the following a short introduction to RD functions is given. It is shown how FRFs can be estimated from the RD functions, and finally the bias reduction is illustrated on a simulated case with two degrees-of-freedom. THE RANDOM DECREMENT ALGORITHM Some initial considerations about using the RD technique for modal data processing have been given in [6] and [7]. An introduction to the general RD technique is given in [8]. The Random decrement function is defined as the conditional expectation of a stochastic Process ( ) x t ) ( ( )) ( ) E ( k k XX x t T x t D (3) and from a time series the corresponding estimate is found as the conditional mean N k k k XX x t T x t N D 1 ) ( ( )) ( 1 ( ) ˆ (4) where the triggering condition ( ( )) k T x t is given by for instance x t a T x t x t a T x t k k k k ( ( )): ( ) ( ( )): ( ) (5) This defines an auto RD function where the averaging and the triggering are performed on the same time series. Corresponding cross RD functions can be defined and estimated as N k k k XY k k XY N k k k YX k k YX x t T y t N x t T y t D D y t T x t N y t T x t D D 1 1 ) ( ( )) ( 1 ( ) ˆ ) ( ( )) ( ) E ( ) ( ( )) ( 1 ( ) ˆ ) ( ( )) ( ) E ( (6) ESTIMATING FRF’s FROM RANDOM DECREMENT FUNCTIONS If we assume the classical linear input-output relation between ( ) x t and ( ) y t t y t h t x d ) ( ) ( ( ) (7) then it is easy to show that the following relations exist between the RD functions h t D d D h t D d D XY YY XX YX ( ) ) ( ( ) ( ) ) ( ( ) (8) Now taking the Fourier transform of the RD functions defines the Fourier transform pairs 499
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