I'm looking for a way to get a wavelet-like resolution in the frequency domain. So I thought of applying the Fourier transform on the results of a wavelet transform.
Say we have a time-domain signal with a length of 8.
From the viewpoint of tiling of the time and frequency axis, this signal would have vertical lines creating 8 bins, meaning that there is no frequency resolution in any of the coefficients. If we take the Fourier transform, then the vertical lines would be transposed into horizontal lines, having no time resolution but a frequency resolution of 8 bins.
We now perform a wavelet transform with a 3-level decomposition on the time-domain signal, so that the coefficients are [4,2,1,1], where the level with 4 coefficients would have a resolution of 2 temporal bins and 4 spectral bins each, level with 2 coefficients would have a resolution of 4 temporal bins and 2 spectral bins each, and so on. (Similar to half of the image on the right)
If we take the Fourier transform of these individual levels, is the result mathematically meaningful?
Edit: Modulated complex lapped transform (which I just found out) has the tiling that I expected my question to have.