Estimating frequencies of two dimensional harmonics Gaussian noise with hypercomplex

The complex signal of two dimensional harmonics is common. The pairing steps were always needed when we estimated the frequency pairs of harmonics which were described by complex signals. We introduced hypercomplex signal in this paper, and used it to study two dimensional harmonics in additive Gaussian white noise. First, we constructed hypercomplex signal using the original two dimensional harmonics and its Hubert transform. Then, we presented our algorithm of estimating frequencies of hypercomplex signal taking advantage of the properties of Hamilton´s quaternion. Some simulations illustrated the prospect of using hypercomplex signal in estimating parameters of two dimensional harmonics in additive Gaussian white noise without pairing steps.