musicdsp.org source code archive - WSInf

Coefficients for Daubechies wavelets 1-38 (click this to go back to the index)
Type : wavelet transform
References : Computed by Kazuo Hatano, Compiled and verified by Olli Niemitalo
Linked file : daub.h (this linked file is included below)
Linked files
/* Coefficients for Daubechies wavelets 1-38
* -----------------------------------------
* 2000.08.10 - Some more info added.
*
* Computed by Kazuo Hatano, Aichi Institute of Technology.
* ftp://phase.etl.go.jp/pub/phase/wavelet/index.html
*
* Compiled and verified by Olli Niemitalo.
*
* Discrete Wavelet Transformation (DWT) breaks a signal down into
* subbands distributed logarithimically in frequency, each sampled
* at a rate that has a natural proportion to the frequencies in that
* band. The traditional fourier transformation has no time domain
* resolution at all, or when done using many short windows on a
* longer data, equal resolution at all frequencies. The distribution
* of samples in the time and frequency domain by DWT is of form:
*
* log f
* |XXXXXXXXXXXXXXXX X = a sample
* |X X X X X X X X f = frequency
* |X X X X t = time
* |X X
* |X
* ----------------t
*
* Single
* subband decomposition and reconstruction:
*
* -> high -> decimate -------------> dilute -> high
* | pass by 2 high subband by 2 pass \
* in | + out
* | / =in
* -> low -> decimate -------------> dilute -> low
* pass by 2 low subband by 2 pass
*
* This creates two subbands from the input signal, both sampled at half
* the original frequency. The filters approximate halfband FIR filters
* and are determined by the choice of wavelet. Using Daubechies wavelets
* (and most others), the data can be reconstructed to the exact original
* even when the halfband filters are not perfect. Note that the amount
* of information (samples) stays the same throughout the operation.
*
* Decimation by 2: ABCDEFGHIJKLMNOPQR -> ACEGIKMOQ
* Dilution by 2: ACEGIKMOQ -> A0C0E0G0I0K0M0O0Q0
*
* To get the logarithmic resolution in frequency, the low subband is
* re-transformed, and again, the low subband from this transformation
* gets the same treatment etc.
*
* Decomposition:
*
* -> high -> decimate --------------------------------> subband0
* | pass by 2
* in | -> high -> decimate ---------------> subband1
* | | pass by 2
* -> low -> decim | -> high -> decim -> subband2
* pass by 2 | | pass by 2
* -> low -> decim |
* pass by 2 | . down to what suffices
* -> . or if periodic data,
* . until short of data
* Reconstruction:
*
* subband0 -----------------------------------> dilute -> high
* by 2 pass \
* subband1 ------------------> dilute -> high + out
* by 2 pass \ / =in
* subband2 -> dilute -> high + dilute -> low
* by 2 pass \ / by 2 pass
* + dilute -> low
* Start . / by 2 pass
* here! . -> dilute -> low