Fourth-order spectra of Gaussian amplitude-modulated sinusoids

Abstract

Propagating underwater signals are sometimes amplitude modulated by the medium or other physical effects. Since this is a multiplicative phenomenon, the resulting spectrum is a convolution of the spectra of the modulation and the desired signal. A new method based on the spectrum of a special case of the fourth-order cumulant is proposed to extract the desired signal. It is shown that the special case of the fourth-order cumulant is independent of the covariance of a Gaussian modulating function, and therefore it can extract the desired signal even when the modulating process is Gaussian white noise. Convergence equations are derived which show that the fourth-order cumulant and its special case will converge to their asymptotic forms as the data length increases. The spectrum of the special case of the fourth-order cumulant is also derived at the output of a low-pass filter. To demonstrate these theoretical results, an experiment was conducted. A sinusoid was modulated by white Gaussian noise and transmitted through the water and received on an omnidirectional hydrophone. The second-order spectrum, the spectrum of the special case of the fourth-order moment and the spectrum of the special case of the fourth-order cumulant were estimated from the filtered data. The experiment corroborated the theoretical results by showing that the second-order spectrum could not extract the sinusoidal frequency, but the spectrum of the special case of the fourth-order moment and the spectrum of the special case of the fourth-order cumulant could. Simulations are included which further corroborate the theoretical results.