Preface
Many processes in nature arise from the interaction of periodic phenomena with random phenomena. The results are processes which are not periodic, but whose statistical functions are periodic functions of time. These processes are called cyclostationary and are an appropriate mathematical model for signals encountered in telecommunications, radar, sonar, telemetry, astronomy, mechanics, econometric, biology. In contrast, the classical model of stationary processes considers statistical functions which do not depend on time. More generally, if different periodicities are present in the generation mechanism of the process, the process is called almost cyclostationary (ACS). Almost all modulated signals adopted in communications, radar, and sonar can be modeled as ACS. Thus, in the past twenty years the exploitation of almost-cyclostationarity properties in communications and radar has allowed the design of signal processing algorithms for detection, estimation, and classification that significantly outperform classical algorithms based on a stationary description of signals. The gain in performance is due to a proper description of the nonstationarity of the signals, that is, the time variability of their statistical functions.
In this book, mathematical models for two general classes of nonstationary processes are presented: generalized almost-cyclostationary (GACS) processes and spectrally correlated (SC) processes. Both classes of processes include cyclostationary and ACS processes as special cases. SC and GACS processes are appropriate models for the received signal in mobile communications or radar scenarios when the transmitted signal is ACS and the propagation channel is a (possibly multipath) Doppler channel due to the relative motion between transmitter, receiver, and/or surrounding scatters or targets. SC processes are shown to be useful in the description of processes encountered in multirate systems and spectral analysis with nonuniform frequency spacing. GACS processes find application in the description of communications signals with slowly varying parameters such as carrier frequency, baud rate, etc.
The problem of statistical function estimation is addressed for both GACS and SC processes. This problem is challenging and of great interest at the applications level. In fact, once the nonstationary behavior of the observed signal has been characterized, statistical functions need to be estimated to be exploited in applications. The existence of reliable statistical function estimators for ACS processes is one of the main motivations for the success of signal processing algorithms based on this model. The results presented in this book extend most of the techniques used for ACS signals to the more general classes of GACS and SC signals. Mean-square consistency and asymptotic Normality properties are proved for the considered statistical function estimators. Both continuous-and discrete-time cases are considered and the problem of sampling and aliasing is addressed. Extensive simulation results are presented to corroborate the theoretical results.
How to use this book
The book is organized so that it can be used by readers with different requirements. Chapter 1 contains background material for easy reference in the subsequent chapters. Chapters 2 and 4 contain the main results presented in the form of theorems with sketches of proofs and illustrative examples. Thus, these chapters can be used by the non-specialist who is only interested in recipes or results and wants to grasp the main ideas. Each of these two chapters is followed by a chapter containing complements and proofs (Chapters 3 and 5). Each proof is divided into two parts. The first part consists of the formal manipulations to find the result. This part is aimed at advanced readers with a background of graduate students in engineering. The second part of the proof consists of the justification of the formal manipulations and is therefore aimed at specialists (e.g., mathematicians).
Book outline
In Chapter 1, the statistical characterization of persistent (finite-power) nonstationary stochastic processes is presented. Both strict-sense and wide-sense characterizations are considered. Harmonizability and time-frequency representations are treated. Definition and properties of almost periodic functions are provided. A brief review of ACS processes is also presented. The chapter ends with some properties of cumulants.
In Chapter 2, GACS processes are presented and characterized. GACS processes have multivariate statistical functions that are almost-periodic function of time. The (generalized) Fourier series of these functions have both coefficients and frequencies, named lag-dependent cycle frequencies, which depend on the lag shifts of the processes. ACS processes are obtained as special case when the frequencies do not depend on the lag parameters. The problems of linear filtering and sampling of GACS processes are addressed. The cyclic correlogram is shown to be, under mild conditions, a mean-square consistent and asymptotically Normal estimator of the cyclic autocorrelation function. Such a function allows a complete second-order characterization in the wide-sense of GACS processes. Numerical examples of communications through Doppler channels due to relative motion between transmitter and receiver with constant relative radial acceleration are considered. Simulation results on statistical function estimation are carried out to illustrate the theoretical results. Proofs of the results in Chapter 2 are reported in Chapter 3.
In Chapter 3, complements and proofs for the results presented in Chapter 2 are reported. Each proof consists of two parts. The first part contains formal manipulations that lead to the result. The second part contains the justifications of the mathematical manipulations of the first part. Thus, proofs can be followed with two different levels of rigor, depending on the background and interest of the reader.
In Chapter 4, SC processes are presented and characterized. SC processes have the Loève bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane. ACS processes are obtained as a special case when the curves are lines with a unit slope. The problems of linear filtering and sampling of SC processes are addressed. The time-smoothed and the frequency-smoothed cross-periodograms are considered as estimators of the spectral correlation density. Consistency and asymptotic Normality properties are analyzed. Illustrative examples and simulation results are presented. Proofs of the results in Chapter 4 are reported in Chapter 5.
In Chapter 5, complements and proofs for the results presented in Chapter 4 are reported. The system used is the same as in Chapter 3.
In Chapter 6, the problem of signal modeling and statistical function estimation is addressed in the functional or fraction-of-time (FOT) approach. Such an approach is an alternative to the classical one where signals are modeled as sample paths or realizations of a stochastic process. In the FOT approach, a signal is modeled as a single function of time and a probabilistic model is constructed by this sole function of time. Nonstationary models that can be treated in this approach are discussed.
In Chapter 7, applications in mobile communications and radar/sonar systems are presented. A model for the wireless channel is developed. It is shown how, in the case of relative motion between transmitter and receiver or between radar and target, the ACS transmitted signal is modified into a received signal with a different kind of nonstationarity. Conditions under which the GACS or SC model are appropriate for the received signal are derived.
In Chapter 8, citations are classified into categories and listed in chronological order.