WPD = Wavelet Packet Decomposition
Hello, dear Stack Overflow. I have questions for my time-series data. My data is a vibration of bearing in a machine or machine tool.
We know that WPD works as a filter and is divided into 8 frequency bands:
(ex - sampling rate = 4000Hz
1. 0 ~ 500Hz
2. 500 ~ 1000Hz
3. 1000 ~ 1500Hz
4. 1500 ~ 2000Hz
5. 2000 ~ 2500Hz
6. 2500 ~ 3000Hz
7. 3000 ~ 3500Hz
8. 3500 ~ 4000Hz );
However, if reconstruction is applied to this 8 frequency band, the only 0~2000Hz can be applied by the Nyquist theorem (according to Nyquist theorem only less than half frequency is valid).
Since 2000Hz above frequency becomes meaningless, then we only use 1, 2, 3, 4 frequency band to reconstruction. Is that right?
I have 2 questions that are:
first is it possible to use only 0 ~ 2000Hz which is half of the sampling frequency when reconstruction is performed after applying WPD
when I reconstruction process after WPD,
second Is it reasonable to use FFT after applying WPD??
1. I think that there is a confusion about the frequency bands returned by the WPD. If your vibration time series are sampled at 4,000Hz, Nyquist theorem tells you that the highest frequency contained in your signal is 2,000Hz, so you highest frequency subband should go from 1,750Hz to 2,000Hz, not from 3,500 to 2,000Hz!
Please check out this Mathworks page to get a better understanding of how the subbands are divided when using a WPD.
2. Yes, you can apply a FFT to your reconstructed data. I guess that if you intend to low-pass filter your time series, you plan to plot the spectra of both the original and reconstructed data; is that why you want to compute the FFT?
Last but not least, if your goal is simply to low-pass filter your data, you might want to consider other, simpler tools than the WPD. As explained in the aforementioned Mathworks page, the WPD allows for a better separation than conventional DWT, but depending on the frequencies that you trying to filter out, it might be overkill.