Communication efficiency is one of the essential elements in the Internet of Things. While most existing works mainly present energy-efficient methods for distributed estimation under ideal conditions, real-world environments involve imbalanced noise variances, such as input noise and measurement noise. In addition, disturbances from impulsive noise drive the system into instability. Thus, conventional methods result in degraded estimation performance when applied to practical environments. To address this issue, this work proposes an extended partial diffusion least-mean-square algorithm based on mean-square-deviation (MSD) analysis. Theoretical analysis shows that the MSD-based formulation extends a scalar-based combination coefficient to a matrix-based one, enabling flexible combinations that allow the network to achieve consensus with low communication cost. Furthermore, it presents an analytical design of an MSD-based optimal variable step size, considering practical noise components in adaptive networks. The simulation results validate the communication efficiency and estimation performance of the proposed algorithm under various noisy conditions.

This article aims to investigate relaxed local stabilization for discrete-time Takagi-Sugeno fuzzy systems with structural relaxation under guaranteed resilience. To mitigate the conservatism and computational burden associated with conventional multiple summation-type approaches for exploiting high-degree membership information, a novel Lyapunov function and nonparallel distributed compensation (non-PDC) control law are developed within an augmented membership-quadratic framework, which relaxes the symmetry constraints on the intertemporal cross terms. To overcome the limitations of existing resilient stabilization methods that rely heavily on a user-defined hyperparameter, a matrix-type threshold condition is introduced, enhancing both practicality and numerical efficiency. Based on orthogonal complements, new structural relaxation lemmas within the membership-quadratic framework are proposed for guaranteeing resilient stabilization. Finally, the effectiveness and reduced conservatism of the proposed method are validated through benchmark examples, demonstrating its computational efficiency and improved performance compared to existing approaches.

This article studies the issue of stability in memristor-based neural network (MNN) systems with time-varying delays. First, a novel matrix-separation Legendre inequality is proposed to achieve a tight hierarchical bound on augmented-type integral terms. To derive implementable inequality conditions, several delay-dependent matrices are introduced to eliminate the reciprocal terms associated with time-varying delay. Furthermore, a new Lyapunov-Krasovskii (L-K) functional is proposed by incorporating augmented-type double integrals and delay-product terms. A series of free-weighting matrices are introduced into the L-K functional, leveraging the zero-sum equations and the S-procedure pertaining to both the delay and its derivative. Based on the proposed matrix-separation Legendre inequality and L-K functional, the derived stability conditions exhibit reduced conservatism, as validated by three numerical cases and simulation results.

The incorporation of matrix polynomial terms in the Lyapunov-Krasovskii (L-K) functional presents new challenges for the delayed discrete-time systems. This article proposes a novel delay-cubed-dependent L-K functional to alleviate the conservatism of the stability criteria for the discrete-time systems with time-varying delay. A novel matrix convex method is proposed to cope with the nonaffine terms introduced by the delay-cubed-dependent functional. Furthermore, a generalized matrix convex method is developed to address the L-K functional with varying orders of delay. Then, the executable stability and stabilization constraints with less conservatism are derived-based the proposed delay-cubed-dependent L-K functional and the matrix convex method. Finally, three case studies are given to elucidate the efficacy of the obtained stability and stabilization method.

Due to the lack of state dimension optimization methods, deep state space models (SSMs) have sacrificed model capacity, training search space, or stability to alleviate computational costs caused by high state dimensions. In this work, we provide a structured pruning method for SSMs, Layer-Adaptive STate pruning (LAST), which reduces the state dimension of each layer in minimizing model-level output energy loss by extending modal truncation for a single system. LAST scores are evaluated using the $\mathcal{H}_{\infty}$ norms of subsystems and layer-wise energy normalization. The scores serve as global pruning criteria, enabling cross-layer comparison of states and layer-adaptive pruning. Across various sequence benchmarks, LAST optimizes previous SSMs, revealing the redundancy and compressibility of their state spaces. Notably, we demonstrate that, on average, pruning 33\% of states still maintains performance with 0.52\% accuracy loss in multi-input multi-output SSMs without retraining. Code is available at https://github.com/msgwak/LAST.

This article presents an investigation of the stability analysis and$H_\infty$control synthesis for Takagi–Sugeno fuzzy systems based on a quadratic fuzzy Lyapunov function, which incorporates second-degree information of membership functions. Instead of a multiple summation expression, the Lyapunov function and controller are designed within the structure of themembership-quadratic framework, which is a quadratic form with membership-dependent outfactors. A structural relaxation lemma is established based on zero-equality conditions through orthogonal complements and characteristics of the matrix of the quadratic form. On the basis of the proposed structural relaxation approach, the stability conditions for the analysis and$H_\infty$synthesis are achieved through linear matrix inequalities with elaborate matrix manipulation techniques. Conceptual foundations employing high-degree membership functions are provided to generalize the structural relaxation approach. Comprehensive numerical examples that demonstrate the efficacy of the proposed approach are detailed.

This article concentrates on the distributed fault-tolerant control (FTC) issue for a herd of delayed unmanned aerial vehicles (UAVs) subject to actuator faults, sensor faults, and full-state prescribed performance (FSPP). A novel distributed event-triggered observer is devised to reconstruct the leader’s states within a finite time, and a Zeno-free dynamic triggering condition is put forward to schedule information transmission between neighboring UAVs. Unknown nonlinear dynamics are handled via radial basis function neural networks. By cleverly combining the presented observer and backstepping control techniques, a distributed FTC paradigm is constructed such that the estimated leader states can be tracked. Besides, finite-time performance functions are introduced to ensure that the tracking errors converge to the specified regions within a bounded time. It is shown by Lyapunov theory that all closed-loop signals are practically finite-time stable, and the FSPP restrictions are strictly obeyed. Finally, case studies are reported to corroborate the practical merits of the presented methodology.

This paper studies the secure aperiodic sampling control issue for Micro-Grids(MGs) under abnormal deception cyber attacks (ADCAs). Firstly, a novel relaxed condition that depends on the time delay (TD) is constructed, further reducing the existing constraint condition. Secondly, a new bilateral delay-dependent looped-functional (BDDLF)$V_{c}(x_{t})$is developed. Thirdly, based on the characteristics of the aperiodic sampling control, another improved BDDLF$V_{d}(x_{t})$is developed for acquiring more state information. Furthermore, an optimized control algorithm is designed using proper integral inequalities and the convex combination method. Then, a new secure aperiodic sampled data (SASD) controller under ADCAs is achieved to guarantee real power-sharing between distributed generators (DGs) and energy storage systems (ESSs) in MG. Finally, simulation experiments are carried out on MG to verify the effectiveness and feasibility of the designed control algorithm.

This article proposes an improved security-based event-triggered fuzzy control (ETFC) method for studying the asymptotic stability problem of nonlinear networked control systems (NCSs) under denial-of-service (DoS) attacks. The nonlinear NCSs are linearized using the central mean fuzzy method to obtain Takagi-Sugeno (T-S) fuzzy NCSs. Lyapunov–Krasovskii functionals (LKFs) with nonlinear problems are constructed based on fuzzy membership functions. A relaxed condition including nonlinear parameters is given, which considers the nonlinear optimization problem and reduces the positive-definiteness constraint problem of LKFs. An improved double-closed delay correlation function is constructed to obtain more sampling information. Furthermore, a quadratic scaling method is used to obtain a tighter upper bound, making the criterion less conservative. A DoS attack causes the channel's throughput to drop and collapse. Thus, in this article, we design ETFC based on the improved descent gradient algorithm to ensure the communication security of the helicopter system (HS). Finally, the communication security and stability of HS are verified using the Simulink platform.

This article investigates the stability and stabilization issues of Takagi–Sugeno (T–S) fuzzy systems with time-varying delay via a generalized reciprocally convex inequality (GRCI). The enhanced stability criteria with less conservatism are achieved by employing a novel GRCI and establishing an asymmetric Lyapunov–Krasovskii (L–K). The proposed reciprocally convex method encompasses some previous methods as exceptional examples by introducing a matrix-valued polynomial. Moreover, an asymmetric L–K functional with delay-product terms is proposed for the stability of the T–S fuzzy systems. Then, a stabilization method is produced based on the parallel distributed compensation. Finally, case studies are carried out to indicate the effectiveness and advantages of the stability and stabilization criteria.

Robustness of neural network models is important in fault diagnosis (FD) because uncertainty in operating conditions varies the power spectral densities of vibration data; however, it is unknown to users due to the limited explainability of the models. This article proposes an FD framework with a power-perturbation-based decision boundary analysis (POBA) to explain the decision boundaries of vibration classification models. In the POBA, perturbed data are obtained from training data by power perturbation on frequency bands centering on dominant class-discriminative frequencies. The decision boundary of a model is then evaluated and visualized to users by testing the model on the perturbed data. Furthermore, the decision boundary information can be used to define a robustness score per class, and a robust model can be obtained by ensembling trained models using their robustness score per class. Demonstration using two vibration datasets verifies the explainability and robustness of the proposed FD framework.

This article proposes a new end-to-end deep learning model for fault diagnosis using three-axis vibration signals measured from facilities. The three-axis vibration signals measured in the time domain are used without domain transformations to train the end-to-end model. The proposed model is designed to effectively extract feature maps of eachX-,Y-, andZ-axis individually from the three-axis vibration signals using a grouped 1-D convolution. The feature maps extracted from each axis are composed of specific frequencies of each axis. Accordingly, the proposed model classifies faults based on the frequency characteristics of each axis. In addition, this article proposes a method to visualize the decision criteria of the proposed model in the frequency domain. Using the proposed model and the proposed visualization method, it is possible to grasp the degree to which specific frequencies of eachX-,Y-, andZ-axis affect the decision criteria of the proposed model. The experiment is conducted by applying the proposed model to an open dataset with three-axis vibration signals measured from a rotary machine. Experimental results demonstrate that the proposed model achieves high accuracy and provides the explainability in eachX-,Y-, andZ-axis using the visualized decision criteria in the frequency domain.

This article investigates the sampled-data control for a class of Takagi-Sugeno (T-S) fuzzy Markovian jump systems via a novel looped functional approach. By refining the fractional delayed state concept, the looped functional is established to include more sampling information of the intervals not only from x(tk) to x(t) but also from x(t) to x(tk+1), which would contribute to a reduction in the conservatism. Then, a sufficient criterion is presented to ensure the stochastic stability of the resulting closed-loop system, based on which the sampled-data controller is further designed for the T-S fuzzy Markovian jump system under the influence of asynchronous premise variables. Finally, the obtained theorems are applied to a numerical example and a practical application to show the effectiveness and superiority of the proposed method.

A linear motion (LM) guide is a mechanical tool for requiring linear motion in a system. Repeating linear movements can cause cracking and deterioration of the LM guide, which can lead to a decrease in productivity. Therefore, predicting the status of the LM guide and diagnosing faults are essential for systems including the LM guide. In this article, we propose a novel framework of fault diagnosis model based on deep learning using a vibration sensor signal mounted on the LM guide. This framework contains the learning vibration signal in the time domain using the proposed 1-D convolutional neural network model and the visualization of the classification criteria in the frequency domain using the learned model in the time domain. To utilize the visualization in the frequency domain, the proposed model is designed to maintain the frequency information in the learning process. With the learned model, we propose a frequency domain-based grad-CAM to visualize the classification criteria in the frequency domain to help to explain the characteristics of normal and fault data. Using LM guide data under various conditions, we visualize the classification criteria of the learned model in the frequency domain.

The normalised subband adaptive filter (NSAF) is a useful adaptive filter, which improves the convergence rate compared with the normalised least mean‐square algorithm. Most analytical results of the NSAF set the regularisation parameter set to zero or present only steady‐state mean‐square error performance of the regularised NSAF ( ‐NSAF). This study presents a mean‐square performance analysis of ‐NSAF, which analyses not only convergence behaviour but also steady‐state behaviour. Furthermore, a novel adaptive regularisation for NSAF (AR‐NSAF) is also developed based on the proposed analysis approach. The proposed AR‐NSAF selects the optimal regularisation parameter that leads to improving the performance of the adaptive filter. Simulation results comparing the proposed analytical results with the results achieved from the simulation are presented. In addition, these results verify that the proposed AR‐NSAF outperforms the previous algorithms in a system‐identification and acoustic echo‐cancellation scenarios.

This paper proposes a fast and precise adaptive filtering algorithm for online estimation under a non‐negativity constraint. A novel variable step‐size (VSS) non‐negative normalised least‐mean‐square (NLMS)‐type algorithm based on the mean‐square deviation (MSD) analysis with a non‐negativity constraint is derived. The NLMS‐type algorithm under the non‐negativity constraint is derived by using the gradient descent of the given cost function and the fixed‐point iteration method. Furthermore, the VSS derived by minimising the MSD yields improvement of the filter performance in the aspects of the convergence rate and the steady‐state estimation error. Simulation results show that the proposed algorithm outperforms existing algorithms.

This paper proposes a new variable step‐size sign algorithm through the minimisation of mean‐square deviation (MSD). As it is difficult to obtain the MSD accurately, the upper bound of the MSD is derived for calculating the step size at each iteration. The proposed algorithm is not only robust to impulsive noises, but also has improved filter performance in aspects of convergence rate and steady‐state estimation error owing to the proposed variable step‐size strategy. The simulation results verify that the proposed algorithm has better performance than the existing algorithms in a system‐identification scenario in the presence of impulsive noises.

This letter proposes a new algorithm for efficiently finding an unbiased RLS estimate of FIR models with noisy inputs. The unbiased estimate is obtained without knowing any a priori information via a new cost. Furthermore, to reduce computational complexity, the estimate is updated along the current input-vector direction and the corresponding gain is efficiently computed. In addition, to increase the convergence rate, the algorithm is extended to update the estimate along not only current but also past input-vector directions. Simulation results show that the proposed algorithm exhibits a fast convergence rate and an enhanced tracking performance with noisy correlated inputs.

This letter proposes a variable step-size sign subband adaptive filter (SSAF) based on the minimization of mean-square deviation (MSD). In the process of minimizing the MSD, because it is not feasible to know the exact value of the MSD, the step size is derived by minimizing the upper bound of the MSD in each iteration. The proposed algorithm uses this step size in the SSAF update equation so as to improve the filter performance in terms of the convergence rate and the steady-state estimation error. The proposed algorithm is tested in a system-identification scenario that includes impulsive noise. Simulation results show that the proposed algorithm performs better than the previous algorithms.

This letter describes how to set up the step size of the affine projection algorithm (APA) based on mean-square deviation analysis. The analysis considers the cross-correlation between the current weight error vector and the prior measurement noises associated with the reused inputs vectors for better prediction of the learning behavior of the APA. With the predetermined step size based on the analysis, the proposed approach eliminates the parameter-tuning process and the derived algorithm achieves both the fast convergence rate and the low steady-state error. Simulation results show that the proposed algorithm performs better than previous algorithms.

This letter presents a variable-step-size normalized least-mean-square algorithm, where the step size is updated only when the current input vector is innovative from the last updated input vector. The instant innovativeness of the two input vectors is investigated through the relation between the angle of the two input vectors and the condition number of the input covariance matrix. Once the condition number is obtained, the resulting algorithm performs an excellent transient and steady-state behavior with different correlations in inputs. To reduce the computational burden of obtaining the condition number, this letter also presents a simple method to determine the condition number based on the power method.

This letter proposes an affine projection algorithm (APA) based on the concept of reuse time of the current input vector. Reuse time is defined as the survival period of an input vector, during which the input vector is continuously reused in the subsequent update equations. The algorithm consists of two key procedures: assignment and reduction. The assignment procedure assigns a fundamental reuse-time or zero to the individual reuse time of each current input vector only once by checking whether the current input vector has enough information for update, which eliminates the repetitive selection procedure for input vectors. The reduction procedure gradually decreases the fundamental reuse time by examining, from a stochastic point of view, whether the current error reaches the steady-state value, which indirectly controls the number of input vectors; this leads to fast convergence and small estimation errors. Through these two procedures, the proposed algorithm achieves not only improved performance but also extremely low computational complexity.

In this study, we establish a less-conservativeH_ stabilization condition for a class of discrete-time Takagi-Sugeno (T-S) fuzzy systems. To this end, we propose a useful relaxation technique that can be used to transform parameterized-linear-matrix inequalities (PLMIs) with homogeneous polynomial dependence on current-time parameters and preceding parameters into a finite set of LMIs. The main feature of this technique is that all possible slack variables can be included in the relaxation process to fully exploit the convexity of the PLMIs. By applying the relaxation technique, we establish a less-conservative LMI-basedH_ stabilization condition.

This paper presents a method of scheduling stepsizes for the normalized least-mean-squares (SS-NLMS) algorithm. Geometrically interpreting the mean square deviation (MSD) learning curve leads to establishing an objective curve and to constructing a lookup table of stepsizes in order for the MSD to follow the curve. The SS-NLMS shows not only good performance but also robustness with respect to different signal-to-noise ratio (SNR) in measurement noise and different correlation in input signals with a very small number of online computations. Moreover, the scalability of the tabled stepsize with respect to the number of taps is described. For the efficient memory usage in practice, a modified version replaces the tabled stepsizes by down-sampled stepsizes with no performance degradation.

This short paper proposes a method of designing a fuzzy observer-based${{\cal H}}_{\infty }$controller for discrete-time Takagi–Sugeno (T--S) fuzzy systems. To enhance the applicability of the output-feedback controller and improve its performance, this short paper first builds a set of fuzzy control rules with premise variables different from those of the T--S fuzzy system, and sets the overall controller to be dependent on not only the current time but also the one-step-past information on the estimated fuzzy weighting functions. Then, based on the fuzzy control rules, this short paper establishes a less conservative${{\cal H}}_{\infty }$stabilization condition incorporated with a multiple Lyapunov function dependent on the estimated fuzzy weighting functions. Through a two-step design procedure, the${{\cal H}}_{\infty }$stabilization condition is formulated in terms of parameterized linear matrix equalities (PLMIs), which are reconverted into LMIs with the help of an efficient and effective relaxation scheme.

To derive less-conservative delay- and range-dependentHinfinstabilization conditions for discrete-time Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delays, the use of a fuzzy-weighting-dependent Lyapunov-Krasovskii functional (FWLKF), in which all variables are set to be affinely or quadratically dependent on fuzzy weighting functions, is proposed. Subsequently, parameterized linear matrix inequality (PLMI)-basedHinfinstabilization conditions are derived by following the free-weighting matrix approach. To fully exploit the convexity of fuzzy weighting functions, the derived PLMIs are sequentially replaced by a finite set of LMIs by considering all possible conditions associated with fuzzy weighting functions.

Based on fuzzy weighting-dependent Lyapunov-Krasovskii functional (FWLKF), this paper proposes a method of designing a delay-dependent fuzzy controller for discrete-time Takagi-Sugeno (T-S) fuzzy systems with time delays. To derive a less conservative delay-dependent stabilization condition, the method enhances the interactions among the fuzz subsystems using FWLKF with quadratic dependence on fuzzy weighting functions, where all the decision variables of FWLKF are set to be associated with the fuzzy weighting functions. Besides, this paper sets the overall controller to be dependent on not only the current-time but also the delay-time information on the fuzzy weighting functions. In the derivation, the stabilization condition is first formulated in terms of parameterized linear matrix inequalities (PLMIs), and then is reconverted LMIs based on several relaxation techniques.

Nowadays, computational power of graphics hardware is strong enough to be used in many applications requiring parallel computing. It can be used any other areas even unrelated to the graphics displays. Following these trends, we adapted this general purpose graphics processing unit (GPGPU) for thin film transistor liquid crystal displays (TFT-LCD) inspection. In this paper, we describe practical system components and the implementation including compute unified device architecture (CUDA) programming. An experimental result compares to the previous system which uses CPU as a main processor.

This paper proposes a method for designing an${\cal H}_{\infty}$state-feedback fuzzy controller for discrete-time Takagi–Sugeno (T-S) fuzzy systems. To derive less conservative${\cal H}_{\infty}$stabilization conditions, this paper enhances the interactions among the fuzzy subsystems using a multiple Lyapunov function with quadratic dependence on fuzzy weighting functions. Besides, for more allocation of the nonlinearity to the fuzzy control system, this paper introduces a slack variable that is quadratically dependent on the one-step-past fuzzy weighting functions as well as the current ones. In the derivation, the${\cal H}_{\infty}$stabilization conditions are formulated in terms of parameterized linear matrix inequalities (PLMIs), which are reconverted into LMI conditions with the help of an efficient relaxation technique.

Frequency-domain decision-feedback equalization (FD-DFE) for single carrier modulation avoids interblock interferences (IBIs) by using guard intervals located in between two data blocks. If the period of a guard interval is shorter than that of the channel impulse response, the IBIs cannot be removed by the FD-DFE, which significantly deteriorates its performance. To handle this IBI problem, this letter introduces the adaptive self-orthogonalizing per-tone FD-DFE without channel estimation, where time dispersiveness, one weakness of the per-tone equalization, is compensated by the self-orthogonalization coming from the lattice structure of the Levinson-Durbin algorithm, which dramatically reduces bit-error rates of the proposed FD-DFE, comparing with the existing FD-DFE.

For emerging multiple-input multiple-output communication systems, we investigate an algorithm that compensates intersymbol interference and suppresses interference among channels as well. In general, such an algorithm requires more complicated computation than single channel algorithms, and this causes many numerical errors, such as truncation errors, when running the algorithm on digital machines. We develop a multi-channel least squares order recursive lattice smoother that suppresses the intersymbol and interchannel interference while being numerically stable in the finite precision computation.

With two level weighting functions, namely, crisp switching-region weighting functions and local fuzzy weighting functions, this paper introduces a discrete-time switching fuzzy system, which inherently contains the features of switched hybrid systems and Takagi-Sugeno (TS) fuzzy systems, and then, for this discrete-time switching fuzzy system, this paper proposes two new guaranteed cost state-feedback controllers minimizing an upper bound of state and input energy called LQ performance under all admissible grades of time-varying fuzzy weighting functions. The first one, associated with a piecewise quadratic Lyapunov function (PQLF), uses time-varying information on the switching-region weighting functions. The second one, associated with a new piece-wise fuzzy weighting-dependent Lyapunov function (PFWLF), uses time-varying information on the local fuzzy weighting functions as well as on the switching-region functions. Especially with a new special structure of the candidate of PFWLF, the PFWLF-based controller employs not only the current-time but also the one-step-past information on the time-varying local fuzzy weighting functions.

This paper suggests a new H/sub /spl infin// output-feedback controller for discrete-time switching fuzzy systems, which have high- and low-level weighting functions, namely, crisp switching-region weighting functions and local fuzzy weighting functions. Based on a new piecewise fuzzy weighting-dependent Lyapunov function (PFWLF) consisting of current-time states and a set of one-step-past local fuzzy weighting-dependent Lyapunov matrices, the new controller directly uses the current-time information on the high-level weighting functions as well as the current-time and the one-step-past information on the low-level weighting functions. This resulting controller is formulated in terms of parametric linear matrix inequalities (PLMIs), which are local fuzzy weighting-dependent conditions.

In this paper, we propose a new kind of dynamic output-feedback variable structure controller to stabilize Takagi-Sugeno (TS) fuzzy systems that have nonlinear disturbances. Based on the new kind of controller, which consists of a general full structure of weighting-dependent dynamic controller and an additional variable structure control part, this paper directly suggests new global asymptotic stability criteria for the uncertain TS fuzzy systems using Lyapunov stability. Furthermore, this paper presents simple solvable conditions for the criteria in terms of linear matrix inequalities (LMIs).

For discrete-time Takagi-Sugeno (TS) fuzzy systems, we propose an H/sub /spl infin// state-feedback fuzzy controller associated with a fuzzy weighting-dependent Lyapunov function. The controller, which is designed via parameterized linear matrix inequalities (PLMIs), employs not only the current-time but also the one-step-past information on the time-varying fuzzy weighting functions. Appropriately selecting the structures of variables in the PLMIs allows us to find an LMI formulation as a special case.

This study sets out to identify a real-world system by employing binary-coded and real-coded genetic algorithms (GAs). However, in a nascent stage of setting configurations of GAs, it is difficult to determine the feasible boundaries of each parameter of the system. In this paper, both GAs are implemented based on the similar mechanisms of crossover and mutation, and performed on discretized linear, logarithmic, and hybrid search spaces with corresponding encoding methods. Through simple probabilistic analysis, it follows that logarithmic space and hybrid space searches are far more advantageous to general linear space search in wide bound searching. The identification results of a simple electrical circuit support this expectation and confirm that real-coded GA (RCGA) of logarithmic and hybrid space searches are the best way to tackle the real-world problem.

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(2)p) computations per iteration, its fast version called a new hybrid QR/LLS algorithm requires O(Mp(2)) 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 so-called inverse QR algorithm was recently introduced for recursive adaptive filtering under the exponentially weighted least-squares criterion. It has some attractive features, including the absence of inversions. The extension to the multi-channel case does require inversion however. We present a new derivation of the inverse QR algorithm, based on the technique of Sayed and Kailath, for reformulating the above adaptive filtering problem as a state-space estimation problem. A well-known square-root covariance algorithm for the latter problem is shown to directly give (a multi-channel version of) the inverse QR algorithm. A new extended square-root covariance algorithm is then applied to get a new inversion-free ‘extended inverse QR’ algorithm, even in the multi-channel case. Kürzlich wurde ein sogenannter inverse QR-Algorithms für rekursive adaptive Filter bei Verwendung des Kriteriums der exponentiell gewichteten kleinsten Quadrate eingeführt. Er weist einige attraktive Merkmale auf, darunter das Fehlen von Matrixinversionen. Die Erweiterung auf den mehrkanaligen Fall benötigt jedoch Inversionen. Wir stellen eine Herleitung des inversen QR-Algorithmus vor, die auf der Methode von Sayed und Kailath beruht, das obige adaptive Filterproblem als Schätzproblem im Zustandsraum umzuformulieren. Es wird gezeigt, daß ein allgemein bekannter Wurzel-Kovarianzalgorithmus für das letztere Problem unmittelbar (eine mehrkanalige Version für) den inversen QR-Algorithmus liefert. Ein neuer erweiterter Wurzel-Kovarianzalgorithmus wird danach angewendet, um sogar für den mehrkanaligen Fall einen neuen inversionsfreien ‘erweiterten inversen QR-Algorithmus’ zu erhalten. Un algorithme appelé QR inverse a récemment été introduit pour le filtrage adaptatif récursif utilisant le critère des moindres carrés pondérés exponentiellement. Cet algorithme a quelques propriétés intéressantes, incluant l'absence d'inversions. Toutefois, l'extension au cas multi-canal réclame l'inversion. Nous présentons une nouvelle dérivation de l'algorithme QR inverse, basés sur la technique de Sayed et Kailath, pour reformuler le problème, de filtrage adaptatif ci-dessus comme un problème d'estimation dans l'espace d'état. Nous montrons qu'un algorithme de la racine carrée de la covariance, bien connu pour la résolution de ce dernier problème, donne directement (une version multi-canal de) l'algorithme QR inverse. Un nouvel algorithme de racine carr'ee de la covariance est alors appliqué pour obtenir un nouvel algorithme ‘QR inverse étendu’ ne nécessitant pas d'inversion, même dans le cas multi-canal.