MATLAB Wavelet Analysis, the principle of discrete wavelets in MATLAB

Go to the next toolkit that includes wavelet analysis.

"I feel that the discrete principles of MATLAB seem to have nothing to do with the principles in the book? "Laughing, wow, you just found out? This is a common problem of wavelet reference books, and it is also a difficult point that is easy to make people mad, that is, the schematic understanding of wavelet theory and the actual implementation algorithm are always confused, and in fact, sometimes the two really have nothing to do with each other, and are completely a problem in two fields.

In MATLAB, the understanding of CWT theory is explained by the description in the first paragraph above, but the algorithm is implemented using the formula in the CWT function help document, that is, first calculate the integral difference of the wavelet function, and then multiply by the square of the scale of 1, and the calculation of this formula is actually used to complete the translation of the wavelet by convolution operation, and the expansion and contraction of the wavelet is completed by multiplying the square of different scales of 1. Your description in the first paragraph above is a theoretical explanation, which is schematic, and to achieve it is to convert it into that formula, and to calculate that formula is mainly to integral, differential, convolution and multiplication of 1 scale to operate, this series of operations and its theoretical explanation is to achieve your description in the first paragraph above.

If wavelet analysis ends here, then wouldn't DWT just take a discrete scale based on CWT? The actual application of DWT is far more than that of CWT, and it is more complex, and it is not just discrete from CWT, the reason for this is the introduction of the Mallat algorithm, which turns the application of wavelets to the study of filters. The Mallat algorithm is the most brilliant part of DWT, which can divide the signal into high-frequency detail and low-frequency approximation, which is proposed to accommodate some applications of signal processing.

CWT Mesoscale 2, 4, 8... The information of the wavelet coefficient corresponds to dwt1,2,3... The information of the high-frequency wavelet detail factor, that is, there is no low-frequency approximation, so the application of CWT is very limited.

Therefore, DWT introduced the Mallat algorithm, and when it came to the frequency problem, it was natural to think of filters for signal processing. Thus, with the two-scale equation, the problem of DWT is transformed into a problem of filter design.

The implementation of DWT is completed by a filter designed according to the wavelet function and the scale function, the convolution of the filter is used to complete the translation of the wavelet, and the elongation of the scale is completed by halving the data volume. You still have to refer to the book on filters for these questions. For the question of DWT frequency calculation, please refer to it.

There are very detailed answers, which should be divided into approximation and detail.

It's a long road, and you still have to go up and down!

Question 1 is right.

Target; Question 2 is also true;

In the first half of question 3, the calculation of the frequency band is also correct, but the statement that "the frequency component of the signal can be determined by comparing the amplitude of each frequency band" is wrong. The wavelet transform is not a pure frequency domain method, so the usual application is not suitable for frequency description analysis, and the frequency band of the calculated results has been completed. If you want to get the frequency values of the results of each level, you need to do FFT for the results of each level, and then calculate the frequency value after FFT according to the frequency bands in front of you, and use the centfrq function to calculate the center frequency of the wavelet basis used in the decomposition of each layer, and then identify the frequency values with higher amplitude that are not caused by the center frequency in the results of FFT, and identify all these frequency values of all FFT results of all results of all decomposition levels are the frequency components of the signal. And most of them will have false frequencies that the original signal does not have, these frequencies are generated during the wavelet packet decomposition operation, so you see that beginners always like to use wavelet decomposition to calculate the frequency of the signal, in fact, wavelet analysis is not used like this at all, wavelet analysis in MATLAB is rarely linked to frequency, it is recommended that you stop competing with frequency, that is the concept of pure frequency domain, I don't think it is even suitable to describe the concept of wavelet.

The last problem, except for the 0 fn 2 n band is an approximate coefficient, the other frequency bands used are detail coefficients, but at this time it may not be possible to use the word "high frequency" to describe it, because the frequency of these detail coefficients may not be high, so it is more appropriate to use the detail coefficient.