多贝西小波
性質

分類方式
多貝西小波的分類是以消失動量（vanishing moment）的值A（亦為消失動量的個數）為依據（A稱為tap），調整函式（scaling function）及小波函式（wavelet function）的平滑度（smoothness）皆會隨著消失動量的值（tap）增加而增加：例如，當A=1時，多貝西小波即是哈爾小波（Haar wavelet），調整函式及小波函式都是不連續的；當A=2時，多貝西小波的調整函式及小波函式為不能平滑微分的連續函式；當A=3時，調整函式及小波函式已經是連續可微的函式了。以此類推，當A愈大時，多貝西小波的兩個函式平滑度會愈來愈高。以下為多貝西小波跟不同A的調整及小波函式圖：
| 0 | 1 | 2 | 3 |
| --- | --- | --- | --- |
尺度函数（蓝色）和小波函数（红色）: 
对应函数的频谱幅度: 
長度
多貝西小波的長度為消失動量（vanishing moment）值A的兩倍；所以當消失動量為A時，多貝西小波的小波濾波器（wavelet filter）及調整濾波器（scaling filter）長度皆為2A（N=2A）。一般而言，我們仍是以N來形容多貝西小波的長度：例如，當A=1時，有一個消失動量，多貝西小波寫成D2，長度為2（也是Haar小波）；當A=2時，有兩個消失動量，多貝西小波寫成D4，長度為4；以此類推。但是，在MATLAB的使用上是以dbA描述多貝西小波，以下則為調整濾波器的係數及A的關係表：
| Scaling Coefficient    p_{k} | db1（Haar） | db2 | db3 | db4 | db5 | db6 | db7 | db8 | db9 | db10 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  p_{0} | 1 | 0.6830127 | 0.47046721 | 0.32580343 | 0.22641898 | 0.15774243 | 0.11009943 | 0.07695562 | 0.05385035 | 0.03771716 |
|  p_{1} | 1 | 1.1830127 | 1.14111692 | 1.01094572 | 0.85394354 | 0.69950381 | 0.56079128 | 0.44246725 | 0.34483430 | 0.26612218 |
|  p_{2} |  | 0.3169873 | 0.650365 | 0.8922014 | 1.02432694 | 1.06226376 | 1.03114849 | 0.95548615 | 0.85534906 | 0.74557507 |
|  p_{3} |  | -0.1830127 | -0.19093442 | -0.03957503 | 0.19576696 | 0.44583132 | 0.66437248 | 0.82781653 | 0.92954571 | 0.97362811 |
|  p_{4} |  |  | -0.12083221 | -0.26450717 | -0.34265671 | -0.31998660 | -0.20351382 | -0.02238574 | 0.18836955 | 0.39763774 |
|  p_{5} |  |  | 0.0498175 | 0.0436163 | -0.04560113 | -0.18351806 | -0.31683501 | -0.40165863 | -0.41475176 | -0.35333620 |
|  p_{6} |  |  |  | 0.0465036 | 0.10970265 | 0.13788809 | 0.1008467 | 6.68194092e-4 | -0.13695355 | -0.27710988 |
|  p_{7} |  |  |  | -0.01498699 | -0.00882680 | 0.03892321 | 0.11400345 | 0.18207636 | 0.21006834 | 0.18012745 |
|  p_{8} |  |  |  |  | -0.01779187 | -0.04466375 | -0.05378245 | -0.02456390 | 0.043452675 | 0.13160299 |
|  p_{9} |  |  |  |  | 4.71742793e-3 | 7.83251152e-4 | -0.02343994 | -0.06235021 | -0.09564726 | -0.10096657 |
|  p_{10} |  |  |  |  |  | 6.75606236e-3 | 0.01774979 | 0.01977216 | 3.54892813e-4 | -0.04165925 |
|  p_{11} |  |  |  |  |  | -1.52353381e-3 | 6.07514995e-4 | 0.01236884 | 0.03162417 | 0.04696981 |
|  p_{12} |  |  |  |  |  |  | -2.54790472e-3 | -6.88771926e-3 | -6.67962023e-3 | 5.10043697e-3 |
|  p_{13} |  |  |  |  |  |  | 5.00226853e-4 | -5.54004549e-4 | -6.05496058e-3 | -0.01517900 |
|  p_{14} |  |  |  |  |  |  |  | 9.55229711e-4 | 2.61296728e-3 | 1.97332536e-3 |
|  p_{15} |  |  |  |  |  |  |  | -1.66137261e-4 | 3.25814671e-4 | 2.81768659e-3 |
|  p_{16} |  |  |  |  |  |  |  |  | -3.56329759e-4 | -9.69947840e-4 |
|  p_{17} |  |  |  |  |  |  |  |  | -5.5645514e-5 | -1.64709006e-4 |
|  p_{18} |  |  |  |  |  |  |  |  |  | 1.32354367e-4 |
|  p_{19} |  |  |  |  |  |  |  |  |  | -1.875841e-5 |
濾波器
這邊列出4到10點的filter，這些在實務上已經很夠用，消失點(vanish moment)也就是k/2。
g會是低頻的濾波器，h會是高頻的濾波器。
可以看出 h[n]=(-1)^{n}g[-n]，這邊採用python的語法，h會是g的反序，且n為基數時要乘上-1。
| Coiflet filter   g_{k},h_{k} | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  h_{4} | 0.4829629131445 | 0.836516303 | 0.22414386 | -0.129409522 |  |  |  |  |  |  |
|  g_{4} | -0.129409522551 | -0.22414386 | 0.83651630 | -0.482962913 |  |  |  |  |  |  |
|  h_{6} | 0.3326705529500 | 0.80689150 | 0.45987750 | -0.135011020 | -0.08544127 | 0.03522629 |  |  |  |  |
|  g_{6} | 0.0352262918857 | 0.08544127 | -0.13501102 | -0.459877502 | 0.806891509 | -0.33267055 |  |  |  |  |
|  h_{8} | 0.2303778133088 | 0.71484657 | 0.63088076 | -0.027983769 | -0.187034811 | 0.030841381 | 0.032883011 | -0.0105974017 |  |  |
|  g_{8} | -0.0105974017850 | -0.03288301 | 0.03084138 | 0.1870348117 | -0.027983769 | -0.630880767 | 0.714846570 | -0.2303778133 |  |  |
|  h_{10} | 0.1601023979741 | 0.60382926 | 0.72430852 | 0.1384281459 | -0.242294887 | -0.032244869 | 0.077571493 | -0.0062414902 | -0.012580751 | 0.003335725 |
|  g_{10} | 0.0033357252854 | 0.012580751 | -0.0062414 | -0.077571493 | -0.032244869 | 0.2422948870 | 0.138428145 | -0.7243085284 | 0.6038292697 | -0.16010239 |