Past, Prediction Error, Prediction with Finite Past, Solution of the Normal Equation, The Triplet a,c,k, Lattice Filters for Prediction, Positivity and Stability, S-Step Prediction, The Maximum Entropy

Past, Prediction Error, Prediction with Finite Past, Solution of the Normal Equation, The Triplet a,c,k, Lattice Filters for Prediction, Positivity and Stability, S-Step Prediction, The Maximum Entropy Method, Wold Decomposition and Related Problems, Appendix on Stability, Problem. It deals with the prediction theory providing one of the most illuminating applications of the results given in the previous two chapters. The first reason is that the calculations, which are rather complex in the general case, are greatly simplified in prediction, mainly due to the fact that the signal being estimated is only a shifted version of the observed signal. The second reason is that prediction theory of stationary signals is a very powerful method for the analysis and classification of these signals. Finally, the calculation of filters introduced in prediction theory, also called predictors, introduces a very powerful class of filters known as lattice filters. The thirteenth chapter, entitled “Time Recursive Methods,” discusses: Time Recursions for Normal Equation, Recursion Least Square Methods, Introduction to Adaptive Filtering, Kalman Filtering. This chapter deals with the time recursive methods, which are very interesting in signal processing problems because they are easy to implement on computers or on digital signal processors. Therefore, there is a strong incentive to give recursive form to many signal processing methods. The fourteenth chapter, entitled “Matched Filters,” has sections entitled The Classical Matched Filter, Amplitude Estimation, and Generalized Matched Filter. The purpose of this chapter is to show that several problems usually presented independently have in reality the same nature, which is why the term matched filtering has been used, although this expression is commonly restricted to signal-to-noise ratio maximization. Finally, it is worth noting that suitable problems are set out at the end of each chapter. According to the French tradition: a problem is like a story requiring various steps, the first elementary and the last often complex. It is possible to leave the story at any step and to return to finalize it later.