Using the Wavelet Packet Transform to evaluate harmonics through a lookup table technique

Abstract

The paper is concerned with an intuitive approach of harmonics evaluation by using the Wavelet Packet Transform. Conceptual argumentation of simple and reliable algorithms is provided firstly. Two wavelet mothers (WM) were analyzed in order to see which of them allows for a simpler and faster implementation, involving less computer resources simultaneously with improved accuracy. The first one (using a Daubechy WM relying on 28 coefficients - known as db14 in Matlab) involves smaller runtimes for decomposition, but the per/node weights of “node-dominant” harmonics have often smaller values when compared to those yielded by db20. For both analyzed WM, 8 nodes from the final level could be grouped in pairs considering the principle: energies of nodes in a group are affected only by a pair of harmonics and the paired corresponding harmonics do not affect any other node except those from that group. The weights of energies generated by signals polluted by a single harmonic were found to be almost identical for a certain node, irrespective to the magnitude or phase of the analyzed signals. On the other hand, when pairs of harmonics (correlated by the nodes where their energies can be found) act simultaneously, variations of the above mentioned weights were recorded. Therefore a harmonic evaluation based on the solving of 2×2 linear systems yields errors and sometimes negative energies leading to values for harmonic magnitudes out of the real numbers' space. In this context we conceived and implemented another algorithm, based on a lookup table technique. For the “non-paired” harmonics another rule was detected. Groups of 4 nodes (quadruples) are affected only by groups of at most 4 harmonics. Three quadruples were found. For one of them only the energies of 3 nodes a required, because only 3 harmonics influence the nodes from the quadruple. Good mean values of errors related to the harmonic magnitudes evaluation were obtained after running sets of 50 randomly polluted signals for all pairs of odd harmonic orders from the range 3...40. Most of the absolute percent errors higher then 1.5% from the RMS value of the polluted signal in the case when the lookup table technique was used were found to be associated to inversions in paired harmonics (that is the magnitudes of harmonics were computed as being switched between the members in a harmonic pair). This is due to the symmetry of some components of the matrices used for calculation and perhaps can be solved by using more accurate values for them.