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Graphons are infinite-dimensional objects that represent the limit of convergent sequences of graphs as their number of nodes goes to infinity. This paper derives a theory of graphon signal processing centered on the notions of graphon Fourier transform and linear shift invariant graphon filters, the graphon counterparts of the graph Fourier transform and graph filters. It is shown that for convergent sequences of graphs and associated graph signals: (i) the graph Fourier transform converges to the graphon Fourier transform when the graphon signal is bandlimited; (ii) the spectral and vertex responses of graph filters converge to the spectral and vertex responses of graphon filters with the same coefficients. These theorems imply that for graphs that belong to certain families, i.e., that are part of sequences that converge to a certain graphon, graph Fourier analysis and graph filter design have well defined limits. In turn, these facts extend applicability of graph signal processing to graphs with large number of nodes -- since signal processing pipelines designed for limit graphons can be applied to finite graphs -- and to dynamic graphs -- since we can relate the result of SP pipelines designed for different graphs from the same convergent graph sequence., Accepted for publication in IEEE TSP

Here we consider the problem of designing finite-impulse-response (FIR) graph filter (GF) in a fully distributed way. For a directed graph with $\mathsf {N}$ nodes, each node designs filter coefficients in a distributed manner, when the knowledge of the graph structure, recognized as global information, is unavailable to each node. By modeling graph signal shifting with observations at a node as a linear dynamical system, we establish fundamental connections between local response of shifting at a node, concerned in the graph signal processing (GSP) field, and the observability of the system, investigated in control theory. The observability, as a measure of how well internal states of a dynamical system can be learned from node's observations, is reflected by the minimal polynomial of a matrix pair related to the system. Specifically, by introducing a notion of observable graph frequencies to a node, we show that the output signals (observations) at a node only contain the spectral components of its so-called observable graph frequencies. Furthermore, we unveil that the observability of a node to the spectral components of a GS is related to the rank of its observability matrix from the perspective of control theory. Our work reveals that partial signal outputs at a node are sufficient to design the FIR GF locally in terms of node-variant (NV) GFs. These findings further enable us to characterize the minimum-degree NV GF, where a minimum number of its shifted GSs are involved in the filter's output, and devise a distributed GF design algorithm for it.

We consider Fair Principal Component Analysis (FPCA) and search for a low dimensional subspace that spans multiple target vectors in a fair manner. FPCA is defined as a non-concave maximization of the worst projected target norm within a given set. The problem arises in filter design in signal processing, and when incorporating fairness into dimensionality reduction schemes. The state of the art approach to FPCA is via semidefinite relaxation and involves a polynomial yet computationally expensive optimization. To allow scalability, we propose to address FPCA using naive sub-gradient descent. We analyze the landscape of the underlying optimization in the case of orthogonal targets. We prove that the landscape is benign and that all local minima are globally optimal. Interestingly, the SDR approach leads to sub-optimal solutions in this simple case. Finally, we discuss the equivalence between orthogonal FPCA and the design of normalized tight frames.

In this study, a novel security-enhanced filter (SEF) is proposed for the system state and malicious attack signal estimation of the stochastic jump-diffusion systems with the external disturbance, measurement noise and malicious attack signal on system and sensor. To efficiently estimate the system state and malicious attack signal by the traditional Luenberger-type filter, a novel smoothed signal model of malicious attack signals is embedded in the system model so that the attack signals in the augmented system do not corrupt the augmented states estimation of SEF again. For the optimal filtering robustness and security, the stochastic multi-objective (MO) $H_{2}/H_{\infty }$ SEF scheme is proposed to achieve optimal disturbance and noise filtering performance and the optimal security enhancement under malicious attack. By using the suboptimal method, the stochastic MO $H_{2}/H_{\infty }$ SEF design could be equivalently transformed to linear matrix inequalities (LMIs)-constrained multi-objective optimization problem (MOP). In the case of nonlinear stochastic system, the MO $H_{2}/H_{\infty }$ SEF design problem could be converted to a Hamilton-Jacobi inequalities (HJIs)-constrained MOP. In order to overcome the difficulty in solving the HJIs-constrained MOP, based on the global linearization technique, the HJIs-constrained MOP for SEF design of nonlinear stochastic systems could be transformed to an LMIs-constrained MOP. Further, an LMIs-constrained multi-objective evolution algorithm (MOEA) is proposed to efficiently solve the LMIs-constrained MOP for the design of SEF. Two simulation examples including the missile trajectory estimation problem by ground radar system under the malicious attack signals and estimation of netwoked-based mass spring system are given to validate the effectiveness of the proposed method.

In this paper we develop an adaptive transform-domain technique based on rational function systems. It is of general importance in several areas of signal theory, including filter design, transfer function approximation, system identification, control theory etc. The construction of the proposed method is discussed in the framework of a general mathematical model called variable projection. First we generalize this method by adding dimension type free parameters. Then we deal with the optimization problem of the free parameters. To this order, based on the well-known particle swarm optimization (PSO) algorithm, we develop the multi-dimensional hyperbolic PSO algorithm. It is designed especially for the rational transforms in question. As a result, the system along with its dimension is dynamically optimized during the process. The main motivation was to increase the adaptivity while keeping the computational complexity manageable. We note that the proposed method is of general nature. As a case study the problem of electrocardiogram (ECG) signal compression is discussed. By means of comparison tests performed on the PhysioNet MIT-BIH Arrhythmia database we demonstrate that our method outperforms other transformation techniques., Comment: Codes are available online at https://codeocean.com/capsule/0308835/tree/v1 Simulation data is available online at http://numanal.inf.elte.hu/~kovi/docs/pubs/Generalized_Rational_VarPro.rar

Generalized frequency division multiplexing (GFDM) has been regarded as a candidate for new multicarrier modulation schemes in future wireless communication systems (e.g., 5G systems). However, GFDM systems still suffer from the problem of high peak-to-average power ratio (PAPR). In this paper, theoretical analysis of the PAPR distribution for GFDM signals is performed. Exact closed-form expressions for the complementary cumulative distribution function (CCDF) of PAPR are firstly derived for critically sampled and oversampled GFDM signals, respectively, which tend to be sufficiently accurate as the number of subcarriers K is large enough (e.g., K ≥ 64). With aid of theoretical analysis results, an optimization criterion for the design of pulse shaping filters is obtained, which can nearly achieve the globally minimum CCDF results and is appropriate for practical applications. This criterion can also achieve the globally optimal result for minimizing the variance of instantaneous power. Based on this criterion, a measurement for the overall deviation of filter coefficients is proposed, which can help to roughly and efficiently evaluate the ability of PAPR reduction for pulse shaping filters. Although the PAPR reduction effect is limited with pulse shaping filters, it is found that large deviation may also lead to higher PAPR. Additionally, with the aid of closed-form CCDF expressions, PAPR characteristics of GFDM and other multicarrier systems (e.g., OFDM and FBMC/OQAM) are compared thoroughly. Simulation results verify the validity and accuracy of derived theoretical results, and demonstrate the effectiveness of the proposed optimization design criterion.

The linear canonical transform (LCT), which includes many classical transforms, has increasingly emerged as a powerful tool for optics and signal processing. Signal reconstruction associated with the LCT has blossomed in recent years. However, many existing reconstruction algorithms for the LCT can only handle noise-free measurements, and when noise is present, they will become ill posed. In this paper, we address the problem of reconstructing an analog signal from noise-corrupted measurements in the LCT domain. A general methodology is proposed to solve this problem in which the analog signal is recovered from ideal samples of its filtered version in a unified way. The proposed methodology allows for arbitrary measurement and reconstruction schemes in the LCT domain. We formulate signal reconstruction in an LCT-based function space, which is the span of integer translates and chirp-modulation of a generating function, with coefficients derived from digitally filtering noise corrupted measurements in the LCT domain. Several alternative methods for designing digital filters in the LCT domain are also suggested using different criteria. The validity of the theoretical derivations is demonstrated via numerical simulation.

In this paper, we consider the joint design of both transmit waveforms and receive filters for a colocated multiple-input-multiple-output (MIMO) radar with the existence of signal-dependent interference and white noise. The design problem is formulated into a maximization of the signal-to-interference-plus-noise ratio (SINR), including various constraints on the transmit waveforms. Compared with the traditional alternating semidefinite relaxation approach, a general and flexible algorithm is proposed based on the majorization-minimization method with guaranteed monotonicity, lower computational complexity per iteration and/or convergence to a B-stationary point. Many waveform constraints can be flexibly incorporated into the algorithm with only a few modifications. Furthermore, the connection between the proposed algorithm and the alternating optimization approach is revealed. Finally, the proposed algorithm is evaluated via numerical experiments in terms of SINR performance, ambiguity function, computational time, and properties of the designed waveforms. The experiment results show that the proposed algorithms are faster in terms of running time and meanwhile achieve as good SINR performance as the the existing methods.

In graph signal processing, filters arise from polynomials in shift matrices that respect the graph structure, such as the graph adjacency matrix or the graph Laplacian matrix. Hence, filter design for graph signal processing benefits from knowledge of the spectral decomposition of these matrices. Often, stochastic influences affect the network structure and, consequently, the shift matrix empirical spectral distribution. Although the joint distribution of the shift matrix eigenvalues is typically inaccessible, deterministic functions that asymptotically approximate the matrix empirical spectral distribution can be found for suitable random graph models using tools from random matrix theory. We employ this information regarding the density of eigenvalues to develop criteria for optimal graph filter design. In particular, we consider filter design for distributed average consensus and related problems, leading to improvements in short-term error minimization or in asymptotic convergence rate.

The maximally flat variable-delay (VD) digital filter can be derived by using the discrete-Pascal-transform (DPT) and the DPT-based Pascal-polynomial interpolation. In this paper, we develop an efficient structure for the low-complexity and high-modularity implementation of this Pascal-type VD filter. As compared to other existing filter structures, this new structure greatly reduces the multiplication operations required in the filtering process. Roughly speaking, the proposed new structure can reduce the multiplication operations by almost one-third. Thus, this new filter structure has the lowest filter complexity among all the existing filter structures for the implementation of the maximally flat VD filter. As a result, signal interpolation can be efficiently performed through utilizing this new filter structure. As opposed to other existing structure with transient problem, we will show that the proposed new structure does not involve any transient problem. That is, it is a transient-free structure. Consequently, the proposed new filter structure can achieve more accurate signal-interpolation results than other structures that have the transient problem. Finally, we show that this proposed structure also has high modularity and it is suited for modularized hardware implementation.

In this paper we derive a multi-sensor multi-Bernoulli (MS-MeMBer) filter for multi-target tracking. Measurements from multiple sensors are employed by the proposed filter to update a set of tracks modeled as a multi-Bernoulli random finite set. An exact implementation of the MS-MeMBer update procedure is computationally intractable. We propose an efficient approximate implementation by using a greedy measurement partitioning mechanism. The proposed filter allows for Gaussian mixture or particle filter implementations. Numerical simulations conducted for both linear-Gaussian and non-linear models highlight the improved accuracy of the MS-MeMBer filter and its reduced computational load with respect to the multi-sensor cardinalized probability hypothesis density filter and the iterated-corrector cardinality-balanced multi-Bernoulli filter especially for low probabilities of detection., 21 pages, 8 figures, 2 table

We study the optimal design of graph filters (GFs) to implement arbitrary linear transformations between graph signals. GFs can be represented by matrix polynomials of the graph-shift operator (GSO). Since this operator captures the local structure of the graph, GFs naturally give rise to distributed linear network operators. In most setups, the GSO is given so that GF design consists fundamentally in choosing the (filter) coefficients of the matrix polynomial to resemble desired linear transformations. We determine spectral conditions under which a specific linear transformation can be implemented perfectly using GFs. For the cases where perfect implementation is infeasible, we address the optimization of the filter coefficients to approximate the desired transformation. Additionally, for settings where the GSO itself can be modified, we study its optimal design as well. After this, we introduce the notion of a node-variant GF, which allows the simultaneous implementation of multiple (regular) GFs in different nodes of the graph. This additional flexibility enables the design of more general operators without undermining the locality in implementation. Perfect and approximate designs are also studied for this new type of GFs. To showcase the relevance of the results in the context of distributed linear network operators, this paper closes with the application of our framework to two particular distributed problems: finite-time consensus and analog network coding.

This paper is concerned with the distributed filtering problem for a class of discrete time-varying systems with stochastic nonlinearities and sensor degradation over a finite horizon. A two-step distributed filter algorithm is proposed where the sensor nodes collaboratively estimate the states of the plant by exploiting the information from both the local and the neighboring nodes. The goal of this paper is to design the distributed filters over a wireless sensor network subject to given sporadic communication topology. Moreover, a resilient operation is guaranteed to suppress random perturbations on the actually implemented filter gains. An upper bound is first derived for the filtering error covariance by utilizing an inductive method and such an upper bound is subsequently minimized via iteratively solving a quadratic optimization problem. To account for the topological information of the sensor networks, a novel matrix simplification technique is utilized to preserve the sparsity of the gain matrices in accordance with the given topology, and the analytical parameterization is obtained for the gain matrices of the desired suboptimal filter. Furthermore, a sufficient condition is established to guarantee the mean-square boundedness of the estimation errors. Numerical simulation is carried out to verify the effectiveness of the proposed filtering algorithm.

The analysis of signals on complex topologies modeled by graphs is a topic of increasing importance. Decompositions play a crucial role in the representation and processing of such information. Here, we propose a new tight frame design that is adapted to a class of signals on a graph. The construction starts from a prototype Meyer-type system of kernels with uniform subbands. The ensemble energy spectral density is then defined for a given set of signals defined on the graph. The prototype design is then warped such that the resulting subbands capture the same amount of energy for the signal class. This approach accounts at the same time for graph topology and signal features. The proposed frames are constructed for three different graph signal sets and are compared with non-signal-adapted frames. Vertex localization of a set of resulting atoms is studied. The frames are then used to decompose a set of real graph signals and are also used in a setting of signal denoising. The results illustrate the superiority of the designed signal-adapted frames, over frames blind to signal characteristics, in representing data and in denoising.

This paper considers the design of robust filters for radar pulse-Doppler processing when the interference is a wide sense stationary random process. The figure of merit which is optimized is the signal-to-interference-plus-noise ratio (SINR) at the filter output under a multitude of constraints accounting for Doppler filter sidelobes as well as uncertainties both in the received useful signal component and interference covariance matrix. The design is analytically formulated as a constrained optimization problem whose solvability is thoroughly studied. Precisely, a polynomial-time solution technique to get the optimal filter is proposed exploiting the representation of non-negative trigonometric polynomials via linear matrix inequalities, the spectral factorization theorem, and the duality theory. Last but not least, a detailed analysis of the optimum filter performance is provided showing the tradeoffs involved in the design and the gain achievable over some already known counterparts.

The topic of probing waveform design has received considerable attention due to its numerous applications in active sensing. Apart from having the desirable property of constant magnitude, it is also anticipated that the designed sequence possesses low sidelobe autocorrelation and/or specified spectral shape. In this paper, the alternating direction method of multipliers (ADMM), which is a powerful variant of the augmented Lagrangian scheme for dealing with separable objective functions, is applied for synthesizing the probing sequences. To achieve impulse-like autocorrelation, we formulate the design problem as minimizing a nonlinear least-squares cost function in the frequency domain subject to the constraint that all sequence elements are of unit modulus. Via introducing auxiliary variables, we are able to separate the objective into linear and quadratic functions where the unimodular constraint is only imposed on the former, which results in an ADMM-style iterative procedure. In particular, fast implementation for the most computationally demanding step is investigated and local convergence of the ADMM method is proved. To deal with the spectral shape requirement, we borrow the concept in frequency-selective filter design where passband and stopband magnitudes are bounded to formulate the corresponding optimization problem. In this ADMM algorithm development, unit-step functions are utilized to transform the multivariable optimization into a quadratic polynomial problem with a single variable. The effectiveness of the proposed approach is demonstrated via computer simulations.

Orthogonal-frequency-division-multiplexing/offset quadrature amplitude modulation (OFDM/OQAM) based multicarrier system achieves better spectral efficiency in comparison with the widely used multicarrier OFDM system. However, the OFDM/OQAM system suffers from intersubchannel interference and intersymbol interference in frequency-selective channels, which cause poor bit error rate (BER) performance. To overcome these challenging issues, in this paper, we propose a per-subchannel joint equalizer and receiver filter design method based on maximizing signal-to-interference-plus-noise ratio (SINR) of each subchannel. An iterative two-step algorithm is developed to design equalizer and receiver filter jointly. At the first step, by assuming that the receiver filter impulse response is identified, an equalizer is designed for each subchannel by maximizing its SINR. At the second step, by assuming that the channel and equalizer impulse responses are known, the SINR of each subchannel is derived based on the receiver filter impulse response coefficients. In this way, the receiver filter of each subchannel is designed independently by maximizing the SINR of that subchannel. The convergence of the proposed iterative design algorithm is proven and also the designing procedure is extended for multi-input–multi-output channels. Simulation results show that the proposed OFDM/OQAM system, in addition of its superior bandwidth efficiency, significantly achieves better performance in comparison with the OFDM system. Moreover, the convergence speed of the iterative SINR maximization algorithm is evaluated by numerical assessments under different scenarios.

This paper proposes two new recursive filtering structures based on the nonlinear myriad operator. First, we develop the general class of recursive weighted myriad (RWMy) filters as a robust filtering structure against impulsive noise that includes, as particular cases, a normalized version of the linear infinite impulse response (IIR) filter, the recursive mode-type filter, and the non-recursive weighted myriad filter. Secondly, considering the fact that the additive noise that contaminates the previous filter's outputs is no longer impulsive, we introduce a novel class of recursive hybrid myriad (RHMy) filters whose structure gathers the advantages of both the weighted myriad and the weighted mean in a single cost function to be minimized. Some properties of the RHMy filter are derived and a fast algorithm to compute the RHMy filter output is proposed. Furthermore, adaptive algorithms for designing the proposed recursive structures, under the equation error formulation framework, are developed. Finally, extensive numerical simulations are shown to evaluate both the iterative update of the adaptive algorithms and the performance of the proposed recursive filters against impulsive noise.

Designing and implementing systems as an interconnection of smaller subsystems is a common practice for modularity and standardization of components and design algorithms. Although not typically cast in this framework, many of these approaches can be viewed within the mathematical context of functional composition. This paper re-interprets and generalizes within the functional composition framework one such approach known as filter sharpening, i.e., interconnecting filter modules which have significant approximation error in order to obtain improved filter characteristics. More specifically, filter sharpening is approached by determining the composing polynomial to minimize the infinity-norm of the approximation error, utilizing the First Algorithm of Remez. This is applied both to sharpening for FIR, even-symmetric filters and for the more general case of subfilters that have complex-valued frequency responses including causal IIR filters and for continuous-time filters. Within the framework of functional composition, this paper also explores the use of functional decomposition to approximate a desired system as a composition of simpler functions based on a two-norm on the approximation error. Among the potential advantages of this decomposition is the ability for modular implementation in which the inner component of the functional decomposition represents the subfilters and the outer the interconnection.

This paper proposes a new recursive implementation of the Gaussian filter. Previous recursive implementations of the Gaussian filter suffer from one or several drawbacks. Some methods require employment of both causal and anti-causal filters. Implementing such filters requires both forward and backward recursions, and thus, significant amounts of memory, in the case of long sequences, to store and combine both recursion results. Moreover, most existing techniques do not ensure that the sum of impulse response weights equals unity, or that the standard deviation of the filter is exactly the desired one. The proposed technique is based on a recursive implementation of cosine-based finite impulse response filters and only requires forward recursions. The sum of filter weights and the standard deviation of the filter are constrained to the desired values during the design process. Experimental results confirm that the proposed filter provides a more suitable tradeoff between computational efficiency and filtering accuracy compared to existing recursive implementations. The performance and advantages of the proposed filter are demonstrated on signal and image processing examples.

In this paper, an optimal unbiased finite impulse response (OUFIR) filter is proposed as a linking solution between the unbiased FIR (UFIR) filter and the Kalman filter (KF). We first derive the batch OUFIR estimator to minimize the mean square error (MSE) subject to the unbiasedness constraint and then find its fast iterative form. It is shown that the OUFIR filter is full horizon (FH) and that its estimate converges to the KF estimate by increasing the horizon length $N$ . As a special feature, we note that the FH OUFIR filter operates almost as fast as the KF. Several other critical properties of the OUFIR filter are illustrated based on simulations and practical applications. Similar to the UFIR filter, and contrary to the KF, the OUFIR filter is highly insensitive to the initial conditions. It has much better robustness than KF against temporary model uncertainties. Finally, the OUFIR filter allows for ignoring system noise, which is typically not well known to the engineer.

An improved optimal factoring and zero-grouping algorithm is presented that reduces the hardware needed to implement FIR filters in a cascade structure, with better factors obtained via an optimal grouping of both off-unit-circle as well as on-unit-circle transfer-function zeros. The concept, additional design techniques, and benefits are examined here using practical examples. It is shown that the extended approach offers access to more features that can be exploited to achieve a greater reduction in coefficient quantization levels, thereby creating filters having minimal hardware complexity.

MIMO radar receivers need to reliably separate waveforms originating from different transmitters while suppressing interference for effective operation. In order to achieve this goal, mismatched filters may be employed at the receivers. By using mismatched filters, it is possible to reduce the power of the interference while strictly controlling the autocorrelation sidelobe and peak cross-correlation levels of the received waveforms. In this paper, we propose a mismatched filter design method that minimizes interference and jamming power at the filter output while the peak sidelobe and cross-correlation levels for all Doppler frequencies are constrained to desired values. The proposed design method is formulated as an optimization problem employing sum-of-squares representation of non-negative polynomials and solved using semidefinite relaxation. It is demonstrated that good interference suppression performance is achieved even when using an estimated covariance matrix.

We detail in this paper an $L_1$ -norm Linearly constrained normalized least-mean-square ( $L_1$ -CNLMS) algorithm and its weighted version ( $L_1$ -WCNLMS) applied to solve problems whose solutions have some degree of sparsity, such as the beamforming problem in uniform linear arrays, standard hexagonal arrays, and (non-standard) hexagonal antenna arrays. In addition to the linear constraints present in the CNLMS algorithm, the $L_1$ -WCNLMS and the $L_1$ -CNLMS algorithms take into account an $L_1$ -norm penalty on the filter coefficients, which results in sparse solutions producing thinned arrays. The effectiveness of both algorithms is demonstrated via computer simulations. When employing these algorithms to antenna array problems, the resulting effect due to the $L_1$ -norm constraint is perceived as a large aperture array with few active elements. Although this work focuses the algorithm on antenna array synthesis, its application is not limited to them, i.e., the $L_1$ -CNLMS is suitable to solve other problems like sparse system identification and signal reconstruction, where the weighted version, the $L_1$ -WCNLMS algorithm, presents superior performance compared to the $L_1$ -CNLMS algorithm.

In this paper, a general event-triggered framework is developed to deal with the finite-horizon $H_{\infty}$ filtering problem for discrete time-varying systems with fading channels, randomly occurring nonlinearities and multiplicative noises. An event indicator variable is constructed and the corresponding event-triggered scheme is proposed. Such a scheme is based on the relative error with respect to the measurement signal in order to determine whether the measurement output should be transmitted to the filter or not. The fading channels are described by modified stochastic Rice fading models. Some uncorrelated random variables are introduced, respectively, to govern the phenomena of state-multiplicative noises, randomly occurring nonlinearities as well as fading measurements. The purpose of the addressed problem is to design a set of time-varying filter such that the influence from the exogenous disturbances onto the filtering errors is attenuated at the given level quantified by a $H_{\infty}$ -norm in the mean-square sense. By utilizing stochastic analysis techniques, sufficient conditions are established to ensure that the dynamic system under consideration satisfies the $H_{\infty}$ filtering performance constraint, and then a recursive linear matrix inequality (RLMI) approach is employed to design the desired filter gains. Simulation results demonstrate the effectiveness of the developed filter design scheme.

New insights suggest that the most efficient FIR digital filters can be created by using a scaled sequence of stages, each representing a factor of the filter's transfer function. A crucial capability for building such filters concerns finding the best FIR filter factors, then carefully scaling and sequencing them. The efficiency of the resulting structure depends heavily upon obtaining such optimal factors. We offer an algorithm to find, scale and sequence optimally factored FIR filters.

We propose an optimization-based method for designing a lower order Luenberger-type state estimator, while providing L 2 -gain guarantees on the error dynamics when the estimator is used with the higher order system. Suitable filter parameters can be computed by modelling the bilinear system as a linear differential inclusion and solving a set of bilinear matrix inequality constraints. Since these constraints are nonconvex, in general, we also show that one can solve a suitably defined semi-definite program to compute a bound on the level of suboptimality. The design method also allows one to explicitly take account of linear parameter uncertainties in order to provide a priori robustness guarantees. The H-infinity estimator not only has lower real-time computational requirements compared with a Kalman filter, but also does not require knowledge of the noise spectrum. For a numerical example, we consider the estimation of the radiation force for a wave energy converter, where a low-order model is used to approximate the radiation dynamics.

The filter design optimization (FDO) problem is defined as finding a set of filter coefficients that yields a filter design with minimum complexity, satisfying the filter constraints. It has received a tremendous interest due to the widespread application of filters. Assuming that the coefficient multiplications in the filter design are realized under a shift-adds architecture, the complexity is generally defined in terms of the total number of adders and subtractors. In this paper, we present an exact FDO algorithm that can guarantee the minimum design complexity under the minimum quantization value, but can only be applied to filters with a small number of coefficients. We also introduce an approximate algorithm that can handle filters with a large number of coefficients using less computational resources than the exact FDO algorithm and find better solutions than existing FDO heuristics. We describe how these algorithms can be modified to handle a delay constraint in the shift-adds designs of the multiplier blocks and to target different filter constraints and filter forms. Experimental results show the effectiveness of the proposed algorithms with respect to prominent FDO algorithms and explore the impact of design parameters, such as the filter length, quantization value, and filter form, on the complexity and performance of filter designs.

This paper investigates the problem of frequency-specific (FS) model approximation of transfer functions using a min-max approach. The objective is to find an approximation model for a transfer function such that the maximum error gain over a specific frequency range is minimized. First, a linear matrix inequality condition characterizing the FS gain of a transfer function is derived by using the generalized Kalman-Yakubovich-Popov lemma, and then a simple iterative approach is proposed to optimize the approximation model. Numerical experiments show that the proposed approach can produce better approximation models over a specific frequency range than some existing approaches. Moreover, it is indicated how to apply the proposed approximation approach to the design problem of infinite impulsive response digital filters, and design examples clearly illustrate that the proposed design flow can generate filters comparable with the latest design method.

This paper investigates the problem of reduced-order generalized $H_{\infty}$ filtering for linear discrete-time systems. Generalized $H_{\infty}$ filtering covers standard $H_{\infty}$ filtering as a special case and can make use of the frequency-domain characteristics of practical signals. Moreover, reduced-order filters are more attractive than full-order ones regarding implementability. By virtue of the generalized bounded real lemma, a new necessary and sufficient condition is proposed for analyzing the generalized $H_{\infty}$ performance of the filtering error system. To compute reduced-order generalized $H_{\infty}$ filters, a necessary and sufficient condition is then derived in terms of matrix inequalities, to solve which iterative linear matrix inequality algorithms are further developed. In addition, the proposed filter design method is applied to cope with the channel equalization problem. Examples are finally presented for illustrating the proposed method.

This paper proposes $M$ -channel oversampled filter banks for graph signals. The filter set satisfies the perfect reconstruction condition. A method of designing oversampled graph filter banks is presented that allows us to design filters with arbitrary parameters, unlike the conventional critically sampled graph filter banks. The oversampled graph Laplacian matrix is also introduced with a discussion of the entire redundancy of the oversampled graph signal processing system. The practical performance of the proposed filter banks is validated through graph signal denoising experiments.

Signals and datasets that arise in physical and engineering applications, as well as social, genetics, biomolecular, and many other domains, are becoming increasingly larger and more complex. In contrast to traditional time and image signals, data in these domains are supported by arbitrary graphs. Signal processing on graphs extends concepts and techniques from traditional signal processing to data indexed by generic graphs. This paper studies the concepts of low and high frequencies on graphs, and low-, high- and band-pass graph signals and graph filters. In traditional signal processing, these concepts are easily defined because of a natural frequency ordering that has a physical interpretation. For signals residing on graphs, in general, there is no obvious frequency ordering. We propose a definition of total variation for graph signals that naturally leads to a frequency ordering on graphs and defines low-, high-, and band-pass graph signals and filters. We study the design of graph filters with specified frequency response, and illustrate our approach with applications to sensor malfunction detection and data classification.