THE STATISTICAL PERFORMANCE OF SOME INSTANTANEOUS FREQUENCY ESTIMATORS

We examine the class of smoothed central finite difference (SCFD) instantaneous frequency (IF) estimators which are based on finite differencing of the phase of the analytic signal. These estimators are of particular interest since they are closely related to IF estimation via the (periodic) first moment, with respect to frequency, of discrete time-frequency representations (TFR's) in Cohen's class (TFR moment IF estimators). Cohen's class includes representations such as the spectrogram and Wigner-Ville distribution. Indeed, in the case of a monocomponent signal, the variance of a TFR moment IF estimator is bounded from below by the variance of the corresponding SCFD estimator. We determine the distribution of this class of estimators and establish a framework which allows the comparison of several other estimators such as the zero-crossing estimator and a recently proposed estimator based on linear regression on the signal phase. We find the regression IF estimator is biased and exhibits a large threshold for much of the frequency range because it does not account for the circular nature of discrete-time frequency estimates. By replacing the linear convolution operation in the regression estimator with the appropriate convolution operation for circular data we obtain the parabolic SCFD (PSCFD) estimator. This estimator is unbiased and has a frequency independent variance and vet still retains the optimal performance and simplicity of the original estimator. The PSCFD estimator would be suitable for use as a real-time line or bearing tracker. In this paper, we propose a number of mathematical operations suitable for circular data which should be used in preference to the conventional linear operations.