Daubechies wavelet

Among the 2A−1 possible solutions of the algebraic equations for the moment and orthogonality conditions, the one is chosen whose scaling filter has extremal phase.

The graphs below are generated using the cascade algorithm, a numeric technique consisting of inverse-transforming [1 0 0 0 0 ... ] an appropriate number of times.

Note that the spectra shown here are not the frequency response of the high and low pass filters, but rather the amplitudes of the continuous Fourier transforms of the scaling (blue) and wavelet (red) functions.

A vanishing moment limits the wavelets ability to represent polynomial behaviour or information in a signal.

For example, D2, with one vanishing moment, easily encodes polynomials of one coefficient, or constant signal components.

This ability to encode signals is nonetheless subject to the phenomenon of scale leakage, and the lack of shift-invariance, which arise from the discrete shifting operation (below) during application of the transform.

Sub-sequences which represent linear, quadratic (for example) signal components are treated differently by the transform depending on whether the points align with even- or odd-numbered locations in the sequence.

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.

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 deg(p) = A − 1, one can write the orthogonality condition as or equally as with the Laurent-polynomial generating all symmetric sequences and

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.

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.

The periodization is accomplished in the forward transform directly in MATLAB vector notation, and the inverse transform by using the circshift() function: It is assumed that S, a column vector with an even number of elements, has been pre-defined as the signal to be analyzed.

It was shown by Ali Akansu in 1990 that the binomial quadrature mirror filter bank (binomial QMF) is identical to the Daubechies wavelet filter, and its performance was ranked among known subspace solutions from a discrete-time signal processing perspective.

[5][6] The magnitude square functions of Binomial-QMF filters are the unique maximally flat functions in a two-band perfect reconstruction QMF (PR-QMF) design formulation that is related to the wavelet regularity in the continuous domain.