From this point on, main issues in system theory are tackled. The very first, considered in this chapter, is the all-important question of system identification. This is perhaps the most basic question in system theory and related linear algebra, with a large pedigree starting from Kronecker's characterization of rational functions to its elegant solution for time-variant systems presented here. Identification, often also called realization, is the problem of deriving the internal system’s equations (called state-space equations) from input–output data. In this chapter, we only consider the causal, or block-lower triangular case, although the theory applies just as well to an anti-causal system, for which one lets the time run backward, applying the same theory in a dual form.

What is a system? What is a dynamical system? Systems are characterized by a few central notions: their state and their behavior foremost, and then some derived notions such as reachability and observability. These notions pop up in many fields, so it is important to understand them in nontechnical terms. This chapter therefore introduces what people call a narrative that aims at describing the central ideas. In the remainder of the book, the ideas presented here are made mathematically precise in concrete numerical situations. It turns out that a sharp understanding of just the notion of state suffices to develop most if not the whole mathematical machinery needed to solve the main engineering problems related to systems and their dynamics.

In digital frequency modulation, in particular frequency-shift keying (FSK), information is represented solely by the instantaneous frequency, whereas the amplitude of the ECB signal and thus the envelope of the RF signal are constant. Therefore, efficient power amplification is possible, an important advantage of digital frequency modulation. Even though the frequency and phase of a carrier signal are tightly related (the instantaneous frequency is given by the derivative of the phase), differentially encoded PSK and FSK fall into different families. Moreover, in FSK, the continuity of the carrier phase plays an important role, resulting in continuous-phase FSK (CPFSK). A generalization of CPFSK leads to continuous-phase modulation (CPM), similar to the generalization of MSK to Gaussian MSK discussed in Chapter 4. A brief introduction to CPM is presented and we especially enlighten the inherent coding of CPFSK and CPM. For the characterization and analysis, the general signal space concept derived in Chapter 6 is applied.

This chapter considers the Moore–Penrose inversion of full matrices with quasi-separable specifications, that is, matrices that decompose into the sum of a block-lower triangular and a block-upper triangular matrix, whereby each has a state-space realization given. We show that the Moore–Penrose inverse of such a system has, again, a quasi-separable specification of the same order of complexity as the original and show how this representation can be recursively computed with three intertwined recursions. The procedure is illustrated on a 4 ? 4 (block) example.

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.