Numerical Optimization Problems of Sine-Wave Fitting Algorithms in the Presence of Roundoff Errors

Abstract

In this paper, the effect of roundoff errors on sine-wave fitting algorithms is investigated. It is shown that the standard calculation of sine-wave parameters may result in unexpectedly large errors, even with floating-point number representation. The three- and four-parameter least squares, the maximum likelihood, and the quantile-based estimator (QBE) methods are investigated. It is pointed out that imprecise phase storage and summation affect almost every sine-wave fitting algorithm. Furthermore, the necessary solution for sometimes ill-conditioned systems and the imprecisely evaluated distribution of the observation noise, as additional error sources, are also shown to influence a part of the fitting methods. Besides error descriptions, compensation techniques are also suggested in order to mitigate the effect of error sources. The enhancement of precision and robustness due to these suggestions is demonstrated, while keeping the given limited precision number representation platform. The QBE is shown to overcome roundoff error problems when its applicability conditions are fulfilled. In addition, its performance in comparison with that of the least squares estimator is highlighted. Finally, it is pointed out that the investigated methods show similar sensitivity to the inaccurate knowledge of the frequency of the sine wave.