Recursive Least Squares Lattice Research Articles

Recursive Least-Squares Lattice Algorithm Combined With Secondary-Path Innovation and Lattice-Order Decision Algorithms for Active Noise Control

This paper proposes a new active noise control (ANC) system based on a recursive least-squares lattice (RLSL) algorithm by designing secondary-path innovation (SPI) and lattice-order decision (LOD) algorithms. The SPI algorithm associated with the power spectral factorization of the secondary path is designed to apply the RLSL algorithm to the ANC system without the filtered-input structure. The SPI algorithm whitens the error microphone signals into the virtual error signals just ahead of the secondary path to construct the virtual desired signals corresponding to the outputs of the lattice filter. The LOD algorithm is designed to operate the ANC system without knowing the tap length of the optimal finite impulse response (FIR) filter. The decision algorithm using the initial and final estimation errors of the lattice filter estimates the tap length of the optimal FIR filter by increasing the lattice order one-by-one until the final estimation error reaches below a threshold value. Even in environments where the tap length of the optimal FIR filter is unknown, the proposed ANC system with low computational complexity shows almost the same performance as the ANC system based on a filtered-input recursive least squares algorithm.

FPGA implementation of adaptive segmentation for non-stationary biomedical signals

A novel field programmable gate array (FPGA) implementation of adaptive segmentation for non-stationary biomedical signals is presented. The design uses Simulink-to-FPGA methodology and has been successfully implemented onto Xilinx Virtex II Pro device. The implementation is based on the recursive least-squares lattice (RLSL) algorithm using double-precision floating-point arithmetic and is programmable for users providing data length, system order and threshold selection functions. The implemented RLSL design provides very good performance in obtaining accurate conversion factor values with a mean correlation above 99% and detecting segment boundaries with high accuracy for both synthesised and real-world biomedical signals.

Optimal Estimation of the Acceleration of a Car Under Performance Tests

In this paper, a recursive least-squares lattice (RLSL) adaptive filter was used to carry out the optimal estimation of the relevant signal coming from an accelerometer placed in car under performance tests. Here, the signal of interest is buried in a broadband noise background where we have little knowledge of the noise characteristics. In addition, due to the fact that the noise and the relevant information sometimes share the same or a very similar frequency spectrum, it is very difficult to cancel the noise that corrupts the relevant information without causing that information to deteriorate. The results of the experiment are satisfactory and, in order to compare classical filtering with optimal adaptive filtering, the signal coming from the accelerometer was also filtered by using a third-order lowpass digital Butterworth filter. The results of comparing the aforementioned filters show that the optimal adaptive filter is superior to the classical filter. Here, a significant improvement of 22.4 dB in the signal-to-noise ratio (SNR) at the RLSL adaptive filter output was achieved. However, the improvement in the SNR at the Butterworth filter output was 6.1 dB, which shows very clear that the optimal adaptive filter performs much better than the classical one

Improving the Response of a Rollover Sensor Placed in a Car under Performance Tests by Using a RLS Lattice Algorithm

In this paper, a sensor to measure the rollover angle of a car under performance testsis presented. Basically, the sensor consists of a dual-axis accelerometer, analog-electronicinstrumentation stages, a data acquisition system and an adaptive filter based on a recursiveleast-squares (RLS) lattice algorithm. In short, the adaptive filter is used to improve theperformance of the rollover sensor by carrying out an optimal prediction of the relevant signalcoming from the sensor, which is buried in a broad-band noise background where we have littleknowledge of the noise characteristics. The experimental results are satisfactory and show asignificant improvement in the signal-to-noise ratio at the system output.

A reliable modeless mobile multi-sensor integration technique based on RLS-lattice

The last two decades have witnessed an increasing trend in integrating different navigation sensors for the overall purpose of overcoming the limitation of stand-alone systems. An example of this integration is the fusion of the global positioning system (GPS) and inertial navigation system (INS) for several navigation and positioning applications. Both systems have their unique features and shortcomings. Therefore, their integration offers a robust navigation solution. This paper introduces a novel multi-sensor system integration using a recursive least-squares lattice (RLSL) filter. The proposed system has a similar structure to the widely used KF. However, it has the major advantage of working without the need of either dynamic or stochastic models. Furthermore, no prior information about the covariance information of INS and GPS is required. The proposed RLSL filter parameters, similar to Kalman gain, are tuned recursively in the update mode utilizing the GPS velocity components. The RLSL, in turn, can filter out the high frequency noise associated with the INS. To test the capabilities of the proposed architecture, a field test was conducted in a land vehicle using a tactical grade INS system (the Honeywell HG1700) integrated with differential GPS measurements collected by a NovAtel OEM4 GPS receiver. The proposed system is examined during the availability of the GPS signal and with intentionally introduced GPS signal outages. The results indicate that the proposed RLSL system is robust in providing a reliable modeless INS/GPS integration module.

Frequency shift keyed narrowband interference rejection: optimal exponential weighting factor for the RLS algorithm

Previous work has shown that co-channel narrowband interference can limit the performance of direct sequence spread spectrum (DSSS) and high frequency (HF) systems. Narrowband interference (NBI) can be single tone, chirped or frequency shift keyed (FSK) in nature and numerous techniques for its removal have been proposed. Linear adaptive prediction filters based on autoregressive modelling have been suggested owing to their ability to perform in a non-stationary environment. In the FSK narrowband interference case, adaptive filters are susceptible to excess residual errors owing to instantaneous frequency step changes and the finite convergence time required for the filter to adapt to a new interference frequency. The signal degradation owing to this type of interference becomes greater in high SNR regimes and has been found to be a function of the frequency parameters of the FSK interference signal. This paper discusses the convergence and frequency tracking properties of the recursive least squares (RLS) adaptive lattice filter using a posteriori estimation errors in the presence of FSK narrowband interference. An optimal exponential weighting factor that balances convergence time and steady state error is derived for this case of NBI. Results are compared to those of the previously proposed fast converging minimum frequency error (FCMFE) RLS lattice filter.

Fast-converging minimum frequency error RLS lattice filter for narrowband interference with discrete frequency steps

This paper discusses the fundamental convergence and frequency tracking properties of the recursive-least-squares (RLS) lattice filter in the presence of narrowband interference (NBI) whose frequency varies in discrete steps. It is shown for filters of this type, that the residual forward energy (RFE) after a frequency transition is a function of the input signal-to-noise ratio (SNR), separation of the sequential frequencies and the filter time constant and is exponentially decaying in nature. Reducing the RFE is important in removing unwanted transient artefacts from the desired signal. The convergence behaviour of the RLS algorithm based on a posteriori estimation errors is analysed under a number of conditions by varying the SNR and frequency step size. In order to limit the impact of the RFE while maintaining a minimum frequency tracking error in steady conditions, a fast-converging minimum frequency error (FCMFE) RLS lattice filter is suggested. For comparison, a least-mean-square (LMS) based gradient-adaptive lattice (GAL) filter is also analysed for this class of narrowband interference.

A New Approach to the Generalized Transfer Function and Its Application to Biomedical Signal Analysis

In this paper, we present a method for modeling an electrocardiogram (ECG) signal using time-varying parameters by considering that the signal is generated by a linear, time-varying (LTV) system with a stationary white noise input, and then by estimating the time-varying coefficients of the LTV system. In effect, since the ECG signal is considered to be non-stationary, this method is based on the Wold-Cramer representation of a non-stationary signal. The relationship between the generalized transfer function of an LTV system and the time-varying coefficients of the difference equation of a discrete-time system is not addressed clearly in the literature; in this paper, we propose a new approach to this problem and apply it for modeling a human ECG signal. We first derive a relationship between the system generalized transfer function and the time-varying parameters of the system. Then we develop an algorithm to solve for the system time-varying parameters from the time-frequency kernel of the system output using the time-varying autocorrelation function (TVACF) and by modifying the modified least-square (MLS) and Durbin's approximation algorithms. A comparison between the proposed algorithm and the recursive least-square (RLS) and RLS lattice (RLSL) algorithms is considered. Computer simulations illustrating the effectiveness of our algorithm are presented when the signal is embedded in noise.

Tracking ill-conditioning for the RLS-lattice algorithms

The numerical performance of lattice-based adaptive signal processing algorithms is shown to involve the conditioning of a 2/spl times/2 matrix whose off-diagonal elements contain reflection coefficients. Degraded algorithmic performance for the a posteriori recursive least squares lattice (RLSL) is shown to be attributed to the ill-conditioning of this matrix, Theoretical results are given which may be used to separate the conditioning of the underlying problem from issues concerning algorithmic stability. Although the results are not restricted to the all-pole case, for simplicity the authors make use of this well understood example since the condition number of the autocorrelation matrix will become arbitrarily close to singularity as the poles of an all-pole filter approach the unit circle. For a second-order prediction problem, four case studies of varying conditioning are provided which demonstrate the appropriateness of the theoretical bounds which analytically describe the sensitivity to perturbations in the residual update recursions. The paper illustrates the use of numerical linear algebra analysis techniques to better understand the numerical performance of algorithms in signal processing as well as emphasising that numerical performance is a function of the problem's conditioning as well as algorithmic stability.

Tracking ill-conditioning for the RLS-lattice algorithms

The numerical performance of lattice-based adaptive signal processing algorithms is shown to involve the conditioning of a 2 × 2 matrix whose off-diagonal elements contain reflection coefficients. Degraded algorithmic performance for the a posteriori recursive least squares lattice (RLSL) is shown to be attributed to the ill-conditioning of this matrix. Theoretical results are given which may be used to separate the conditioning of the underlying problem from issues concerning algorithmic stability. Although the results are not restricted to the all-pole case, for simplicity the authors make use of this well understood example since the condition number of the autocorrelation matrix will become arbitrarily close to singularity as the poles of an all-pole filter approach the unit circle. For a second-order prediction problem, four case studies of varying conditioning are provided which demonstrate the appropriateness of the theoretical bounds which analytically describe the sensitivity to perturbations in the residual update recursions. The paper illustrates the use of numerical linear algebra analysis techniques to better understand the numerical performance of algorithms in signal processing as well as emphasising that numerical performance is a function of the problem's conditioning as well as algorithmic stability.

Adaptive filtering, modelling and classification of knee joint vibroarthrographic signals for non-invasive diagnosis of articular cartilage pathology.

Interpretation of vibrations or sound signals emitted from the patellofemoral joint during movement of the knee, also known as vibroarthrography (VAG), could lead to a safe, objective, and non-invasive clinical tool for early detection, localisation, and quantification of articular cartilage disorders. In this study with a reasonably large database of VAG signals of 90 human knee joints (51 normal and 39 abnormal), a new technique for adaptive segmentation based on the recursive least squares lattice (RLSL) algorithm was developed to segment the non-stationary VAG signals into locally-stationary components; the stationary components were then modelled autoregressively, using the Burg-Lattice method. Logistic classification of the primary VAG signals into normal and abnormal signals (with no restriction on the type of cartilage pathology) using only the AR coefficients as discriminant features provided an accuracy of 68.9% with the leave-one-out method. When the abnormal signals were restricted to chondromalacia patella only, the classification accuracy rate increased to 84.5%. The effects of muscle contraction interference (MCI) on VAG signals were analysed using signals from 53 subjects (32 normal and 21 abnormal), and it was found that adaptive filtering of the MCI from the primary VAG signals did not improve the classification accuracy rate. The results indicate that VAG is a potential diagnostic tool for screening for chondromalacia patella.

A fast transversal filtering algorithm for pole-zero modeling

This paper describes a pole-zero (ARMA) modeling of discrete time linear systems ùsing a recursive least-squares (RLS) fast transversal filter (FTF) algorithm. This ARMA FTF algorithm is derived using geometric projections. The geometric projection approach gives insight and useful interpretation of various filters that form the algorithm. This algorithm can exactly estimate unknown filter coefficients. Simulation results are presented to show the rapid convergence speed and the numerical accuracy of the algorithm. We show that this algorithm converges more accurately than the ARNA FTF algorithm of Ardalan and Faber (1988). This algorithm also requires less computations than RLS lattice filters and the algorithm of Ardalan and Faber. Dieser Beitrag beschreibt eine Pol-Nullstellen (ARMA) Modellierung zeitdiskreter linearer Systeme mit einem schnellen rekursiven Least Squares (RLS) Transversalfilteralgorithms (FTF). Die Herleitung dieses ARMA-FTF-Algorithmus geschieht mit Hilfe geometrischer Projektionen. Der Ansatz basierend auf geometrischen Projektionen ermöglicht die Einsicht und eine nützliche Interpretation verschiedener Filter, welche im Algorithmus zusammenwirken. Der vorgestellte Algorithmus ermöglicht eine genaue Schätzung der unbekannten ARMA-Systemparameter. Die Simulationsergebnisse belegen die rasche Konvergenz und die numerische Stabilität des Verfahrens. Wir zeigen, daβ dieser Algorithmus genauer konvergiert als das ARMA-FTF-Verfahren von Ardalan und Faver (1988). Darüber hinaus erfordert das neue Verfahren einen geringeren Rechenaufwand als RLS-Lattice-Algorithmen und ebenfalls einen geringeren Rechenaufwand als das Verfahren von Ardalan und Faber. Cet article décrit un modèle pôles-zéros (ARMA) de systèmes linéaires à temps discret utilisant un algorithme rapide de filtre transversal (FTF) aux moindres carrés récursif (RLS). Cet algorithme ARMA FTF est dérivé à l'aide de projections géométriques. Cette approche de projection géométrique donne une meilleure vision et des interprétations utiles des divers filtres qui forment l'algorithme. Cet algorithme est capable d'estimer exactment des coefficients de filtre inconnus. Nous présentons des résultats de simulation pour montrer la rapidité de convergence et la précision numérique de l'algorithme. Nous montrons qu'il converge plus précisément que l'algorithme de filtrage ARMA FTF de Ardalan et Faber (1988). Il requièrt également moins de calculs que les algorithmes de filtre en treillis RLS et de Ardalan et Faber.

A numerically stable fast transversal filtering algorithm for pole-zero modelling

A numerically stable pole-zero (ARMA) modelling of discrete time linear systems is described using a recursive-least-squares (RLS) fast transversal filter (FTF) algorithm. This numerically stable ARMA FTF algorithm is derived using geometric projections and the error feedback procedure. This algorithm can exactly estimate unknown filter coefficients. Simulation results are presented to show the rapid convergence speed, numerical accuracy and stability of the algorithm. This algorithm is shown to converge more accurately and rapidly than the ARMA FTF algorithm of Ardalan and Faber (1988). The computaticnal requirements of this algorithm are compared with RLS lattice filters and the Ardalan-Faber algorithm.

Pole-zero modeling of speech using fast transversal filtering algorithm

This paper describes a pole-zero (ARMA) modeling of speech using a recursive-least-squares (RLS) fast transversal filter (FTF) algorithm. This ARMA FTF algorithm can estimate unknown input excitation and the estimated input is used to determined the parameters of the pole-zero model. This algorithm is derived using geometric projections. The geometric projection approach gives insight and useful interpretation of various filters that form the algorithm. We give a performance evaluation of the proposed algorithm by applying to synthetic and natural speech spectral estimations. This algorithm accurately represents spectral peaks and valleys of speech and requires less computations than RLS lattice filters and ARMA FTF algorithm of Ardalan and Faber (1988). Additionally, this algorithm can also be applied to other signal processing areas where the input is unknown.

Dual RLS lattice joint process estimation algorithm for a time-varying ARMA speech model

An algorithm which estimates both the input excitation and the parameters of a time-varying autoregressive moving-average (ARMA) speech production model is proposed. Here, the AR and MA coefficients are estimated by two independent recursive least squares (RLS) lattice joint process estimators, respectively, and the input excitation is estimated by a bootstrap from one of the two estimators. From the experimental spectral envelope estimations for the natural speech signals, it is shown that the proposed algorithm yields accurate parameter estimation of a slowly time-varying ARMA speech model.

Constrained-input/constrained-output stability for adaptive RLS lattice filters

A new approach to characterizing the stability of adaptive filters is presented. The notion of constrained-input/constrained-output (CICO) stability is introduced as a generalization of the standard notion of bounded-input/bounded-output (BIBO) stability. This new notion involves a set of constraints on the filter data (i.e., signals and parameters) that, unlike boundedness, are specific to the filter in consideration. The set of all data that satisfy the constraints is the feasibility domain of the adaptive filter. The feasibility domain of two particular adaptive filters is established: (i) the normalized recursive-least-squares (RLS) lattice, and (ii) the unnormalized RLS lattice/.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Detection of coronary occlusions using autoregressive modeling of diastolic heart sounds

Previous studies have indicated that diastolic heart sounds may contain information useful in the detection of occluded coronary arteries. In this study, recordings of diastolic heart sound segments were modeled by autoregressive (AR) methods including the adaptive recursive least-square lattice (RLSL) and the gradient lattice predictor (GAL). Application of the Akaike criterion demonstrated that between 5 and 15 AR coefficients are required to completely describe a diastolic segment. The reflection coefficients, prediction coefficients, zeros of the polynomial of the inverse filter, and the AR spectrum were determined over a number (N = 20-30) of diastolic segments. Preliminary results indicate that the averaged AR spectrum and the zeros of the inverse filter polynomial can be used to distinguish between normal patients and those with coronary artery disease.

Convergent and roundoff error properties of reflection coefficients in adaptive spatial recursive least squares lattice algorithm

The spatial recursive least-squares lattice (RLSL) algorithm is considered and the convergent properties as well as the finite-precision roundoff effects of the reflection coefficients are studied in detail. It is shown that when the forgetting factor lambda is set equal to one, the reflection coefficients can converge to a constant with probability one with the rate of convergence being O(1/T/sup 3/2/) where T is the number of terms involved. When O< lambda <1, the reflection coefficients converge weakly. Furthermore, the variable of the reflection coefficient is decreased by increasing lambda . Thus, under stationary conditions lambda should be taken as large as possible, while under nonstationary conditions lambda should be taken such that the depth of memory in the system matches the rate of change of nonstationarity. Under a finite-wordlength constraint, lambda should be taken to be less than some threshold, which is a function of the number of significant digits, such that the system can continuously adapt. In addition, it was found that the recursive form of computation for the reflection coefficient is better than direct form of computation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>