Random Signals, Estimation, and Filtering

Abstract

Electrical, mechanical, or atmospheric noise causes disturbances in the measurement of signals. The observed signals cannot be predicted deterministically. These signals are to be characterized by statistical methods. We will call these nondeterministic signals random signals. We will study the properties of random signals, known as stochastic processes, in Sections 1 through 4. The various classifications are given in Section 1. Correlation covariance functions of random signals are treated in Section 2. Section 3 is devoted to Gaussian and Brownian processes. A very important class of random processes, stationary processes, is covered in Section 4. Power spectral density is defined for these processes. We derive the second-order statistics of linear systems with stationary signal input in Section 5. A brief discussion of the narrow band process is given in Section 6. We discuss estimation of parameters of random signals and various estimation criteria in Section 7. Various estimation methods, such as maximum likelihood estimation, least-square estimation, linear meansquare estimation, and the Cramer-Rao bound, are covered in Section 8. Estimation of parameters in the presence of colored noise is treated in Section 9. We have derived the recursive estimator. Optimum linear filters are derived in Section 10. A three-step algorithm is given for Wiener filter using spectral factorization. A short bibliographical note is given in Section 11. Spectral estimation methods and Kaiman filtering are given in Appendices 3.A and 3.B.