1,717 research outputs found
A Unifying Approach to Quaternion Adaptive Filtering: Addressing the Gradient and Convergence
A novel framework for a unifying treatment of quaternion valued adaptive
filtering algorithms is introduced. This is achieved based on a rigorous
account of quaternion differentiability, the proposed I-gradient, and the use
of augmented quaternion statistics to account for real world data with
noncircular probability distributions. We first provide an elegant solution for
the calculation of the gradient of real functions of quaternion variables
(typical cost function), an issue that has so far prevented systematic
development of quaternion adaptive filters. This makes it possible to unify the
class of existing and proposed quaternion least mean square (QLMS) algorithms,
and to illuminate their structural similarity. Next, in order to cater for both
circular and noncircular data, the class of widely linear QLMS (WL-QLMS)
algorithms is introduced and the subsequent convergence analysis unifies the
treatment of strictly linear and widely linear filters, for both proper and
improper sources. It is also shown that the proposed class of HR gradients
allows us to resolve the uncertainty owing to the noncommutativity of
quaternion products, while the involution gradient (I-gradient) provides
generic extensions of the corresponding real- and complex-valued adaptive
algorithms, at a reduced computational cost. Simulations in both the strictly
linear and widely linear setting support the approach
Finding Octonionic Eigenvectors Using Mathematica
The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some
surprises, which we have reported elsewhere. In particular, the eigenvalues
need not be real, there are 6 rather than 3 real eigenvalues, and the
corresponding eigenvectors are not orthogonal in the usual sense. The
nonassociativity of the octonions makes computations tricky, and all of these
results were first obtained via brute force (but exact) Mathematica
computations. Some of them, such as the computation of real eigenvalues, have
subsequently been implemented more elegantly; others have not. We describe here
the use of Mathematica in analyzing this problem, and in particular its use in
proving a generalized orthogonality property for which no other proof is known.Comment: LaTeX2e, 22 pages, 8 PS figures (uses included PS prolog; needs
elsart.cls and one of epsffig, epsf, graphicx
Polygon spaces and Grassmannians
We study the moduli spaces of polygons in R^2 and R^3, identifying them with
subquotients of 2-Grassmannians using a symplectic version of the
Gel'fand-MacPherson correspondence. We show that the bending flows defined by
Kapovich-Millson arise as a reduction of the Gel'fand-Cetlin system on the
Grassmannian, and with these determine the pentagon and hexagon spaces up to
equivariant symplectomorphism. Other than invocation of Delzant's theorem, our
proofs are purely polygon-theoretic in nature.Comment: plain TeX, 21 pages, submitted to Journal of Differential Geometr
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