# The obvious definition of the psd of a stationary

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preferred to another. The obvious definition of the PSD of a stationary random process is the ensemble average of the power spectral densities for all the members in the ensemble (compare with (1.13)): S x ( ω ) = lim T →∞ | X T ( ω ) | 2 T where the bar denotes an ensemble average and X T ( ω ) is the Fourier transform of the truncated random process x ( t )Π( t/T ) : X T ( ω ) = integraldisplay −∞ x ( t ) e jωt dt For a complex process, we have | X T ( ω ) | 2 = X T ( ω ) X T ( ω ) = integraldisplay T/ 2 T/ 2 x ( t 1 ) e jωt 1 dt 1 integraldisplay T/ 2 T/ 2 x ( t 2 ) e jωt 2 dt 2 = integraldisplay T/ 2 T/ 2 integraldisplay T/ 2 T/ 2 x ( t 1 ) x ( t 2 ) e ( t 2 t 1 ) dt 1 dt 2 so that S x ( ω ) = lim T →∞ 1 T integraldisplay T/ 2 T/ 2 integraldisplay T/ 2 T/ 2 x ( t 1 ) x ( t 2 ) e ( t 2 t 1 ) dt 1 dt 2 The ensemble averaging and integrating operations may be interchanged, yielding S x ( ω ) = lim T →∞ 1 T integraldisplay T/ 2 T/ 2 integraldisplay T/ 2 T/ 2 ρ x ( t 2 t 1 ) e ( t 2 t 1 ) dt 1 dt 2 where the general complex form of the autocorrelation function ρ x ( τ ) has been used with the implicit assumption of wide-sense stationarity, so that the particular time interval span in question is irrelevant. The integral can be evaluated using a change of variables, with τ t 2 t 1 and τ = ( t 2 + t 1 ) / 2 . The corresponding region of integration will be diamond shaped, with only the τ coordinate appearing within the integrand. S x ( ω ) = lim T →∞ 1 T integraldisplay T T integraldisplay | τ | / 2 −| τ | / 2 ρ x ( τ ) e jωτ = lim T →∞ integraldisplay T T ρ x ( τ )(1 − | τ | /T ) e jωτ = integraldisplay −∞ ρ x ( τ ) e jωτ Consequently, we find that the PSD for a wide-sense stationary random process is the Fourier transform of its autocor- relation. This is known as the Wiener-Khinchine relation. It illustrates the fundamental equivalence of the PSD and the autocorrelation function for characterizing wide-sense stationary random processes. Even moment theorem for GRVs There is a very useful theorem that applies to the even moments of Gaussian random variables. According to this theorem, any 2 n th moment of a set of GRVs can be expressed in terms of all the permutations of the n th moments. This is best illustrated by example. Consider the fourth moment ( x 1 x 2 x 3 x 4 ) . According to the theorem, we may write ( x 1 x 2 x 3 x 4 ) ( x 1 x 2 )( x 3 x 4 ) + ( x 1 x 3 )( x 2 x 4 ) + ( x 1 x 4 )( x 2 x 3 ) 25

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The theorem holds for complex GRVs such as are represented by the outputs of quadrature radio receivers and is especially useful for predicting the mean and variance of quantities derived from them, like the signal power and autocorrelation function. In the next chapter, we begin the main material of the text, starting with the basic tenets of antenna theory. More complicated aspects of the theory emerge in subsequent chapters. Material from this introductory chapter will mainly be useful in the second half of the text, which covers noise, signals, and radar signal processing.
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