Zak Transform And Its Application In ZCZ Sequence Design

The Zak transformation is essentially a Unitary transformation from L2(R)to L2(Q)where Q is the unit quare on R2.It has important applications in many fields,such as the Gabor representation problem in analysis,the study of coherent states in physics,the time-frequency representation of continuous signals in the field of signal processing,and Sequence design,etc.In 2013,Brozik proposed a suboptimal IF-ZCZ sequence set based on finite Zak transform[21].Inspired by Brozik's work,A new construction of optimal(KM,K,M-1)IF-ZCZ sequence set is given.The parameters of our newly constructed sequence set are the same as those in Brozik's construction.Brozik's construction is only optimal in some special cases,but our construction does not have this limitation.The structure of this thesis is organized as follows:Chapter One:Preliminaries.The basic concepts of the Zak transform and its discrete form are introduced.Their properties are systematically summarized.Frames commonly used when using the Zak transform to study function representation problems are briefly discussed.Chapter Two:Extension of the Zak transform on a space of generalized functions.The Boehmians,which is more general than the Schwartz distributions,is introduced.Operations between Boehmians and the convergence of a Boehmians sequence are introduced.Then,we explain how to extend the Zak transform to a generalized function space composed of Boehmians.Chapter Three:Construction of optimal Interference-Free ZCZ sequence set by finite Zak transform.This chapter focuses on the application of finite Zak transform in the design of ZCZ sequence set.First,the basic concepts of ZCZ sequence set and IF-ZCZ sequence set are introduced.Then,A new construction of IF-ZCZ sequence set is given.The correctness of this construction is demonstrated and it is optimal in the sense of Tang-Fan-Matsufuji bounds.Finally,the Zak spectra analysis and comparison of the known construction are given,the advantages of the sequence constructed by this method are summarized:sparse and highly structured Zak and Fourier spectra.Therefore,it is beneficial to reduce the calculation complexity related to the sequence operation.