Some questions about the intuition of the DWT

Question

Assuming a DWT of a signal of length 8 with Haar filter taps. At the lowest level, I end up with a3 and d3 both of length 1, d2 of length 2 and d1 of length 4 which is the same number of coefficients of the original signal and which I can plot on a dyadic grid.

In contrast to the WPT in the DWT there is only basis representation for a given decomposition level ([a3,d3,d2,d1] for the 3 level decomposition in the example above). From this sequence one can choose to recover the original signal or zero out some coefficients and then reconstruct for denoising purposes and the likes.

The dyadic grid representation is analogous to the CWT and so displays signal energy over a non-uniformly discretized 2-dimensional grid of time and scale/frequency.

I often see in books a representation in which the coefficients are just concatenated and plotted on a 1-dimensional frequency axis just next to the original signal axis.

I find this representation quite confusing since it suggests to directly compare both graphs although they are displayed over time (figure a) and frequency (figure b) respectively.

Now my questions:

1. What is the intuition behind the stacked frequency plot of the coefficients, i.e. what is the best way to read it? Going from right to left there is a decreasing time-resolution but increasing freq resolution. In reference to the signal above, I would have the 4 d1 coefficients on the most right followed by the 2 d2 coefficients and so on. Is the correct way of interpreting this to say, that in the spectrum of [pi/2T, pi/T] I have frequency information of the original signal at just 4 different points in time (due to downsampling with a factor of 2) for [pi/4T, pi/2T] one has freq information at only 2 points in time but being narrowed down to half the frequency interval than before and so on?

2. Does this mean, that the signal representation I described above has now been transformed from a time-series (the original signal) to a pseudo tf-representation and only becomes a pure time-signal again by sending it again through a synthesis filter bank? Can both representations still be considered equivalent? If one would not have the axis description, how would it be possible to recognize, if one is represented the original signal or a decomposition?

3. As for a WPT there are now a large number of possible signal representations and one has to keep track of the chosen one, if a later reconstruction is desired. Limiting ourselves to a level basis of level 3 in the above example, what is now the meaning of the approx and detail coefficients in every freq subband? For the DWT one can argue, that d3 are the signal details at the coarsest scale that are necessary to go from a3 to a2 and so forth. So the details capture the missed-out fine-grained signal details that were neglected along the decomposition. Is there a similar meaning to a WPT decomposition? I have a hard time transferring this intuition to the more general signal deconstruction of a WPT

4. In all these transforms, usually boundary problems arise whenever using a filter that has a larger number of taps than 2. So decomposing a length 8 signal with dB2 filters already yields more than 4 values both for a1 and d1 at the first decomposition level. If I were to plot the signal at this level, would I have to cut off the additionally introduced values somehow or would the signal at this level just naturally comprise then more values than the original time signal?

Thanks a lot for any help to deepen my understanding of this complex matter

Answer 1

First, one should be cautious about processing short signals like these. I am unsure about the length of 3 for a3 and d3. Now, the many questions:

1. I would not call them "frequency plot", but stacked subband plots. They are just illustrations of the behavior of the wavelet coefficients. For each subband, different scalings may be used. Another rendering is stacking the intensity of coefficients in a image. For the interpretation of the points: the coefficients in each subband at located in time and scale, so the interpretation in $[\pi/2T, \pi/T]$, $[\pi/4T, \pi/2T]$ is only approximative (because of aliasing, subsampling, etc.).
2. No, one is concatenating chunks of coefficients in different subbands and at different rates
3. Yes, subbands of a given WPT can generally be interpreted in dyadic portions of the time-scale axis
4. Haar wavelets (2-tap filters) have the rare property (among standard wavelet filters) they don't overlap at a given scale. For longer filters with 2-band wavelets, those filters overlap, and yield troubles at the extremities of a finite-length signal. There is a lot of works on non-expensive expansions, relaying on the symmetry/antisymmetry of the wavelet.