An Improved LSF-based Algorithm for Sinusoidal Frequency Estimation that Achieves Maximum Likelihood Performance
19 Jul 2020-pp 1-5
TL;DR: Improvements are proposed for sinusoidal frequency estimation that reduced the number of candidate frequencies to at most 2p points, reduced the method’s threshold to equal that of ML, and reduced the computational burden by switching to methods like ESPRIT when the SNR is above threshold.
Abstract: In this paper we propose a method for sinusoidal frequency estimation that improves upon our previously proposed LSF-based algorithm that used at most 5p candidate points, where p is the number of sinusoids present. In this paper we propose the following improvements: (i) reduced the number of candidate frequencies to at most 2p points, (ii) reduced the method’s threshold to equal that of ML, and (iii) reduced the computational burden by switching to methods like ESPRIT when the SNR is above threshold. Since neither the SNR nor the threshold is known, we estimate them from the data. The proposed reduction-in-threshold step can be applied to EPUMA (proposed Qian et al.), with which we compare our results. For the well-known two-sinusoid example the proposed method has the same threshold as that of ML; ML performance is also achieved when tested on a new, three-sinusoid example.
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TL;DR: In this paper, the authors proposed a new algorithm that carries out zero-padding and removal and re-estimation to achieve a threshold SNR that is lower than that of the MLE for noisy signals with random parameters and up to five components.
Abstract: Estimating the frequencies of multiple sinusoids in the presence of AWGN and when the data record is short is commonly accomplished by subspace-based methods such as ESPRIT, MUSIC, Min-Norm, etc. These methods do not assume that the data are zero outside the observation interval. If we assume otherwise, the threshold SNR is lowered significantly, but the price paid is unacceptable bias. Among all known unbiased estimators, the maximum-likelihood estimator (MLE) has the lowest threshold, but is computationally the most expensive. We propose a new algorithm that carries out, when needed, (i) zero-padding, and (ii) removal and re-estimation. These added steps result in a threshold SNR that is lower than that of the MLE for the examples considered herein, viz., noisy signals containing sinusoids with random parameters and up to five components. The maximum improvement in threshold was 10 dB for the two-sinusoid case. The bias of the estimates is also either equal to or lower than MLE's. Unlike the MLE, the proposed method is very much computationally feasible.
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TL;DR: An improved atomic search optimization (IASO) algorithm is proposed based on the idea of speed update in particle swarm optimization (PSO), which holds great potential for not only guaranteeing the estimation accuracy but also greatly reducing the computational complexity of multidimensional nonlinear optimization of ML estimator.
Abstract: The atom search optimization (ASO) algorithm has the characteristics of fewer parameters and better performance than the traditional intelligent optimization algorithms, but it is found that ASO may easily fall into local optimum and its accuracy is not higher. Therefore, based on the idea of speed update in particle swarm optimization (PSO), an improved atomic search optimization (IASO) algorithm is proposed in this paper. Compared with traditional ASO, IASO has a faster convergence speed and higher precision for 23 benchmark functions. IASO algorithm has been successfully applied to maximum likelihood (ML) estimator for the direction of arrival (DOA), under the conditions of the different number of signal sources, different signal-to-noise ratio (SNR) and different population size, the simulation results show that ML estimator with IASO algorithum has faster convergence speed, fewer iterations and lower root mean square error (RMSE) than ML estimator with ASO, sine cosine algorithm (SCA), genetic algorithm (GA) and particle swarm optimization (PSO). Therefore, the proposed algorithm holds great potential for not only guaranteeing the estimation accuracy but also greatly reducing the computational complexity of multidimensional nonlinear optimization of ML estimator.
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04 Jun 2023
TL;DR: In this article , the Wirtinger derivatives of a complex exponential surrogate and any first order gradient-based optimizer are used for joint sinusoidal frequency and amplitude estimation, enabling end-to-end training of neural network controllers.
Abstract: Sinusoidal parameter estimation is a fundamental task in applications from spectral analysis to time-series forecasting. Estimating the sinusoidal frequency parameter by gradient descent is, however, often impossible as the error function is non-convex and densely populated with local minima. The growing family of differentiable signal processing methods has therefore been unable to tune the frequency of oscillatory components, preventing their use in a broad range of applications. This work presents a technique for joint sinusoidal frequency and amplitude estimation using the Wirtinger derivatives of a complex exponential surrogate and any first order gradient-based optimizer, enabling end-to-end training of neural network controllers for unconstrained sinusoidal models.
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26 Oct 2022
TL;DR: This work presents a technique for joint sinusoidal frequency and amplitude estimation using the Wirtinger derivatives of a complex exponential surrogate and any gradient-based optimizer, enabling end-to-end training of neural network controllers for unconstrained sinusoid models.
Abstract: Sinusoidal parameter estimation is a fundamental task in applications from spectral analysis to time-series forecasting. Estimating the sinusoidal frequency parameter by gradient descent is, however, often impossible as the error function is non-convex and densely populated with local minima. The growing family of differentiable signal processing methods has therefore been unable to tune the frequency of oscillatory components, preventing their use in a broad range of applications. This work presents a technique for joint sinusoidal frequency and amplitude estimation using the Wirtinger derivatives of a complex exponential surrogate and any first order gradient-based optimizer, enabling end to-end training of neural network controllers for unconstrained sinusoidal models.
1 citations
TL;DR: A new algorithm is proposed that carries out, when needed, zero-padding, and removal and re-estimation that results in a threshold SNR that is lower than that of the MLE for the examples considered herein, viz., noisy signals containing sinusoids with random parameters and up to five components.
Cites methods from "An Improved LSF-based Algorithm for..."
...Our previously proposed Γβ-based method [17] is quite effective for making this switch....
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...In [17] we proposed a method for estimating both the SNR and threshold, and introduced a parameter Γβ = Estimated SNR− Estimated Threshold....
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References
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Book•
01 Jan 2005
TL;DR: 1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra.
Abstract: 1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra. 4. Parametric Methods for Line Spectra. 5. Filter Bank Methods. 6. Spatial Methods. Appendix A: Linear Algebra and Matrix Analysis Tools. Appendix B: Cramer-Rao Bound Tools. Appendix C: Model Order Selection Tools. Appendix D: Answers to Selected Exercises. Bibliography. References Grouped by Subject. Subject Index.
2,620 citations
"An Improved LSF-based Algorithm for..." refers background or methods in this paper
...1) Find Root-MUSIC estimates f̌k, k = 1, 2, . . . , p 2) Initialization: i = 1, Ω1 = Ω2 = ∅ 3) Find LSP (θl, θl+1) near to f̌i such that θl ≤ f̌i ≤ θl+1....
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...For this example Root-MUSIC is unable to resolve the two closely spaced signal frequencies f1 and f2 (f̌1 = 0.75 and f̌2 ≈ 0.5(f1 + f2))....
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...The first step in the method proposed in [6] is estimating an M -th order linear prediction polynomial A(z) = 1 +a1z + · · · + aMz using Root-MUSIC [1], which has, ideally, p roots on the unit circle at ωk = 2πfk, with the remaining roots lying well inside the unit circle....
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...The problem of estimating the parameters of multiple sinusoids is a well-studied one [1], and arises in many diverse fields such as radar, sonar, etc....
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...In [6], [7], we used the Line Spectral Frequency (LSF) representation of the linear prediction polynomial obtained using Root-MUSIC [1] for estimating the fk’s (these methods are superior to the LSF-based method of Stoica and Nehorai [8])....
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01 Sep 1982
TL;DR: In this paper, the frequency estimation performance of the forward-backward linear prediction (FBLP) method was improved for short data records and low signal-to-noise ratio (SNR) by using information about the rank M of the signal correlation matrix.
Abstract: The frequency-estimation performance of the forward-backward linear prediction (FBLP) method of Nuttall/Uhych and Clayton, is significantly improved for short data records and low signal-to-noise ratio (SNR) by using information about the rank M of the signal correlation matrix. A source for the improvement is an implied replacement of the usual estimated correlation matrix by a least squares approximation matrix having the lower rank M. A second, related cause for the improvement is an increase in the order of the prediction filter beyond conventional limits. Computationally, the recommended signal processing is the same as for the FBLP method, except that the vector of prediction coefficients is formed from a linear combination of the M principal eigenvectors of the estimated correlation matrix. Alternatively, singular value decomposition can be used in the implementation. In one special case, which we call the Kumaresan-Prony (KP) case, the new prediction coefficients can be calculated in a very simple way. Philosophically, the improvement can be considered to result from a preliminary estimation of the explainable, predictable components of the data, rather than attempting to explain all of the observed data by linear prediction.
1,072 citations
"An Improved LSF-based Algorithm for..." refers background in this paper
...2 shows the effectiveness of the steps taken to lower the threshold for the well-known two sinusoids example [11]: N = 25, f1 = 0....
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TL;DR: The application of a subspace invariance approach (ESPRIT) to the estimation of parameters (frequencies and powers) of cisoids in noise is described, which has several advantages including improved resolution over Pisarenko's technique for harmonic retrieval.
Abstract: The application of a subspace invariance approach (ESPRIT) to the estimation of parameters (frequencies and powers) of cisoids in noise is described. ESPRIT exploits an underlying rotational invariance of signal subspaces spanned by two temporally displaced data sets. The new approach has several advantages including improved resolution over Pisarenko's technique for harmonic retrieval.
1,040 citations
"An Improved LSF-based Algorithm for..." refers methods in this paper
...Subspace-based methods such as ESPRIT [2] are computationally less burdensome but have a higher threshold than ML....
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TL;DR: A novel pseudo-noise resampling (PR) based unitary root-MUSIC algorithm for direction-of-arrival (DOA) estimation is derived and a distance detection strategy which exploits the information contained in the estimated root estimator to help determine the final DOA estimates when all the DOA estimators fail to pass the reliability test is proposed.
Abstract: A novel pseudo-noise resampling (PR) based unitary root-MUSIC algorithm for direction-of-arrival (DOA) estimation is derived in this letter. Our solution is able to eliminate the abnormal DOA estimator called outlier and obtain an approximate outlier-free performance in the unitary root-MUSIC algorithm. In particular, we utilize a hypothesis test to detect the outlier. Meanwhile, a PR process is applied to form a DOA estimator bank and a corresponding root estimator bank. We propose a distance detection strategy which exploits the information contained in the estimated root estimator to help determine the final DOA estimates when all the DOA estimators fail to pass the reliability test. Furthermore, the proposed method is realized in terms of real-valued computations, leading to an efficient implementation. Simulations show that the improved MUSIC scheme can significantly improve the DOA resolution at low signal-to-noise ratios and small samples.
81 citations
"An Improved LSF-based Algorithm for..." refers methods in this paper
...In [6] the beamformer function [10] was used to identify...
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...Further improvements were obtained by eliminating outliers using the beamformer function [10]....
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TL;DR: A two-step method is proposed that improves the performance of the DOA estimation by modifying the sample covariance matrix such that the amount of the subspace leakage is reduced and is combined with the pseudo-noise resampling method to further improve the performance.
Abstract: Classical methods of DOA estimation such as the MUSIC algorithm are based on estimating the signal and noise subspaces from the sample covariance matrix. For a small number of samples, such methods are exposed to performance breakdown, as the sample covariance matrix can largely deviate from the true covariance matrix. In this paper, the problem of DOA estimation performance breakdown is investigated. We consider the structure of the sample covariance matrix and the dynamics of the root-MUSIC algorithm. The performance breakdown in the threshold region is associated with the subspace leakage where some portion of the true signal subspace resides in the estimated noise subspace. In this paper, the subspace leakage is theoretically derived. We also propose a two-step method that improves the performance by modifying the sample covariance matrix such that the amount of the subspace leakage is reduced. Furthermore, we introduce a phenomenon named as root-swap which occurs in the root-MUSIC algorithm in the low sample size region and degrades the performance of the DOA estimation. A new method is then proposed to alleviate this problem. Numerical examples and simulation results are given for uncorrelated and correlated sources to illustrate the improvement achieved by the proposed methods. Moreover, the proposed algorithms are combined with the pseudo-noise resampling method to further improve the performance.
70 citations