PSD of DWT details coefficnet

Hi,

I am working on a project involves sensor data analysis. I have a data for entire year that is sampled every 15 minutes. I used DWT (db4 up to three levels) to analyze the high-frequency content of the main signal.

After applying PSD on the reconstructed details signal, I can see some patterns between the levels, which I don't have a good explanations for them. According to the attached figure, level 3 has two cycles, level 2 one cycle, and level one half cycle. Why is that?

Also, as I remember DWT isolate frequency bands per each level, while there are some overlap in frequencies here. Is it due to the filter nature?

I really appreciate if you can give me your thoughts or suggestions.

Thank you in advance.

Answer 1

[Abhimenyu]

Hello,

I understand that you are trying to use "Discrete Wavelet Transform (DWT)" to analyze the high-frequency content of your main signal. Your observation of different cycles in the "Power Spectral Density (PSD)" of reconstructed detail signals from different levels of DWT decomposition is related to the nature of wavelet decomposition and the characteristics of the signal that is analyzed.

When "DWT" is applied to a signal, especially with a wavelet like 'db4' (Daubechies wavelet with 4 vanishing moments), the signal is passed through a series of high-pass and low-pass filters. Each level of decomposition splits the signal into two parts:

1. Approximation (Low-Frequency Component): This is the output of the low-pass filter, representing the trend or the slow-changing components of the signal.
2. Detail (High-Frequency Component): This is the output of the high-pass filter, capturing the fast-changing components or the "details" of the signal.

As the levels move deeper from 1 to 3, the frequency band covered by each subsequent detail signal is halved, effectively zooming into lower frequency bands for the approximation signals. The cycles observed in the "PSD" plots of the detail components at different levels can be explained by the frequency content isolated by each level. Level 3, being a higher level of decomposition, isolates lower frequency components compared to levels 2 and 1. Thus, more pronounced cycles (or slower cycles) are seen in level 3 compared to level 1, which isolates higher frequency components and shows faster cycles.

While "DWT" aims to decompose the signal into distinct frequency bands, the exact cut-off frequencies are influenced by the characteristics of the wavelet (like 'db4') and the decomposition level. The filters used in "DWT" are not ideal. They have transition bands and do not have a sharp cutoff. This results in some overlap between the frequency bands isolated by adjacent levels of decomposition. The extent of overlap depends on the wavelet's properties and the filter design inherent to that wavelet.

Please refer below to some possible reasons that might influence the cycles and frequency overlaps in "DWT" composition:

• Signal Characteristics: The patterns might also be influenced by the inherent characteristics of your sensor data as natural or human-made phenomena can introduce periodicities.
• Wavelet Choice: Different wavelets might provide different insights into the frequency content of the signal. Experimenting with other wavelets could help to understand if the observed patterns are consistent or wavelet-dependent.
• Analysis Approach: Please consider analyzing the approximation components alongside the detail components. Sometimes, the trends captured in approximation components can provide context for the cycles observed in the detail components.

For more information on "Discrete Wavelet Transform (DWT)", please refer to this MATLAB R2024A documentation link: https://www.mathworks.com/help/wavelet/gs/continuous-and-discrete-wavelet-transforms.html

I hope this helps!