Integral Equations, Biorthonormal Expansions, and Noise

Previous article Next article Integral Equations, Biorthonormal Expansions, and NoiseRoy LeipnikRoy Leipnikhttps://doi.org/10.1137/0107002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] H. Cramér, Random Variables and Probability Distributions, Cambridge Tracts in Mathematics, No. 36, Cambridge Univ. Press, 1937 0016.36304 Google Scholar[2] Kari Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys., 1947 (1947), 79– MR0023013 0030.16502 Google Scholar[3] Paul Lévy, Processus Stochastiques et Mouvement Brownien. Suivi d'une note de M. Loève, Gauthier-Villars, Paris, 1948, 365–, See in particular the supplement by Loève, M. Fonctions aléatoires du second ordre. MR0029120 0034.22603 Google Scholar[4] Ulf Grenander and , Murray Rosenblatt, Statistical analysis of stationary time series, John Wiley & Sons, New York, 1957, 300– MR0084975 0080.12904 Google Scholar[5] D. Hilbert, Integralgleichungen, 1912, Reprinted 1953 by Chelsea, New York. Google Scholar[6] K. Franz, Die Amplituden von Geranchshanunger, Elektr. Nachr. Techn., 19 (1942), 166–167 Google Scholar[7] Mark Kac and , A. J. F. Siegert, On the theory of noise in radio receivers with square law detectors, J. Appl. Phys., 18 (1947), 383–397, Note on the theory of noise in receivers with square law detector, Phys. Rev., 70 (1946), p. 449 10.1063/1.1697662 MR0020229 CrossrefISIGoogle Scholar[8] A. J. F. Siegert, Passage of stationary processes through linear and non-linear devices, Trans. I.R.E., PG IT-3 (1954), 4–25 0058.34702 Google Scholar[9] D. A. Darling and , A. J. F. Siegert, On the distribution of certain functionals of Markoff chains and processes, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 525–529, On the distribution of certain functionals of Markoff processes, RAND Corp., paper p-429, 1953, A systematic approach to a class of problems in the theory of noise and other random phenomena, Trans. I.R.E., vol. IT-3, No. 1 (1957), pp. 32–43. MR0079846 0074.34001 CrossrefISIGoogle Scholar[10] J. F. Barrett and , D. G. Lampard, An expansion for some second-order probability distributions and its application to noise problems, Trans. I.R.E., IT-1 (1955), 10–15 Google Scholar[11] D. G. Lampard, The probability distribution for the filtered output of a multiplier whose inputs are correlated, stationary, Gaussian time series, Trans. I.R.E., IT-2 (1956), 4–11 Google Scholar[12] J. L. Brown, Jr., On a cross-correlation property for stationary random processes, Trans. I.R.E., IT-3 (1957), 28–31 Google Scholar[13A] M. Kac, On the average of a certain Wiener functional and a related limit theorem in calculus of probability, Trans. Amer. Math. Soc., 59 (1946), 401–414 MR0016570 CrossrefISIGoogle Scholar[13B] M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc., 65 (1949), 1–13 MR0027960 CrossrefISIGoogle Scholar[14] Norbert Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications, The Technology Press of the Massachusetts Institute of Technology, Cambridge, Mass, 1949ix+163, Originally issued in 1942 as a National Defense Research Committee report MR0031213 CrossrefGoogle Scholar[15] Julius Bendat, A general theory of linear prediction and filtering, J. Soc. Indust. Appl. Math., 4 (1956), 131–151 10.1137/0104008 MR0081020 LinkISIGoogle Scholar[16] Roy B. Leipnik, Moment generating functions of quadratic forms in serially correlated normal variables, Biometrika, 45 (1958), 198–210 MR0092303 CrossrefISIGoogle Scholar[17] Roy Leipnik, The effect of instantaneous nonlinear devices on cross-correlation., IRE Trans., IT-4 (1958), 73–76 10.1109/TIT.1958.1057445 MR0147321 CrossrefISIGoogle Scholar[18] Paul R. Halmos, Introduction to Hilbert Space and the theory of Spectral Multiplicity, Chelsea Publishing Company, New York, N. Y., 1951, 114– MR0045309 Google Scholar[19] E. Schmidt, Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener, Math. Ann., 63 (1907), 473–476 10.1007/BF01449770 CrossrefGoogle Scholar[20] E. W. Hobson, Theory of Functions of a Real Variable, Vol. 2, Cambridge Univ. Press, 1950, Ch. 8, 9, 10. Reprinted by Dover in 1957 Google Scholar[21] I. P. Natanson, Theory of functions of a real variable, Frederick Ungar Publishing Co., New York, 1955, 178–, (translated from the Russian edition of 1941) MR0067952 Google Scholar[22] Adriaan Cornelis Zaanen, Linear analysis. Measure and integral, Banach and Hilbert space, linear integral equations, Interscience Publishers Inc., New York, 1953vii+601, Part III MR0061752 Google Scholar[23] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xvii+396, The formulas we require are found on pp. 188, 189, 193, 194, 207 MR0058756 Google Scholar[24] J. J. Bussgang, Gross-correlation functions of amplitude distorted Gaussian signals, Tech. Rep., 216, Research Laboratory for Electronics, Mass. Inst. of Technology, 1952 Google Scholar[25] A. Sommerfeld, Masters Thesis, Die willkürlichen Funktionen der mathematischen Physik, dissertation, Koenigsberg, 1891 Google Scholar[26] E. Borel, Leçons sur les Séries Divergentes, Gauthier Villars, Paris, 1928 Google Scholar[27A] S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J., 23 (1944), 282–332 MR0010932 CrossrefGoogle Scholar[27B] S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J., 24 (1945), 46–156 MR0011918 CrossrefISIGoogle Scholar[28] S. O. Rice, Statistical properties of a sine wave plus random noise, Bell System Tech. J., 27 (1948), 109–157 MR0023483 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 7, Issue 1| 1959Journal of the Society for Industrial and Applied Mathematics History Submitted:07 November 1957Accepted:07 August 1958Published online:10 July 2006 InformationCopyright © 1959 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0107002Article page range:pp. 6-30ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics