Statistical characterization of sample fourth-order cumulants of a noisy complex sinusoidal process

The paper deals with the statistical characterization of sample estimates of the fourth-order cumulants of a random process consisting of multiple complex sinusoids and additive colored Gaussian noise. In particular, it presents necessary and sufficient conditions for strong consistency of the sample cumulants of arbitrary orders, and derives expressions for the asymptotic covariance of the sample estimates of the fourth-order cumulants. It is shown that the fourth-order cumulant C_4y (τ₁,...,τ₄) can be written as a function of a single argument τ=τ₃+τ₄ -τ₁-τ₂, which implies large flexibility in estimating the cumulant. It is recommended that the estimate be based upon lags such that τ₁ is distant from τ₂ and τ₃ is distant from τ₄, and/or as a linear combination of such terms. The asymptotic variance of a cumulant-based frequency estimator is shown to have the form c₂·SNR^-2+c₃·SNR^-3 +c₄·SNR^-4, where the coefficient c ₂ may possibly vanish. The theory is illustrated via numerical examples. The results of this paper will be useful in analyzing the performance of various cumulant-based frequency estimation algorithms