Zastosowanie H-rozkładów i rozkładów macierzowo-wykładniczych do modelowania zakłóceń radiolokacyjnych

• Autorzy: Nałęcz M.

Warianty tytułu

EN

Radar clutter modelling using H-distributions and matrix-exponential distributions

• Języki publikacji: PL

Abstrakty

PL

W pracy zaproponowano dwie niezależne zaawansowane matematycznie metody opisu zakłóceń radiolokacyjnych i analizy systemów radiolokacyjnych z uwzględnieniem wpływu tych zakłóceń. Pierwsza z nich wynika ze spostrzeżenia, że powszechnie stosowane modele zakłóceń są opisywane tzw. H-rozkładami i można do ich analizy wykorzystać teorię H-funkcji Foxa. Druga z metod polega na zastosowaniu rozkładów macierzowo-wykładniczych (rozkładów ME), a w szczególności pewnej ich podklasy, a mianowicie rozkładów typu fazowego (rozkładów PH). Podejście to pozwala na stosowanie stosunkowo prostej algebry macierzowej do analizy wpływu zakłóceń. Na wstępie omówiono zasadę działania radaru impulsowego, podstawy statystycznej teorii detekcji i optymalną strukturę detektora, a także nowoczesne metody realizacji bloku przetwarzania sygnałów radiolokacyjnych. Przedstawiono klasyfikację rozkładów zakłóceń, ze szczególnym podkreśleniem roli tzw. rozkładów złożonych. Znane z literatury podejścia do rozkładów złożonych usystematyzowano i uporządkowano. Pokazano, że niemal wszystkie znane w literaturze rozkłady zakłóceń są przypadkami szczególnymi lub granicznymi tzw. uogólnionego rozkładu złożonego, a wiele z nich sprowadza się do jednego z tych przypadków szczególnych, a mianowicie do uogólnionego rozkładu gamma. Oba te rozkłady dokładnie przebadano dzięki potraktowaniu ich jako H-rozkłady. Wyprowadzono nieznane dotąd w probabilistyce jawne wzory na gęstość, dystrybuantę, momenty i transformatę Laplace' a gęstości dla wielu najważniejszych rozkładów amplitudy i mocy zakłóceń radiolokacyjnych. Dla wszystkich badanych rozkładów zaproponowano wspólną parametryzację, dzięki czemu możliwe stało się pokazanie w innym świetle znanych i odkrycie nowych związków między tymi rozkładami. Wiele miejsca poświęcono analizie asymptotycznych właściwości funkcji gęstości prawdopodobieństwa rozkładów amplitudy i mocy zakłóceń radiolokacyjnych. Zaproponowano zastosowanie teorii wartości ekstremalnych (Gumbela) do formalizacji właściwości tzw. ogona rozkładu. Zauważono związek gęstości uogólnionego rozkładu złożonego ze znaną z matematyki całką Faxena. Znaleziono nowe, prostsze od znanych dotąd, pełne rozwinięcie asymptotyczne tej całki w nieskończoności. Dzięki temu pokazano, że dowolny uogólniony rozkład złożony zachowuje się asymptotycznie identycznie jak uogólniony rozkład gamma, czyli należy do rozkładów typu I w klasyfikacji Gumbela. Z kolei wyprowadzenie pełnego rozwinięcia w zerze całki Faxena pozwoliło na znalezienie istotnych błędów w znanych do tej pory w literaturze przedstawieniach gęstości uogólnionego rozkładu złożonego w postaci szeregów nieskończonych. W dalszej części pracy przedstawiono pojęcie rozkładu macierzowo-wykładniczego i jego reprezentacji, a następnie omówiono najważniejsze właściwości takiego rozkładu znane z literatury. Podano nową metodę wyznaczania reprezentacji rozkładu ME na podstawie momentów za pomocą algorytmu QD Rutishausera. Zaproponowano nowatorską koncepcję połączenia teorii rozkładów kołowo symetrycznych i teorii rozkładów typu fazowego. W ramach tej koncepcji wyprowadzono nieznane dotąd wzory na funkcję charakterystyczną (wyrażającą się przez transformatę Hankela) rozkładu kołowo symetrycznego zmiennej losowej, której amplituda lub moc ma rozkład PH. Podano przykłady dowodzące efektywności algebraizacji obliczeń probabilistycznych związanych z rozkładami zakłóceń, możliwej dzięki zastosowaniu rozkładów typu fazowego. Przedstawiono także metodę dokładnej analizy rozkładów prawdopodobieństwa z uwzględnieniem kwantowania, za pomocą dyskretnych rozkładów typu fazowego, opartą na podanej w pracy reprezentacji PH słusznej dla dowolnego rozkładu dyskretnego. Wykorzystując teorię łańcuchów Markowa zaproponowano ujednolicone podejście do ciągłych i dyskretnych rozkładów PH przez ich reprezentacje grafami przepływu sygnału. Dla obu klas rozkładów podano nowe algorytmy otrzymywania takich grafów bezpośrednio na podstawie reprezentacji macierzowej rozkładu, z uwzględnieniem nietrywialnego wektora prawdopodobieństw początkowych. W skazano sposób wyznaczania funkcji charakterystycznej rozkładu (albo L-transformaty gęstości czy Z-transformaty funkcji prawdopodobieństwa) jako transmitancji tak otrzymanego grafu. Dzięki temu podejściu znaleziono interesujący związek między rozkładami ciągłymi i dyskretnymi typu fazowego przez przybliżenie Eulera (transformację FD). W zakończeniu rozprawy wskazano na możliwe kierunki dalszych badań i potencjalne obszary zastosowań uzyskanych wyników.

EN

In the monograph there are proposed two independent advanced mathematical methods for the radar clutter modeling and for the analysis of radar systems including the influence of the clutter. The first one is based on the observation that most commonly used radar clutter distributions belong to the class of the so-called H-distributions. Such distributions can be analyzed by using the theory of Fox H-functions. The other method utilizes matrix-exponential (ME) distributions, and especially their most important proper subset, namely phase-type (PH) distributions. This approach results in a simplified analysis of the influence of the clutter, due to a straightforward matrix algebra. The introductory chapters provide basic background concerning the principles of pulsed radar and statistical theory of optimal detection. The modern techniques for the implementation of the radar processors are also described. A detailed classification of radar clutter distributions is presented, with a special emphasis put on the role of compound distributions. Most approaches to the compound distributions known in the literature are compared and generalized. It is shown that almost all widely used clutter distributions are special or limiting cases of the Generalized Compound (GC) distribution. Furthermore, most of them can be regarded as one particular special case, namely the Generalized Gamma (Gr) distribution. Both GC and Gr distributions are thoroughly analyzed by treating them as H -distributions. The novel closed-form formulas for the density, distribution function, moments and the Laplace transform of the density are derived for a number of most popular radar clutter distributions. All investigated distributions share the common parameterization, which makes it possible to show some new relationships between these distributions. A significant part of the monograph is devoted to the asymptotic analysis of the densities of the amplitude and intensity of the radar clutter. The application of Gumbel's Extreme Values Theory (EVT) is proposed to formalize the properties of the tail of distributions. The author has found a close relationship between the density of the GC distribution and the Faxen integral. Due to the derivation of a new, closed-form, full asymptotic expansion of the Faxen integral at infinity it was shown that the tail of the GC distribution is the same as that of the Gr distribution, i.e., it belongs to the Type I distributions by Gumbel. The closed-form full asymptotic expansion of the Faxen integral at the origin was also derived, which made it evident that infinite series representations for the GC den sity known in the literature are erroneous. Regarding the second approach, the definition, basic properties and matrix representations of ME distributions are described, based on the available literature. The new method is given for computing the matrix representation based on the moments. This method utilizes a variant of the QD algorithm by Rutishauser. Anovel approach to Spherically Invariant Random Vectors (SIRV) via phase-type distributions is also proposed. New formulas based on the Hankel transform are derived for the characteristic function of the SIRV distribution, provided that the corresponding amplitude or intensity is PH-distributed. A number of examples are given that illustrate the advantages of the phase-type approach to the analysis of radar clutter distributions in radar systems. Moreover, the method for the exact analysis of probability distributions in digital systems, taking into account the quantization effects, is proposed. The method utilizes discrete PH distributions based on the representation of the arbitrary discrete distribution presented in the monograph. A unified approach to continuous and discrete PH distributions via signal flow graphs is proposed, based on the Markov chain theory. The algorithms for obtaining such graphs directly from the corresponding matrix representations are derived for both continuous and discrete PH distributions. The non-trivial initial probability vectors are allowed. The characteristic function (or L-transform of the density, or Z-transform of the probability function) can be computed as the transfer function of the graph, using the Mason rule. The unified approach resulted in obtaining an interesting relationship between continuous and discrete PH distributions via Euler's approximation (also known as the FD transformation). In the concluding section of the monograph, the directions for the future research and some potential application areas are pointed out.

Słowa kluczowe

PL

zakłócenia radiolokacyjne; modelowanie matematyczne

EN

mathematical models; radar clutter modelling

• Wydawca: Oficyna Wydawnicza Politechniki Warszawskiej
• Czasopismo: Prace Naukowe Politechniki Warszawskiej. Elektronika
• Rocznik: 2007
• Tom: z. 158
• Strony: 5--273
• Opis fizyczny: Bibliogr. 351, schem., wykr.

Twórcy

autor

Nałęcz M.

• Instytut Systemów Elektronicznych Politechnika Warszawska

Bibliografia

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