Lattice Algorithm Research Articles

New schemes and theoretical analysis of the master-slave family of recursive identification algorithms

An analysis of the Master-Slave family of system identification algorithms is developed for the white noise input case. These methods try to avoid the biased parameter estimation problem of equation error methods in the presence of noisy data, while keeping the quadratic nature of the error variances to be minimized (a feature that is lost with the output error formulation). The Master-Slave techniques may be seen as off-line schemes that construct a sequence of equation error optimization problems, each one based on the solution to the previous one. Some conditions for the stationary points of the iteration are given. Examples are presented for plants with poles close to z = 1 where undesirable attractive convergent points arise. An alternative scheme based on interpolation conditions rather than minimizing error variances is proposed and shown to have a single stationary point corresponding to the plant parameters. Finally, a simplified structure and a lattice algorithm are presented for on-line implementation.

Enhanced detectability of small objects in correlated clutter using an improved 2-D adaptive lattice algorithm

Two-dimensional (2-D) adaptive filtering is a technique that can be applied to many image processing applications. This paper will focus on the development of an improved 2-D adaptive lattice algorithm (2-D AL) and its application to the removal of correlated clutter to enhance the detectability of small objects in images. The two improvements proposed here are increased flexibility in the calculation of the reflection coefficients and a 2-D method to update the correlations used in the 2-D AL algorithm. The 2-D AL algorithm is shown to predict correlated clutter in image data and the resulting filter is compared with an ideal Wiener-Hopf filter. The results of the clutter removal will be compared to previously published ones for a 2-D least mean square (LMS) algorithm. 2-D AL is better able to predict spatially varying clutter than the 2-D LMS algorithm, since it converges faster to new image properties. Examples of these improvements are shown for a spatially varying 2-D sinusoid in white noise and simulated clouds. The 2-D LMS and 2-D AL algorithms are also shown to enhance a mammogram image for the detection of small microcalcifications and stellate lesions.

STAR recursive least square lattice adaptive filters

The recursive least square lattice (LSL) algorithm based on the newly developed scaled tangent rotations (STAR) is derived. Similar to other recursive least square lattice algorithms for adaptive filtering, this algorithm requires only O(N) operations. This algorithm also preserves the desired properties of the STAR recursive least square (STAR-RLS) algorithm. Specifically, it can be pipelined at fine-grain level. To this end, a pipelined version of the STAR-LSL (referred to as PSTAR-LSL) is also developed. Computer simulations show that the performance of the STAR-LSL algorithm is as good as the QRD-LSL algorithm. The finite precision error properties of the STAR-LSL algorithm are also analyzed. The mean square error expressions show that the numerical error propagates from stage to stage in the lattice, and the numerical error of different quantities in the algorithm varies differently with /spl lambda/. This suggests that different word lengths need to be assigned to different variables in the algorithm for best performance. Finally, finite word length simulations are carried out to compare the performances of different topologies.

Zero-Phase Filter Bank and Wavelet Code r Matrices: Properties, Triangular Decompositions, and a Fast Algorithm

The class of digital filter banks (DFB‘s) and wavelets composed of zero-phase filters is particularly useful for image processing because of the desirable filter responses and the possibility of using the McClellan transformation for two-dimensional design. In this paper unimodular polyphase matrices with the identity matrix for Smith canonical form are introduced, and then decomposed to a product of unit upper and lower triangular or block-triangular matrices which define ladder structures. A fundamental approach to obtaining suitable unimodular matrices for one and two dimensions is to focus on the shift (translation) operators, as is done in the harmonic analysis discipline. Several matrix shift operators of different dimensions are introduced and their properties and applications are presented, most notable of which is that the McClellan transformation can be effected by a simple substitution of a 2\times 2 circulant matrix for the polynomial variable, w = (z + z^{-1})/2. Unimodular matrix groups and pertinent subgroups are identified, and these are observed to be subgroups of the special linear group over polynomials,SL(k[w]) . A class of coiflet-like wavelets containing the well-known wavelet, based on the Burt and Adelson filter, is decomposed by these methods and is seen to require only 3/2 multiplications/sample if a scaling property, introduced herein, is satisfied. Making use of certain paraunitary wavelets, coiflets, that are closely comparable to the zero-phase wavelets of this class, it is seen that, in these cases, the zero-phase ladder algorithm is twice as fast as the paraunitary lattice algorithm.

QRD-based multichannel adaptive lattice algorithms for the parameter identification problem

A pair of multichannel recursive least squares (RLS) adaptive lattice algorithms based on the order recrusive of lattice filters and the superior numerical properties of Givens algorithms is derived in this paper. The derivation of the first algorithm is based on QR decomposition of the input data matrix directly, and the Givens rotations approach is used to compute the QR decomposition. Using first a prerotation of the input data matrix and then a repetition of the single channel Givens lattice algorithm, the second algorithm can be obtained. Both algorithms have superior numerical properties, particularly the robustness to wordlength limitations. The parameter vector to be estimated can be extracted directly from internal variables in the present algorithms without a backsolve operation with an extra triangular array. The results of computer simulation of the parameter identification of a two-channel system are presented to confirm efficiently the derivation.

Lattice algorithms for recursive instrumental variable methods

We develop recursive lattice algorithms for the estimation of the AR parameters of an ARMA process using an instrumental variable (IV) such as z(n)=y2(n), which leads to AR estimates based on the third-order cumulants, or z(n)=y(n-q), which leads to AR estimates based on the higher-order Yule-Walker equations. In the non-recursive mode the resulting set of linear equations involves the cross-correlation matrix of the observed process y(n), and the instrumental process z(n) and hence this matrix is generally not Hermitian. Our time- and order-recursive algorithm leads to a pair of lattices, one excited by y(n) and the other by z(n), the lattices being coupled through order and time update equations. We show that IV-based AR model fitting is equivalent to linear prediction with non-conventional orthogonality conditions. Extensions to ARMA processes are discussed. The joint-process estimation problem is also treated. Some statistical analysis is carried out and convergence results are given when y(n) and z(n) can be modelled as stationary processes. The only assumption that we make about z(n), the associated process, is that the cross-correlation of y(n) and z(n) suffices for the estimation of the AR parameters, i.e. we assume that z(n) is an instrumental process. Particular choices of z(n) legal to estimates based on the higher-order cumulants, which are useful when the non-Gaussian signal is corrupted by additive coloured Gaussian noise.

Spherical trigonometry, Yule's PARCOR identity, and FRLS fully normalized lattice

Yule's PARCOR identity is recognized as the cosine law of spherical trigonometry. The six PARCORs propagated by the fully normalized FRLS lattice filter are the cosines of the six elements of a spherical triangle, and this lattice algorithm is one solution to a spherical triangle problem that arises naturally in navigation and astronomy. Exploiting this new geometric interpretation yields unnoticed (and potentially useful) recursions among FRLS quantities.

A lattice algorithm dual to the extended inverse QR algorithm

This paper presents a new lattice algorithm dual to the extended Inverse QR (IQR) algorithm recently derived by the authors. The original IQR and our extended IQR algorithms are transversal filter algorithms. We found that by assuming a spatial displacement property on the initial correlation matrix of reference data consistent with a time-shifting property of reference data stored in taps of a transversal filter, we could develop a fast version of the extended IQR algorithm. This fast IQR algorithm turns out to have a lattice filter structure instead of a transversal filter structure. Whereas the extended IQR algorithm requires O( M 2p ) computations per iteration, its fast version called a new hybrid QR/LLS algorithm requires O( Mp 2p ) computations per iteration, where M, the number of taps, is usually much greater than p, the number of channels. This new hybrid QR/LLS algorithm differs from Regalia and Bellanger's hybrid QR/LLS algorithm, because the former is dual to the IQR algorithm and is based on the use of variance-normalized a priori backwards prediction errors, while the latter is dual to the QR algorithm and is based on the use of variance-normalized a posteriori backwards prediction errors.

A highly modular adaptive lattice algorithm for multichannel least squares filtering

In this paper a highly modular adaptive lattice algorithm for multichannel least squares FIR filtering and multivariable system identification is presented. Multichannel filters with different number of delay elements per input channel are allowed. The main features of the proposed multichannel adaptive lattice least squares algorithm is the use of scalar only operations, multiplications/divisions and additions, and the local communication which enables the development of a fully pipelining architecture. The tracking capability and the numerical stability and accuracy of the proposed technique are illustrated by simulations.

Lattice Models for Pricing American Interest Rate Claims

This article establishes efficient lattice algorithms for pricing American interest-sensitive claims in the Heath, Jarrow, Morton paradigm, under the assumption that the volatility structure of forward rates is restricted to a class that permits a Markovian representation of the term structure. The class of volatilities that permits this representation is quite large and imposes no severe restrictions on the structure for the spot rate volatility. The algorithm exploits the Markovian property of the term structure and permits the efficient computation of all types of interest rate claims. Specific examples are provided.

Lattice Models for Pricing American Interest Rate Claims

ABSTRACTThis article establishes efficient lattice algorithms for pricing American interest‐sensitive claims in the Heath, Jarrow, and Morton paradigm, under the assumption that the volatility structure of forward rates is restricted to a class that permits a Markovian representation of the term structure. The class of volatilities that permits this representation is quite large and imposes no severe restrictions on the structure for the spot rate volatility. The algorithm exploits the Markovian property of the term structure and permits the efficient computation of all types of interest rate claims. Specific examples are provided.

Fractionally-spaced equalization using the lattice structure

Fractionally-spaced equalizers (FSE) are designed using the adaptive lattice algorithm. The design is compared with the lattice structure synchronous equalizer (SE). The effect of the sampler phase on the mean square error (MSE) is investigated. It is shown that the FSE is not affected by the sampler phase. Improved performance of the FSE over the SE design as applied to severe amplitude distortion channel is obtained when its time span is made equal to that of the synchronous equalizer.

Split Burg and covariance lattice algorithms

The number of multiplications in split lattice algorithms, related to the autocorrelation matrix, is reduced by 30 percent in comparison with classical lattice algorithms. Split version of the Burg method (BM) and the covariance lattice method (CLM) are presented. As in previously published split algorithms the number of multiplications in the proposed algorithms is reduced. >

Cascade lattice IIR adaptive filters

The feedback lattice filter forms, including the two-multiplier form and the normalized form, are examined with respect to their relationships to the feedback direct form filter. Specifically, the transformation matrix between the lattice forms and the direct form is derived; parameter and state relationships between the lattice forms and the direct form are therefore obtained. An IIR filter structure-the cascade lattice IIR structure-is constructed. Based on this structure, three IIR adaptive filtering algorithms in the two-multiplier form can then be developed following the gradient approach, the Steiglitz-McBride approach and the hyperstability approach. Convergence of these algorithms is theoretically analyzed using either the ODE approach or the hyperstability theorem. These algorithms are then simplified into forms computationally as efficient as their corresponding direct form algorithms. Relationships of the simplified algorithms to the direct form algorithms are also studied, which disclose a consistency in algorithm structure regardless of the filter form. Three normalized lattice algorithms can also be derived from the two-multiplier lattice algorithms. Experimental results show much improved performance of the normalized lattice algorithms over the two-multiplier lattice algorithms and the direct form algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Lattice algorithms for recursive least squares adaptive second-order Volterra filtering

This paper presents two computationally efficient recursive least-squares (RLS) lattice algorithms for adaptive nonlinear filtering based on a truncated second-order Volterra system model. The lattice formulation transforms the nonlinear filtering problem into an equivalent multichannel, linear filtering problem and then generalizes the lattice solution to the nonlinear filtering problem. One of the algorithms is a direct extension of the conventional RLS lattice adaptive linear filtering algorithm to the nonlinear case. The other algorithm is based on the QR decomposition of the prediction error covariance matrices using orthogonal transformations. Several experiments demonstrating and comparing the properties of the two algorithms in finite and "infinite" precision environments are included in the paper. The results indicate that both the algorithms retain the fast convergence behavior of the RLS Volterra filters and are numerically stable.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A space-and-time-efficient coding algorithm for lattice computations

We present an encoding algorithm for lattices that significantly reduces space requirements while allowing fast computations of least upper bounds and greatest lower bounds of pairs of elements. We analyze the algorithms for encoding, LUB and GLB computations, and prove their correctness. Empirical experiments reveal that our method is significantly more space efficient than the transitive closure method, and the saving becomes increasingly more important as the size of the lattice increases.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multichannel ARMA modeling by least squares circular lattice filtering

The paper makes an attempt to develop least squares lattice algorithms for the ARMA modeling of a linear, slowly time-varying, multichannel system employing scalar computations only. Using an equivalent scalar, periodic ARMA model and a circular delay operator, the signal set for each channel is defined in terms of circularly delayed input and output vectors corresponding to that channel. The orthogonal projection of each current output vector on the subspace spanned by the corresponding signal set is then computed in a manner that allows independent AR and MA order recursions. The resulting lattice algorithm can be implemented in a parallel architecture employing one processor per channel with the data flowing amongst them in a circular manner. The evaluation of the ARMA parameters from the lattice coefficients follows the usual step-up algorithmic approach but requires, in addition, the circulation of certain variables across the processors since the signal sets become linearly dependent beyond certain stages. The proposed algorithm can also be used to estimate a process from two correlated, multichannel processes adaptively allowing the filter orders for both the processes to be chosen independently of each other. This feature is further exploited for ARMA modeling a given multichannel time series with unknown, white input. >

The gradient adaptive split lattice algorithm

The authors present a new algorithm for adaptive filtering of autoregressive processes, based on a three-term recurrence that replaces the lattice equations of the gradient adaptive lattice (GAL) algorithm of Griffiths (1977) and requires only half as many multiplications per recursion. They also present a single-stage convergence analysis and a numerical example comparing the new algorithm to the GAL. Its performance is similar to the GAL, while requiring less than 2/3 as many multiplications and additions. >

A bounded-distance decoding algorithm for lattices obtained from a generalized code formula

A multistage decoding algorithm is given for lattices obtained from a multilevel code formula. The algorithm is shown to have the same effective error-correcting radius as maximum-likelihood decoding, so that the performance loss is essentially determined by the increase in the effective error coefficient, for which an expression is given. The code formula generalizes some previous multilevel constructions to constructions of known single-level binary lattices with many levels, and then to decoders for them with the proposed algorithm. The trade-off between complexity reduction and performance loss is evaluated for several known lattices and two new ones, indicating that the approach is effective provided the binary codes involved in the code formula are not too short. All codes used in our constructions are binary Reed-Muller codes. >

Prediction of Sound Pressure in the Application of Least Squares Lattice Algorithm to Active Noise Control for Automobile

Prediction of Sound Pressure in the Application of Least Squares Lattice Algorithm to Active Noise Control for Automobile