Line Spectrum Estimation Research Articles

A Dual-Stream Deep Learning-Based Acoustic Denoising Model to Enhance Underwater Information Perception

Estimating the line spectra of ship-radiated noise is a crucial remote sensing technique for detecting and recognizing underwater acoustic targets. Improving the signal-to-noise ratio (SNR) makes the low-frequency components of the target signal more prominent. This enhancement aids in the detection of underwater acoustic signals using sonar. Based on the characteristics of low-frequency narrow-band line spectra signals in underwater target radiated noise, we propose a dual-stream deep learning network with frequency characteristics transformation (DS_FCTNet) for line spectra estimation. The dual streams predict amplitude and phase masks separately and use an information exchange module to swap learn features between the amplitude and phase spectra, aiding in better phase information reconstruction and signal denoising. Additionally, a frequency characteristics transformation module is employed to extract convolutional features between channels, obtaining global correlations of the amplitude spectrum and enhancing the ability to learn target signal features. Through experimental analysis on ShipsEar, a dataset of underwater acoustic signals by hydrophones deployed in shallow water, the effectiveness and rationality of different modules within DS_FCTNet are verified.Under low SNR conditions and with unknown ship types, the proposed DS_FCTNet model exhibits the best line spectrum enhancement compared to methods such as SEGAN and DPT_FSNet. Specifically, SDR and SSNR are improved by 14.77 dB and 13.58 dB, respectively, enabling the detection of weaker target signals and laying the foundation for target localization and recognition applications.

Bayesian 3D User Localization and Channel Reconstruction with Planar Extremely Large-Scale Antenna Array

An extremely large-scale antenna array (ELAA) can potentially provide significantly increased spatial multiplexing and beamforming gains, as well as enhanced localization capability. While presenting new potential, its near-field propagation and spatial non-stationary properties also impose a great challenge on the receiver design. This paper focuses on the receiver design in an uplink orthogonal frequency-division multiplexing system with a planar ELAA deployed at the base station. To solve the challenging problem of 3D user localization and channel estimation with the planar ELAA, a space-frequency user localization and channel reconstruction (SF-ULCR) receiver is proposed. Under the Bayesian framework, an extended probability model is first established, to capture the channel structural information comprehensively, based on which an iterative receiver consisting of three modules is derived: element-wise line spectrum estimation (ELSE), distance parameter estimation (DPE), and near-field localization (NFL). In particular, the ELSE module handles the line spectrum relationships among multiple subcarriers in the frequency domain, the DPE module extracts and integrates the distance information from the line spectrum parameters, and the NFL module utilizes the messages of distances for user localization based on near-field spatial characteristics. Our numerical results demonstrate that the proposed SF-ULCR algorithm outperforms existing baselines in terms of channel estimation and localization performance, and that it approaches the Cramèr–Rao bound.

A note on spike localization for line spectrum estimation

A note on spike localization for line spectrum estimation

A Novel Gradient Descent Least-Squares (GDLSs) Algorithm for Efficient Gridless Line Spectrum Estimation With Applications in Tomographic SAR Imaging

This paper presents a novel efficient method for gridless line spectrum estimation problem with single snapshot and sparse signals, namely the gradient descent least squares (GDLS) method. Conventional single snapshot (a.k.a. single measure vector or SMV) line spectrum estimation methods either rely on smoothing techniques that sacrificing the range and/or azimuth resolution, or adopt the sparsity constraint and utilize compressed sensing (CS) method by defining prior grids and resulting in the off-grid problem. Recently emerged atomic norm minimization (ANM) methods achieved gridless SMV line spectrum estimation, but its computational complexity is extremely high; thus it is practically infeasible in real applications with large problem scales. Our proposed GDLS method reformulates the line spectrum estimations problem into a least squares (LS) estimation problem and solves the corresponding objective function via gradient descent algorithm in an iterative fashion with efficiency. The convergence guarantee, computational complexity, as well as performance analysis for evenly distributed antenna array case are discussed in this paper. Numerical simulations show that the proposed GDLS algorithm outperforms the state-of-the-art methods e.g., CS and ANM, in terms of estimation performances. It can completely avoid the off-grid problem, and its computational complexity is significantly lower than ANM. Our method has been tested in tomographic SAR (TomoSAR) imaging applications via simulated and real experiment data. Results show great potential of the proposed method in terms of better cloud point performance and eliminating the gridding effect.

CFAR based NOMP for Line Spectral Estimation and Detection

The line spectrum estimation and detection problem are considered in this paper. We propose a CFAR-based Newtonized OMP (NOMP-CFAR) method which can maintain a desired false alarm rate without the knowledge of the noise variance. The NOMP-CFAR consists of two steps, namely, an initialization step and a detection step. In the initialization step, NOMP is employed to obtain candidate sinusoidal components. In the detection step, CFAR detector is applied to detect each candidate frequency, and provides the “soft” thresholds. After removing the most unlikely target, the Newton refinements are used to refine the remaining parameters. The relationship between the false alarm rate and the required threshold is established. Compared with NOMP, NOMP-CFAR has only 1 dB performance loss in additive white Gaussian noise scenario with false alarm probability <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-2}$</tex-math></inline-formula> and detection probability 0.8 without knowledge of noise variance. For varied noise variance scenario, NOMP-CFAR still preserves its CFAR property, while NOMP violates the CFAR. Besides, real experiments are also conducted to demonstrate the detection performance of NOMP-CFAR, compared to CFAR and NOMP.

Joint CFO, Gridless Channel Estimation and Data Detection for Underwater Acoustic OFDM Systems

In this article, we propose an iterative receiver based on gridless variational Bayesian line spectra estimation (VALSE), named JCCD-VALSE, that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</i> ointly estimates the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> arrier frequency offset (CFO), the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> hannel with high resolution and carries out <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ata decoding. Based on a modularized point of view and motivated by the high resolution and low-complexity gridless VALSE algorithm, three modules named the VALSE module, the CFO estimation (CFO est.) module, and the decoder module are built. Soft information is exchanged between the modules to progressively improve the channel estimation and data decoding accuracy. Since the delays of multipath of the channel are treated as continuous parameters instead of on a grid, the leakage effect is avoided. Beside, the proposed approach is a more complete Bayesian approach as all the nuisance parameters, such as the noise variance, the parameters of the prior distribution of the channel, and the number of paths are automatically estimated. Numerical simulations and sea test data are utilized to demonstrate that the proposed approach performs better than the existing grid-based joint channel and data decoding approaches. Furthermore, it is also verified that joint processing, including CFO estimation, provides performance gain.

Two-dimensional multi-snapshot Newtonized orthogonal matching pursuit for DOA estimation

Estimating the azimuth and elevation angles of targets is referred to as two-dimensional direction of arrival (2D-DOA) estimation problem, which is of vital importance in array signal processing and multiple input multiple output (MIMO) millimeter wave communication fields. Inspired by the Newtonized orthogonal matching pursuit (NOMP) for line spectrum estimation (LSE), this paper proposes the two-dimensional multi-snapshot NOMP (2D-MNOMP) to deal with the 2D-DOA estimation. Specifically, two-dimensional fast Fourier transform (FFT) is utilized to significantly reduce the computation complexity, and a Newton refinement step and feedback strategy are proposed to improve performance of DOA estimation. Based on the generalized likelihood ratio test (GLRT), the stopping criterion is established. The near-optimal performance of 2D-MNOMP is also demonstrated by comparing against other methods and the Cramér-Rao bound (CRB), both in terms of simulations and real data.

Line spectrum extraction method based on hidden Markov model

To solve the difficult problem of line spectrum detection with a low signal-to-noise ratio, a line spectrum extraction algorithm based on a hidden Markov model (HMM) is proposed. The forward–backward algorithm was used to track single and multiple spectral lines in a lofargram, and a new algorithm for the state transition probability matrix was proposed to handle the large amount of HMM calculation and the uncertainty difficulty in processing real signals. Finally, a median fitting algorithm was used to correct the few outlier points in the line spectrum estimation process. Simulation and sea trial data showed that the algorithm had good line spectrum tracking ability for a low signal-to-noise ratio.

On the Performance of the SPICE Method

This letter considers the performance of the SPICE method in the context of line spectrum estimation by exploiting its equivalent square-root lasso (SR-LASSO) representation established in previous work. The inefficiencies of existing analyses are that the guarantees are not applicable to deterministic Fourier measurement matrices, and the studied range of the tuning parameter excludes the SR-LASSO formulation of the SPICE method. The key observation of this letter is that, for a particular range of tuning parameter, the solutions which overfit the noisy measurements are valid SR-LASSO solutions. Based on this observation, we propose to analyze these overfitting solutions and derive the condition of exact on-grid source localization that is applicable to the SPICE method. The numerical experiments demonstrate our theoretical claims.

Recovery analysis of damped spectrally sparse signals and its relation to MUSIC

Abstract One of the classical approaches for estimating the frequencies and damping factors in a spectrally sparse signal is the MUltiple SIgnal Classification (MUSIC) algorithm, which exploits the low-rank structure of an autocorrelation matrix. Low-rank matrices have also received considerable attention recently in the context of optimization algorithms with partial observations, and nuclear norm minimization (NNM) has been widely used as a popular heuristic of rank minimization for low-rank matrix recovery problems. On the other hand, it has been shown that NNM can be viewed as a special case of atomic norm minimization (ANM), which has achieved great success in solving line spectrum estimation problems. However, as far as we know, the general ANM (not NNM) considered in many existing works can only handle frequency estimation in undamped sinusoids. In this work, we aim to fill this gap and deal with damped spectrally sparse signal recovery problems. In particular, inspired by the dual analysis used in ANM, we offer a novel optimization-based perspective on the classical MUSIC algorithm and propose an algorithm for spectral estimation that involves searching for the peaks of the dual polynomial corresponding to a certain NNM problem, and we show that this algorithm is in fact equivalent to MUSIC itself. Building on this connection, we also extend the classical MUSIC algorithm to the missing data case. We provide exact recovery guarantees for our proposed algorithms and quantify how the sample complexity depends on the true spectral parameters. In particular, we provide a parameter-specific recovery bound for low-rank matrix recovery of jointly sparse signals rather than use certain incoherence properties as in existing literature. Simulation results also indicate that the proposed algorithms significantly outperform some relevant existing methods (e.g., ANM) in frequency estimation of damped exponentials.

Multi-snapshot Newtonized orthogonal matching pursuit for line spectrum estimation with multiple measurement vectors

In this paper, multi-snapshot Newtonized orthogonal matching pursuit (MNOMP) algorithm is proposed to deal with the line spectrum estimation with multiple measurement vectors (MMVs). MNOMP has the low computation complexity and state-of-the-art performance advantage of NOMP, and also includes two key steps: Detecting a new sinusoid on an oversampled discrete Fourier transform (DFT) grid and refining the parameters of already detected sinusoids to avoid the problem of basis mismatch. We provide a stopping criterion based on the overestimating probability of the model order. In addition, the convergence of the proposed algorithm is also proved. Finally, numerical results are conducted to show that the performance of MNOMP benefits from MMVs, and the effectiveness of MNOMP when compared against the state-of-the-art algorithms in terms of frequency estimation accuracy and computation complexity.

Efficient two-dimensional line spectrum estimation based on decoupled atomic norm minimization

This paper presents an efficient optimization technique for gridless 2-D line spectrum estimation, named decoupled atomic norm minimization (D-ANM). The framework of atomic norm minimization (ANM) is considered, which has been successfully applied in 1-D problems to allow super-resolution frequency estimation for correlated sources even when the number of snapshots is highly limited. The state-of-the-art 2-D ANM approach vectorizes the 2-D measurements to their 1-D equivalence, which incurs huge computational cost and may become too costly for practical applications. We develop a novel decoupled approach of 2-D ANM via semi-definite programming (SDP), which introduces a new matrix-form atom set to naturally decouple the joint observations in both dimensions without loss of optimality. Accordingly, the original large-scale 2-D problem is equivalently reformulated via two decoupled one-level Toeplitz matrices, which can be solved by simple 1-D frequency estimation with pairing. Compared with the conventional vectorized approach, the proposed D-ANM technique reduces the computational complexity by several orders of magnitude with respect to the problem size, at no loss of optimality. It also retains the benefits of ANM in terms of precise signal recovery, small number of required measurements, and robustness to source correlation. The complexity benefits are particularly attractive for large-scale antenna systems such as massive MIMO, radar signal processing and radio astronomy.

Correlation Awareness in Low-Rank Models: Sampling, Algorithms, and Fundamental Limits

The role of correlation awareness in low-rank compressive inverse problems is studied in this article. In such inverse problems, the ultimate goal is to estimate certain physically meaningful parameters from measurements collected across space and time. The spatiotemporal correlation structure of the data can be judiciously exploited to design highly efficient samplers that allow reliable parameter estimation from compressed measurements. For a large class of spectrum estimation problems (including source localization and line spectrum estimation), certain structured samplers, based on the idea of difference sets, will be shown to be optimal and outperform random samplers. Using these samplers, it is even possible to localize more sources than the number of physical sensors. The underlying principles are also extended to sparse estimation problems under the Bayesian framework. Fundamental performance limits in terms of Cramer-Rao bounds (CRBs) are studied in a new underdetermined regime, and certain saturation effects that only occur in this regime are discussed.

Gridless Line Spectrum Estimation and Low-Rank Toeplitz Matrix Compression Using Structured Samplers: A Regularization-Free Approach

This paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank Toeplitz covariance matrix. Certain families of structured deterministic samplers are shown to efficiently compress a high-dimensional Toeplitz matrix of size $N\times N$ , producing a compressed sketch of size $O(\sqrt{r})\times O(\sqrt{r})$.The reconstruction problem can be cast as that of line spectrum estimation, whereby, in absence of noise, Toeplitz matrices of any size $N$ can be exactly recovered from compressive sketches of size $O(\sqrt{r})\times O(\sqrt{r})$, no matter how large $N$ is. In the presence of noise and finite data, the line spectrum estimation algorithm is combined with a novel denoising technique that only exploits a positive semidefinite (PSD) Toeplitz constraint to denoise the compressed sketch using a simple least-squares minimization framework. A major advantage of the algorithm is that it does not require any regularization parameter. It also enjoys lower computational complexity owing to its ability to predict the unobserved entries of the low-rank Toeplitz matrix. Explicit bounds on the reconstruction error are established and it is shown that the PSD constraint on the denoiser is sufficient to ensure stable reconstruction from a sketch of size $O(\sqrt{r})\times O(\sqrt{r})$. Extensive simulations demonstrate that the proposed algorithm provides better performance over random samplers and algorithms that use nuclear-norm-based regularizers.

Performance of Uniform and Sparse Non-Uniform Samplers In Presence of Modeling Errors: A Cramér-Rao Bound Based Study

This paper evaluates the performance of uniform and sparse nonuniform sampling techniques (namely nested and coprime sampling) for line spectrum estimation, in presence of nonideal conditions such as perturbation in sampling instants, and limited data for computing statistical averages. Coprime and nested sampling are well-known deterministic sampling techniques that operate at rates significantly lower than Nyquist, and yet allow perfect reconstruction of the spectra of wide sense stationary signals. However, theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling, and ability to perfectly compute statistical expectations. This paper studies the performance of coprime and nested samplers when these assumptions are violated. Using a general grid-based signal model that applies to both spatial and temporal line spectrum estimation, the effect of perturbations in sampling instants is evaluated by deriving fundamental Cramer-Rao Bounds (CRB) for line spectrum estimation with perturbed samplers. For the first time, simplified expressions for the Fisher Information matrix for perturbed coprime and nested samplers are derived, which explicitly highlight the role of coarray. Even in presence of perturbations, it is possible to resolve O(M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) spectral lines under appropriate conditions on the size of the grid. The effect of finite data on the CRB is also studied, and necessary and sufficient conditions are derived to ensure that the CRB decreases monotonically to zero with the number of measurements, even when there are more sources than sensors. Finally, the theoretical results derived in this paper are supported by extensive numerical experiments.

A general sequential delay-Doppler estimation scheme for sub-Nyquist pulse-Doppler radar

Sequential estimation of the delay and Doppler parameters for sub-Nyquist radars by analog-to-information conversion (AIC) systems has received wide attention recently. However, the estimation methods reported are AIC-dependent and have poor performance for off-grid targets. This paper develops a general estimation scheme in the sense that it is applicable to all AICs regardless whether the targets are on or off the grids. The proposed scheme estimates the delay and Doppler parameters sequentially, in which the delay estimation is formulated into a beamspace direction-of- arrival problem and the Doppler estimation is translated into a line spectrum estimation problem. Then the well-known spatial and temporal spectrum estimation techniques are used to provide efficient and high-resolution estimates of the delay and Doppler parameters. In addition, sufficient conditions on the AIC to guarantee the successful estimation of off-grid targets are provided, while the existing conditions are mostly related to the on-grid targets. Theoretical analyses and numerical experiments show the effectiveness and the correctness of the proposed scheme.

Off-the-Grid Line Spectrum Denoising and Estimation With Multiple Measurement Vectors

Compressed Sensing suggests that the required number of samples for reconstructing a signal can be greatly reduced if it is sparse in a known discrete basis, yet many real-world signals are sparse in a continuous dictionary. One example is the spectrally-sparse signal, which is composed of a small number of spectral atoms with arbitrary frequencies on the unit interval. In this paper we study the problem of line spectrum denoising and estimation with an ensemble of spectrally-sparse signals composed of the same set of continuous-valued frequencies from their partial and noisy observations. Two approaches are developed based on atomic norm minimization and structured covariance estimation, both of which can be solved efficiently via semidefinite programming. The first approach aims to estimate and denoise the set of signals from their partial and noisy observations via atomic norm minimization, and recover the frequencies via examining the dual polynomial of the convex program. We characterize the optimality condition of the proposed algorithm and derive the expected convergence rate for denoising, demonstrating the benefit of including multiple measurement vectors. The second approach aims to recover the population covariance matrix from the partially observed sample covariance matrix by motivating its low-rank Toeplitz structure without recovering the signal ensemble. Performance guarantee is derived with a finite number of measurement vectors. The frequencies can be recovered via conventional spectrum estimation methods such as MUSIC from the estimated covariance matrix. Finally, numerical examples are provided to validate the favorable performance of the proposed algorithms, with comparisons against several existing approaches.

Modal Analysis Using Co-Prime Arrays

Let a measurement consist of a linear combination of damped complex exponential modes, plus noise. The problem is to estimate the parameters of these modes, as in line spectrum estimation, vibration analysis, speech processing, system identification, and direction of arrival estimation. Our results differ from standard results of modal analysis to the extent that we consider sparse and co-prime samplings in space, or equivalently sparse and co-prime samplings in time. Our main result is a characterization of the orthogonal subspace. This is the subspace that is orthogonal to the signal subspace spanned by the columns of the generalized Vandermonde matrix of modes in sparse or co-prime arrays. This characterization is derived in a form that allows us to adapt modern methods of linear prediction and approximate least squares, such as iterative quadratic maximum likelihood (IQML), for estimating mode parameters. Several numerical examples are presented to demonstrate the validity of the proposed modal estimation methods, and to compare the fidelity of modal estimation with sparse and co-prime arrays, versus SNR. Our calculations of Cram\'{e}r-Rao bounds allow us to analyze the loss in performance sustained by sparse and co-prime arrays that are compressions of uniform linear arrays.

Compressive Two-Dimensional Harmonic Retrieval via Atomic Norm Minimization

This paper is concerned with estimation of two-dimensional (2-D) frequencies from partial time samples, which arises in many applications such as radar, inverse scattering, and super-resolution imaging. Suppose that the object under study is a mixture of r continuous-valued 2-D sinusoids. The goal is to identify all frequency components when we only have information about a random subset of n regularly spaced time samples. We demonstrate that under some mild spectral separation condition, it is possible to exactly recover all frequencies by solving an atomic norm minimization program, as long as the sample complexity exceeds the order of rlogrlogn. We then propose to solve the atomic norm minimization via a semidefinite program and provide numerical examples to justify its practical ability. Our work extends the framework proposed by Tang for line spectrum estimation to 2-D frequency models.

Line spectrum estimation with probabilistic priors

For line spectrum estimation, we derive the maximum a posteriori probability estimator where prior knowledge of frequencies is modeled probabilistically. Since the spectrum is periodic, an appropriate distribution is the circular von Mises distribution that can parameterize the entire range of prior certainty of the frequencies. An efficient alternating projections method is used to solve the resulting optimization problem. The estimator is evaluated numerically and compared with other estimators and the Cramér-Rao bound.