Linear prediction: mathematics and engineering

We present an introduction to some aspects of digital signal processing and time series analysis which are not always covered in classical textbooks. One of the objectives is to illustrate how mathematics and engineering can be combined in a fruitful interplay, resulting in many new techniques and methods in many different fields. We shall illustrate how the prediction problem is related to linear algebra, orthogonal polynomials, classical interpolation problems, inverse scattering theory, Toeplitz operators, network theory etc. Contents 1. Introduction 2. Discrete signals and digital filters 3. Stochastic processes 4. Linear prediction 5. The filter model 6. Toeplitz systems and orthogonal polynomials 7. Construction of the predictor 8. Example: seismic signal processing 9. Orthogonal polynomials 10. J-inner matrices 11. Stability test 12. Fast (inverse) Cholesky factorisation 13. The Schur algorithm 14. The Gohberg-Semencul formula 15. The general pole-zero model 16. Relation with c...