Non-efficiency of the non-linear least squares estimator of polynomial phase signals in colored noise

The focus of this paper is on the estimation of the parameters of a constant amplitude polynomial phase signal (PPS) observed in circularly symmetric colored complex Gaussian noise. We derive a closed-form expression for the large-sample Cramer-Rao bound and show that it depends upon an average SNR defined in the frequency domain (as opposed to the time-domain averaged SNR, i.e., the variance). The non-linear least squares estimator (NLLSE) is derived, and its performance studied. We show that the NLLSE is not asymptotically efficient. This is in contrast with the case of harmonic signals in colored noise. The asymptotic relative efficiency (ARE) of the NLLSE is studied both analytically and through simulations. It is seen that the larger the bandwidth of the noise, the larger the ARE. Although the NLLSE is not efficient, it provides a good compromise between computational complexity and estimation accuracy.