A Vector Representation for Multicomplex Numbers and its Application to Radio Frequency Signals

Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial for 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 highlighted. Finally, a multicomplex polar representation is derived. These properties are used to extend the standard complex baseband signal representation to the multi-dimensional case. It is shown that a set of 2^n 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.

BORIO Daniele; 

2024-08-12

MDPI

JRC136625

2075-1680 (online),   

https://www.mdpi.com/2075-1680/13/5/324,    https://publications.jrc.ec.europa.eu/repository/handle/JRC136625,   

10.3390/axioms13050324 (online),