TAILIEUCHUNG - Nguyên tắc phân tích tín hiệu ngẫu nhiên và thiết kế tiếng ồn thấp P8

Linear System Theory In this chapter, the fundamental relationships between the input and output of a linear time invariant system, as illustrated in Figure , are detailed. Specifically, the relationships between the input and output time signals, Fourier transforms and power spectral densities, are established. Such relationships are fundamental to many aspects of system theory, including analysis of noise in linear systems, and low noise amplifier design. | 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 8 Linear System Theory INTRODUCTION In this chapter the fundamental relationships between the input and output of a linear time invariant system as illustrated in Figure are detailed. Specifically the relationships between the input and output time signals Fourier transforms and power spectral densities are established. Such relationships are fundamental to many aspects of system theory including analysis of noise in linear systems and low noise amplifier design. The relationships between the parameters defined in Figure and proved in this chapter are 1 y t J x X h t A d . o Y T f H T f X T f where X and Y are the respective Fourier transforms evaluated on the interval 0 T of the signals x and y. However as will be shown in this chapter the relationship defined in Eq. is an approximation. If both x h e L then the relative error in this approximation can be made arbitrarily small by making T sufficiently large. However stationary random signals are not Lebesgue integrable on the interval 0 oo and hence this convergence is not guaranteed. However it is shown for a broad class of signals and random processes including periodic signals and stationary random processes that the corresponding relationship between the input and output power spectral densities namely G T f X H T f 2GX T f becomes exact as T increases without bound. Establishing the relationships as per Eqs. - for a linear time invariant system requires the system impulse response to be defined and this is the subject of the next section. 229 230 LINEAR SYSTEM THEORY x e Ex GX h H y e ey gy Figure Schematic diagram of a linear system. EX and EY respectively represent the ensemble of input and output signals. H is the Fourier transform of the impulse response function h. GX and GY respectively are

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