APPENDIX A - REVIEW OF DIGITAL SIGNAL PROCESSING AND FILTERING METHODS (DSP 101)
- A.1 General DSP Review
Time/Frequency Relationships and their Annotation. Tighter Sampling vs. Longer Sampling. Periodic, Aperiodic, Continuous and Discrete Relationships in Time and Frequency. The Fourier Transform, Discrete Time Fourier Transform (DTFT), Fourier Series, and the Discrete (in both Time and Frequency) Fourier Transform (DFT or the Functionally Equivalent FFT).
- A.2 Filtering Review
Overview of Basic DSP Filtering Concepts. Finite Impulse Response (FIR) Filters. Circular Convolution. Zero Padding. Equivalence of Convolution in Time to Multiplication in Frequency. Magnitude and Phase. Symmetry and Linear Phase. Shifting in Time for Zero Phase. Causal and Anticausal Relationships. Z-Transforms, Z-Planes and Unit Circles. Parks-McClellen, REMEZ, and Least Squares Algorithms.
- A.3 Brickwall Filters
The Lure of an Ideal Filter. Proper and Improper Use of “Brick-Wall” Filters. Special Cases of Ideal Filtering Without a Transition Band. The Problem of Bin Leakage . When to Persevere with Conventional Filtering Methods.
- A.4 LTI Systems
Short Review of Linear Time Invariant (LTI) Systems and Unit Impulse Response. Tests for Linearity and Time Invariance. Diagrams and Examples. Demonstrating Perfect Reconstruction in the DWT to Show its LTI Properties.
- A.5 Interpolation
The Upsampling and LowPass Filtering Process Seen as Simple Interpolation from Conventional DSP. Comparison to FFT/IFFT Interpolation Methods. Diagrams and Examples.
- A.6 Eigenvalues
Short Tutorial Review of Eigenvalues and Eigenvectors for use with some Advanced Concepts. Examples and Demonstrations.
APPENDIX B - ADDITIONAL INFORMATION AND CLARIFICATIONS
- B.1 Detailed Walk-Through of the DWT
A Closer Look at the DWT Approximations, Details, and their Coefficients. Following a Split-Sine Signal Through the HighPass and LowPass Paths. Setting Thresholds for De-Noising. Single and Multi-Level DWTs and their Frequency Allocations. MATLAB Display Idiosyncrasies and how to Understand Them.
- B.2 Complex Vectors
Complex Vectors in General. Magnitude and Phase. Application to Alias Nullification in Wavelets. Review of In-Phase Addition and Out-Of-Phase Cancellation.
- B.3 Orthogonality
Orthogonal Vectors and Right Angles. Orthogonality of Sines and Cosines. Phase Relationships. Wavelets and Scaling Function Orthogonality and Orthonormality. Orthogonality in Stretched Wavelets. Mathematical Relationships.
- B.4 Haar Filters seen as Simple Block Averagers and Differentiators
Getting a “Feel” for Wavelets by Comparing the Scaling Function (LowPass) Filters to the Familiar Block Averagers from Conventional DSP. Smoothing the Signal by Block
Averaging Compared to the “Approximations” in Wavelets. Haar Wavelet Function (HighPass) Filters Compared to Familiar Differentiators from DSP. Differences of Successive Points Compared to the “Details” in Wavelets.
- B.5 Perfect Reconstruction in the UDWT
Demonstrating how the Simple Undecimated DWT (UDWT, RDWT or A’ Trous) Achieves
Perfect Reconstruction. HighPass and LowPass Filter Paths Combining to Reproduce the Input Signal to within a Delay and a Constant of Multiplication.
APPENDIX C - ADDITIONAL WAVELET OPTIONS, APPLICATIONS AND DEMONSTRATIONS
- C.1 Additional Image Processing and Display Options
MATLAB Image Processing Displays and How to Interpret Them. Extending a 1-D Signal to 2-D for Ease in Understanding. 1-D DWT vs. 2-D Displays. Vertical, Horizontal, and Diagonal Details all used with the 2-D Approximations for Reconstruction. Various MATLAB 2-D Display Methods (“Tree”, “Nested”, etc.).
- C.2 Additional Examples of Wavelet Processing
Multiscale Feature Detection using a Stairstep Function. Noisy Signal with an Embedded Pulse. Finding the Time Corresponding to a Frequency Variation.
APPENDIX D - ADVANCED CONCEPTS
- D.1 Conventional DFT (FFT) Seen as Either Weighted Sums or Correlations
The Discrete Fourier Transform (DFT) with Correlations to Sinusoids. The Discrete Wavelet Transform (DWT) with Correlations to Wavelets. Differences and Similarities for Better Conceptual Understanding. Nomenclature Found in Conventional Wavelet Literature.
- D.2 In-Depth Look at Aliasing in a DWT and how Frequency Locations are Manipulated
“Sliding” of Signals. Switching of Frequency Locations by Downsampling and Upsampling. Frequency Switching in Wavelet Packet Transforms and in Transmultiplexers. TDMA, CDMA, FDMA Applications.
- D.3 Introduction to Complex Wavelets
Uses of Complex Wavelets in Basebanding, Resampling, Bandshifting, Downconverting, and Bin Alignment. Uses of Complex Wavelets with a Complex Signal. How to Make a Real Signal Complex. Complex Morlet and Shannon Wavelets. Displays Showing the Real, Imaginary and Modulus Components of a Complex CWT.
- D.4 Building Scaling and Wavelet Functions from Eigenvalues and Eigenvectors
An Alternative Method Often Found in Conventional Literature. Producing an Estimation of the Scaling Function and Wavelet Function Using the Dilation Equation.
- D.5 Spectral Factorization
An Alternative Method to Obtain the “Magic Numbers” Filter Coefficients. Using Software such as MATLAB’s ROOTS.M to Obtain the Roots of the Z Transform Polynomial of the HalfBand filters. Using the Two Haar 3-point HalfBand Filters to Obtain the Four Haar Scaling and Wavelet Filters. Using the 7-point Db4 HalfBand filters to Obtain the Scaling and Wavelet filters. Adding a Constant to the Frequency Domain Values to Obtain Real Roots for Spectral Factorization. Additional Methods From Wavelet Literature.
- D.6 Discussion of the Dual-Tree Complex Wavelet Method
A Method Incorporating Elements of Both the DWT and UDWT. Shift Variance and Alias Problems Addressed. Change of Filters at Various Levels. Half-Sample Shifts, Q-Shifts, Common Factors, Filter Swapping and Other Methods. Software Resources.
APPENDIX E - FURTHER RESOURCES
- The Best of Wavelet books, Papers, Software, and Websites with Real-World Applications. References and Suggestions for the Student. Synopses of the Contents and Teaching Methods of Recommended Books and Papers. Wavelet Websites and an Overview of their Contents and Applicability. Wavelet-Related Software and Other Resources.
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