Without conceding a blemish in the first edition, I think I had best come clean and admit that I embarked on a second edition largely to adopt a more geometric approach to the detection of signals in white Gaussian noise. Equally rigorous, yet more intuitive, this approach is not only student-friendly, but also extends more easily to the detection problem with random parameters and to the radar problem.

The new approach is based on the projection of white Gaussian noise onto a finite-dimensional subspace (Section 25.15.2) and on the independence of this projection and the difference between noise and projection; see Theorem 25.15.6 and Theorem 25.15.7. The latter theorem allows for a simple proof of the sufficiency of the matched-filters’ outputs without the need to define sufficient statistics for continuous-time observables. The key idea is that—while the receiver cannot recover the observable from its projection onto the subspace spanned by the mean signals—it can mimic the performance of any receiver that bases its decision on the observable using three steps (Figure 26.1 on Page 623): use local randomness to generate an independent stochastic process whose law is equal to that of the difference between the noise and its projection; add this stochastic process to the projection; and feed the result to the original receiver.

But the new geometric approach was not the only impetus for a second edition. I also wanted to increase the book's scope. This edition contains new chapters on the radar problem (Chapter 30), the intersymbol interference (ISI) channel (Chapter 32), and on the mathematical preliminaries needed for its study (Chapter 31). The treatment of the radar problem is fairly standard with two twists: we characterize all achievable pairs of false-alarm and missed-detection probabilities (pFA, pMD) and not just those that are Pareto-optimal. Moreover, we show that when the observable has a density under both hypotheses, all achievable pairs can be achieved using deterministic decision rules.

As to ISI channels, I adopted the classic approach of matched filtering, discretetime noise whitening, and running the Viterbi Algorithm. I only allow (boundedinput/ bounded-output) stable whitening filters, i.e., filters whose impulse response is absolutely summable; others often only require that the impulse response be square summable.