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Biorthogonal decomposition: Difference between revisions
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e.g., travelling waves have a structure such as ''cos(kx-ωt)''; however, most propagating waves can still be recognised clearly by their distinct footprint in the biorthogonal modes (provided there are not too many): a travelling wave will produce a pair of modes with similar amplitude and a 90° phase difference. | e.g., travelling waves have a structure such as ''cos(kx-ωt)''; however, most propagating waves can still be recognised clearly by their distinct footprint in the biorthogonal modes (provided there are not too many): a travelling wave will produce a pair of modes with similar amplitude and a 90° phase difference. | ||
== Relation with signal | == Relation with signal covariance == | ||
Assuming the signals ''j<sub>1</sub>'' and ''j<sub>2</sub>'' have zero mean (their temporal average is zero), their [[:Wikipedia:Covariance|covariance]] is defined as: | |||
:<math>C(j_1,j_2) = \sum_i {Y(i,j_1)Y(i,j_2)},\!</math> | :<math>C(j_1,j_2) = \sum_i {Y(i,j_1)Y(i,j_2)},\!</math> | ||
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:<math>C(j_1,j_2) = \sum_k {\lambda_k^2 \phi_k(j_1)\phi_k(j_2)}</math> | :<math>C(j_1,j_2) = \sum_k {\lambda_k^2 \phi_k(j_1)\phi_k(j_2)}</math> | ||
The technique is therefore ideally suited to perform cross | The technique is therefore ideally suited to perform cross covariance analyses of multipoint measurements. | ||
By multiplying this expression for the | By multiplying this expression for the covariance matrix ''C'' with the vector φ<sub>k</sub> it is easy to show that the topos φ<sub>k</sub> are the eigenvectors of the covariance matrix ''C'', and λ<sub>k</sub><sup>2</sup> the corresponding eigenvalues. | ||
== See also == | == See also == | ||