A Quantitative Study of the Estimation Error of the DFT in Estimating Noisy Sinusoids

A study is presented on the implications of the use of the discrete Fourier transform (DFT) in estimating sinusoidal signals of fixed frequency under large amounts of noise. Although the DFT technique has been extensively investigated and heavily employed in diverse areas, no systematic study seems to have reported the effects of various parameters, such as the signal-to-noise ratio (SNR), sampling rate, constant phase of the sinusoid of interest, and the length of the available data, on the error incurred in the process of estimation of sinusoids under noise. A quantitative study of the estimation error as a function of the process variables is performed. While the dependency of the error on the value of the SNR and the length of the available data is a pretty obvious observation, the study sheds light on some other dependencies which are not as intuitive. More particularly, it is observed that under noisy conditions the estimation error is notably dependent on the constant phase of the sinusoidal signal of interest and the sampling rate beyond that implied by the Nyquist criterion