# wavelet packet transform and lifting scheme?

so the lifting scheme is basically an alternative to performing the discrete wavelet transform with several advantages.

But here are three questions which I did not find an answer to:

1. is it possible to use lifting also for calculation of the wavelet packet transform or does it just apply to the normal wavelet transform?
2. is the wavelet packet transform also critically sampled like the dwt or is it redundant?
3. if using a level basis in the packet transform, i.e. a representation which uses only elements from the same decomposition level, the time-frequency plot is regularly discretized. How does this representation then differ, if so, from a STFT which also exhibits constant Heisenberg boxes in the time-frequency plane.

Thank you for helping in advance

A lifting scheme is a method for splitting a sequence of discrete samples into downsampled subsequences, so that you can predict a subsequence from the other, and update the former later, using possibly nonlinear predict and update operators. This corresponds to a generalization of one level of discrete wavelet transform, usually obtained fully linearly by convolution (first-generation wavelet).

Lifting wavelets were called second-generation wavelets, and proved to be a generalization of first-generation wavelets.

This being said, for question 1:

• yes, a wavelet packet being iterations of several one level DWT on other subbands.

For question 2:

• A lifting discrete wavelet packet can be critically sampled, like the DWT, but also can be undecimated or oversampled if needed. And they are interesting questions about what kind of stationary lifted wavelets can be built.

For question 3:

Generally, the Heisenberg boxes are considered to be of constant area. With the STFT, titles have the same shape, whereas with wavelet they are dilated/stretched in each direction. With a uniform wavelet packet decomposition (where the time-scale plot is regularly discretized), the time-scale bins can look similar to that of the STFT. If the packet transform is critical, the result is often poorer with wavelets packets (because of the lack of redundnacy).

At the same redundancy, aliasing and constraints on wavelet shapes often yield less degrees of freedom that full-fledged oversampled filter banks, but the filter are generally easier to optimize. Nevertheless, if you want uniform frequency bin, wavelets are rarely very beneficial.

• thank you, I appreciate your detailed answer. I am just now delving into the field of Wavelet theory and was wondering, if I could ask you some questions occasionally since you seem to be an expert in these matters
– CD86
Jun 8, 2020 at 7:02