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# Where the input cross spectrum matrix gff may be

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Unformatted text preview: × 1 [H] { {F} − {υ } } {X} H = {X } {X } H H H [H] N o × Ni {F } Ni × 1 {X }1 × N o = {X } N o × 1 {X }1 × N o (5.29) (5.30) (5.31) (5-21) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + One problem with using the H 2 algorithm is that the solution for [H] can only be found directly using an inverse when the number of inputs N i and number of outputs N o are equal. Then: [ H ] [ GFX ] = [ GXX ] [ H ] = [ GXX ] [ GFX ]−1 (5.32) (5.33) H v Algorithm: Minimize Noise on Input and Output ( υ and η ) [H] N o × Ni { {F} Ni × 1 − {υ } Ni × 1 } = {X } N o × 1 − {η } N o × 1 [H] { {F} − {υ } } = {X } − {η } (5.34) (5.35) The solution for [H] is found by the eigenvalue decomposition of one of the following two matrices: [GFF] [GFFX p ] = [GXF p ] [GXX p ] [GXFF p ] = H [GXF p ] [GXF p ] H [GXX p ] [GXF p ] [GFF] (5.36) (N i +1) × (N i +1) (5.37) (N i +1) × (N i +1) Therefore, the eigenvalue decomposition would be: [GFFX p ] = [V ] Λ [V ] H Or: [GXFF p ] = [V ] Λ [V ] H where: • (5.38) (5.39) Λ = diagonal matrix of eigenvalues (5-22) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + Solution for the p − th row of the [H] matrix is found from the eigenvector associated with the smallest (minimum) eigenvalue. Note that the size of the eigenvalue problem is N i + 1 and that the eigenvalue solution must be repeated for each frequency. Note also that the complete solution process must be repeated for each response point X p . The frequency response functions associated with a single output p and all inputs is found by normalizing the eigenvector associated with the smallest eigenvalue. If [GFFX p ] is used, the eigenvector associated with the smallest eigenvalue must be normalized as follows: H p1 H p2 . = . H pNi −1 {V }λ min (5.40) If [GXFF p ] is used, the eigenvector associated with the smallest eigenvalue must be normalized as follows: −1 H p1 H = p2 . . H pNi {V }λ min (5.41) The concept of the coherence function, as deﬁned for single-input measurement, needs to be expanded to include the variety of relationships that are possible for multiple inputs. Ordinary coherence is deﬁned in this general sense as the correlation coefﬁcient describing the linear relationship between any two spectra. This is consistent with the ordinary coherence function that is deﬁned for single input, single output measurements. Great care must be taken in the interpretation of ordinary coherence when more than one input is present. The ordinary coherence of an output with respect to an input can be much less than unity even though the (5-23) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + linear relationship between inputs and outputs is valid, because of the inﬂuence of the other inputs. The ordinary coherence function can be formulated in terms of the elements of the matrices deﬁned previously. The ordinary coherence function between the pth output and the q th input can be computed from the follo...
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