In my Point-of-View article, entitled “Cognitive dynamic systems,” Proceedings of the IEEE, November 2006, I included a footnote stating that a new book on this very topic was under preparation. At long last, here is the book that I promised then, over four years later.

Just as adaptive filtering, going back to the pioneering work done by Professor Bernard Widrow and his research associates at Stanford University, represents one of the hallmarks of the twentieth century in signal processing and control, I see cognitive dynamic systems, exemplified by cognitive radar, cognitive control, and cognitive radio and other engineering systems, as one of the hallmarks of the twenty-first century.

The key question is: How do we define cognition? In this book of mine, I look to the human brain as the framework for cognition. As such, cognition embodies four basic processes:

• perception–action cycle,

• memory,

• attention, and

• intelligence,

each of which has a specific function of its own. In identifying this list of four processes. I have left out language, the fifth distinctive characteristic of human cognition, as it is outside the scope of this book. Simply put, there is no better framework than human cognition, embodying the above four processes, for the study of cognitive dynamic systems, irrespective of application.

Interest in a new generation of engineering systems enabled with cognition, started with cognitive radio, a term that was coined by Mitola and McGuire (1999). In that article, the idea of cognitive radio was introduced within the software-defined radio (SDR) community. Subsequently, Mitola (2000) elaborated on a so-called “radio knowledge representation language” in his own doctoral dissertation. Furthermore, in a short section entitled “Research issues” at the end of his doctoral dissertation, Mitola went on to say the following:

Then, in Haykin (2005a), the first journal paper on cognitive radio, detailed expositions of signal processing, control, learning and adaptive processes, and game-theoretic ideas that lie at the heart of cognitive radio were presented for the first time. Three fundamental cognitive tasks, embodying the perception–action cycle of cognitive radio, were identified in that 2005 paper:

• radio-scene analysis of the radio environment performed in the receiver;

• transmit-power control and dynamic spectrum management, both performed in the transmitter; and

• global feedback, enabling the transmitter to act and, therefore, control data transmission across the forward wireless (data) channel in light of information about the radio environment fed back to it by the receiver.

Perception of the environment, viewed as a problem in spectrum estimation as discussed in Chapter 3, is most appropriate for applications where spectrum sensing of the environment is crucial to the application at hand; cognitive radio is one such important application. A distinctive aspect of spectrum estimation is the fact that it works directly on environmental measurements (i.e. observables). However, in many other environments encountered in the study of cognitive dynamic systems, perception of the environment boils down to state estimation, which is the focus of attention in this chapter.

We defer a formal definition of the state to Section 4.4, where the issue of state estimation is taken up. For now, it suffices to say that estimating the state of a physical environment is compounded by two practical issues:

1. (1) The state of the environment is hidden from the observer, with information about the state being available only indirectly through dependence of the observables (measurements) on the state.

2. (2) Evolution of the state across time and measurements on the environment are both corrupted by the unavoidable presence of physical uncertainties in the environment.

To tackle perception problems of the kind just described, the first step is to formulate a probabilistic model that accounts for the underlying physics of the environment. Logically, the model consists of a pair of equations:

• a system equation, which accounts for evolution of the environmental state across time, and

• a measurement equation, which describes dependence of the measurements on the state.

In this chapter we discuss some basic concepts of Hilbert space and the related operations and properties as the mathematical foundation for the topics of the subsequent chapters. Specifically, based on the concept of unitary transformation in a Hilbert space, all of the unitary transform methods to be specifically considered in the following chapters can be treated from a unified point of view: they are just a set of different rotations of the standard basis of the Hilbert space in which a given signal, as a vector, resides. By such a rotation the signal can be better represented in the sense that the various signal processing needs, such as noise filtering, information extraction and data compression, can all be carried out more effectively and efficiently.

Inner product space

Vector space

In our future discussion, any signal, either a continuous one represented as a time function x(t), or a discrete one represented as a vector x = […, x[n], …]T, will be considered as a vector in a vector space, which is just a generalization of the familiar concept of N-dimensional (N-D) space, formally defined as below.

Definition 2.1.A vector space is a set v with two operations of addition and scalar multiplication defined for its members, referred to as vectors.

Why wavelet?

Short-time Fourier transform and Gabor transform

In Chapter 3, we learned that a signal can be represented as either a time function x(t) as the amplitude of the signal at any given moment t, or, alternatively and equivalently, as a spectrum X(f) = F[x(t)] representing the magnitude and phase of the frequency component at any given frequency f. However, no information in terms of the frequency contents is explicitly available in the time domain, and no information in terms of the temporal characteristics of the signal is explicitly available in the frequency domain. In this sense, neither x(t) in the time domain nor X(f) in the frequency domain provides complete description of the signal. In other words, we can have either temporal or spectral locality regarding the information contained in the signal, but never both at the same time.

To address this dilemma, the short-time Fourier transform (STFT), also called windowed Fourier transform, can be used. The signal x(t) to be analyzed is first truncated by a window function w(t) before it is Fourier transformed to the frequency domain. As all frequency components in the spectrum are known to be contained in the signal segment inside this particular time window, certain temporal locality in the frequency domain is achieved.