# A singular value decomposition updating algorithm for subspace tracking

8 A method of claim 3 comprising a step of estimating a rank of the matrix transfer function from one of the orthogonal projections of the output vector onto the left singular vectors and the orthogonal projections of the input vector onto the right singular vectors for detecting a change of said rank.

11 A method of claim 3 wherein the step of determining updated estimates of singular vectors comprises the steps of:computing an orthogonal correction vector for each singular vector; and, adding the orthogonal correction vector to the initial estimate of a corresponding singular vector.

The superscript ~ ]H in equation (2) denotes the complex conjugate transpose operation.[ 11 ] The left and right singular vectors form two orthogonal sets of vectors, which define Mdimensional receiver and K-dimensional transmitter spaces, respectively, with M typically equal to or exceeding K.

Transmission of a j'" right singular vector by the transmitter results in the reception of the j'" left singular vector scaled by a gain coefficient equal to the j'" singular value 6; and some additive noise: [l~] S=U~ ~ P=6~V~ u~ . discloses a wireless MIMO system employing the SVD at the transmitter to determine eigen-modes, i.e., spatial subchannels, of the MIMO channel and to derive a first set of steering vectors used to "precondition" modulation symbols.

The transfer matrix H of a M1M0 wireless channel can change with time and in dependence on channel frequency.[08] SVD of the transfer matrix H has the form [09] H = VEU~' .

(2) [10] Columns of the unitary matrices V - [v ~ ~ ~ ~ v," ~ and U - ~u, ~ ~ ~ u,~ ~ are commonly referred to as left and right singular vectors v; and u;, respectively, corresponding to singular values o-; .

These algorithms typically require computation of the full SVD and have complexity of at least O(M'), i.e. 1 is a flowchart of an SVD updating algorithm using orthogonal expansion coefficients.[35] FIG.

20 A method of adaptive signal processing in multiple-input multiple-output communication systems comprising the steps of:transmitting information signal in multiple signal streams with a multiple-output transmitter;sampling multiple data streams received by a multiple-input receiver to form an output vector;obtaining an input vector related to the output vector and to a transfer matrix;obtaining initial estimates of left singular vectors, right singular vectors and singular values of a singular-value decomposition of the transfer matrix;determining updated estimates of the left singular vectors, right singular vectors and the singular values using the initial estimates thereof and projections of the input and output vectors on the initial estimates of the right and left singular vectors; and, using the updated estimates of the left and right singular vectors and the singular values for adaptively extracting the information signal from the multiple signal streams received by the receiver and/or for adaptively emitting the information signal in multiple signal streams by the transmitter.17 A method of claim 16 wherein the steps of recursively computing an orthogonal correction vector for each of the p left and p right singular vectors comprise the steps of recursively calculating weighted projections of the input and output vectors onto subspaces of increasing and decreasing dimensions.18 A method of claim 3 wherein the step of determining updated estimates of the singular values includes, for each of the p singular values, the steps of weighting the initial estimate of the singular value with a first weighting factor to obtain a weighted initial estimate of the singular value, wherein the first weighting factor is less or equal to one;calculating a weighted correction factor from the orthogonal projections of the input and output vectors onto the initial estimates of the left and right singular vectors corresponding to the particular singular value using a second weighting factor of less than one; and adding together the weighted initial estimate of the singular value and the weighted correction factor.12 A method of claim 11 wherein the step of computing an orthogonal correction vector for each singular vector further comprises the step of computing (p-1) orthogonal expansion coefficients for each singular vector from the orthogonal projections of the output and input vectors onto initial estimates of the right and left singular vectors using the initial estimates of the singular values.13 A method of claim 11 wherein p is less than min(K, M), further comprising the steps of:projecting the input vector and the output vector onto subspaces orthogonal to the right and left signal subspaces respectively to form residual input and output vectors; and normalizing the residual input and output vectors to form normalized residual input and output vectors.