Next: Quaternionic Transform Up: Witten: Quaternion-based Signal Processing Previous: Introduction

# Hypercomplex Mathematics

The set of hypercomplex numbers is defined as
 (1)
Hypercomplex numbers define a n+1-dimensional complex space with il orthonormal to im, for l m. For all cases presented in this paper l will be limited to 3. Such numbers are quaternions, which can be represented as q=q0+iq1+jq2+kq3, where i,j,k are imaginary numbers that satisfying the following relations:
 (2)

 1 i j k 1 1 i j k i i -1 k -j j j -k -1 i k k j -i -1

The multiplication table for quaternion unit vectors is shown in Table 1.
With these definitions, quaternionic addition between two quaternions, q and p, can be defined as
 (3)
and multiplication as
 (4)
Notice that multiplication in equation 4, is not commutative due to the quaternionic algebra rules defined in table 1. Quaternions are often separated into two parts, q0 and , respectively called the scalar and vector part of the quaternion. Using this definition, the conjugate of q, , is
 (5)
and the norm of a quaternion is defined by
 (6)

It is useful to formulate a polar representation of the quaternion, as this will be the primary notation throughout this paper. For any complex number, , the argument or phase-angle is defined as atan2(b,a). If is written it the form , then is the phase (argument) of , denoted arg()=.Quaternions contain three complex subfields and, correspondingly, three phases that are the projections onto the i, j, or k-complex plane,
 (7) (8) (9)