Discrete Wavelets with Applications

122a - The Coeffcients of the (9,7) Cohen-Daubechies-Feauveau Filter Pair (CDF97)

Helmut  Knaust - Department of Mathematical Sciences, University of Texas at El Paso, hknaust@utep.edu
04/28/09

The Lowpass Conditions

h={h4,h3,h2,h1,h0,h1,h2,h3,h4}

H[x_]=h.Table[Exp[I k x],{k,-4,4}]

H[π]

H'[π]

H''[π]

H'''[π]

j={j3,j2,j1,j0,j1,j2,j3}

J[x_]=j.Table[Exp[I k x],{k,-3,3}]

J[π]

J'[π]

J''[π]

J'''[π]

We will also impose the condition H(0)=(Sqrt[2]).  (In the end, we will get the condition J(0)=(Sqrt[2]) for "free".)

lp={J[π]==0,J''[π]==0,H[π]==0,H''[π]==0,H[0]==Sqrt[2]};
TableForm[lp]

Length[lp]

The Orthogonality Conditions

ortho=
{2h3 j3+2 h2 j2+2 h1 j1+h0 j0==1,h1 j3+ h0 j2+h1 j1+h2 j0 + h3 j1+h4 j2==0,h1 j3+h2 j2+h3 j1+h4 j0==0,h3 j3 +h4 j2==0};
TableForm[ortho]

Length[ortho]

Solving the Complete System

eqn=Union[lp,ortho];
TableForm[eqn]

sol=Solve[eqn];
TableForm[N[sol]]

sol=sol[[1]]//N

h={h4,h3,h2,h1,h0,h1,h2,h3,h4}/.sol

j={j3,j2,j1,j0,j1,j2,j3}/.sol

As a "bonus", we obtain also that J (0) =(Sqrt[2]).

Total[j]