References & Further Reading
References
- J. A. Gubner, Probability and Random Processes for Electrical and Computer Engineers, Cambridge University Press, 2006
The primary textbook for Caire's FSP course. Chapter 6 covers random vectors and the multivariate Gaussian with clear examples.
- T. W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley, 3rd ed., 2003
The definitive reference for multivariate Gaussian theory. Chapters 2-3 cover the distribution theory; Chapters 5-7 cover hypothesis testing and the Wishart distribution. Rigorous and comprehensive.
- A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 4th ed., 2002
An excellent engineering-oriented treatment. Chapter 6 on random vectors and Chapter 12 on the Karhunen-Loeve expansion are particularly relevant to this chapter.
- S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall, 1993
Chapter 4 covers the multivariate Gaussian in the context of estimation. The whitening pre-filter and its role in detection are thoroughly covered.
- G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, 4th ed., 2013
The standard reference for numerical linear algebra. Chapters 3-4 cover Cholesky factorization and eigendecomposition — the computational backbone of whitening and PCA.
- P. J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, 2010
The definitive treatment of complex-valued random vectors and the proper/improper distinction. Essential for understanding the complex Gaussian model used in wireless communications.
- N. R. Goodman, Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution, 1963
The foundational paper for complex Gaussian statistics. Introduces the complex Wishart distribution and associated tests.
- A. Fengler, S. Haghighatshoar, P. Jung, and G. Caire, Non-Bayesian Activity Detection, Large-Scale Fading Coefficient Estimation, and Unsourced Random Access with a Massive MIMO Receiver, 2021
Applies complex Gaussian channel models and Wishart-type sample covariance analysis to the massive random access problem.
- D. Tse and P. Viswanath, Fundamentals of Wireless Communications, Cambridge University Press, 2005
Chapter 2 uses the complex Gaussian channel model extensively. The Rayleigh fading model derived from bivariate Gaussians is a direct application of this chapter's results.
- K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis, Academic Press, 1979
A classic reference for multivariate statistics. Chapter 3 on the multivariate normal provides an alternative derivation of the conditional distribution formulas.
- R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 2nd ed., 2012
The standard reference for matrix theory. The Schur complement and positive definiteness results used in this chapter are developed in Chapters 0 and 7.
Further Reading
Resources for deepening your understanding of multivariate Gaussian theory and its applications.
Random matrix theory and the Marchenko-Pastur law
Book FSP, Ch. 21 — Random Matrix Theory
When the dimension $n$ and sample size $m$ grow together, the eigenvalues of the sample covariance follow the Marchenko-Pastur distribution. This has profound implications for massive MIMO.
LMMSE estimation and the Wiener filter
Book FSI, Ch. 3 — MMSE Estimation
The conditional Gaussian formulas of this chapter are exactly the LMMSE estimator. The FSI book develops the estimation framework systematically.
Graphical models and the precision matrix
S. L. Lauritzen, Graphical Models, Oxford University Press, 1996
The zeros of the precision matrix $\ntn{covmat}^{-1}$ encode conditional independence in Gaussian graphical models — a connection that is central to modern statistics and machine learning.
Improper complex random vectors
P. J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge, 2010
When the pseudo-covariance is nonzero (improper signals), the standard complex Gaussian formulas must be modified. This book develops the full theory for both proper and improper cases.