Next: Wavelets of different orders Up: WAVELET TRANSFORM OF A Previous: The butterfly

## The lowpass filter coefficients

In the section Wavelet transform of a vector'' I assumed the matrix C to be orthogonal. Additionally, I stated lowpass and highpass characteristics of the filter coefficients ck. In this chapter I will derive the simple nonlinear equations for ck which guarantee these conditions.

To be orthogonal the matrix C in equation (1) multiplied by its transpose in equation (2) has to yield a unit matrix. The multiplication provides the following orthogonality relation for the wavelet filter coefficients ck:

 c02 + c12 + c22 + c32 = 2 (4)

 c0 c2 + c1 c3 = 0 (5)

To determine the four coefficients, two additional relations are required. The conditions

 c0 - c1 + c2 - c3 = 0 (6)

 0 c0 - 1 c1 + 2 c2 - 3 c3 = 0 (7)

ensure that any inner product of (c0,-c1,c2,-c3) and a series k1 (1,1,1,1) + k2 (1,2,3,4) yields . This annihilation is called an approximation condition of polynomial order 1'' or the vanishing of the zeroth and first momentum.'' It is important to note that these rejected polynomials are the low-order and therefore smooth polynomials. These local rejections justify the imaginative visualisation of the odd row multiplications as local highpass filtering. The even row multiplication destroys high-frequency events by weighted adding.

I have coded the conditions in a mathematica script coef.ma, which solves numerically for the coefficient set ck.

(* coef.ma *)
(* number of coefficients *)
(* tested for n=4 and n=6 *)
n  = 4 ;
nh = n / 2 ;
(* definitions kronecker & Power *)
kron[i_,j_] := If[ i == j, 1, 0] ;
Unprotect[Power] ;
Power[0,0] = 1 ;
Protect[Power] ;
(* approximation condition *)
Do[aus[i] = Sum[(-1)^k (n-k)^(i-1) c[k], {k,1,n}] == 0, {i,1,nh}]
(* orthogonality condition *)
Do[aus[i+nh] = Sum[ c[k] c[k-2(i-1)] , {k,2(i-1)+1,n}] ==  kron[i,1], {i,1,nh}]
(* solving Equation *)
eqsys = Table[aus[i],{i,1,n}] ;
coef = Table[c[i],{i,1,n}] ;
sol = N[Solve[eqsys,coef]] ;
(* use NSolve[] for numerical results *)
sol >>> sol.ma ;


The conditions (4), (5), (6), and (7) determine the coefficients ck's amplitude but not their phase spectrum. Daubechies defines her wavelets as the minimum phase solution to these equations. The 4-point wavelet coefficients are
 (8)
 (9)
 (10)
 (11)

Next: Wavelets of different orders Up: WAVELET TRANSFORM OF A Previous: The butterfly
Stanford Exploration Project
11/18/1997