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 \sqrt{2}, 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, :a(Z)=2^{1-A}(1+Z)^A p(Z), with
N = 2
A,
p having real coefficients,
p(1) = 1 and deg(
p) =
A − 1, one can write the orthogonality condition as :a(Z)a \left (Z^{-1} \right )+a(-Z)a \left (-Z^{-1} \right )=4, or equally as :(2-X)^A P(X)+X^A P(2-X)=2^A \qquad (*), with the Laurent-polynomial :X:= \frac{1}{2}\left (2-Z-Z^{-1} \right ) generating all symmetric sequences and X(-Z)=2-X(Z). Further,
P(
X) stands for the symmetric Laurent-polynomial :P(X(Z))=p(Z)p \left ( Z^{-1} \right ). Since :X(e^{iw})=1-\cos(w) :p(e^{iw})p(e^{-iw})=|p(e^{iw})|^2
P takes nonnegative values on the segment [0,2]. Equation (*) has one minimal solution for each
A, which can be obtained by division in the ring of truncated
power series in
X, :P_A(X)=\sum_{k=0}^{A-1} \binom{A+k-1}{A-1} 2^{-k}X^k. Obviously, this has positive values on (0,2). The homogeneous equation for (*) is antisymmetric about
X = 1 and has thus the general solution :X^A(X-1)R \left ((X-1)^2 \right ), with
R some polynomial with real coefficients. That the sum :P(X)=P_A(X)+X^A(X-1)R \left ((X-1)^2 \right ) shall be nonnegative on the interval [0,2] translates into a set of linear restrictions on the coefficients of
R. The values of
P on the interval [0,2] are bounded by some quantity 4^{A-r}, maximizing
r results in a linear program with infinitely many inequality conditions. To solve :P(X(Z))=p(Z)p \left (Z^{-1} \right) for
p one uses a technique called spectral factorization resp. Fejér-Riesz-algorithm. The polynomial
P(
X) splits into linear factors :P(X)=(X-\mu_1)\cdots(X-\mu_N), \qquad N=A+1+2\deg(R). Each linear factor represents a Laurent-polynomial :X(Z)-\mu =-\frac{1}{2}Z+1-\mu-\frac12Z^{-1} that can be factored into two linear factors. One can assign either one of the two linear factors to
p(
Z), thus one obtains 2
N 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. For Daubechies wavelet transform, a pair of linear filters is used. Each filter of the pair should be a
quadrature mirror filter. Solving the coefficient of the linear filter c_i using the quadrature mirror filter property results in the following solution for the coefficient values for filter of order 4. :c_0 = \frac{1+\sqrt{3}}{4\sqrt{2}}, \quad c_1 = \frac{3+\sqrt{3}}{4\sqrt{2}}, \quad c_2 = \frac{3-\sqrt{3}}{4\sqrt{2}}, \quad c_3 = \frac{1-\sqrt{3}}{4\sqrt{2}}. == The scaling sequences of lowest approximation order ==