Noise Power Estimation by the Three-Parameter and Four-Parameter Sine-Fit Algorithms

This paper analyzes the performance provided by the three-parameter sine-fit (3PSF) and the four-parameter sine-fit (4PSF) algorithms when estimating the noise power of a sine wave corrupted by a white Gaussian noise. In the former case, the frequency parameter is extracted from the available data by using the interpolated discrete Fourier transform (IpDFT) method. The related procedure is called the 3PSF-IpDFT algorithm. Simple expressions for the expected sum-squared fitting and the expected sum-squared residual errors are derived for both the 3PSF and 4PSF algorithms, which agree with previously published results. These expressions show that the sum-squared fitting error of the 4PSF algorithm is smaller than the corresponding value associated with the 3PSF algorithm when the uncertainty of the sine-wave frequency employed by the latter algorithm is greater than the related Cramér-Rao lower bound. From this point of view, the 4PSF algorithm outperforms the 3PSF-IpDFT algorithm. However, since the frequency estimator provided by the IpDFT method is consistent, the sum-squared fitting error associated with both the 3PSF-IpDFT and 4PSF algorithms can be made negligible as compared with the sum-squared residual errors, when the number of analyzed samples is large enough. Moreover, several simulation results show that the 3PSF-IpDFT algorithm requires a much lower computational effort than the 4PSF algorithm. Therefore, it represents the best alternative when estimating the noise power of a sine wave embedded in white noise.