Exponential-Plus-Constant Fitting based on Fourier Analysis
Hodgkinson, Matthieu (2011) Exponential-Plus-Constant Fitting based on Fourier Analysis. Proceedings of the 17èmes Journées d'Informatique Musicale Saint-Etienne, France, 2011 .
A closed form method for Exponential Plus Constant (EPC) curve fitting is introduced. The method, referred to as Fourierbased Exponential Plus Constant Fitting (FEPCF), returns estimates of the EPC coefficients for user-input sets of equidistant time values and corresponding data points. An analytical expression for the exponent’s coefficient is found through algebraic manipulation of the Fourier series of the EPC, taken over a finite time interval. The unknown is therefore approximated through the Discrete Fourier Transform (DFT) of the input sequence, itself used as an approximation of Fourier series. The discrepancy between Fourier series and DFT, byproduct of aliasing, is reduced with sine windowing. This is the essential contribution of the paper – the exponent coefficient found, what remains of the problem can be approached with standard linear least squares. FEPCF was motivated by the need for a measuring tool for the time-modeling of fundamental frequency and inharmonicity coefficient of tension-modulated strings. Care is however taken in the mathematical formulation of the method to facilitate its transportation to other fields of science. MATLAB function implementation and a demonstration of FEPCF can be downloaded from the address http://www.cs.nuim.ie/~matthewh/.
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