Ideal sequence design in time-frequency space. Applications to radar, sonar, and communication systems (Q2465374)

The book under review develops a new theory leading to systematic sequence design in time-frequency space. The essential tool of this interesting method is the finite Zak transform, which provides sparse representations for the discrete Fourier transform, cyclic convolution, and correlation. Using this Zak representation, the authors construct a large class of sequence sets satisfying pairwise ideal correlation. Note that the importance of continuous Zak transform in signal processing was first pointed out by \textit{A.J.E.M. Janssen} [Philips J. Res. 43, No. 1, 23--69 (1988; Zbl 0653.94002)]. After a short introduction, Chapters 2-3 present a brief review of matrix algebras, tensor products, and permutations. In Chapters 3-4, the discrete and fast Fourier transforms, cyclic convolutions, and correlations are developed. Chapters 6-7 introduce discrete chirps and the finite Zak transform. Zak space representations of shifts and discrete Fourier transforms are presented too. Chapters 8-9 develop the main results on Zak space representations of discrete chirps and of correlations of discrete chirps. It is shown that the finite Zak transform of a discrete chirp is the product of several copies of a permutation matrix and a diagonal matrix whose nonzero entries have absolute value one. Therefore the finite Zak transform plays a central role in this approach to sequence design. Chapters 10-11 analyze special permutation sequences. Chapter 12 investigates properties of modulation. In Chapter 13, the authors develop design strategies for constructing collections of permutation sequences with pairwise ideal correlations. Chapters 14-15 form the core of this book and develop the Zak space design framework of ideal sequences. In the last Chapter 16, open problems and directions of further research are discussed. This interesting and well-written book will be a useful reference text for graduate students, researchers, and engineers interested in radar/sonar signal processing or communication systems.