Schur Recursions Research Articles

A Reduced-Complexity Partial-Interference-Cancellation Group Decoder for STBCs

In this letter, we propose a reduced-complexity implementation of partial interference cancellation group decoder with successive interference cancellation (PIC-GD-SIC) by employing the theory of displacement structures. The proposed algorithm exploits the block-Toeplitz structure of the effective matrix and chooses an ordering of the groups such that the zero-forcing matrices associated with the various groups are obtained through Schur recursions without any approximations. We show using an example that the proposed implementation offers a significantly reduced computational complexity compared to the direct approach without any loss in performance.

Multidimensional Nonlinear Schur Parametrization of NonGaussian Stochastic Signals, Part Two: Generalized Schur Algorithm

In the first part l1r of this paper we stated the multidimensional nonlinear Schur parametrization problem for higher-order (and non-Gaussian) stochastic sequences as a generalization of the linear prediction/innovations filter problem for second-order stochastic sequences. The subject of this paper is a multidimensional nonlinear generalization of the linear Schur parametrization procedure for second-order stochastic sequences. We state and solve the problem geometrically, in the space of generalized (multi-indexed) matrices. We derive multidimensional nonlinear Levinson and Schur recursions, resulting in a complete parametrization schema for higher-order nonstationary stochastic sequences and show its connection to solution the non-linear orthogonal approximate filter problem of the Volterra–Wiener class. In the third part l2r we propose a low-complexity nonlinear Schur parametrization procedure, following from a multidimensional nonlinear generalization of the staircase matrix extension problem.

Convergence behavior of the Schur recursion in the Krein space for the J-spectral factorization

We present a "Krein-space version" of the Schur recursion for the J-spectral factorization which arises in H/sup /spl infin//-related problems. The most notable difference of the proposed Schur recursion from the ordinary one is that the proposed recursion can handle temporary changes of the inertia during the process. We show that the Schur recursion in the Krein-space converges to a J-spectral factor exponentially under a suitable condition.

Effects of finite-precision conversion on linear predictive coefficients

Two novel methods are presented for improving the finite-precision conversion of linear prediction coefficients in the form of reflection coefficients. The first method is based on the improvement of the dynamic range of the intermediate variables in the Schur recursion using block exponential format. For the second method, improvement is achieved by changing the statistical properties of the quantisation noise incurred during conversion. This is achieved with the addition of low-power uniform uncorrelated noise during the conversion from lattice filter structure to direct-form filter structure. Statistical tests performed over a 1000 test frames reveal a significant reduction in both the variance and mean of the error compared with the standard fixed-point LPC filter.

Two-dimensional schur algorithm

In this paper, a novel 2-D Schur algorithm is developed as a natural extension of the 1-D Schur recursion. This lattice structure is based on Parker and Kayran‘s four-field lattice approach. Starting with given 2-D autocorrelation samples, four quarter-plane gapped functions are generated. Their linear combination is used to satisfy gap conditions and calculate 2-D lattice parameter reflection factors for the first stage. In order to determine the growing number of 2-D reflection coefficients at succesive stages, appropriately defined auxiliary gapped functions are introduced after the first order. The theory has been confirmed by computer simulations. In addition to developing the basic theory, the presentation includes a comparison between the proposed 2-D lattice structure and other existing four-field lattice structures.

Two-Dimensional Schur Algorithm

In this paper, a novel 2-D Schur algorithm is developed as a natural extension of the 1-D Schur recursion. This lattice structure is based on Parker and Kayran‘s four-field lattice approach. Starting with given 2-D autocorrelation samples, four quarter-plane gapped functions are generated. Their linear combination is used to satisfy gap conditions and calculate 2-D lattice parameter reflection factors for the first stage. In order to determine the growing number of 2-D reflection coefficients at succesive stages, appropriately defined auxiliary gapped functions are introduced after the first order. The theory has been confirmed by computer simulations. In addition to developing the basic theory, the presentation includes a comparison between the proposed 2-D lattice structure and other existing four-field lattice structures.

COMPARISON OF INVARIANT IMBEDDING AND LAYER-PEELING METHODS IN INVERSE PROBLEM

This paper examines an implementation of a layer-peeling method to inverse scattering problems in the time domain, which is applied here to the case of one-dimensional lossless discontinuous medium. The method is based on the fast Schur recursion applied directly to the discrete problem. The method is tested on several numerical examples in order to compare its performance to the standard invariant imbedding algorithm. The reconstructions of media from both synthetic data and measured data are presented. The layer-peeling method is shown to be considerably faster than the invariant imbedding method without the loss of precision.

Multiscale signal processing: isotropic random fields on homogeneous trees

In this paper we consider isotropic processes indexed by the nodes of a homogeneous tree of order q. An (hanging) version of the qth order tree appears naturally when we consider successive filtering-and-decimation (by a factor of q) operations, as in multirate filtering and wavelet transforms. We derive Levinson and Schur recursions which provide us with a parametrization of an isotropic process via its reflection (or PARCOR) coefficient sequence. This can be done in an elegant way in the non-oriented setting, by making use of some general prediction errors. We state the counterpart of these results for the case of the tree, where we have forward and backward prediction errors, defined according to a notion of causality from coarse to fine scales. These oriented results represent generalizations of the Levinson and Schur recursions derived in for the case of the dyadic tree (i.e., q=2), which were used in to develop modeling and whitening filters for isotropic processes. >

Generalized Chandrasekhar recursions from the generalized Schur algorithm

Presents a new approach to the Chandrasekhar recursions and some generalizations thereof. The derivation uses the generalized Schur recursions, which are O(N/sup 2/) recursions for the triangular factorization of N/spl times/N matrices having a certain Toeplitz-like displacement structure. It is shown that when the extra structure provided by an underlying state-space model is properly incorporated into the generalized Schur algorithm, it reduces to the Chandrasekhar recursions, which are O(Nn/sup 2/) recursions for estimating the n-dimensional state of a time-invariant (or constant-parameter) system from N measured outputs. It is further noted that the generalized Schur algorithm factors more general structured matrices, and this fact is readily used to extend the Chandrasekhar recursions to a class of time-variant state-space models, special cases of which often arise in adaptive filtering.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Superresolution frequency estimation by alternating notch periodogram

A novel periodogram-based maximum-likelihood algorithm is proposed for a frequency estimation problem. It is called an alternating notch-periodogram algorithm (ANPA), since the original multidimensional maximum likelihood problem is decomposed into a sequence of much simpler one-dimensional problems of finding the peaks of notch periodograms. The ANPA achieves superresolution and a very low SNR threshold and can be computed and implemented in several efficient ways. First, with FFT and a concurrent Gram-Schmidt procedure using Schur's recursions, the notch periodogram can be computed without any costly eigendecomposition and matrix inversion. This approach can further lead to a mapping of the notch periodogram onto a VLSI architecture consisting mainly of a highly pipelined notch processor and two FFT processors. Second, without degrading the excellent performance of ANPA, the notch periodogram can be simplified and approximated to provide further computational reduction and implementational simplicity. >

A combined detection-estimation algorithm for the harmonic-retrieval problem

Unlike usual separated detection-estimation methods, an efficient algorithm that can estimate both the number and the parameters of multiple complex sinusoids is proposed. The combined detection-estimation operation is accomplished by the joint applications of the Maximum Likelihood (ML) estimation approach and Zhao's efficient detection criterion (EDC). The algorithm is iterative; at each iteration it needs simply to find the peak of a notch periodogram. With the aid of FFT and a concurrent Gram-Schmidt procedure using Schur's recursions, the notch periodogram can be efficiently computed without any costly matrix operation. This also facilitates a real-time implementation of the notch periodogram in terms of a highly pipelined notch processor cascaded by two FFT processors. Simulation results show that the algorithm not only has much higher detection rate than Wax's Minimum Description Length (MDL) method, but also achieves the CR bound at a lower threshold SNR than that of the Modified Forward Backward Linear Prediction (MFBLP) method proposed by Tufts and Kumaresan. Thus, our algorithm offers both good performance and high speed.

Multiscale autoregressive processes. I. Schur-Levinson parametrizations

In many applications it is of interest to analyze and recognize phenomena occurring at different scales. The recently introduced wavelet transforms provide a time-and-scale decomposition of signals that offers the possibility of such analysis. A corresponding statistical framework to support the development of optimal, multiscale statistical signal processing algorithms is described. The theory of multiscale signal representation leads naturally to models of signals on trees, and this provides the framework for investigation. In particular, the class of isotropic processes on homogeneous trees is described, and a theory of autoregressive models is developed in this context. This leads to generalizations of Schur and Levinson recursions, associated properties of the resulting reflection coefficients, and the initial pieces in a system theory for multiscale modeling. >

Hardware implementation of partitioned-parallel algorithms in linear prediction

Abstract In this paper, a partitioned-parallel strategy for the solution of the Toeplitz equations appearing in the linear prediction case is introduced. From the one extreme, i.e., the use of the Schur recursions for an order recursive implementation with O(1) processors which perform O(p2) MADs (multiplications and divisions), to the other extreme, i.e., the use of the same recursions for a fully parallel implementation with O(p) processors which perform O(p) MADs, there exist a compromise: The hardware designer can ‘cut’ the computational scheme into suitable partitions, which are executed one after the other, with all the computations of each partition organized in a parallel manner. This way he can achieve increased flexibility, especially in relation to the model order, which can become totally independent of the available number of processors. Moreover, in this paper an abatement methodology is introduced which significantly reduces the number of multiplications of the above computational schemes, as well as the overall algorithm complexity in the case of the parallel design.

Real-time computation of the maximum likelihood criterion for some superimposed signals problems

A highly concurrent Gram–Schmidt procedure based on the Schur recursions is proposed for computing the ML criterion of some superimposed signals problems. This approach greatly reduces the computational burden and leads to a pipelined combiner-lattice structure which can compute the ML criterion in real time.

Reconstruction of finite one-dimensional objects from long- and short-exposure images

A new procedure is described for reconstructing one-dimensional discrete finite objects. The procedure uses both long-exposure and short-exposure images rather than only long-exposure or short-exposure images. It reconstructs the minimum-phase part of the object in two steps. First, it estimates the true autocorrelation function of the object by using the noisy correlation data that can be derived from the short-exposure images. Next, it invokes the fast Levinson or Schur recursions of one-dimensional linear least-squares prediction to estimate the minimum-phase part of the object from the estimated object correlation data. The procedure reconstructs the all-pass part of the object by initially estimating the number of factors in that part by using the noisy long-exposure data. A least-squares approach is then used to find those factors. The object is finally reconstructed by combining the estimates of its minimum-phase and all-pass factors.

Fixed-point error analysis of the lattice and the Schur algorithms for the autocorrelation method of linear prediction

Levinson recursions, Schur recursions, and lattice recursions provide three different ways of computing reflection coefficients for a stationary time series. Of these three approaches, only the Schur and lattice algorithms are scaled algorithms that can be implemented in fixed-point arithmetic. The authors study the finite arithmetic properties of the Schur and lattice recursions. They derive error variances for the reflection coefficients and present experimental results which agree very closely with the analytical results. They show analytically and experimentally that lattice recursions have a significant accuracy advantage over Schur recursions when the problem is ill-conditioned or, equivalently, when the absolute values of the reflection coefficients are close to one. This result is not surprising since the lattice recursions, which compute reflection coefficients by QR-factoring a Toeplitz data matrix, are 'square-root' recursions with respect to Schur recursions, which compute reflection coefficients by Cholesky-factoring a Toeplitz correlation matrix that is quadratic in the data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fast algorithms for computing QR and Cholesky factors of Toeplitz operators

Levinson, Lattice and Schur recursions provide alternative ways of computing reflection coefficients for stationary time series. The authors show that when a Toeplitz correlation matrix is a product of two Toeplitz data matrices, as in the correlation method of linear prediction, then the Levinson recursions can be used to derive the Lattice recursions and the Lattice recursions can be used to derive the Schur recursions. They explore the close relation between QR and Cholesky algorithms in the Toeplitz case, and present the underlying matrix factorization. >

Linear statistical models for stationary sequences and related algorithms for Cholesky factorization of Toeplitz matrices

We review the Levinson-Durbin recursions for computing the Cholesky factors of the inverse of a Toeplitz correlation matrix R, and the Schur recursions for computing the Cholesky factors of R itself. We show that both algorithms may be imbedded in a single vector lattice recursion. Then, depending only on how the initial conditions are set, one gets either the Levinson-Durbin or the LeRoux-Gueguen recursions for computing reflection coefficients. We extend the analysis to treat the Toeplitz nonsymmetric case, and place the corresponding algorithms into the same framework. We also review the relation between the vector lattice recursions and their implementations in a serial lattice structure. Then we show how the factorizations may be run backwards to compute the Cholesky factors and correlations from the reflection coefficients. This generalizes a result usually attributed to Robinson and Treitel. One of our main purposes is to emphasize that the Levinson-Durbin recursions for going back-and-forth between correlations, reflection coefficients, and autoregressive filter parameters may be replaced with a dual set of recursions for going back-and-forth between correlations, reflection coefficients, and moving average filter parameters. When the correlation sequence is ARMA (p, p), then we show how the Schur recursions simplify to the Morf-Sidhu-Kailath algorithm for computing Kalman gains. Finally, the MSK algorithm may be specialized to the Rissanen algorithm for factoring a pure MA correlation sequence.

Some properties of the Schur recursion for the direct computation of the matricial spectral factor

The Schur recursion is equivalent to the Levinson-Wiggins-Robinson algorithm and computes the reflection matrices (or Schur parameters) associated with the autocorrelation function of a stationary process. It produces a sequence that converges towards the causal minimum phase spectral factor of the corresponding matricial power spectrum density. In this paper, we intend to extend some of its properties already known in the case of scalar signals to the case of vectorial signals: firstly, it is shown that a minimum phase estimation of the spectral factor is computed at each step of the recursion, this result is based on the canonical form of the Schur recursion; secondly, it is shown that all the data computed during the recursion are bounded which may be useful for a fixed point implementation, and a method for performing this fixed point implementation is proposed.

The Schur algorithm and its applications

The Schur algorithm and its time-domain counterpart, the fast Cholseky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems. These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves. In this paper, the Schur and fast Chokesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.