随机过程分析
§2 功率谱密度与自相关函数的关系2 j t 由 X T (ω , ξ i ) = ∫ T xT (t , ξ i )e ω dt 与 | X T (ω , ξ i ) | = X T (ω , ξ i ) X T ( ω , ξ i ) 可证： 可证： 1 G X ( ω ) = lim E [ | X T ( ω , ξ i ) |2 ] T →∞ 2T T 1 T jω t1 xT ( t1 , ξ i )e dt1 ∫ xT ( t 2 , ξ i )e jω t 2 dt 2 ] = lim E [ T T →∞ 2T ∫ T 1 T T E [ X T ( t1 ) X T ( t 2 )]e jω ( t 2 t1 )dt1dt 2 = lim T → ∞ 2T ∫ T ∫ TT
∴ RXT ( t1 ,t2 ) = E[ XT ( t1 )XT ( t2 )], T ≤ ( t1 ,t2 ) ≤ T1 G X ( ω ) = lim T → ∞ 2T
∫ ∫ T
+T
+T
T
R X T ( t 1 , t 2 )e jω ( t 2 t1 ) dt 1 dt 2