direction of arrival estimation and tracking of narrowband and wideband signals

182 Pages·2013·2.89 MB·English

Purdue University
Purdue e-Pubs
ECE Technical Reports Electrical and Computer Engineering
3-1-1995
DIRECTION OF ARRIVAL ESTIMATION
AND TRACKING OF NARROWBAND AND
WIDEBAND SIGNALS
A. Satish
Purdue University School of Electrical Engineering
Rangasami L. Kashyap
Purdue University School of Electrical Engineering
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Satish, A. and Kashyap, Rangasami L., "DIRECTION OF ARRIVAL ESTIMATION AND TRACKING OF NARROWBAND AND
WIDEBAND SIGNALS" (1995).ECE Technical Reports.Paper 112.
http://docs.lib.purdue.edu/ecetr/112
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additional information.
D ARRIVAL
IRECTION OF
E T
STIMATION AND RACKING OF
N WIDEBAND
ARROWBAND AND
S
IGNALS
TR-EE 95-5
MARCH 1995
DIRECTION OF ARRIVAL ESTIMATION AND TRACKING OF
NARROWBAND AND WIDEBAND SIGNALS'
A. Satish and Rangasami L. Kashyap
School of Electrical Engineering
1285 Electrical Engineering Building
Purdue University
West Lafayette, IN 47907-1285
l~hisw ork was supported by the Innovation Science and Technology (IST) Program of the BMDO
monitored by the Office of Naval Research under contract ONR N00014-85-K-0611 and ONR N00014-91-
5-4 126.

TABLE OF CONTENTS
Page
LIST OF TABLES .................................................................................................... ix
...................................................................................................
LIST OF FIGURES xi
..............................................................................................................
ABSTRACT xix
1. INTRODUCTION AND OVERVIEW ................................................................ 1
.........................................................
1.1 Direction of Amval (DOA) Estimation 2
.............................................................................
1.1.1 Narrowband signals 2
1.2 Wideband Signals. DOA Estimation and Tracking ...................................... 5
1.3 Multiple Signal Tracking and Data Association ............................................ 6
1.4 Layout of the Thesis. ...................................................................................... 10
2. MAXIMUM LIKELIHOOD ESTIMATION AND CRAMER-RAO BOUNDS
OF DIRECTION OF ARRIVAL PARAMETERS OF A LARGE SENSOR
ARRAY ................................................................................................................. 11
2.1 Introduction .................................................................................................... 11
2.1.1 Outline of the Problem ........................................................................ 11
........................................................................................
2.1.2 Contribution 12
2.1.2.1 ML estimation method ........................................................... 12
2.1.2.2 Cramer-Rao lower bounds ..................................................... 13
. .
2.1.2.3 Resolution cntenon ................................................................ 13
2.1.3 Chapter overview ................................................................................. 13
2.2 Signal Model and Proposed Estimation Method ........................................... 14
Page
...............................................................................................
2.2.1 Signal model 14
.........................................................
2.2.2 Maximum likelihood estimation 15
2.2.3 Proposed likelihood expression ........................................................... 16
...........................................................................
2.2.3.1 Theorem2.1 17
2.2.4 Relation of proposed likelihood with other criterion functions .......... 18
2.2.5 Benefits of making approximation ...................................................... 18
2.2.6 Necessary conditions for maximization .............................................. 18
2.2.7 Estimation of signal power and noise variance ................................... 19
2.2.7.1 First order expressions ........................................................... 19
.......................................................
2.2.7.2 Second order expressions 19
2.2.7.3 Closed form expressions for estimators. ............................... 20
2.3 Explicit Expressions for Cramer-Rao Bounds on the DOA Estimates ..........
...........................................................................................
2.3.1 Motivation
2.3.2 General form of the Fisher information mamx ...................................
2.3.2.1 Notation and Symbols ............................................................
2.3.3 Expressions for the CR bounds ...........................................................
2.3.3.1 Theorem 2.2 ...........................................................................
2.3.3.2 Observations from first order bounds ....................................
2.3.3.3 Theorem2.3 ...........................................................................
2.3.3.4 Observations from second order bounds. ...............................
2.4 Properties of ML Estimates ..........................................................................
2.5 Performance of Proposed Method .................................................................
2.5.1 Experimental results ............................................................................
2.5.2 Comparison with stochastic maximum likelihood ..............................
2.5.3 Comparison with signal subspace methods .........................................
2.6 Discussion on the Cramer - Rao Bounds .......................................................
2.6.1 Comparison of Fisher information matrices and computation of
....................................................................................................
CRB
2.6.2 Effect of parameters on first and second order bounds .......................
2.6.3 Summary of observations. ...................................................................
2.7 DOA Resolution Criterion .............................................................................
2.7.1 Description of the criterion .................................................................
2.7.2 Illustration ...........................................................................................
2.7.3 Usefulness of the proposed criterion. ..................................................
Page
2.8 Summary ........................................................................................................ 39
.
3 A MAXIMUM LIKELIHOOD APPROACH FOR ESTIMATING TRACK
............................................
PARAMETERS OF NARROWBAND SIGNALS 41
....................................................................................................
3.1 Introduction
3.1.1 Motivation for the tracking problem ..................................................
...........................
3.1.2 Existing approaches for angle parameter estimation
3.1.3 Proposed method for track parameter estimation ................................
3.1.4 Crux of the tracking problem ..............................................................
3.1.5 Subspace methods and estimate association .......................................
3.1.6 Current approaches ..............................................................................
........................................................................................
3.1.7 Contribution
3.2 Signal Model and Estimation Method ..........................................................
........................................................................................
3.2.1 Signal model
3.2.1.1 Fresnel Approximation .........................................................
.........................................................................
3.2.2 Likelihood Expression
3.2.2.1 Simplification of likelihood expression .................................
.......................................
3.2.2.2 Upshot of likelihood simplification
3.3. Development of the Proposed Tracking ALgori thm (TAL) .........................
3.3.1 Parameter Estimation and Estimate Association .................................
3.3.2 Maximum Likelihood estimation of DOA and range parameters ......
3.3.3 The proposed Tracking ALgorithm (TAL) ........................................
3.3.4 Features of proposed method ..............................................................
.................................................................
3.4 Performance of Proposed Method
3.4.1 Comparison with Swindlehurst and Kailath's method .......................
...................................................
3.4.2 Comparison with Sword's algorithm
......................................................................
3.4.3 Crossing target scenario
3.4.3.1 Experiment 1 (two targets, constant velocity, high SNR &
small sampling interval) ........................................................
3.4.3.2 Experiment 2 (two targets moving with constant velocity,
low SNR and large sampling interval) ..................................
3.4.3.3 Experiment 3 (two targets and an interfering decoy). ............
........
3.4.3.4 Experiment 4 (three targets with parabolic trajectories)
Page
..................................
3.4.3.5 Experiment 5 (statistical performance) 62
..........................................................................................
3.4.4 Extensions 70
3.5 Asymptotic Cramer-Rao Lower Bound (CRB) Expressions for the Track
.....................................................................................................
Parameters
........................................................................................
3.5.1 Theorem3.1
. .......................................................................
3.5.1.1 Corollary 3 1
3.6 Information on System Parameters from CR Bounds ....................................
3.6.1 Theorem 3.2 ........................................................................................
...........................................................................................
3.6.2 Discussion
3.6.3 Effect of angle on the CR bounds .......................................................
3.6.4 Effect of array spacing ........................................................................
.....................................................................................................
3.7 Conclusions
.
4 A MAXIMUM LIKELIHOOD APPROACH FOR TRACKING MULTIPLE
WIDEBAND SIGNALS .................................................................................... 79
4.1 Introduction ....................................................................................................
..................................................................................................
4.2 Signal Model
......................................................
4.2.1 Narrowband signal approximation
......................................................................
4.2.2 Wideband Signal model
4.3 Maximum Likelihood Parameter Estimation .................................................
............................................................
4.3.1 Direction of amval estimation
4.3.2 Estimation of Spectral density matrix .................................................
4.4 Wideband Signal Model and Estimate Association .......................................
4.5 Bayes Classification to get Updated Target Estimates ..................................
4.6 The Proposed Wideband Tracking Algorithm ...............................................
4.7 Performance of Proposed Method .................................................................
4.7.1 A simulation result ..............................................................................
4.7.2 Discussion ...........................................................................................
4.8 Conclusion .....................................................................................................
5. ESTIMATION OF SINGULARITIES FOR INTERCEPT POINT
.................................................................................................
FORECASTING 91
. vii .
Page
....................................................................................................
5.1 Introduction 91
5.1.1 Focus of the problem ........................................................................ 92
......................................................................................................
5.2 Motivation
.................................................
5.2.1 Choice of curve fit for data modeling
5.3 Direction of Amval Estimation .....................................................................
...........................................................................................
5.4 Chapter overview
5.5 Algorithm for Estimation of Singularity (AES). ...........................................
5.5.1 Singularity estimation .........................................................................
5.5.2 Recursive least squares estimation ......................................................
.....................................
5.6 The Overall Algorithm for Intercept Point Tracking
5.7 Performance of Proposed Method .................................................................
5.7.1 Experimental results ............................................................................
5.7.1.1 Effect of DOA spacing on intercept point estimates .............
...........................................................................................
5.7.2 Discussion
5.7.2.1 Drawbacks of the method ......................................................
5.8 Summary ........................................................................................................ 107
.
6 TRACKING DIRECTION OF ARRIVAL BY SINGULARITY
....................................................................................................
ESTIMATION 109
6.1 Current Approaches for Data / Estimate Association ....................................
........................................................................................
6.2 Proposed Approach
6.3 Overview of Proposed Method ......................................................................
6.3.1 Direction of arrival estimation ............................................................
6.3.2 Motivation for intercept point forecasting ..........................................
...............................................................................
6.3.3 Choice of curve fit
..............................................................
6.3.4 Tracking algorithm summary
..........................................................
6.4 Singularity Estimation Method (SEM)
6.4.1 Estimation of the intercept point .........................................................
.......................................................
6.5 Algorithm for Data Association (ADA)
6.5.1 Forming initial estimated tracks (Algorithm AA ) ..............................
6.5.2 Selection of appropriate measurement vectors (Algorithm BB) .........
6.5.3 Cross-over detection from intercept point forecast (Algorithm CC) .
6.5.3.1 Idea behind Algorithm CC .....................................................
Page
6.5.3.2 Algorithm CC ....................................................................... 121
6.5.4 Evidence combination for linking (Algorithm DD) ............................ 123
6.5.4.1 Structure of the Link Matrix .................................................. 123
6.5.4.2 Formation of the Ordering Matrix ......................................... 123
6.5.4.3 Forming the link matrix from the ordering matrix ................. 124
6.6 Proposed Tracking Algorithm. ....................................................................... 124
6.7 Performance of Proposed Method ................................................................. 125
............................................................................
6.7.1 Experimental results 125
..........................................................................
6.7.1.1 Experiment 1 125
..........................................................................
6.7.1 -2 Experiment 2
6.7.1.3 Experiment 3 ..........................................................................
6.7.2 Discussion ...........................................................................................
6.7.2.1 Computational complexity of data association ......................
6.8 Conclusion .....................................................................................................
7. CONCLUDING REMARKS ...............................................................................
..........................................................................................................
REFERENCES
APPENDICES ...........................................................................................................
APPENDIX I: PROOF OF THEOREM 2.1 ......................................................
APPENDIX 11: PROOF OF COROLLARY 2.1 .................................................
APPENDIX 111 ......................................................................................................
APPENDIX IV: PROOF OF THEOREM 2.2 .....................................................
IV .1 Determination of Qpp. .....................................................
IV . 2 Determination of Q, .....................................................
APPENDIX V: PROOF OF THEOREM 2.3 .....................................................
V. 1 Determination of Qpp.. .....................................................
V.2 Determination of Q, .......................................................
APPENDIX VI: TRANSFER VECTORS AND THEIR DERIVATIVES FOR
SIMPLIFYING EQUATIONS (3.31 ) . (3.34) ......................
....................................................
APPENDIX VII: PROOF OF THEOREM 3.1
VII.1 Determination of QL ...................................................
k ...................................................
VII.2 Determination of Qn
a",
VII.3 Determination of ...................................................