310 M. Brehm and A. Deraemaeker is derived in the time step interval Œs;e withgwi 2 Rmw andmw De s C1. The symbol ıdenotes the Schur product (e.g., [13]). The vector w2Rmw represents the discretization of the window functionwof the support Œs;e . The vector of the jth degree of freedom of the excitationfj 2 RN is given for all discrete time steps Œ1;N withN De. The matrixqi;j 2 Rmw N is derived from the impulse response function related to the linear combiner i and the excitation degree of freedomj. qi;j D t 2 6 6 6 6 6 6 6 6 6 6 6 6 4 hi;j;s hi;j;s 1 hi;j;s 2 : : : 0 hi;j;sC1 hi;j;s hi;j;s 1 : : : 0 hi;j;sC2 hi;j;sC1 hi;j;s : : : 0 : : : hi;j;sC2 hi;j;sC1 : : : 0 hi;j;e 2 : : : hi;j;sC2 : : : 0 hi;j;e 1 hi;j;e 2 : : : : : : 0 hi;j;e hi;j;e 1 hi;j;e 2 : : : hi;j;e NC1 3 7 7 7 7 7 7 7 7 7 7 7 7 5 8n 0 W hi;j;n D0 (30.11) In a next step the discrete Fourier transformation is applied to Eq. (30.10) through a complex matrix operator B 2 C. mw 2 C1/ mw containing the coefficients [7, p. 50] .B/k;n D t exp # 2 N .k 1/.n 1/ (30.12) with the imaginary unit # D p 1for nD1;2; : : : ;mw andk D1;2; : : : ; mw 2 C1. This yields Fgwi DB 0 @ wı mf X jD1 qi;j fj 1 A (30.13) withFgwi 2 C. mw 2 C1/. Equation (30.13), which represents the Fourier transform of the ith linear combination for windowed responses, can be reformulated to Fgwi D mf X jD1 Z1i;j fj (30.14) using Z1i;j DB 11 N ˝w ıqi;j ; (30.15) where ˝denotes the Kronecker product. The matrix 11 N is an integer matrix of dimension 1 N only filled with the value 1. By combining the evaluations of Eq. (30.14) for all linear combinations i D1;2; : : : ;mg and all excitation degrees of freedomj D1;2; : : : ;mf in the random vector and linear deterministic operator, FNg1 D 2 6 6 6 6 4 Fgw1 Fgw2 : : : Fgwmg 3 7 7 7 7 5 and NZ1 D 2 6 6 6 6 4 Z11;1 Z11;2 : : : Z11;mf Z12;1 Z12;2 : : : Z12;mf : : : : : : : : : : : : Z1mg;1 Z1mg;2 : : : Z1mg;mf 3 7 7 7 7 5 ; (30.16) respectively, a simple linear relation FNg1 D NZ1 Nf with NZ1 2 C. mw 2 C1/mg Nmf (30.17) between the random discrete excitation time series Nf defined in Eq. (30.3) and the discrete Fourier transform of the linearly combined windowed response displacement signals represented by FNg1 can be derived.
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