Fourier series and estimation on the circle with applications to synchronous communication--I: Analysis

A wide variety of continuous- and discrete-time estimation problems on the circle <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S^1</tex> are considered with the aid of Fourier series analysis. Measurement and diffusion update equations are derived for the conditional expectation of certain functions of the parameter to be estimated, and we investigate the use of Fourier series to obtain easily implemented optimal estimation equations. A variety of important examples--phase tracking, frequency demodulation, and phase demodulation in the presence of oscillator instabilities, additive noise, Rayleigh fading, or any combination of these--are considered.