Backward Prediction Errors Research Articles

Two-dimensional orthogonal lattice structures for autoregressive modeling of random fields

Two-dimensional orthogonal lattice filters are developed as a natural extension of the 1-D lattice parameter theory. The method offers a complete solution for the Levinson-type algorithm to compute the prediction error filter coefficients using lattice parameters from the given 2-D augmented normal equations. The proposed theory can be used for the quarter-plane and asymmetric half-plane models. Depending on the indexing scheme in the prediction region, it is shown that the final order backward prediction error may correspond to different quarter-plane models. In addition to developing the basic theory, the article includes several properties of this lattice model. Conditions for lattice model stability and an efficient method for factoring the 2-D correlation matrix are given. It is shown that the unended forward and backward prediction errors form orthogonal bases. A simple procedure for reduced complexity 2-D orthogonal lattice filters is presented. The proposed 2-D lattice method is compared with other alternative structures both in terms of conceptual background and complexity. Examples are considered for the given covariance case.

2D maximum likelihood spectrum estimation using orthogonal 2D lattice filters

A new method to obtain the 2D ML spectrum estimation using recently developed orthogonal 2D lattice filters for AR modelling is reported. It is shown that the reciprocal of the 2D ML estimate is equal to the average of the reciprocals of the AR spectra obtained from the unended orthogonal forward or backward prediction error filters in the 2D AR lattice structure.

Further improved 2-D lattice filter structure employing missing reflection coefficients

A new lattice filter structure to model two-dimensional (2-D) autoregressive (AR) fields is proposed. The proposed structure utilizes and extracts the information contained in the backward prediction error fields and their delayed versions. The main idea is to use two sets of reflection coefficients corresponding to two quadrant filters and to increase the number of reflection coefficients with the order of the lattice filter. Increasing the number of reflection coefficients at each stage produces a sufficient number of independent parameters to model AR fields up to order three, which is an improvement over the existing 2-D lattice filter structures. The improvement is confirmed by computer simulations. In addition, a relationship between the reflection coefficients and the AR coefficients is derived. It is also shown that the entropy contained in the backward prediction error field vector of the proposed structure is closer to the input entropy when compared to those contained in existing 2-D lattice filters.

Split-dimension vector quantization of Parcor coefficients for low bit rate speech coding

A novel split vector quantization (SVQ) scheme for low bit rate coding of speech signals is proposed. In this scheme, the LPC parameter vector, which is represented by Parcor coefficients, is split into small-dimension subvectors, and each subvector is sequentially quantized according to a multistage structure that resembles a segmented lattice filter. The forward and backward prediction residuals in the segmented filter are coupled across VQ stages. The quantizer in each stage operates on the principle of minimizing the forward and backward prediction error energies similar to linear predictive analysis. Simulation results show that the new split VQ scheme can achieve transparent quantization of LPC parameters at 25 b/frame.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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. >

A note on recursive maximum likelihood for autoregressive modeling

Rederives recursive maximum likelihood (RML) for an autoregressive (AR) time series using the Levinson decomposition. This decomposition produces a recursive update of the likelihood function for the AR parameters in terms of the reflection coefficients, prediction error variances, and forward and backward prediction errors. A fast algorithm for this recursive update is presented and compared with the recursive updates of the Burg (1975) algorithm. The comparison clarifies the connection between Burg's algorithm and RML.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An improved 2-D lattice filter and its entropy relations

Abstract In this paper, an improved lattice filter structure to model two-dimensional (2-D) autoregressive (AR) fields is presented. This work is the generalization of the three-parameter lattice filter developed by Parker and Kayran. The proposed structure generates four prediction error fields (one forward and three backward prediction error fields) at the first stage. After the first stage, two additional prediction error fields are generated using two of the backward prediction error fields at the output of the first stage. This leads to six prediction error fields whose linear combination defines the successive lattice stages and the reflection coefficients. A recursive relationship between the reflection coefficients of the lattice filter and the AR coefficients is derived. In addition, the new structure and the three-parameter lattice filter are compared from information-theoretic point of view. The entropy calculations are carried out for Gaussian distributed data. It is concluded that the new structure approximates the maximum entropy more closely compared to the three-parameter structure. The increase in entropy naturally leads to a more reliable and better modelling of AR data fields.

New multistage scheme for vector quantisation of parcor coefficients

A new multistage scheme for vector quantisation of PARCOR coefficients for low bit rate speech coding is proposed. The novelty of this scheme is that the lattice filter structure is preserved after the quantisation. This is achieved by coupling the forward and backward prediction errors across vector quantisation (VQ) stages. Simulation results show that the new scheme can achieve lower average distortion than other multistage VQ schemes.

On the duality between fast QR methods and lattice methods in least squares adaptive filtering

The authors show that fast QR methods and lattice methods in least squares adaptive filtering are duals and follow from identical geometric principles. Whereas the lattice methods compute the residuals of a projection operation via the forward and backward prediction errors, the QR methods compute instead the weights used in the projections. Within this framework, the parameter identification problem is solved using fast QR methods by showing that the reflection coefficients and tap parameters of a least squares lattice filter operating in the joint process mode are immediately available as internal variables in the fast QR algorithms. This parameter set can be readily exploited in system identification, signal analysis, and linear predictive coding, for example. The relations derived also lead to a fast least squares algorithm of minimal complexity that is a hybrid between a QR and a lattice algorithm. The algorithm combines the order recursive properties of the lattice approach with the robust numerical behavior of the QR approach. >

Adaptive spatial domain forward-backward predictors for bearing estimation

Adaptive algorithms are proposed for computing the coefficients of spatial domain predictors. The method presented uses the least-mean-squares approach to compute the coefficients of the predictors realized by using the tapped-delay-line and the escalator structures. The predictors presented differ from the conventional ones in that the relevant weights are updated such that the sum of the mean-squared values of the forward and the backward prediction errors is minimized. Using the coefficients of such spatial domain predictors yields improved linear predictive spatial spectrums. The algorithms are applied to the problem of estimating incident angles of multiple narrowband signals received by a linear array of sensors. Simulation results demonstrating the performances of the proposed methods are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Forward-backward LMS adaptive line enhancer

A new LMS (least-mean-square) adaptive line enhancer algorithm is presented. It makes use of forward and backward prediction errors to update the coefficient values. For a given feedback factor, the algorithm converges to the optimal Wiener solution with the same speed as for the LMS algorithm, but requires about twice the number of multiplications and additions. However, in the situation when the order of the enhancer is at least a few times larger than the number of sinusoids to be enhanced, or when the frequencies of the sinusoids to be enhanced are not close to 0 or 0.5, the misadjustment of the new algorithm is approximately half that of the LMS algorithm. >

Coefficient-constrained lattice structures for retrieving single-tone harmonic signals

Two modified lattice structures for retrieving a real, single-frequency harmonic signal from noisy data samples is presented. The first lattice structure comprises a two-stage cascaded lattice filter with the value of the second reflection coefficient constrained to unity. The first reflection coefficient, from which the sinusoidal frequency is estimated, is obtained by a procedure similar to that of Burg's method. In the noiseless case, this filter structure is able to compute exactly the sinusoidal frequency regardless of the initial phase or data record length. With the presence of white noise, however, it will yield a biased frequency estimate. To remedy this problem, a second lattice structure is proposed which imposes a unit-norm constraint on the resulting prediction error filter coefficients before minimising the resulting forward and backward prediction errors. It is shown that theoretically, even with the presence of white noise, this second lattice structure still yields unbiased estimates. A statistical performance analysis of these frequency estimation methods is derived and compared with simulation results.

A three step algorithm for ARMA modeling

A new ARMA estimation algorithm is proposed. It is based on a fundamental relationship which shows that the AR polynomial of an ARMA ( N, M) model belongs to the linear space spanned by the forward and backward linear predictors. This relationship allows us to construct an equivalent linear system with two inputs and the same output of the ARMA system. The inputs of this new system are the forward and backward linear prediction errors. As in this case the inputs and output are known, a least-squares identification algorithm is used to obtain the parameters of the system. These parameters define three polynomials. One of them is the AR polynomial. The other two converge asymptotically to the MA polynomial and to zero. Simple recursions are available to perform such a limiting operation.

Fast adaptive least squares algorithms for power spectral estimation

Power spectrum estimation is of great importance in various applications of signal processing, such as geophysics and communications. In this paper two new fast algorithms are presented that adaptively compute a least squares estimate of the power spectrum of a time series. This is achieved by modeling the input as an AR signal of order m and simultaneous minimization of the sum of the forward and backward prediction error energies. The first algorithm is of the 0 (m2) type, and the second of 0(m) requiring 9m multiplications and additions.

Application of Autoregressive Moving Average Parametric Modeling in Magnetic Resonance Image Reconstruction

The modeling of data is an alternative to conventional use of the fast Fourier transform (FFT) algorithm in the reconstruction of magnetic resonance (MR) images. The application of the FFT leads to artifacts and resolution loss in the image associated with the effective window on the experimentally-truncated phase encoded MR data. The transient error modeling method treats the MR data as a subset of the transient response of an infinite impulse filter (H(z) = B(z)IA(z)). Thus, the data are approximated by a deterministic autoregressive moving average (ARMA) model. The algorithm for calculating the filter coefficients is described. It is demonstrated that using the filter coefficients to reconstruct the image removes the truncation artifacts and improves the resolution. However, determining the autoregressive (AR) portion of the ARMA filter by algorithms that minimize the forward and backward prediction errors (e.g., Burg) leads to significant image degradation. The moving average (MA) portion is determined by a computationally efficient method of solving a finite difference equation with initial values. Special features of the MR data are incorporated into the transient error model. The sensitivity to noise and the choice of the best model order are discussed. MR images formed using versions of the transient error reconstruction (TERE) method and the conventional FFT algorithm are compared using data from a phantom and a human subject. Finally, the computational requirements of the algorithm are addressed.

Some properties of the Givens' reduction applied to linear prediction analysis of speech.

Described here are some physical interpretations for the computational process of the Givens' reduction, an efficient algorithm for obtaining the least-squares solution of a set of over-determined linear equations. These physical interpretations are brought into view by applying the Givens' reduction to the linear predictive analysis of speech, and would not have suggested themselves if the Givens' reduction were considered only in connection with the solution of an ordinary set of over-determined linear equations. The attractive advantages found in the computational process of the Givens' reduction are the following:(1) The forward and the backward prediction errors for each data sample are automatically obtained in the working matrix of the augmented Givens' reduction.(2) The direct time-update recursion for the K-parameter, the reflection coefficients in linear prediction analysis, is implicitly performed sample by sample.(3) An index representing the rank of the covariance matrix of the input data sequence is available in the working vector. These advantages come out from the physical interpretation of the corresponding parts of the working areas for the Givens' reduction. The behavior of these values in the working areas are empirically confirmed by actual analysis of speech.

Nonstationary analysis by bpes (backward prediction error space)

AbstractThis paper describes a method of non‐stationary analysis based on the geometrical orthogonal projection with least‐squares estimation for a linear model. In the past, local stationary analysis and adaptive methods which successively estimate the parameters have been used for this purpose. However, in application to the nonstationarity case as in biomedical signals these methods have the disadvantage that it is difficult to see the physical significance of the parameters. In consideration of this, the present paper defines the backward prediction error space (BPES) consisting of the backward prediction error vector in each dimension and proposes to handle the nonstationarity by expanding the property into the orthogonal states in each dimension. The backward prediction error vectors constitute an orthogonal system in the least‐squares estimation and the reflection coefficient is the parameter directly related to this system. The orthogonality can be interpreted in direct relation to the independence of physical functions. As a measure of the nonstationarity, the angle between the vectors is adopted in the BPES. When the parameters which were optimum in the stationary interval are applied to an arbitrary interval, the angle is an exact expression of the discrepancy due to the nonstationarity. By an example of analysis of biomedical signals it is verified that the angle is useful in evaluation of the nonstationarity, especially of that arising from the change of orders. Thus, it is anticipated that the deterministic nonstationary analysis by BPES will provide new information about biological systems.

On a modified algorithm for the autoregressive recovery of the acoustic impedance

In a recent paper, Walker and Ulrych (1983) presented an algorithm for the recovery of the acoustic impedance from band‐limited seismic reflection records. The approach used is based on the autoregressive (AR) modeling of the band‐limited frequency transform of the data. This modeling procedure allows prediction of both the high and low missing frequencies. The low frequencies, which are particularly important in the inversion for the acoustic impedance, are determined by considering the low‐frequency band as a gap of missing data which is centered at zero frequency. The gap is filled by minimizing the sum of the squared forward and backward prediction errors which result when the known spectral data are modeled as an AR process.

Reflection coefficient estimation using Cholesky decomposition

A discussion of linear prediction analysis methods that employ the Cholesky decomposition is given. Only one method insures stability of the synthesis filters by assuring that estimated reflection coefficients are bounded in magnitude by one. This estimate is generalized to one based on forward and backward prediction errors. An efficient computational algorithm based on Cholesky factorization is derived.