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I understand that approximate and detail coefficient represent the different signal bands. But what do the values mean and how are they used?

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  • $\begingroup$ Answers to your question may fill entire books. Could you please focus a little more, sharing your sources and motivations for understanding the DWT? $\endgroup$ – Laurent Duval Dec 3 '20 at 12:41
  • $\begingroup$ @LaurentDuval I was watching the mathworks video on DWT mathworks.com/videos/understanding-wavelets-part-3-an-example-application-of-the-discrete-wavelet-transform-121284.html. When applying it my own data, I noticed the resulting arrays are of shorter length than the original signal. So I am wondering if the values are narrowbanded signals or represent something else. $\endgroup$ – HVjay Dec 3 '20 at 17:47
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DWTs turn a signal into a list of coefficients, whose combined sizes are equal to or greater than the signal length. Coefficients are grouped in packs called subbands. Each subband gathers coefficients resulting from some band-pass filtering, followed by a subsampling.

They are not exactly narrow-band signals, but projections onto embedded subspaces. The "narrowband" signal for each subband could be recovered by inverse transforming those coefficients only.

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Every number in a Discrete Wavelet Decomposition is an amplitude of a corresponding wavelet. Each of them is localized in time and has its own shape. The original signal is a sum of these wavelets.

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