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Richard Bamler - One of the best experts on this subject based on the ideXlab platform.
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super resolution power and robustness of compressive sensing for Spectral Estimation with application to spaceborne tomographic sar
IEEE Transactions on Geoscience and Remote Sensing, 2012Co-Authors: Richard BamlerAbstract:We address the problem of resolving two closely spaced complex-valued points from N irregular Fourier do- main samples. Although this is a generic super-resolution (SR) problem, our target application is SAR tomography (TomoSAR), where typically the number of acquisitions is N = 10 - 100 and SNR = 0-10 dB. As the TomoSAR algorithm, we introduce "Scale-down by LI norm Minimization, Model selection, and Estimation Reconstruction" (SL1MMER), which is a Spectral Estimation algorithm based on compressive sensing, model order selection, and final maximum likelihood parameter Estimation. We investigate the limits of SLIMMER concerning the following questions. How accurately can the positions of two closely spaced scatterers be estimated? What is the closest distance of two scat- terers such that they can be separated with a detection rate of 50% by assuming a uniformly distributed phase difference? How many acquisitions N are required for a robust Estimation (i.e., for separating two scatterers spaced by one Rayleigh resolution unit with a probability of 90%)? For all of these questions, we provide numerical results, simulations, and analytical approxima- tions. Although we take TomoSAR as the preferred application, the SLIMMER algorithm and our results on SR are generally applicable to sparse Spectral Estimation, including SR SAR focus- ing of point-like objects. Our results are approximately applicable to nonlinear least-squares Estimation, and hence, although it is derived experimentally, they can be considered as a fundamental bound for SR of Spectral estimators. We show that SR factors are in the range of 1.5-25 for the aforementioned parameter ranges of N and SNR.
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super resolution power and robustness of compressive sensing for Spectral Estimation with application to spaceborne tomographic sar
IEEE Transactions on Geoscience and Remote Sensing, 2012Co-Authors: Xiao Xiang Zhu, Richard BamlerAbstract:We address the problem of resolving two closely spaced complex-valued points from N irregular Fourier do- main samples. Although this is a generic super-resolution (SR) problem, our target application is SAR tomography (TomoSAR), where typically the number of acquisitions is N = 10 - 100 and SNR = 0-10 dB. As the TomoSAR algorithm, we introduce "Scale-down by LI norm Minimization, Model selection, and Estimation Reconstruction" (SL1MMER), which is a Spectral Estimation algorithm based on compressive sensing, model order selection, and final maximum likelihood parameter Estimation. We investigate the limits of SLIMMER concerning the following questions. How accurately can the positions of two closely spaced scatterers be estimated? What is the closest distance of two scat- terers such that they can be separated with a detection rate of 50% by assuming a uniformly distributed phase difference? How many acquisitions N are required for a robust Estimation (i.e., for separating two scatterers spaced by one Rayleigh resolution unit with a probability of 90%)? For all of these questions, we provide numerical results, simulations, and analytical approxima- tions. Although we take TomoSAR as the preferred application, the SLIMMER algorithm and our results on SR are generally applicable to sparse Spectral Estimation, including SR SAR focus- ing of point-like objects. Our results are approximately applicable to nonlinear least-squares Estimation, and hence, although it is derived experimentally, they can be considered as a fundamental bound for SR of Spectral estimators. We show that SR factors are in the range of 1.5-25 for the aforementioned parameter ranges of N and SNR.
Wenjing Liao - One of the best experts on this subject based on the ideXlab platform.
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sensor calibration for off the grid Spectral Estimation
Applied and Computational Harmonic Analysis, 2020Co-Authors: Yonina C Eldar, Wenjing Liao, Sui TangAbstract:Abstract This paper studies sensor calibration in Spectral Estimation where the true frequencies are located on a continuous domain. We consider a uniform array of sensors that collects measurements whose spectrum is composed of a finite number of frequencies, where each sensor has an unknown calibration parameter. Our goal is to recover the spectrum and the calibration parameters simultaneously from multiple snapshots of the measurements. In the noiseless case with an infinite number of snapshots, we prove uniqueness of this problem up to certain trivial, inevitable ambiguities based on an algebraic method, as long as there are more sensors than frequencies. We then analyze the sensitivity of this algebraic technique with respect to the number of snapshots and noise. We next propose an optimization approach that makes full use of the measurements by minimizing a non-convex objective which is non-negative and continuously differentiable over all calibration parameters and Toeplitz matrices. We prove that, in the case of infinite snapshots and noiseless measurements, the objective vanishes only at equivalent solutions to the true calibration parameters and the measurement covariance matrix. The objective is minimized using Wirtinger gradient descent which is proven to converge to a critical point. We show empirically that this critical point provides a good approximation of the true calibration parameters and the underlying frequencies.
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music for single snapshot Spectral Estimation stability and super resolution
Applied and Computational Harmonic Analysis, 2016Co-Authors: Wenjing Liao, Albert FannjiangAbstract:Abstract This paper studies the problem of line Spectral Estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the Spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.
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music for single snapshot Spectral Estimation stability and super resolution
arXiv: Information Theory, 2014Co-Authors: Wenjing Liao, Albert FannjiangAbstract:This paper studies the problem of line Spectral Estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurements is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the Spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent much smaller than an upper bound established by Donoho.
Petre Stoica - One of the best experts on this subject based on the ideXlab platform.
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new Spectral Estimation based on filterbank for spectrum sensing
International Conference on Acoustics Speech and Signal Processing, 2008Co-Authors: Miguel Lagunas A Hernandez, M A Rojas, Petre StoicaAbstract:The advance of cognitive radio (CR) technology put in evidence the need of new Spectral Estimation methods for proper labeling of licensed and un-licensed users. We present a new Spectral Estimation procedure for monitoring the radio spectrum. The estimate is derived from a different view point of traditional filter bank approach. The resulting method is able to detect a predetermined Spectral shape forming part or contributing to a given data record, providing at the same time an estimate of its power level and its frequency location. We prove that traditional filter-bank Spectral Estimation reduces to a particular case of our procedure. The specific Spectral shape to detect is named hereafter as the candidate spectrum. The major motivation for this procedure was the proper spectrum labeling of licensed users in cognitive radio scenarios. The performance of the Spectral monitoring procedure is demonstrated in the detection of a BPSK primary user in a wireless scenario containing DVB-T emissions.
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Spectral analysis of signals the missing data case
Synthesis Lectures on Signal Processing, 2006Co-Authors: Yanwei Wang, Petre StoicaAbstract:Spectral Estimation is important in many fields including astronomy, meteorology, seismology, communications, economics, speech analysis, medical imaging, radar, sonar, and underwater acoustics. Most existing Spectral Estimation algorithms are devised for uniformly sampled complete-data sequences. However, the Spectral Estimation for data sequences with missing samples is also important in many applications ranging from astronomical time series analysis to synthetic aperture radar imaging with angular diversity. For Spectral Estimation in the missing-data case, the challenge is how to extend the existing Spectral Estimation techniques to deal with these missing-data samples. Recently, nonparametric adaptive filtering based techniques have been developed successfully for various missing-data problems. Collectively, these algorithms provide a comprehensive toolset for the missing-data problem based exclusively on the nonparametric adaptive filter-bank approaches, which are robust and ac urate, and can provide high resolution and low sidelobes. In this book, we present these algorithms for both one-dimensional and two-dimensional Spectral Estimation problems.
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on nonparametric Spectral Estimation
Circuits Systems and Signal Processing, 1999Co-Authors: Petre Stoica, Tomas SundinAbstract:In this paper the Cramer-Rao bound (CRB) for a general nonparametric Spectral Estimation problem is derived under a local smoothness condition (more exactly, the spectrum is assumed to be well approximated by a piecewise constant function). Further-more, it is shown that under the aforementioned condition the Thomson method (TM) and Daniell method (DM) for power Spectral density (PSD) Estimation can be interpreted as approximations of the maximum likelihood PSD estimator. Finally the statistical efficiency of the TM and DM as nonparametric PSD estimators is examined and also compared to the CRB for autoregressive moving-average (ARMA)-based PSD Estimation. In particular for broadband signals, the TM and DM almost achieve the derived nonparametric performance bound and can therefore be considered to be nearly optimal.
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an adaptive filtering approach to Spectral Estimation and sar imaging
IEEE Transactions on Signal Processing, 1996Co-Authors: Petre StoicaAbstract:We present an adaptive FIR filtering approach, which is referred to as the amplitude and phase Estimation of a sinusoid (APES), for complex Spectral Estimation. We compare the APES algorithm with other FIR filtering approaches including the Welch (1967) and Capon (1969) methods. We also describe how to apply the FIR filtering approaches to target range signature Estimation and synthetic aperture radar (SAR) imaging. We show via both numerical and experimental examples that the adaptive FIR filtering approaches such as Capon and APES can yield more accurate Spectral estimates with much lower sidelobes and narrower Spectral peaks than the FFT method, which is also a special case of the FIR filtering approach. We show that although the APES algorithm yields somewhat wider Spectral peaks than the Capon method, the former gives more accurate overall Spectral estimates and SAR images than the latter and the FFT method.
Albert Fannjiang - One of the best experts on this subject based on the ideXlab platform.
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music for single snapshot Spectral Estimation stability and super resolution
Applied and Computational Harmonic Analysis, 2016Co-Authors: Wenjing Liao, Albert FannjiangAbstract:Abstract This paper studies the problem of line Spectral Estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the Spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.
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music for single snapshot Spectral Estimation stability and super resolution
arXiv: Information Theory, 2014Co-Authors: Wenjing Liao, Albert FannjiangAbstract:This paper studies the problem of line Spectral Estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurements is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the Spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent much smaller than an upper bound established by Donoho.
Zhiwei Xu - One of the best experts on this subject based on the ideXlab platform.
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Grid-less variational Bayesian line Spectral Estimation with multiple measurement vectors
Signal Processing, 2019Co-Authors: Qi Zhang, Mihai-alin Badiu, Peter Gerstoft, Zhiwei XuAbstract:Abstract Line Spectral Estimation (LSE) with multiple measurement vector (MMV) is studied utilizing the Bayesian variational inference. Motivated by the recent grid-less variational line Spectral Estimation (VALSE) method, we develop the MMV VALSE (MVALSE). The MVALSE shares the advantages of the VALSE method, such as automatically estimating the model order, noise variance, weight variance, and providing the uncertainty of the frequency estimates. The MVALSE can be viewed as applying the VALSE with single measurement vector to each snapshot, and combining the intermediate data appropriately. Furthermore, the MVALSE is developed to perform sequential Estimation. Numerical results demonstrate the effectiveness of the MVALSE method, compared to the state-of-the-art MMV methods.
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Variational Bayesian Line Spectral Estimation with Multiple Measurement Vectors
arXiv: Information Theory, 2018Co-Authors: Qi Zhang, Mihai-alin Badiu, Peter Gerstoft, Zhiwei XuAbstract:In this paper, the line Spectral Estimation (LSE) problem with multiple measurement vectors (MMVs) is studied utilizing the Bayesian methods. Motivated by the recently proposed variational line Spectral Estimation (VALSE) method, we develop the multisnapshot VALSE (MVALSE) for multi snapshot scenarios, which is especially important in array signal processing. The MVALSE shares the advantages of the VALSE method, such as automatically estimating the model order, noise variance, weight variance, and providing the uncertain degrees of the frequency estimates. It is shown that the MVALSE can be viewed as applying the VALSE with single measurement vector (SMV) to each snapshot, and combining the intermediate data appropriately. Furthermore, the Seq-MVALSE is developed to perform sequential Estimation. Finally, numerical results are conducted to demonstrate the effectiveness of the MVALSE method, compared to the state-of-the-art methods in the MMVs setting.
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Variational Bayesian Inference of Line Spectral Estimation with Multiple Measurement Vectors.
arXiv: Information Theory, 2018Co-Authors: Qi Zhang, Mihai-alin Badiu, Peter Gerstoft, Zhiwei XuAbstract:In this paper, the line Spectral Estimation (LSE) problem with multiple measurement vectors (MMVs) is studied utilizing the Bayesian methods. Motivated by the recently proposed variational line Spectral Estimation (VALSE) method, we extend it to deal with the MMVs setting, which is especially important in array signal processing. The VALSE method can automatically estimate the model order and nuisance parameters such as noise variance and weight variance. In addition, by approximating the probability density function (PDF) of the frequencies with the mixture of von Mises PDFs, closed-form update equation and the uncertainty degree of the estimates can be obtained. Interestingly, we find that the VALSE with MMVs can be viewed as applying the VALSE with single measurement vector (SMV) to each snapshot, and combining the intermediate data appropriately. Furthermore, the proposed prior distribution provides a good interpretation of tradeoff between grid and off-grid based methods. Finally, numerical results demonstrate the effectiveness of the VALSE method, compared to the state-of-the-art methods in the MMVs setting.
