User soup - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:37:57Z http://mathoverflow.net/feeds/user/29691 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116054/signal-model-classification-between-two-possbile-candidates Signal model classification between two possbile candidates Soup 2012-12-11T07:35:52Z 2013-02-21T21:22:00Z <p>How to decide the most possible signal model between two model candidates besed on the received signal vector?</p> <p>Assume the received signal vector is \$y\$, the possible signal model candidates could be:</p> <p>(1) \$y = Ax+n\$, or</p> <p>(2) \$y = Bx+n\$,</p> <p>in which \$x\$ is the transmitted signal vector, and \$A\$ and \$B\$ are the system matrices for signal model candidate-1 and candidate-2 respectively, and \$n\$ is the Gaussian noise vector.</p> <p>If \$y\$,\$A\$ and \$B\$ are all known, and the noise covariance matrix is \$E[nn^H] = w^2I\$, in which \$w^2\$ stands for noise power, and \$I\$ is the identity matrix, how to decide the most possible signal model candidate between the two. What's the optimal solution?</p> <p>Thanks for any discussions.</p> http://mathoverflow.net/questions/116054/signal-model-classification-between-two-possbile-candidates/116301#116301 Answer by Soup for Signal model classification between two possbile candidates Soup 2012-12-13T17:24:57Z 2012-12-13T17:24:57Z <p>A good reference book could be</p> <p>Hero, A. “Signal Detection and Classification” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999</p> <p>in which different scenarios were discussed. Besides, some reference books are also helpful.</p> http://mathoverflow.net/questions/115476/how-to-calculate-the-inverse-of-the-sum-of-two-eigen-decomposed-matrices How to calculate the inverse of the sum of two eigen-decomposed matrices Soup 2012-12-05T07:34:45Z 2012-12-10T19:02:57Z <p>The are two eigen-decomposed matrices \$A\$ = \$U_1\$\$V_1\$\$U_1\$\$^H\$, \$B\$ = \$U_2\$\$V_2\$\$U_2\$\$^H\$, in which \$V_1\$ and \$V_2\$ are the eigen-matrices formed by the non-negative eigenvalues and the eigenvalues are all less than 1, \$U_1\$ and \$U_2\$ are unitary matrices formed by the eigenvectors. Is there any efficient way (including any efficient iterative solution) to calcualte the following vector?</p> <p>\$y\$ = (\$A\$+\$B\$+\$I\$)\$^{-1}\$\$x\$</p> <p>in which \$I\$ is the identity matrix, and \$x\$ could be an arbitrary vector.</p> <p>Thanks for any discussions.</p> http://mathoverflow.net/questions/115476/how-to-calculate-the-inverse-of-the-sum-of-two-eigen-decomposed-matrices/116008#116008 Comment by Soup Soup 2012-12-11T07:08:28Z 2012-12-11T07:08:28Z So we only need to inverse a matrix with the size = rank(\$B\$), right? This is a good solution. Thank you very much! http://mathoverflow.net/questions/115476/how-to-calculate-the-inverse-of-the-sum-of-two-eigen-decomposed-matrices Comment by Soup Soup 2012-12-07T17:32:19Z 2012-12-07T17:32:19Z What if \$A\$ or \$B\$ have very low rank (but do not share a large eigenspace)? Any solution for this scenario? Thanks. http://mathoverflow.net/questions/115476/how-to-calculate-the-inverse-of-the-sum-of-two-eigen-decomposed-matrices Comment by Soup Soup 2012-12-07T06:42:04Z 2012-12-07T06:42:04Z You are right. x is a given vector. U1 and U2 are the unitary matrices, while V1 and V2 are diagonal matrices, formed by the eigen-vectors of A and B respectively.