The second-order local polynomial Fourier transform as instantaneous frequency and chirp rate estimator

Abstract

Statistical properties, bias and variance, of the second-order local polynomial Fourier transform (LPFT) in the instantaneous frequency (IF) and chirp rate (CR) estimation are considered. We have derived expressions based on the error linearization around the exact value of signal parameters as it is commonly done in this field. However, we have also demonstrated the limits of this methodology. Two particularly important frequency modulated (FM) signals are considered as test cases: quadratic and cubic phase signals. Approximative expression for the bias and variance in the latter case are also derived. It is demonstrated that the optimal design parameter of the LPFT, i.e., window width, must be optimized independently for the IF and CR estimation to produce the minimum mean squared error (MSE). The derived expressions are checked by simulations.