On the effect of nuisance parameters on the threshold SNR value of the Barankin bound

The performance of nonlinear estimators of signal parameters in noise appears to exhibit a threshold phenomenon. Below a critical value of signal-to-noise ratio (SNR), the performance of these estimators deviates significantly from the Cramer-Rao bound. In a large-error (ambiguity-prone) SNR region, the Barankin bound has been proved to be an advantageous tool to assess the attainable performance and the threshold value. Obviously, when the estimation problem involves additional unknown nuisance parameters, the mean square error (MSE) of the estimator does not decrease. However, the impact of these nuisance parameters on the threshold value is not clear. In this correspondence, we discuss the influence of unknown nuisance parameters on the threshold value. The analysis is done for the common problem concerning estimating parameters of a Gaussian process. We confine our scope to a simplified problem concerning only two estimated parameters. However, we explain how this simplified analysis can be used to handle a more complicated problem comprising multiple nuisance parameters. We derive a sufficient condition applied to the structure of the data covariance matrix. When the condition is satisfied, then the need to estimate additional nuisance parameters does not change the threshold value. Using the proposed condition, we prove that the threshold SNR in passive source localization does not increase when the spectral parameters of the source are unknown, whereas the presence of another source at an unknown bearing may change the threshold SNR