Basic concept
- 涉及两个概念
- Timing
- $F_{eq}=\frac{C_f}{S\delta t}$
- $C_f$ is teh center frequency of the wavelet
- $s$ is the waveley scale
- $\delta_t$ is the sampling interval
- when scaling a wavelet by a factor of 2, it results in reducing teh equvalent frequency by an octave
- Shifting
- Timing
Continuous Wavelet Transform (CWT)
- Application
- time frequency analysis
- filtering of time localized frequency components
- Auitbale analytic wavelets:
- Morse Wavelets
- Bump Wavelet
- Analytic Morlet Wavelet
- Output of CWT
- are coefficients which are functions of scales or frequency and time
- Wavelet scales:
- scales per Octave:
- e.g. A signal with 1000 samples analyzed with 20 scales results in 20000 coefficients
Discrete Wavelet Transform (DWT)
- Application
- denoising
- compressing signals and images
- Scaling:
- The basic scale in DWT is set to 2: $2^j (j=1,2,3,…)$
- can obtain different scales by raising this base scale to integers value represented in this way
- Translation:
- dyadic scaling and shifting: $2^jm, (m=1,2,3,…)$
- this kind of sampling eliminates redundancy in coefficients
- The output of the transform yields the same number of coefficients as the length of the input signal
- Process:
- In each level of the computation, the signal will be filtered with a high pass and low pass filters, yielding the coefficients
- dowm through the levels, teh scale will be reduce by 2 times
Coherence
- Time-Varying Coherence
- Fourier-domain coherence measures the linear correlation between two stationary processes as a function of frequency on a scale from 0 to 1.
- Because wavelets provide local information about data in time and scale (frequency), wavelet-based coherence allows you to measure time-varying correlation as a function of frequency.
- In other words, a wavelet coherence measure suitable for nonstationary processes.
- function
wcoherence
to output the wavelet coherence, cross-spectrum, scale-to- frequency, or scale-to-period conversions
Matlab functions
- helper function:
helperPlotScalogram3d
: plot the scalogram as a 3-D surfacehelperPlotScalogram
: plot the scalogram of the signal and the instantaneous frequencieshelperPlotCoherence
: plot the outputs of wcoherence
Reference:
MATLAB 出品: https://www.youtube.com/watch?v=F7Lg-nFYooU
https://au.mathworks.com/help/wavelet/time-frequency-analysis.html
https://au.mathworks.com/help/wavelet/ug/removing-a-time-localized-frequency-component-using-the-inverse-cwt.html