Digital Signal Processing Reference
In-Depth Information
rithm [
116
] extension on Cartan-Hadamard manifold and on Fréchet Metric spaces.
This new mathematical framework will allow developing concept of Ordered Statis-
tic (OS) for Hermitian Positive Definite Covariance Space/Time Toeplitz matrices
or for Space-Time Toeplitz-Block-Toeplitz matrices. We will then define Ordered
Statistic High Doppler Resolution CFAR (OS-HDR-CFAR) and Ordered Statistic
Space-Time Adaptive Processing (OS-STAP). This approach is based on the exis-
tence of a center of mass in the large for manifolds with non-positive curvature that
was proven and used by Cartan back in the 1920s [
35
]. The general case was employed
by Calabi in an unpublished note. In 1977, Karcher [
68
] has proposed intrinsic flow
to compute this barycenter, that we adapt for covariance matrices. This geometric
foundation of Radar Signal Processing is based on general concept of Cartan-Siegel
domains [
36
,
102
,
103
]. We will then give a brief history of Siegel domains studies
in Europe, Russia and China. In 1935, Cartan [
36
] proved that irreducible homo-
geneous bounded symmetric domains could be reduced to six types, included two
exceptional ones. Four non-exceptional Cartans domains are now called classical
models, and their extension by Siegel are considered as the higher dimensional ana-
logues of the Poincaré Unit Disk [
98
] in the complex plane. After these seminal
work of Cartan, in the framework of Sympletic Geometry [
102
,
103
], Siegel has
introduced first explicit descriptions of symmetric domains, where the realization of
bounded domains as unbounded domains played fundamental role (for an important
class of them, these unbounded domains are Siegel domains of the first kind, with
important particular case of Siegel Upper Half Plane). In 1953, Hua [
58
] obtained
the orthonormal system and the Bergman/Cauchy/Poisson kernel functions for each
of the four classical domains using group representation theory.
Cartan proved that all bounded homogeneous complex domains in dimension
2 and 3 are symmetric and conjectured that is true for dimension greater than 3.
Piatetski-Shapiro [
96
], after Hua works, has extended Siegel description to other sym-
metric domains and has disproved the Cartan conjecture that all transitive domains
are symmetric with a counter example. In parallel, Borel showed that if in a bounded
homogeneous region a semi-simple Lie group operates transitively, then that region
is symmetric. These results were strengthened by Hano and obtained by Koszul [
40
,
72
,
73
] who also studied affinely homogeneous regions that are fundamental for
Information Geometry and real Hessian or complex Kählerian geometries (see in
topic [
101
], Koszul's references inside). Piatetski-Shapiro introduced affinally gen-
eral definition of a Siegel domain of the second kind (all symmetric domains allow
a generalization of Siegel tube domains), and has proved in 1963 with Gindikin
and Vinberg that any bounded homogeneous domain has a realization as a Siegel
domain of the second kind with transitive action of linear transformation. In parallel,
Vinberg [
113
] worked on the theory of homogeneous convex cones, as fundamental
construction of Siegels domains (he introduced a special class of generalized matrix
T-algebras), and Gindikin worked on analytic aspects of Siegels domains. More
recently, classical complex symmetric spaces have been studied by Berezin [
26
]in
the framework of quantization. With Karpelevitch [
69
], Piatetski-Shapiro explored
underlying geometry of these complex homogeneous domains manifolds, and more
especially, the fibration of domains over components of the boundary. Let a bounded
