Editing Daubechies wavelet (section)

==Implementation==
While software such as [[Mathematica]] supports Daubechies wavelets directly<ref>
[http://reference.wolfram.com/mathematica/ref/DaubechiesWavelet.html Daubechies Wavelet in Mathematica. Note that in there ''n'' is ''n''/2 from the text.]</ref> a basic implementation is possible in [[MATLAB]] (in this case, Daubechies 4). This implementation uses periodization to handle the problem of finite length signals. Other, more sophisticated methods are available, but often it is not necessary to use these as it only affects the very ends of the transformed signal. The periodization is accomplished in the forward transform directly in MATLAB vector notation, and the inverse transform by using the <code>circshift()</code> function:

===Transform, D4===
It is assumed that ''S'', a column vector with an even number of elements, has been pre-defined as the signal to be analyzed. Note that the D4 coefficients are [1&nbsp;+&nbsp;{{radic|3}}, 3&nbsp;+&nbsp;{{radic|3}}, 3&nbsp;−&nbsp;{{radic|3}}, 1&nbsp;−&nbsp;{{radic|3}}]/4.

<syntaxhighlight lang="matlab">
N = length(S);
sqrt3 = sqrt(3);
s_odd = S(1:2:N-1);
s_even = S(2:2:N);

s = (sqrt3+1)*s_odd + (3+sqrt3)*s_even + (3-sqrt3)*[s_odd(2:N/2);s_odd(1)] + (1-sqrt3)*[s_even(2:N/2);s_even(1)];
d = (1-sqrt3)*[s_odd(N/2);s_odd(1:N/2-1)] + (sqrt3-3)*[s_even(N/2);s_even(1:N/2-1)] + (3+sqrt3)*s_odd + (-1-sqrt3)*s_even
s = s / (4*sqrt(2));
d = d / (4*sqrt(2));
</syntaxhighlight>

===Inverse transform, D4===
<syntaxhighlight lang="matlab">
d1 = d * ((sqrt(3) - 1) / sqrt(2));
s2 = s * ((sqrt(3) + 1) / sqrt(2));
s1 = s2 + circshift(d1, - 1);
S(2:2:N) = d1 + sqrt(3) / 4 * s1 + (sqrt(3) - 2) / 4 * circshift(s1, 1);
S(1:2:N - 1) = s1 - sqrt(3) * S(2:2:N);
</syntaxhighlight>