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.