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Unformatted text preview: 1 For a deterministic signal x (t), the spectrum is well defined: If represents its Fourier transform, i.e., if then represents its energy spectrum. This follows from Parsevals theorem since the signal energy is given by Thus represents the signal energy in the band (see Fig 18.1). ( ) X ( ) ( ) , j t X x t e dt + = 2  ( )  X 2 2 1 2 ( )  ( )  . x t dt X d E + + = = (181) (182) 2  ( )  X ( , ) + Fig 18.1 18. Power Spectrum t ( ) X t PILLAI 2  ( ) X Energy in ( , ) + + 2 However for stochastic processes, a direct application of (181) generates a sequence of random variables for every Moreover, for a stochastic process, E { X ( t )  2 } represents the ensemble average power (instantaneous energy) at the instant t . To obtain the spectral distribution of power versus frequency for stochastic processes, it is best to avoid infinite intervals to begin with, and start with a finite interval ( T , T ) in (181). Formally, partial Fourier transform of a process X ( t ) based on ( T , T ) is given by so that represents the power distribution associated with that realization based on ( T , T ). Notice that (184) represents a random variable for every and its ensemble average gives, the average power distribution based on ( T , T ). Thus . ( ) ( ) T j t T T X X t e dt = 2 2  ( )  1 ( ) 2 2 T j t T T X X t e dt T T = (183) (184) , PILLAI 3 represents the power distribution of X ( t ) based on ( T , T ). For wide sense stationary (w.s.s) processes, it is possible to further simplify (185). Thus if X ( t ) is assumed to be w.s.s, then and (185) simplifies to Let and proceeding as in (1424), we get to be the power distribution of the w.s.s. process X ( t ) based on ( T , T ). Finally letting in (186), we obtain T 1 2 ( ) 1 2 1 2 1 ( ) ( ) . 2 T XX T T j t t T T P R t t e dt dt T = 1 2 1 2 2 ( ) * 1 2 1 2 ( ) 1 2 1 2  ( )  1 ( ) { ( ) ( )} 2 2 1 ( , ) 2 T XX T T j t t T T T T T j t t T T X P E E X t X t e dt dt T T R t t e dt dt T = = = 2 2 2   2 2 1 ( ) ( ) (2  ) 2 ( ) (1 ) T XX XX T j T T j T T P R e T d T R e d = = 1 2 1 2 ( , ) ( ) XX XX R t t R t t = 1 2 t t = (185) (186) PILLAI 4 to be the power spectral density of the w.s.s process X ( t ). Notice that i.e., the autocorrelation function and the power spectrum of a w.s.s Process form a Fourier transform pair, a relation known as the WienerKhinchin Theorem . From (188), the inverse formula gives and in particular for we get From (1810), the area under represents the total power of the process X ( t ), and hence truly represents the power spectrum. (Fig 18.2)....
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 Fall '09
 Jacobs

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