The appendix defines the data model used throughout the book and describes what can best be called an algorithmic design specification, that is, the functional and graphical characterization of an algorithm, chosen so that it can be translated to a computer architecture (be it in soft- or in hardware). We follow hereby a powerful “data flow model” that generalizes the classical signal flow graphs and that can be further formalized to generate the information necessary for the subsequent computer system design at the architectural level (i.e., the assignment of operations, data transfer and memory usage). The model provides for a natural link between mathematical operations and architectural representations. It is, at the same token, well adapted to the generation of parallel processing architectures.

The chapter shows how classical interpolation problems of various types (Schur, Nevanlinna–Pick, Hermite–Fejer) carry over and generalize to the time-variant and/or matrix situation. We show that they all reduce to a single generalized constrained interpolation problem, elegantly solved by time-variant scattering theory. An essential ingredient is the definition of the notion of valuation for time-variant systems, thereby generalizing the notion of valuation in the complex plane provided by the classical z-transform.

In practice, sometimes an estimate of the carrier phase for coherent signal reception is not possible with sufficient accuracy or carrier phase synchronization is not possible at all, in particular, in situations of very fast varying channel conditions (e.g., Doppler effect due to fast-moving transmitters or receivers). For such scenarios, digital transmission schemes have to be applied which are robust to non-perfect carrier frequency and carrier phase estimation. To that end we consider differential PSK which can tolerate phase errors and, to some amount, frequency errors. Then, schemes, which does not require phase (and frequency) synchronization at all, so-called non-coherent demodulation schemes, are developed and analyzed in detail.

This chapter introduces a different kind of problem, namely direct constrained matrix approximation via interpolation, the constraint being positive definiteness. It is the problem of completing a positive definite matrix for which only a well-ordered partial set of entries is given (and also giving necessary and sufficient conditions for the existence of the completion) or, alternatively, the problem of parametrizing positive definite matrices. This problem can be solved elegantly when the specified entries contain the main diagonal and further entries crowded along the main diagonal with a staircase boundary. This problem turns out to be equivalent to a constrained interpolation problem defined for a causal contractive matrix, with staircase entries again specified as before. The recursive solution calls for the development of a machinery known as scattering theory, which involves the introduction of nonpositive metrics and the use of J-unitary transformations where J is a sign matrix.

Two chapters conclude the basic course.

This chapter presents an alternative theory of external and coprime factorization, using polynomial denominators in the noncommutative time-variant shift Z rather than inner denominators as done in the chapter on inner–outer theory. “Polynomials in the shift Z” are equivalent to block-lower matrices with a support defined by a (block) staircase, and are essentially different from the classical matrix polynomials of module theory, although the net effect on system analysis is remarkably similar. The polynomial method differs substantially and in a complementary way from the inner method. It is computationally simpler but does not use orthogonal transformations. It offers the possibility of treating highly unstable systems using unilateral series. Also, this approach leads to famous Bezout equations that, as mentioned in the abstract of Chapter 7, can be derived without the benefit of Euclidean divisibility methods.

A general view on digital modulation schemes beyond the concept of PAM is developed. This is required as many important modulation formats (e.g., digital frequency modulation) do not fall under the umbrella of PAM. To that end, the separation between the operations of coding and modulation is unambiguously defined. The key tool for the analysis and synthesis of transmission schemes is the representation of signals in a signal space. The concept is introduced and discussed in detail. Based on this view, methods for optimum coherent and non-coherent signal reception for any kind of general digital modulation scheme are derived. The principles of maximum-likelihood detection and maximum-likelihood sequence detection are discussed.