Wavelet Toolbox™ User's Guide R2022b.

Acknowledgments
	Acknowledgments
Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters
	Wavelet Families
		Daubechies Wavelets: dbN
		Symlet Wavelets: symN
		Coiflet Wavelets: coifN
		Biorthogonal Wavelet Pairs: biorNr.Nd
		Meyer Wavelet: meyr
		Gaussian Derivatives Family: gaus
		Mexican Hat Wavelet: mexh
		Morlet Wavelet: morl
		Additional Real Wavelets
		Complex Wavelets
	Wavelet Families and Associated Properties — I
	Wavelet Families and Associated Properties — II
	Lifting Method for Constructing Wavelets
		Lifting Background
		Polyphase Representation
		Split, Predict, and Update
		Haar Wavelet Via Lifting
		Bior2.2 Wavelet Via Lifting
		Add Lifting Step To Haar Lifting Scheme
		Integer-to-Integer Wavelet Transform
	Orthogonal and Biorthogonal Filter Banks
	Scaling Function and Wavelet
	Lifting a Filter Bank
	Add Quadrature Mirror and Biorthogonal Wavelet Filters
	Least Asymmetric Wavelet and Phase
Continuous Wavelet Analysis
	Using Wavelet Time-Frequency Analyzer App
	1-D Continuous Wavelet Analysis
	Morse Wavelets
		What Are Morse Wavelets?
		Morse Wavelet Parameters
		Effect of Parameter Values on Morse Wavelet Shape
		Relationship Between Analytic Morse Wavelet and Analytic Signal
		Comparison of Analytic Wavelet Transform and Analytic Signal Coefficients
		Recommended Morse Wavelet Settings for the CWT
		References
	Boundary Effects and the Cone of Influence
	Time-Frequency Analysis and Continuous Wavelet Transform
	Continuous Wavelet Analysis of Modulated Signals
	Remove Time-Localized Frequency Components
	Time-Varying Coherence
	Continuous Wavelet Analysis of Cusp Signal
	Two-Dimensional CWT of Noisy Pattern
	2-D Continuous Wavelet Transform
Discrete Wavelet Analysis
	Critically Sampled and Oversampled Wavelet Filter Banks
		Double-Density Wavelet Transform
		Dual-Tree Complex Wavelet Transform
		Dual-Tree Double-Density Wavelet Transforms
	1-D Decimated Wavelet Transforms
		Analysis-Decomposition Functions
		Synthesis-Reconstruction Functions
		Decomposition Structure Utilities
		Denoising and Compression
		1-D Analysis Using the Command Line
		1-D Analysis Using the Wavelet Analyzer App
		Importing and Exporting Information from the Wavelet Analyzer App
	Fast Wavelet Transform (FWT) Algorithm
		Filters Used to Calculate the DWT and IDWT
		Algorithms
		Why Does Such an Algorithm Exist?
		1-D Wavelet Capabilities
		2-D Wavelet Capabilities
	Border Effects
		Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding
		Practical Considerations
		Arbitrary Extensions
		Image Extensions
	Nondecimated Discrete Stationary Wavelet Transforms (SWTs)
		ε-Decimated DWT
		How to Calculate the ε-Decimated DWT: SWT
		Inverse Discrete Stationary Wavelet Transform (ISWT)
		More About SWT
	1-D Stationary Wavelet Transform
		Analysis-Decomposition Functions
		Synthesis-Reconstruction Functions
		1-D Analysis
	Wavelet Changepoint Detection
	Scale-Localized Volatility and Correlation
	R Wave Detection in the ECG
	Wavelet Cross-Correlation for Lead-Lag Analysis
	1-D Multisignal Analysis
		1-D Multisignal Analysis
	2-D Discrete Wavelet Analysis
		Analysis-Decomposition Functions
		Synthesis-Reconstruction Functions
		Decomposition Structure Utilities
		Denoising and Compression
		Wavelet Image Analysis and Compression
	2-D Stationary Wavelet Transform
		Analysis-Decomposition Function
		Synthesis-Reconstruction Function
		2-D Analysis
	Shearlet Systems
		Shearlets
		Transform Type
		References
	Dual-Tree Complex Wavelet Transforms
	Analytic Wavelets Using the Dual-Tree Wavelet Transform
	Multifractal Analysis
Time-Frequency Gallery
	Time-Frequency Gallery
		Short-Time Fourier Transform (Spectrogram)
		Continuous Wavelet Transform (Scalogram)
		Wigner-Ville Distribution
		Reassignment and Synchrosqueezing
		Constant-Q Gabor Transform
		Data-Adaptive Methods and Multiresolution Analysis
Wavelet Packets
	About Wavelet Packet Analysis
	Wavelet Packets
		From Wavelets to Wavelet Packets
		Wavelet Packets in Action: An Introduction
		Building Wavelet Packets
		Wavelet Packet Atoms
		Organizing the Wavelet Packets
		Choosing the Optimal Decomposition
		Some Interesting Subtrees
		Wavelet Packets 2-D Decomposition Structure
		Wavelet Packets for Compression and Denoising
	Introduction to Object-Oriented Features
	Objects in the Wavelet Toolbox Software
	Examples Using Wavelet Packet Tree Objects
		plot and wpviewcf
		Change Terminal Node Coefficients
		Thresholding Wavelet Packets
	Description of Objects in the Wavelet Toolbox Software
		WTBO Object
		NTREE Object
		Private
		DTREE Object
		WPTREE Object
	Build Wavelet Tree Objects
		Building a Wavelet Tree Object (WTREE)
		Working With Wavelet Tree Objects (WTREE)
		Building a Right Wavelet Tree Object (RWVTREE)
		Working With Right Wavelet Tree Objects (RWVTREE)
		Building a Wavelet Tree Object (WVTREE)
		Working With Wavelet Tree Objects (WVTREE)
		Building a Wavelet Tree Object (EDWTTREE)
		Working With Wavelet Tree Object (EDWTTREE)
Denoising, Nonparametric Function Estimation, and Compression
	Wavelet Denoising and Nonparametric Function Estimation
		Denoising Methods
		Soft or Hard Thresholding
		Dealing with Unscaled Noise and Nonwhite Noise
		Wavelet Denoising in Action
		Extension to Image Denoising
		1-D Wavelet Variance Adaptive Thresholding
		Wavelet Denoising Analysis Measurements
	Wavelet Denoising
	Denoise a Signal with the Wavelet Signal Denoiser
	Translation Invariant Wavelet Denoising with Cycle Spinning
		1-D Cycle Spinning
	Multivariate Wavelet Denoising
		Multivariate Wavelet Denoising — Command Line
	Wavelet Multiscale Principal Components Analysis
	Wavelet Data Compression
		Compression Scores
	Wavelet Compression for Images
		Effects of Quantization
		True Compression Methods
		Quantitative and Perceptual Quality Measures
		More Information on True Compression
	2-D Wavelet Compression
		Compression by Global Thresholding and Huffman Encoding
		Uncompression
		Compression by Progressive Methods
		Handling Truecolor Images
Matching Pursuit
	Matching Pursuit Algorithms
		Redundant Dictionaries and Sparsity
		Nonlinear Approximation in Dictionaries
		Basic Matching Pursuit
		Orthogonal Matching Pursuit
		Weak Orthogonal Matching Pursuit
	Matching Pursuit
		Sensing Dictionary Creation and Visualization
		Orthogonal Matching Pursuit on 1-D Signal
		Electricity Consumption Analysis Using Matching Pursuit
Code Generation from MATLAB Support in Wavelet Toolbox
	Code Generation Support, Usage Notes, and Limitations
	Denoise Signal Using Generated C Code
	Generate Code to Denoise a Signal
	CUDA Code from CWT
Special Topics
	Wavelet Scattering
		Wavelet Scattering Transform
		Invariance Scale
		Quality Factors and Filter Banks
		In Practice
	Wavelet Scattering Invariance Scale and Oversampling
	Empirical Wavelet Transform
	Tunable Q-factor Wavelet Transform
		Frequency-Domain Scaling
		TQWT Algorithm
		Redundancy and Q-factor
		Example: MRA of Audio Signal
Featured Examples — Time-Frequency Analysis
	Practical Introduction to Continuous Wavelet Analysis
	CWT-Based Time-Frequency Analysis
	Time-Frequency Reassignment and Mode Extraction with Synchrosqueezing
	Frequency- and Time-Localized Reconstruction from the Continuous Wavelet Transform
	Compare Time-Frequency Content in Signals with Wavelet Coherence
	Continuous and Discrete Wavelet Analysis of Frequency Break
	Wavelet Packets: Decomposing the Details
	Wavelet Analysis of Physiologic Signals
	Visualize and Recreate EWT Decomposition
	Visualize and Recreate VMD Decomposition
Featured Examples — Discrete Multiresolution Analysis
	Practical Introduction to Multiresolution Analysis
	Wavelet Analysis for 3-D Data
	Multisignal 1-D Wavelet Analysis
	Detecting Discontinuities and Breakdown Points
	Haar Transforms for Time Series Data and Images
	Wavelet Analysis of Financial Data
	Image Fusion
	Boundary Effects in Real-Valued Bandlimited Shearlet Systems
	Wavelet Packet Harmonic Interference Removal
	Visualize and Recreate TQWT Decomposition
Featured Examples — Denoising and Compression
	Denoising Signals and Images
	Wavelet Interval-Dependent Denoising
	Multivariate Wavelet Denoising
	Multiscale Principal Components Analysis
	Data Compression using 2-D Wavelet Analysis
	Smoothing Nonuniformly Sampled Data
	Two-Dimensional True Compression
	Signal Deconvolution and Impulse Denoising Using Pursuit Methods
Featured Examples — Machine Learning and Deep Learning
	Signal Classification Using Wavelet-Based Features and Support Vector Machines
	Wavelet Time Scattering for ECG Signal Classification
	Wavelet Time Scattering Classification of Phonocardiogram Data
	Wavelet Time Scattering with GPU Acceleration — Spoken Digit Recognition
	Spoken Digit Recognition with Wavelet Scattering and Deep Learning
	Classify Time Series Using Wavelet Analysis and Deep Learning
	Texture Classification with Wavelet Image Scattering
	Digit Classification with Wavelet Scattering
	Acoustic Scene Recognition Using Late Fusion
	GPU Acceleration of Scalograms for Deep Learning
	Deploy Signal Classifier Using Wavelets and Deep Learning on Raspberry Pi
	Crack Identification from Accelerometer Data
	Deploy Signal Classifier on NVIDIA Jetson Using Wavelet Analysis and Deep Learning
	Code Generation for a Deep Learning Simulink Model to Classify ECG Signals
	Modulation Classification Using Wavelet Analysis on NVIDIA Jetson
	Parasite Classification Using Wavelet Scattering and Deep Learning
	Air Compressor Fault Detection Using Wavelet Scattering
	Fault Detection Using Wavelet Scattering and Recurrent Deep Networks
	Anomaly Detection Using Autoencoder and Wavelets
	Detect Anomalies Using Wavelet Scattering with Autoencoders
	Classify ECG Signals Using DAG Network Deployed to FPGA
	Human Health Monitoring Using Continuous Wave Radar and Deep Learning
	Signal Recovery with Differentiable Scalograms and Spectrograms
Wavelet Analyzer Topics
	Matching Pursuit Using Wavelet Analyzer App
		Matching Pursuit 1-D Interactive Tool
		Interactive Matching Pursuit of Electricity Consumption Data
	1-D Wavelet Packet Analysis
		Starting the Wavelet Packet 1-D Tool
		Importing a Signal
		Analyzing a Signal
		Computing the Best Tree
		Compressing a Signal Using Wavelet Packets
		De-Noising a Signal Using Wavelet Packets
	2-D Wavelet Packet Analysis
		Starting the Wavelet Packet 2-D Tool
		Compressing an Image Using Wavelet Packets
	Importing and Exporting from Wavelet Analyzer App
		Saving Information to Disk
		Loading Information into the Graphical Tools
	drawtree and readtree
	2-D CWT App Example
	Importing and Exporting Information
		Loading Signals
		Saving Wavelet Coefficients
	1-D Adaptive Thresholding of Wavelet Coefficients
		1-D Local Thresholding Using the Wavelet Analyzer App
		Importing and Exporting Information from the Wavelet Analyzer App
	Multiscale Principal Components Analysis Using the Wavelet Analyzer App
	Importing and Exporting PCA from the Wavelet Analyzer App
	2-D Wavelet Compression using the Wavelet Analyzer App
	Importing and Exporting Compressed Image from the Wavelet Analyzer App
	Univariate Wavelet Regression
		Regression for Equally-Spaced Observations
		Regression for Randomly-Spaced Observations
		Importing and Exporting Information from the Wavelet Analyzer App
	Multivariate Wavelet Denoising Using the Wavelet Analyzer App
	Importing and Exporting Denoised Signal from the Wavelet Analyzer App
	Interactive 1-D Stationary Wavelet Transform Denoising
		Selecting a Thresholding Method
	Importing and Exporting 1-D SWT Denoised Image from the Wavelet Analysis App
	3-D Discrete Wavelet Analysis
		Performing 3-D Analysis Using the Command Line
		Performing 3-D Analysis Using the Wavelet Analyzer App
		Importing and Exporting Information from the Wavelet Analyzer App
	2-D Wavelet Analysis Using the Wavelet Analyzer App
	Importing and Exporting 2-D Information from the Wavelet Analyzer App
		Saving Information to Disk
		Loading Information into the Wavelet 2-D Tool
	Interactive 2-D Stationary Wavelet Transform Denoising
	Importing and Exporting 2-D SWT Information from the Wavelet Analyzer App
	Comparing 1-D Extension Differences
	1-D Multisignal Analysis Using the Wavelet Analyzer App
		Manual Threshold Tuning
		Statistics on Signals
		Clustering Signals
		Partitions
		More on Clustering
	Importing and Exporting Multisignal Information from the Wavelet Analyzer App
		Saving Information to Disk
		Saving Decompositions
		Loading Information into the Wavelet 1-D Multisignal Analysis Tool
		Loading and Saving Partitions.
Generating MATLAB Code from Wavelet Toolbox Wavelet Analyzer App
	Generate MATLAB Code for 1-D Decimated Wavelet Denoising and Compression
		Wavelet 1-D Denoising
	Generate MATLAB Code for 2-D Decimated Wavelet Denoising and Compression
		2-D Decimated Discrete Wavelet Transform Denoising
		2-D Decimated Discrete Wavelet Transform Compression
	Generate MATLAB Code for 1-D Stationary Wavelet Denoising
		1-D Stationary Wavelet Transform Denoising
	Generate MATLAB Code for 2-D Stationary Wavelet Denoising
		2-D Stationary Wavelet Transform Denoising
	Generate MATLAB Code for 1-D Wavelet Packet Denoising and Compression
		1-D Wavelet Packet Denoising
	Generate MATLAB Code for 2-D Wavelet Packet Denoising and Compression
		2-D Wavelet Packet Compression
Wavelet Analyzer App Features Summary
	General Features
		Color Coding
		Connection of Plots
		Using the Mouse
		Controlling the Colormap
		Using Menus
		Using the View Axes Button
	Wavelet 2-D Tool Features
	Wavelet Packet Tool Features (1-D and 2-D)
		Coefficients Coloration
		Node Action
		Node Label
		Node Action Functionality