Cramer-Rao Research Articles

Complex-Valued Noisy Sinewaves Frequency Estimator Based on DTFT Linearization and Interpolation

This paper proposes an Interpolated Discrete-Time Fourier Transform (LIpDTFT) estimator for complex-valued noisy sinewave frequency based on the Linearization of the DTFT module behavior around the spectrum peak. It belongs to a class of LIpDTFT estimators that compensate the poor Discrete Fourier Transform (DFT) frequency resolution by interpolating two DTFT samples located very close to an initial frequency estimate. As compared with other DTFT-based frequency estimators recently proposed in the literature, the proposed LIpDTFT estimator ensures a smaller interpolation error and a lower processing effort. Moreover, if the rectangular window is adopted, it almost attains the unbiased Cramér-Rao Lower Bound (CRLB) even if a very small number of samples is analyzed. The accuracies ensured in different operating conditions by the proposed frequency estimator and other state-of-the-art DTFT-based algorithms are compared to each other through computer simulations.

Enhanced Moving Source Localization with Time and Frequency Difference of Arrival: Motion-Assisted Method for Sub-Dimensional Sensor Networks

Localizing a moving source by Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) commonly requires at least N+1 sensors in N-dimensional space to obtain more than N pairs of TDOAs and FDOAs, thereby establishing more than 2N equations to solve for 2N unknowns. However, if there are insufficient sensors, the localization problem will become underdetermined, leading to non-unique solutions or inaccuracies in the minimum norm solution. This paper proposes a localization method using TDOAs and FDOAs while incorporating the motion model. The motion between the source and sensors increases the equivalent length of the baseline, thereby improving observability even when using the minimum number of sensors. The problem is formulated as a Maximum Likelihood Estimation (MLE) and solved through Gauss–Newton (GN) iteration. Since GN requires an initialization close to the true value, the MLE is transformed into a semidefinite programming problem using Semidefinite Relaxation (SDR) technology, while SDR results in a suboptimal estimate, it is sufficient as an initialization to guarantee the convergence of GN iteration. The proposed method is analytically shown to reach the Cramér–Rao Lower Bound (CRLB) accuracy under mild noise conditions. Simulation results confirm that it achieves CRLB-level performance when the number of sensors is lower than N+1, thereby corroborating the theoretical analysis.

Reduced-complexity multiple parameters estimation via toeplitz matrix triple iteration reconstruction with bistatic MIMO radar

In this advanced exploration, we focus on multiple parameters estimation in bistatic Multiple-Input Multiple-Output (MIMO) radar systems, a crucial technique for target localization and imaging. Our research innovatively addresses the joint estimation of the Direction of Departure (DOD), Direction of Arrival (DOA), and Doppler frequency for incoherent targets. We propose a novel approach that significantly reduces computational complexity by utilizing the Temporal-Spatial Nested Sampling Model (TSNSM). Our methodology begins with a multi-linear mapping mechanism to efficiently eliminate unnecessary virtual Degrees of Freedom (DOFs) and reorganize the remaining ones. We then employ the Toeplitz matrix triple iteration reconstruction method, surpassing the traditional Temporal-Spatial Smoothing Window (TSSW) approach, to mitigate the single snapshot effect and reduce computational demands. We further refine the high-dimensional ESPRIT algorithm for joint estimation of DOD, DOA, and Doppler frequency, eliminating the need for additional parameter pairing. Moreover, we meticulously derive the Cramér-Rao Bound (CRB) for the TSNSM. This signal model allows for a second expansion of DOFs in time and space domains, achieving high precision in target angle and Doppler frequency estimation with low computational complexity. Our adaptable algorithm is validated through simulations and is suitable for sparse array MIMO radars with various structures, ensuring higher precision in parameter estimation with less complexity burden.

Simultaneous multi-transient linear-combination modeling of MRS data improves uncertainty estimation.

The interest in applying and modeling dynamic MRS has recently grown. Two-dimensional modeling yields advantages for the precision of metabolite estimation in interrelated MRS data. However, it is unknown whether including all transients simultaneously in a 2D model without averaging (presuming a stable signal) performs similarly to one-dimensional (1D) modeling of the averaged spectrum. Therefore, we systematically investigated the accuracy, precision, and uncertainty estimation of both described model approaches. Monte Carlo simulations of synthetic MRS data were used to compare the accuracy and uncertainty estimation of simultaneous 2D multitransient linear-combination modeling (LCM) with 1D-LCM of the average. A total of 2,500 data sets per condition with different noise representations of a 64-transient MRS experiment at six signal-to-noise levels for two separate spin systems (scyllo-inositol and gamma-aminobutyric acid) were analyzed. Additional data sets with different levels of noise correlation were also analyzed. Modeling accuracy was assessed by determining the relative bias of the estimated amplitudes against the ground truth, and modeling precision was determined by SDs and Cramér-Rao lower bounds (CRLBs). Amplitude estimates for 1D- and 2D-LCM agreed well and showed a similar level of bias compared with the ground truth. Estimated CRLBs agreed well between both models and with ground-truth CRLBs. For correlated noise, the estimated CRLBs increased with the correlation strength for the 1D-LCM but remained stable for the 2D-LCM. Our results indicate that the model performance of 2D multitransient LCM is similar to averaged 1D-LCM. This validation on a simplified scenario serves as a necessary basis for further applications of 2D modeling.

Advancements in wideband source localization with an acoustic vector sensor line array

A method is presented for localizing wideband acoustic sources using an array of acoustic vector sensors (AVS). The method focuses on leveraging the lack of correlation between acoustic pressure and particle velocity in the underwater environmental noise field. Coherent processing is introduced in the signal processing stage to suppress noise components in the observed data. By employing a focused transform that maps various frequency components of the wideband signal into the same signal subspace, an asymptotically unbiased Direction of Arrival (DOA) estimation is achieved using a multidimensional subspace method. A concise expression for the Cramér-Rao Bound (CRB) on the estimation errors is provided within the framework of the wideband multi-vector-sensor model. The performance of the proposed method for localizing sources is assessed through simulation and experimental data analysis. The research indicates that coherent processing effectively suppresses isotropic components in background noise, the proposed method achieves a lower estimation error, closer to the CRB, compared to the scalar array counterpart under identical signal-to-noise ratio (SNR) conditions.

Layout of Detection Array Based on Multi-Strategy Fusion Improved Adaptive Mayfly Algorithm in Bearing-Only Sensor Network.

The various applications of bearing-only sensor networks for detection and localization are becoming increasingly widespread and important. The array layout of the bearing-only sensor network seriously impacts the detection performance. This paper proposes a multi-strategy fusion improved adaptive mayfly algorithm (MIAMA) in a bearing-only sensor network to perform layout planning on the geometric configuration of the optimal detection. Firstly, the system model of a bearing-only sensor network was constructed, and the observability of the system was analyzed based on the Cramer-Rao Lower Bound and Fisher Information Matrix. Then, in view of the limitations of the traditional mayfly algorithm, which has a single initial population and no adaptability and poor global search capabilities, multi-strategy fusion improvements were carried out by introducing Tent chaos mapping, the adaptive inertia weight factor, and Random Opposition-based Learning. Finally, three simulation experiments were conducted. Through comparison with the Particle Swarm Optimization (PSO) algorithm, Mayfly Algorithm (MA), and Genetic Algorithm (GA), the effectiveness and superiority of the proposed MIAMA were validated.

Maximum likelihood estimation of probabilistically described loads in beam structures

In recent years, focus has been shifted towards predictive maintenance in an effort to improve the reliability of operating structures. Processing structural response data obtained from in-situ sensors during operation can provide added value towards this direction. Structural Health Monitoring (SHM) methods are uniquely suited for this task; however, accounting for the effect of stochastic structural loads is critical for their robustness. In this work, a framework based on Maximum Likelihood Estimation (MLE) is presented, whose goal is to obtain inferences on typically unobservable quantities that describe stochastic structural loading. A structural beam is employed as a demonstrative case study, that is subjected to point loads with stochastic magnitude and application points. The hyperparameters that govern their underlying probability distribution functions (pdf) are the quantities of inferential interest. The inverse (load) identification process is performed using a marginalized MLE objective, where stochastic Monte Carlo (MC) integration is employed to perform the marginalization and Genetic Algorithms (GAs) are used as the optimizer. The Cramer–Rao (CR) lower bound is used to produce 95 % Confidence Intervals (CIs) to quantify estimation uncertainty.

Photon Phase Delay Sensing with Sub-Attosecond Uncertainty.

The application of statistical estimation theory to Hong-Ou-Mandel interferometry led to enticing results in terms of the detection limit for photon reciprocal delay and polarisation measurement. In the following paper, a fully fibre-coupled setup operating in the telecom wavelength region proves to achieve, for the first time, in common-path Hong-Ou-Mandel-based interferometry, a detection limit for photon phase delay at the zeptosecond scale. The experimental results are then framed in a theoretical model by calculating the Cramer-Rao bound (CRB) and, after comparison with the obtained experimental results, it is shown that our setup attains the optimal measurement, nearly saturating CRB.

Underlying Topography Estimation over Forest Using Maximum a Posteriori Inversion with Spaceborne Polarimetric SAR Interferometry

This paper presents a method for extracting the digital elevation model (DEM) of forested areas from polarimetric interferometric synthetic aperture radar (PolInSAR) data. The method models the ground phase as a Von Mises distribution, with a mean of the topographic phase computed from an external DEM. By combining the prior distribution of the ground phase with the complex Wishart distribution of the observation covariance matrix, we derive the maximum a posterior (MAP) inversion method based on the RVoG model and analyze its Cramer–Rao Lower Bound (CRLB). Furthermore, considering the characteristics of the objective function, this paper introduces a Four-Step Optimization (FSO) method based on gradient optimization, which solves the inefficiency problem caused by exhaustive search in solving ground phase using the MAP method. The method is validated using spaceborne L-band repeat-pass SAOCOM data from a test forest area. The test results for FSO indicate that it is approximately 5.6 times faster than traditional methods without compromising accuracy. Simultaneously, the experimental results demonstrate that the method effectively solves the problem of elevation jumps in DEM inversion when modeling the ground phase with the Gaussian distribution. ICESAT-2 data are used to evaluate the accuracy of the inverted DEM, revealing that our method improves the root mean square error (RMSE) by about 23.6% compared to the traditional methods.

Cooperative sensing, communication and computation resource allocation in mobile edge computing-enabled vehicular networks

The combination of integrated sensing and communication (ISAC) with mobile edge computing (MEC) enhances the overall safety and efficiency for vehicle to everything (V2X) system. However, existing works have not considered the potential impacts on base station (BS) sensing performance when users offload their computational tasks via uplink. This could leave insufficient resources allocated to the sensing tasks, resulting in low sensing performance. To address this issue, we propose a cooperative power, bandwidth and computation resource allocation (RA) scheme in this paper, maximizing the overall utility of Cramér-Rao bound (CRB) for sensing accuracy, computation latency for processing sensing information, and communication and computation latency for computational tasks. To solve the RA problem, a twin delayed deep deterministic policy gradient (TD3) algorithm is adopted to explore and obtain the effective solution of the RA problem. Furthermore, we investigate the performance tradeoff between sensing accuracy and summation of communication latency and computation latency for computational tasks, as well as the relationship between computation latency for processing sensing information and that of computational tasks by numerical simulations. Simulation demonstrates that compared to other benchmark methods, TD3 achieves an average utility improvement of 97.11% and 27.90% in terms of the maximum summation of communication latency and computation latency for computational tasks and improves 3.60 and 1.04 times regarding the maximum computation latency for processing sensing information.

Performance bounds for source localization using tetrahedral microphones

The Cramér-Rao Bound (CRB) quantifies the minimum variance on a given unbiased estimation parameter. It can be used to derive a lower bound on source localization error for arrays, of which there are several probabilistic models in the literature. However, existing work only considers source localization using arrays of omnidirectional sensors. This presentation describes a model of the CRB as a minimum variance estimation of localization using multiple tetrahedral microphones which have cardioid microphones. Tetrahedral microphones are commonly used to record soundfields for virtual audio reproduction. The proposed method uses the cross spectral density of signals in order to calculate the CRB. This allows for more signal parameters to be included in the model. The model of the CRB was validated with an experiment in an anechoic room with a single sound source. Experimental results show the CRB forms a lower bound at low signal-to-noise ratios.

Direction of arrival estimation with convolutional neural networks and multiple signal classification

Recently, there has been a proliferation of applied machine learning (ML) research, including the use of convolutional neural networks (CNNs) for direction of arrival (DoA) estimation. With the large increase of research in this area, it is important to balance the performance and computational costs of CNNs with classical methods of DoA estimation such as Multiple Signal Classification (MUSIC). We outline the performance of both methods of DoA estimation for single source and two source cases and compare them to the Cramer-Rao lower bound (CRLB). For each source case, a CNN was trained for a perfect uniform line array (ULA), a perturbed ULA, and a ULA with missing sensors. The three cases are interesting studies as classical methods for DoA estimation such as MUSIC assume perfect array conditions, whereas the CNNs assume nothing about the structure of the data. The training data for each network was created by leveraging a signal model to create synthetic data at different SNRs. The results for each network are then compared to the results for MUSIC using the same signal case and array condition. The results indicate that for the single source case the CNN only performs significantly better than MUSIC for the perturbed array case. For the two-source case, the CNNs significantly outperform MUSIC for all ULA conditions, however, when compared to the CRLB it is shown that the CNN typically produces a biased estimate.

An Underwater Source Localization Method Using Bearing Measurements.

Angle-of-arrival (AOA) measurements are often used in underwater acoustical localization. Different from the traditional AOA model based on azimuth and elevation measurements, the AOA model studied in this paper uses bearing measurements. It is also often used in the Ultra-Short Baseline system (USBL). However, traditional acoustical localization needs additional range information. If the range information is unavailable, the closed-form solution is difficult to obtain only with bearing measurements. Thus, a localization closed-form solution using only bearing measurements is explored in this article. A pseudo-linear measurement model between the source position and the bearing measurements is derived, and considering the nonlinear relationship of the parameters, a weighted least-squares optimization equation based on multiple constraints is established. Different from the traditional two-step least-squares method, the semidefinite programming (SDP) method is designed to obtain the initial solution, and then a bias compensation method is proposed to further minimize localization errors based on the SDP result. Numerical simulations show that the performance of the proposed method can achieve Cramer-Rao lower bound (CRLB) accuracy. The field test also proves that the proposed method can locate the source position without range measurements and obtain the highest positioning accuracy.

Обобщённое неравенство Крамера – Рао в случае совместного пеленгования по азимуту и углу места в условиях сложной электромагнитной обстановки

In this work, a generalized expression is obtained for the Cramer–Rao lower bound in the case of a joint estimation of azimuth and elevation angle in an arbitrary coordinate system in which the vector complex radiation pattern is determined, under conditions of a complex electromagnetic environment, including the presence of noise correlation in spatial channels and differences in its intensity. A theorem is formulated for the Fisher information of joint estimates of azimuth and elevation angle in the case of a multi-channel detector-direction finder with an antenna system described in the selected coordinate system by a vector complex radiation pattern. Dependences of the root-mean-square error of direction finding in azimuth and elevation on the signal-to-noise ratio and true directions to the radio source are obtained, calculated using known and generalized expressions for the Cramer–Rao lower bound. The condition for the applicability of the existing and proposed in the article expression for the Fisher information matrix is obtained. The concept of the “energy” center of the antenna system is given. To confirm the reliability of the obtained generalized expressions for the lower bound of the dispersion of the Cramer–Rao inequality, statistical modeling is carried out. The use of the obtained expressions in practice will allow the most optimal synthesis of antenna devices depending on the structure and characteristics of the direction finder antenna system, the number of antennas and the characteristics of the radio receiving path. Generalized expressions for the lower Cramer–Rao bound are the foundation for organizing the operation of intelligent radio monitoring systems, allowing for a well-founded choice of the optimal antenna array structure that provides the required direction-finding efficiency.

Spectral information content of Compton scattering events in silicon photon counting detectors.

Silicon (Si) is a possible sensor material for photon counting detectors (PCDs). A major drawback of Si is that roughly two-thirds of x-ray interactions in the diagnostic energy range are Compton scattering. Because Compton scattering is an energy-insensitive process, it is commonly assumed that Compton events retain little spectral information. To quantify how much information can be recovered from Compton scattering events in models of Si PCDs. We built a simplified model of Si interactions including two interaction mechanisms: photoelectric effect and Compton scattering. We considered three different binning options that represent strategies for handling Compton events: in Compton censoring, all events under 38keV (the maximum energy possible from Compton scattering for a 120keV incident photon) were discarded; in Compton counting, all events between 1 and 38keV were placed into a single bin; in Compton binning, all events were placed into energy bins of uniform width. These were compared to the ideal detector, which always recorded the correct energy (i.e., 100% photoelectric effect). Every photon was assumed to interact once and only once with Si, and the energy bin width was 5keV. In the primary analysis, the Si detector was irradiated with a 120kV spectrum filtered by 30cm of water, with 99.5% of the arriving spectrum above 38keV so that there was good separation between photoelectric effect and Compton scattering, and the figures of merit were the Cramér-Rao lower bound (CRLB) of the variance of iodine and water basis material decomposition images, as well as the CRLB of virtual monoenergetic images (i.e., linear combinations of material images) that maximize iodine CNR or water CNR. We also constructed a local linear estimator that attains the CRLB. In secondary analyses, we applied other sources of spectral distortion: (1) a nonzero minimum energy threshold; (2) coarser, 10keV energy bins; and (3) a model of charge sharing. With our chosen spectrum, 67% of the interactions were Compton scattering. Consistent with this, the material decomposition variance for the Compton censoring model, averaged over both basis materials, was 258% greater than the ideal detector. If Compton events carried no spectral information, the Compton counting model would show similar variance. Instead, its basis material variance was 103% greater than the ideal detector, implying that Compton counts indeed carry significant spectral information. The Compton binning model had a basis material variance 60% greater than the ideal detector. The Compton binning model was not affected by a 5keV minimum energy threshold, but the variance increased from 60% to 107% when charge sharing was included and to 78% with coarser energy bins. For optimized CNR images, the average variance was 149%, 12%, and 10% higher than the ideal detector for the Compton censoring, counting, and binning models, reinforcing the hypothesis that Compton counts are useful for detection tasks and that precise energy assignments are not necessary. Substantial spectral information remains after Compton scattering events in silicon PCDs.

On-scalp Yttrium-Iron Garnet sensor arrays for brain source localization: Cramér-Rao bound analysis

A new type of highly sensitive Magnetometers based on Yttrium-Iron Garnet films (YIGM) is being designed and applied to brain signals recording. We modelled sensor arrays based on YIGM and well-established OPM (QuSpin Inc.) and SQUID (Elekta Neuromag) arrays used for magnetoencephalography (MEG). In this simulation study we investigate the inverse problem accuracy bound depending on the sensor arrays configuration, types of magnetometers and instrumental noise level. We estimate the variance of the probability of dipole source localization using the Cramér-Rao Lower Bound (CRLB); the CRLB is calculated with use of the numerical approximation of the forward model and its partial derivative parametrization. Our models confirm that the development of multi-channel YIG-MEG systems requires noise reduction, rather than an increase in number of sensors, to outperform OPM and SQUID arrays. The modelling showed the advantages of YIG-based gradiometry compared to conventional SQUID-system.

STARS-ISAC: How Many Sensors Do We Need?

A simultaneously transmitting and reflecting surface (STARS) enabled two-phase integrated sensing and communications (ISAC) framework is proposed, where a novel bi-directional sensing-STARS architecture is devised to facilitate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">full-space</i> communication and sensing in a time-switching manner. Based on the proposed framework, a joint optimization problem is formulated, where the Cramér-Rao bound (CRB) for estimating the 2-dimension direction-of-arrival of the sensing target is minimized. Two cases are considered for sensing performance enhancement. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1) For the two-user case</i> with the fixed number of sensors, an alternating optimization algorithm is proposed. In particular, the maximum number of deployable sensors is obtained in the closed-form expressions, where the maximum number of sensors is revealed to be only relevant to the QoS requirements of communications. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2) For the multi-user case</i> with the variable number of sensors, an extended CRB (ECRB) metric is proposed to characterize the impact of the number of sensors on the sensing performance. A generic decoupling approach is proposed to convexify the non-convex ECRB expression. Based on this, a novel penalty-based double-loop (PDL) algorithm is proposed. Simulation results reveal that 1) the proposed PDL algorithm achieves a near-optimal performance with consideration of sensor deployment; 2) it is preferable to deploy more passive elements than sensors in terms of achieving optimal sensing performance.

ZPW-2000 frequency-shift signal detection algorithm based on nonlinear least squares method

Aiming at the problems of spectrum leakage, difficulty in guaranteeing demodulation accuracy, and endangering driving safety in signal detection algorithm based on periodogram with limited sampling duration, this paper proposes for the first time a kind of ZPW-20000 frequency-shift signal detection algorithm based on nonlinear least squares, so as to improve the detection performance of ZPW-2000 frequency-shift signal, to ensure the safe operation of trains, and to improve the operational efficiency. The mathematical model of ZPW-2000 frequency-shift signal is established, the Cramér-Rao bound (CRB) for carrier-frequency and low-frequency estimation are derived, and a nonlinear least squares algorithm containing three processes, namely, the coarse estimation, the grid search based on Fast Fourier Transform (FFT), and the exact search, is designed, which is applied to the detection of ZPW-2000 frequency-shift signal. The research results show that the accuracy of carrier-frequency and low-frequency estimation of the proposed algorithm is better than that of the periodogram method for different sampling durations, and the number of decoding errors of the proposed algorithm and the periodogram method is reduced to zero when the sampling durations are greater than or equal to 0.11 s and 0.19 s, respectively, and the proposed algorithm reduces the limitation on the sampling durations. As the signal-to-noise ratio (SNR) increases, the root mean square error (RMSE) of carrier-frequency and low-frequency estimation of the proposed algorithm can approach the CRB. when the SNR of −2 dB, the number of decoding errors of the proposed algorithm is reduced to zero, while the periodogram method requires the SNR of 2 dB to achieve the same effect. The proposed algorithm performs well in detection accuracy, real-time and anti-interference ability, which is of great significance for ensuring driving safety and improving the reliability of the vehicle-ground communication mode based on ZPW-2000 track circuit.

Compartment-based reconstruction of acquisition-weighted 31P cardiac MRSI reduces sensitivity to cardiac motion and scan planning.

Motivation: 31P magnetic resonance spectroscopic imaging (31P MRSI) is a powerful technique for investigating the metabolic effects of treatments for heart failure in vivo, allowing a better understanding of their mechanism of action in patient cohorts. Unfortunately, cardiac 31P MRSI is fundamentally limited by low SNR, which leads to compromises in acquisition, such as no cardiac or respiratory gating or low spatial resolution, in order to achieve reasonable scan times. Spectroscopy with linear algebra modeling (SLAM) reconstruction may be able to address these challenges and therefore improve repeatability by incorporating a segmented localizer into the reconstruction. Methods: Six healthy volunteers were scanned twice in a test-retest procedure to allow quantification of repeatability. Each scan consisted of anatomical localizers and two acquisition-weighted (AW) 31P MRSI acquisitions, which were acquired with and without cardiac gating. Five patients with heart failure with a preserved ejection fraction were then scanned with the same 31P MRSI sequence without cardiac gating. All 31P MRSI datasets were reconstructed with both conventional Fourier transform (FT)-based reconstruction and SLAM reconstruction, which were compared statistically. The effect of shifting the 31P MRSI acquisition field of view was also investigated. Results: In the healthy volunteer cohort, the spectral fit of the SLAM reconstructions had significantly improved Cramer-Rao lower bounds (CRLBs) compared to the FT-based reconstruction of non-cardiac gated data, as well as improved coefficients of variability and repeatability. The SLAM reconstruction found a significant difference in the PCr/ATP ratio between the healthy volunteer and patient cohorts, which the FT-based reconstruction did not find. Furthermore, the SLAM reconstruction was less influenced by the placement of the field of view (FOV) of the 31P MRSI acquisition in post hoc analysis. Discussion: The experimental benefits of the SLAM reconstruction for AW data were demonstrated by the improvements in fit confidence and repeatability seen in the healthy volunteer cohort and post hoc FOV analysis. The benefit of SLAM reconstruction of AW data for clinical studies was then illustrated by the patient cohort, which suggested improved sensitivity to clinically significant changes in the PCr/ATP ratio.

Graph signal recovery using variational Bayes in Fourier pairs with Cramér–Rao bounds

In this paper, the graph signal recovery problem is addressed by employing an aggregation of samples in the vertex domain and the Fourier graph transform domain. The statistical graph signal is modeled using a Gaussian Markov Random Field (GMRF). The reconstruction process involves employing a variational Bayes (VB) algorithm, which is a fully Bayesian method that iteratively estimates all unknown parameters by computing the posteriors in a closed-form. Furthermore, the closed-form of the Cramér–Rao lower bound (CRLB) for graph signal estimation is also derived. Simulation results demonstrate the superiority of the proposed algorithm over some of state-of-the-art algorithms in the literature.