References & Further Reading

References

1. H. V. Poor, An Introduction to Signal Detection and Estimation, Springer, 2nd ed., 1994

   §II.C covers $M$-ary hypothesis testing, MAP decisions, and the structure of signal-space receivers.

2. J. G. Proakis and M. Salehi, Digital Communications, McGraw-Hill, 5th ed., 2007

   Chapter 4 develops $M$-PSK and $M$-QAM detection in detail, with exact error-probability derivations.

3. M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels, Wiley, 2nd ed., 2005

   The canonical reference for MGF-based error analysis and Craig's formula. Every SER-in-fading formula used here is derived there.

4. J. W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations, 1991

   Original paper introducing Craig's formula, enabling MGF-based averaging over fading.

5. H. L. Van Trees, K. L. Bell, and Z. Tian, Detection, Estimation, and Modulation Theory, Part I, Wiley, 2nd ed., 2013

   Classical treatment of $M$-ary detection, signal-space decomposition, and union bounds.

6. J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering, Wiley, 1965

   Original signal-space development of $M$-ary digital modulation.

7. E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses, Springer, 3rd ed., 2005

   General $M$-ary decision theory and asymptotic error exponents.

8. T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley, 2nd ed., 2006

   Chapter 11 develops the KL-divergence-based error exponents that underlie our §5 asymptotic results.

9. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, 2nd ed., 2010

   Rigorous large-deviations foundation for Chernoff-type error exponents.

10. G. D. Forney, Jr., and G. Ungerboeck, Modulation and coding for linear Gaussian channels, 1998

    Survey paper connecting constellation geometry to shaping, coding gain, and error probability.

11. C. E. Shannon, Probability of error for optimal codes in a Gaussian channel, 1959

    Classical sphere packing bound relating $M$-ary error probability to channel capacity.

12. G. Caire, F. Liu, MGF-Based Sensing-Communication Tradeoffs Under Generalized Fading, 2023

Further Reading

• Tight error bounds beyond the union bound

  Sason and Shamai, 'Performance Analysis of Linear Codes under ML Decoding: A Tutorial', Foundations and Trends in Communications and Information Theory, 2006.

  Surveys the improved bounding techniques (tangential-sphere, DS2) that sharpen the union bound at low SNR.

• Asymptotic geometry of good constellations

  Forney, 'Coset Codes I & II', IEEE Trans. IT, 1988.

  Introduces shaping and coding gains for lattice and coset-based constellations, quantifying how much $d_{\min}$ can be squeezed out of a given dimension.

• Computer simulation of BER curves

  Jeruchim et al., 'Simulation of Communication Systems', Kluwer 2000.

  Practical guidance for Monte-Carlo estimation of very low error probabilities via importance sampling.