Power Spectral Density Analysis In this chapter, general results for the power spectral density that facilitate evaluation of the power spectral density of specific random processes are given. First, the nature of the Fourier transform on the infinite interval is discussed and a criterion is given for the power spectral density to be bounded on this interval. Second, the use of an alternative power spectral density function that can be defined for the case where a signal consists of a sum of orthogonal or disjoint waveforms is discussed. . | 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 4 Power Spectral Density Analysis INTRODUCTION In this chapter general results for the power spectral density that facilitate evaluation of the power spectral density of specific random processes are given. First the nature of the Fourier transform on the infinite interval is discussed and a criterion is given for the power spectral density to be bounded on this interval. Second the use of an alternative power spectral density function that can be defined for the case where a signal consists of a sum of orthogonal or disjoint waveforms is discussed. Third a theorem is proved that specifies when signal components outside of the interval 0 T can be included when evaluating the power spectral density. Including such signal components can greatly simplify analysis. Fourth the cross power spectral density is defined and bounds on its level are established. Fifth the power spectral density of the sum of an infinite number of random processes is derived. Sixth the power spectral density of a periodic signal is derived and is shown to have the expected form namely impulsive component at integer multiples of the fundamental frequency. Finally the power spectral density of a random process containing a periodic and a nonperiodic component is derived and it is shown for the infinite interval that the periodic and nonperiodic components can be treated separately. BOUNDEDNESS OF POWER SPECTRAL DENSITY To prove subsequent results it is necessary to demarcate those random processes that have a bounded power spectral density on the interval 0 oo from those that do not. Clearly if for all f e R there exists k To e R such 92 BOUNDEDNESS OF POWER SPECTRAL DENSITY 93 that VT To X T f kJT then lim Gx T f lim X T f k2 f eR T Note any signal that is periodic or contains a periodic component .
