This paper investigates the positioning of the pilot symbols, as well as the power distribution between the pilot and the communication symbols for the orthogonal time frequency space (OTFS) modulation scheme. We analyze the pilot placements that minimize the mean squared error (MSE) in estimating the channel taps. This allows us to identify two new pilot allocations for OTFS that save approximately 50% of the pilot overhead compared to existing allocations. In addition, we optimize the average channel capacity by adjusting the power distribution. We show that this leads to a significant increase in average capacity. The results provide valuable guidance for designing the OTFS parameters to achieve maximum capacity. Numerical simulations are performed to validate the findings.

In the paper, we consider the line spectral estimation problem in an unlimited sensing framework (USF), where a modulo analog-to-digital converter (ADC) is employed to fold the input signal back into a bounded interval before quantization. Such an operation is mathematically equivalent to taking the modulo of the input signal with respect to the interval. To overcome the noise sensitivity of higher-order difference-based methods, we explore the properties of the first-order difference of modulo samples, and develop two line spectral estimation algorithms based on the first-order difference, which are robust against noise. Specifically, we show that, with a high probability, the first-order difference of the original samples is equivalent to that of the modulo samples. By utilizing this property, line spectral estimation is solved via a robust sparse signal recovery approach. The second algorithms is built on our finding that, with a sufficiently high sampling rate, the first-order difference of the original samples can be decomposed as a sum of the first-order difference of the modulo samples and a sequence whose elements are confined to three possible values. This decomposition enables us to formulate the line spectral estimation problem as a mixed integer linear program that can be efficiently solved. Simulation results show that both proposed methods are robust against noise and achieve a significant performance improvement over the higher-order difference-based method. methods.

It is with great pleasure that we introduce this special issue of Signal Processing, published to mark the 45th anniversary of the journal. Since its inception in 1979, Signal Processing has played a foundational role in shaping the evolution of the field. As the flagship journal of the European Association for Signal Processing (EURASIP), it has served not only as a premier venue for scientific dissemination but also as a pillar of the signal processing research community. [...]

In this paper, we design Graph Neural Networks (GNNs) with attention mechanisms to tackle an important yet challenging nonlinear regression problem: massive network localization. We first review our previous network localization method based on Graph Convolutional Network (GCN), which can exhibit state-of-the-art localization accuracy, even under severe Non-Line-of-Sight (NLOS) conditions, by carefully preselecting a constant threshold for determining adjacency. As an extension, we propose a specially designed Attentional GNN (AGNN) model to resolve the sensitive thresholding issue of the GCN-based method and enhance the underlying model capacity. The AGNN comprises an Adjacency Learning Module (ALM) and Multiple Graph Attention Layers (MGALs), employing distinct attention architectures to systematically address the demerits of the GCN-based method, rendering it more practical for real-world applications. Comprehensive analyses are conducted to explain the superior performance of these methods, including a theoretical analysis of the AGNN's dynamic attention property and computational complexity, along with a systematic discussion of their robust characteristic against NLOS measurements. Extensive experimental results demonstrate the effectiveness of the GCN-based and AGNN-based network localization methods. Notably, integrating attention mechanisms into the AGNN yields substantial improvements in localization accuracy, approaching the fundamental lower bound and showing approximately 37% to 53% reduction in localization error compared to the vanilla GCN-based method across various NLOS noise configurations. Both methods outperform all competing approaches by far in terms of localization accuracy, robustness, and computational time, especially for considerably large network sizes.

Ultrasonography could allow operator-independent examination and continuous monitoring of the carotid artery (CA) but normally requires complex and expensive transducers, especially for 3-D. By employing computational ultrasound imaging (cUSi), using an aberration mask and model-based reconstruction, a monitoring device could be constructed with a more affordable simple transducer design comprising only a few elements. We aim to apply the cUSi concept to create a CA monitoring system. The system’s possible configurations for the 2-D imaging case were explored using a linear array setup emulating a cUSi device in silico, followed by in vitro testing and in vivo CA imaging. Our study shows enhanced reconstruction performance with the use of an aberrating mask, improved lateral resolution through proper choice of the mask delay variation, and more accurate reconstructions using least-squares with QR (LSQR) decomposition compared to matched filtering (MF). Together, these advancements enable B-mode reconstruction and power Doppler imaging (PDI) of the CA with sufficient quality for monitoring using a configuration of 12 transceivers coupled with a random aberration mask with a maximum delay variation of four wave periods (WPs).

In this letter, we propose an analytical solution for recovering a low-rank positive semi-definite (PSD) matrix from its rank-one measurements. We show that by utilizing a set of structured measurement vectors, we can analytically determine the null space of this low-rank PSD matrix. Based on the result, the PSD matrix can be efficiently recovered. Our analysis shows that the proposed method only requires (N - K)(2K + 1)+ K ² measurements to guarantee exact recovery of the PSD matrix, where N and K respectively denote the dimension and the rank of the PSD matrix. Numerical results show that the proposed method achieves a considerable improvement over existing state-of-the-art methods in terms of both sample complexity and computational efficiency. Specifically, the proposed method helps improve the computational efficiency by an order of magnitude as compared with existing methods.

This paper addresses graph topology identification for applications where the underlying structure of systems like brain and social networks is not directly observable. Traditional approaches based on signal matching and spectral templates have limitations, particularly in handling scale issues and sparsity assumptions. We introduce a novel covariance matching methodology that efficiently reconstructs the graph topology using observable data. For the structural equation model (SEM) using an undirected graph, we demonstrate that our method can converge to the correct result under relatively soft conditions. Furthermore, we extend our methodology to polynomial models and any known distribution of latent variables, broadening its applicability and utility in diverse graph-based systems.

Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of the fields of graph signal processing (GSP) and graph machine learning (GML). Key graph-aware methods include Fourier transform, filtering, sampling, as well as topology identification and spatiotemporal processing. Although versatile, graphs can model only pairwise dependencies in the data. To this end, topological structures such as simplicial and cell complexes have emerged as algebraic representations for more intricate structure modeling in data-driven systems, fueling the rapid development of novel topological-based processing and learning methods. This paper first presents the core principles of topological signal processing through the Hodge theory, a framework instrumental in propelling the field forward thanks to principled connections with GSP-GML. It then outlines advances in topological signal representation, filtering, and sampling, as well as inferring topological structures from data, processing spatiotemporal topological signals, and connections with topological machine learning. The impact of topological signal processing and learning is finally highlighted in applications dealing with flow data over networks, geometric processing, statistical ranking, biology, and semantic communication.

Ultrafast imaging, which uses unfocussed transmissions to form images, provides very high frame rates at the cost of low signal-to-noise ratio (SNR). This loss of SNR becomes especially apparent when imaging deeper structures. Ultrafast imaging is mostly used in combination with Doppler processing. Even if we apply tissue-separation filters, they lead to significant energy loss and decrease the SNR. Previous work showed that this loss in SNR and, hence, penetration depth can be partially regained using coded transmissions. However, these codes are mostly either standard or randomly generated and can be improved with a design rooted in an optimization scheme. To address this limitation, we design an optimized code tailored to ultrasound imaging with unfocused transmissions represented by a generalized encoding matrix in a linear signal model. We employ the minimization of the Cramér-Rao lower bound (CRB) over the unknown coding matrix as a way to optimize the code. Due to the high computational cost of the resulting optimization problems, we also introduce a trace-constraint optimization problem based on the Fisher information matrix (FIM). Simulation results show that the optimized code provides higher SNR in deep image regions than previously tested coding schemes such as the Barker code, albeit with a trade-off for decreased resolution. On the other hand, the application of least-squares QR (LSQR) mitigates this resolution degradation. Lastly, the optimized code was tested in simulations using a numerical model of a clinical transducer setting, demonstrating its potential for higher SNR in ultrafast Doppler imaging.

Neural networks on simplicial complexes (SCs) can learn representations from data residing on simplices such as nodes, edges, triangles, etc. However, existing works often overlook the Hodge theorem that decomposes simplicial data into three orthogonal characteristic subspaces, such as the identifiable gradient, curl and harmonic components of edge flows. This provides a universal tool to understand the machine learning models on SCs, thus, allowing for better principled and effective learning. In this paper, we study the effect of this data inductive bias on learning on SCs via the principle of convolutions. Particularly, we present a general convolutional architecture that respects the three key principles of uncoupling the lower and upper simplicial adjacencies, accounting for the inter-simplicial couplings, and performing higher-order convolutions. To understand these principles, we first use Dirichlet energy minimizations on SCs to interpret their effects on mitigating simplicial oversmoothing. Then, we show the three principles promote the Hodge-aware learning of this architecture, through the lens of spectral simplicial theory, in the sense that the three Hodge subspaces are invariant under its learnable functions and the learning in two nontrivial subspaces is independent and expressive. Third, we investigate the learning ability of this architecture in optic of perturbation theory on simplicial topologies and prove that the convolutional architecture is stable to small perturbations. Finally, we corroborate the three principles by comparing with methods that either violate or do not respect them. Overall, this paper bridges learning on SCs with the Hodge theorem, highlighting its importance for rational and effective learning from simplicial data, and provides theoretical insights to convolutional learning on SCs.

CANDECOMP/PARAFAC (CP) decomposition is the mostly used model to formulate the received tensor signal in a massive MIMO system, as the receiver generally sums the components from different paths or users. To achieve accurate and low-latency channel estimation, good and fast CP decomposition (CPD) algorithms are desired. The CP alternating least squares (CPALS) is the workhorse algorithm for calculating the CPD. However, its performance depends on the initializations, and good starting values can lead to more efficient solutions. Existing initialization strategies are decoupled from the CPALS and are not necessarily favorable for solving the CPD. This paper proposes a deep-learning-aided CPALS (DL-CPALS) method that uses a deep neural network (DNN) to generate favorable initializations. The proposed DL-CPALS integrates the DNN and CPALS to a model-based deep learning paradigm, where it trains the DNN to generate an initialization that facilitates fast and accurate CPD. Moreover, benefiting from the CP low-rankness, the proposed method is trained using noisy data and does not require paired clean data. The proposed DL-CPALS is applied to millimeter wave MIMO-OFDM channel estimation. Experimental results demonstrate the significant improvements of the proposed method in terms of both speed and accuracy for CPD and channel estimation.

Affine formation control (AFC) is a distributed networked control system that has recently received increasing attention in various applications. AFC is typically achieved using a generalized consensus system where the stress matrix, which encodes the graph structure, is used instead of a graph Laplacian. Universally rigid frameworks (URFs) guarantee the existence of the stress matrix and have thus become the guideline for such a network design. In this work, we propose a convex optimization framework to design the stress matrix for AFC without predefining a rigid graph. We aim to find a resulting network with a reduced number of communication links, but still with a fast convergence speed. We show through simulations that our proposed solutions can yield a more sparse graph, while admitting a faster convergence compared to the state-of-the-art solutions.

Sparse array design is used to help reduce computational, hardware, and power requirements compared to uniform arrays while maintaining acceptable performance. Although minimizing the Cramér-Rao bound has been adopted previously for sparse sensing, it did not consider multiple targets and unknown target directions. To handle the unknown target directions when optimizing the Cramér-Rao bound, we propose to use the worst-case Cramér-Rao bound of two uncorrelated equal power sources with arbitrary angles. This new worst-case two-target Cramér-Rao bound metric has some resemblance to the peak sidelobe level metric which is commonly used in unknown multi-target scenarios. We cast the sensor selection problem for 3-D arrays using the worst-case two-target Cramér-Rao bound as a convex semi-definite program and obtain the binary selection by randomized rounding. We illustrate the proposed method through numerical examples, comparing it to solutions obtained by minimizing the single-target Cramér-Rao bound, minimizing the Cramér-Rao bound for known target angles, the concentric rectangular array and the boundary array. We show that our method selects a combination of edge and center elements, which contrasts with solutions obtained by minimizing the single-target Cramér-Rao bound. The proposed selections also exhibit lower peak sidelobe levels without the need for sidelobe level constraints.

The use of digital sequences in automotive radars provides better support for multiple antennas in imaging radar applications. However, a challenge in such digital radars is the higher complexity in the receiver processing chain, starting from the bank of correlators used to estimate the range of targets. State-of-the-art correlators are implemented using using fast Fourier transforms (FFTs), which have log linear complexity in the FFT length used in correlating the digital sequence with the received sequence. This Results in high complexity due to the large sequence lengths needed to achieve high sensing range and fine velocity resolution. We propose an adaptive block FFT-based correlator processing method that exploits sparsity in the range domain. In comparison to conventional FFT-based correlator processing, the proposed method provides a significant reduction in complexity.

This paper investigates jointly optimal array geometry and waveform designs for active sensing. Specifically, we focus on minimizing the Cramér-Rao lower bound (CRB) of the angle of a single target in white Gaussian noise. We first find that several array-waveform pairs can yield the same CRB by virtue of sequences with equal sums of squares, i.e., solutions to certain Diophantine equations. Furthermore, we show that under physical aperture and sensor number constraints, the CRB-minimizing receive array geometry is unique, whereas the transmit array can be chosen flexibly. We leverage this freedom to design a novel sparse array geometry that not only minimizes the single-target CRB given an optimal waveform, but also has a nonredundant and contiguous sum co-array—a desirable property when launching independent waveforms, with relevance also to the multi-target case.

Phase-modulated continuous-wave (PMCW) radars offer better multiple antenna support and flexible waveform design. These radars, however, suffer from the high computational complexity of correlation processing used to obtain the range of targets, particularly due to the use of long code sequences. To address this, we exploit signal sparsity and propose an adaptive sliding block-based fast Fourier transform (FFT) correlator that selectively processes only the relevant range bins and dynamically aligns processing blocks using a time-shift parameter. This sliding mechanism minimizes boundary effects and reduces the number of blocks required for processing. The framework also incorporates external sensor data for context-aware range-bin selection and includes an analytical formulation for optimal block size selection. Simulations demonstrate that the proposed method preserves the signal-to-noise ratio (SNR) and target peaks while significantly reducing processing time. The effectiveness of the proposed method is demonstrated through numerical simulations.

The main focus of this paper is an active sensing application that involves selecting transmit and receive sensors to optimize the Cramér-Rao bound (CRB) on target parameters. Although the CRB is non-convex in the transmit and receive selection, we demonstrate that it is convex in the virtual array weight vector, which describes the multiplicity of the virtual array elements. Based on this finding, we propose a novel algorithm that optimizes the virtual array weight vector first and then finds a matching transceiver array. This greatly enhances the efficiency of the transmit and receive sensor selection problem.

In this paper, we present a novel convolution theorem which encompasses the well known convolution theorem in (graph) signal processing as well as the one related to time-varying filters. Specifically, we show how a node-wise convolution for signals supported on a graph can be expressed as another node-wise convolution in a frequency domain graph, different from the original graph. This is achieved through a parameterization of the filter coefficients following a basis expansion model. After showing how the presented theorem is consistent with the already existing body of literature, we discuss its implications in terms of non-stationarity. Finally, we propose a data-driven algorithm based on subspace fitting to learn the frequency domain graph, which is then corroborated by experimental results on synthetic and real data.

Hearing impairment is a prevalent problem with daily challenges like impaired speech intelligibility and sound localisation. One of the shortcomings of spatial filtering in hearing aids is that speech intelligibility is often not optimised directly, meaning that different auditory processes contributing to intelligibility are often not considered. One example is the perceptual phenomenon known as spatial release from masking (SRM). This paper develops a signal model that explicitly considers SRM in the beamforming design, achieved by transforming the binaural intelligibility prediction model (BSIM) into a signal processing framework. The resulting extended signal model is used to analyse the performance of reference beamformers and design a novel beamformer that more closely considers how the auditory system perceives binaural sound. It can be shown that the binaural minimum variance distortionless response (BMVDR) beamformer is also an optimal solution for the extended, perceived model, suggesting that SRM does not play a significant role in intelligibility enhancement after optimal beamforming. However, the optimal beamformer is no longer unique in the extended signal model. The additional secondary degrees of freedom can be used to preserve binaural cues of interfering sources while still achieving the same perceived performance of the BMVDR beamformer, though with a possible high sensitivity to intelligibility model mismatch errors.

In this work, we deal with the problem of reconstructing a complete bandlimited graph signal from partially sampled noisy measurements. For a known graph structure, an efficient greedy algorithm is presented to partition the graph nodes into disjoint subsets such that sampling the graph signal from any subset leads to a sufficiently accurate reconstruction. Furthermore, we consider a scenario where the graph is massive and data processing centrally is no longer practical. To overcome this issue, a distributed framework is proposed that allows us to implement partitioning algorithms in a parallelized fashion. Finally, we provide numerical simulation results on synthetic and real-world data to show that our proposals outperform the state-of-the-art.