TAILIEUCHUNG - Nguyên tắc phân tích tín hiệu ngẫu nhiên và thiết kế tiếng ồn thấp P2

Background: Signal and System Theory INTRODUCTION The power spectral density arises from signal analysis of deterministic signals, and random processes, and is required to be evaluated over both the finite and infinite time intervals. While signal analysis for the finite case, for example, the integral on a finite interval of a finite summation of bounded signals, causes few problems, signal analysis for the infinite case is more problematic. For example, it can be the case that the order of the integration and limit operators cannot be interchanged. With the infinite case, careful attention to detail and a reasonable knowledge of. | Principles of Random Signal Analysis and Low Noise Design The Power Spectral Density and Its Applications. Roy M. Howard Copyright 2002 John Wiley Sons Inc. ISBN 0-471-22617-3 2 Background Signal and System Theory INTRODUCTION The power spectral density arises from signal analysis of deterministic signals and random processes and is required to be evaluated over both the finite and infinite time intervals. While signal analysis for the finite case for example the integral on a finite interval of a finite summation of bounded signals causes few problems signal analysis for the infinite case is more problematic. For example it can be the case that the order of the integration and limit operators cannot be interchanged. With the infinite case careful attention to detail and a reasonable knowledge of underlying mathematical theory is required. Clarity is best achieved for integration for example through measure theory and Lebesgue integration. This chapter gives the necessary mathematical background for the development and application of theory related to the power spectral density that follows in subsequent chapters. First a review of fundamental results from set theory real and complex analysis signal theory and system theory is given. This is followed by an overview of measure and Lebesgue integration and associated results. Finally consistent with the requirements of subsequent chapters results from Fourier theory and a brief introduction to random process theory are given. BACKGROUND THEORY Set Theory Set theory is fundamental to mathematical analysis and the following results from set theory are consistent with subsequent analysis. Useful references for set theory include Sprecher 1970 Lipschutz 1998 and Epp 1995 . 3 4 BACKGROUND SIGNAL AND SYSTEM THEORY Definition Set A set is a collection of distinct entities. The notation at a2 . aN is used for the set of distinct entities acj CJ I n n A rj l ATI J V I lv 10 11 04141 TAI 1141 041 AT 411 4HY14111

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