∑k=0
J−1∑j=0
[W−1m ]kjTΩβkmQΩOmM j
ΩK
J−1∑l=0
T−ΩβlmT− jΩKf
=M−1∑m=0
J−1∑k=0
J−1∑j=0
[W−1m ]kjTΩβkmQΩOmM j
ΩK
∞∑n=−∞
T−ΩβnT− jΩKf
= 1Ω
M−1∑m=0
J−1∑k=0
J−1∑j=0
[W−1m ]kjTΩβkmQΩOmM j
ΩK
∞∑n=−∞
T− jΩKf( n
Ω)M n
Ω1R
= 1Ω
∞∑n=−∞
J−1∑j=0
T− jΩKf( n
Ω)M−1∑m=0
J−1∑k=0
[W−1m ]kjTΩβkmM j+nK
ΩK1ΩOm
= 1Ω
∞∑n=−∞
J−1∑j=0
T− jΩKf( n
Ω)M−1∑m=0
J−1∑k=0
[W−1m ]kj[Wm]jkM j+nK
ΩK1Ω(Om+βkm)
111
Theorem 4.9 is based on the fact that the functions e−2πi(j+nK)ω
ΩK 1Σ form a frame
for L2(Σ). The functionsM∑m=1
J−1∑k=0
[W−1m ]kj[Wm]jke
− 2πi(j+nK)ωΩK 1Imk form a dual frame.
Equation 57 describes reconstruction of a band-limited function after uniform
sampling at the Landau rate ΩJ .
For comparison, we will rewrite Equations 17 and 18 from [24] using i for
the imaginary number, starting index sets at 0, choosing C = 0, ..., p − 1 and
making the following definitions: Ω = 1TL
, J = p and K = L. With these changes
Equations 17 and 18 of [24] may be written as follows.
f(t) =∞∑
n=−∞
J−1∑j=0
f( j+nKΩK
)φj(t− j+nKΩK
)
φj(ω) = 1Ω
M−1∑m=0
J−1∑l=0
[A−1m ]lj[Am]jl1Gm+Ωkm(l)
Our presentation has several differences from that of [24]. First of all, we give
contraints for spectral slices in terms of tiling sets so that the periodic nonuniform
sampling formula is actually a generalization of the Shannon sampling formula for
tiling sets. Second, the presentation is more rigorous in the sense that it is based
on the distribution theory presented in Chapter 2, giving results in L2(R). In
terms of notation, three other adjustments are made. We use Wm = Am to use
the DFT matrix instead of its inverse. The spectral slices Gm in [24] partition
[0,Ω] while the spectral slices ΩOm here partition [−Ω2, Ω
2) to facilitate comparisons
with the original Shannon formula. Finally, the shifts km(l) are written as βml.
112
Example 4.10. If f is band-limited to the intervals of Figure 53, then we may
use the parameters of Table 12 to show that f satisfies the reconstruction formula
of Table 12. The function f has support in a 40-tiling set which contains 3 10-
tiling sets, each consisting of two intervals (instead of one as in Figure 52). Each
interval from a 10-tiling set is a complete alias of exactly one interval from each
of the remaining 10-tiling sets. Here, Ω = 10, K = 4, J = 3 and M = 2.
Figure 53: Support for f from Example 4.10
Table 12: Uniform Sampling Example
Ω = 10, K = 4, O0 = [−.5, .25), O1 = [.25, .5)
f(t) = 110
∞∑n=−∞
J−1∑j=0
f( j+4n40
)M−1∑m=0
J−1∑k=0
[W−1m ]kj[Wm]jk1Imk(
j+4n40− t)
m = 0 m = 1
j = 0 I00 = Ω(O0 − 4) = [−45, 2− 7.5) I01 = Ω(O1 − 3)I01 = [−27.5,−25)
j = 1 I10 = Ω(O0 + 2) = [15, 22.5) I11 = Ω(O1 − 1) = [−7.5,−5)
j = 2 I20 = Ω(O0 + 3) = [25, 32.5) I21 = Ω(O1 + 2) = [22.5, 25)
113
4.7 More on ΩK-Tiling Sets
In this section, we discuss how to break up an ΩK-tiling set into intervals of
the form Ω(Om + βkm). This process is referred to as spectral slicing.
Theorem 4.11. Suppose Σ is the union of M disjoint intervals, each having finite
measure. Then, there exists Ω, K such that Ω is contained in an ΩK-tiling set Σ.
Furthermore, there exists a partition of [−Ω2, Ω
2), denoted by O0, ..., OM−1 where
M ≤ 2M + 1 such that the following two properties are satisfied.
1. Σ =K−1∪k=0
M−1∪m=0
Ω(Om + βkm) for some βkm ∈ N
2. Σ =J−1∪j=0
Mj−1
∪m=0
Ω(Om +βjm) for some J ≤ K and 0 ≤MJ−1 ≤ ... ≤M0 ≤ M .
We omit the proof of this theorem, since this process is described in detail in
[24]. However, let us summarize the process. First of all, one possible choice for
Σ is the convex hull of Σ. Once the ΩK-tiling set Σ is chosen, Σ may be broken
up into intervals of the form Ikm = Ω(Om + βkm) where the union over a subset
of these intervals forms Σ.
If we place the intervals Ikm into a K by M matrix, as we did in Table 12, we
will obtain a matrix satisfying several properties. First of all, the union over all
intervals is the ΩK-tiling set Σ. Second, the union over all intervals in each row
is an Ω-tiling set. Finally, the entries within a given column are all aliases of each
other with respect to the sampling rate Ω. In fact, we may permute the columns
114
of this matrix or the entries within a column without changing these properties.
Therefore, we may make assumptions about the locations of the intervals whose
union is Σ (ie 0 ≤MJ−1 ≤ ... ≤M0 ≤ M).
Consider the spectral slicing representation shown in Table 13. Notice, for
example, each interval in the third column is an alias of [.1, .25) under the sampling
rate Ω = .5. Since Σ is contained in an ΩK-tiling set, Theorem 4.2 states that any
function which is band-limited to Σ may be reconstructed after sampling at the
rate ΩK which is the sum of the measures of all the intervals in Table 13. Theorem
4.9 reduces this rate to ΩJ which is the sum of the measures of all intervals in the
first J rows (each of these rows has an interval which is contained in Σ). Finally,
the Landau rate is |Σ|, which is the sum of the measures of all intervals which are
contained in Σ.
In Table 13, the double horizontal line separates the first J rows from the
remaining and the ∗ marks the intervals contained in Σ. We have ΩK = 2.5,
ΩJ = 2 and the Landau rate is 1.2. Notice that the efficiency gain between
Shannon sampling and Uniform sampling is ΩK − ΩJ , represented by the rows
which do not have an interval contained in Σ. There still may be an efficiency
gain between Uniform sampling and some other type of sampling. The maximum
efficiency gain possible is ΩJ − |Σ| which is represented by the intervals in the
first J rows which are not contained in Σ.
115
Table 13: Spectral Splicing
Σ = [−2.75,−2.4) ∪ [−2.25,−2) ∪ [−1.75,−1.5) ∪ [1.5, 1.6) ∪ [1.75, 2)
Σ = [−2.75,−1.25) ∪ [1.25, 2.25)
Ω = .5, K = 5, J = 4, O0 = [−.5, 0), O1 = [0, .2), O2 = [.2, .5)
m = 0 m = 1 m = 2
k = 0 I∗00 = Ω(O0 − 5) I∗01 = Ω(O1 − 5) I02 = Ω(O2 − 5)
I∗00 = [−2.75,−2.5) I∗01 = [−2.5,−2.4) I02 = [−2.4,−2.25)
k = 1 I∗10 = Ω(O0 − 4) I∗11 = Ω(O1 + 3) I12 = Ω(O2 − 4)
I∗10 = [−2.25,−2) I∗11 = [1.5, 1.6) I12 = [−1.9,−1.75)
k = 2 I∗20 = Ω(O0 − 3) I21 = Ω(O1 − 4) I22 = Ω(O2 − 3)
I∗20 = [−1.75,−1.5) I21 = [−2,−1.9) I22 = [−1.4,−1.25)
k = 3 I∗30 = Ω(O0 + 4) I31 = Ω(O1 − 3) I32 = Ω(O2 + 3)
I∗30 = [1.75, 2) I31 = [−1.5,−1.4) I32 = [1.6, 1.75)
k = 4 I40 = Ω(O0 + 3) I41 = Ω(O1 + 4) I42 = Ω(O2 + 4)
I40 = [1.25, 1.5) I41 = [2, 2.1) I42 = [2.1, 2.25)
116
5 TIME-FREQUENCY LOCALIZATION OPERATORS
Many sampling formulas use infinitely many samples. However, applications
typically use samples which are finite in number. In order, to reduce the associated
error, one might explore options presented by time-frequency localizations. In this
section, we discuss the time-frequency localization operator PΣQS.
We start with basic properties of these operators and their eigenvalues. We
move on to discuss related sampling formulas and discuss why understanding
the eigenvalues is critical to using these formulas. We then discuss results of
Landau which describe the behavior of the eigenvalues and provide some additional
improvements and discussion. Then, we build on the work of [16], presenting a
theorem which may be used to approximate eigenvalues numerically. Finally, we
discuss the error associated with these numerical approximations.
5.1 Prolate Spheriodal Wave Functions
Here, we will analyze the operator PΣQS where S is a union of M intervals
of total measure T and Σ is a single interval of measure Ω. We refer the reader to
[27] for a similar discussion regarding the case of one time interval. First of all,
using Theorem 2.13, we have PΣQSf = F−1(F (f1S)1Σ) = (f1S) ∗ (F−11Σ) for
any f ∈ L2(R). This expression may be expanded into the following equation.
PΣQSf(t) =∫S1Σ(x− t)f(x)dx (59)
117
Since we will be discussing the eigenvalues and eigenfunctions of PΣQS, let us
quickly define what it means to be an eigenvalue or eigenfunction.
Definition 5.1. Let T be an operator. A non-zero function φ is an eigenfunction
with eigenvalue λ if Tφ = λφ.
We know that neither PΣ or QS, by themselves, are compact operators, since
the unit ball in an infinite dimensional space is not compact. However, the compo-
sition PΣQS satisfies Equation 59 which is an integral equation over a compact set
with a continuous positive definite symmetric kernel. By Mercer’s theorem, PΣQS
is compact, its eigenvalues (non-negative) sum to the integral of the kernel and its
eigenfunctions form an orthogonal basis for L2(S). If we denote the eigenvalues
as λ0, λ1, ... and place them in decreasing order, we have the following equation.
∞∑n=0
λn =∫S1Σ(x− t)dx
The prolate spheroidal wave functions (PSWF) are the eigenfunctions of the
operator PΣQS. We will use φn to represent the normalized nth eigenfunction
corresponding to the eigenvalue λn. Using the fact that the PSWFs are band-
limited to Σ and are eigenfunctions of Equation 59, we have the following two
equations.
φn(t) =∫∞−∞ 1Σ(x− t)φn(x)dx (60)
λnφn(t) =∫S1Σ(x− t)φn(x)dx (61)
118
We may use properties of PΣ andQS to compute the inner product between two
eigenfunctions over the interval S. That is, using the fact that PΣ is idempotent
we have the following equalities.∫Sφm(x)φn(x)dx = 1
λn
∫Sφm(x)PΣQSφn(x)dx
= 1λn〈QSφm, PΣQSφn〉
= 1λn〈PΣQSφm, PΣQSφn〉
= λm〈φm, φn〉= λmδmn
We already know that the eigenfunctions are orthogonal over the whole real
line so combining the previous result yields the following dual orthogonality.
∫∞−∞ φm(x)φn(x)dx = δmn (62)
∫Sφm(x)φn(x)dx = λmδmn (63)
In a sense, the eigenfunctions with the largest eigenvalues are the functions
which are band-limited to Σ and have the most localization in S. That is, as a
special case of Equation 63, we have the following equation.
∫S|φn(x)|2dx = λn (64)
Finally, the PSWFs form an orthonormal basis for the space R(PΣ) and are
band-limited to Σ. Therefore, 1Σ(k− t) has the following orthonormal expansion.
1Σ(k − t) =∞∑n=0
∫∞−∞ 1Σ(k − x)φn(x)dx φn(t)
=∞∑n=0
φn(k)φn(t)(65)
119
5.2 Related Operators
In this section, we will discuss operators whose eigenfunctions and eigenvalues
are related to those of PΣQS. Following the theorems, we present the results of
the section in Table 14.
Theorem 5.2. Let n ≥ 1. Then, the following are equivalent.
1. φ is an eigenfunction of (PΣQS)n with eigenvalue λ.
2. Fφ is an eigenfunction of (QΣP−S)n with eigenvalue λ.
3. Dαφ is an eigenfunction of (PαΣQ 1αS)n with eigenvalue λ for any real α 6= 0.
4. Tβφ is an eigenfunction of (PΣQS+β)n with eigenvalue λ for any real β.
5. Mγφ is an eigenfunction of (PΣ−γQS)n with eigenvalue λ for any real γ.
6. Rφ is an eigenfunction of (P−ΣQ−S)n with eigenvalue λ.
Proof. The fact that F , Dα, Tβ, Mγ and R are all unitary operators shows that
if one of the given functions is a non-zero function, then all of the given functions
are non-zero functions. Let α 6= 0, β and γ be any real values. Then, the following
equations may be used to obtain the properties given above.
1. (QΣP−S)nF = F (PΣQS)n.
2. (PαΣQ 1αS)nDα = Dα(PΣQS)n.
120
3. (PΣQS+β)nTβ = Tβ(PΣQS)n.
4. (PΣ−γQS)nMγ = Mγ(PΣQS)n.
5. (P−ΣQ−S)nR = R(PΣQS)n.
Theorem 5.3. Let n ≥ 1. Suppose φ is an eigenfunction of (PΣQS)n with non-
zero eigenvalue λ. Then,
1. QSφ is an eigenfunction of (QSPΣ)n with eigenvalue λ.
2. PΣφ is an eigenfunction of (PΣQS)n with eigenvalue λ.
Proof. 1. Since λ is non-zero real and φ is a non-zero function, λφ is a non-zero
function. Thus, (PΣQS)nφ is a non-zero function which implies that QSφ is a
non-zero function. Further, (QSPΣ)nQS = QS(PΣQS)n
2. First of all, (PΣQS)nφ = λφ. Since λ is non-zero, φ is in the range of PΣ.
Since PΣ is idempotent, φ = PΣφ.
Theorems 5.2 and 5.3 may be generalized to make similar statments. For
example, suppose φ is an eigenfunction of (PΣQS)n with non-zero eigenvalue λ.
Then, FQSφ is an eigenfunction of (P−SQΣ)n with eigenvalue λ. Also, for any
real value γ, FMγφ is an eigenfunction of (QΣ−γP−S)n with eigenvalue λ. Finally,
for any m ∈ N, (PΣQS)mPΣφ is an eigenfunction of (PΣQS)n with eigenvalue λ.
121
Theorem 5.4. Let n ≥ 1. The following are equivalent.
1. φ is an eigenfunction of (PΣQS)n with non-zero eigenvalue λ.
2. φ is an eigenfunction of (PΣQSPΣ)n with non-zero eigenvalue λ.
Proof. 1⇔ 2. Suppose φ is an eigenfunction of (PΣQS)n with non-zero eigenvalue
λ. Then, (PΣQS)nφ = λφ. Since λ is non-zero, φ is in the image of PΣ. Since PΣ
is idempotent, this implies that φ = PΣφ. Thus,
(PΣQSPΣ)nφ = (PΣQS)nPΣφ
= (PΣQS)nφ
= λφ
Now, suppose φ is an eigenfunction of (PΣQSPΣ)n with non-zero eigenvalue λ.
Then, (PΣQSPΣ)nφ = λφ. Again, since λ is non-zero, φ = PΣφ. Therefore,
(PΣQS)nφ = (PΣQS)nPΣφ
= (PΣQSPΣ)nφ
= λφ
Theorem 5.5. Let m ≥ 1. Suppose φ is an eigenfunction of PΣQS with non-zero
eigenvalue λ. Then,
1. φ is an eigenfunction for PΣQR/S with eigenvalue 1− λ.
2. φ is an eigenfunction for (PΣQS)m with eigenvalue λm.
122
Proof. 1. Again, since λ is non-zero, φ = PΣφ. We have,
PΣQR/Sφ = (PΣ − PΣQS)φ
= (1− λ)φ
2. (PΣQS)mφ = (PΣQS)m−1λφ = ... = λmφ.
Theorem 5.3 shows how PΣQS and QSPΣ are related. Using arguments similar
to that of Theorem 5.4, we know that the non-zero eigenvalues and corresponding
eigenfunctions for both (QSPΣ)n and (QSPΣQS)n are the same. In a similar
fashion, notice that PΣQS, QΣP−S and P−SQΣ all have the same eigenvalues.
In particular, PΣQS and P−SQΣ have the same eigenvalues. See Table 14 for a
summary of related operators.
Table 14: Related Operators
Fφ, (QΣP−S)n, λ QSφ, (QSPΣ)n, λ
Dαφ, (PαΣQ 1αS)n, λ PΣφ, (PΣQS)n, λ
Tβφ, (PΣQS+β)n, λ φ, (PΣQSPΣ)n, λ
Mγφ, (PΣ−γQS)n, λ φ, PΣQR/S, 1− λ
Rφ, (P−ΣQ−S)n, λ φ, (PΣQS)m, λm
This table shows eigenfunctions and eigenvalues for operators related to PΣQS.
We assume that φ is an eigenfunction for (PΣQS)n with eigenvalue λ where n ∈ N.
We also assume α 6= 0, β and γ are real whereas m ∈ N.
123
Many of the proofs in this chapter involve multiple time intervals and a single
frequency interval but the theorems also hold for a single time interval and multiple
frequency intervals. In fact, when S is a single interval and Σ is the union of M
intervals, the corresponding PSWFs form an orthonormal basis for R(PΣQS).
5.3 PSWF Sampling Formula
In this section, we will derive a sampling formula based on the PSWFs.
This sampling formula differs from the Walter-Shen formula (see [27]) because we
assume Σ to be a union of M intervals rather than a single interval. One may
consider a multiband signal to be the sum of several single band signals. However,
one would encounter the same issues discussed in Section 4.4. We aim at providing
a formula which does not require the complexity and error gained when band-
limiting the signal before sampling. In this section, we will use φΣ,n to denote an
eigenfunction of PΣQS to distinguish between different sets of eigenfunctions.
Theorem 5.6. Suppose S is an interval of measure T and Σ is a union of M
intervals of total measure Ω. Then, if f is band-limited to Σ, f satisfies the
following reconstruction formula in L2(R).
〈f, φIm,n〉 =∞∑
ν=−∞
1|Im|(PImf)( ν
|Im|)φIm,n( ν|Im|)
(66)
f(t) =M−1∑m=0
∞∑n=0
〈f, φIm,n〉φIm,n(t) (67)
124
Proof. Assume S is an interval of measure T , Σ is a union of M intervals of total
measure Ω and f is band-limited to Σ. Further, suppose m ∈ 0, ...,M − 1 and
n ∈ N. Then, from Equations 4.4 and 65, we have the following calculation.
〈f, φIm,n〉 =∫∞−∞ f(x)φIm,n(x)dx
=∫∞−∞
M−1∑µ=0
1|Iµ|
∞∑ν=−∞
(PIµf)( ν|Iµ|)1Iµ( ν
|Iµ| − x)φIm,n(x)dx
= 1|Im|
∞∑ν=−∞
(PImf)( ν|Im|)
∫∞−∞ 1Im( ν
|Im| − x)φIm,n(x)dx
= 1|Im|
∞∑ν=−∞
(PImf)( ν|Im|)φIm,n( ν
|Im|)
This calculation shows Equation 66. Notice that the eigenfunctions from the
set n ∈ N : φIm,n form an orthonormal basis for R(PImQS), the space of func-
tions bandlimited to Im. Furthermore, if m 6= µ, for any n, ν ∈ N, we have
〈φIm,n, φIµ,ν〉 = 0. This follows from Parseval’s equality and the fact that φIm,n
has frequency support in Im. Since Σ =M−1∪m=0
Im, the eigenfunctions from the set
φIm,n : m = 0, ...,M − 1, n ∈ N form an orthonormal basis for R(PΣQS). Using
basic Hilbert Space theory (see [23]), we obtain Equation 67.
Theorem 5.6 takes the approach of band-limiting f to each interval indepen-
dently and then combining several reconstruction formulas. However, as men-
tioned in Section 4.4, this is not very useful in practice due to the complexity
of band-limiting f . Instead, we may build upon Theorem 4.9 and use periodic
nonuniform sampling to gain the following result.
125
Theorem 5.7. Suppose Σ is an ΩK-tiling set which is the union of J different Ω-
tiling sets Σj. Further, suppose that each Ω-tiling set Σj is the union of M disjoint
intervals of the form Ijm = Ω(Om + βjm), where the union of the M intervals Om
is the unit interval Σ0 and βjm ∈ Z for j = 0, ..., J − 1, m = 0, ...,M − 1. Let
Wm represent the J by J matrix whose entry in the jth row and kth column is
[Wm]jk = e−2πijβkm
K . For simplicity, define Cjkm = [W−1m ]kj[Wm]jk/Ω. If f is
band-limited to Σ, then f satisfies the following reconstruction formulas in L2(R).
〈f, φIkm,n〉 =J−1∑j=0
∞∑ν=−∞
Cjkmf( j+νKΩK
)φIkm,n( j+νKΩK
) (68)
f(t) =M−1∑m=0
J−1∑k=0
∞∑n=0
〈f, φIkm,n〉φIkm,n(t) (69)
f(t) =M−1∑m=0
J−1∑k=0
∞∑n=0
∞∑l=0
〈f, φIkm,n〉〈φΣ,l, φIkm,n〉φΣ,l (70)
Proof. Assume f satisfies the assumptions stated in Theorem 5.7. Further, sup-
pose m ∈ 0, ...,M − 1, k ∈ 0, ..., J − 1 and n ∈ N. Then, from Equations 4.9
and 65 we have the following calculation.
〈f, φIkm,n〉 =∫∞−∞ f(x)φIkm,n(x)dx
=J−1∑j=0
∞∑ν=−∞
Cjkmf( j+νKΩK
)∫∞−∞ 1Ikm( j+νK
ΩK− x)φIkm,n(x)dx
=J−1∑j=0
∞∑ν=−∞
Cjkmf( j+νKΩK
)φIkm,n( j+νKΩK
)
Notice that the eigenfunctions from the set φIkm,n : n ∈ N form an orthonor-
mal basis for R(PIkmQS). For reasoning stated in the proof of Theorem 5.6, the
eigenfunctions from the set φIkm,n : k = 0, ..., J − 1,m = 0, ...,M − 1, n ∈ N
form an orthonormal basis for R(PΣQS) and Equation 69 is satisfied.
126
Now, both f and φΣ,l satisfy the assumptions of Theorem 5.7. Therefore,
they satisfy Equation 69. Using Parseval’s Equality, we have the following inner
product computation.
〈f, φIΣ,l〉 =M−1∑m=0
J−1∑k=0
∞∑n=0
〈f, φIkm,n〉〈φΣ,l, φIkm,n〉
Recall that the eigenfunctions from the set φΣ,l : l ∈ N form an orthonormal
basis for R(PΣQS). Therefore, from basic Hilbert Space theory and using the fact
that both f and φΣ,l satisfy Equation 69, we obtain Equation 70.
5.4 Eigenvalues Greater Than 1/2
In Section 5.1, we learned that the PSWFs are the band limited functions
with greatest localization in S where the amount of localization is directly tied to
the eigenvalue associated with the PSWF. In Section 5.3, we demonstrated how a
band-limited signal could be reconstructed from the PSWFs. Therefore, in order
to reconstruct a signal which is well localized in S, we may use only the PSWFs
with high eigenvalues.
In order to explore the behavior of eigenvalues, we follow the work of Landau,
which is distributed over a series of papers (see [17]-[19]). We will quantify which
eigenvalues are greater than and less than 1/2. Before presenting these estimates,
let us recall some properties from Hilbert Space theory. For the proof of the next
theorem, we refer the reader to [23] and [17].
127
Theorem 5.8. Suppose T is a compact self-adjoint operator on L2(R) so that its
positive eigenvalues λ0, λ1, ... may be listed in non-increasing order. If we let H
denote the set of subspaces of L2(R) with dimension d, then we have
1. λd−1 = maxH∈H
minf∈H
〈Tf,f〉‖f‖2 .
2. λd−1 ≥ minf∈H
〈Tf,f〉‖f‖2 for every H ∈H .
3. λd = minH∈H
maxf∈H⊥
〈Tf,f〉‖f‖2 .
4. λd ≤ maxf∈H⊥
〈Tf,f〉‖f‖2 for every H ∈H .
We now apply this theorem to the operator PΣQS where Σ is a single interval
of measure Ω and S is a union of M intervals of total measure T . Since PΣQS is
compact, PΣQSPΣ is also compact by Theorem 5.4. Using the fact that PΣ and
QS are both self adjoint, we may replace 〈PΣQSPΣf, f〉 with ‖QSf‖2 to obtain
the following corollary.
Corollary 5.9. Let λ0, λ1,... denote the eigenvalues of PΣQS in non-increasing
order. Let H denote the set of subspaces of L2(R) with dimension d. Then,
1. λd−1 = maxH∈H ,H∈R(PΣ)
minf∈H
‖QSf‖2
‖f‖2 .
2. λd−1 ≥ minf∈H
‖QSf‖2
‖f‖2 for every H ∈H such that H ⊂ R(PΣ).
3. λd = minH∈H
maxf∈H⊥∩R(PΣ)
‖QSf‖2
‖f‖2 .
4. λd ≤ maxf∈H⊥∩R(PΣ)
‖QSf‖2
‖f‖2 for every H ∈H .
128
In order to estimate the minimum and maximum number of eigenvalues greater
than 1/2, we will apply Property 2 of Corollary 5.9 to a specific subspace H1 and
apply Property 4 of Corollary 5.9 to a separate subspaceH2. Our next two lemmas
concern the dimensions of these subspaces and a function which will be used to
define H1 and H2.
Lemma 5.10. Suppose Σ is an interval of measure Ω and S is a union of M
disjoint intervals of total measure T . Then, N1 ≥ bΩT c − 2M + 2 and N2 ≤
dΩT e+ 2M − 2 where N1 and N2 are defined by the following equations.
N1 = supβ∈R
∣∣∣N1(β)∣∣∣, N1(β) = n ∈ Z : ΩS + β contains Σ0 + n (71)
N2 = infβ∈R
∣∣∣N2(β)∣∣∣, N2(β) = n ∈ Z : ΩS + β intersects Σ0 + n (72)
Proof. Express ΩS + β in the formM∪m=1
[am, am + qm) using β = −12− Ω min
t∈St.
Notice thatM∑m=1
qm = ΩT and a1 = −12
by the choice of β. Analogous to the
definition of N1 and N2, let N1m and N2m represent the number of intervals of the
form Σ0 + n for which [am, am + qm) contains and intersects Σ0 + n, respectively.
Let brc represent the greatest integer less than or equal to r and dre = −b−rc.
Also, recall that for r ∈ R and n ∈ N, we have the following properties.
1. r ≤ n⇔ dre ≤ n and n ≤ r ⇔ n ≤ brc
2. n < r ⇔ n ≤ dr − 1e and r < n⇔ br + 1c ≤ n
3. dre+ n = dr + ne and brc+ n = br + nc
129
Now, [am, am+qm) will contain Σ0+n if both am ≤ −12+n and 1
2+n ≤ am+qm.
From Property 1, this is equivalent to dam + 12e ≤ n and n ≤ bam + qm − 1
2c.
Similarly, [am, am + qm) will intersect Σ0 + n if am < 12
+ n or −12
+ n < am + qm.
From Property 2, this is equivalent to bam + 12c ≤ n or n ≤ dam + qm − 1
2e.
Therefore, by Property 3, we have the following two equations:
N1m = bam + qm −1
2c − dam +
1
2e+ 1 = bam + qm +
1
2c − dam +
1
2e
N2m = dam + qm −1
2e − bam +
1
2c+ 1 = dam + qm +
1
2e − bam +
1
2c
In addition to the properties above, if r ∈ R, we have brc > r − 1 and
dre < r + 1. Therefore, using the fact that a1 = −12
yields N11 > q1 − 1,
N1m > qm − 2, N21 < q1 + 1 and N2m < qm + 2 for m = 2, ...,M . We have
N1 ≥∣∣∣N1(−1
2− Ω min
t∈St)∣∣∣ =
M∑m=1
N1m > ΩT − 2M + 1
N2 ≤∣∣∣N2(−1
2− Ω min
t∈St)∣∣∣ =
M∑m=1
N2m < ΩT + 2M − 1
From Properties 2 and 3, we obtain N1 ≥ bΩT − 2M + 2c = bΩT c − 2M + 2
and N2 ≥ dΩT + 2M − 2e = dΩT e+ 2M − 2.
Lemma 5.11. Let h(t) = 2 cos(πt)1Σ0(t) where Σ0 = [−12, 1
2). Then, ‖h‖2 = 2
and if f is band-limited to Σ0, then we have the following inequality.
‖f‖2 ≤∞∑
n=−∞|∫
Σ0+nf(t)h(n− t)dt|2 (73)
130
Proof. First of all, from the trigonometric identity cos2(t) = 12
+ 12
cos(2t) we know
h2(t) =(
2 + 2 cos(2πt))1Σ0(t). Therefore, by direct integration, ‖h‖2 = 2.
Now, h = (M− 12
+ M 12)1Σ0 so h = (T− 1
2+ T 1
2)sinc. If we use V to denote the
function |ω| − 1 due to its shape, then sinc ≥ −V on [−1, 1] which implies that
h ≥ −(T− 12
+ T 12)V = 1 on Σ0. Using the fact that h is supported on Σ0 along
with Theorems 2.12, 4.2 and Bessel’s Inequality, we have
‖f‖2 ≤ ‖f ∗ h‖2
=∞∑
n=−∞|f ∗ h(n)|2
=∞∑
n=−∞|∫
Σ0+nf(t)h(n− t)dt|2
(74)
We are now ready to present Landau’s eigenvalue estimate (see [17]). However,
we give a slight improvement, since if ΩT = 7 and M = 3, our estimate implies
λ2 ≥ .5 and λ12 ≤ .5 while Landau’s estimate implies λ1 ≥ .5 and λ14 ≤ .5.
Theorem 5.12. Suppose Σ is a single interval of measure Ω and S is a union of
M intervals of total measure T . Let λ0, λ1, ... denote the eigenvalues of PΣQS in
non-increasing order. Then λN1−1 ≥ 12
and λN2 ≤ 12
where N1 and N2 are defined
as in Lemma 5.10. In particular, λbΩT c−2M+1 ≥ 12
and λdΩT e+2M−2 ≤ 12.
Proof. Let α = 1Ω
, β = −12− Ω min
t∈St and γ = 1
2+ 1
Ωminω∈Σ
ω. By Theorem 5.2,
the eigenvalues of PΣQS and PαΣ−γQ 1αS+β = PΣ0QΩS+β are the same. We will use
Corollary 5.9 to estimate the eigenvalues of PΣ0QΩS+β.
131
Define N1(β) and N1 as in Lemma 5.10 and h as in Lemma 5.11. For n ∈
N1(β), let g1n satisfy g1n ∗ h = Tn(sinc). Let H = spang1n : n ∈ N1(β) and
f ∈ H. We will show that‖QΩS+βf‖2
‖f‖2 ≥ 12
so that by Corollary 5.9, λbΩT c−2M+1 ≥
λN1−1 ≥ 12. Notice that g1n ∗ h(n) = 0 if n 6∈ N1(β). Therefore,
‖f‖2 ≤∞∑
n=−∞|∫
Σ0+nf(t)h(n− t)dt|2
=∑
n∈N1(β)
|∫
Σ0+nf(t)h(n− t)dt|2
≤∑
n=N1(β)
∫Σ0+n
|f(t)|2dt∫
Σ0+n|h(n− t)|2dt
≤ ‖h‖2 ∫t∈ΩS+β
|f(t)|2dt
= 2 ‖QΩS+β‖2
Define N2(β) and N2 as in Lemma 5.10 and h as in Lemma 5.11. For n ∈
N2(β), let g2n(t) = h(n − t). Let H = spang2n : n ∈ N2(β) and f ∈ R(PΣ0) ∩
H⊥. We will show that‖QΩS+βf‖2
‖f‖2 ≤ 12
so that by Corollary 5.9, λdΩT e+2M−2 ≤
λN2 ≤ 12. Notice that 〈f, g2n〉 = 0, if n ∈ N2(β). Therefore,
‖f‖2 ≤∞∑
n=−∞|∫
Σ0+nf(t)h(n− t)dt|2
=∑
n6∈N2(β)
|∫
Σ0+nf(t)h(n− t)dt|2
≤∑
n 6∈N2(β)
∫Σ0+n
|f(t)|2dt∫
Σ0+n|h(n− t)|2dt
≤ ‖h‖2 ∫t6∈ΩS+β
|f(t)|2dt
= 2(‖f‖2 − ‖QΩS+β‖2)
132
It turns out that if the M intervals have endpoints which lie on the integer
lattice, then we may greatly improve the estimate given above. This is due to the
fact that both N1 and N2 will be equal to ΩT .
Corollary 5.13. Suppose Σ is a single interval of measure Ω and S is a union of
M intervals whose total measure is T . Further suppose, each of the M intervals
has the form 1Ω
(Σ0 +n+β0) for some β0 ∈ R. Let λ0, λ1, ... denote the eigenvalues
of PΣQS in non-increasing order. Then, λΩT−1 ≥ 12
and λΩT ≤ 12.
5.5 When No Significant Eigenvalues Exist
Theorem 5.12 suggests that if Σ is a pairwise disjoint union of a large number
of short intervals, PΣQS might fail to have an eigenvalue larger than 1/2, even
though the product area ΩT could be much larger than one. This is due to the
fact that if Σ is a union of M intervals which are highly separated, then the time
localization operators for these intervals are almost orthogonal.
In this section, we will give a bound on the largest eigenvalue λ0 for PΣQS0
where S0 = [−12, 1
2) and Σ is a union of M intervals. However, this bound will
only be useful for the case in which the M intervals all have measure less than 1
and are well separated. At the end of the section, we will give a Corollary that
describes how to contstruct a set Σ based on an arbitrarily large product area ΩT
such that PΣQS0 has no eigenvalues greater than 1/2. In order to prove the bound
on λ0, we will require several lemmas.
133
Lemma 5.14. Suppose S0 = [−12, 1
2) and I is an interval of measure Ω. Further,
suppose ψ ∈ R(PIQS0) where ‖ψ‖ = 1. Then, we have the following estimate.
∫S0|ψ|2dt ≤ |I| (75)
Proof. Suppose ψ ∈ R(PIQS0) where ‖ψ‖ = 1. Then, ψ is the inverse Fourier
Transform of ψ. Using the Cauchy-Schwartz inequality yields the following.
∫S0|ψ|2dt =
∫S0|∫Iψ(ω)e2πiωtdω|2dt
≤∫S0
∫I|ψ(ω)|2dω
∫I|e2πiωt|2dωdt
=∫S0|I|dt
= |I|
Lemma 5.15. Suppose that f1, f2 ∈ L2(R) are band-limited to [−Ω/2,Ω/2]. Then
f1f2 = f1 ∗ f2 is supported in [−Ω,Ω] and |f1f2| ≤ ‖f1‖ ‖f2‖.
Proof. First of all, f1f2(ω) = (f1∗f2)(ω) =∫∞−∞ f1(ξ)f2(ω−ξ)dξ. Suppose |ω| > Ω
and ξ ∈ R. In the case where |ξ| > Ω/2, we have f1(ξ) = 0. In the other case
where |ξ| ≤ Ω/2, we have |ω − ξ| > Ω/2 and f2(ω − ξ) = 0. Therefore, f1 ∗ f2
is supported in [−Ω,Ω]. In addition, from the Cauchy-Schwartz inequality and
Parseval’s equality, we have the following.
|∫∞−∞ f1(ξ)f2(ω − ξ)dξ| ≤
( ∫∞−∞ |f1(ξ)|2dν
)1/2( ∫∞−∞ |f2(ω − ξ)|2dν
)1/2
= ‖f1‖ ‖f2‖
134
Lemma 5.16. Suppose S0 = [−12, 1
2) and I1, I2 are two intervals of measure Ω.
Further, suppose ψ1 ∈ R(PS0QI1) and ψ2 ∈ R(PS0QI2) where ‖ψ1‖ = ‖ψ2‖ = 1.
Finally, assume the distance ω0 between the midpoints I1 and I2 is greater than
Ω. Then, we have the following estimate.
∫S0ψ1(t)ψ2(t)dt ≤ 2Ω
ω0−Ω(76)
Proof. Define f1 = MI1ψ1 and f2 = MI2
ψ2. Then, from Table 1, f1 and f2 are
bandlimited to [−Ω/2,Ω/2]. According to Lemma 5.15 and Theorem 2.12, we
have the following inequalities.
|∫S0ψ1(t)ψ2(t)dt| = |
∫∞−∞ f1(t)f2(t)1S0(t)e−2πiω0t dt|
=∣∣∣ ∫∞−∞ f1f2(ξ)1S0(ω0 − ξ)dξ
∣∣∣≤
∫ Ω
−Ω|sinc(ω0 − ξ)|dξ
≤ 2Ωω0−Ω
Theorem 5.17. Assume that S0 = [−12, 1
2) and Σ is the union of K intervals
I0,...,IK−1 each having measure Ω. Further suppose that the minimum distance
ω0 between any two midpoints I0, ..., IK−1 is greater than Ω. Then, we have the
following estimate.
λ0 ≤ Ω + 2Ωω0−Ω
(K2 −K) (77)
135
Proof. Let φ0 denote the eigenfunction of PΣQS0 which has the largest eigenvalue
λ0 and assume ‖φ0‖ = 1. Write φ0 =K−1∑k=0
akψk where ψk ∈ R(PS0QIk) and
‖ψk‖ = 1. Notice thatK−1∑k=0
|ak|2 = 1 and λ0 =∫S0|φ0(t)|2dt. Using Lemma 5.14
and Lemma 5.16, we have the following equalities.∫S0|φ0(t)|2dt =
∣∣∣K−1∑j=0
K−1∑k=0
ajak∫S0ψj(t)ψk(t)dt
∣∣∣=
K−1∑k=0
|ak|2∫S0|ψk(t)|2dt+
K−1∑j=0
K−1∑k=0,k 6=j
ajak∫S0ψj(t)ψk(t)dt
≤ ΩK−1∑k=0
|ak|2 +K−1∑j=0
K−1∑k=0,k 6=j
2Ωω0−Ω
≤ Ω + 2Ωω0−Ω
(K2 −K)
Corollary 5.18. For every N ∈ N, there exists a K and a set Σ such that Σ has
measure N , Σ is the union of K intervals of discrete measure Ω and PΣQS0 has
no eigenvalues larger than 1/2.
Proof. Choose Ω = 14
and K = 4N . Next, choose ω0 such that 2Ωω0−Ω
(K2−K) ≤ 14.
Finally, choose K intervals I0, ..., IK−1 each of measure Ω such that Ik = 14
+ kω0
for k = 0, ..., K − 1. For Σ =K−1∪k=0
Ik, Theorem 5.17 shows that λ0 ≤ 12
where λ0 is
the largest eigenvalue of PΣQS0 .
One interpretation of Theorem 5.17 is that a signal cannot be well localized in
time and be bandlimited to highly separated intervals of small measure. However,
even for such a case, the PSWFs have the best time localization and their use will
require the fewest samples for reconstruction.
136
5.6 Eigenvalue Calculations
In this section, we give a method for numerically estimating the eigenvalues
of the time-frequency localization operator PΣQS. Following the theorem, we
present an example demonstrating this method. One should take note of how this
algorithm was coded by referencing the Matlab code given in the Appendix.
Theorem 5.19. Suppose Σ and S are subsets of R. Suppose we have the sampling
formula f(t) =∑n∈N
f( nΩ
)hn(t) for all f band-limited to Σ. Define the matrix A
(possibly infinite) such that Amn =∫S1Σ(x− m
Ω)hn(x)dx for m ∈ Z and n ∈ N .
Then, if φ is an eigenfunction of the operator PΣQS with eigenvalue λ, then the
vector φ( nΩ
) : n ∈ N is an eigenvector of the matrix A with eigenvalue λ.
Conversely, if the vector φ( nΩ
) : n ∈ N is an eigenvector of the matrix A
with eigenvalue λ and the function φ(t) =∑n∈N
φ( nΩ
)hn(t) converges, then φ is an
eigenfunction of the operator PΣQS with eigenvalue λ.
Proof. Now, suppose φ is an eigenfunction of the operator PΣQS with eigenvalue
λ. Then, φ is band-limited to Σ. Therefore, φ(t) =∑n∈N
φ( nΩ
)hn(t). Plugging
this into the expression for PΣQSφ, yields λφ(mΩ
) = PΣQSφ(mΩ
) =∑n∈N
Amnφ( nΩ
).
Therefore, φ( nΩ
) : n ∈ N is an eigenvector of the matrix A with eigenvalue λ.
On the other hand suppose φ( nΩ
) : n ∈ N is an eigenvector of the matrix A
with eigenvalue λ and the function φ(t) =∑n∈N
φ( nΩ
)hn(t) converges. Then, φ and
PΣQS are both band-limited to Σ. Therefore,
137
PΣQSφ(t) =∑m∈N
PΣQSφ(mΩ
)hm(t)
=∑m∈N
∑n∈N
Amnφ( nΩ
)hm(t)
= λ∑m∈N
φ(mΩ
)hm(t)
= λφ(t)
Notice that it is possible to obtain more accurate estimates by using a sampling
formula with better efficiency. However, using small spectral slices may cause the
DFT matrix to be ill conditioned.
Example 5.20. Let S = [−.5, .5) and Σ = [0, 1.3)∪ [4.5, 5). We will calculate the
eigenvalues of PΣQS. For these sets, the parameters may be chosen as in Table
5.20 or as in Table 5.20. The eigenvalues calculated for each case are shown in
Tables 5.20. In both cases, we truncated the matrix A to a 21 by 21 matrix. One
should take particular note of how the algorithm was coded up to depend on the
parameters chosen (see Appendix).
Table 15: Eigenvalues for Example 5.20
n = 0 n = 1 n = 2 n = 3 n = 4
λn, Set 1 0.88342 0.47761 0.32722 0.03670 0.02495
λn, Set 2 0.86616 0.50594 0.32567 0.04507 0.01162
138
Table 16: First Set of Parameters for Example 5.20
Ω = 5, K = 1, O0 = [0, .26), O1 = [.26, .5), O2 = [−.5,−.1), O3 = [−.1, 0)
hn(x) = 1Ω
(1[0,1.3) + 1[4.5,5)(t)
)|t= n
Ω−x
m = 0 m = 1 m = 2 m = 3
j = 0 I0 = Ω(O0 + 0) I1 = Ω(O1 + 0) I2 = Ω(O2 + 1) I3 = Ω(O3 + 1)
I0 = [0, 1.3) I1 = [1.3, 2.5) I2 = [2.5, 4.5) I3 = [4.5, 5)
Table 17: Second Set of Parameters for Example 5.20
Ω = .5, K = 10, O0 = [0, .5), O1 = [−.5,−.4), O2 = [−.4, 0)
hj+nK(x) = 1Ω
2∑m=0
3∑k=0
[W−1m ]kj[Wm]jk1Ikm(t)|t= j+nK
ΩK−x
m = 0 m = 1 m = 2
j = 0 I00 = Ω(O0 + 0) I01 = Ω(O1 + 1) I02 = Ω(O2 + 1)
I00 = [0, .25) I01 = [.25, .3) I02 = [.3, .5)
j = 1 I10 = Ω(O0 + 1) I11 = Ω(O1 + 2) I12 = Ω(O2 + 2)
I10 = [.5, .75) I11 = [.75, .8) I12 = [.8, 1)
j = 2 I20 = Ω(O0 + 2) I21 = Ω(O1 + 3) I22 = Ω(O2 + 3)
I20 = [1, 1.25) I21 = [1.25, 1.3) I22 = [1.3, 1.5)
j = 3 I30 = Ω(O0 + 9) I31 = Ω(O1 + 10) I32 = Ω(O2 + 10)
I30 = [4.5, 4.75) I31 = [4.75, 4.8) I32 = [4.8, 5)
139
5.7 Matrix Error Bounds
In section 5.6 we established a very general result relating projections onto
Paley-Wiener spaces with samples of corresponding interpolating functions. In
order to give the result practical value, one would like to know how accurately
one can approximate the samples φ( nΩ
) : n ∈ N of the eigenvectors from
finite dimensional approximations of the matrix Amn =∫SR1Σ(m
Ω− x)hn(x)dx.
Since the Amn are defined in terms of integrals, there is a secondary question
of how accurately one can estimate these entries. We will ignore this issue. In
practical applications, the singular value decomposition of A would be a one time
calculation. On the other hand, slow decay of interpolating functions suggests
that a large truncation is needed to obtain accurate sample estimates. In this
section we will not address truncation errors for the general case. Instead we
will focus specifically on the case of the matrix corresponding to the traditional
truncations to time and frequency interval pairs with the goal of obtaining sharp
estimates on the norm of the truncation remainder.
Our method will not address the accuracy of truncations of A when restricted
to specific eigenspaces. In the case of the prolate spheroidal wave functions this
question was already addressed, in effect, in the work of Walter and Shen (see [27]
and [26]). Their Fourier transform is normalized as f(ξ) =∫f(t)eitξ dt. With ϕn
140
the λn-eigenfunction of P[−π,π]Q[−T,T ], Walter and Shen proved that
∑|k|>T
ϕn(k)2 ≤ (1− λn) + 4πT(1− λn
3λn
)1/2
≡ (1− λn)1/2C(n, T ) (78)
As a consequence, they showed that if f ∈ span ϕ1, . . . , ϕN, N ≤ 2T , then∫ T
−T
∣∣∣f(t)−∑|k|≤T
f(k) [2T ]∑n=0
ϕn(k)ϕn(t)1[−T,T ]
∣∣∣2 ≤ ‖f‖2
N∑n=0
λn(1− λn)1/2C(n, T )
Thus, one can approximate f very effectively in L2 using a finite number of samples
if f is in the subspace of time-frequency localized signals.
Now we are ready to consider approximations via truncations of the matrix
A. Specifically, let A denote the “time-frequency localization matrix” with entries
Ak` =∫ T/2−T/2 sinc(x−k) sinc(x−`) dx. We have seen that the eigenvectors of A are
the samples of the eigenfunctions of PΣQT . Thus one can compute the projection
of any f ∈ R(P[−π,π]Q[−T,T ]) onto the eigenfunctions by means of sample inner
products in `2(Z). This requires all samples of f . One would like, instead, to
estimate the projections from finitely many samples as Shen and Walter did for
eigenspaces. Since the PSWFs with eigenvalues close to one are also the most
localized on [−T/2, T/2] one expects these also to be the ones whose samples
decay fastest off the time support. However, this leaves open the problem of
estimating the projections onto the PSWFs in the transition region. Ideally one
would like direct estimates on the decay of the eigenvectors of A and estimates
on some truncation of A that respects these decay estimates. What we propose
in this section is really more basic: we want to prove `2-bounds on the remainder
141
A = AN obtained by truncating the entry Ak` to zero if max|k|, |`| > N .
Obtaining bounds actually is not difficult. It takes a little more work to show that
the bounds are essentially sharp and to provide a formula from which numerical
approximations of the eigenfunctions can be computed efficiently.
In the following we denote by A = A(T ) the matrix with entries
Ak` =
∫ T
−Tsinc(x− k) sinc(x− `) dx, k, ` ∈ Z (79)
Unless specifically stated otherwise we will assume that T ∈ N though, again,
this is really only for notational convenience. Now if |x| ≤ T and |k| > T , then
x−k ∈ [−T −k, T −k] so that |sinc(x−k)| ≤ 1π
max 1|T+k| ,
1|T−k| = 1
π1
|k|−T . Since
the same holds for `, if |k|, |`| > T , we have
|Ak`| ≤ 2T B(k)B(`) where B(k) =1
π
1
|k| − T(80)
If one of k or ` is no bigger than T then we can get a corresponding dependence
logarithmic in T . For example, if |k| < T then
|Ak`| =∣∣∣∫ T
−Tsinc(x− k) sinc(x− `) dx
∣∣∣≤ 2
πB(`)
∫ T
−T
1
1 + |x− k|dx
=2
πB(`)
∫ T−k
−T−k
1
1 + |u|dx
≤ 2
πB(`) maxln |T − k|, ln |T + k| ≤ CB(`) ln(T ) (81)
There will always be essentially 2T values of k and ` that are less than T in
absolute value.
142
Proposition 5.21. Let A = A(T ) be as in (79) and let Atr = AtrN be such that
Atrk` = Ak` if max|k|, |`| ≤ NT and Atr
k` = 0 otherwise. Then A = A− Atr is `2
bounded with norm at most C ln(T )N−1/2 where C is independent of N and T .
Proof. We use (80) and (81) to obtain elementwise bounds. Fix s = s` ∈ `2(Z).
Case 1: |k| ≤ T .
The only nonzero terms of Ak` then correspond to |`| > NT . If |k| ≤ T then
|Ak`| ≤ C ln(T )|`|−T with |`| ≥ NT . In this case, we have
∣∣∑`
Ak`s`∣∣ ≤ C ln(T )
(−NT−1∑−∞
+∞∑
NT+1
1
|`| − T|s`|)
We can consider the cases ` > 0 and ` < 0 separately. For the positive terms,
∞∑NT+1
1
|`| − T|s`| ≤
( ∞∑NT+1
1
(|`| − T )2
)1/2( ∞∑NT+1
|s`|2)1/2
Together with the terms for ` < 0, when |k| ≤ T , one obtains
∣∣∑`
Ak`s`∣∣ ≤ C ln(T )
( ∞∑NT+1
1
(|`| − T )2
)1/2
‖s‖`2 ≤ Cln(T )√NT‖s‖`2
Case 2: T < |k| ≤ NT .
That Ak` = 0 unless |`| > NT still pertains, but now the Ak` satisfy the
inequality |Ak`| ≤ 2T B(k)B(`) ≤ 2T (|k|−T )−1(|`|−T |)−1 and otherwise arguing
just as before, when T < |k| ≤ NT , we obtain
∣∣∑`
Ak`s`∣∣ ≤ C
T
(|k| − T )√NT‖s‖`2
143
Case 3: |k| > NT . In this case Ak` can be nonzero for any ` but we have∑|`|<T +
∑|`|≥T min
ln(T ), T
|`|−T
|s`|
≤ ln(T )(∑
|`|<T 1)1/2(∑
|`|<T |s`|2)1/2
+(∑
|`|≥TT
(1+|`|−T )2
)1/2(∑|`|≥T |s`|2
)1/2
≤ C√T ln(T )‖s‖`2
so that for |k| > NT , we have
∣∣∑` Ak`s`
∣∣ ≤ C ln(T )√T
(|k|−T )‖s‖`2
To estimate the norm of A we then have
‖As‖2 =∑|k|≤T
+∑
T<|k|≤NT
+∑|k|>NT
∣∣∑`
Ak`s`∣∣2
≤ C ln(T )2
NT‖s‖2
`2 +C
N‖s‖2
`2
( ∑T<|k|≤NT
1
1 + (|k| − T )2
)+ CT ln(T )2‖s‖2
`2
∑|k|>NT
1
(|k| − T )2
≤ C ′ ln(T )2
N‖s‖2
`2
If the bounds A ≈ T B(k)B(`) are sharp then one can improve the `2 bound
at most by removal of the factor ln(T ). This is because in the case of separable
bounds one cannot get any faster decay than B(k). However, showing that the
bounds are effectively separable requires more precise estimates of the entries of A
that will be supplied below. Removal of the logarithmic factor ln(T ) in the norm
bound for A would require additional cancellation in the integral defining Ak`. We
will leave the question of whether this logarithmic improvement is possible open.
144
The error should be smaller when Atr is applied to estimate sample vectors of
PSWFs that are localized on [−T, T ]. Part of the reason comes from the case in
which k ≤ NT . There, only contributions from ` ≥ NT apply and the PSWF
samples decay fast away from [−T, T ]. On the other hand, when k is large one is
effectively getting the k-th entry of the eigenvector which is known to decay.
We now wish to provide an alternative formula for the entries of A from which
it can be inferred that the only potential improvement on the error bounds given
by Proposition 5.21 would be removal of the factor ln(T ) in the norm estimate for
A. As such, we will focus primarily on the entries Ak` in which K ≥ 0 and ` ≥ 0.
We will see that the matrix has a great deal of symmetry as well. But, in order
to show that the estimate of Proposition 5.21 cannot be substantially improved,
it is enough to show that it cannot be substantially improved when operating on
s supported in N = Z+.
Diagonal terms. Akk =∫ T−T
sin2 π(x−k)π2(x−k)2 dx. Setting u = π(x− k) we can write
Akk =1
π
∫ π(T−k)
−π(T+k)
sin2 u
u2du
Using the half angle formula one replaces this with
Akk =1
2π
∫ π(T−k)
−π(T+k)
1− cos 2x
x2dx
First we assume that k 6= ±T .
145
Case 1. 0 ≤ k < T . One can integrate by parts, setting u = 1 − cos 2x and
v = −1/x, to get
12π
[cos 2x−1
x
∣∣∣π(T−k)
−π(T+k)+ 1
π
∫ π(T−k)
−π(T+k)sin 2xx
dx
= 1π
[cosx−1
x
∣∣∣2π(T−k)
−2π(T+k)+ 1
π
∫ 2π(T−k)
−2π(T+k)sinxxdx
= T (cos 2πT−1)π2(T 2−k2)
+ 1π
(Si(2π(T + k)) + Si(2π(T − k))
)by replacing x for 2x. Here Si(x) =
∫ x0
sin ttdt is the sine integral function.
Case 1′. −T < k ≤ 0. Here we have k = −|k| and the same integration by
parts, keeping track of signs, gives
Akk =T (cos 2πT − 1)
π2(T 2 − k2)+
1
π
(Si(2π(T − |k|)) + Si(2π(T + |k|))
)(82)
Case 2. |k| > T . We use integration by parts just as before, but rearrange
terms to insure the argument of the sine integral is positive. This yields
Akk =T cos(2πT − 1)
π2(T 2 − k2)+
1
π
(Si(2π(T + k))− Si(2π(k − T ))
), (k > T )
Akk =T (cos 2πT − 1)
π2(T 2 − k2)+
1
π
(Si(2π(|k|+ T ))− Si(2π(|k| − T ))
), (k < −T )
Note that, just as in case 1, A−k,−k = Ak,k which can also be seen directly from
the symmetry of the integral defining A.
Case 3. |k| = T . In this case T ∈ Z and one can easily check that the evaluation
of cosx−1 at the given endpoints yields zero. One is left simply with the estimate
Akk = ATT = 1πSi(4πT ).
146
Off diagonal terms.
Lemma 5.22. If k 6= ` then
sinc(x− k) sinc(x− `) = 2(−1)k−`
k − `
(1− cos 2π(x− k)
2π(x− k)− 1− cos 2π(x− `)
2π(x− `)
)(83)
Proof. First we recall that
sin π(x− k) sin π(x− `) =1
2
[cosπ(`− k)− cos π((x− k) + (x− `))
]=
1
2
[(−1)k−` − cos π((2x− (k + `))
]=
(−1)k−`
2
[1− cos 2π(x− k)
]=
(−1)k−`
2
[1− cos 2π(x− `)
]
where we used that fact that cosπ(2x− n) = (−1)n cos 2πx. We also have
1
(x− k)(x− `)=
1
(k − `)
[ 1
(x− k)− 1
(x− `)
]Putting these together we get
sinc(x− k) sinc(x− `)
= (−1)k−`
2
[1− cos 2π(x− k)
](1
π2(x−k)(x−`)
)= (−1)k−`
2π2(k−`)
[1− cos 2π(x− k)
][1
(x−k)− 1
(x−`)
]= 2 (−1)k−`
(k−`)
(1−cos 2π(x−k)
2π(x−k)
)−(
1−cos 2π(x−`)2π(x−`)
)which is what was to be proved.
147
The cosine integral Ci(x) is ordinarily defined as
Ci(x) = −∫ ∞x
cos t
tdt = γ + lnx+
∫ x
0
cos t− 1
tdt
where γ is Euler’s number.
In view of (83) it is possible to express the matrix elements in terms of cosine
integral values. Assume, as above, that we are truncating in time over [−T, T ].
Then, we have the following equalities.
Ak` =
∫ T
−Tsinc(t− k) sinc(t− `) dt
= 2(−1)k−`
(k − `)
∫ T
−T
(1− cos 2π(t− k)
2π(t− k)
)−∫ T
−T
(1− cos 2π(t− `)2π(t− `)
)Setting u = 2π(t− k) in the first integral and u = 2π(t− `) in the second,
Ak` =
∫ T
−Tsinc(t− k) sinc(t− `) dt
=(−1)k−`
π(k − `)
∫ 2π(T−k)
−2π(T+k)
(1− cosu
u
)du−
∫ 2π(T−`)
−2π(T+`)
(1− cosu
u
)du
In the following we will assume k < ` since Ak` is symmetric. There are different
cases that we need to consider depending on whether k or ` is between −T and
T . We will first phrase the cases in terms of k. Then, we will consider both k and
` together.
148
Case 1. T + k > 0, T − k < 0 (k > T ). Since the integrand is odd, we may
switch the limits of integration using∫ BA
= −∫ AB
to obtain∫ 2π(T−k)
−2π(T+k)
(1−cosu
u
)du
=∫ 0
−2π(T+k)
(1−cosu
u
)du+
∫ 2π(T−k)
0
(1−cosu
u
)du
= −∫ 2π(T+k)
0
(1−cosu
u
)du−
∫ 0
2π(T−k)
(1−cosu
u
)du
= −∫ 2π(T+k)
0
(1−cosu
u
)du+
∫ 2π(k−T )
0
(1−cosu
u
)du
= −Ci(2π(T + k)) + γ + ln(2π(T + k)) + (Ci(2π(k − T ))− γ − ln(2π(k − T )))
= Ci(2π(k − T ))− Ci(2π(k + T )) + ln(k+Tk−T
).
Case 1′. T + k < 0, (k < −T ). We may use a calculation similar to the one
shown previously in Case 1 to show that if M(k;T ) =∫ 2π(T−k)
−2π(T+k)
(1−cosu
u
)du then
M(−|k|;T ) = −M(|k|;T ) when |k| > T .
Case 2. k ≥ 0, T − k > 0 (k ∈ [0, T )). Since the integrand is odd,∫ 2π(T−k)
−2π(T+k)
(1−cosu
u
)du
=∫ 0
−2π(T+k)
(1−cosu
u
)du+
∫ 2π(T−k)
0
(1−cosu
u
)du
= −∫ 2π(T+k)
0
(1−cosu
u
)du+
∫ 2π(T−k)
0
(1−cosu
u
)du
= Ci(2π(T − k))− γ − ln(2π(T − k))− (Ci(2π(T + k))− γ − ln(2π(T + k)))
= Ci(2π(T − k))− Ci(2π(T + k)) + ln(T+kT−k
)Case 2′. k ≤ 0, T + k > 0 (k ∈ (−T, 0]). Using the oddness of the integrand
again, yields the following equalities.∫ 2π(T−k)
−2π(T+k)
(1−cosu
u
)du =
∫ 0
−2π(T+k)
(1−cosu
u
)du+
∫ 2π(T−k)
0
(1−cosu
u
)du
=∫ 0
−2π(T−|k|)
(1−cosu
u
)du+
∫ 2π(T+|k|)0
(1−cosu
u
)du
= −∫ 2π(T−|k|)
0
(1−cosu
u
)du+
∫ 2π(T+|k|)0
(1−cosu
u
)du
so, once again, we have M(−|k|;T ) = −M(|k|;T ).
149
Case 3. T = k. In this case the definition of the cosine integral gives
∫ 2π(T−k)
−2π(T+k)
(1− cosu
u
)du = −
∫ 4πT
0
(1− cosu
u
)du
= −Ci(4πT ) + γ + ln(4πT )
Case 3′. T = −k. Just as in Case 2, the integral is odd in the argument k.
Putting all of these cases together yields the following proposition.
Proposition 5.23. Define M(k;T ) =∫ 2π(T−k)
−2π(T+k)
(1−cosu
u
)du. In addition, define
C(k;T ) = Ci(2π|k − T |) − Ci(2π|k + T |) and L(k;T ) = ln∣∣∣T+kT−k
∣∣∣. When k 6= T ,
we can write M(k;T ) = C(k;T ) + L(k;T ). On the other hand, when k = T , we
can write M(k;T ) = −Ci(4πT ) + γ + ln(4πT ) for k > 0.
The terms M(k;T ) provide a recipe for writing the terms of Ak` as follows.
Corollary 5.24. Let M(k;T ) =∫ 2π(T−k)
−2π(T+k)
(1−cosu
u
)du. Then, for k 6= `, we can
write Ak` = (−1)k−`
π(k−`) (M(k)−M(`)).
Size estimates on Ak`. We are concerned with the decay of Ak` so we are
concerned with the cases in which both k, ` are large and positive, or both large and
negative – symmetric to the large positive case – or when one is large and positive
and the other is large and negative. We can express the differences M(k)−M(`)
in terms of integrals involving (cosu)/u (cosine terms) and involving 1/u (log
terms). We are justified in separating these terms and treating them as principal
150
value integrals because limu→0
(1− cosu)/u = 0. Our goal is to show that the cosine
terms in M(k)−M(`) are relatively insignificant when k or ` and k − ` is large.
Then we want to show that the log terms look like the products B(k)B(`) with
B(k) ∼ (|k| − T ), or a close variant of them, as discussed above. This will allow
us to show that the norm bound for the remainder term A as discussed above is
essentially sharp.
We have (for k, ` both larger than T and positive, and with ` > k, say)
M(k)−M(`) = M(k;T )
=
∫ 2π(T−k)
−2π(T+k)
−∫ 2π(T−`)
−2π(T+`)
(1− cosu
u
)du
=
∫ 2π(T+`)
0
−∫ 2π(T+k)
0
+
∫ 2π(T−k)
0
−∫ 2π(T−`)
0
(1− cosu
u
)du
=
∫ 2π(T+`)
2π(T+k)
−∫ 0
2π(T−k)
+
∫ 0
2π(T−`)
(1− cosu
u
)du
=
∫ 2π(T+`)
2π(T+k)
+
∫ 2π(k−T )
0
−∫ 2π(`−T )
0
(1− cosu
u
)du
=
∫ 2π(`+T )
2π(k+T )
−∫ 2π(`−T )
2π(k−T )
(1− cosu
u
)du
We can analyze the logarithmic term and the term∫
cosuudu separately. In
particular, we have the following equalities.
151
∫ 2π(n+1)
2πncosuudu
=∫ π
0cosu
(1
(2πn+u)− 1
(2πn+π+u)
)du
=∫ π
0cosu
(π
(2πn+u)(2πn+π+u)
)du
=∫ π/2
0cosu
(π
(2πn+u)(2πn+π+u)− π
(2πn+π−u)(2πn+2π−u)
)du
=∫ π/2
0cosu
(π[(2πn+π−u)(2πn+2π−u)−(2πn+u)(2πn+π+u)]
(2πn+u)(2πn+π+u)(2πn+π−u)(2πn+2π−u)
)du
=∫ π/2
0cosu
(4π2(n+1)(π−u)
(2πn+u)(2πn+π+u)(2πn+π−u)(2πn+2π−u)
)du
Since each term in the denominator of the large factor is at least 2πn it follows
that the integrand is bounded by (n + 1) cosu/(4πn4) so the integral on [0, π/2]
is bounded by (n + 1)/(4πn4). This indicates that the contribution of the cosine
part decays like 1/n3 and the sum over those n larger than some fixed N decays
like 1/N2. We summarize this as follows.
Proposition 5.25. For ` ≥ k > T the contribution of the cosine term found in
M(k)−M(`) is bounded in modulus by a constant (essentially 1/(4π)) times 1/k2
independent of `. Consequently, by Corollary 5.24 the cosine contribution to Ak`
is essentially bounded by 1/(4π2k2(`− k)) when ` > k ≥ T .
The logarithmic terms. We can treat the logarithmic terms in M(k)−M(`) as
principal value integrals, since the integrals defining M(k)−M(`) are absolutely
convergent and the parts involving an integrand 1/u with u small cancel upon
subtraction. As such, bounding M(k)−M(`) reduces to bounding the logarithm
parts of the integrals above. Using these ideas, along with the assumption that
152
` > k > T yields the following approximation.
∫ 2π(`+T )
2π(k+T )
−∫ 2π(`−T )
2π(k−T )
(1
u
)du = log
((`+ T )(k − T )
(`− T )(k + T )
)= log
(k − Tk + T
)− log
(`− T`+ T
)= log
(1− 2T
k + T
)− log
(1− 2T
`+ T
)≈ 2T
`+ T− 2T
k + T
= 2Tk − `
(k + T )(`+ T )
where we have applied the Taylor estimate log(1 + t) = t + o(t2) as t → 0. That
is, the approximation of the log term log((k − T )/(k + T )) is accurate up to a
constant times (2T/(k+T ))2 when k > T and similarly for `. In view of Corollary
5.24 the total contribution of the logarithmic terms to Ak` will be essentially
Lk` =(−1)k−`
π(k − `)(L(k)− L(`))
≈ 2T(−1)k−`
π(k − `)k − `
(k + T )(`+ T )
=(−1)k−`
π
2T
(k + T )(`+ T ).
Putting all these estimates together yields the following.
Proposition 5.26. When ` > k ≥ T one has
Ak` =(−1)k−`
π
2T
(k + T )(`+ T )+O
( 1
k2(`− k)
), as k, `→∞.
153
A closer inspection of the cosine term shows that the second term above is
actually more like 1k2`
. In particular, this error term will always be at most a
fraction of the term 2T(k+T )(`+T )
. This shows that `2-bound for the error matrix A
given by Proposition 5.21 is sharp except possibly for the factor ln(T ).
154
6 CONCLUSION
In our discussion we have learned several methods for dealing with analog
signals in the digital setting. In Chapter 2, we learned that band-limited signals
could be represented by countably infinite samples due to the Fourier Series and
that a good approximation for the Fourier Transform of a Schwartz function can
be found using the Discrete Time Fourier Transform. In addition, we learned
that for f in specific classes of functions, f may be represented by N samples
whose Discrete Fourier Transform consists of N samples of f and also represents
f . For these specific classes of functions, sample trunctation does not lead to error
in the representations of f or f implying that only quantization is necessary to
completely tie the analog signal to its digital counterpart.
In addition to relating four different settings of the Fourier Transform, we
included the distribution theory which is often missing in the literature. This
included Poisson summation formulas for the class of Schwartz functions S (R),
the square integrable functions L2(R) and the tempered distributions S ′(R).
In Chapters 3 and 4, we explored the idea that generalized bases lead to
generalized sampling formulas. Several generalized bases were brought into one
framework giving a new type of characterization. In addition, the gaps between
Shannon sampling and periodic nonuniform sampling were bridged. Familiar con-
cepts in these areas were extended to include tiling sets. An additional level of
155
rigor, often missing in the literature, was used to apply concepts from distribution
theory to show equality in L2(R) for the sampling formulas presented. Finally,
new methods for applying this theory were also introduced.
In Chapter 5, we discussed time-frequency localization operators and the ad-
vantages of using time-frequency localized eigenfunctions. We developed a new
sampling formula which may be applied to a function which is band-limited to
several intervals. Since the key to using this sampling formula is an understand-
ing of eigenvalues, we included an additional discussion regarding eigenvalues.
We reviewed several known concepts, improved certain results and provided new
estimates.
156
7 APPENDIX
This appendix contains Matlab code (Version 7.3) used to generate the figures
presented here. For each section, two procedure based script files were created to
generate the plots. The code to generate the signals and matrices is shown in the
first section and is separated from the code to generate the plots which is shown
in the last section. One reason for separating this code is so that the plots may
be configured without repeating calculations.
Each script file accesses three main types of methods. The first type of method
returns a digital signal representation. The second type of method performs an
algorithm based on theory. These methods where used to create a small library
for the sake of modularization and code reuse. The final type of method is a
configuration method used to help configure the plots. Examples of how to use
latex along with how to save the workspace or figures are included.
In writing code, we attempted to make clear the theory being used. For
example, there is a simple algorithm to compute the FT of a periodic function
such as f(t) = sin(t). This algorithm is demonstrated by Line 50 in the first
Computation Script. First, we time limit the function to one period. Then, we
use the DTFT to compute a Riemann sum approximation to the FT. In many
cases, the DTFT converges uniformly to the FT. Finally, we impulse sample.
One last item to point out is that the parameters for the script files are placed
157
at the beginning of the script. Therefore, one can easily change these parameters
to explore other examples.
7.1 Computation Scripts
1 % This script calculates functions satifying various FT relations
2 % Author S. Izu, NMSU, 2009
3
4 close all; clear;
5 t = −5:.01:5; omega = −60:.01:60;
6 T = .5; Omega = 6;
7 vector = [4 1 2]; %Dimensions are 2 by T*Omega
8
9 %Basic Schwartz Functions
10 rhoA = rho(16*t+6); %Uses rho
11
12 rho = rho(t); %Method creates rho
13 rhohat = DTFT(rho, 100, omega);
14
15 psi = unity(1,0,t); %Partition of Unity
16 psihat = DTFT(psi, 100, omega);
17
18 psi1 = psi1(t); %Method creates psi1
19 psi1hat = DTFT(psi1, 100, omega);
158
20
21 psi2hat = psi2hat(omega); %Method creates psi2hat
22 psi2 = reverse(DTFT(psi2hat, 100, t));
23
24 %psi1 and f satisfy the FSII Equations
25 %f = f1+f2, f1 is a Schwartz function, f2 is an Impulse
26 f1 = rhoA; f1(1,:) = t; f2 = [3/8; .02];
27 fhat = add(DTFT(f1, 100, omega), impulseFT(f2, omega));
28
29 %a and b satisfy FS1 Equations
30 %b, c and d satisfy DFTI Equations
31 a = a(t,Omega*T*ifft(vector),Omega,T); %Method creates a (vector)
32 A = DTFT(a,100,omega);
33
34 b = periodize(a,.5,t);
35 B = Isample(A,T,1);
36
37 c = Isample(b,Omega,1);
38 C = periodize(B, Omega, −8/T:1/T:8/T);
39
40 D = D(omega,T*vector,T,Omega); %Method creates D (vector)
41 d = reverse(DTFT(D,100,t));
42
43 %g0g1 satisfies FSII Equations
44 %g0, c and d satisfy DFTI Equations
159
45 %g0 and g0g1 satisfy DFTII Equations
46 %g0 and g0g2 satsify DFTII Equations
47 %g0g2 satisfies DFTIII Equations
48 g0 = g0(t, vector, Omega, T); %Method creates g0 (vector)
49 g0hat = Isample(DTFT(time limit(g0, 0, T), 100, omega), T, 1);
50
51 g0g1 = multiply(g0, unity(T,2,t));
52 g0g1hat = DTFT(g0g1, 100, omega);
53
54 g0g2 = multiply(g0, g2(t, Omega, T)); %Method creates g2
55 g0g2hat = DTFT(g0g2, 100, omega);
56
57 %Shifts of psi
58 psiRshift = shift(psi, 1);
59 psiLshift = shift(psi, −1);
60
61 %Approximations for f and fhat
62 fApp = reverse(multiply(DTFT(fhat, T, t), psi1));
63 fhatApp = interpolate(fhat, T, psi1hat,1);
64
65 %Alternate function for a
66 Alta = time limit(b, 0, T);
67 AltAhat = DTFT(Alta, 100, omega);
68
69 %Approximations for c and C
160
70 cApp1 = reverse(DTFT(D, T, t));
71 CApp1 = DTFT(b, Omega, omega);
72
73 cApp2 = periodize(d, T, t);
74 CApp2 = periodize(B, Omega, −8/T:1/T:8/T);
75
76 %Approximations for g0 and g0hat
77 g0App1 = interpolate(g0, Omega, psi2, 1);
78 g0hatApp = multiply(DTFT(g0, Omega, omega), psi2hat);
79
80 g0App2 = reverse(DTFT(g0g1hat, T, t));
81 g0App3 = periodize(g0g1, T, t);
82 save('FourierTransform.mat');
1 % This script applies a filtering algorithm to a signal s
2 % Author S. Izu, NMSU, 2009
3
4 clear; close all;
5 N = 7; skip = 1;
6 [x, y] = textread('lappFile001.dat','%f %f');
7 s = x(1:10000+2ˆN); %s comes from the input file above
8 h = [ 0.03771716; 0.26612218; 0.74557507; ...
9 0.97362811; 0.39763774; −0.35333620; ...
10 −0.27710988; 0.18012745; 0.13160299; ...
161
11 −0.10096657; −0.04165925; 0.04696981; ...
12 5.10043697e−3; −0.01517900; 1.97332536e−3; ...
13 2.81768659e−3; −9.69947840e−4; −1.64709006e−4; ...
14 1.32354367e−4; −1.875841e−5]; %Daubechies 20 filter
15
16 %Compute Filter and denoise signal
17 W = wavelet packet matrix(h, N, N);
18 beforeS = scalogram(s, W, skip);
19 r = denoising filter(beforeS, W);
20 filtered = conv(s,r);
21 afterS = scalogram(filtered, W, skip);
22 save('Wavelet.mat');
1 % This script calculates eigenvalues
2 % Author S. Izu, NMSU, 2009
3
4 clear; close all; tic;
5 Sigma = [ 0.0 1.3 ;
6 4.5 5.0 ];
7 Omega = 5; K = 1; %Omega = .5; K = 10;
8 beta = [ 0 0 1 1 ] ; %beta = [ 0 1 1 ;
9 % 1 2 2 ;
10 % 2 3 3 ;
11 % 9 10 10 ];
162
12 O = [ 0.0 0.26 ; %O = [ 0.0 0.5 ;
13 0.26 0.5 ; % −0.5 −0.4 ;
14 −0.5 −0.1 ; % −0.4 0.0 ];
15 −0.1 0.0 ];
16 save('g.mat', 'Sigma'); %Used by g.m
17
18 %Step 1: Calculate necessary Intervals and Matrices
19 %I jm = Omega(O m+ beta jm)
20 %[W m] jk = exp(−(2 pi i j beta jm)/K)
21 %(W m)ˆ(−1)
22 J = size(beta,1); M = size(beta,2); %M = size(O,1);
23 const = −2*pi*i/K;
24 for m=0:(M−1)
25 for j=0:(J−1)
26 I(j+1,m+1,1:2) = Omega*(O(m+1,1:2)+beta(j+1,m+1));
27 W(j+1,1:J,m+1) = exp(const*j*beta(1:J,m+1));
28 end
29 Winv(1:J,1:J,m+1) = W(1:J,1:J,m+1)ˆ(−1);
30 end
31 save('h.mat', 'J', 'M', 'I', 'W', 'Winv'); %Used by h.m
32
33 %Step 2: Calculate A
34 %A mn = integral S(FT 1 Sigma(x−m/Omega) * h n(x) dx)
35 M = 2*ceil(Omega*K); N = 2*ceil(Omega*K);
36 A = zeros(2*M+1, 2*N+1);
163
37 Kst = ['(' num2str(Omega) ',' num2str(K) ','];
38 for m=−M:M
39 for n=−M:M
40 %st = h(Omega, K, n, x).*g(Omega, K, m, x)
41 st = ['h' Kst num2str(n) ',x).*g' Kst num2str(m) ',x)'];
42 A(m+M+1,n+N+1) = quadl(st, −.5, .5, 1e−10);
43 end
44 end
45 [V, D] = eig(A); d=real(sort(diag(D)));
46 save('Eigenvalue.mat');
7.2 Fourier Transform Digital Signal Representations
1 function rho = rho(t)
2 % Shwartz function
3 % rho(t) = eˆ(tˆ2/(tˆ2−1)) if −1<t<1
4 % Author S. Izu, NMSU, 2009
5
6 ind = logical((−1<t).*(t<1));
7 tsquared = t(ind).ˆ2;
8 rho(1,:) = t;
9 rho(2,ind) = exp(tsquared./(tsquared − 1));
164
1 function psi1 = psi1(t)
2 % psi1(t) = psi0(8t+3) + psi0(8t−3)
3 % Author S. Izu, NMSU, 2009
4
5 psi1(1,:) = t;
6 y1 = unity(.5,2,8*t+3);
7 y2 = unity(.5,2,8*t−3);
8 psi1(2,:) = 3*y1(2,:) + 3*y2(2,:);
1 function psi2hat = psi2hat(omega)
2 % psi2hat(omega)
3 % = rho(omega/2+2)+rho(omega/2−3)+rho(omega/2−5)
4 % Author S. Izu, NMSU, 2009
5
6 psi2hat(1,:) = omega;
7 y1 = rho(omega/2+2);
8 y2 = rho(omega/2−3);
9 y3 = rho(omega/2−5);
10 psi2hat(2,:) = y1(2,:) + y2(2,:) + y3(2,:);
1 function a = a(t, v, Omega, T)
2 % M = Omega*T
3 % a(t) = sum(m=0:M−1) v(m)*rho(Omega*t−m+mod(m,2)*M)
165
4 % Assumes size(v,2) = M and M is an integer
5 % a(t, [24 6 12], 6, .5) = 24rho(6t) + 6rho(6t+2) + 12rho(6t−2)
6 % Author S. Izu, NMSU, 2009
7
8 M = Omega*T;
9 a = zeros(2,size(t,2));
10 a(1,:) = t;
11 for m=0:M−1
12 temp = rho(Omega*t−m+mod(m,2)*M);
13 a(2,:) = a(2,:)+ v(m+1)*temp(2,:);
14 end
1 function D = D(omega,v,T,Omega)
2 % M = Omega*T
3 % D(t)
4 % = sum(n=−infty:infty) sum(m=1:M)
5 % v(m)*rho(2ˆ|n |(T*omega−m−N*n)
6 % Assumes size(v,2) = N and N is an integer
7 % D(omega,[24 6 12],6,.5)
8 % = sum(n=−infty:infty) 24 rho(2ˆ|n|(6omega−0−3n))
9 % + 6 rho(2ˆ|n|(6omega−1−3n))
10 % + 12 rho(2ˆ|n|(6omega−2−3n))
11 % Author S. Izu, NMSU, 2009
12
166
13 M = Omega*T;
14 D = zeros(2,size(omega,2));
15 D(1,:) = omega;
16 flr = floor(omega/Omega);
17 for m=0:M−1
18 temp = rho(2.ˆabs(flr).*(T*omega−m−M*flr));
19 D(2,:) = D(2,:)+v(m+1)*temp(2,:);
20 end
21 temp = rho(2.ˆabs(flr+1).*(T*omega−M−M*flr));
22 D(2,:) = D(2,:)+v(1)*temp(2,:);
1 function g0 = g0(t, v, Omega, T)
2 % M = Omega*T
3 % g0(t) = sum(m=0:M−1) v(m)eˆ(−2*pi*i(m+(−1)ˆm*M)t/T)
4 % Assumes size(v,2) = M and M is an integer
5 % g0(t, [24 6 12], 6, .5)
6 % = 6eˆ(−8*pi*i*t)+24eˆ(12*pi*i*t)+12eˆ(20*pi*i*t)
7 % Author S. Izu, NMSU, 2009
8
9 M = Omega*T;
10 C1 = 2*pi*i/T;
11 g0 = zeros(2,size(t,2));
12 g0(1,:) = t;
13 for m=0:M−1
167
14 g0(2,:) = g0(2,:) + v(m+1)*exp(C1*(m+(−1)ˆm*M)*t);
15 end
1 function g2 = g2(t, Omega, T)
2 % M = Omega*T
3 % g2(t) = sum(m=0:M−1) psi(Omega*t−m+mod(m,2)*M)
4 % Assumes size(v,2) = M and M is an integer
5 % g2(t, 6, .5) = psi(6t) + psi(6t+2) + psi(6t−2)
6 % Author S. Izu, NMSU, 2009
7
8 M = Omega*T;
9 g2 = zeros(2,size(t,2));
10 g2(1,:) = t;
11 for m=0:M−1
12 temp = unity(1,0,Omega*t−m+mod(m,2)*M);
13 g2(2,:) = g2(2,:) + temp(2,:);
14 end
7.3 Fourier Transform Library
1 function added = add(f1,f2)
2 % Adds functions
3 % Assumes domains are equal, ie f1(1,:) = f2(1,:);
168
4 % Author S. Izu, NMSU, 2009
5
6 added(1,:) = f1(1,:);
7 added(2,:) = f1(2,:) + f2(2,:);
1 function F = DTFT(f,Omega,omega)
2 % Calculates DTFT and may be used to approximate FT
3 % F(omega)
4 % = (1/Omega) sum(n=−N:N) f(n/Omega) eˆ(−2 pi i omega n/Omega)
5 % Author S. Izu, NMSU, 2009
6
7 F(1,:) = omega;
8 F(2,:) = zeros(1,size(omega,2));
9 deltat = f(1,2) − f(1,1); %Assume evenly spaced sample locations
10 N = floor(max(f(1,:))*Omega);
11 for n=−N:N
12 samp = sample(f,n/Omega,deltat);
13 F(2,:) = F(2,:) + samp*exp(−2*pi*i*omega*n/Omega);
14 end
15 F(2,:) = (1/Omega)*F(2,:);
1 function F = impulseFT(f,omega)
2 % Returns FT for an Impulse Set
169
3 % Author S. Izu, NMSU, 2009
4
5 F(1,:) = omega;
6 F(2,:) = zeros(1,size(omega,2));
7 for k=1:size(f,2)
8 F(2,:) = F(2,:) + f(2,k).*exp(−2*pi*i*f(1,k).*omega);
9 end
1 function answer = interpolate(f,Omega,psi,multiply)
2 % Interpolates samples of f using translates of psi
3 % Returns sum(n=−N:N) f(n/Omega)*psi(t − n/Omega)
4 % multipy == 1 multiplies summation by 1/Omega
5 % Author S. Izu, NMSU, 2009
6
7 answer = zeros(size(f));
8 answer(1,:) = psi(1,:);
9 N = floor(max(f(1,:))*Omega);
10 deltat = f(1,2) − f(1,1);
11 for n=−N:N
12 samp = sample(f, n/Omega, deltat);
13 psi shift = shift(psi, n/Omega);
14 answer(2,:) = answer(2,:) + samp.*psi shift(2,:);
15 end
16 answer(2,:) = (1/Omega)ˆ(multiply)*answer(2,:);
170
1 function answer = intervalFT(a,b,t)
2 % Calculates FT of the characteristic function 1 [a,b)
3 % Author S. Izu, NMSU, 2009
4
5 len = b−a;
6 mid = (a+b)/2;
7 answer = len*exp(−2*pi*i*mid*t).*sinc(len*t);
1 function impulses = Isample(f,Omega,multiply)
2 % Impulse samples f, Returns f(n/Omega)
3 % multiply == 1 multiplies set values by 1/Omega
4 % Author S. Izu, NMSU, 2009
5
6 N = floor(max(f(1,:))*Omega);
7 deltat = f(1,2) − f(1,1);
8 impulses(1,:) = (1/Omega)*(−N:N);
9 C1 = (1/Omega)ˆ(multiply);
10 for n=−N:N
11 impulses(2, n + N + 1) = C1*sample(f,n/Omega,deltat);
12 end
1 function multiplied = multiply(f1,f2)
2 % Multiplies functions
171
3 % Assumes domains are equal, ie f1(1,:) = f2(1,:);
4 % Author S. Izu, NMSU, 2009
5
6 multiplied(1,:) = f1(1,:);
7 multiplied(2,:) = f1(2,:).*f2(2,:);
1 function answer = periodize(f, periodL, t)
2 % Periodizes f, Returns sum(−N:N) f(t−n*periodL)
3 % t holds the domain for answer
4 % Author S. Izu, NMSU, 2009
5
6 %Find location of zero
7 tol = 10ˆ−10;
8 deltat = f(1,2) − f(1,1);
9 samps per period = ceil(periodL/deltat + tol);
10 zeroind = find( abs(f(1,:)) <= deltat/2 + tol);
11
12 %Calculate one period
13 temp(1,:) = f(1,zeroind(1):zeroind(1)+samps per period−1);
14 temp(2,:) = zeros(1,samps per period);
15 for l=1:size(f,2)
16 ind = mod(l−zeroind(1),samps per period)+1;
17 temp(2,ind) = temp(2,ind) + f(2,l);
18 end
172
19
20 %Extend the period to all of t
21 answer(1,:) = t;
22 for l=1:size(answer,2)
23 ind = mod(answer(1,l)+deltat/2,periodL)−deltat/2;
24 answer(2,l) = sample(temp,ind,deltat);
25 end
1 function reversed = reverse(f)
2 % Reverses f, f(−t)
3 % Author S. Izu, NMSU, 2009
4
5 reversed(1,:) = −1*f(1,size(f,2):−1:1);
6 reversed(2,:) = f(2,size(f,2):−1:1);
1 function samp = sample(f,t,deltat)
2 % Samples f
3 % Returns f(t0) where |t−t0|<deltat
4 % t cannot be a vector
5 % Author S. Izu, NMSU, 2009
6
7 tol = 10ˆ−10; %Account for machine tolerance
8 samp = f(2,logical( abs(f(1,:)−t) <= deltat/2+tol ));
173
9 samp = samp(1);
1 function shifted = shift(f,shift)
2 % Shifts f, f(t−shift)
3 % Author S. Izu, NMSU, 2009
4
5 L = size(f,2);
6 shifted(1,:) = f(1,:);
7 shifted(2,:) = zeros(1,L);
8 shiftind = floor(−shift/(f(1,2)−f(1,1)));
9 if shiftind >= 0 %shift left
10 shifted(2, 1:L−shiftind) = f(2, 1+shiftind:L);
11 else
12 shifted(2, 1−shiftind:L) = f(2, 1:L+shiftind);
13 end
1 function y=sinc(x)
2 % y=sin(pi*x)/(pi*x)
3 % Author D. Menemenlis, MIT, 7 feb 94 (289B)
4 % Modified S. Izu, NMSU, 2009
5
6 ix=find(x==0);
7 x(ix)=ones(size(ix));
174
8 y=sin(pi*x)./(pi*x);
9 y(ix)=ones(size(ix));
1 function time limited = time limit(f,a,b)
2 % Time limits f to [a,b]
3 % Author S. Izu, NMSU, 2009
4
5 time limited(1,:) = f(1,:);
6 time limited(2,:) = (a<=f(1,:)).*(f(1,:)<b).*f(2,:);
1 function unity = unity(T,N,t)
2 % Creates a T−partition of unity which is flat over N periods
3 % The trailing edges have a specific form
4 % y1 = rho(t/T+N/2), y2 = rho(t/T+N/2+1)
5 % y3 = rho(t/T−N/2), y4 = rho(t/T−N/2−1)
6 % unity(t) = y1/((N+1)*(y1+y2)) if −T*N/2−T < t < −T*N/2
7 % unity(t) = 1/(N+1) if −T*N/2 <= t <= T*N/2
8 % unity(t) = y3/((N+1)*(y3+y4)) if T*N/2 < t < T*N/2+T
9 % Author S. Izu, NMSU, 2009
10
11 unity(1,:) = t;
12 x2 = T*N/2; x3 = x2+T; x1 = −x2; x0 = −x3;
13 y1 = rho(t/T+N/2); y2 = rho(t/T+N/2+1);
175
14 y3 = rho(t/T−N/2); y4 = rho(t/T−N/2−1);
15
16 ind = logical((x0<t).*(t<x1));
17 unity(2,ind) = y1(2,ind)./(y1(2,ind)+y2(2,ind));
18
19 ind = logical((x1<=t).*(t<=x2));
20 unity(2,ind) = 1;
21
22 ind = logical((x2<t).*(t<x3));
23 unity(2,ind) = y3(2,ind)./(y3(2,ind)+y4(2,ind));
24
25 unity(2,:) = unity(2,:)/(N+1);
7.4 Wavelet Library
1 function W = wavelet packet matrix(h, N, N 0)
2 % Generates the 2ˆN by 2ˆN Wavelet Packet matrix W
3 % h is the scaling filter
4 % N 0 <= N is the number of levels in the wavelet tree
5 % Author S. Izu, NMSU, 2009
6
7 W = eye(2ˆN);
8 for n=0:N 0−1
9 %Place wavelet matrices along the diagonal of W level n
176
10 C1 = 2ˆ(N−n);
11 indeces = 1:C1;
12 W level n = zeros(2ˆN);
13 W level n(1:C1,1:C1) = wavelet matrix(h, N−n);
14 for i=2:2ˆn
15 indeces = indeces + C1;
16 W level n(indeces, indeces) = W level n(1:C1,1:C1);
17 end
18 W = W level n*W;
19 end
20
21 %Account for spectral flipping
22 W = W(spectral flip(N),:);
1 function W N = wavelet matrix(h, N)
2 % Generates the 2ˆN by 2ˆN Wavelet Transform matrix W N
3 % h is the scaling filter whose indeces are 0 through L−1
4 % W N combines Approximation H N and Detail G N
5 % Author S. Izu, NMSU, 2009
6
7 %Zeropad h so its length is a multiple of 2ˆN
8 L = size(h,1); C1 = 2ˆN; cols = ceil(L/C1);
9 h = [h; zeros(cols*C1−L,1)];
10
177
11 %Calculate first row of Approximation H N and Detail G N
12 H N = zeros(C1/2,C1); G N = H N;
13 H N(1,:) = sum(reshape(h, C1, cols), 2);
14 G N(1,:) = (−1).ˆ(0:C1−1).* conj(H N(1,[2 1 C1:−1:3]));
15
16 %Shift each row two indeces to right to obtain next row
17 indeces = [C1−1 C1 1:C1−2];
18 for m = 2:C1/2
19 H N(m,:) = H N(m−1,indeces);
20 G N(m,:) = G N(m−1,indeces);
21 end
22 W N = [H N; G N];
1 function indeces = spectral flip(N)
2 % Calculates indeces according to spectral flipping
3 % N is the number of levels in the tree
4 % Author S. Izu, NMSU, 2009
5
6 indeces = 1;
7 for i=1:N
8 J = size(indeces,2);
9 indeces(2*J:−1:J+1) = indeces+J;
10 end
11 indeces = indeces(size(indeces,2):−1:1);
178
1 function r = denoising filter(S, W)
2 % Calculates the filter for the denoising algorithm
3 % S is a Scalogram matrix
4 % W is a Wavelet matrix
5 % Author S. Izu, NMSU, 2009
6
7 N = log2(size(W,1));
8 MeanS = mean(S,2);
9 Winv = pinv(W);
10 r = zeros(2ˆ(N+1)−1,1);
11 for n=1−2ˆN:2ˆN−1
12 if( n <= 0)
13 lvect = 1−n:2ˆN;
14 else
15 lvect = 1:2ˆN−n;
16 end
17 outersum = 0;
18 for m=1:2ˆN
19 innersum = 0;
20 for l=lvect
21 innersum = innersum + Winv(l,m)*W(m,l+n);
22 end
23 outersum = outersum + (1/MeanS(m))*innersum;
24 end
179
25 r(n+2ˆN) = 1/(2ˆN)*outersum;
26 end
1 function S = scalogram(s, W, skip)
2 % Calculates the scalogram matrix
3 % s is the input signal
4 % W is the Wavelet matrix
5 % skip >= 1 gives the number of blocks to skip
6 % Author S. Izu, NMSU, 2009
7
8 C1 = size(W,1);
9 C2 = size(s,1)−C1+1;
10 S = zeros(C1, C2);
11 for i = 1:skip:C2
12 indeces = i:(i+C1−1);
13 S(:,i) = abs(W*s(indeces));
14 end
7.5 Eigenvalue Calculations
1 function h = h(Omega, K, N, x)
2 % Assumes h.mat contains J,M,I,W and Winv
3 % Author S. Izu, NMSU, 2009
180
4
5 load('h.mat');
6 h = 0;
7 j = mod(N,K); %N = j + nK = mod(N,K) + floor(N/K)*K
8 if(0 <= j && j < J)
9 t = N/(Omega*K)−x;
10 for m=0:(M−1)
11 for k=0:(J−1)
12 C = conj(Winv(k+1,j+1,m+1)*W(j+1,k+1,m+1));
13 h = h + C*intervalFT(I(k+1,m+1,1),I(k+1,m+1,2),t);
14 end
15 end
16 h = (1/Omega)*h;
17 end
1 function g = g(Omega, K, m, x)
2 % Assumes g.mat contains Sigma
3 % Author S. Izu, NMSU, 2009
4
5 load('g.mat');
6 g = 0;
7 t = x−m/(Omega*K);
8 for l=1:size(Sigma)
9 g = g + intervalFT(Sigma(l,1),Sigma(l,2), t);
181
10 end
7.6 Plots and Configuration
1 % This script plots functions satifying various FT relations
2 % Author S. Izu, NMSU, 2009
3
4 close all; clear;
5 flag = 1; %0−use sub plots, 1−separate figures
6
7 load('FourierTransform.mat');
8 taxis = [−.6 .6 −1.5 1.5]; Omegaaxis = [−12 12 −5 5];
9 taxisI = taxis.*[1 1 Omega Omega]; %Time Impulse Sample
10 OmegaaxisI = Omegaaxis.*[1 1 T T]; %Frequency Impulse Sample
11
12 %Basic Schwartz functions
13 figure(1); if(˜flag); subplot(2,1,1); end; hold on;
14 plotf(rho,['−b';'−r'],0); %rho
15 label([−1.1 1.1 −.1 1.1], '$t$', '$\rho(t)$');
16 if(flag); saveas(gcf, 'images/rho.png'); end
17
18 if(flag); figure(2); else subplot(2,1,2); end; hold on;
19 plotf(rhohat,['−b';'−r'],0); %rhohat
20 label([−6 6 −.2 1.4], '$\omega$', '$\widehat\rho(\omega)$');
182
21 if(flag); saveas(gcf, 'images/rhohat.png'); end
22
23 figure(11); if(˜flag); subplot(3,1,1); end; hold on;
24 plotf(psi,['−b';'−r'],0); %psi
25 label([−1.1 1.1 −.1 1.1], '$t$', '$\psi(t)$');
26 if(flag); saveas(gcf, 'images/psi.png'); end
27
28 if(flag); figure(12); else subplot(3,1,2); end; hold on;
29 plotf(psihat,['−b';'−r'],0); %psihat
30 label([−6 6 −.2 1.4], '$\omega$', '$\widehat\psi(\omega)$');
31 if(flag); saveas(gcf, 'images/psihat.png'); end
32
33 if(flag); figure(13); else subplot(3,1,3); end; hold on;
34 plot([−2.1 2.1], [1 1], 'k');
35 plotf(psi,['−b';'−r'],0); %psi(t)
36 plotf(psiRshift,['−g';'−r'],0); %psi(t−1)
37 plotf(psiLshift,['−m';'−r'],0); %psi(t+1)
38 label([−2.1 2.1 −.1 1.1], '$t$', '$\psi(t+1),\psi(t),\psi(t−1)$');
39 if(flag); saveas(gcf, 'images/psi shifts.png'); end
40
41 %FSII Equations
42 figure(21); if(˜flag); subplot(2,1,1); end; hold on;
43 plotf(psi1,['−b';'−r'],0); %psi
44 label([−1 1 −.1 1.1], '$t$', '$\psi(t)$');
45 if(flag); saveas(gcf, 'images/fs psi.png'); end
183
46
47 if(flag); figure(22); else subplot(2,1,2); end; hold on;
48 plotf(psi1hat,['−b';'−r'],0); %psihat
49 label([−30 30 −.4 .4], '$\omega$', '$\widehat\psi(\omega)$');
50 if(flag); saveas(gcf, 'images/fs psihat.png'); end
51
52 figure(26); if(˜flag); subplot(2,2,1); end; hold on;
53 plotf(f1,['−b';'−r'],0); %Schwartz component of f
54 plotf(f2,['−b';'−r'],1); %Impulse component of f
55 label([−1 1 −.1 1.1], '$t$', '$f(t)$');
56 if(flag); saveas(gcf, 'images/fs f.png'); end
57
58 if(flag); figure(27); else subplot(2,2,3); end; hold on;
59 plotf(fhat,['−b';'−r'],0); %fhat = f1hat+f2hat
60 label([−30 30 −.1 .1], '$\omega$', '$\widehatf(\omega)$');
61 if(flag); saveas(gcf, 'images/fs fhat.png'); end
62
63 if(flag); figure(28); else subplot(2,2,2); end; hold on;
64 plotf(fApp,['−b';'−r'],0); %f approx
65 label([−1 1 −.4 1.4], '$t$', '$\tildef(t)$');
66 if(flag); saveas(gcf, 'images/fs fapp.png'); end
67
68 if(flag); figure(29); else subplot(2,2,4); end; hold on;
69 plotf(fhatApp,['−b';'−r'],0); %fhat approx
70 label([−30 30 −.1 .1],'$\omega$','$\tilde\widehatf(\omega)$');
184
71 if(flag); saveas(gcf, 'images/fs fhatapp.png'); end
72
73 %First Plot, FS Equations
74 figure(31); if(˜flag); subplot(2,4,1); end; hold on;
75 plotf(a,['−b';'−r'],0); %a
76 label(taxisI, '$t$', '$a(t)$');
77 if(flag); saveas(gcf, 'images/a.png'); end
78
79 if(flag); figure(32); else subplot(2,4,5); end; hold on;
80 plotf(A,['−b';'−r'],0); %A
81 plotf(Isample(A,T,0),['−−b';'−−r'],1); %A sampled
82 label(OmegaaxisI, '$\omega$', '$A(\omega)$');
83 if(flag); saveas(gcf, 'images/ahat.png'); end
84
85 if(flag); figure(33); else subplot(2,4,2); end; hold on;
86 plotf(b,['−b';'−r'],0); %b
87 plotf(Isample(b,Omega,0),['−−b';'−−r'],1); %b sampled
88 label(taxisI, '$t$', '$b(t)$');
89 if(flag); saveas(gcf, 'images/b.png'); end
90
91 if(flag); figure(34); else subplot(2,4,6); end; hold on;
92 plotf(B,['−b';'−r'],1); %B
93 label(Omegaaxis, '$\omega$', '$B(\omega)$');
94 if(flag); saveas(gcf, 'images/bhat.png'); end
95
185
96 if(flag); figure(35); else subplot(2,4,3); end; hold on;
97 plotf(c,['−b';'−r'], 1); %c
98 label(taxis, '$t$', '$c(t)$');
99 if(flag); saveas(gcf, 'images/c.png'); end
100
101 if(flag); figure(36); else subplot(2,4,7); end; hold on;
102 plotf(C,['−b';'−r'],1); %C
103 label(Omegaaxis, '$\omega$', '$C(\omega)$');
104 if(flag); saveas(gcf, 'images/chat.png'); end
105
106 if(flag); figure(37); else subplot(2,4,4); end; hold on;
107 plotf(d,['−b';'−r'],0); %d
108 label([−.75 .75 −30 30], '$t$', '$d(t)$');
109 if(flag); saveas(gcf, 'images/d.png'); end
110
111 if(flag); figure(38); else subplot(2,4,8); end; hold on;
112 plotf(D,['−b';'−r'],0); %D
113 plotf(Isample(D,T,0),['−−b';'−−r'],1); %D sampled
114 label(OmegaaxisI, '$\omega$', '$D(\omega)$');
115 if(flag); saveas(gcf, 'images/dhat.png'); end
116
117 %Secont Plot, FS Equations
118 figure(41); if(˜flag); subplot(2,4,1); end; hold on;
119 plotf(g0g1,['−b';'−r'],0); %g0g1
120 label(taxisI, '$t$', '$g 0 g 1(t)$');
186
121 if(flag); saveas(gcf, 'images/g0g1.png'); end
122
123 if(flag); figure(42); else subplot(2,4,5); end; hold on;
124 plotf(g0g1hat,['−b';'−r'],0); %g0g1hat
125 plotf(Isample(g0g1hat,T,0),['−−b';'−−r'],1); %g0g1hat sampled
126 label(OmegaaxisI, '$\omega$', '$\widehatg 0 g 1(\omega)$');
127 if(flag); saveas(gcf, 'images/g0g1hat.png'); end
128
129 if(flag); figure(43); else subplot(2,4,2); end; hold on;
130 plotf(g0,['−b';'−r'],0); %g0
131 plotf(Isample(g0,Omega,0) ,['−−b';'−−r'],1); %g0 sampled
132 label(taxisI, '$t$', '$g 0(t)$');
133 if(flag); saveas(gcf, 'images/g0.png'); end
134
135 if(flag); figure(44); else subplot(2,4,6); end; hold on;
136 plotf(g0hat,['−b';'−r'],1); %g0
137 label(Omegaaxis, '$\omega$', '$\widehatg 0(\omega)$');
138 if(flag); saveas(gcf, 'images/g0hat.png'); end
139
140 if(flag); figure(45); else subplot(2,4,3); end; hold on;
141 plotf(g0g2,['−b';'−r'],0); %g0g2
142 plotf(Isample(g0g2,Omega,0),['−−b';'−−r'],1); %g0g2 sampled
143 label(taxisI, '$t$', '$g 0 g 2(t)$');
144 if(flag); saveas(gcf, 'images/g0g2.png'); end
145
187
146 if(flag); figure(46); else subplot(2,4,7); end; hold on;
147 plotf(g0g2hat,['−b';'−r'],0); %g0g2hat
148 plotf(Isample(g0g2hat,T,0),['−−b';'−−r'],1); %g0g2hat sampled
149 label(OmegaaxisI, '$\omega$', '$\widehatg 0 g 2(\omega)$');
150 if(flag); saveas(gcf, 'images/g0g2hat.png'); end
151
152 if(flag); figure(47); else subplot(2,4,4); end; hold on;
153 plotf(Isample(g0g2,Omega,1),['−b';'−r'], 1); %g0g2 Discrete
154 label(taxis, '$t$', '$(1/\Omega)g 0 g 2(n/\Omega)$');
155 if(flag); saveas(gcf, 'images/g0g2D.png'); end
156
157 if(flag); figure(48); else subplot(2,4,8); end; hold on;
158 plotf(Isample(g0g2hat,T,1),['−b';'−r'],1); %g0g2hat Discrete
159 label(Omegaaxis, '$\omega$', '$(1/T)\widehatg 0 g 2(m/T)$');
160 if(flag); saveas(gcf, 'images/g0g2hatD.png'); end
161
162 %Third Plot, FS Equations, Alternate A
163 figure(51); if(˜flag); subplot(2,1,1); end; hold on;
164 plotf(Alta,['−b';'−r'],0); %Alta
165 label(taxisI, '$t$', '$A(t)$');
166 if(flag); saveas(gcf, 'images/alta.png'); end
167
168 if(flag); figure(52); else subplot(2,1,2); end; hold on;
169 plotf(AltAhat,['−b';'−r'],0); %Ahat
170 plotf(Isample(AltAhat,T,0),['−−b';'−−r'],1); %Ahat sampled
188
171 label(OmegaaxisI, '$\omega$', '$A(\omega)$');
172 if(flag); saveas(gcf, 'images/altahat.png'); end
173
174 %Fourth Plot, FS Equations, Approximate c and C
175 figure(61); if(˜flag); subplot(2,2,1); end; hold on;
176 plotf(cApp1,['−b';'−r'],0); %D samples interpolated
177 label([−.75 .75 −150 150], '$t$', '$c(t)$');
178 if(flag); saveas(gcf, 'images/capp.png'); end
179
180 if(flag); figure(62); else subplot(2,2,3); end; hold on;
181 plotf(CApp1,['−b';'−r'],0); %b samples interpolated
182 label([−12 12 −45 45],'$\omega$','$C(\omega)$');
183 if(flag); saveas(gcf, 'images/chatapp.png'); end
184
185 if(flag); figure(63); else subplot(2,2,2); end; hold on;
186 plotf(cApp2,['−b';'−r'],0); %d periodized
187 label([−.75 .75 −90 90], '$t$', '$c(t)$');
188 if(flag); saveas(gcf, 'images/capp2.png'); end
189
190 if(flag); figure(64); else subplot(2,2,4); end; hold on;
191 plotf(CApp2,['−b';'−r'],1); %B periodized
192 label(Omegaaxis , '$\omega$', '$C(\omega)$');
193 if(flag); saveas(gcf, 'images/chatapp2.png'); end
194
195 %Fifth Plot, FS Equations, Approximate g0 and g0hat
189
196 figure(71); if(˜flag); subplot(2,2,1); end; hold on;
197 plotf(g0App1,['−b';'−r'],0); %g0 samples interpolated
198 label(taxisI, '$t$', '$g 0(t)$');
199 if(flag); saveas(gcf, 'images/g0app.png'); end
200
201 if(flag); figure(72); else subplot(2,2,3); end; hold on;
202 plotf(g0hatApp,['−b';'−r'],0); %g0 samples interpolated
203 label([−12 12 −50 50],'$\omega$','$\widehatg 0(\omega)$');
204 if(flag); saveas(gcf, 'images/g0hatapp.png'); end
205
206 if(flag); figure(73); else subplot(2,2,2); end; hold on;
207 plotf(g0App2,['−b';'−r'],0); %g0g1hat samples interpolated
208 label(taxisI, '$t$', '$g 0(t)$');
209 if(flag); saveas(gcf, 'images/g0app2.png'); end
210
211 if(flag); figure(74); else subplot(2,2,4); end; hold on;
212 plotf(g0App3,['−b';'−r'],0); %g0*g 1 periodized
213 label(taxisI, '$t$', '$g 0(t)$');
214 if(flag); saveas(gcf, 'images/g0app3.png'); end
1 function plotf(f,color,impulses)
2 % Plots distribution f
3 % color(1) is the color to plot real(f)
4 % color(2) is the color to plot imag(f)
190
5 % impulses == 1 will treat f like a set of impulses
6
7 if(impulses) %Distribution is set of impulses
8 for i=1:size(f,2)
9 loc = f(1,i); weight = f(2,i);
10 wt = [real(weight) imag(weight)];
11 %Plot longer impulse first
12 imagFirst = wt(1)*wt(2)*(abs(wt(2))−abs(wt(1)))>0;
13 ind = mod(imagFirst,2)+1;
14 plot([loc loc], [0, wt(ind)], color(ind,:));
15 ind = mod(imagFirst+1,2)+1;
16 plot([loc loc], [0, wt(ind)], color(ind,:));
17 end
18 else %Distribution is regular function
19 plot(f(1,:),real(f(2,:)),color(1,:));
20 plot(f(1,:),imag(f(2,:)),color(2,:));
21 end
1 function label(Axis,xstring,ystring)
2 % Use latex to place x−axis/y−axis labels
3 % Author S. Izu, NMSU, 2009
4
5 axis(Axis);
6
191
7 xloc = (Axis(1)+Axis(2))/2;
8 yloc = (Axis(3)−(Axis(4)−Axis(3))/10);
9 H = text(xloc,yloc,xstring);
10 set(H, 'Interpreter', 'latex');
11 set(H, 'FontSize', 14);
12 set(H, 'HorizontalAlignment','Center');
13
14 % xloc = (Axis(1)−(Axis(2)−Axis(1))/5); %Alt option
15 xloc = (Axis(1)−(Axis(2)−Axis(1))/10);
16 yloc = (Axis(3)+Axis(4))/2;
17 H = text(xloc,yloc,ystring);
18 set(H, 'Rotation', 90);
19 set(H, 'Interpreter', 'latex');
20 set(H, 'FontSize', 14);
21 set(H, 'HorizontalAlignment','Center');
1 % This script plots a signal s before and after denoising
2 % Author S. Izu, NMSU, 2009
3
4 load('Wavelet.mat');
5 figure(1); %Plot the scaling filter h
6 plot(h); xlabel('n'); ylabel('h n');
7 saveas(gcf, 'images/h.png');
8
192
9 figure(2); %Calculate the Wavelet packet matrix W
10 imagesc(W);
11 saveas(gcf, 'images/W.png');
12
13 figure(3); %Plot the original signal
14 plot(s); xlabel('n'); ylabel('s n');
15 saveas(gcf, 'images/s.png');
16
17 figure(4); %Plot the scalogram of the original signal
18 imagesc(beforeS); colorbar;
19 saveas(gcf, 'images/sScalogram.png');
20
21 figure(5); %Calculate the filter r
22 plot(r); xlabel('n'); ylabel('r n');
23 saveas(gcf, 'images/filter.png');
24
25 figure(6); %Plot the filtered signal
26 plot(filtered); xlabel('n'); ylabel('r*s n');
27 saveas(gcf, 'images/filtereds.png');
28
29 figure(7); %Plot the scalogram of the filtered signal
30 imagesc(afterS); colorbar;
31 saveas(gcf, 'images/filteredsScalogram.png');
193
1 % This script plots eigenvalues
2 % Author S. Izu, NMSU, 2009
3
4 load('Eigenvalue.mat');
5 figure(1); imagesc(abs(A));
6 figure(2); plot(1:size(D,2),d,'x');
7 display(['sum = ' num2str(sum(d)) ' time = ' num2str(toc)]);
194
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[3] J. L. Jr. Brown. Sampling expansions for multiband signals. IEEE. Trans.Acoustics., 33(1):312–315, 1985.
[4] Peter G. Casazza and Ole Christensen. Frames and Schauder bases. InApproximation theory, volume 212 of Monogr. Textbooks Pure Appl. Math.,pages 133–139. Dekker, New York, 1998.
[5] Peter G. Casazza, Ole Christensen, and Nigel J. Kalton. Frames of translates.Collect. Math., 52(1):35–54, 2001.
[6] Peter G. Casazza and Gitta Kutyniok. Frames of subspaces. In Wavelets,frames and operator theory, volume 345 of Contemp. Math., pages 87–113.Amer. Math. Soc., Providence, RI, 2004.
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