A Vector Representation of Multicomplex Numbers and Its Application to Radio Frequency Signals

Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce a vector representation allowing one to isolate the hyperbolic real and imaginary parts of a multicomplex number. The orthogonal decomposition of a multicomplex number is also discussed, and its connection with Hadamard matrices is highlighted. Finally, a multicomplex polar representation is provided. These properties are used to extend the standard complex baseband signal representation to the multi-dimensional case. It is shown that a set of 2n Radio Frequency (RF) signals can be represented as the real part of a single multicomplex signal modulated by several frequencies. The signal RFs are related through a Hadamard matrix to the modulating frequencies adopted in the multicomplex baseband representation. Moreover, an orthogonal decomposition is provided for the obtained multicomplex baseband signal as a function of the complex baseband representations of the input RF signals.