Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis.Discrete wavelet transform (continuous in time) of a discrete-time (sampled) signal by using discrete-time filterbanks of dyadic (octave band) configuration is a wavelet. Wavelet Toolbox™ provides functions and apps for analyzing and synthesizing signals and images. The toolbox includes algorithms for continuous wavelet analysis, wavelet coherence, synchrosqueezing, and data-adaptive time-frequency analysis.
Analyze signals and images in the wavelet domain
Wavelet transforms are mathematical tools for analyzing data where features vary over different scales. For signals, features can be frequencies varying over time, transients, or slowly varying trends. For images, features include edges and textures. Wavelet transforms were primarily created to address limitations of the Fourier transform.
While Fourier analysis consists of decomposing a signal into sine waves of specific frequencies, wavelet analysis is based on decomposing signals into shifted and scaled versions of a wavelet. A wavelet, unlike a sine wave, is a rapidly decaying, wave-like oscillation. This enables wavelets to represent data across multiple scales. Different wavelets can be used depending on the application. Wavelet Toolbox™ for use with MATLAB® supports Morlet, Morse, Daubechies, and other wavelets used in wavelet analysis.
Audio signals, time-series financial data, and biomedical signals typically exhibit piecewise smooth behavior punctuated by transients. Similarly, images typically include homogenous, piecewise smooth regions separated by transients, which appear as edges. For both signals and images, the smooth regions and transients can be sparsely represented with wavelet transforms.
Capturing transient behavior in signals using a MATLAB wavelet transform.
Wavelet transforms can be classified into two broad classes: the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT).
Matlab Wavelet Pdf
The continuous wavelet transform is a time-frequency transform, which is ideal for analysis of non-stationary signals. A signal being nonstationary means that its frequency-domain representation changes over time. CWT is similar to the short-time Fourier transform (STFT). The STFT uses a fixed window to create a local frequency analysis, while CWT tiles the time-frequency plane with variable-sized windows. The window widens in time, making it suitable for low-frequency phenomena, and narrows for high-frequency phenomena. The continuous wavelet transform can be used to analyze transient behavior, rapidly changing frequencies, and slowly varying behavior.
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Analyzing a hyperbolic chirp signal (left) with two components that vary over time in MATLAB. The short-time Fourier transform (center) does not clearly distinguish the instantaneous frequencies, but the continuous wavelet transform (right) accurately captures them. See the MATLAB code.
With the discrete wavelet transform scales are discretized more coarsely than with CWT. This makes DWT useful for compressing and denoising signals and images while preserving important features. You can use discrete wavelet transforms to perform multiresolution analysis and split signals into physically meaningful and interpretable components.Original (left) and denoised (right) images. The image was denoised while preserving the edges using a wavelet denoising function. See the MATLAB code.
Wavelet Transform Matlab Tutorial'>
Original (left) and denoised (right) images. The image was denoised while preserving the edges using a wavelet denoising function. See the MATLAB code.
For more information on applying wavelet techniques and selecting the right wavelets for your application in MATLAB, see Wavelet Toolbox.
Examples and How To
- Practical Introduction to Continuous Wavelet Analysis - Example
- Practical Introduction to Multiresolution Analysis - Example
- Machine Learning and Deep Learning with Wavelets - Documentation
- Continuous and Discrete Wavelet Transforms - Documentation
- Continuous Wavelet Transform and Scale-Based Analysis - Documentation
- Wavelet Packets - Documentation
- Lifting Method for Constructing Wavelets - Documentation
- Critically-Sampled Discrete Wavelet Transform - Documentation
- Matching Pursuit Algorithms - Documentation
- Function Reference for Wavelet Analysis - Documentation
See also: Signal Processing Toolbox, DSP System Toolbox, wavelet transforms videos, empirical mode decomposition
Definition of the Continuous Wavelet Transform
Like the Fourier transform, the continuous wavelet transform (CWT) uses inner products to measure the similarity between a signal and an analyzing function. In the Fourier transform, the analyzing functions are complex exponentials, . The resulting transform is a function of a single variable, ω. In the short-time Fourier transform, the analyzing functions are windowed complex exponentials, , and the result is a function of two variables. The STFT coefficients, represent the match between the signal and a sinusoid with angular frequency ω in an interval of a specified length centered at τ.
In the CWT, the analyzing function is a wavelet, ψ. The CWT compares the signal to shifted and compressed or stretched versions of a wavelet. Stretching or compressing a function is collectively referred to as dilation or scaling and corresponds to the physical notion of scale. By comparing the signal to the wavelet at various scales and positions, you obtain a function of two variables. The 2-D representation of a 1-D signal is redundant. If the wavelet is complex-valued, the CWT is a complex-valued function of scale and position. If the signal is real-valued, the CWT is a real-valued function of scale and position. For a scale parameter, a>0, and position, b, the CWT is:
where denotes the complex conjugate. Not only do the values of scale and position affect the CWT coefficients, but the choice of wavelet also affects the values of the coefficients.
By continuously varying the values of the scale parameter, a, and the position parameter, b, you obtain the cwt coefficientsC(a,b). Note that for convenience, the dependence of the CWT coefficients on the function and analyzing wavelet has been suppressed.
Multiplying each coefficient by the appropriately scaled and shifted wavelet yields the constituent wavelets of the original signal.
There are many different admissible wavelets that can be used in the CWT. While it may seem confusing that there are so many choices for the analyzing wavelet, it is actually a strength of wavelet analysis. Depending on what signal features you are trying to detect, you are free to select a wavelet that facilitates your detection of that feature. For example, if you are trying to detect abrupt discontinuities in your signal, you may choose one wavelet. On the other hand, if you are interested in finding oscillations with smooth onsets and offsets, you are free to choose a wavelet that more closely matches that behavior.