A singular value decomposition updating algorithm for subspace tracking

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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.wherein the step of computing an orthogonal correction vector for each singular vector further comprises the step of computing (p1) 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.using the normalized residual input and output vectors scaled by the corresponding residual orthogonal expansion coefficients to compute the orthogonal correction vectors for the p left and p right singular vectors. Provisional Patent Application No: 60/534,189 filed Jan.The left and right singular vectors form two orthogonal sets of vectors, which define M-dimensional receiver and K-dimensional transmitter spaces, respectively, with M typically equal to or exceeding K. (3) Therefore, knowledge of the SVD of the transfer matrix enables the establishment of up to K orthogonal independent communication channels between the receiver and the transmitter, and the deciphering of the transmitted signal at the receiver site. with frequency, the SVD needs to be periodically updated to maintain channel orthogonality and avoid inter-channel interference. In many applications, the MIMO channel and the corresponding transfer matrix changes slowly, so that the sets of the singular vectors do not change much between consecutive updates. 452-463, May 2003, disclosed an adaptive algorithm for use in MIMO wireless communication systems, wherein weight arrays are used at the transmitter and the receiver for establishing the orthogonal communications channels. In accordance with another aspect of this invention, the step of computing an orthogonal correction vector for each singular vector comprises the steps of recursively computing an orthogonal correction vector for each of the p left singular vectors and recursively computing an orthogonal correction vector for each of the p right singular vectors using the orthogonal projections of the output and input vectors onto initial estimates of the right and left singular vectors and the initial estimates of the singular values.Therefore, instead of performing a full SVD computation each time an SVD update is required, tracking techniques may be employed when the singular vectors and singular values at an nstep. Zeidler, in an article entitled “Feedback assisted transmission subspace tracking for MIMO systems,” IEEE Sel. The weight arrays are periodically jointly updated to adapt to changing transmission conditions. Wolniansky, entitled “Detection algorithm and initial laboratory results using VBLAST space-time communication architecture,” Elect. In accordance with another aspect of the invention, a method is provided for adaptive signal processing in multiple-input multiple-output communication systems comprising the steps of: a) transmitting information signal in multiple signal streams with a multiple-output transmitter; b) sampling multiple data streams received by a multiple-input receiver to form an output vector; c) obtaining an input vector related to the output vector and to a transfer matrix; d) obtaining initial estimates of left singular vectors, right singular vectors and singular values of a singular-value decomposition of the transfer matrix, e) 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; f) 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. The algorithm employs one-sided transformations, and therefore provides a cheap alternative to earlier-developed updating algorithms based on two-sided transformations.The present algorithm as well as the corresponding systolic implementation is therefore rpughly twice as fast as the former method, while the tracking properties are preserved.

The method can be applied to any system that can be described by the linear matrix equation (1) with a transfer matrix H that is either constant or varies in dependence on time or some other parameter, e.g.

In many cases, for example when some of the singular values are too small for reliable detection, only a subset of the singular vectors and singular values needs to be updated; such periodic updating of a subset of the singular vectors is sometimes referred to in the art as subspace tracking. However, the Kavić method does not provide the left singular vectors, which again can be computed from equation (2) requiring a matrix multiplication of complexity O(K), which can be unacceptable for applications requiring frequent updates. The Banister method requires computation of a binary feedback decision, which is used to increment or decrement the transmitter weight array. The present invention provides a recursive, low complexity method for adaptively estimating some or all of the left and right singular vectors and the corresponding singular values of a transfer matrix H.

However, most prior-art subspace tracking techniques are based on eigenvalue decomposition (EVD), wherein only the left singular vectors v). A method for updating the right singular vectors and eigenvalues, i.e., the eigendecomposition of RI, was disclosed in an article by A. Yang entitled “A new efficient subspace tracking algorithm based on singular value decomposition,” Proc. Furthermore, this method has an advantage of low feedback throughput requirements, at the cost of complexity at the transmitter and receiver; it requires at least one matrix multiplication at the receiver, as well as one matrix addition and a Gram-Schmidt orthogonalization at the transmitter, and has the complexity of at least O(M). Neither the transfer matrix H, which relates an input, or signal, vector s to an output, or received vector r in accordance with equation (1), nor its SVD defined by equation (2) are explicitly computed.

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