Intersymbol interference (ISI) occurs for linear dispersive channels (i.e., channels where the transfer function is not flat within the transmission band). Hence, an obvious strategy to avoid ISI is to divide the transmission band into a large number of subbands, which are used individually in parallel. If these bands are small enough, such fluctuations of the channel transfer function can be ignored and no linear distortions occur that would have to be equalized. In this chapter, we study this idea in the particular form of orthogonal frequency-division multiplexing (OFDM). It is shown that even starting from the frequency-division multiplexing idea, the key principle behind OFDM is blockwise transmission and the use of suitable transformations at transmitter and receiver. We analyze OFDM in detail and show how the resulting parallel data transmission can be used in an optimum way. OFDM is compared with the equalization schemes discussed in the previous chapter, and incorporated in the unified description framework.

In this chapter, we consider the central issue of minimality of the state-space system representation, as well as equivalences of representations. The question introduces important new basic operators and spaces related to the state-space description. In our time-variant context, what we call the Hankel operator plays the central role, via a minimal composition (i.e., product), of a reachability operator and an observability operator. Corresponding results for LTI systems (a special case) follow readily from the LTV case. In a later starred section and for deeper insights, the theory is extended to infinitely indexed systems, but this entails some extra complications, which are not essential for the main, finite-dimensional treatment offered, and can be skipped by students only interested in finite-dimensional cases.

The set of basic topics then continues with a major application domain of our theory: linear least-squares estimation (llse) of the state of an evolving system (aka Kalman filtering), which turns out to be an immediate application of the outer–inner factorization theory developed in Chapter 8. To complete this discussion, we also show how the theory extends naturally to cover the smoothing case (which is often considered “difficult”).

Several types of factorizations solve the main problems of system theory (e.g., identification, estimation, system inversion, system approximation, and optimal control). The factorization type depends on what kind of operator is factorized, and what form the factors should have. This and the following chapter are, therefore, devoted to the two main types of factorization: this chapter treats what is traditionally called coprime factorization, while the next is devoted to inner–outer factorization. Coprime factorization, here called “external factorization” for more generality, characterizes the system’s dynamics and plays a central role in system characterization and control issues. A remarkable result of our approach is the derivation of Bezout equations for time-variant and quasi-separable systems, obtained without the use of Euclidean divisibility theory. From a numerical point of view, all these factorizations reduce to recursively applied QR or LQ factorizations, applied on appropriately chosen operators.

In carrier-modulated (digital) communication, the transmit signal has spectral components in a band around a so-called carrier frequency. Here, a baseband transmit signal is upconverted to obtain the radio-frequency (RF) transmit signal and the RF receive signal is downconverted to obtain the baseband receive signal. The processing of transmit and receive signals is done as far as possible in the baseband domain. The aim of the chapter is to develop a mathematically precise compact representation of real-valued RF signals independent of the actual center frequency (or carrier frequency) by equivalent complex baseband (ECB) signals. In addition, transforms of corresponding systems and stochastic processes into the ECB domain and back are covered in detail. Conditions for wide-sense stationary and cyclic-stationary stochastic processes in the EBC domain are discussed.

This chapter starts developing our central linear time-variant (LTV) prototype environment, a class that coincides perfectly with linear algebra and matrix algebra, making the correspondence between system and matrix computations a mutually productive reality. People familiar with the classical approach, in which the z-transform or other types of transforms are used, will easily recognize the notational or graphic resemblance, but there is a major difference: everything stays in the context of elementary matrix algebra, no complex function calculus is involved, and only the simplest matrix operations, namely addition and multiplication of matrices, are needed. Appealing expressions for the state-space realization of a system appear, as well as the global representation of the input–output operator in terms of four block diagonal matrices and the (now noncommutative but elementary) causal shift Z. The consequences for and relation to linear time-invariant (LTI) systems and infinitely indexed systems are fully documented in *-sections, which can be skipped by students or readers more interested in numerical linear algebra than in LTI system control or estimation.