Editing Daubechies wavelet (section)

== The scaling sequences of lowest approximation order ==

Below are the coefficients for the scaling functions for D2-20. The wavelet coefficients are derived by reversing the order of the [[Wavelet#Scaling function|scaling function]] coefficients and then reversing the sign of every second one, (i.e., D4 wavelet <math>\approx</math> {−0.1830127, −0.3169873, 1.1830127, −0.6830127}). Mathematically, this looks like <math>b_k = (-1)^k a_{N-1-k} </math> where ''k'' is the coefficient index, ''b'' is a coefficient of the wavelet sequence and ''a'' a coefficient of the scaling sequence. ''N'' is the wavelet index, i.e., 2 for D2.

<div style="font-size:83%">
{| class="wikitable"
|+'''Orthogonal Daubechies coefficients (normalized to have sum 2)'''
!D2 ([[Haar wavelet|Haar]])
!D4
!D6
!D8
!D10
!D12
!D14
!D16
!D18
!D20
|----
|1
|0.6830127
|0.47046721
|0.32580343
|0.22641898
|0.15774243
|0.11009943
|0.07695562
|0.05385035
|0.03771716
|----
|1
|1.1830127
|1.14111692
|1.01094572
|0.85394354
|0.69950381
|0.56079128
|0.44246725
|0.34483430
|0.26612218
|----
|
|0.3169873
|0.650365
|0.89220014
|1.02432694
|1.06226376
|1.03114849
|0.95548615
|0.85534906
|0.74557507
|----
|
| −0.1830127
| −0.19093442
| −0.03957503
|0.19576696
|0.44583132
|0.66437248
|0.82781653
|0.92954571
|0.97362811
|----
|
|
| −0.12083221
| −0.26450717
| −0.34265671
| −0.31998660
| −0.20351382
| −0.02238574
|0.18836955
|0.39763774
|----
|
|
|0.0498175
|0.0436163
| −0.04560113
| −0.18351806
| −0.31683501
| −0.40165863
| −0.41475176
| −0.35333620
|----
|
|
|
|0.0465036
|0.10970265
|0.13788809
|0.1008467
|6.68194092&nbsp;×&nbsp;10<sup>−4</sup>
| −0.13695355
| −0.27710988
|----
|
|
|
| −0.01498699
| −0.00882680
|0.03892321
|0.11400345
|0.18207636
|0.21006834
|0.18012745
|----
|
|
|
|
| −0.01779187
| −0.04466375
| −0.05378245
| −0.02456390
|0.043452675
|0.13160299
|----
|
|
|
|
|4.71742793&nbsp;×&nbsp;10<sup>−3</sup>
|7.83251152&nbsp;×&nbsp;10<sup>−4</sup>
| −0.02343994
| −0.06235021
| −0.09564726
| −0.10096657
|----
|
|
|
|
|
|6.75606236&nbsp;×&nbsp;10<sup>−3</sup>
|0.01774979
|0.01977216
|3.54892813&nbsp;×&nbsp;10<sup>−4</sup> 
| −0.04165925
|----
|
|
|
|
|
| −1.52353381&nbsp;×&nbsp;10<sup>−3</sup>
|6.07514995&nbsp;×&nbsp;10<sup>−4</sup>
|0.01236884
|0.03162417
|0.04696981
|----
|
|
|
|
|
|
| −2.54790472&nbsp;×&nbsp;10<sup>−3</sup>
| −6.88771926&nbsp;×&nbsp;10<sup>−3</sup>
| −6.67962023&nbsp;×&nbsp;10<sup>−3</sup>
|5.10043697&nbsp;×&nbsp;10<sup>−3</sup>
|----
|
|
|
|
|
|
| 5.00226853&nbsp;×&nbsp;10<sup>−4</sup>
| −5.54004549&nbsp;×&nbsp;10<sup>−4</sup>
| −6.05496058&nbsp;×&nbsp;10<sup>−3</sup>
| −0.01517900
|----
|
|
|
|
|
|
|
|9.55229711&nbsp;×&nbsp;10<sup>−4</sup>
|2.61296728&nbsp;×&nbsp;10<sup>−3</sup>
|1.97332536&nbsp;×&nbsp;10<sup>−3</sup>
|----
|
|
|
|
|
|
|
| −1.66137261&nbsp;×&nbsp;10<sup>−4</sup>
|3.25814671&nbsp;×&nbsp;10<sup>−4</sup>
|2.81768659&nbsp;×&nbsp;10<sup>−3</sup>
|----
|
|
|
|
|
|
|
|
| −3.56329759&nbsp;×&nbsp;10<sup>−4</sup>
| −9.69947840&nbsp;×&nbsp;10<sup>−4</sup>
|----
|
|
|
|
|
|
|
|
| 5.5645514&nbsp;×&nbsp;10<sup>−5</sup>
| −1.64709006&nbsp;×&nbsp;10<sup>−4</sup>
|----
|
|
|
|
|
|
|
|
|
|1.32354367&nbsp;×&nbsp;10<sup>−4</sup>
|----
|
|
|
|
|
|
|
|
|
| −1.875841&nbsp;×&nbsp;10<sup>−5</sup>
|}
</div>

Parts of the construction are also used to derive the biorthogonal [[Cohen–Daubechies–Feauveau wavelet]]s (CDFs).