An overview of digital communications techniques is given. The notions of source, transmitter, channel, receiver, and sink are explained. Examples of digital communication schemes and respective applications are given. The main quantities and performance measures are introduced and summarized. The fundamental trade-off between both power efficiency and bandwidth efficiency is characterized.

The following five chapters exhibit further contributions of the theory of time-variant and quasi-separable systems to matrix algebra. This chapter treats LU factorization, or, equivalently, spectral factorization, which is another, often occurring type of factorization of a quasi-separable system. This type of factorization does not necessarily exist and, when it exists, could traditionally not be computed in a stable numerical way (Gaussian elimination). Here we present necessary and sufficient existence conditions and a stable numerical algorithm to compute the factorization using orthogonal transformations applied to the quasi-separable representation.

In practice, channels often cause linear dispersive signal distortions (e.g., due to low-pass properties of cables or multipath propagation in wireless communications). Consequently, in this chapter we study PAM transmission over time-invariant linear dispersive channels, where so-called intersymbol interference (ISI) occurs. First, receiver-side equalization strategies for linear dispersive channels are introduced and analyzed. Besides the optimum procedure, which follows immediately from the general signal space concept, we assess low-complexity receivers, specifically linear equalization and decision-feedback equalization. In each case, we are interested in the achievable error performance; the loss caused by ISI is quantified. In addition, transmitter-side techniques for pre-equalization are addressed. The duality between receiver-side and transmitter-side schemes is highlighted. A unified theoretic framework for filter design and the calculation of the error performance of the various strategies for digital transmission over linear dispersive channels is presented.

The book starts out with a motivating chapter to answer the question: Why is it worthwhile to develop system theory? To do so, we jump fearlessly in the very center of our methods, using a simple and straight example in optimization: optimal tracking. Although optimization is not our leading subject– which is system theory– it provides for one of the main application areas, namely the optimization of the performance of a dynamical system in a time-variant environment (for example, driving a car or sending a rocket to the moon). The chapter presents a recursive matrix algebra approach to the optimization problem, known as dynamic programming. Optimal tracking is based on a powerful principle called “dynamic programming,” which lies at the very basis of what ”dynamical” means.

This chapter provides for a further extension of constrained interpolation that is capable of solving the constrained model reduction problem, namely the generalization of Schur–Takagi-type interpolation to the time-variant setting. This remarkable result demonstrates the full power of time-variant system theory as developed in this book.

The final chapter completes the scattering theory with an elementary approach to inner embedding of a contractive, quasi-separable causal system (in engineering terms: the embedding of a lossy or passive system in a lossless system, often called Darlington synthesis). Such an embedding is always possible in the finitely indexed case but does not generalize to infinitely indexed matrices. (This last issue requires more advanced mathematical methods and lies beyond the subject matter of the book.)

This chapter introduces and develops the scattering formalism, whose usefulness for interpolation has been demonstrated in Chapter 13, for the case of systems described by state-space realizations. This is in preparation for the next three chapters that use it to solve various further interpolation and embedding problems.

In some situations, it is convenient to apply modifications to the conventional digital PAM scheme, in order to achieve desired properties of the transmit signal and/or in order to modify the demodulation process. First, we have a look at the crest factor or peak-to-average power ratio of the transmit signal, which should be as low as possible. In this context, offset QAM, minimum-shift keying, and Gaussian minimum-shift keying are studied. Moreover, the replacement of the coherent I/Q demodulator by different principles is addressed. First, “carrierless” amplitude and phase modulation is treated as an alternative approach to PAM. Here, no explicit mixing of the pulse-shaped continuous-time baseband signal to the RF domain is required. Second, in some cases (e.g., fiber-optical transmission), coherent reception is possible in principle but very costly. Here it is desired that even when demodulating without phase information (i.e., by conducting energy detection), a performance close to a coherent receiver is enabled. We study in detail an advanced scheme, called the Kramers–Kronig coherent receiver, which meets this aim by performing more complex operations at the digital part.

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.