Table Of Contents

8. Bibliography on Cyclostationarity

8.4 Comprehensive Bibliography of Contributions to the Subject Fraction-of-Time Probability and Generalized Harmonic Analysis

This bibliography was created by the new Website Content Manager, Professor Antonio Napolitano on 2 August 2023. Unlike the bibliographies on pages 8.1 and 8.2, which are in reverse Chronological order, this one is in forward chronological order, and spans 110 years.

A. Einstein, “Method for the determination of the statistical values of observations concerning quantities subject to irregular fluctuations,” 1914, IEEE ASSP Mag., vol. 4, no. 4, p. 6, Oct. 1987.

A. Wintner, Zur Theorie beschrankten Bilinearformen, Math. Z. 30 (1929) 228–282.

N. Wiener, Generalized harmonic analysis, Acta Math. 55 (1930) 117–258.

H. Bohr, Kleinere Beitrage zur Theorie der fastperiodischen Funktionen. II, Det Kgl. Danske Videnskabernes Selskab, Mathematisk-Fysiske 10 (6) (1930) 12–17.

A. Wintner, Remarks on the ergodic theorem of Birkhoff, Proc. Natl. Acad. Sci. USA 18 (1932) 248–251.

A. Wintner, On the asymptotic repartition of the values of real almost periodic functions, Amer. J. Math. 54 (2) (April 1932) 339–345.

E.K. Haviland, On statistical methods in the theory of almost-periodic functions, Proc. Natl. Acad. Sci. USA 19 (1933) 549–555.

N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge University Press, London, 1933, Dover Publications, New York, 1958.

A. Wintner, On the asymptotic differential distribution of almost-periodic and related functions, Amer. J. Math. 56 (1/4) (1934) 401–406.

S. Bochner, B. Jessen, Distribution functions and positive definite functions, Ann. Math. 35 (2) (April 1934) 252–257.

H. Bohr, B. Jessen, Mean-value theorems for the Riemann zeta-function, Quart. J. Math. 5 (1934) 43–47.

E.K. Haviland, A. Wintner, On the Fourier–Stieltjes transform, Amer. J. Math. 56 (1/4) (1934) 1–7.

E.K. Haviland, A. Wintner, A note on the Kronecker–Weyl theorem, Amer. J. Math. 56 (1/4) (1934) 17–24.

B. Jessen, A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1) (July 1935) 48–88.

R. Kershner and A. Wintner, “On the asymptotic distribution of almost periodic functions with linearly independent frequencies,” Amer. J. Math., vol. 58, no. 1, pp. 91-94, 1936.

M. Kac and H. Steinhaus, “Sur les fonctions indépendantes (II),” Studia Mathematica, vol. 6, pp. 59–66, 1936.

J. Marcinkiewicz and A. Zygmund, “Sur les fonctions indépendantes,” Fundamenta Math., vol. 29, pp. 60-90, Jan. 1937.

J. Marcinkiewicz and A. Zygmund, “Quelques théorémes sur les fonctions indépendantes,” Fundamenta Math., vol. 7, pp. 104-120, 1938.

M. Kac, H. Steinhaus, Sur les foncions independantes IV, Studia Math. 7 (1938) 1–15.

P. Hartman, E. R. van Kampen, and A.Wintner, “Asymptotic distributions and statistical independence,” Amer. J. Math., vol. 61, no. 2, pp. 477-486, 1939.

H. Steinhaus, Sur les foncions inde´pendantes VI, Studia Math. 9 (1940) 121–132.

M. Frechet, Les fonctions asymptotiquement presque periodiques, Rev. Sci. 79 (1941) 341–354.

R.P. Agnew, M. Kac, Translated functions and statistical independence, Bull. Am. Math. Soc. 47 (2) (1941) 148–154.

P. Hartman and A. Wintner, “The (L2)-space of relative measure,” Proc. Nat. Acad. Sci. USA., vol. 33, pp. 128-132, May 1947.

H. Steinhaus, Sur les foncions inde´pendantes VII, Studia Math. 10 (1948) 1–20.

H.O.A. Wold, On prediction in stationary time series, Ann. Math. Statist. 19 (1948) 558–567.

J. Bass, Cours de Mathe´matiques, Tome III, Masson & Cie, Paris, 1971. W.M. Brown, Time statistics of noise, IRE Trans. Inform. Theory (December 1958) 137–144.

K. Urbanik, Effective processes in the sense of H. Steinhaus, Studia Math. 17(1958).

M. Kac, Statistical Independence in Probability, Analysis and Number Theory, The Mathematical Association of America, USA, 1959.

J. Bass, Suites uniformement denses, moyennes trigonometriques, fonctions pseudo-aleatoires, Bull. Soc. Math. France 87 (1959) 1–64.

H. Furstenberg, Stationary Processes and Prediction Theory, Princeton University Press, Princeton, NJ, 1960.

P. Masani, “Review: H. Furstenberg, Stationary processes and prediction theory,” Bull. Amer. Math. Soc., vol. 69, no. 2, pp. 195-207, 1963.

J. Bass, “Espaces de Besicovitch, fonctions presque-périodiques, fonctions pseudo-aléatories,” Bull. de la Société Mathématique de France, vol. 91, pp. 39-61, Jan. 1963.

E.M. Hofstetter, Random processes, in: H. Margenau, G.M. Murphy (Eds.), The Mathematics of Physics and Chemistry, vol. II, Van Nostrand, Princeton, NJ, 1964 (Chapter 3).

Y. Meyer, Le spectre de Wiener, Studia Math. 27 (1966) 189–201.

J.-P. Bertrandias, “Espaces de fonctions bornées et continues en moyenne asymptotique d’orde p,” Bull. Soc. Math. France (Mémories de La S. M. F.), vol. 5, pp. 3-106, Jan. 1966

Y.W. Lee, Statistical Theory of Communication, Wiley, New York, 1967.

W.M. Brown, C.J. Palermo, Random Processes, Communications, and Radar, McGraw-Hill, New York, 1969.

J. Bass, “Fonctions stationnaires. Fonctions de corrélation. Application à la représentation spatio-temporelle de la turbolence,” Annales de l’Inst. Henri Poincaré, Sect. B, vol. 5, no. 2, pp. 135-193, 1969.

J. Henniger, “Functions of bounded mean square, and generalized Fourier-stieltjes transforms,” Can. J. Math., vol. 22, no. 5, pp. 1016-1034, Oct. 1970.

P.P. Hien, Sur les mesures asymptotiques, Ph.D. Dissertation, Universite Paris VI, 1972.

J. Bass, “Stationary functions and their applications to the theory of turbolence: I. Stationary functions,” J. Math. Anal. Appl., vol. 47, no. 2, pp. 354-399, Aug. 1974.

J. Bass, “Stationary functions and their applications to the theory of turbolence: II. Turbolent solutions of the Navier-Stokes equations,” J. Math. Anal. Appl., vol. 47, no. 3, pp. 458#503, Sep. 1974.

P. P. Hien, “Mesure asymptotique définie par une fonction à valeurs dans Rn ou dans un espace vectoriel topologique,” Annales de l’Institut Henri Poincaré Sect. B, vol. 11, no. 1, pp. 23-107, 1975.

E. Pfaffelhuber, Generalized harmonic analysis for distributions, IEEE Trans. Inform. Theory IT-21 (November 1975) 605–611.

R.B. Davies, Inference from non-ergodic time series, Adv. Appl. Probab. 11 (2) (1979) 261–262 .

W.A. Gardner, Introduction to Random Processes with Applications to Signals and Systems, Macmillan, New York, 1985 (second ed., McGraw-Hill, New York, 1990).

W.A. Gardner, The spectral correlation theory of cyclostationary time series, Signal Processing 11 (July 1986) 13–36.

W.A. Gardner, Statistical Spectral Analysis: A Nonprobabilistic Theory, Prentice-Hall, Englewood Cliffs, NJ, 1987.

W.A. Brown, On the theory of cyclostationary signals, Ph.D. Dissertation, Department of Electrical Engineering and Computer Science, University of California, Davis, CA, 1987

W. Gardner, “Introduction to Einstein’s contribution to time-series analysis,” IEEE ASSP Mag., vol. 4, no. 4, pp. 4#5, Oct. 1987.

K. Fukuyama, “Some limit theorems of almost periodic function systems under the relative measure,” J. Math. Kyoto Univ. (JMKYAZ), vol. 28, no. 3, pp. 557–577, 1988.

J. Bass, “Fonctions de corrélation des fonctions pseudo-aléatories,” Annales de l’Inst. Henri Poincaré, Section B, vol. 25, no. 4, pp. 503-515, 1989

W.A. Gardner, Two alternative philosophies for estimation of the parameters of time series, IEEE Trans. Inform. Theory 37 (January 1991) 216–218.

W.A. Gardner, W.A. Brown, Fraction-of-time probability for time-series that exhibit cyclostationarity, Signal Processing 23 (June 1991) 273–292.

J.J. Benedetto, A multidimensional Wiener–Wintner theorem and spectrum estimation, Trans. Amer. Math. Soc. 327 (2) (October 1991) 833–852.

W.A. Gardner, An introduction to cyclostationary signals, in: W.A. Gardner (Ed.), Cyclostationarity in Communications and Signal Processing, IEEE Press, New York, 1994, pp. 1–90 (Chapter 1).

W.A. Gardner, C.M. Spooner, The cumulant theory of cyclostationary time-series, part I: foundation, IEEE Trans. Signal Process. 42 (December 1994) 3387–3408.

C.M. Spooner, W.A. Gardner, The cumulant theory of cyclostationary time-series, part II: development and applications, IEEE Trans. Signal Process. 42 (December 1994) 3409–3429.

J. J. Benedetto, Harmonic Analysis and Applications. New York, NY, USA: CRC Press, 1996

J. Mari, A counterexample in power signal space, IEEE Trans. Automat. Control 41 (January 1996) 115–116.

W. A. Gardner, “History and equivalence of two methods of spectral analysis,” IEEE Signal Process. Mag., vol. 13, pp. 20-23, Jul. 1996.

P. Hughett, Linearity and sigma-linearity in discrete-time linear shift-invariant systems, Signal Processing 59 (1997) 329–333.

P. Hughett, A representation theorem for local LSI operators on two-sided sequences, Signal Processing 63 (1997) 169–173.

P.M. Makila, J.P. Partington, T. Norlander, Bounded power signal spaces, in: Proceedings of the 36th IEEE Conference on Decision & Control, San Diego, CA, USA, December 1997.

P.M.Makila, J.P. Partington, T. Norlander, Bounded power signal spaces for robust control and modeling, SIAM J. Control Optim. 37 (1) (1998) 92–117.

P. Hughett, Representation theorems for semilocal and bounded linear shift-invariant operators on sequences, Signal Processing 67 (1998) 199–209.

A. B. Nobel, G. Morvai, and S. R. Kulkarni, “Density estimation from an individual numerical sequence,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 537-541, Mar. 1998

L. Izzo, A. Napolitano, The higher-order theory of generalized almost-cyclostationary time-series, IEEE Trans. Signal Process. 46 (November 1998) 2975–2989.

A.B. Nobel, First order predictive sequences and induced transformations, Technical Report No. 2367, Department of Statistics, University of North Carolina, Chapel Hill, NC, USA, 1999.

I. Sandberg, A note on the convolution scandal, IEEE Signal Process. Lett. 8 (7) (July 2001) 210–211.

J. Leskow, A. Napolitano, Quantile prediction for time series in the fraction-of-time probability framework, Signal Processing 82 (November 2002) 1727–1741.

L. Izzo, A. Napolitano, Linear time-variant transformations of generalized almost-cyclostationary signals, part I: Theory and method, IEEE Trans. Signal Process. 50 (December 2002) 2947–2961.

L. Izzo, A. Napolitano, Linear time-variant transformations of generalized almost-cyclostationary signals, part II: Development and applications, IEEE Trans. Signal Process. 50 (December 2002) 2962–2975.

P.M. Makila, On chaotic and random sequences, Physica D 198 (2004) 309–318.

A.B. Nobel, Some stochastic properties of memoryless individual sequences, IEEE Trans. Inform. Theory 50 (7) (July 2004) 1497–1505.

P.M. Makila, J.P. Partington, Least-squares LTI approximation of nonlinear systems and quasistationarity analysis, Automatica 40 (July 2004) 1157–1169.

H.L. Hurd, T. Koski, The Wold isomorphism for cyclostationary sequences, Signal Processing 84 (May 2004) 813–824.

C. Zhang, “New limit power function spaces,” IEEE Trans. Autom. Control, vol. 49, no. 5, pp. 763-766, May 2004.

C. Zhang and W. Liu, “Uniform limit power-type function spaces,” Int. J. Math. Math. Sci., vol. 2006, pp. 1-14, Dec. 2006.

J. Leskow and A. Napolitano, “Foundations of the functional approach for signal analysis,” Signal Process., vol. 86, no. 12, pp. 3796–3825, Dec. 2006.

J. Leskow and A. Napolitano, “Non-relatively measurable functions for secure-communications signal design,” Signal Process., vol. 87, no. 11, pp. 2765–2780, Nov. 2007

G. Casinovi, “L1-norm convergence properties of correlogram spectral estimates,” IEEE Trans. Signal Process., vol. 55, no. 9, pp. 4354-4365, Sep. 2007

C. Zhang and C. Meng, “Two new spaces of vector-valued limit power functions,” Studia Scientiarum Mathematicarum Hungarica, vol. 44, no. 4, pp. 423-443, Dec. 2007.

A. Napolitano, Generalizations of Cyclostationary Signal Processing: Spectral Analisys and Applications. Hoboken,NJ, USA:Wiley-IEEE Press, 2012.

D. Dehay, J. Leskow, A. Napolitano, Central limit theorem in the functional approach, IEEE Trans. Signal Process. 61 (16) (2013) 4025–4037.

D. Dehay, J. Leskow, and A. Napolitano, “Time average estimation in the fraction-of-time probability framework,” Signal Process., vol. 153, pp. 275-290, Dec. 2018.

W. A. Gardner, “Statistically inferred time warping: Extending the cyclostationarity paradigm from regular to irregular statistical cyclicity in scientic data,” EURASIP J. Adv. Signal Process., vol. 2018, no. 1, p. 59, Sep. 2018

W. A. Gardner. (2018). Cyclostationarity.Com. [Online]. Available:

A. Napolitano, Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations. Amsterdam, The Netherlands: Elsevier, 2019.

H. Miao, F. Zhang, and R. Tao, “A general fraction-of-time probability framework for chirp cyclostationary signals,” Signal Process., vol. 179, Feb. 2021, Art. no. 107820

A. Napolitano and W.A. Gardner, “Fraction-of-time probability: Advancing beyond the need for stationarity and ergodicity assumptions”, IEEE Access, vol. 10, pp. 34591-34612, 2022

W. A. Gardner, “Transitioning away from stochastic process models”, Journal of Sound and Vibration, Volume 565, 2023, 117871