Optimal Filtering

by Fomin, V. N.

• ISBN13: 9780792357346
• ISBN10: 0792357345
• Format: Hardcover
• Copyright: 1999-05-01
• Publisher: Kluwer Academic Pub

Summary

This book considers methods for the optimal processing of random fields. In particular, it studies spatio-temporal filtering problems such as the problem of optimal signal detection (Bayes' approach) and estimating angles of arrival of local signals. The exposition of the problem of optimal filtering is presented with the help of insights from probability theory, functional analysis and mathematical physics. An algorithmic form of the net results facilitates computer-aided applications. Audience: This volume will be of interest to experts in the design of signal processing and theorists in functional analysis, probability theory, functional analysis and mathematical physics.

Table of Contents

Preface	xi
Fields and means of describing them	1	(42)
Regular fields	2	(5)
Preliminary information on fields	2	(1)
Fields as elements of Hilbert space	3	(3)
Stochastic fields	6	(1)
Generalized fields	7	(13)
Completion of Hilbert space	8	(1)
Fields as generalized elements of Hilbert space	9	(11)
Spatio-temporal fields and frequency-wave fields	20	(6)
Spatio-temporal fields	20	(2)
Frequency-wave fields	22	(4)
Stochastic discrete fields	26	(10)
Fields on discrete lattices	26	(4)
Multi-variable difference equations	30	(3)
`Cauchy problem' for regressive equation	33	(3)
Proofs of Lemmas and Theorems	36	(5)
Proof of Lemma 1.1	36	(1)
Proof of Lemma 1.2	36	(1)
Proof of Lemma 1.3	37	(1)
Proof of Lemma 1.4	38	(1)
Proof of Theorem 1.1	38	(2)
Proof of Lemma 1.5	40	(1)
Proof of Theorem 1.2	40	(1)
Bibliographical Comments	41	(2)
Models of continuous fields and associated problems	43	(118)
Fields in electrodynamics	45	(49)
Initial boundary value problem	46	(11)
Electrostatic boundary value problem	57	(13)
Electrodynamics of hollow systems	70	(24)
Acoustic fields	94	(26)
Acoustic waves in the world's ocean	94	(5)
Acoustic fields in cylindrical waveguides	99	(21)
Parametric vibrations of distributed systems	120	(19)
A general idea of parametric resonance	120	(5)
Examples of modeles of parametric vibrations	125	(7)
Collection of some results on parametric vibrations	132	(7)
Proof of Lemmas and Theorems	139	(17)
Proof of Lemma 2.1	139	(1)
Proof of Lemma 2.2	140	(1)
Proof of Lemma 2.3	140	(1)
Proof of Lemma 2.4	141	(1)
Proof of Lemma 2.5	142	(1)
Proof of Lemma 2.6	142	(1)
Proof of Theorem 2.1	143	(1)
Proof of Lemma 2.7	144	(1)
Proof of Lemma 2.8	144	(1)
Proof of Theorem 2.2	145	(1)
Proof of Lemma 2.9	146	(1)
Proof of Lemma 2.10	147	(1)
Proof of Lemma 2.11	147	(2)
Proof of Lemma 2.12	149	(1)
Proof of Lemma 2.13	149	(1)
Proof of Lemma 2.14	150	(1)
Proof of Theorem 2.3	150	(2)
Proof of Lemma 2.15	152	(1)
Proof of Theorem 2.5	153	(1)
Proof of Theorem 2.6	153	(1)
Proof of Theorem 2.7	153	(3)
Bibliographical Comments	156	(5)
Filtering of spatio-temporal fields	161	(60)
Linear filters and antenna arrays	161	(12)
Linear filtering of fields	161	(2)
Antenna arrays	163	(7)
Tunable filters with neuron type of structure	170	(3)
Signal optimal detection	173	(15)
Bayes' approach to the problem of decision making	173	(2)
Simplification of Bayes' decision rule	175	(5)
Bayes' decision rule for Gaussian signals	180	(5)
Factorization of the quadratic form operator	185	(3)
Estimation of angles of arrival of local signals	188	(30)
Signal to noise model of spatio-temporal signal	189	(6)
Solubility of the angles of arrival problem	195	(8)
Subspace rotation approach of signal parameter estimation	203	(3)
Moving antenna array	206	(4)
Adaptive filtering	210	(8)
Proofs of Lemmas and Theorems	218	(1)
Proof of Theorem 3.1	218	(1)
Proof of Lemma 3.1	219	(1)
Bibliographical Comments	219	(2)
Optimal filtering of discrete homogeneous fields	221	(38)
Optimal filtering of discrete homogeneous fields	221	(8)
Linear filter and mean square performance criterion	222	(1)
Filtering problem (stability and realizability)	223	(2)
Optimal filtering problem in `frequency terms'	225	(1)
Optimization of stationary filters	226	(1)
Optimization of stable non-stationary filters	227	(1)
Optimization of physically realizable filters	228	(1)
Synthesis of optimal physically realizable stationary filter	229	(7)
General scheme	229	(5)
Example: Optimal filtering of stationary time series	234	(2)
Optimal prediction of two-dimensional regressive fields	236	(7)
Optimal prediction scheme	236	(1)
Stable autoregressive equation	237	(1)
Separation of rational functions	238	(2)
Recurrence representation of optimal filter	240	(1)
Structure of optimal filter	240	(1)
Special case of unstable autoregressive equation	241	(2)
Multi-dimensional factorization and its attendant problems	243	(7)
Factorization of spectral density	245	(1)
Cepstrum in the factorization problem	246	(2)
Formative filter for homogeneous field	248	(2)
Proofs of lemmas and theorems	250	(8)
Proof of Theorem 4.1	250	(1)
Proof of Theorem 4.3	251	(1)
Proof of Theorem 4.4	252	(1)
Proof of Theorem 4.5	253	(1)
Proof of Lemma 4.1	253	(2)
Proof of Theorem 4.6	255	(3)
Bibliographical Comments	258	(1)
A Appendix: Fields in electrodynamics	259	(26)
A.1 Self-conjugate Laplace operator	259	(14)
A.1.1 Laplace operator in invariant subspace	260	(3)
A.1.2 Invariant subspaces of Laplace operator	263	(5)
A.1.3 Continuous spectrum of Laplace operator	268	(5)
A.2 Electrodynamic problem in tube domain	273	(6)
A.2.1 Eigenfields in tube domain	273	(4)
A.2.2 Example: Oscillations in rectangular resonator	277	(1)
A.2.3 Example: Rectangular semi-infinite waveguide	278	(1)
A.3 Proofs of Lemmas and Theorems	279	(4)
A.3.1 Proof of Lemma A.1	279	(1)
A.3.2 Proof of Lemma A.2	279	(1)
A.3.3 Proof of Lemma A.3	280	(1)
A.3.4 Proof of Lemma A.4	281	(1)
A.3.5 Proof of Lemma A.5	282	(1)
A.3.6 Proof of Lemma A.6	282	(1)
A.3.7 Proof of Theorem A.1	283	(1)
A.4 Bibliographical Comments	283	(2)
B Appendix: Spectral analysis of time series	285	(36)
B.1 Reconstruction of spectral densities	286	(11)
B.1.1 Quasi-stationary signals and their power spectra	286	(7)
B.1.2 Optimal estimation of power spectrum	293	(4)
B.2 Pade approximation	297	(5)
B.2.1 Pade approximation of analytic function	298	(2)
B.2.2 Pade approximation of spectral density	300	(2)
B.3 Identification of regressive equation	302	(8)
B.3.1 Optimal prediction	304	(4)
B.3.2 Estimation of coefficients of regressive equation	308	(2)
B.4 Proofs of Lemmas and Theorems	310	(7)
B.4.1 Proof of Lemma B.1	310	(1)
B.4.2 Proof of Theorem B.1	311	(1)
B.4.3 Proof of Theorem B.2	311	(1)
B.4.4 Proof of Theorem B.3	312	(1)
B.4.5 Proof of Lemma B.2	312	(1)
B.4.6 Proof of Theorem B.4	313	(1)
B.4.7 Proof of Lemma B.3	314	(1)
B.4.8 Proof of Lemma B.4	315	(1)
B.4.9 Proof of Lemma B.5	316	(1)
B.4.10 Proof of Lemma B.6	316	(1)
Bibliographical Comments	317	(4)
C Appendix: Spectral analysis of discrete homogeneous fields	321	(24)
C.1 Latticed cones and functions	321	(6)
C.1.1 Latticed cones	321	(2)
C.1.2 Latticed fields	323	(4)
C.2 Discrete fields	327	(6)
C.2.1 Generalized discrete fields	327	(2)
C.2.2 Stochastic fields	329	(4)
C.3 Latticed cone filters	333	(8)
C.3.1 Stable linear filters	333	(4)
C.3.2 Multi-variate analog of Pade approximation	337	(4)
C.4 Proofs of Lemmas and Theorems	341	(2)
C.4.1 Proof of Lemma C.1	341	(1)
C.4.2 Proof of Theorem C.1	342	(1)
C.4.3 Proof of Lemma C.2	342	(1)
C.4.4 Proof of Theorem C.2	343	(1)
C.5 Bibliographical Comments	343	(2)
References	345	(8)
Notation	353	(4)
Index	357