OUTLINE OF THE UPCOMING BOOK...

Chapters currently available for free download in PDF format indicated by asterisk (*)

• CHAPTER 1* - Preview of Wavelets, Wavelet Filters and Wavelet Transforms in Digital Signal Processing (DSP)
  What is a Wavelet?
  What is a Wavelet Filter and how is it Different from a Wavelet?
  The Value of Transforms and Examples of Everyday Use
  Short-Time Transforms, Sheet Music, and a First Look at Wavelet Transforms
  An Example of the Fast Fourier Transform (FFT) with an Embedded Pulse Signal
  Examples Using the Continuous Wavelet Transform
  A First Glance at the Undecimated Discrete Wavelet Transform (UDWT)
  Why the UDWT is Sometimes Called a “Redundant” DWT (RDWT) or “A’ Trous” (“with holes”)
  A First Glance at the Conventional Discrete Wavelet Transform (DWT)
  Examples of Use of the Conventional DWT
  Summary
• CHAPTER 2* - Walk-Through of the Continuous Wavelet Transform (CWT) using the Haar Wavelet Filters
  Simple Scenario: Comparing Exam Scores Using the Haar Wavelet
  Above Comparison Process Seen as Simple Correlation or Convolution
  Display of the Continuous Wavelet Transform (CWT) of the Exam Scores Using the Haar Wavelet Filter
  Summary
• CHAPTER 3* - Walk-Through of the Undecimated Discrete Wavelet Transform (UDWT) using the Haar Wavelet Filters
  Single-Level UDWT of Exam Data
  Frequency Allocation of a Single-Level UDWT
  Multi-Level UDWT
  Frequency Allocation of a Multi-Level UDWT
  The Haar UDWT as a Moving Averager
  Summary
• CHAPTER 4* - Walk-Through of the Conventional (Decimated) Discrete Wavelet Transform (DWT) using the Haar Wavelet Filters
  Single-Level (Decimated/Downsampled) Discrete Wavelet Transform (DWT) of Exam Data
  Additional Example of Perfect Reconstruction in a Single-Level DWT
  Compression and Denoising Example using the Single-Level DWT
  Multi-Level Conventional (Decimated) Discrete Wavelet Transform (DWT) of Exam Data using Haar Wavelet Filters
  Frequency Allocation in a (Conventional, Decimated) DWT
  Final Approximations and Details and How to Read the DWT Display
  Denoising Using a Multi-Level DWT
  Summary
• CHAPTER 5 - Filters from Wavelets—Obtaining Real-World Discrete Filters from Wavelets with Explicit Mathematical Expressions (“Crude Wavelets”)
  Review of Familiar DSP Truncated Sinc Function
  Adding More Points at the Ends for Better Filter Performance
  Adding More Points by Interpolation for Lower Cutoff Frequency
  Multi- Point “Stretched Filters” Derived from Explicit Mathematical Equations of “Crude” Wavelets
  Mexican Hat Wavelet and Morlet Wavelet as Examples of Stretched Filters
  How the Passband is Changed due to Stretching
  CWT Display of the Results of Using these Stretched Filters on a Split-Sine Test Signal
  Summary
• CHAPTER 6 - Wavelets from Filters—Fixed Length Filters to Continuous Wavelet Estimation to Variable Length Filters
  Interpolation of Original Wavelet Filter by Upsampling and LowPass Filtering
  Building an Estimation (Approximation) of a “Continuos” Wavelet Function
  Building a Filter of Any Desired Length from the “Continuous” Estimation
  Example of Building a 258-Point “Continuous” Haar Wavelet Function from the 2-Point Filter
  Building a Haar Filter of Arbitrary Length from the 258 Point Estimation
  Numerical Integration to Handle the Discontinuities in the Haar
  Frequency Response of the Original and Stretched Haar Filters
  Haar and Shannon Wavelet Filters as “Duals” of Time/Frequency Precision
  Example of Building a 768-Point “Continuous” Daubechies 4 (Db4) Wavelet Function from the 4-Point Filter
  Building a Db4 Wavelet Function Filter of Arbitrary Length from the 768 Points
  Perfect Overlay of Four Equispaced Db4 Filter Points on the “Continuous” Wavelet Estimation
  Two Additional Equispaced Zero Points (at the end) Complete the Overlay
  Physical Meaning of the “Length” of a Wavelet
  Frequency Response of the Original and Stretched Db4 Wavelet Function Filters
  Perfect Overlay of the Db6, Db8, Coiflet, and Biorthogonal Filter Points and Zero Points on their Estimations
  Summary
• CHAPTER 7 - Overview and Comparison of the Four Major Types of Wavelet Transforms
  Advantages and Disadvantages of the Continuous Wavelet Transform (CWT)
  Stretching the Wavelet—The Undecimated Discrete Wavelet Transform (UDWT)
  Shrinking the Signal Instead—The Conventional (Downsampled) Discrete Wavelet Transform (DWT)
  Decomposing More of the Signal—The Wavelet Packet Transform (WPT)
  The Problem of Aliasing Caused by Downsampling in the WPT and the Conventional DWT
  How to Cancel Out the Aliasing —If Done Correctly
  Summary
• CHAPTER 8 - Perfect Reconstruction Quadrature Mirror Filters (PRQMF) and their Relationships
  The Four PRQMF Filters—HighPass (HP) Decomposition, HP Reconstruction, LowPass (LP) Decomposition and LP Reconstruction
  Close Relationships of the Four PRQMF Filters to Each Other
  Building the Wavelet Function from the HP Reconstruction Filter
  Building the Scaling Function from the LP Reconstruction Filter
  Summary
• CHAPTER 9 - Another Look at the DWT and UDWT Displays—Important Relationships
  DWT and UDWT Displays of a Split Sine Signal with the Db4 Scaling Function
  DWT and UDWT Displays of a Split Sine Signal with the Db4 Wavelet Function
  DWT and UDWT Displays of a Decaying Exponential Signal with the Haar Scaling Function.
  DWT and UDWT Displays of a Decaying Exponential Signal with the Haar Wavelet Function
  Summary
• CHAPTER 10 - Reconciling the DWT and UDWT to the CWT—All are Basic Correlations
  Locations on the Signal Diagram where the UDWT and the CWT are Exactly the Same Using Haar Wavelet Filters
  Locations on the Signal Diagram where the Conventional DWT and the CWT are the Same When Downsampled Using Haar Wavelet Filters
  Locations on the Signal Diagram where the Conventional DWT and the CWT are Very Similar Using Four-Point Db4 Filters
  Summary
• CHAPTER 11 - Looking at the Major Wavelet Families—Their Strengths, Weaknesses, and Popular Uses
  Vanishing Moments
  Time vs. Frequency Resolution
  Orthogonality
  Symmetry
  Suitability for use in a DWT or UDWT
  Table of Desired Properties and Best Choice of Wavelet
  The “Sport of Basis Hunting” and Best Basis
  The Lifting Scheme
  Summary
• CHAPTER 12 - Case studies of Applications of Wavelets to Real-Life Problems
  Removing White Noise Using the CWT
  Extracting a Weak Binary QPSK Signal from 80 dB of Noise Using Time-Dependant Thresholding
  Noise Identification Using the CWT and the DWT
  Compression of Images Using a 2-D DWT with the Biorthogonal Wavelet
  JPEG Compression Using Wavelets
  Example showing Superior Denoising Capability Using the UDWT
  Summary
• CHAPTER 13 - Building the Scaling Function (phi) using the PRQMF Filters and the Dilation Equation
  Review of Building Wavelet Function (Estimation) Using Wavelet Filter Points at Known Locations to Produce Additional Points
  Building Scaling Function (Estimation) Using Scaling Function Filter Points at Known Locations
  Repeated Upsampling and Filtering to Produce Multi-Point Estimation of a “Continuous” Scaling Function
  Overlaying Original Scaling Function Filter Points on the “Continuous” Scaling Function Estimation for Perfect Fit
  Historical Perspective of Roman Numerals, Negative and Irrational Numbers, Imaginary Numbers, etc.
  The 2-Scale Difference Equation and the Scaling Function Dilation Equation
  Relating the Scaling Function Dilation Equation to the Conventional Discrete Wavelet Transform
  Recursive Wavelet Processing as a “Child of the Digital Age”
  Summary
• CHAPTER 14 - Building the Scaling Function (phi) using the Perfect Reconstruction Quadrature Mirror Filters and Simple Convolution
  Replication of the Dilation Equation by Dyadic Upsampling and Convolution
  The Scaling Function is Built from the Filters, not the Other Way Around
  Upsampling and Filtering can be Observed in DWT Signal Flow Charts—Miniature Scaling Function Replicas Seen as a Result
  Summary
• CHAPTER 15 - Building the Wavelet Function (psi) from Both the Dilation Equation and by Simple Convolution
  Wavelet Function Dilation Equation uses BOTH Scaling Function AND Wavelet Function Basic Filter Points (LowPass and HighPass) to Produce Additional Points
  Other Similarities and Differences to the Scaling Function Dilation Equation
  Another Look at Repeated Upsampling and Filtering to Build the Multi-Point “Continuous” Wavelet Estimation
  Upsampling and Filtering can be Observed in DWT Signal Flow Charts
  Miniature Wavelet (Function) Replicas Also Observed as Artifacts from Repeated Upsampling and Filtering in Signal Diagrams.
  Summary
• CHAPTER 16 - Perfect Reconstruction Begins with the HalfBand Filters
  A First Look at the HalfBand (HB) Filters
  HighPass HB Filter Factored into Wavelet Decomposition and Reconstruction Filters
  LowPass HB Filter Factored into Scaling Function Decomposition and Reconstruction Filters
  HB Filters in the Simple Undecimated Discrete Wavelet Transform
  HB Filters in the Conventional (Downsampled) Discrete Wavelet Transform
  Summary
• CHAPTER 17 - Alias Cancellation Demonstrated in the Time Domain
  Step-By-Step Look at Convolution (Filtering) on both the HP and LP Paths of a DWT
  Cancellation of HP and LP Terms Except When They Align
  Perfect Reconstruction to Within a Delay and a Constant of Multiplication
  Linear Time Invariant (LTI) Properties of the DWT
  Signal Flow Diagrams of the LTI DWT to Extract Needed Approximations and Details
  Summary
• CHAPTER 18 - Alias Cancellation Demonstrated in the Frequency Domain
  Filtering of the Signal in the DWT by the Decomposition Filters
  Downsampling the Result Which Can Cause Aliasing
  Equivalence of Downsampling to “Sliding” Higher Frequencies to the Left
  Upsampling Now Causes Imaging or “Unfolding” of Frequencies
  Complex Addition of HP and LP Paths
  Alias Cancellation on Because Same Magnitude but Opposite Phase from the HP and LP Paths
  Signal Reconstruction Because In Phase Components from the HP and LP Paths
  Summary
• CHAPTER 19 - Relating Alias Cancellation Concepts to Equations Found in Traditional Literature
  Paraunitary Conditions
  “No-Distortion” Equations and In-Phase Addition
  Alias Cancellation Equations and Out-Of-Phase Cancellations
  Summary
• CHAPTER 20 - Creating “Fake” Wavelets
  Appreciating the Elegant Properties of the Classic Wavelets
  Single Cycle of a Sine Wave as a “Crude” Fake Wavelet
  CWT Display Comparison Using Haar and Db4 Wavelets
  Using a GPS Signal as a “Fake Wavelet” in a CWT to Account for Doppler, Slew, and Chirp
  Truncated Db4 Wavelet Filters—Valid HalfBand Filters but Fewer Vanishing Moments
  Vanishing Moments Revisited
  Summary
• CHAPTER 21 - Deriving the “Magic Numbers” of the Wavelet Filters from the Desired Capabilities
  Vanishing Moments Properties
  Desired Orthogonality Properties
  Wavelet Filter Values Sum to Zero
  Squares of Wavelet Filter Values Sum to Unity (Unit Energy)
  Simple Substitution to Produce the “Magic Numbers” for all Four Basic Wavelet Filters
  Summary
• CHAPTER 22 - Trading Pure Orthogonality for “Biorthogonality”, Symmetry, and Linear Phase
  Another Way to Factor the HalfBand Filters
  Decomposition and Reconstruction Filters with Different Lengths
  Symmetry and Linear Phase
  Comparing Db4 Filters (4 Points Each) with BIOR 3/5 Filters (3 or 5 Points Each)
  Limited Orthogonality or “Bi”-Orthogonality
  Use of the BIOR 7/9 Filters for JPEG and FBI Fingerprint Compression
  Summary
• CHAPTER 23 - The Undecimated Discrete Wavelet Transform (UDWT) Revisited
  Extra Storage and Computational Burden vs. Alias-Free Processing
  Stretching the Filters Correctly in Octaves of Frequency
  Comparison of UDWT and Conventional DWT Displays
  Pathological Noise Cases Which Favor the use of the UDWT
  The UDWT as Sanity Check for the DWT
  Hybrid Methods using Both UDWT and DWT
  Summary
• APPENDICES

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