If I understand your question correctly:

In time series analysis, using the discrete wavelet transform, the maximum wavelet level is limited by the number of time points, N; for the DWT, J0 (the maximum number of levels), cannot be greater than j, where N=2^j. For redundant discrete transforms, J0 can be greater than j, but the information tends to zero for these higher levels. The exact choice of J0 will depend on the application, e.g. you can define J0 conservatively as J0

Always the wavelet of effectively limited duration that it starts from zero and ends to zero.

Daniel I wanted to know about the level of decomposition no the mother wavelet function. I apologize for any confusions caused by my question statement.

Ultrasound images: An adaptive method that terminates automatically the wavelet packet decomposition at the optimal level by assessing the fractal characteristics of the subbands at each resolution level. The open access paper "Quantification of ultrasonic texture intra-heterogeneity via volumetric stochastic modeling for tissue characterization" can be accessed from the following url: http://dx.doi.org/10.1016/j.media.2014.12.004

Microscopic images: The paper "A multiresolution clinical decision support system based on fractal model design for classification of histological brain tumours" deals with same problem can be accessed from the following link: doi:10.1016/j.compmedimag.2014.05.013

Also the overview of previous work in the 2 papers above can give a good idea on the most well know ways for wavelet decomposition level selection.

Based on the frequency content that you want to detect or process , you write your condition in the algorithm.

If I understand your question correctly:

In time series analysis, using the discrete wavelet transform, the maximum wavelet level is limited by the number of time points, N; for the DWT, J0 (the maximum number of levels), cannot be greater than j, where N=2^j. For redundant discrete transforms, J0 can be greater than j, but the information tends to zero for these higher levels. The exact choice of J0 will depend on the application, e.g. you can define J0 conservatively as J0

Muhammad, 

what data are you talking about? What do you want ot achieve? 

Mr. Hofmann it is actually a series of thermal images captured during the photo-thermal analysis of artworks. The aim was Non-Destructive detection of faults.

Muhammad,

sometimes it is important to know what mother wavelet function is considered, because this, for instance, may affect the number of levels of decomposition. When you are considering images, and depending on the application, it is enough to consider four or five wavelet levels of decomposition, where an energy criterion will help us. 

Hello

I can suggest you to view this publications

https://www.researchgate.net/publication/235933497_Noise_cancellation_in_Doppler_moment_estimation_of_array_weather_radar_signals_using_lifted_biorthogonal_wavelets

https://www.researchgate.net/publication/233966030_De-noised_estimation_of_theweather_Doppler_spectrum_by_the_wavelet_method

Cordially

Article Noise cancellation in Doppler moment estimation of array wea...

Article De-noised estimation of theweather Doppler spectrum by the w...

Selection of decomposition levels in DWT is depended on Sample frequency and analysis process of the signal. For suppose, to analyze the EEG signal we split the EEG signal into sub-bands such as 

DELTA (0.5 to 4 Hz) 

THETA(4  to 8  Hz), 

APLA( 8 to 12  Hz),

BETA( 12 to 30  Hz),

GAMMA(  >30   Hz)  

For this, we use sample frequency for finding decomposition levels. 

For example, if the sampling frequency is 256Hz then the decomposition levels are six. At Decomposition level CA6 (approximate coefficient) we get delta and at CD6, CD5, CD4, CD3 (Detailed coefficients) levels we Theta, Alpha, Beta and Gamma respectively. The remaining decomposition considers as noise that is at CD2 and CD1.

Matlab code:

Fs=256;

l=ceil(log2(Fs/4))

Hi Venkata, 

I think something is wrong with your answer. For a signal with sampling frequency of 256 Hz, the frequency band corresponding to each wavelet coefficient is as follows:

CA1: 0 - 64 Hz        |         CA2: 0 - 32 Hz         |      CA3: 0 - 16 Hz

CD1: 64-128 Hz     |         CD2: 32 - 64 Hz       |      CD3: 16 - 32 Hz

----------------------------------------------------------------------------------------------

CA4: 0 - 8 Hz          |         CA5: 0 - 4 Hz            |       CA6: 0 - 2 Hz

CD4: 8 - 16 Hz       |          CD5: 4 - 8 Hz           |       CD6: 2 - 4 Hz

Based on the above frequency bands, at CD6 (2 - 4 Hz) we do not get Theta ( 4- 8 HZ). Also, CD5 (4 - 8 Hz) is not equivalent to Alpha (8 -12 Hz) and so on.

Moreover, the level of noise can be detected by performing PSD (Power Spectral Density) analysis for the stationary signal. To me, we cannot say that the decomposition levels 1 and 2 of a signal with sampling frequency of 256 Hz can surely be considered as the noise. For noise detection, PSD analysis is essential. Also, we need to first make sure that the signal is stationary, as applying FFT and PSD to a non-stationary signal may result in wrong detection of the frequency components of that signal.

Hi ALi Maddahi.

Thanks for viewing my answer and replying it. I am a Computer Science student. My task extracts the features and classifies the signal. For feature extraction from EEG through DWT, we mining the sub-bands (such delta, theta, alpha, beta, and gamma). For this, I studied some papers on EEG. According to that, I decomposing the EEG signal based on the sampling frequency.

For sampling frequency 256, CA1 start at 0-128 and CD1 start at 128-256 (ie., in the first level).

CA1: 0-128      |  CA2: 0 - 64  |CA3: 0 - 32  | CA4: 0 - 16

CD1: 128-256   |  CD2: 64-128 |CD3: 32 - 64  |      CD4: 16 - 32

----------------------------------------------------------------------------------------------

CA5: 0 - 8 Hz          |         CA6: 0 - 4 Hz  

CD5: 8 - 16 Hz       |          CD6: 4 - 8 Hz  

but according to your way  CA1 starting at 0-64 and CD1 starting at 64-128 (ie., at lelve-2).

CA1: 0 - 64 Hz        |         CA2: 0 - 32 Hz         |      CA3: 0 - 16 Hz

CD1: 64-128 Hz     |         CD2: 32 - 64 Hz       |      CD3: 16 - 32 Hz

----------------------------------------------------------------------------------------------

CA4: 0 - 8 Hz          |         CA5: 0 - 4 Hz            |       CA6: 0 - 2 Hz

CD4: 8 - 16 Hz       |          CD5: 4 - 8 Hz           |       CD6: 2 - 4 Hz

Please provide the reason to start at  second level. For getting required things for classification we extract the sub-bands. I am considering sample frequency

Is there any ways to find the number of levels for decomposing the signal. 

Hi Venkata,

Please refer to the following references:

1) Amin Y. Goharrizi, Nariman Sepehri, "Application of Fast Fourier and Wavelet Transforms towards Actuator Leakages Diognosis: A Comparative Study", Int. J. of Fluid Power, ... .

2) R.J.E. Merry, "Iterative Learning Control with Wavelet Filtering", Mater's Thesis, Eindhoven University of Technology, 2005.

For your convenience, take a look at pages 13-17 of second reference (Mater's Thesis). You will also find a formula for the decomposition level. 

The first level of decomposition contains frequency range of 0 - fs/4 Hz for approximation coefficient (CA1), and fs/4 - fs/2 Hz for detailed coefficient (CD1), not that, CA1: 0 - fs/2 and CD1: fs/2 - fs. It relates to the concept of downsampling by a factor 2. As you are working with Wavelet Transform, I think the above references and the first three chapters of the following thesis can be helpful to your research:

3) M.G.E. Schneiders, "Wavelets in Control Engineering", Mater's Thesis, Eindhoven University of Technology, 2001.

Good Luck

Hello!

I have one question about this discussion.

Suppose that after decomposition I have located specific bands of frequencies that interest me and that I wish to reconstruct the signal using only those specific bands. How to do this?

It is clearer: I intend to decompose a signal s (t) into two others: one containing only a certain range of low frequencies and another containing only a certain low frequency range. I want to separate this into the reconstructed signal. How can I do this using dwt?

Thank you.

@Luciano Aparecido Magrini:

I suppose that you are using matlab. Instead of using the function dwt, I would suggest using these two functions, wavedec (for decomposition) and waverec (for reconstruction). The waverec allows for selective reconstruction. 

The idea is to reconstruct the specific levels by setting all other coefficients to 0

I can suggest you to view this publications

https://www.degruyter.com/view/j/bmte.ahead-of-print/bmt-2018-0001/bmt-2018-0001.xml