The two volumes of Signal Processing are based on lectures delivered during a six week program held at the IMA from June 27 to August 5, 1988. The first two weeks of the program dealt with general areas and methods of Signal Pro­ cessing. The problem areas included imaging and analysis of recognition, x-ray crystallography, radar and sonar, signal analysis and 1-D signal processing, speech, vision, and VLSI implementation. The methods discussed included harmonic anal­ ysis and wavelets, operator theory, algorithm complexity, filtering and estimation, and inverse scattering. The topics of weeks three and four were digital filter, VLSI implementation, and integrable circuit modelling. In week five the concentration was on robust and nonlinear control with aerospace applications, and in week six the emphasis was on problems in radar, sonar and medical imaging. Because of the large overlap between the various one-week and two-week seg­ ments of the program, we found it more convenient to divide the material somewhat differently. Part I deals with general signal process theory and Part II deals with (i) application of signal processing, (ii) control theory related themes. We are grateful to the scientific organizers: Tom Kailath (Chairman), Louis Aus­ lander, F. Alberto Grunbaum, J. William Helton, Pramod P. Khargonekar and Sanjoy K. Mitter. We are also grateful for the generous support given to the IMA program by the Office of Naval Research, the Air Force Office of Scientific Research, the Army Research Office and the National Security Agency.

I: Signal Processing Theory.- Wide-band ambiguity function and ax + b group.- On finite Gabor expansion of signals.- Two dimensional FFT algorithms on data admitting 90°-rotational symmetry.- Displacement structure for Hankel- and Vandermonde-like matrices.- Wavelet analysis and signal processing.- Estimating interesting portions of the ambiguity function.- The band method for extension problems and maximum entropy.- On the complexity of pattern recognition algorithms on a tree-structured parallel computer.- Soliton mathematics in signal processing.- Selective 'complex' reflectionless potentials.- Wavelets and frames.- Linear and polynomial methods in motion estimation.- Positive definite completions: A guide to selected literature.- An existence theorem and lattice approximations for a variational problem arising in computer vision.- Extension problems under the displacement structure regime.- Recent extension of the sampling theorem.- Generalized split Levinson, Schur, and lattice algorithms for estimation and inverse scattering.