By R. Lynn Kirlin, William J. Done
This quantity is meant to offer the geophysical sign analyst adequate fabric to appreciate the usefulness of knowledge covariance matrix research within the processing of geophysical signs. A historical past of uncomplicated linear algebra, facts, and basic random sign research is thought. This reference is exclusive in that the information vector covariance matrix is used all through. instead of facing just one seismic information processing challenge and featuring a number of equipment, the focus during this e-book is on just one primary technique - research of the pattern covariance matrix offering many seismic facts difficulties to which the technique applies. This quantity will be of curiosity to many researchers, offering a style amenable to many detailed functions. It deals a various sampling and dialogue of the idea and the literature built to this point from a standard viewpoint.
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They point out that any perturbation to signal subspace eigenvectors vi that lie inside the signal subspace ⑀s introduces no error in the estimation of signal subspace. Rather, only the component ␦v i֊ of the perturbation of vi that is orthogonal to ⑀s needs consideration. Following the notation of Clergeot et al. 15) iϭ1 be the projection operator that projects a vector x onto the signal subspace. 16) i ϭ rϩ1 be the orthogonal subspace projection operator. From the sample covariance matrix Cx, we obtain an eigenvector vˆ i ϭ v i ϩ ␦v i , where vi is associated with a distinct eigenvalue i of R, the true covariance matrix.
1989, Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bounds: IEEE Trans. , Speech and Sig. , 37, 1703-1720. Kirlin, R. , 1991, A note on the effects of narrowband and stationary signal model assumptions on the covariance matrix of sensor array data vectors: IEEE Trans. Signal Processing, 503-506. Pillai, S. , 1989, Array signal processing: Springer-Verlag Inc. Scharf, L. , 1991, Statistical signal processing: Addison-Wesley Publ. Corp. 50. org/ 50 Chapter 5 Temporal and Spatial Spectral Analysis R.
Org/ 27 tor is composed of a scaled constant plus a spatially and temporally white noise vector: x i ϭ s ( t i ) 1 ϩ n i , i ϭ 1, 2, … , p , where 1 ϭ (1 1... 1)T of length M. With all xi as columns of X, we find the eigenstructure of XXH, an M ϫ M matrix. , p. , 1 ϭ E s ϩ n2 , i ϭ n2 , i ϭ 2 , … , M . 4, v1 ϭ s. Now note that SVD would have found both eigenstructures. 32) where V are the eigenvectors of XHX(p ϫ p), and U are the eigenvectors of XXH(M ϫ M). Then U 1 ϭ s, v 1 ϭ 1 , and ⌳ 11 ϭ ( E s ϩ n2 ) 1 / 2 ; these are the singular vector u1 of X for the first example, the eigenvector v1 of XHX for the current example, and the first singular value 1 of the SVD of X.