The book presents the necessary mathematical basis to obtain and rigorously use likelihoods for detection problems with Gaussian noise. To facilitate comprehension the text is divided into three broad areas – reproducing kernel Hilbert spaces, Cramér-Hida representations and stochastic calculus – for which a somewhat different approach was used than in their usual stand-alone context.
One main applicable result of the book involves arriving at a general solution to the canonical detection problem for active sonar in a reverberation-limited environment. Nonetheless, the general problems dealt with in the text also provide a useful framework for discussing other current research areas, such as wavelet decompositions, neural networks, and higher order spectral analysis.
The structure of the book, with the exposition presenting as many details as necessary, was chosen to serve both those readers who are chiefly interested in the results and those who want to learn the material from scratch. Hence, the text will be useful for graduate students and researchers alike in the fields of engineering, mathematics and statistics.
Author(s): Antonio F. Gualtierotti
Publisher: Springer
Year: 2016
Language: English
Pages: 1198
Tags: Probability Theory and Stochastic Processes; Functional Analysis; Information and Communication, Circuits
Front Matter....Pages i-xxxiv
Front Matter....Pages 1-1
Reproducing Kernel Hilbert Spaces: The Rudiments....Pages 3-123
The Functions of a Reproducing Kernel Hilbert Space....Pages 125-215
Relations Between Reproducing Kernel Hilbert Spaces....Pages 217-305
Reproducing Kernel Hilbert Spaces and Paths of Stochastic Processes....Pages 307-327
Reproducing Kernel Hilbert Spaces and Discrimination....Pages 329-430
Front Matter....Pages 431-431
Cramér-Hida Representations from “First Principles”....Pages 433-504
Cramér-Hida Representations via Direct Integrals....Pages 505-527
Some Facts About Multiplicity....Pages 529-702
Cramér-Hida Representations via the Prediction Process....Pages 703-791
Front Matter....Pages 793-794
Bench and Tools....Pages 795-850
Calculus for Cramér-Hida Processes....Pages 851-901
Sample Spaces....Pages 903-925
Likelihoods for Signal Plus “White Noise” Versus “White Noise”....Pages 927-960
Scope of the Signal Plus “White Noise” Model (I)....Pages 961-971
Scope of the Signal Plus “White Noise” Model (II)....Pages 973-992
Scope of Signal Plus “White Noise” Model (III)....Pages 993-1085
Likelihoods for Signal Plus Gaussian Noise Versus Gaussian Noise....Pages 1087-1160
Back Matter....Pages 1161-1176
