Wavelet packet transformation

The wavelet packet transformation is an extension of the fast wavelet transformation (FWT) and, like this, is used in digital signal processing for the analysis and compression of digital signals . In the FWT, a time-discrete input signal with a sampling rate F is split into a low-pass channel L and a band-pass channel H with half the sampling rate F / 2 using a wavelet filter bank (e.g. the Daubechies wavelets ) and this procedure is repeated recursively for the low-pass channel . In the next step, the channels LL and LH with sampling rate F / 4 are created from the channel L , the channels LLL and LLH from the channel LL in the next step and so on.

In the case of the wavelet packet transformation, the bandpass channels are now also split, so that in the second recursion step not only LL and LH , but also the channels HL and HH arise. In the third step, eight sub-channels are created, etc. The sub-channels of the result and the intermediate steps can be arranged in a binary tree .

Package tree with filter g for the L channel and h for the H channel

This transformation can be used to convert a 2-channel DWT such as B. the Daubechies wavelets to get an M -channel DWT, where M is a power of two, the exponent is called the depth of the packet tree. This method is used in broadband data transmission as DWT-OFDM or DWPT-OFDM as an alternative to the fast Fourier transformation in FFT- OFDM .

If the underlying wavelet transformation has a scaling function φ with a low-pass filter (L-channel) and band-pass filter (H-channel), then the wavelets of the channels result in ${\displaystyle a(Z)}$ ${\displaystyle b(Z)}$

${\displaystyle {\begin{aligned}\psi _{\text{L}}(x/2)&:=a(S)\phi (x)=\sum _{n}a_{n}\phi (x-n)=\phi (x/2)\,,\\\psi _{\text{H}}(x/2)&:=b(S)\phi (x)=\sum _{n}b_{n}\phi (x-n)=\psi (x/2)\,,\end{aligned}}}$

where the operator is the shift by 1 in the direction of increasing values, i.e. H. . Powers of are then shifts by the exponent of the power, Laurent polynomials in correspond to the respective linear combinations of the shifted functions. ${\displaystyle S}$ ${\displaystyle x}$ ${\displaystyle (Sf)(x)=f(x-1)}$ ${\displaystyle S}$ ${\displaystyle S}$

Up to here the functions and are identical to those occurring in the FWT. In the second step, new functions arise ${\displaystyle \phi }$ ${\displaystyle \psi }$

${\displaystyle {\begin{aligned}\psi _{\text{LL}}(x/4)&:=a(S^{2})a(S)\phi (x)=\phi (x/4),\\\psi _{\text{LH}}(x/4)&:=b(S^{2})a(S)\phi (x)=\psi (x/4),\\\psi _{\text{HL}}(x/4)&:=a(S^{2})b(S)\phi (x)\,,\\\psi _{\text{HH}}(x/4)&:=b(S^{2})b(S)\phi (x)\,.\end{aligned}}}$

Is the spectrum of near-optimal to baseband limited and and good frequency-selective digital filter for which one-periodically repeating intervals or , as the spectrum is of to be concentrated that of on which of on , d. H. the frequency bands of the channels are arranged in , each with width 1/2, in the order LL, LH, HH, HL. ${\displaystyle \phi (x)=\psi _{\text{LL}}(x)}$ ${\displaystyle \left[0,1/2\right]}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \left[-1/4,1/4\right]}$ ${\displaystyle \left[1/4,3/4\right]}$ ${\displaystyle \psi (x)=\psi _{\text{LH}}(x)}$ ${\displaystyle \left[1/2,1\right]}$ ${\displaystyle \psi _{\text{LH}}(x)}$ ${\displaystyle (\left[-1/2,1/2\right]\cup \left[3/2,5/2\right])\cap \left[1,3\right]\cap \left[0,2\right]=\left[3/2,2\right]}$ ${\displaystyle \psi _{\text{HH}}(x)}$ ${\displaystyle (\left[1/2,3/2\right]\cup \left[5/2,7/2\right])\cap \left[1,3\right]\cap \left[0,2\right]=\left[1,3/2\right]}$ ${\displaystyle \left[0,2\right]}$

Then in the third step

${\displaystyle {\begin{aligned}\psi _{\text{LLL}}(x/8)&:=a(S^{4})a(S^{2})a(S)\phi (x)=\phi (x/8),\\\psi _{\text{LLH}}(x/8)&:=b(S^{4})a(S^{2})a(S)\phi (x)=\psi (x/8)\,,\\\psi _{\text{LHL}}(x/8)&:=a(S^{4})b(S^{2})a(S)\phi (x)\,,\\\psi _{\text{LHH}}(x/8)&:=b(S^{4})b(S^{2})a(S)\phi (x)\,,\\\psi _{\text{HLL}}(x/8)&:=a(S^{4})a(S^{2})b(S)\phi (x)\,,\\\psi _{\text{HLH}}(x/8)&:=b(S^{4})a(S^{2})b(S)\phi (x)\,,\\\psi _{\text{HHL}}(x/8)&:=a(S^{4})b(S^{2})b(S)\phi (x)\,,\\\psi _{\text{HHH}}(x/8)&:=b(S^{4})b(S^{2})b(S)\phi (x)\,.\end{aligned}}}$

etc.

In the following graphic, the wavelets of the third stage, which result from the Daubechies-12-tap wavelet D12, are shifted in whole numbers for the sake of clarity. In addition, the amplitudes of the Fourier transforms of the individual wavelets. From the spectra in the amplitude range above 0.7 one can read the division of the frequency band into the eight sub-channels of width 1/2 with the order LLL, HLL, HHL, LHL, LHH, HHH, HLH, LLH. This corresponds to a variant of a Gray code . ${\displaystyle \left[0,4\right]}$

Web links

• Wavelet Packet Modulation for Wireless Communications (PDF, English; 677 kB)
• Implementation of the wavelet packet transformation in Java (English; software project)