Estimating frequencies of two dimensional harmonics with hypercomplex

The complex signal of two dimensional harmonics is common. Pairing steps are always needed when we estimate the frequency pairs of harmonics which are described by complex signals. We introduce the hypercomplex signal, and use it to study two dimensional harmonics. First, we construct the hypercomplex signal using the signal´s original two dimensional harmonics and its Hilbert transform. Then, we present our algorithm for estimating the frequencies of the hypercomplex signal through taking advantage of the properties of Hamilton´s quaternion. Some simulations illustrate the prospect of using hypercomplex signals in estimating the parameters of two dimensional harmonics without pairing steps.