Deep learning-based feature matching has showcased great superiority for point cloud registration. While coarse-to-fine matching architectures are prevalent, they typically perform sparse and geometrically inconsistent coarse matching. This forces the subsequent fine matching to rely on computationally expensive optimal transport and hypothesis-and-selection procedures to resolve inconsistencies, leading to inefficiency and poor scalability for large-scale real-time applications. In this paper, we design a consistency-aware spot-guided Transformer (CAST) to enhance the coarse matching by explicitly utilizing geometric consistency via two key sparse attention mechanisms. First, our consistency-aware self-attention selectively computes intra-point-cloud attention to a sparse subset of points with globally consistent correspondences, enabling other points to derive discriminative features through their relationships with these anchors while propagating global consistency for robust correspondence reasoning. Second, our spot-guided cross-attention restricts cross-point-cloud attention to dynamically defined "spots"-the union of correspondence neighborhoods of a query's neighbors in the other point cloud, which are most likely to cover the true correspondence of the query ensured by local consistency, eliminating interference from similar but irrelevant regions. Furthermore, we design a lightweight local attention-based fine matching module to precisely predict dense correspondences and estimate the transformation. Extensive experiments on both outdoor LiDAR datasets and indoor RGB-D camera datasets demonstrate that our method achieves state-of-the-art accuracy, efficiency, and robustness. Besides, our method showcases superior generalization ability on our newly constructed challenging relocalization and loop closing benchmarks in unseen domains.

The increasing penetration of distributed energy resources in active distribution networks poses significant challenges to voltage regulation and system stability. This article proposes a distributed optimal voltage control algorithm that jointly coordinates active and reactive power regulation to maintain voltage stability while satisfying operational constraints. To enhance convergence speed, a momentum-based acceleration mechanism is incorporated into the traditional primal–dual framework. Furthermore, an event-triggered communication strategy is developed to substantially reduce data exchange among neighboring agents by activating transmissions only when necessary. Rigorous theoretical analysis establishes the convergence of the proposed algorithm, and comprehensive simulation studies verify its effectiveness. The results demonstrate that the proposed algorithm can accurately regulate voltage profiles with faster convergence and significantly lower communication overhead.

Human activity is intermittent, and social interaction changes over time. It embodies the time-varying nature of social networks. However, due to the complexity and dynamics of time-varying networks, the analysis opinion dynamics with decision-making over time-varying social networks is still a challenging problem. In this paper, we study the time-varying social network opinion dynamics under one-step ahead optimal decision-making mechanism. An explicit relationship between the supremum/infimum of the ultimate opinions, the maximum desired opinion and the maximum/minimum intensity of the influence of players is given. We provide the criteria for determining that individual achieves the desired opinion. Moreover, an explicit relationship between the ultimate opinions and the influence weight of the decision-making mechanism is presented. Besides, we employ our theory framework to analyze time-varying Friedkin-Johnsen opinion dynamics under one-step ahead optimal decision-making mechanism. Based on the real networks (Dolphin network and western US power grid) and the datasets, simulation experiments applying our theory illustrate that the dolphins achieve the desired performance by the keepers and the substations achieve the desired voltage regulation rates by the technicians.

Lyric-to-melody generation is a highly challenging task in the field of AI music generation. Due to the difficulty of learning strict yet weak correlations between lyrics and melodies, previous methods have suffered from weak controllability, low-quality and poorly structured generation. To address these challenges, we propose CSL-L2M, a controllable song-level lyric-to-melody generation method based on an in-attention Transformer decoder with fine-grained lyric and musical controls, which is able to generate full-song melodies matched with the given lyrics and user-specified musical attributes. Specifically, we first introduce REMI-Aligned, a novel music representation that incorporates strict syllable- and sentence-level alignments between lyrics and melodies, facilitating precise alignment modeling. Subsequently, sentence-level semantic lyric embeddings independently extracted from a sentence-wise Transformer encoder are combined with word-level part-of-speech embeddings and syllable-level tone embeddings as fine-grained controls to enhance the controllability of lyrics over melody generation. Then we introduce human-labeled musical tags, sentence-level statistical musical attributes, and learned musical features extracted from a pre-trained VQ-VAE as coarse-grained, fine-grained and high-fidelity controls, respectively, to the generation process, thereby enabling user control over melody generation. Finally, an in-attention Transformer decoder technique is leveraged to exert fine-grained control over the full-song melody generation with the aforementioned lyric and musical conditions. Experimental results demonstrate that our proposed CSL-L2M outperforms the state-of-the-art models, generating melodies with higher quality, better controllability and enhanced structure.

Stability analysis of multi-terminal HVDC systems is an important and challenging issue for the safety and stable operation of power systems with large-scale renewable energy generations. It is quite difficult to analyze the system stability caused by the interactions among different VSCs with the coupling of DC and AC networks. This paper aims to present a method to analyze the relative stability of different interactions in multi-terminal HVDC systems. By the$\nu $-gap metric based robust stability theory, firstly we propose a stability index to quantify the stability margin with respect to different paths of interactions. The influence of the interactions among different VSCs on the stability margin can be analyzed quantitatively. Then we present a method to easily calculate the stable region of parameters in the interactions. Extensive examples are given to demonstrate the application and the effectiveness of the proposed method. Note to Practitioners—This paper is motivated by the problem of improving the small-signal stability of multi-terminal high voltage direct current (MTDC) systems, which are widely used to transmit large-scale renewable energy generations. In recent years, many small-signal instability accidents have occurred in actual HVDC projects. The dynamic behavior of MTDC systems is related to the complex interactions among different voltage source converters (VSCs) through the coupling of AC and DC networks. Therefore, we aim to explain the mechanism of oscillations caused by the interactions and the relation between the dynamics and the control parameters. In this paper, we employ the$\nu $-gap metric based robust control theory to quantify the stability of an MTDC system with respect to different interactions and to calculate the stable region of a particular parameter in the interactions. The results in this paper can explain the influence of the interactions among different VSCs on the stability margin quantitatively. Moreover, the results provide new ideas for the setting and tuning of control parameters in MTDC systems and even in other multi-equipment power systems.

This article aims to propose a general control strategy for coordination of multiple nonholonomic agents in three dimensional space. For real-world applications, since the field sensor equipped on the mobile agents for local information detection and estimation has some limited detecting range, it is necessary to guarantee that the nearby agents must stay within this range of the onboard sensor. This is called sensory connectivity maintenance. A function termed as coordination function with sensory connectivity maintenance (CFSCM) is defined to describe the performances of the coordination as well as the sensory connectivity status between the agents. Then, a general control strategy is designed based on the proposed CFSCM for spatial coordination of multiple nonholonomic agents with sensory connectivity maintenance. Moreover, the applications of the proposed control strategy for formation with omnidirectional sensors and flocking with directional sensors are shown, respectively. Finally, some numerical examples are conducted to validate the theoretical analysis.

Broad learning system (BLS), a tri-layer feedforward neural network, has gained widespread recognition for its exceptional scalability and computational efficiency. However, BLS and its derivatives encounter several challenges: 1) overlooking the uncertainty introduced by numerous nonlinear random mappings; 2) failing to cope with the misalignment of model inputs with the output sampling rate; 3) lack of attention to nonstationary scenarios; and 4) absence of theoretical optimization for incremental learning. To overcome these obstacles, we propose a regression modeling and anomaly detection scheme rooted in a temporal kernel stationary BLS (TKS-BLS). We first create a nonlinear kernel broad representation (NKBR) extraction strategy, providing a robust nonlinear foundation for random feature mapping via kernel technology. Following this, we probe the mechanism of temporal matching between model inputs and outputs through a temporal alignment parameter, interpretable under a latent variable relationship. In the integration phase, we establish a Kullback-Leibler divergence objective function to facilitate the capture of stationary relationships within time-series data, in conjunction with the regression error. Subsequently, a double-loop parameter optimization algorithm and an independent incremental learning mechanism are put forth, both backed by comprehensive theoretical analyses. Our method’s superiority is thoroughly confirmed by experimental outcomes from extensive case studies across seven real ironmaking process datasets.

It is well known that adding the momentum term can accelerate the convergence rate of optimization algorithms. In this paper, the momentum term is exploited to double accelerate the distributed Nesterov gradient descent algorithm without increasing the communication burden, and a novel momentum-based distributed Nesterov gradient descent (DNGD-M) algorithm is proposed. Using the method of inequality transformation, the convergence problem of the DNGD-M algorithm is converted into the stability problem of the extended system. Then, by analyzing the state transition matrix of the extended system, the convergence conditions of the proposed algorithm and the corresponding ranges of parameter values are obtained. Finally, numerical experiments are provided to verify the effectiveness of the proposed algorithm.

For the centralized optimization, it is well known that adding one momentum term (also called the heavy-ball method) can obtain a faster convergence rate than the gradient method. However, for the distributed counterpart, there is quite few results about the effect of added momentum terms on the convergence rate. This article is aimed at studying the issue in the distributed setup, where N agents minimize the sum of their individual cost functions using local communication over a network. The cost functions are twice continuously differentiable. We first study the algorithm with one momentum term and develop a distributed heavy-ball (D-HB) method by adding one momentum term on to the distributed gradient algorithm. By borrowing tools from the control theory, we provide a simple convergence proof and an explicit expression of the optimal convergence rate. Furthermore, we consider adding two momentum terms case and propose a distributed double-heavy-ball (D-DHB) method. We show that adding one momentum term allows faster convergence while adding two momentum terms does not perform any superiorities. Finally, simulation examples are given to illustrate our findings.

With the rapid development of distributed energy resources, an increasing number of residential and commercial users have been switched from pure electricity consumers to prosumers who can both consume and produce energy. To properly manage these emerging prosumers, the peer-to-peer (P2P) electricity market has been explored and extensively studied. In this article, a scalable energy management mechanism is proposed for the P2P electricity market. First, the multibilateral economic dispatch problem that maximizes social welfare is formulated, considering product differentiation and network constraints. Then, an energy management mechanism is devised to improve the scalability from two aspects as follows. i) An accelerated distributed clearing algorithm with fewer optimization variables and communication messages and a faster convergence rate. ii) A novel selection strategy to reduce the computational and communication overheads per player. Finally, the convergence rate of the proposed clearing algorithm is given, and the effectiveness of the proposed energy management mechanism is illustrated through numerical simulations.

DC microgrids with battery energy storage systems are being widely implemented for integrating renewable energy. The convergence performance of the battery controller is an important index in the evaluation of the microgrids performance. However, the convergence time of existing finite-time control, fixed-time control, and predefined-time control cannot be preset explicitly. Moreover, existing state-of-charge (SoC)-equalization-based accelerating control algorithms will make the batteries suffer excessive voltage and current. To deal with these problems, this article presents a distributed prescribed-time control scheme embedded with a prescribed-time dynamic average consensus (DAC) algorithm for both discharging mode and charging mode. In discharging mode, the droop coefficient is designed such that all batteries keep the same relative SoC variation rate. With the proposed secondary control input, theoretical analysis shows that the voltage regulation and accurate current sharing can be obtained within any physically allowable user-preassigned time, which is independent of any other control parameters and initial states. SoC balancing is also achieved within the preassigned time. All batteries can keep the same relative SoC variation rate, which is more reasonable than simple SoC equalization. In charging mode, an SoC-based virtual resistance and a prescribed-time virtual voltage compensation control are proposed, with which the current sharing and the same relative SoC variation rate of each battery is achieved within the preassigned time. Simulation studies are conducted to demonstrate the effectiveness of the proposed control scheme

The vaccination has played a significant role in government departments to control the spread of infectious diseases. Therefore, it is interesting to theoretically analyse the impact of vaccination on the disease spreading. In this article, we propose a discrete-time epidemic-willingness dynamics model to analyse the influence of vaccine willingness on epidemic spreading. Sufficient conditions are provided to guarantee that the proportion of the infected population exponentially converges to zero. The explicit relationship between the trend of epidemic spreading and the willingness-based reproduction number is presented. Based on the real data from a survey conducted on a sample of Italian population, we employ the proposed epidemic-willingness dynamics model to reproduce the social phenomenon that increasing the willingness to vaccinate can reduce and delay the maximum proportion of infected communities. Additionally, simulation experiments validate the effectiveness of the proposed epidemic-willingness dynamics model by utilizing the real data of COVID-19 infections from 28 February to 31 May 2022 in Shanghai. It is shown that the higher the level of infection, the greater the willingness to vaccinate. Moreover, we find that the willingness-based reproduction number is not monotonically decreasing and differs from the classical reproduction number.

The cognitive inertia of sociology, similar to the inertia of physics, plays a significant role on the opinions formulation. In a general way, each individual possesses both a private and an expressed opinion in social networks. In this article, we propose a novel model with expressed and private opinions over issue sequences to capture cognitive inertia. The persistent disagreement and consensus for opinion dynamics over issue sequences are studied by the network structure and the social power evolutions. Moreover, we consider the opinion evolutions over issue sequences with bounded confidence. It is shown that the private opinions and the expressed opinions can also achieve consensus. By our proposed issue-sequence-based opinion dynamics model, the social phenomenon of cognitive freezing can be reproduced. The simulation analysis reveals that the convergence rates for the discrepancy of individuals are inversely proportional to the social powers of individuals, and the number of issue sequences for opinions achieving consensus is inversely proportional to the confidence thresholds and directly proportional to the number of individuals.

In this article, a deployment method of spotlights is studied for visual coverage task in pure dark indoor scene. To characterize the illumination performance of target in visual sensor, a criterion called illumination strength of spotlight with combining the diffuse reflection and specular reflection on the basis of distance/angle illumination attenuation of light source as well as shading is proposed. Then, an optimization problem is formulated for a number of spotlights with the aim of covering the target with a desired luminance value in camera network as uniformly as possible. An elimination genetic algorithm possessing faster convergence rate and better convergence performance than standard genetic algorithm is utilized to solve the optimization problem. Finally, numerical simulations and experiments are conducted to validate the proposed deployment method of multiple spotlights for visual coverage task in pure dark indoor scene.

Since high-order relationships among multiple brain regions-of-interests (ROIs) are helpful to explore the pathogenesis of neurological diseases more deeply, hypergraph-based brain networks are more suitable for brain science research. Unlike the existing hypergraph based brain network (brain hypernetwork), where hyperedges containing the same number of ROIs are assumed to have equal weights (to some extent, the network is unweighted), and the underlying structure is described only by an incidence/adjacency matrix, in this paper, we propose a framework for constructing a truly weighted brain hypernetwork described by an adjacency tensor. Considering the relationships among vertices within a hyperedge, we propose a novel hyperedge weight estimation method and convert the incidence matrix into a weighted adjacency tensor. On the basis of tensor decomposition, we apply hypergraph signal processing tools, such as hypergraph Fourier transform, to analyze and compare the spectrum between schizophrenia patients and normal controls. It is found that there are more high frequency components in the spectrum of patients than controls, and the average amplitude is significantly greater than that of controls. Instead of extracting some simple topological features from brain hypernetworks for classification, we innovatively use the hypergraph spectrum and the spectral signal as classification features, and the classification results on two public datasets demonstrate the effectiveness of our proposed method.

Led by the kernelized expectation maximization (KEM) method, the kernelized maximum-likelihood (ML) expectation maximization (EM) methods have recently gained prominence in PET image reconstruction, outperforming many previous state-of-the-art methods. But they are not immune to the problems of non-kernelized MLEM methods in potentially large reconstruction variance and high sensitivity to iteration numbers, and the difficulty in preserving image details and suppressing image variance simultaneously. To solve these problems, this paper derives, using the ideas of data manifold and graph regularization, a novel regularized KEM (RKEM) method with a kernel space composite regularizer for PET image reconstruction. The composite regularizer consists of a convex kernel space graph regularizer that smooths the kernel coefficients, a concave kernel space energy regularizer that enhances the coefficients' energy, and a composition constant that is analytically set to guarantee the convexity of composite regularizer. The composite regularizer renders easy use of PET-only image priors to overcome KEM's difficulty caused by the mismatch of MR prior and underlying PET images. Using this kernel space composite regularizer and the technique of optimization transfer, a globally convergent iterative algorithm is derived for RKEM reconstruction. Tests and comparisons on the simulated and in vivo data are presented to validate and evaluate the proposed algorithm, and demonstrate its better performance and advantages over KEM and other conventional methods.

Quantitative relationship between the activity/property and the structure of compound is critical in chemical applications. To learn this quantitative relationship, hundreds of molecular descriptors have been designed to describe the structure, mainly based on the properties of vertices and edges of molecular graph. However, many descriptors degenerate to the same values for different compounds with the same molecular graph, resulting in model failure. In this paper, we design a multidimensional signal for each vertex of the molecular graph to derive new descriptors with higher discriminability. We treat the new and traditional descriptors as the signals on the descriptor graph learned from the descriptor data, and enhance descriptor dissimilarity using the Laplacian filter derived from the descriptor graph. Combining these with model learning techniques, we propose a graph signal processing based approach to obtain reliable new models for learning the quantitative relationship and predicting the properties of compounds. We also provide insights from chemistry for the boiling point model. Several experiments are presented to demonstrate the validity, effectiveness and advantages of the proposed approach.

With the increasing integration of intermittent energy sources into the smart grid, distributed battery energy storage systems (DBESSs) are employed to balance power generation and demand. Power allocation among DBESSs plays an important role in maintaining the stability of energy systems. So far, the control of DBESSs has focused on either continuous-time control for continuous-time battery dynamics or discrete-time control for discretized battery dynamics. However, in realistic industrial applications, DBESSs have continuous-time dynamics in nature, and their control is implemented on digital controllers. To tackle this issue, a distributed multirate control system is designed in this article for continuous-time DBESSs. It allocates power by keeping the same relative State-of-Charge (SoC) variation rate for all DBESSs. For the accurate computation of the output/input power without global information, the control strategy consists of distributed multirate estimators each for a DBESS. The operating rate of the estimation algorithm is designed multiple times higher than the sampling rate of the measurement of battery and the sampling rate of the battery controller. With the proposed multirate estimator, a smaller estimation error is achieved. The same relative SoC variation rate and the stability of closed-loop system can be guaranteed. Simulation studies are given to demonstrate the effectiveness of the proposed control strategy.

The cyber-physical system is highly desirable for real-time perception and dynamic control, especially in task scheduling and network communication for large and complex engineering systems. In most of the applications, the resource scheduling of cyber and physical network are independent on each other, which may cause severe deterioration of the network performance. To implement the cognitive cooperation between the upper layer of industrial services and the underlying network, we propose a cooptimization method in this article. We adopt the software defined networking technology to extract the network control logic, which is integrated with the upper-level task scheduling. This cooptimization scheme can dynamically calculate optimal end-to-end virtual paths over the underlying network infrastructures. Meanwhile, it can also adjust the task scheduling of the upper-level network by signals derived from the underlying network. Simulation results show that our proposed scheme outperforms the traditional method in terms of task acceptance rate, average end-to-end delay, and network load balance degree.

Uncertainty principles in finite dimensional vector space have been studied extensively, however they cannot be applied to sparse representation of rational functions. This paper considers the sparse representation for a rational function under a pair of orthonormal rational function bases. We prove the uncertainty principle concerning pairs of compressible representation of rational functions in the infinite dimensional function space. The uniqueness of compressible representation using such pairs is provided as a direct consequence of uncertainty principle.

The performance limitation of linear equalizers is studied for the structural similarity (SSIM) criteria. Given a blurring filter and an image with zero mean, the closed form formula is obtained to compute the maximal SSIM index and the corresponding optimal linear equalizer. The formula shows that the equalizer with maximal SSIM index is equal to the one with minimal mean square error (MSE) multiplied by a positive real number. Numerical examples are given to demonstrate the theoretical results.

The optimization of multichannel equalizers is studied for the structural similarity (SSIM) criteria. The closed-form formula is provided for the optimal equalizer when the mean of the source is zero. The formula shows that the equalizer with maximal SSIM index is equal to the one with minimal mean square error (MSE) multiplied by a positive real number, which is shown to be equal to the inverse of the achieved SSIM index. The relation of the maximal SSIM index to the minimal MSE is also established for given blurring filters and fixed length equalizers. An algorithm is also presented to compute the suboptimal equalizer for the general sources. Various numerical examples are given to demonstrate the effectiveness of the results.

The optimal synthesis filter bank (FB) is designed for a given analysis FB by using the structural similarity (SSIM) criteria. Under the assumption that the source signal is a wide sense stationary (WSS) process with known power spectral density (PSD), that the noise is Gaussian white, and that the filter length is equal to the decimation number, the optimization problem is formulated. The closed-form solution is obtained when the mean of the source is zero. It is shown that the optimal FB designed by SSIM criteria and the one by mean square error (MSE) criteria differs only from a scalar factor. Finally, numerical example is given to compare the performance of different synthesis FBs.

Shift variance bounds of critically sampled multirate filter banks are usually defined by using eigensystem analysis of the commutator operator. Computational methods are provided for two-channel FIR filter banks by Aach and Fuhr (2009). This paper investigates shift variance bounds in oversampled multi-channel filter banks and derives some general results which include some of Aach and Fuhr (2009) as special cases. Using polyphase representation, numerical methods are provided to compute the shift variance bounds for arbitrary filter banks. Examples are given to demonstrate the effectiveness of the obtained results.

In one-dimensional signal processing, the perfect reconstruction (PR) synthesis FB is not unique for a given analysis LP oversampled filter bank (FB). Optimal Design methods have been developed to choose the optimal noise reduction linear phase (LP) synthesis FB. This paper deals with the optimal design of LP synthesis FBs for separable 2D image processing. With the same analysis LP oversampled FB for row and column processing separately, we show that the optimal synthesis FBs for row and column processing are generally not the same. A design method is developed when the spectrum of the subband noises is known. Numerical examples are given to demonstrate the effectiveness of our results.

Frame (upper and lower) bound ratio is a key factor of numerical stability of frame systems. While tight frames, of which the minimal ratio 1 is achieved, are much preferred in many practical situations, they may not be easily (even impossible) designed due to other criteria (for example, good subband frequency shape). In these situations, we may find alternative methods for improving the frame-bound-ratio. This paper studies the minimization of frame-bound-ratio for filter bank frames by using periodic precoding. A convex optimization based algorithm is proposed. Examples show the effective ness of our results.

This paper presents a frame analysis of Zero Forcing (ZF) precoding problem. It shows that from a frame point of view, the filter bank (FB) channel is a synthesis frame and its canonical dual analysis frame is the optimal ZF precoder under total power constraint. Based on this analysis, a new precoder design is presented. The new design removes the minimum phase assumption of polyphase FB channel in the existing methods and gives rise to optimal noncausal (stable) precoder. It provides a general framework for optimal zero-forcing precoding under total power constraint. The numerical results show that the new design outperforms existing causal stable design.

This paper presents a simple method to improve the frame-bounds-ratio of perfect reconstruction (PR) oversampled filter banks (FBs) by adjusting the gain of each subband filter. For a given analysis PRFB, a finite convex optimization algorithm is presented to redesign the subband gains such that the frame-bounds-ratio of the FB is minimized. The algorithm also provides an effective way to compute the frame bounds. Examples show the effectiveness of the presented method.

This paper is concerned with signal predictive subband quantization in oversampled filter banks. It uses the analysis tools of robust filtering and control to obtain a worst case power spectral analysis of the prediction error system. The analysis removes the unrealistic assumptions on the quantization noises in the present literature and leads to the definition of Hinfinoptimal signal predictor. Based on this analysis, an LMI optimization based method is obtained to design the Hinfinoptimal signal predictor with guaranteed inverse stability. Simulation results are presented to show the advantages of the Hinfinoptimal signal predictor over the conventional signal predictor.

This paper presents a filter bank (FB) analysis of parallel magnetic resonance imaging (PMRI). The underlying image reconstruction strategies of the most widely used PMRI reconstruction methods are unified within the framework and their fundamental perfect reconstruction (PR) constraints are analyzed. Based on this analysis, an improved reconstruction method, called H∞optimal sense, is developed and its advantage is demonstrated by an example.

In this paper, we present an explicit and numerically efficient formulae to construct a tight (paraunitary) FB frame from a given un-tight (non-paraunitary) FB frame. The derivation uses the well developed techniques from modern control theory, which results in the unified formulae for generic IIR and FIR FBs. These formulae involve only algebraic matrix manipulations and can be computed efficiently and reliably without the approximation required in the existing literature.

This paper studies the noise reduction design problem for oversampled filter banks (FBs) with perfect reconstruction (PR) constraint. Both the optimal design and worst case design are considered, where the former method caters for the noise with known power spectral density (PSD) and the latter one for the noise with unknown PSD. Explicit formulae involving only algebraic Riccati equation and matrix manipulations are provided for the general (IIR or FIR) oversampled PR FBs.

This paper studies the frames corresponding to stable oversampled filter banks (FBs). For this class of frames, we present explicit and numerically efficient formulae to compute the tightest frame bounds, to obtain the dual frame and to construct a paraunitary FB for a given non-paraunitary FB. The derivation uses the well developed techniques from modern control theory, which results in the formulae that involve only algebraic matrix manipulation and can be performed efficiently and reliably without the approximation required in the existing methods