XXIX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES - SBrT 2011, DE 02 A 05 DE OUTUBRO DE 2011, CURITIBA, PR Abstract— A new Hyperspherical trigonometry based on Quaternion´s Algebra is introduced. Spatial curves on the surface of the unit-quaternion hypersphere are described in the parametric form. A draft of a new Continuous Quaternion Modulation System is presented, in which two modulation signals are used to drive paths in the normalized 4D- hypersphere. The outline of a PLL-based demodulation is also sketched. Resumo— Uma nova trigonometria hiperesférica baseada na Álgebra de Quatérnions é introduzida. Curvas espaciais sobre a superfície da hiperesfera unitária de quatérnions são descritas sob forma paramétrica. O esboço de um projeto para um novo sistema de Modulação Analógica com Quatérnions é apresentado, no qual dois sinais moduladores são utilizados para traçar caminhos na hiperesfera normalizada 4-D. Uma demodulação com base em PLL é apresentada. Index Terms— Quaternion´s algebra, hyperpherical trigonometry, quaternion modulation system. I. INTRODUCTION landmark on the modern algebra was achieved in 1843 by William Hamilton [1], who was the first to invent an algebra in which the commutative law of multiplication does not hold. The elements of such noncommutative algebra were called quaternions by Hamilton [2]. Since then, unit quaternions have provided a convenient mathematical notation for representing different spatial scenarios, rotations in three-dimensional space, topological curves and matrices. A quaternion is a mathematical entity denoted by = c + x̆ + y̆ + z , (1) with c, x, y, z: real numbers, and ̆ , ̆ , : imaginary numbers. Quaternions form a division ring, the elements of which can be viewed as a scalar plus a vector. They can be seen as some sort of hypercomplex numbers (that is, extended complex numbers). Equipped with the addition and multiplication of quaternions [3], they form a division ring. The addition of quaternions has an identity element: 0 = 0 + 0̆ + 0̆ + 0 , and an inverse element: - = - c - x̆ - y̆ - z . The multiplication of quaternions has an identity element: 1 =1+0̆ +0̆ +0 . Fundamental formulae are: Authors are with the Signal Processing Group, Federal University of Pernambuco (UFPE), C.P. 7800, CEP: 50711-970, Recife-PE. (e-mail: danielblast@gmal.com , {hmo, ricardo}@ufpe.br). ̆ 2 =̆ 2 = 2 =̆̆ =-1. (2) Among many applications, besides quantum mechanics [4], the quaternions have been used in the coding of movements in a 3-D space [5] and in estimating the position and orientation of objects [6]. More related to this work, quaternion-based geodesic [7] and geometry of spherical curves [8] have also been introduced. There are also a loaded relationship among topological surfaces, codes and modulation [9-10]. We have particular interest in new special trigonometries as the trigonometry over finite fields recently introduced [11]. This paper presents the foundations of a “unit quaternion”-based spherical trigonometry. Given a point P on the surface of a unit sphere of coordinates (r,δ,λ), setting r=1, it can be associated with a unit quaternion in terms of the latitude and longitude according with := sin(δ).cos(λ)+ cos(δ).sin(λ).̆ +sin(δ).sin(λ).̆ +cos(δ).cos(λ). , (3) with these unit quaternions confined to be on the surface of a hypersphere of unity radius. Indeed, if n(.) denotes the magnitude of (norm of the quaternion), then n( )= 1. A matrix representation of quaternions can also be derived. In fact, one of the pioneers of the geometry of more than three dimensions, Arthur Cayley (1821-1895), offered matrices to represent noncommutative algebras. He was also able to link determinants with straight lines and planes, sin . cos . sin sin . sin . cos sin . sin . cos sin . cos . sin (4) Another keypoint is the inner product between two quaternions, which can be computed by . ! = sin . sin ! . cos . cos ! cos . cos ! . sin . sin ! sin . sin ! . sin . sin ! cos . cos ! . cos . cos ! . (5) Rearranging this expression, we get the symmetric and nice- looking expression: . ! =cos(∆δ).cos(∆λ), (6) where ∆δ:=δ 2 -δ 1 and ∆λ:=λ 2 -λ 1 . A Unit Quaternion-based Spherical Trigonometry and a New Two-carrier Phase- quadrature Quaternion Modulation System H.M. de Oliveira, D.R. de Oliveira, R.M. Campello de Souza A