GENwave
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## GENwave

"wave" — Generates a compactly supported wavelet function.

## Description

Creates a compactly supported wavelet, scaling function or wavelet packet. The output function is obtained by deconvolution of corresponding mirror filter impulse response. This procedure is applied in an iterative fashion.

The filter bank used in classic discrete wavelet transform grows only towards low frequencies. Meanwhile wavelet packet transform introduces any possible direction of tree's growth. The sequence of mirror filters used in deconvolution is determined by binary form of seq value. "0" corresponds to low-pass filter and "1" to high-pass.

The number of iteration steps is determined by filter's length and size of function table. I.e. for filter length 8 and table size 256 there are log2(256/8) = 5 iterations available.

## Syntax

    f # time size "wave" fnsf seq rescale

## Initialization

size -- number of points in the table. Must be a power of 2 or power-of-2 plus 1 (see f statement).

fnsf -- pre-existing table with scaling function coefficients.

seq -- non-negative integer number which corresponds to sequence of low-pass and high-pass mirror filters during deconvolution procedure.

rescale -- if not zero the table is not rescaled

## Examples

Here is an example of the GENwave routine. It uses the file genwave.csd.

Example 1223. Example of the GENwave routine.

See the sections Real-time Audio and Command Line Flags for more information on using command line flags.

    <CsoundSynthesizer>
    <CsOptions>
    ; Select audio/midi flags here according to platform
    ; Audio out   Audio in    No messages
    -odac           -iadc     -d     ;;;RT audio I/O
    ; For Non-realtime ouput leave only the line below:
    ; -o 0dbfs.wav -W ;;; for file output any platform
    </CsOptions>
    <CsInstruments>

    sr = 44100
    kr = 4410
    nchnls = 1
    0dbfs = 1

    zakinit 3,1

    instr 1 ; wavelet synth instrument
    iamp  =	 p4; scaling factor of wavelets
    ifreq =	 p5; frequency of wavelets
    itab  =	 p6; selected wavelet function
    inum  =  p7; number of wavelets to be created
    a1 osciln p4, p5, p6, p7
    out a1
    endin

    instr 2 ; wavelet analysis intrument
    a1 soundin "fox.wav"
    ; Decomposition Structure:
    ;     1 LEVEL  2 LEVEL
    ;     HP->ah1
    ; a1->|	       HP(up2)->ah2
    ;     LP->al1->|
    ;	       LP(up2)->al2
    ;
    ain = a1*.5; attenuate input signal
    	   ; since wavelet coefficients
    	   ; could reach big values
    ah1 dconv ain,ftlen(8),8
    al1 dconv ain,ftlen(7),7
    ah2 dconv al1,ftlen(10),10
    al2 dconv al1,ftlen(9),9

    zaw ah1,0
    zaw al1,1
    zaw ah2,2
    zaw al2,3

    aout zar p4
    out aout
    zacl 0,3
    endin

    </CsInstruments>
    <CsScore>

    ; First of all, we need several FIR filters which are capable
    ; to produce wavelet families.
    ; One can input filter coefficients manualy using GEN02
    ; or read them from text file.
    ; Most of compact-supported wavelet coefficients can be obtained from
    ; Wavelet Browser by PyWavelets wavelets.pybytes.com
    ; You can select family and order of filter
    ; then copy desired coefficients into txt file.
    ; Notice that for correct interpretation of results you should use
    ; coeffs of Decomposition low-pass filter.

    ; Daubechies 2
    f 1 0 4 -2 -0.1294095226 0.2241438680 0.8365163037 0.4829629131
    ; Symlet 10
    f 2 0 0 -23 "sym10.txt"

    ; Now we want to produce some wavelet granules.
    ; They can be used in wavelet synthesis etc.
    ; Tables of large sizes should produce smoother wavelets.
    ; We take array of filter coefficients from ftable 1
    ; and deconvolve it until output length of 16384.
    ; The order of filters through the deconvolution process
    ; is given by 14 which is 1110 in binary.
    ; So the first filter is LP ('0') and others are HP ('1').
    f 3 0 16384 "wave" 1 14 0
    f 4 0 16384 "wave" 2 1  0
    f 5 0 16384 "wave" 2 7  0
    f 6 0 16384 "wave" 2 6  0
    ; The main purpose of using wavelets is wavelet transform.
    ; It is not that easy to perform a classic DWT in Csound since downsampling
    ; of audio signal is needed at each step of wavelet decomposition.
    ; Anyway, using GENwave it is possible to create a number of upsampled
    ; wavelets and perform a so-called undecimated wavelet transform
    ; aka stationary wavelet transform (and it is even better).
    ; So we need some upsampled childs of mother wavelet.
    f 7 0 16 "wave" 1 0 -1 ;db2 scaling function for 1st iteration
    f 8 0 16 "wave" 1 1 -1 ;db2 wavelet function for 1st iteration
    f 9 0 32 "wave" 1 0 -1 ;db2 scaling function for 2nd iteration
    f 10 0 32 "wave" 1 1 -1 ;db2 wavelet function for 2nd iteration

    ; Let's hear how some wavelets could sound..
    ;		 amp frq wave	 times
    i 1 0	 1	 0.6 15	 3	 8
    i 1 0.5	 .	 0.9 20  4	 5
    i 1 0.9  .	 0.7 8	 5	 .
    i 1 1.1	 .	 0.4 30  6	 9

    ; Now try to decompose input file using wavelets
    i 2 2 4  1; approximation 1st level
    i 2 5 .  2; details 2nd level
    </CsScore>
    </CsoundSynthesizer>

## See Also

Ingrid Daubechies. Ten Lectures on Wavelets, SIAM 1992.

Pedro A.G. Batista. An Introduction to Sound Synthesis with Wavelet Packets, Csound Book CD Chapters, MIT 2000.

Victor M. Wickerhauser. Acoustic Signal Compression with Wavelet Packets, Yale 1992.

## Credits

Written by Gleb Rogozinsky

Saint-Petersburg University of Film and Television

Saint-Petersburg, Russia

April 2012

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