# 维纳－辛钦定理

## 连续时间过程的情形

${\displaystyle r_{xx}(\tau )=\int _{-\infty }^{\infty }e^{2\pi i\tau f}dF(f)}$

（星号表示复共轭，当随机过程是过程时可以将其省去。）

${\displaystyle r_{xx}(\tau )=\int _{-\infty }^{\infty }S(f)e^{2\pi i\tau f}df.}$

${\displaystyle S(f)=\int _{-\infty }^{\infty }r_{xx}(\tau )e^{-2\pi if\tau }d\tau .}$

## 离散时间过程的情形

${\displaystyle S_{xx}(f)=\sum _{k=-\infty }^{\infty }r_{xx}[k]e^{-j2\pi kf}}$

${\displaystyle r_{xx}[k]=\operatorname {E} {\big [}\,x[n]x^{*}[n-k]\,{\big ]}\ }$

${\displaystyle S_{xx}(f)\ }$

## 参考文献

1. C. Chatfield. The Analysis of Time Series—An Introduction fourth. Chapman and Hall, London. 1989: 94–95. ISBN 0-412-31820-2.
2. ^ Norbert Wiener. Time Series. M.I.T. Press, Cambridge, Massachusetts. 1964: 42.
3. ^ Hannan, E.J., "Stationary Time Series", in: John Eatwell, Murray Milgate, and Peter Newman, editors, The New Palgrave: A Dictionary of Economics. Time Series and Statistics, Macmillan, London, 1990, p. 271.
4. ^ Dennis Ward Ricker. Echo Signal Processing. Springer. 2003. ISBN 1-4020-7395-X.
5. ^ Leon W. Couch II. Digital and Analog Communications Systems sixth. Prentice Hall, New Jersey. 2001: 406–409. ISBN 0-13-522583-3.
6. ^ Krzysztof Iniewski. Wireless Technologies: Circuits, Systems, and Devices. CRC Press. 2007. ISBN 0-8493-7996-2.
7. ^ Joseph W. Goodman. Statistical Optics. Wiley-Interscience. 1985. ISBN 0-471-01502-4.
8. ^ Wiener, Norbert. Generalized Harmonic Analysis. Acta Mathematica. 1930, 55: 117–258.
9. ^ Nahin, Paul J. Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press. 2011: 225. ISBN 9780691150376.
10. ^ Khintchine, A. Korrelationstheorie der stationären stochastischen Prozesse. Mathematische Annalen. 1934, 109 (1): 604–615. doi:10.1007/BF01449156.
11. ^ Jerison, David; Singer, Isadore Manuel; Stroock, Daniel W. The Legacy of Norbert Wiener: A Centennial Symposium (Proceedings of Symposia in Pure Mathematics). American Mathematical Society. 1997: 95. ISBN 0-8218-0415-4.
12. ^ Hannan, E. J. Stationary Time Series. (编) Eatwell, John; Milgate, Murray; Newman, Peter. The New Palgrave: A Dictionary of Economics. Time Series and Statistics. London: Macmillan. 1990: 271.
13. ^ Chatfield, C. The Analysis of Time Series—An Introduction Fourth. London: Chapman and Hall. 1989: 96. ISBN 0-412-31820-2.
14. ^ Champeney, D. C. A Handbook of Fourier Theorems. Cambridge Univ. Press. 1987: 20–22.
15. ^ Shlomo Engelberg. Random signals and noise: a mathematical introduction. CRC Press. 2007: 130. ISBN 978-0-8493-7554-5.
16. ^ C. Chatfield. The Analysis of Time Series—An Introduction fourth. Chapman and Hall, London. 1989: 98. ISBN 0-412-31820-2.

## 延伸阅读

• Brockwell, Peter A.; Davis, Richard J. Introduction to Time Series and Forecasting Second. New York: Springer-Verlag. 2002. ISBN 038721657X.
• Chatfield, C. The Analysis of Time Series—An Introduction Fourth. London: Chapman and Hall. 1989. ISBN 0412318202.
• Fuller, Wayne. Introduction to Statistical Time Series. Wiley Series in Probability and Statistics Second. New York: Wiley. 1996. ISBN 0471552399.
• Wiener, Norbert. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Cambridge, Massachusetts: Technology Press and Johns Hopkins Univ. Press. 1949. (a classified document written for the Dept. of War in 1943).
• Yaglom, A. M. An Introduction to the Theory of Stationary Random Functions. Englewood Cliffs, New Jersey: Prentice–Hall. 1962.