12 Identification of Independent Inputs and Their Spatial Positions 131 Fig. 12.2 Rank definition as a function of the window size Pos. 1 1.6 a b Pos. 2 Pos. 3 Pos. 4 Pos. 5 Pos. 6 Pos. 7 Pos. 8 Pos. 9 Pos.10 Pos.2 and 5 1.4 1.2 1 0.8 Subpeca angle [rad] 0.6 0.4 0.2 0 0 5 10 15 Frequency [Hz] 0 101 100 10-1 10-2 10-3 Mean subspace angle [rad] 2 4 6 Positions 8 10 20 25 Fig. 12.3 (a) Subspace angle vs. frequency and (b) mean subspace angle vs. position 12.3 On the Location Having identified the number of independent inputs the next question on their characterization is to determine their distribution in space. The localization can be ascertained by noting that the span of the matrices BFFT (!) is the same as that of the columns of the transfer matrix for the columns associated with the position of the inputs. While is evident that one can locate the position of the inputs by finding the combination of the columns of the transfer matrix of a model that has the same span of BFFT (!) this approach is combinatorial and is thus not scalable for models with a large number of dof when there are several inputs. Fortunately, however, it is not necessary to check all the possible combinations as it suffices to test each position individually. Figure 12.3a illustrates the subspace angle that the columns of the transfer matrix of the model
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