Wavelet packet decomposition
The wavelet packet transformation is an extension of the fast wavelet transform ( FWT ), and is used like this in the digital signal processing and compression of the analysis of digital signals. In the FWT a discrete-time input signal with a sampling rate F by means of a wavelet filter bank (for example, the Daubechies wavelets ) is split into a low-pass channel L and a band-pass duct H with a half sampling rate F / 2, and repeats this procedure for the low pass channel recursively. This results in the following step out of the channel L is the LL and LH channels with sampling rate F / 4, from the channel LL in the next step the channels LLL and LLH and so on.
In the wavelet packet transform and the band-pass channels now be split so that the second recursion step arise not only LL, and LH, but also the channels HL and HH. In the third step arise as eight sub- channels, etc. The sub-channels of the result and the intermediate steps can be arranged in a binary tree.
This transformation can serve as example to get from a 2-channel DWT the Daubechies wavelets an M -channel DWT, where M is a power of two, the exponent is called the depth of the package tree. This method is used in the broadband data transmission as DWT OFDM or OFDM DWPT as an alternative to the fast Fourier transformation in the FFT OFDM.
Has the underlying wavelet transform is a scaling function φ with low-pass filter a ( Z) (L- channel) and band-pass filter b ( Z ) (H - channel), then the wavelets of the channels found to
Where S is the displacement (shift ) around 1 in the direction of increasing x values operator, i.e., (SF) (x) = f (x -1). Powers of S are then shifts to the exponent of the power of Laurent polynomials in S correspond to the respective linear combinations of the shifted functions.
Up to here, the functions φ and ψ are the same as those occurring in the FWT. In the second step, new features arise
Is the spectrum of φ (x ) = ψLL (x ) limited almost perfectly to the baseband [0,1 / 2] and a and b are good frequency selective digital filters for which 1- periodically repeating intervals [ -1 / 4,1 / 4 ] or [1/ 4, 3/ 4], as will be concentrated the spectrum of ψ (x ) = ψLH (x ) on [1/ 2, 1], that of ψLH (x ) to ([ - 1/2, 1/2] ∪ [3/ 2, 5 /2] ) ∩ [1,3] ∩ [0,2] = [ 3/2, 2] of the ψHH (x) to ([ 1 / 2,3 / 2] ∪ [ 5/2, 7/ 2] ) ∩ [1,3] ∩ [0,2] = [1,3 / 2], i.e. the frequency bands of the channels are in [ 0,2], each arranged with width of 1/2, in order LL, LH, HH, HL.
In the third step,
Etc.
The following graphic shows the third stage wavelets were shown which differs Daubechies 12 -tap wavelet D12 result from the shifted integer for clarity. In addition, the amplitudes of the Fourier transforms of the individual wavelets. One can read the distribution of the frequency band [ 0.4 ] into the eight sub-channels of width 1 /2, the order LLL, HLL, HHL, LHL, LHH, HHH, HLH, LLH of the amplitude spectra in the range above 0.7. This corresponds to a variation of a gray code.