Optimal detection of digital modulations in AWGN
Book chapter, 2016
The material developed in this chapter can be found in many other books, but perhaps presented in a different manner. The approach here is inspired by several classical books and countless other sources, including discussions with teachers, students, and colleagues. It would be impossible to list all of them. However, the choice to start the exposition with the minimum-distance detector is inspired by Lee and Messerschmidt . Another important source of inspiration is the classical book from
1965 by Wozencraft and Jacobs , which is very readable even today. An accessible introduction to estimation and detection theory is found in Kay’s two books [3,4], and a more advanced text is by Poor .
Authors are always faced with a dilemma when selecting notation and the level of mathematical rigor to employ. Too little rigor leaves the reader unsure about what exactly is claimed, while too much rigor will obscure the big picture. For readers who would enjoy a more precise and rigorous approach than the one taken here, we recommend the excellent books by Lapidoth  and by Gallager . In particular, the shortcuts taken here (including the use of white noise and Dirac delta functions, the fearless faith in that all integrals and sums converge, and that it is always allowed to change the order of expectation, integration, and summation) are criticized, explained, and (sometimes) justified in these books. Readers can, however, rely on the fact that the developed detectors and analysis methods in this chapter are indeed the same as would have been obtained with a more rigorous approach.