References & Further Reading
References
- A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill, 4th ed., 2002
Ch. 9 covers Gaussian processes; Ch. 10 covers LTI systems with random inputs.
- G. Caire, Fundamentals of Stochastic Processes: Lecture Notes, TU Berlin, 2024
Ch. 8: Second-order processes, Gaussian processes, and sampling.
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 2nd ed., 1991
The standard graduate reference for the Wiener process and stochastic calculus.
- J. G. Proakis and M. Salehi, Digital Communications, McGraw-Hill, 5th ed., 2008
Ch. 4 on matched filters and Gaussian noise in communication systems.
- H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I, John Wiley & Sons, 1968
Ch. 4: definitive treatment of detection in Gaussian noise.
- S. Haykin, Adaptive Filter Theory, Prentice Hall, 4th ed., 2001
Ch. 2--3 cover Wiener filtering and colored noise generation.
- C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006. [Link]
The standard reference for GP regression and classification.
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, 2nd ed., 1999
Ch. 8: Donsker's invariance principle and functional CLT.
- A. N. Shiryaev, Essentials of Stochastic Finance, World Scientific, 1999
Historical notes on Kolmogorov and the foundations of stochastic processes.
- J. B. Johnson, Thermal Agitation of Electricity in Conductors, 1928
Original experimental paper on thermal noise.
- K. Vu, R. Cavalcante, and G. Caire, LMMSE Channel Estimation with Spatial Covariance Side Information, 2018
LMMSE estimation exploiting spatial structure in massive MIMO.
- A. Leon-Garcia, Probability, Statistics, and Random Processes for Electrical Engineering, Pearson, 3rd ed., 2008
Accessible treatment of Gaussian processes and Brownian motion.
Further Reading
These resources extend the Gaussian process framework to stochastic calculus, machine learning, and advanced communication system analysis.
Stochastic calculus and Ito's formula
Karatzas and Shreve (1991), Ch. 3
The Wiener process is the building block of stochastic calculus. Ito's formula generalizes the chain rule to stochastic integrals — essential for analyzing stochastic differential equations in phase noise and channel models.
Gaussian process regression and kernel methods
Rasmussen and Williams (2006), Ch. 2--5
GP regression provides a principled Bayesian framework for nonparametric estimation. The choice of kernel encodes prior assumptions about smoothness and correlation structure — directly applicable to channel prediction and spatial interpolation.
Information-theoretic optimality of Gaussian noise
Book ITA, Chapter 5
The Gaussian channel achieves extremal capacity properties: among all noise distributions with a given variance, Gaussian noise minimizes channel capacity. This connects the Gaussian process framework to fundamental limits of communication.
Phase noise analysis in OFDM systems
Proakis and Salehi (2008), Ch. 5, §5.4
The Wiener model of phase noise leads to common phase error (CPE) and inter-carrier interference (ICI) in OFDM — understanding these effects requires the tools developed in this chapter.