The generalized triangular decomposition

Abstract

Abstract. Given a complex matrix H, we consider the decomposition H = QRP * , where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diagonal for R can be achieved that satisfies Weyl's multiplicative majorization conditions:where K is the rank of H, σ i is the i-th largest singular value of H, and r i is the i-th largest (in magnitude) diagonal element of R. Given a vector r which satisfies Weyl's conditions, we call the decomposition H = QRP * , where R is upper triangular with prescribed diagonal r, the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

ZF-DFE transceiver design for time-varying MIMO channels using space-time generalized triangular decomposition

“…The generalized triangular decomposition [8]: Let H ∈ C J×N have rank M with non-zero singular values…”

“…, than the complexity of directly applying GTD on H, which is O(K 2 M (M + J)) [8]. Moreover, ST-GTD is more suitable for the design of ST-GTD transceivers without channel prediction which will be discussed in section 5.…”

“…Based on the GTD [8], Work supported in parts by the ONR grant N00014-08-1-0709 and the California Institute of Technology.…”

“…diag(x) is a diagonal matrix with the entries of x on the diagonal. u ≺ + v and u ≺ × v denote additive and multiplicative majorization, respectively [5], [8].…”

Resource Allocation in Correlated MIMO Systems

“…MIMO systems have been the subject of intensive research over the past years due to the fact that it supports rather higher data rates and reliability than SISO systems [24]. In this doctoral thesis, the MIMO system has been modelled as follows:…”