Multi-sensor estimation and detection of phase-locked sinusoids

This paper proposes a method to compute the likelihood function for the amplitudes and phase shifts of noisily observed phase-locked and amplitude-constrained sinusoids. The sinusoids are assumed to be coupled based on a set of parameters as, e.g., measurements of a monochromatic wave field. A factor graph is used to formulate the probability density function of the observations given the parameters. The factor graph consists of one second-order state-space model per signal and one additional factor connecting all the final states. Because the parameters appear only in this latter factor, we are able to formulate a sufficient statistic for parameter estimation and signal detection in terms of messages in the factor graph. In special cases, the general form of the sufficient statistic reduces to the discrete Fourier transform. As extensions we provide iterative algorithms for approximate maximum likelihood estimation of the noise variances and the parameters of superposed waves.