Fourier Transforms, Filtering, Probability and Random Processes

This book provides the background and the mathematical methods necessary to understand the basic transforms in signal processing and linear systems and probability and random processes to prepare for in depth study of analog and digital communications systems. This tutorial presentation provides developments of Fourier series and other orthogonal series, including trigonometric and complex exponential Fourier series, least squares approximations and generalized Fourier series, and the spectral content of periodic signals. This text thoroughly covers Fourier transform pairs for continuous time signals, Fourier transform properties, and the magnitude and phase of Fourier transforms. The author includes discussions of techniques for the analysis of continuous time linear systems in the time and frequency domains with particular emphasis on the system transfer function, impulse response, system/filter bandwidth, power and energy calculations, and the time domainsampling theorem. The basics of probability and random processes, including the key concepts of expected value, variance, characteristic functions, common probability distributions, autocorrelation, power spectral densities, wide sense stationarity, and ergodicity, are all developed in some detail.  Many examples and problems are included to illustrate and examine these topics. • Provides developments of Fourier series and other orthogonal series• Presents fundamental Fourier transform properties and example applications• Discusses techniques for the analysis of continuous time linear systems in the time and frequency domains• Presents a fundamental development of probability and random variables• Develops the basic ideas of random processes including autocorrelation, power spectral densities, stationarity, and ergodicity