The Statistical Performance of Some Instantaneous Frequency Estimators

We examine the class of smoothed central finite differences (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 (periodic) first moment , with respect to frequency, of discrete time frequency representations (TFRs) in Cohen's class (TFR moment IF estimators). Cohen's class includes representation 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 bound from below by the variance of the corresponding SCFD estimator. We determine the distribution of these 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 can 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 on 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 yet still remains 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.