By Athanassios Manikas
In view of the importance of the array manifold in array processing and array communications, the position of differential geometry as an analytical software can't be overemphasized. Differential geometry is especially limited to the research of the geometric homes of manifolds in third-dimensional Euclidean area R3 and in genuine areas of upper size.
Extending the theoretical framework to complicated areas, this precious publication provides a precis of these result of differential geometry that are of functional curiosity within the examine of linear, planar and 3-dimensional array geometries.
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Extra info for Differential geometry in array processing
D−1 (s), Starting with Eq. 13) and then using Eq. 15) July 6, 2004 9:29 WSPC/Book Trim Size for 9in x 6in Diﬀerential Geometry of Array Manifold Curves chap02 31 This is a ﬁrst order diﬀerential equation of the frame matrix F(s) with initial condition F(0) = Id . 16) where expm(·) denotes the matrix exponential. 16) provides the relationship between the frame matrix F(s) at the running point s and the curvatures (Cartan matrix) of the manifold attached to this point, which can also be used to rewrite Eq.
64) where p = 0 (s = 0) is taken along the array axis. 64) is associated with a real N -dimensional hyperhelix having the same length and identical curvatures with those of the complex N -dimensional manifold and this is formally described below. 65) having diﬀerential geometry properties equivalent to those of the complex N -dimensional manifold of the array. The vector areal (p) can be regarded as the real representation of the manifold of a symmetric array in RN . Thus, the manifolds of symmetrical arrays are shown to admit real representation.
The manifold is parametrized in terms of a directional parameter p with the sensor locations given by the vector r in half-wavelengths. 27) where r and v are two constant vectors and p is a generic parameter (parameter of interest). The main results are presented in the form of two theorems but, ﬁrstly, it is easy to show using Eq. e. 29) where the initial condition s(0) = 0 has been assumed. It is worth noting that for a linear array of N sensors with locations r (in units of half-wavelengths) the rate of change of the arc length is a non-linear function of the directional parameter p and depends on the norm of the vector of sensor locations.
Differential geometry in array processing by Athanassios Manikas