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Biorthogonal decomposition: Difference between revisions
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This is not always the most appropriate assumption: | This is not always the most appropriate assumption: | ||
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 correlation == | |||
The correlation between signals ''j<sub>1</sub>'' and ''j<sub>2</sub>'' is defined as: | |||
:<math>C(j_1,j_2) = \sum_i {Y(i,j_1)Y(i,j_2)},\!</math> | |||
Using the above expansion of ''Y'' and the orthogonality relations, it is easy to show that the topos ''φ<sub>k</sub>'' are the eigenvectors of the correlation matrix ''C'', and ''λ<sub>k</sub><sup>2</sup>'' the corresponding eigenvalues. | |||
== See also == | == See also == | ||