As far a I understand WPT is a generalization of DWT in a sense that you get a whole binary tree of possible orthogonal decompositions where DWT is only a one branch. And WPT also downsamples a signal at every level by the factor of 2.

  • Am I correct?

But at the same time there are statements that WPT gives better precision for high frequencies compared to DWT.

  • Why is it so? Is it because it contains possible orthogonal decompositions where we decompose details of the previous level?

It seems that two-scale equations for WPT imply dyadic scales ($\frac{t}{2}$ on the lhs)

$$2^{-\frac{1}{2}}W_{2m}(\frac{t}{2} - k) = \sum^{\infty}_{l=-\infty}h_{l-2k}W_m(t-k)$$ $$2^{-\frac{1}{2}}W_{2m+1}(\frac{t}{2} - k) = \sum^{\infty}_{l=-\infty}g_{l-2k}W_m(t-k)$$

  • So, is it correct to assume that WPT like DWT works on dyadic scales?
  • $\begingroup$ Wavelet packets iterates the high-pass filter bank on selected bands, yielding more frequencies (combinations of low-high) but less spatial information at least if you stick with performing the subsampling on the new high passes performed. The DWT is usually only iterated on the low-pass band up to the level required. $\endgroup$ – mathreadler Feb 22 '17 at 12:53

I suppose you are talking about $2$-channel DWT. WPT may have slightly different interpretations. With DWT, one only allows to decompose frequencies "downward ", ie after lowpass and subsampling. WPT allows to further decompose, at each level, after (lowpass or highpass) and subsampling. In that sense, WPT are a generic scheme, of which the DWT denotes a specific branch. The decision to decompose or stop at each level is governed by rules "external" to wavelets: frequency target, entropy, etc.

Another vision is the uniform WPT, or full wavelet packet, where at each level, both outputs are further decomposed, generating a uniform frequency splitting.

So basically, you are correct at question 1. For question 2, when a DWT subband is approximately located (in reduced frequencies) between two successive dyadic bounds $[2^{-j-1},2^{-j}]$, a WPT can be located between any two dyadic rational bounds $[k_1 2^{-j_1},k_2 2^{-j_2}]$, with $j, j_1,j_2$ nonnegative integers, $k_1 \le 2^{j_1}$ and $k_2 \le 2^{j_2}$.

For instance, the upper DWT subband only (approximately) localizes stuff in $[1/2,1]$, while you can reach $[5/8,3/4]$ with a WPT. However, the aliasing comes into play, and this $[5/8,3/4]$ is disturbed by aliased components from other frequencies. Aliasing is considered more severe for high frequency wavelet packet coefficients.

With $M$-band DWT and WPT, both structures remains $M$-adic at each level, but the WPT tends to split the subband axis onto $M$-adic fractions, with form $k/M^j$.

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