Memoryless Transformations of Random Processes This chapter uses the fact that a memoryless nonlinearity does not affect the disjointness of a disjoint random process to illustrate a procedure for ascertaining the power spectral density of a signaling random process after a memoryless transformation. Several examples are given, including two illustrating the application of this approach to frequency modulation (FM) spectral analysis. Alternative approaches are given in Davenport (1958 ch. 12) and Thomas (1969 ch. 6). . | 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 7 Memoryless Transformations of Random Processes INTRODUCTION This chapter uses the fact that a memoryless nonlinearity does not affect the disjointness of a disjoint random process to illustrate a procedure for ascertaining the power spectral density of a signaling random process after a memoryless transformation. Several examples are given including two illustrating the application of this approach to frequency modulation FM spectral analysis. Alternative approaches are given in Davenport 1958 ch. 12 and Thomas 1969 ch. 6 . POWER SPECTRAL DENSITY AFTER A MEMORYLESS TRANSFORMATION The approach given in this chapter relies on a disjoint partition of signals on a fixed interval. The following section gives the relevant results. Decomposition of Output Using Input Time Partition Consider a signal f which based on a set of disjoint time intervals It . IN can be written as a summation of disjoint waveforms according to f t y f. t f t Jf t t e I 71 J t a Jlw Jl t 0 plcpwhprp i o elsewhere 206 POWER SPECTRAL DENSITY AFTER A MEMORYLESS TRANSFORMATION 207 If such a signal is input into a memoryless nonlinearity characterized by an operator G then the output signal g G f can be written as a summation of disjoint waveforms according to g t gt t gt t i L _ g t t e I 0 11 I where as detailed in Section W - o t e I 1tIi Implication If all signals from a signaling random process can be written as a summation of disjoint signals then this result can be used to define each of the corresponding output signals after a memoryless transformation and hence define a signaling random process for the output random process. As the power spectral density of a signaling random process is well defined see Theorem such an approach allows the output power spectral density to be .
