DISPLACEMENT STRUCTURES OF COVARIANCE MATRICES, LOSSLESS SYSTEMS, ANDNUMERICAL ALGORITHM DESIGN

Low displacement rank theory underlies many fast algorithms designed f or structured covariance matrices. Some of these have gained notoriety for their numerical instability problems, particularly fast least-squ ares algorithms. Recent studies have shown that instability is not inh erent to fast algorithms, but rather comes from the violation of backw ard consistency constraints. This paper thus details the connection be tween covariance matrices of a given displacement inertia and lossless rational matrices, as well as the role of this connection in numerica lly consistent algorithms. This basic connection allows displacement s tructures to be parametrized via a sequence of rotation angles obtaine d from a lossless system. The utility of this approach is that, irresp ective of errors in the rotation parameter set, they remain consistent with a positive definite matrix of a prescribed displacement inertia. This property in turn may be rephrased as meaningful forms of backwar d consistency in numerical algorithms. The rotation parameters then ta ke the form of Givens or Jacobi angles applied to data, in contrast to classical approaches which directly manipulate dyadic decompositions of the displacement structure. The concepts are illustrated in popular signal processing applications. In particular, these connections lend clear insight into the stable computation of reflection coefficients of Toeplitz, systems, and also serve to resolve the numerical instabil ity problem of fast least-squares algorithms.