Bias of amplitude estimation using three-parameter sine fitting in the presence of additive noise

The estimation of the amplitude of a sine wave using traditional sine fitting algorithms which are based on square error minimization is biased in the presence of additive noise contrary to what happens generally in linear regression problems. An approximate closed form expression for the estimation error as a function of sine wave amplitude, additive noise standard deviation and number of data points is derived here. It is demonstrated that although the estimator is biased, it is asymptotically unbiased, that is, the estimation error vanishes as the number of data points increase to infinity. It is shown that in practical conditions the relative error in the amplitude estimation is very small – lower than 0.5% for a signal to noise ratio as low as 0 dB (with 100 data points). Only the three-parameter algorithm in the case of coherent sampling is studied.