References & Further Reading

References

  1. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill, 4th ed., 2002

    Chapters 7-8 cover conditional expectation and LMMSE estimation with many worked examples.

  2. P. Billingsley, Probability and Measure, Wiley, 3rd ed., 1995

    Chapter 34 gives the rigorous measure-theoretic treatment of conditional expectation.

  3. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford University Press, 4th ed., 2020

    Chapter 7 provides a clear intermediate-level treatment of conditional expectation and total variance.

  4. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall, 1993

    Chapters 10-12 develop MMSE and LMMSE estimation with engineering applications. The standard reference for signal processing students.

  5. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I, Wiley, 1968

    The classical reference for Bayesian estimation theory. Chapter 2 covers MMSE estimation.

  6. S. Haykin, Adaptive Filter Theory, Prentice Hall, 4th ed., 2002

    Chapters 1-3 cover Wiener filtering and LMMSE from a signal processing perspective.

  7. A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, 1933

    The foundational monograph establishing the axiomatic probability theory that underpins conditional expectation.

  8. G. Caire, Foundations of Statistical Inference (FSI Lecture Notes), TU Berlin, 2018

    Caire's treatment of LMMSE estimation with wireless communication applications.

  9. K. Ito and G. Caire, LMMSE Channel Estimation for Massive MIMO-OFDM with Sounding Reference Signals, 2021

    Structured LMMSE channel estimator exploiting Kronecker factorization for massive MIMO-OFDM.

  10. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, MIT Press, 1949

    Wiener's wartime monograph establishing optimal linear filtering theory.

  11. H. V. Poor, An Introduction to Signal Detection and Estimation, Springer, 2nd ed., 2013

    Chapters 2-4 cover MMSE and LMMSE estimation with clear proofs.

Further Reading

For readers who want to go deeper into estimation theory and its applications.

  • Measure-theoretic conditional expectation

    Billingsley, *Probability and Measure*, Ch. 34

    For the rigorous treatment using Radon-Nikodym derivatives and sigma-algebras — essential for understanding martingale theory.

  • LMMSE in wireless communications

    Kay, *Fundamentals of Statistical Signal Processing*, Ch. 12-15

    Extends the LMMSE theory to Wiener filtering, Kalman filtering, and adaptive algorithms — the workhorses of modern receivers.

  • Bayesian estimation beyond MMSE

    Van Trees, *Detection, Estimation, and Modulation Theory*, Part I, Ch. 2-4

    Covers MAP estimation, minimax estimation, and general Bayesian cost functions — the broader framework that contains MMSE as a special case.

  • MMSE estimation in massive MIMO

    Bj{\"o}rnson, Hoydis, and Sanguinetti, *Massive MIMO Networks*, Cambridge, 2017

    Shows how the LMMSE channel estimator from this chapter is used in every cell of a massive MIMO network.

ExercisesCh 13 →