Autonomous consumption
Coiflets are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman, to have scaling functions with vanishing moments. The wavelet is near symmetric, their wavelet functions have vanishing moments and scaling functions , and has been used in many applications using Calderón-Zygmund Operators.[1][2]
Coiflet coefficients
Both the scaling function (low-pass filter) and the wavelet function (High-Pass Filter) must be normalised by a factor . Below are the coefficients for the scaling functions for C6-30. The wavelet coefficients are derived by reversing the order of the scaling function coefficients and then reversing the sign of every second one (i.e. C6 wavelet = {−0.022140543057, 0.102859456942, 0.544281086116, −1.205718913884, 0.477859456942, 0.102859456942}).
Mathematically, this looks like where k is the coefficient index, B is a wavelet coefficient and C a scaling function coefficient. N is the wavelet index, i.e. 6 for C6.
| k | C6 | C12 | C18 | C24 | C30 |
|---|---|---|---|---|---|
| -10 | -0.0002999290456692 | ||||
| -9 | 0.0005071055047161 | ||||
| -8 | 0.0012619224228619 | 0.0030805734519904 | |||
| -7 | -0.0023044502875399 | -0.0058821563280714 | |||
| -6 | -0.0053648373418441 | -0.0103890503269406 | -0.0143282246988201 | ||
| -5 | 0.0110062534156628 | 0.0227249229665297 | 0.0331043666129858 | ||
| -4 | 0.0231751934774337 | 0.0331671209583407 | 0.0377344771391261 | 0.0398380343959686 | |
| -3 | -0.0586402759669371 | -0.0930155289574539 | -0.1149284838038540 | -0.1299967565094460 | |
| -2 | -0.1028594569415370 | -0.0952791806220162 | -0.0864415271204239 | -0.0793053059248983 | -0.0736051069489375 |
| -1 | 0.4778594569415370 | 0.5460420930695330 | 0.5730066705472950 | 0.5873348100322010 | 0.5961918029174380 |
| 0 | 1.2057189138830700 | 1.1493647877137300 | 1.1225705137406600 | 1.1062529100791000 | 1.0950165427080700 |
| 1 | 0.5442810861169260 | 0.5897343873912380 | 0.6059671435456480 | 0.6143146193357710 | 0.6194005181568410 |
| 2 | -0.1028594569415370 | -0.1081712141834230 | -0.1015402815097780 | -0.0942254750477914 | -0.0877346296564723 |
| 3 | -0.0221405430584631 | -0.0840529609215432 | -0.1163925015231710 | -0.1360762293560410 | -0.1492888402656790 |
| 4 | 0.0334888203265590 | 0.0488681886423339 | 0.0556272739169390 | 0.0583893855505615 | |
| 5 | 0.0079357672259240 | 0.0224584819240757 | 0.0354716628454062 | 0.0462091445541337 | |
| 6 | -0.0025784067122813 | -0.0127392020220977 | -0.0215126323101745 | -0.0279425853727641 | |
| 7 | -0.0010190107982153 | -0.0036409178311325 | -0.0080020216899011 | -0.0129534995030117 | |
| 8 | 0.0015804102019152 | 0.0053053298270610 | 0.0095622335982613 | ||
| 9 | 0.0006593303475864 | 0.0017911878553906 | 0.0034387669687710 | ||
| 10 | -0.0001003855491065 | -0.0008330003901883 | -0.0023498958688271 | ||
| 11 | -0.0000489314685106 | -0.0003676592334273 | -0.0009016444801393 | ||
| 12 | 0.0000881604532320 | 0.0004268915950172 | |||
| 13 | 0.0000441656938246 | 0.0001984938227975 | |||
| 14 | -0.0000046098383254 | -0.0000582936877724 | |||
| 15 | -0.0000025243583600 | -0.0000300806359640 | |||
| 16 | 0.0000052336193200 | ||||
| 17 | 0.0000029150058427 | ||||
| 18 | -0.0000002296399300 | ||||
| 19 | -0.0000001358212135 |