Roadmap to the book
Michael A. Unser, Pouya D. Tafti · Cambridge University Press eBooks · 2014
The writing of this book was motivated by our desire to formalize and extend the ideas presented in Section 1.3 to a class of differential operators much broader than the derivative D. Concretely, this translates into the investigation of the family of stochastic processes specified by the general innovation model that is summarized in Figure 2.1. The corresponding generator of random signals (upper part of the diagram) has two fundamental components: (1) a continuous-domain noise excitation w , which may be thought of as being composed of a continuum of independent identically distributed (i.i.d.) random atoms (innovations), and (2) a deterministic mixing procedure (formally described by L −1 ) which couples the random contributions and imposes the correlation structure of the output. The concise description of the model is L s = w , where L is the whitening operator. The term “innovation” refers to the fact that w represents the unpredictable part of the process. When the inverse operator L −1 is linear shift-invariant (LSI), the signal generator reduces to a simple convolutional system which is characterized by its impulse response (or, equivalently, its frequency response). Innovation modeling has a long tradition in statistical communication theory and signal processing; it is the basis for the interpretation of a Gaussian stationary process as a filtered version of a white Gaussian noise [Kai70, Pap91]. In the present context, the underlying objects are continuously defined.