Nonlinear system theory(BOOK)

# Ω 1 ω n m f r uu n m t 1 t n m 103 and using the

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) ( ω 1 , . . . , ω n +m ) = F [ R uu ( n +m ) ( t 1 , . . . , t n +m )] (103) and using the system function defined previously gives S ˆ y n y m ( ω 1 , . . . , ω n +m ) = H nsym ( ω 1 , . . . , ω n ) H msym ( ω n + 1 , . . . , ω n +m ) S uu ( n +m ) ( ω 1 , . . . , ω n +m ) (104) Repeating the derivation leading to (79) gives, in the present setting S y n y m ( ω 1 , ω 2 ) = (2 π ) n +m 2 1 _________ −∞ S ˆ y n y m ( γ 1 , . . . , γ n +m ) δ 0 ( ω 1 −γ 1 . . . −γ n ) δ 0 ( ω 2 −γ n + 1 . . . −γ n +m ) d γ 1 . . . d γ n +m (105) Furthermore, for stationary random process inputs, the partial output power spectral density is given by S y n y m ( ω ) = (2 π ) n +m 1 1 _________ −∞ S ˆ y n y m ( γ 1 , . . . , γ n +m ) δ 0 ( ω−γ 1 . . . −γ n ) d γ 1 . . . d γ n +m (106) Of course, this formula checks with (84) for the case m = n . Now assume that the input is real, stationary, zero-mean, Gaussian, and with power spectral density S uu ( ω ). Substituting (87) and (104) into (106) gives S y n y m ( ω ) = 0 , n +m odd (107) and S y n y m ( ω ) = (2 π ) ( n +m 2) / 2 1 ___________ p Σ −∞ H nsym ( γ 1 , . . . , γ n ) H msym ( γ n + 1 , . . . , γ n +m ) δ 0 ( ω−γ 1 . . . −γ n ) j , k Π n +m S uu ( γ j ) δ 0 ( γ j + γ k ) d γ 1 . . . d γ n +m , n +m even (108) The reduction of this expression to more explicit form is a combinatorial problem of some complexity because the integrand lacks symmetry. I will be content to work out the terms that give the output power spectral density for polynomial systems of degree 3. 230
Example 5.12 To compute S yy ( ω ) for the case of a degree-3 polynomial system, the terms S y n y m ( ω ) must be computed for n , m = 1,2,3. But it is evident that S y 1 y 2 ( ω ) = S y 2 y 1 ( ω ) = S y 2 y 3 ( ω ) = S y 3 y 2 ( ω ) = 0 For n = m = 1,2,3 the partial output power spectral densities have been calculated previously, and are given in (64), (93), and (94). For n = 1 and m = 3, (108) gives S y 1 y 3 ( ω ) = 2 π 1 ___ p Σ −∞ H 1 ( γ 1 ) H 3 sym ( γ 2 , γ 3 , γ 4 ) δ 0 ( ω−γ 1 ) j , k Π 4 S uu ( γ j ) δ 0 ( γ j + γ k ) d γ 1 . . . d γ 4 = 2 π 1 ___ −∞ H 1 ( γ 1 ) H 3 sym ( γ 2 , γ 3 , γ 4 ) δ 0 ( ω−γ 1 ) S uu ( γ 1 ) S uu ( γ 3 ) δ 0 ( γ 1 + γ 2 ) δ 0 ( γ 3 + γ 4 ) d γ 1 . . . d γ 4 + 2 π 1 ___ −∞ H 1 ( γ 1 ) H 3 sym ( γ 2 , γ 3 , γ 4 ) δ 0 ( ω−γ 1 ) S uu ( γ 1 ) S uu ( γ 2 ) δ 0 ( γ 1 + γ 3 ) δ 0 ( γ 2 + γ 4 ) d γ 1 . . . d γ 4 + 2 π 1 ___ −∞ H 1 ( γ 1 ) H 3 sym ( γ 2 , γ 3 , γ 4 ) δ 0 ( ω−γ 1 ) S uu ( γ 1 ) S uu ( γ 2 ) δ 0 ( γ 1 + γ 4 ) δ 0 ( γ 2 + γ 3 ) d γ 1 . . . d γ 4 Performing the integrations gives S y 1 y 3 ( ω ) = 2 π 3 ___ H 1 ( ω ) S uu ( ω ) −∞ H 3 sym ( −ω , γ , −γ ) S uu ( γ ) d γ In a similar fashion S y 3 y 1 ( ω ) can be computed. Alternatively, the easily proved fact that S y 3 y 1 ( ω ) = S y 1 y 3 ( −ω ) can be used to obtain S y 3 y 1 ( ω ) = 2 π 3 ___ H 1 ( −ω ) S uu ( ω ) −∞ H 3 sym ( ω , γ , −γ ) S uu ( γ ) d γ Now, collecting together all the terms gives the expression 231
S yy ( ω ) = H 1 ( ω ) H 1 ( −ω ) S uu ( ω ) + 2 π 3 ___ H 1 ( ω ) S uu ( ω ) −∞ H 3 sym ( −ω , γ , −γ ) S uu ( γ ) d γ + 2 π 3 ___ H 1 ( −ω ) S uu ( ω ) −∞ H 3 sym ( ω , γ , −γ ) S uu ( γ ) d γ + 2 π 1 ___ δ 0 ( ω ) −∞ −∞ H 2 sym ( γ 1 , −γ 1 ) H 2 sym ( γ 2 , −γ 2 ) S uu ( γ 1 ) S uu ( γ 2 ) d γ 1 d γ 2 + π 1 __ −∞ H 2 sym ( ω−γ , γ ) H 2 sym ( −ω + γ , −γ ) S uu ( γ ) S uu ( ω−γ ) d γ + (2 π ) 2 6 _____ −∞ −∞ H 3 sym ( ω−γ 1 −γ 2 , γ 1 , γ 2 ) H 3 sym ( −ω +

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• Spring '11
• Jung
• The Land, LTI system theory, Linear system, Nonlinear system, Homogeneous Systems, kernel, triangular kernel

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