AD-753 125 REPRESENTATION AND ESTIMATION OF CYCLO- STATIONARY PROCESSES William Allen Gardner Massachusetts Uiiiversity Prepared for: Air Force Office of Scientific Research August 1972 DISTRIBUTED BY: National Technical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road, Springfield Va. 22151 AFOSR TR-72-2379 if UNIVERSITY OF MASSACEIUSETTS AMHERST, MASSACHUSETTS u'" LCDEC26 1T Approved for publia r Illail distribution unlimited. i~ REPRESENTATION AND FSTIMATION O1F CYCLOSTAT IONARY PROCESSES | / William Allen GardnerD T7 ~Ph.D., Unvrst of Masahset 'DA Amherst, Massachusetts August 1972 I secunts C~essficeflon DOCUMEHi CONTROL DATA- R &D Ii. thM ($SocuIV1 closafOsetian of title. bdy of abotroct at$ Inndexing annotatio ewS efnriotd -hm ove.all rtppert cl osaettled) I ORIG.INATING ACT&ViTY (CoipqrtM*8.54ho) 20. REPORT SECURITY CLASSSUFICATION Engineering Research Institute UNCLASSIFIED University of Massachusetts Zb. GRouP Amherst, Massachusetts 3 REPORT TITFLE REPRESENTATION AND ESTIMATION OF CYCLOSTATIONARY PROCESSES 2 4. •ESCAIPTIVE NOTES ( Tipe of •port ind nciusile datese Scientific Interim S AUTNORIS (Fitlet 0i1m0. middle Mitiol. Iset Reemv) William A. Gardner 6 REPORT DATE 70. TOT4L NO. OF PAGES 7b. NO. O- %v•s August 1972 353,I 63 Ce.CONTRACT OR GRANT NO Se. ORIGINATOR'S REPORT NUMOCRIS) 71-2111 6. ,PRO JECT NO0 AFOSR9 7 6 9 " C. 112F b.OTHEI REPORT N01M (An~y olhor ,rumbere tMet masy be aeeIl-sed c. 110F O. tle *pt d. 681304 ,,'o,.il 137 10 DISTRIBUTION STATEMENT AppL'ved for public release; distribution unlimited. II SUPPLEMENTARY NOYhS ]Ai1roIX. STON F4or~ceIt •lutlNGOffi IT.$Tv ce oScientificACTIVe., Research/hsM T1400 Wilson Boulevard TECH., OTHER IArlington, Virginia 22209 13 AGSTRACT Random signal processes which have been subjected to some form of repetitive operation such as sampling, scanning or multiplexing will usually exhibit statistical properties which vary periodically with time. In many cases, the repetitive operation is introduced intentionally to put the signal in a format which is easily manipulated and which preserves the time-position integrity of the events which the signal is representing. Familiar examples are radar antenna scanning patterns, raster formats for scanning video fields, synchronous miltiplexing schemes, and synchronizing and framing tech- niques employed in data transmission. In fact, in all forms of data transmiss•ion, it seems that some form of periodicity is imposed on the signal format. Random processes with statistical properties that vary periodically with time are encountered frequently, not only in electrical communication systems, but in biological systems, chemical processes, and studies concerr.cd with meteorology, ecology, and other physical and natural sciences. Systems analysts have tended, for the most part, to treat these "cyclostationary"' processes as though they were stationary. This is done simply by averaging the statistical parameters (mean, variance, etc.) over one cycle. This averaging is equivalent to modelling the time-reierence or-phase of the process as a random variable uniformly distributed over one cycle. Thi• type of analysis is appropriate in situations,! where the process is not observed in synchronism with its periodic structure. However, in a receiver that is intended for a cyclostationary signal, there is usually provided a great deal of information--in the form of a synchronizing pulse-stream or a sinusoida' timing signal--about the exact phase of the signal format. Most systems are, in fact, inoperative without this information. DD N.1473 S. FSecusrity Cte~sication ACKNOWLEDGEMENT U I would like to express my gratitude to Professor L.E. Franks for suggesting to me the area of research to which this report is devoted. I would also like to acknowledge pFrtial support of the research reported herein by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-71-2111. U U U II Li Iiv ABSTRACT Random signal processes which have been subjected to some form of 5 repetitive operation sL(h as sampling, scanning or multiplexing will usually exhibit statistical properties which vary periodically with time. In many cases, the repetitive operation is introduced intentionally to [ put the signal in a format which is easily manipulated and which preserves the time-position integrity of the events which the signal is representing. j Familiar examples are radar antenna scanning patterns, raster formats for scanning video fields, synchronous multiplexing schemes, and synchronizing and framing techniques employed in data transmission. In fact, in all L forms of data transmission, it seems that some form of periodicity is imposed on the signal format. processes with statistical properties that vary periodically S1Random with time are encountered frequently, not only in electrical communicatiol1 systems, but in biological systems, chemical processes, and studies 71. concerned with meteorology, ecology, and other physical and natural sciences. Systems analysts have tended, for the most part, to treat these T "cyclostationary" processes as though they were stationary. This is done simply by averaging the statistical parameters (mean, variance, etc.) over one cycle. This averaging is equivalent to modelling the time-reference or phase of the process as a random variable uniformly distributed over one cycle. This type of analysis is appropriate in situations where the T process is not observed in synchronism with its periodic structure. However, in a receiver that is intended for a cyclostationary signal, there is usually provided a great deal of information--in the form of a I |v synchronizing pulse-stream or a sinusoidal timing signal--about the exact phase of the signal format. Most systems are, in fact, inoperative without this information. The first chapter of this dissertation-is introductory and features IIa detailed historical account of the meager development and application of the theory--still in its infancy--'of cyclostationary processes. (as it jappears in the engineering literature). The second chapter is an extensive treatment of the topics of 14 transformation, generation, and modelling of cyclostationary processes; iiand, among other things, serves to introduce a large number of models for cyclostationary processas. These models are used throughout the dissertation for illustrating various theoretical results. The third chapter is an in-depth treatment of series representations [Ifor cyclostationary processes, and their autocorrelatior -unctions, and other periodic kernels. These representations are apriied to the problems of analysing cyclostationary processes, solving linear integral equations with periodic kernels, realizing periodically time-varying linear systems, and defining a generalized Fourier transform for cyclostationary (and I1; stationary) processes. The fourth chapter addresses itself to the problem of least-mean- squared-error linear estimation (optimum filtering) of cyclostationary processes, and employs the representations of Chapter III to obtain solutions, and the models of Chapter II to illustrate these solutions. Previous analyses of optimum filtering operations have assumed the stationary model for these processes and result in time-invariant filters. One of the major results in this thesis is the demonstration of the improvement in 1 ,vi I performance that can be obtained by recognizing that--by virtue of timing information at the receiver--the received process is actually cyclostationary and the optimum filter is a periodically time-varying system. Numerous illustrative examples including amplitude-modulation, 3 frequency-shift-keying, pulse-amplitude-modulation, frequency-division- multiplexing, and time-division-multiplexing are worked out in detail I and include realizations of the optimum time-varying filters. I U U I; IT i (Vii 5 TABLE OF CONTENTS' I Title Page i I Acceptance Page ii Acknowledgement iii I Abstract iv List of Figures ix I. INTRODUCTION 1 u1.Motivation and Brief Description 1 2. Definition of Cyclostationarity 5 3. Hiszorical Notes 12 4. Preliminary Comments on Mathematical Rigor 17 II. TRANSFORMATION, GENERATION, AND MODELING OF CYCLOSTATIONARY PROCESSES 25 1. Introduction 25 2. Linear Transformations 27 Multi-dimensional Linear Transformations S3. (Scanning) 52 4. Nonlinear Transformations 57 S. Random Linear Transformations 66 6. Random Multi-dimensional Linear Transformations (Random Scanning) 91 7. Nonlinear Random Transformations (Jitter) 93 A Viii 111. SERIES REPRESENTATIONS FOR CYCLOSTMONARY PROCESSES (AND THEIR AUTOCORRELATION FUNCTIONS) 114 1. Introduction 114 2. Translation Series Representations 119 3. Harmonic Series Representations 165 4. Fourier Series Representation for Autocorrelation Functions 179 IV. LEAST-MEAN-SQUARED-ERROR LINEAR ESTMATION OF CYCLOSTATIONARY PROCESSES 210 1. Introduction 210 2. The Orthogonality Condition 210 3. Optimum Time-Invariant Filters for Stationary Processes 226 4. Optimum Time-Invariant Filters for Cyclostationary Frocesses 230 S. Optimum Time-varying Filters for Cyclostationary Process 233 6. Examples of Optimum Filters and Improvements in Performance 266 V. SUMARY 324 1. Summary of Chapters II, III, IV 325 2. Suggested Topics for Further Research 344 Bibliography 348 U ix I 'I LIST OF FIGURES Figure Page (1-1) Venn diagram for random processes 19 S(1-2) Sampled and held process 20 (1-3) Autocorrelation function for a sampled and held process 21 S(1-4) Time-division-multiplexed signal 22