Daubechies Wavelet - Construction

Construction

Both the scaling sequence (Low-Pass Filter) and the wavelet sequence (Band-Pass Filter) (see orthogonal wavelet for details of this construction) will here be normalized to have sum equal 2 and sum of squares equal 2. In some applications, they are normalised to have sum, so that both sequences and all shifts of them by an even number of coefficients are orthonormal to each other.

Using the general representation for a scaling sequence of an orthogonal discrete wavelet transform with approximation order A,

, with N=2A, p having real coefficients, p(1)=1 and degree(p)=A-1,

one can write the orthogonality condition as

, or equally as (*),

with the Laurent-polynomial generating all symmetric sequences and . Further, P(X) stands for the symmetric Laurent-polynomial . Since and, P takes nonnegative values on the segment .

Equation (*) has one minimal solution for each A, which can be obtained by division in the ring of truncated power series in X,

.

Obviously, this has positive values on (0,2)

The homogeneous equation for (*) is antisymmetric about X=1 and has thus the general solution, with R some polynomial with real coefficients. That the sum

shall be nonnegative on the interval translates into a set of linear restrictions on the coefficients of R. The values of P on the interval are bounded by some quantity, maximizing r results in a linear program with infinitely many inequality conditions.

To solve for p one uses a technique called spectral factorization resp. Fejer-Riesz-algorithm. The polynomial P(X) splits into linear factors, N=A+1+2deg(R). Each linear factor represents a Laurent-polynomial that can be factored into two linear factors. One can assign either one of the two linear factors to p(Z), thus one obtains 2N possible solutions. For extremal phase one chooses the one that has all complex roots of p(Z) inside or on the unit circle and is thus real.