Auto-regressive spectral estimation of noisy sinusoids

Analytical expressions for spectral smoothing and frequency shifting are derived for complex and real noisy sinusoids. Spectral smoothness is measured in terms of the half-power bandwidth (BW). For a complex sinusoid in white noise, bandwidth is related to the distance (a) of signal zero to the origin, which is in turn a function of signal-to-noise ratio (SNR) and the prediction error filter order (P). It is also shown that there is no frequency shifting. The BW computed from the derived formula agrees very well with the simulation results. For noisy real sinusoids there is frequency shifting in addition to the spectral smoothing. Peak spectrum and BW are expressed in terms of the signal zero distance (a). Expressions for frequency shifting have also been obtained for two equal-amplitude complex sinusoids in white noise.