By Athanassios Manikas
In view of the importance of the array manifold in array processing and array communications, the function of differential geometry as an analytical instrument can't be overemphasized. Differential geometry is especially limited to the research of the geometric homes of manifolds in 3-dimensional Euclidean house R3 and in actual areas of upper measurement.
Extending the theoretical framework to complicated areas, this priceless publication provides a precis of these result of differential geometry that are of functional curiosity within the learn of linear, planar and three-d array geometries.
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Additional resources for Differential geometry in array processing
July 6, 2004 9:29 WSPC/Book Trim Size for 9in x 6in Diﬀerential Geometry of Array Manifold Curves chap02 25 Fig. e. speed = a(p) ˙ = s(p) ˙ = variable = 1. moving object while the total distance travelled, s(p), can be obtained by integrating speed with respect to “time” p. e. a(s), indicates that the “velocity” vector (tangent vector) a (s) always has constant length equal to 1 (moving object with unity speed). Note that, in general, it is not necessary to mention the origin of the arc length (s = 0) since most concepts are deﬁned only in terms of the derivatives of a(s).
9 “Moving frame” U(0) and U(s). July 6, 2004 9:29 WSPC/Book Trim Size for 9in x 6in chap02 Diﬀerential Geometry of Array Manifold Curves 29 Properties of the frame matrix F(s). 3 Frame Matrix and Curvatures The question is, how the frame matrix F(s), at the running point s, is related to the curvatures of the manifold attached to this point. The initial part of the mechanism to answer this question is to deﬁne the coordinate vectors in CN by exploiting and extending the procedure used in real spaces where a basis of up to N orthonormal vectors is necessary to fully deﬁne a curve in RN .
However, this is by no means unique and, furthermore, this is not the most suitable parametrization for the study of the behavior of the array manifold . Therefore in Chapters 4 and 5, based on the material presented in Chapters 2 and 3, the following manifold-surface parametrizations • the (azimuth, elevation), or (θ, φ), parametrization • the cone-angles, or (α, β), parametrization are examined for non-linear arrays (2D and 3D geometries). The signiﬁcance of these parametrizations is demonstrated by a number of representative examples.
Differential geometry in array processing by Athanassios Manikas