Nonmatrix cramér-rao bound expressions for amplitude and phase estimates of two closely spaced complex sinusoids

Abstract

This correspondence presents nonmatrix expressions for the Cramér-Rao (CR) lower bound on the variance of amplitude and phase estimates for the two-signal time-series data model consisting of two complex sinusoids in complex white Gaussian noise. The expressions give the CR bounds in terms of the signal-to-noise ratio (SNR) (or the noise power), the number of data samples, and a function that is dependent on the frequency separation and the initial phase difference between the two signal components of the model. The bounds are examined as the phase difference is varied, and the largest and smallest bounds and the corresponding critical values of the phase difference are obtained. The exact expressions are analyzed for the case of small frequency separations, and simple expressions are developed for the largest and smallest CR bounds and for the critical phase difference values that are valid for this regime. The results are applicable to the general case of sampling where the samples are taken at arbitrary instants. © 1998 IEEE.