The accurate analysis of smoothly fluctuating harmonics applied to the calibration of harmonic analysers.

The aim of this research is to develop an accurate method for the analysis of signals composed of fluctuating harmonics. The results obtained of analysis are applied to the calibration of harmonic analysis instruments. A new method is presented suitable for the accurate analysis of smoothly fluctuating harmonic signals. The method is based on a model of signals with a known period, in which the harmonics are individually modulated by polynomial functions normalised over a sampled signal sequence time. Using this model, a decomposition method is developed such that the modulating polynomials can be recovered from a signal. The polynomial decomposition method leads to a piece-wise analysis of the waveform. Two methods based on least squares and splines respectively, are developed with the aim of giving continuity to the piece-wise analysis. Comparisons of the new method with the short time Fourier transform are given. Having defined a test signal and obtained and accurate analysis of it properties, it can be used to calibrate harmonic analysers. For a given applied signal, analysis with these devices can give rise to variation in results as a function of the phase between the signal and the STFT windows. This result distribution due to variable phase (RDVP) is discussed and examples are given for various signals. The RDVP complicates the calibration process due to the spread of results that occur when testing the device. A method is developed to find the RDVP for an applied signal that uses the polynomial decomposition method to find the modulation functions of each harmonic in the applied calibration signal. Having found the RDVP for an applied signal, it is necessary to fit the results of the analyser under test, to the distribution. The random nature of the phase makes the systematic comparison of the theoretical and measured distributions difficult to achieve. A novel method that uses multiple phase shifted modulated harmonics is presented. By comparing the results of the analyser under test to the distributions of each of the phase-shifted harmonics, a best-fit phase shift can be determined and the required calibration comparison made. Key words: time-frequency analysis, demodulation, harmonic analysis, fluctuating harmonics, waveform metrology, calibration of harmonic analysers