The Kalman Filter (KF) is used to estimate and predict state-space models. It frequently encounters uncertain covariates, such as those derived from noisy measurements of physical signals (e.g., sensor readings, weather data, etc.). Our aim is to account for their uncertainties and develop robust KF approaches when incorporating the uncertainties of the covariates into the model; that is, by treating them as stochastic variables with known variances. To enhance the robustness of the KF viewed as a Bayesian method that estimates the posterior distribution of the system state conditioned on past observations, we propose two methods. The linear Bayes estimator yields an explicit, unbiased forecast with minimal variance. The optimal risk estimator is nonlinear and remains robust even when the covariates are highly noisy. This second method is approximated using a sequential Monte Carlo technique.
This article investigates the problem of deep learning-based state estimation for dynamic systems with partially known dynamics and unknown noise statistics. We propose a Deep Double-Bayesian Filter (DDBF), which leverages Bayesian deep learning for Bayesian filtering, augmenting state estimation while enabling the quantification of both aleatoric and epistemic uncertainties in a deep learning-based filter. First, we introduce a deep neural network augmented mismatch compensation mechanism into the state-space representation to bridge the gap between the nominal model and the actual system dynamics.We model the aleatoric uncertainty in the compensation terms using Bayesian deep learning tools, resulting in an SSM with explicit uncertainty formulation that establishes the probabilistic propagation model, which defines the evolution of the states and measurements. Second, we present a Bayesian filtering algorithm based on this SSM that enables an approximate Bayesian approach for state estimation under partially unknown system dynamics. In contrast to classical deep learning-based filters, the proposed DDBF learns the posterior predictive distribution of states from an approximate Bayesian perspective. This development allows for a more nuanced consideration of learning-based state estimation by combining input-dependent aleatoric uncertainty together with epistemic uncertainty. Finally, the effectiveness of the proposed DDBF is verified through simulations and real-world datasets.
In this paper, we investigate the design of multiple orthogonal frequency-division multiplexing (OFDM) waveforms for joint radar and communications applications. The design objectives are to simultaneously minimize the integrated sidelobe level (ISL) and the peak-to-mean envelope power ratio (PMEPR) of the OFDM waveforms, while enabling high-rate data transmission through efficient information embedding (IE). To this end, we propose an adaptive IE scheme that jointly exploits frequency index and phase modulations to embed communication information while preserving sufficient degrees of freedom for radar-oriented optimization. Based on the proposed hybrid modulation framework, multi-OFDM waveforms are designed by optimizing a balanced trade-off between ISL and PMEPR. Unlike conventional approaches that focus solely on peak-to-average power ratio reduction or power allocation, the proposed method jointly optimizes the amplitudes and phases of complex weights across multiple OFDM subcarriers. The resulting design incorporates multiple constraints, including support for hybrid IE, compliance with the total transmit power budget, and enforcement of minimum amplitude levels for OFDM subcarrier weights. We formulate it as a nonconvex optimization problem, which is inherently challenging to solve. To address this challenge, we reformulate the objective function and apply majorization-minimization techniques. Furthermore, to address the complexity arising from multiple constraints, we reformulate the original problem as a joint optimization framework involving two separable variables, namely, the weight amplitude and phase, which is then solved in a cyclic manner. Closed-form iterative update rules are derived using suitable majorization functions, and the amplitude optimization is addressed via Lagrangian analysis and Karush-Kuhn-Tucker conditions. Finally, an efficient algorithm is proposed, and extensive simulation results demonstrate the effectiveness of the proposed approach.
An analytic para-Hermitian matrix is commonly used to characterize wideband signal models in array signal processing. In this paper, an algorithm for computing the analytic solution of a linear system involving an analytic, point-wise positive-definite, para-Hermitian matrix is proposed. The algorithm is based on the iterative Krylov-subspace method and extends the well-known conjugate gradient (CG) algorithm to an infinite-dimensional Hilbert space that contains the desired analytic solution. The convergence of the resulting para-Hermitian CG (pHCG) algorithm is proved based on the linearity, self-adjointness, boundedness, and positive-definiteness properties of a convolution operator associated with the para-Hermitian matrix. The proposed algorithm converges iteratively to the solution within a predefined accuracy, without involving matrix inversion or decomposition. In addition to the pHCG, which works with sequences (or time-domain signals), an equivalent z-domain algorithm, named z-pHCG, is also proposed. The effectiveness of the proposed algorithm is illustrated through numerical examples, which also reveal the limitations of conventional approaches from the literature.
For vector sensors (VSs) comprising multiple components, quaternion embedding (QE) is a promising modeling approach that provides an elegant algebraic representation and captures cross-component correlations. In particular, the quaternion formalism facilitates diverse signal models. By coupling this advantage with the spatial flexibility provided by multiple VSs within an array, we propose the hybrid QE (HQE) methodology. Unlike conventional fixed representations of array data, we adopt different embedding types across VSs, yielding a novel family of signal models. This motivates us to investigate the following issues in a unified manner. First, by leveraging quaternion orthogonality, we develop the HQE-MUSIC algorithm for VS-array direction-of-arrival estimation and show that mutually complementary HQE patterns yield identical spatial spectra. Furthermore, we derive a closed-form expression for the asymptotic mean square error (MSE) of HQE-MUSIC and reveal its tight lower bound. Building on this, we formulate an MSE-based optimization problem to identify satisfactory HQE models. Considering the numerous candidates resulting from a large array, we introduce a tailored genetic algorithm to alleviate the computational burden. We demonstrate that HQE serves as a remedy for conventional QE schemes, mitigating their potential performance compromise in terms of MSE. Moreover, the optimized HQE-MUSIC, in contrast to its complex-valued counterpart, provides improved estimation accuracy, achieves higher angular resolution, especially in snapshot-limited cases, and exhibits robustness to cross-component correlated noise.
Compressive sensing (CS) enables fast spatial channel estimation in millimeter-wave and terahertz systems by leveraging the sparsity of the channel in the angle-domain. CS measurements, however, are often distorted by in-phase and quadrature-phase (IQ) imbalance at the oscillator, leading to a model mismatch. In this paper, we study how this mismatch impacts the channel estimated with a standard CS algorithm. Next, we develop an augmented CS model to jointly estimate the sparse channel and the IQ imbalance parameter. The sparse vector in our model comprises the channel as well as the IQ imbalance parameter. We show that this vector exhibits group sparsity, which is exploited using our custom paired-support orthogonal matching pursuit (PSOMP) algorithm. Finally, the estimate is decomposed to determine the channel and the IQ imbalance parameter. We provide support recovery guarantees for our PSOMP algorithm, highlighting the impact of IQ imbalance on channel recovery. Numerical results show that our method achieves better support recovery and lower error in the estimated channel than the baselines.
Accurate direction-of-arrival (DoA) estimation is beneficial in array signal processing and directional wireless communications, where improving the spatial resolution is a key challenge. This paper proposes an oversampled discrete Fourier transform (DFT) beamspace framework combined with ESPRIT-type algorithms. By formulating the generalized shift-invariance equations with an arbitrary integer oversampling factor, the proposed DFT-O-UESPRIT algorithm enables high oversampling factor processing and improves the spatial focusing capability of DFT beams. Furthermore, we derive a first-order asymptotic performance analysis for the proposed DFT-O-UESPRIT algorithm to examine its effectiveness. The resulting analytical expressions for the parameter estimation errors hold in the high effective signal-to-noise ratio (SNR) or large-sample-size regime. Notably, these expressions provide theoretical support for resolution enhancement in oversampled beamspace. Numerical simulations validate the theoretical expressions and demonstrate that the proposed approach outperforms both conventional antenna space Unitary ESPRIT and non-oversampled DFT beamspace Unitary ESPRIT, particularly under low SNR conditions and with closely spaced sources.
It often occurs in multistatic localization systems that the direct paths between an unknown position transmitter and the receivers are absent or blocked. Moreover, the receivers may not be able to time-synchronize with one another. To enable the effective localization of a moving object, we introduce calibration objects to generate additional measurements that can help remove the receiver-dependent synchronization offsets and provide information about the transmitter position. First, we analyze the localizability of the object position and velocity, as well as the transmitter position, in relation to the number of calibration objects. Based on the localizability analysis, we formulate two semidefinite programming problems and develop a closed-form solution, and utilize their combinations to effectively solve the localization problems with different numbers of calibration objects. Additionally, the theoretical mean squared errors (MSEs) of these localization solutions are derived, demonstrating that they can achieve the Crámer-Rao lower bound (CRLB) performance. Finally, simulations validate both the theoretical findings and the good performance of the proposed solutions.
State-space models, while fundamental to dynamic system analysis, face significant challenges in handling non-Gaussian outliers characterized by skewness and heavy tails. This paper addresses robust state estimation problems utilizing various asymmetric noise distributions and loss functions. We propose an optimization algorithm based on the Successive Convex Approximation (SCA) scheme. Our paper presents different interpretations of the proposed state estimation algorithm, offering different perspectives on robust filters. Comprehensive experiments demonstrate that our methods provide robust and efficient performance under non-Gaussian noise conditions, validating the algorithm’s effectiveness.
In this paper, a distributed estimation algorithm is proposed to address the H∞-consensus state estimation challenges over binary sensor networks. The algorithm extracts richer information from limited binary data by modeling the time-varying threshold of binary sensors as a linear combination of switching measurements at two consecutive time instants. By incorporating the constraint induced by the switching events, a novel distributed estimator is designed, which integrates data from both the sensor itself and neighboring nodes within its sensing range to enhance estimation accuracy. In addition, a local performance analysis framework is developed based on the vector dissipativity theory such that the proposed distributed estimation algorithm can be independently executed at each node, significantly reducing the computational complexity. Furthermore, sufficient conditions for the desired H∞-consensus performance are derived, and a systematic procedure for directly calculating the estimator gains is formulated as linear matrix inequalities. Both theoretical analysis and simulation results demonstrate the effectiveness of the proposed algorithm.
Conventional uniform Pulse Repetition Frequency (PRF) radars face an inherent tradeoff between maximum unambiguous range and maximum unambiguous velocity, which fundamentally limits the detection of distant moving targets. Non-uniform PRF radar mitigates this tradeoff by breaking sampling periodicity, thereby extending the unambiguous detection region, alleviating velocity ambiguities, and improving Low-Probability-of-Intercept (LPI) performance. In this paper, pulse transmission instants are introduced as an additional design variable, jointly optimized with the waveform to enhance target detection in cluttered environments. The resultant design is formulated as a Signal-to-Interference-plus-Noise Ratio (SINR) maximization problem under practical constraints on waveform dynamics and interpulse transmission intervals. The proposed joinT pulSe-amplitUde aNd trAnsMission-instant optImization (TSUNAMI) algorithm employs an alternating optimization scheme. It is shown that waveform phase does not affect the maximum achievable SINR, enabling a globally optimal solution for the waveform subproblem, whereas the transmission-instant subproblem is solved via a sequence of Quadratic Programming (QP) steps. Simulation results demonstrate that joint optimization of wave-form amplitude and transmission instants provides substantial SINR gains and enables favorable tradeoffs between range and velocity ambiguities for long-range, high-speed targets.
In recent years, integrated sensing and communication (ISAC) has garnered widespread attention from both the academia and wireless industry. Sequences with excellent ambiguity functions are critical in ISAC systems. In this paper, we propose several constructions of Doppler resilient complementary sequence sets with zero ambiguity zone properties (ZAZ-DRCSSs). First, we present two construction frameworks for ZAZ-DRCSSs using orthogonal matrices and designed mapping sets. By providing mapping sets that satisfy the conditions, the parameters of the resulting sequence sets can meet or conditionally approach the theoretical bound. Additionally, the flock size of the sequence set in the first construction can be flexibly specified. We then introduce a class of conditionally optimal ZAZ-DRCSSs based on additive and multiplicative characters over finite fields. To the best of our knowledge, this is the first construction of DRCSS based on the theory of finite fields.
In this paper, we investigate the problem of decentralized online resource allocation in the presence of Byzantine attacks. In this problem setting, some agents may be compromised due to external manipulations or internal failures, causing them to behave maliciously and disrupt the resource allocation process by sending incorrect messages to their neighbors. Given the non-consensual nature of the resource allocation problem, we formulate it under a primal-dual optimization framework, in which the dual variables are aggregated among the agents, enabling the incorporation of robust aggregation rules to mitigate Byzantine attacks. By leveraging the classical Byzantine attack model, we propose a class of Byzantine-resilient decentralized online resource allocation algorithms that judiciously integrate the adaptive robust clipping technique with the existing robust aggregation rules to filter out malicious messages. We establish theoretical guarantees, showing that the proposed algorithms achieve tight linear dynamic regret and accumulative constraint violation bounds, where the constants depend on the properties of robust aggregation rules. Numerical experiments on decentralized online economic dispatch validate the effectiveness of our approach and support our theoretical results.
This paper proposes an interleaved transmission architecture for a multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system that enables continuous data transmission while facilitating receiver processing reminiscent of pulse-based radar. A key feature of this architecture is that we can jointly design each transmitted ISAC block and its sensing filter to mitigate the high sidelobes that arise from the inter-block interference caused by targets at different ranges. We provide an exemplar of such a design in which constructive interference is integrated into the communication aspects, and the integrated mainlobe-to-sidelobe ratio (IMSR) of the beampattern is used to ensure desirable directivity of the sensing. In addition, the power of each time sample is constrained to manage the peak-to-average power ratio (PAPR). The joint transceiver design problem is addressed using an alternating optimization (AO) framework, with the subproblem for transmitted waveform design being solved via the successive convex approximation (SCA) method. To further enhance computational efficiency, the alternating direction penalty method (ADPM) is employed to solve the subproblems within the SCA iterations. The convergence of ADPM is established, with convergence of the case of more than two auxiliary variables being established for the first time. Numerical simulations validate the effectiveness of our proposed approach in achieving desirable performance in both radar sensing and communication, with the fast algorithm achieving comparable performance with greater computational efficiency.
In this paper, we investigate a fundamental connection between prime numbers and the sinc function, which we term the prime-sinc relation (PSR). At the core of our analysis lie two classical facts: (i) every natural number greater than one has a unique prime factorization, and (ii) the normalized sinc function vanishes precisely at all nonzero integers, thereby allowing representation as an infinite product of structured polynomials. Exploiting these facts, we introduce a novel set-theoretic operation termed the universal combinatorial product (UCP), from which we construct two classes of entire functions as infinite products of distinct sinc functions. This construction leads directly to our main result for the PSR. By leveraging this foundational relationship, we introduce a novel expression for well-known functions (such as the Riemann zeta function), highlighting its mathematical significance. From a practical perspective, the proposed PSR theorem provides a unifying representation framework for signal processing. In particular, we introduce a new class of atom sets based on PSR, which enables flexible spectral filter design with arbitrary shapes and offers an efficient representation for implicit neural networks that learn functional mappings. Furthermore, the PSR framework suggests broader potential across signal processing tasks, including waveform shaping, signal denoising, and sequence design, with relevance to interference mitigation, quality enhancement, and signal detection.
It is known that joint channel estimation and soft symbol decoding enhance the performance of multiple-input multiple-output (MIMO) communication systems with low-resolution analog-to-digital converters (ADCs). However, existing techniques require accurate knowledge of channel statistics to be effective. In this paper, we consider an uplink multi-user MIMO system and develop novel algorithms for joint estimation of channel coefficients and statistics along with soft symbol decoding. Specifically, we develop a sequential processing technique, based on variational Bayesian inference, for jointly estimating the covariance matrices of all users’ channels, the corresponding channel matrices, and soft symbols. We also develop an alternative block processing scheme for the same problem. Corresponding algorithms for sequential and block processing of unquantized data can be obtained as simplifications of our algorithms. On the analytical side, we derive a marginalized Bayesian Cramér-Rao Lower Bound (MB-CRLB) for covariance matrix estimation and elucidate the effect of the number of snapshots and SNR on the estimation error. Finally, we empirically evaluate the new algorithms and demonstrate their superior performance in scenarios where the channel statistics are unknown. The results also show that the mean squared error in channel covariance matrix estimation via the proposed algorithm closely follows the behavior of the MB-CRLB as a function of the SNR and number of frames.
This paper extends the widely used probability hypothesis density (PHD) and cardinalized PHD (CPHD) filters to non-standard observation models (NSOMs), such as pixelized track-before-detect and superpositional sensor models. Classical (C)PHD filters are computationally attractive but are derived under the standard point-object observation model. Existing (C)PHD variants for specific NSOMs typically rely on simplified assumptions, such as independent object-generated observations, to obtain closed-form solutions. These assumptions can fail in practical scenarios, such as closely spaced or merged-observation objects, resulting in severe performance degradation. To address this issue, we adapt (C)PHD filtering to the generic observation model (GOM), where the update step directly uses a generic multi-object likelihood. The Bayesian posterior under the GOM is projected back onto the Poisson and independently and identically distributed cluster families via Kullback-Leibler divergence minimization, yielding the proposed GOM-PHD and GOM-CPHD filters. We further show that the proposed filters reduce to existing (C)PHD variants under specific model assumptions and analyze the posterior-projection error. Furthermore, we develop a fast sequential Monte Carlo implementation of the proposed filters that avoids the combinatorial explosion, reduces the number of likelihood evaluations, and includes its convergence analysis. Finally, simulation results, including one diagnostic posterior-projection analysis scenario and two NSOM tracking scenarios, show regimes in which the posterior-projection error is smaller than the error induced by a mismatched likelihood and that the proposed filters improve tracking accuracy over representative NSOM-specific baselines with lower computational cost than labeled random finite set GOM baselines.
This paper develops a novel optimization framework for target localization in a distributed Frequency Diverse Array (FDA) system. A general signal model is firstly established, where widely dispersed nodes each employ a colocated FDA Multiple-Input Multiple-Output radar exploiting intra-node frequency diversity. Closed-form Cramér-Rao Bound (CRB) expression for distributed FDA is derived, revealing that the overall CRB decouples into a sum of per-node contributions, where the node-specific CRB formulation separates into a geometric factor dependent solely on node coordinates and a ratio of quadratic forms capturing the frequency increment dependence. Based on this structure, a joint optimization problem is formulated to minimize the node-specific CRBs subject to circular position constraints and bounded frequency allocations. To tackle this nonconvex problem, a nested Minorization-Maximization (MM)-Maximum Block Improvement (MBI) algorithm is developed. The outer loop greedily selects the node yielding the greatest overall CRB reduction, while the inner loop optimizes the chosen node’s variables via block-wise updates. In particular, a Coordinate Descent -Projection method leverages the simple algebraic form of the geometric factor for topology optimization, and the MM framework constructs a tight minorant for frequency increments. The MBI strategy updates only the block providing the maximum decrease in the objective. Convergence analysis establishes that every cluster point of the iterates satisfies the Karush-Kuhn-Tucker conditions. Numerical results demonstrate significant CRB reduction and validate the effectiveness of the joint optimization through comparisons with Alternating Optimization and simpler MBI-based counterparts.
Sparse representation of signals in overcomplete dictionaries is a powerful tool with applications in denoising, restoration, compression, reconstruction, and more. Classically, these dictionaries are based on transforms such as DCT and wavelets. These so-called analytic dictionaries capture the global structure of a signal and allow a fast implementation. In contrast, modern dictionaries are data-driven, in that the elements are derived from the data. As such, they may adapt to a class of signals of interest, offering better and/or sparser representations. The drawback is a higher computational load, since learned dictionaries are represented by unstructured matrices. In this work, we use the concept of Displacement Structure to introduce some degree of structure on the trained dictionary. The resulting dictionary provides a favorable performance-complexity tradeoff, being less complex than unstructured dictionaries, while typically offering improved performance over analytic ones. Some image denoising experiments are provided to validate the proposed technique. While it does not consistently outperform other structured dictionary learning approaches, its flexibility remains a key advantage, as it does not restrict the structure to a single predefined model and can be adapted through an appropriate choice of displacement operators.
