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Author Q&A: Cyclostationary Processes and Time Series
What got you interested in cyclostationarity?
I became aware of cyclostationarity in 1989. It was the subject of my Master thesis discussed in 1990 and my PhD thesis discussed in 1994. I was hit by the elegance of the mathematical model. It is suitable to describe processes arising from the interaction of periodic and random phenomena. These processes, even if not periodic, have statistical functions that are periodic functions of time.
How has it evolved over the years?
Cyclostationary (or periodically correlated) processes have been introduced in the Russian literature in the early 1960s and then extensively treated in the pioneering works of Harry L. Hurd (1969) and William A. Gardner (1972). Significant advances of theory and applications of cyclostationarity were made in the 1990s as proved by the large number of papers published in those years on the subject. The interest in this model has grown up to now. Applications have been considered in many fields.
Why is it important today and what fields can benefit from it?
The cyclostationary and almost-cyclostationary models are ubiquitous statistical models for science data. In communications, radar, sonar, and telemetry, periodicities in the statistical functions arise from the modulation by random data of carriers or pulse trains. Periodicities are also due to operations such as sampling, scanning, multiplexing, and coding. In monitoring and diagnosis of mechanical machinery, periodicities in the statistics of vibro-acoustic signals are due to rotations of gears, belts, and bearings. In econometrics, the weekly opening and closing of markets and the season-depending supply and demand of products give rise to periodicities in the statistical functions of prices and exchange rates. In radio astronomy, periodicities are due to the revolution and rotation of planets and pulsation of stars. In human biological signals, cyclostationarity is due to heart pulsation or alternation of day and night. Hidden periodicities are present in genome sequences and diffusion processes of molecular dynamics. Evidence of spectral correlation, which is equivalent to cyclostationarity, is present in signals encountered in neuroscience.
An appropriate statistical modeling of the observed processes gives rise to significant advantages in performance of signal detection and prediction and parameter estimation.
What is the relevance of cyclostationarity to communication systems:
The design and analysis of communications systems benefits of the exploitation of cyclostationarity properties exhibited by almost all communications signals.
I discuss in Chapter 9, Communications Systems: Design and Analysis, the signal selectivity property of cyclostationarity-based algorithms. It allows one to separate a signal of interest from disturbance. The problems of minimum mean-squared error filtering (Wiener filtering), synchronization, and system identification and equalization are analyzed in detail. In addition, other applications including parameter estimation, source location, beamforming, and source separation are considered. Performance of cyclostationarity-based algorithms and their fundamental limits are discussed.
I also discuss statistical functions estimators, their performance, and their Matlab/Octave implementations. The problem of signal detection is also extensively treated.
You have just published a reference entitled Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations. Who is the book aimed at and how will it be useful to them?
The intended audience of the book is constituted by researchers in Statistical Signal Processing, Communications, and related disciplines. The book will also be useful to graduate students and PhD students in these fields and to telecommunications engineers and statisticians. The practitioner in these disciplines can use the book as a reference manual. For each chapter, results are reported as theorems and corollaries, proofs are in the last section. Thus, the reader can simply grasp the main ideas or, if needed, go in deep with the proofs. Each chapter ends with a summary where the main results are presented in a concise way. The implementation of the algorithms for cyclic spectral analysis is discussed in detail and Matlab/Octave code is provided. Applications of cyclostationarity in many fields are discussed and the relative references cited. An extensive list of references covers the majority of fields where cyclostationary signal analysis has been exploited.
What do you see as the future direction of cyclostationarity?
There are some limits to the applicability of the cyclostationary model that can be overcame by resorting to generalizations of cyclostationarity These generalizations constitute one of the future directions in the field. Limits to the exploitation of the cyclostationary model arise from possible irregularities or imperfections in the pace of the periodic phenomena underlying the analyzed signal, imperfections that become evident on large observation intervals. These irregularities or imperfections can be caused, for example, by (slight) time variability of timing parameters or by the relative motion between transmitter and receiver Two extensions of the cyclostationary model have been proposed for higher fidelity modeling of communications signals in high mobility scenarios: the generalized almost-cyclostationary signals and the spectrally correlated signals. More recently, a further generalization has been proposed for describing phenomena where statistical cyclicity in the data is irregular as, for example, in the biological signals. Specifically, the oscillatory almost cyclostationary signals have been shown to be a suitable model for processes exhibiting irregular cyclicity. This class of signals includes as a special case the amplitude-modulated and time-warped almost-cyclostationary signals. Part II of the book is devoted to the generalizations of cyclostationarity.
About the book
- Presents the foundations and developments of the second- and higher-order theory of cyclostationary signals
- Performs signal analysis using both the classical stochastic process approach and the functional approach for time series
- Provides applications in signal detection and estimation, filtering, parameter estimation, source location, modulation format classification, and biological signal characterization
- Includes algorithms for cyclic spectral analysis along with Matlab/Octave code
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