Unformatted text preview: + F1 ˆ F2 + F2 ˆ F Ni + F Ni H X1
η1 X2 +
η2 X No +
η No + ˆ X No ˆ X1 ˆ X2 Figure 56. System Model: Multiple Inputs
(519) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + In order to develop an estimation of the frequency response function for the multiple input case, a number of averages N avg will be used to minimize the random errors (variance). This can be easily accomplished through use of intermediate measurment of the auto and cross power spectrums as deﬁned in Equations (5.8) through (5.11). Additional matrices, constructed from the auto and cross power spectrums need to be deﬁned as follows. Note that each function and, therefore, each resulting matrix is a function of frequency. Input/Output Cross Spectra Matrix X1 X GXF 11 2 . * * [GXF] = {X }{F } H = . [F 1 F 2 . . F * i ] = N . . GXF N o 1 X No . . . . . . . . GXF 1Ni . . GXF N o Ni (5.22) Input Cross Spectra Matrix F1 F GFF 11 2 . * * [GFF] = {F }{F } H = . [F 1 F 2 . . F * i ] = N . . GFF Ni 1 F Ni . . . . . . . . GFF 1Ni . . GFF Ni Ni (5.23) The frequency response functions can now be estimated for the three algorithms as follows: H 1 Algorithm: Minimize Noise on Output ( η ) [H] N o × Ni {F } Ni × 1 = {X } N o × 1 − {η } N o × 1 [H] {F } {F } H = {X } {F } H − {η } {F} H (5.24) (5.25) (520) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + H H [H] N o × Ni {F } Ni × 1 {F }1 × Ni = {X } N o × 1 {F }1 × Ni (5.26) The above relationship can be concisely stated as: [H][GFF] = [GXF] [H] = [GXF][GFF]−1 where:
• (5.27) (5.28) [ ] H = Complex conjugate transpose (Hermitian Matrix) In the experimental procedure, the input and response signals are measured, and the averaged cross spectra and auto spectra necessary to create the [GXF] and [GFF] matrices are computed. If the computation of ordinary, multiple, or partial coherence functions will be required, then the diagonal elements of the output cross spectrum matrix [GXX] must be computed also. Equation (5.27) is valid regardless of whether the various inputs are correlated. Unfortunately, there are a number of situations where the input cross spectrum matrix [GFF] may be singular for speciﬁc frequencies or frequency intervals. When this happens, the inverse of [GFF] will not exist and Equation (5.28) cannot be used to solve for the frequency response function at those frequencies or in those frequency intervals. A computational procedure that solves Equation (5.28) for [H] should therefore monitor the rank the matrix [GFF] that is to be inverted, and desirably provide direction on how to alter the input signals or use the available data when a problem exists. The current approach for evaluating whether the inputs are sufﬁciently uncorrelated at each frequency involves determining the principal/virtual forces using principal component analysis [8]. This will be covered in a later section. H 2 Algorithm: Minimize Noise on Input ( υ ) [H] N o × Ni { {F} Ni × 1 − {υ } Ni × 1 } = {X } N o...
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 Fall '11
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 Signal Processing, GFF Ni Ni, GFF Ni

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