1. ANALYTIC SIGNALS AND RADAR PROCESSING Wayne M. Lawton Department of Mathematics National University of Singapore Block S14, 10 Kent Ridge Road Singapore 119260 wlawton@math.nus.edu.sg

2. REVIEW BASIC CONCEPTS OBJECTIVES Analytic signal representation Accuracy of analytic signal representation FORMULATE ISSUES Wigner-Ville distribution Warp transformations and radar echos Matched filtering, radar images, and ambiguity Signal design and ambiguity localization Fourier and Hilbert transform operators

3. FOURIER TRANSFORM OPERATOR Hilbert space Fourier transform operator and extended by continuity to of complex-valued functions with scalar product Unitary property

4. HILBERT TRANSFORM OPERATOR Hilbert transform operator f smooth, compact support and extended by continuity to Unitary, and

5. ANALYTIC SIGNAL REPRESENTATION Construct where is the identity operator Then, where functions,is Hardy subspace of functions f that satisfy is subspace of real-valued f admits an analytic extension to the upper half of the complex plane Furthermore, and

6. WIGNER-VILLE DISTRIBUTION Moyal Describes time/frequency distribution of signal energy

7. group of orientation preserving diffeomorphisms of ~ circle Cayley Unitary representation WARP TRANSFORMATIONS

8. Let an antenna (in an inertial frame) transmit a signal f that propagates at the speed of light c in the direction of a point scatterer whose distance (from the antenna) function d satisfies Then special relativity implies that the echo signal reflected by the point scatterer is proportional to U(g)f where RADAR ECHOS

9. The radar echo signal is the sum of echos from point scatterers MATCHED FILTERING, RADAR IMAGES, AND AMBIGUITY Thus the radar image, computed from matched filtering, is Since U is unitary, the radar image equals the a convolution

10. ACCURACY OF ANALYTIC SIGNAL REPRESENTATION Accuracy depends on transmitted signal f and warp U(x) Error equals the commutator This vanishes for all h if and only if x is a linear fractional transformation Error can be bounded using a wavelet expansion of f

11. SIGNAL DESIGN AND AMBIGUITY LOCALIZATION Ideally, the radar image equals the scattering measure Severe ambiguity constraints makes this impossible This requires However, a priori knowledge about adequate ambiguity localization. enables

12. REFERENCES Sanjay K. Mehta, Signal Design Issues for the Wigner Distribution Function and New Twin Processor for the Measurent of Target and/or Channel Structures, PhD Dissertation, University of Rochester, 1991 Elias Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, New Jersey, 1993