Continuous wavelet transform and linear time-frequency transforms are coefficients of continuous unitary group representations of the affine and Heisenberg groups. Many properties of these transforms that are important for wideband radar and sonar signal processing follow directly from group representation theory. These properties include volume invariance and variance of narrowband and wideband ambiguity functions and wavelet transform domain implementations of detects and signal estimators. For several radar, sonar, and array processing applications, the basic definition of wavelet and time-frequency representations must be generalized by using unitary representations of other groups and using reproducing kernel Hilbert space (RKHS) inner products in the definition of the linear transforms. The general definition then leads to weighted continuous wavelet transforms where the RKHS is determined by a nonstationary covariance function; generalized wideband ambiguity functions also follow from this general definition along with other important generalizations that arise in wideband array processing and model based signal processing in complex scattering and propagation media. This paper presents the generalized wavelet transform along with the weighted wavelet transforms. The classical narrowband and wideband ambiguity functions are then special cases. The application of generalized transform to multidimensional transforms for space-time processing are also presented, along with the application to conformal groups for detections and estimation of accelerating scatterers.

Date Published: 26 March 1998
PDF: 8 pages
Proc. SPIE 3391, Wavelet Applications V, (26 March 1998); doi: 10.1117/12.304899

Published in SPIE Proceedings Vol. 3391:
Wavelet Applications V
Harold H. Szu, Editor(s)