Kernel estimation of the instantaneous frequency

Abstract

Considers kernel estimators of the instantaneous frequency of a slowly evolving sinusoid in white noise. The expected estimation error consists of two terms. The systematic bias error grows as the kernel halfwidth increases while the random error decreases. For a nonmodulated signal g(t), the kernel halfwidth that minimizes the expected error is proportional to h/spl sim/[(/spl sigma//sup 2/)/(N|/spl part//sub t//sup 2/g|2)]/sup 1/5/ where /spl sigma//sup 2/ is the noise variance and N is the number of measurements per unit time. The author shows that estimating the instantaneous frequency corresponds to estimating the first derivative of a modulated signal, A(t)exp(i/spl phi/(t)). For instantaneous frequency estimation, the halfwidth which minimizes the expected error is larger: h/sub 1,3//spl sim/[(/spl sigma//sup 2/)/(A/sup 2/N|/spl part//sub t//sup 3/(e/sup i/spl phi/~(t/))|/sup 2/)]/sup 1//$ u7. Since the optimal halfwidths depend on derivatives of the unknown function, the authors initially estimate these derivatives prior to estimating the actual signal.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>