Editing Daubechies wavelet (section)

==Construction==
{{Confusing section|reason=there is undefined math symbols (e.g. a, p, P)|date=September 2019}} 
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 <math>\sqrt{2}</math>, 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'',

:<math>a(Z)=2^{1-A}(1+Z)^A p(Z),</math>

with ''N'' = 2''A'', ''p'' having real coefficients, ''p''(1) = 1 and deg(''p'')&nbsp;=&nbsp;''A''&nbsp;−&nbsp;1, one can write the orthogonality condition as

:<math>a(Z)a \left (Z^{-1} \right )+a(-Z)a \left (-Z^{-1} \right )=4,</math>

or equally as

:<math>(2-X)^A P(X)+X^A P(2-X)=2^A \qquad (*),</math>

with the Laurent-polynomial

:<math>X:= \frac{1}{2}\left (2-Z-Z^{-1} \right )</math>

generating all symmetric sequences and <math>X(-Z)=2-X(Z).</math> Further, ''P''(''X'') stands for the symmetric Laurent-polynomial

:<math>P(X(Z))=p(Z)p \left ( Z^{-1} \right ).</math>

Since

:<math>X(e^{iw})=1-\cos(w)</math> 
:<math>p(e^{iw})p(e^{-iw})=|p(e^{iw})|^2</math>

''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'',

:<math>P_A(X)=\sum_{k=0}^{A-1} \binom{A+k-1}{A-1} 2^{-k}X^k.</math>

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

The homogeneous equation for (*) is antisymmetric about ''X'' = 1 and has thus the general solution

:<math>X^A(X-1)R \left ((X-1)^2 \right ),</math>

with ''R'' some polynomial with real coefficients. That the sum

:<math>P(X)=P_A(X)+X^A(X-1)R \left ((X-1)^2 \right )</math>

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 <math>4^{A-r},</math> maximizing ''r'' results in a linear program with infinitely many inequality conditions.

To solve

:<math>P(X(Z))=p(Z)p \left (Z^{-1} \right)</math>

for ''p'' one uses a technique called spectral factorization resp. Fejér-Riesz-algorithm. The polynomial ''P''(''X'') splits into linear factors

:<math>P(X)=(X-\mu_1)\cdots(X-\mu_N), \qquad N=A+1+2\deg(R).</math>

Each linear factor represents a Laurent-polynomial

:<math>X(Z)-\mu =-\frac{1}{2}Z+1-\mu-\frac12Z^{-1}</math>

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<sup>''N''</sup> 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 <math>c_i</math> using the quadrature mirror filter property results in the following solution for the coefficient values for filter of order 4.

:<math>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}}.</math>