Problem Of Frequency Estimation Research Articles

Time-Frequency Domain Channel Estimation for OTFS Systems

Orthogonal-time-frequency-space (OTFS) has been developed as a promising modulation scheme for high Doppler channels. In OTFS systems, channel estimation (CE) plays a key role. Most existing CE schemes put pilots in delay-Doppler (DD)-domain, often resulting in low spectrum efficiency and high peak-to-average-power-ratio (PAPR). In this paper, we propose three schemes to solve the problems. Proposed methods have distinct features being able to estimate fractional delays and fractional Dopplers. Different from existing methods, our approach places pilots in time-frequency (TF)-domain. The first scheme overlaps pilots and data in both TF- and DD-domain, while Scheme II overlaps them in TF- but not in DD-domain. To further enhance the performance, the third scheme makes pilots and data non-overlapped in both TF- and DD-domain. To conduct CE, time-domain pilot signals are extracted with a filtering operation and a two-dimensional frequency estimation problem is formulated consequently. Delays are first estimated with root-MUSIC. Then, Dopplers are estimated using a simple line-fitting method. Finally, channel gains are estimated by a least-squares method. To solve the interference between pilots and data, successive-interference-cancellation (SIC) is also applied. With the estimated channel, a minimum-mean-square-error (MMSE) equalizer is used for the compensation of channel effect. Simulation results show that the bit-error-rate (BER) performance with the channel estimated by proposed CE methods can approach to that with known channel. Also, spectrum efficiency can be improved by at most 25% and PAPR can be reduced by at least 19 dB.

Parameter estimation of biased harmonic signal

This paper deals with the problem of asymptotically estimating amplitude, frequency and phase of a sinusoidal signal by adopting the theory of invariant relations proposed in analytical mechanics [8] and further investigated in [9, 10] (see also [11]). The problem of frequency, amplitude and phase estimation of a sinusoidal signal has attracted a remarkable research attention in the past and current literature. The reasons of this interest rely on several engineering applications where an effective and robust solution to this problem is crucial. To mention few, it is worth mentioning problems of harmonic disturbance compensation in automatic control, design of phase-looked loop circuits in telecommunication, adaptive filtering in signal processing, etc. In principle, the method of least squares, Fourier analysis, Laplace transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. The goal of this paper is to suggest a further contribution to this task by showing how to solve the problem at hand through the observer’s theory. The method of invariant relations is used for the asymptotically observation scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns variables can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. The asymptotic convergence of the estimates of the sought phase vector components to their true value is proved. The simulation results demonstrate the effectiveness of the proposed method of solving the state observation problem of the harmonic oscillator. It should be noted that a more general approach, which forms an appropriate method for solving observation problems for nonlinear dynamical systems due to the synthesis of invariant manifold, was proposed as a modification of the I&I method (Input and Invariance) of stabilization of nonlinear systems in [12, 13].

Frequency Detection and Change Point Estimation for Time Series of Complex Oscillation

We consider detecting the evolutionary oscillatory pattern of a signal when it is contaminated by nonstationary noises with complexly time-varying data generating mechanism. A high-dimensional dense progressive periodogram test is proposed to accurately detect all oscillatory frequencies. A further phase-adjusted local change point detection algorithm is applied in the frequency domain to detect the locations at which the oscillatory pattern changes. Our method is shown to be able to detect all oscillatory frequencies and the corresponding change points within an accurate range with a prescribed probability asymptotically. A Gaussian approximation scheme and an overlapping-block multiplier bootstrap methodology for sums of complex-valued high dimensional nonstationary time series without variance lower bounds are established, which could be of independent interest. This study is motivated by oscillatory frequency estimation and change point detection problems encountered in physiological time series analysis. An application to spindle detection and estimation in electroencephalogram recorded during sleep is used to illustrate the usefulness of the proposed methodology. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.

SDPLL-Based Frequency Estimation of a Sinusoid in Colored Noise

The problem of frequency estimation of a single sinusoid observed in colored noise is addressed. Our estimator is based on the operation of the sinusoidal digital phase-locked loop (SDPLL) which carries the frequency information in its phase error after the noisy sinusoid has been acquired by the SDPLL. We show by computer simulations that this frequency estimator beats the Cramer-Rao bound (CRB) on the frequency error variance for moderate and high SNRs when the colored noise has a general low-pass filtered (LPF) characteristic, thereby outperforming, in terms of frequency error variance, several existing techniques some of which are, in addition, computationally demanding. Moreover, the present approach generalizes on existing work that addresses different methods of sinusoid frequency estimation involvingspecific colored noise models such as the moving average (MA) noise model. An insightful theoretical analysis is presented to support the practical findings.

Observer of harmonic oscillator parameters

This paper deals with the problem of asymptotically estimating amplitude, frequency and phase of a sinusoidal signal by adopting the theory of invariant relations proposed in analytical mechanics [8] and further investigated in [9], [10] (see also [11]). The problem of frequency, amplitude and phase estimation of a sinusoidal signal has attracted a remarkable research attention in the past and current literature. The reasons of this interest rely on several engineering applications where an effective and robust solution to this problem is crucial. To mention few, it is worth mentioning problems of harmonic disturbance compensation in automatic control, design of phase-looked loop circuits in telecommunication, adaptive filtering in signal processing, etc. In principle, the method of least squares, Fourier analysis, Laplace transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. The goal of this paper is to suggest a further contribution to this task by showing how to solve the problem at hand through the observer’s theory. The method of invariant relations is used for the asymptotically observation scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns variables can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. The asymptotic convergence of the estimates of the sought phase vector components to their true value is proved. The simulation results demonstrate the effectiveness of the proposed method of solving the state observation problem of the harmonic oscillator. It should be noted that a more general approach, which forms an appropriate method for solving observation problems for nonlinear dynamical systems due to the synthesis of invariant manifold, was proposed as a modification of the I\&I method (Input and Invariance) of stabilization of nonlinear systems in [12, 13].

Control-enhanced quantum metrology under Markovian noise

Quantum metrology is supposed to significantly improve the precision of parameter estimation by utilizing suitable quantum resources. However, the predicted precision can be severely distorted by realistic noises. Here, we propose a control-enhanced quantum metrology scheme to defend against these noises for improving the metrology performance. Our scheme can automatically alter the parameter encoding dynamics with adjustable controls, thus leading to optimal resultant states that are less sensitive to the noises under consideration. As a demonstration, we numerically apply it to the problem of frequency estimation under several typical Markovian noise channels. Through comparing our control-enhanced scheme with the standard scheme and the ancilla-assisted scheme, we show that our scheme performs better and can improve the estimation precision up to around one order of magnitude. Furthermore, we conduct a proof-of-principle experiment in nuclear magnetic resonance system to verify the effectiveness of the proposed scheme. The research here is helpful for current quantum platforms to harness the power of quantum metrology in realistic noise environments.

Breaking the Communication-Privacy-Accuracy Trilemma

Two major challenges in distributed learning and estimation are 1) preserving the privacy of the local samples; and 2) communicating them efficiently to a central server, while achieving high accuracy for the end-to-end task. While there has been significant interest in addressing each of these challenges separately in the recent literature, treatments that simultaneously address both challenges are still largely missing. In this paper, we develop novel encoding and decoding mechanisms that simultaneously achieve optimal privacy and communication efficiency in various canonical settings. In particular, we consider the problems of mean estimation and frequency estimation under <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> -local differential privacy and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> -bit communication constraints. For mean estimation, we propose the SQKR mechanism, a scheme based on Kashin’s representation and random sampling, with order-optimal estimation error under both constraints. We further apply SQKR to distributed SGD and obtain a communication efficient and (locally) differentially private distributed SGD protocol. For frequency estimation, we present the RHR mechanism, a scheme that leverages the recursive structure of Walsh-Hadamard matrices and achieves order-optimal estimation error for all privacy levels and communication budgets. As a by-product, we also construct a distribution estimation mechanism that is rate-optimal for all privacy regimes and communication constraints, extending recent work that is limited to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$b=1$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon =O(1)$ </tex-math></inline-formula> . Our results demonstrate that intelligent encoding under joint privacy and communication constraints can yield a performance that matches the optimal accuracy achievable under either constraint alone. In other words, the optimal performance is determined by the more stringent of the two constraints, and the less stringent constraint can be satisfied for free.

Parameter Estimation of a Noisy Real Sinusoid Using Quartic Polynomial Approach

An analytical polynomial expression, for accurate and computationally efficient frequency estimation of a single real sinusoid under Additive White Gaussian Noise (AWGN), is derived and proposed in this paper. The method, which can be easily adapted for real time frequency estimation, is based on transforming the frequency estimation problem as the solution of a fourth order (quartic) expressed as powers of a trigonometric function containing the unknown frequency. The coefficients of the quartic polynomial can be found using the complex magnitudes of three Discrete Fourier Transform (DFT) bins, centered at the maximum magnitude value of the DFT coefficients. Simulated results illustrate that, the performance of the proposed estimator has a mean squared-error (MSE) performance which is very close to the Cramer Rao Lower Bound (CRLB) for high signal-to-noise ratio (SNR) region, as well as close to previously published estimators in the low SNR region.

A Class of Bayesian Lower Bounds for Parameter Estimation Via Arbitrary Test-Point Transformation

In this paper, a new class of global mean-squared-error (MSE) lower bound for Bayesian parameter estimation is derived. First, it is shown that under the non-Bayesian framework, the Hammersley-Chapman-Robbins (HCR) for the problem of single-source parameter estimation, is related to the corresponding ambiguity function. This result is achieved by judicious choice of signal test-point. This result implies that optimal shift test-points may be parameter-dependent, but this approach cannot be imitated in test-point based Bayesian bounds, where the test-points should be independent of the random parameters. Based on this observation, a new class Bayesian MSE lower bound is derived. In the proposed class, the shift test-points in the Cauchy-Schwarz based bounds are substituted with arbitrary transformations. The proposed class generalizes the Weiss-Weinstein bound (WWB) to arbitrary transformations. For the problem of single source parameter estimation, a scaling transformation for the signal test-point is proposed based on the structure of the optimal test-point in the HCR bound in the non-Bayesian framework. The test-point scale parameters are analytically optimized and a closed-form expression for the bound, which depends on the ambiguity function, is derived. For the problem of direction-of-arrival (DOA) estimation using a linear array, a nonlinear transformation for the parameter of interest is proposed. In the problems of frequency estimation and DOA estimation, the proposed bound accurately predicts the threshold phenomenon of the maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a-posteriori</i> probability estimator, and is much tighter than the WWB in both the threshold prediction and the asymptotic performance. In addition, an asymptotic version of the bound was derived and it is analytically shown to be tighter than the Bayesian Cramér-Rao bound.

A High-Accuracy, High Anti-Noise, Unbiased Frequency Estimator Based on Three CZT Coefficients for Deep Space Exploration Mission.

Deep space exploration navigation requires high accuracy of the Doppler measurement, which is equivalent to a frequency estimation problem. Because of the fence effect and spectrum leakage, the frequency estimation performances, which is based on the FFT spectrum methods, are significantly affected by the signal frequency. In this paper, we propose a novel method that utilizes the mathematical relation of the three Chirp-Z Transform (CZT) coefficients around the peak spectral line. The realization, unbiased performance, and algorithm parameter setting rule of the proposed method are described and analyzed in detail. The Monte Carlo simulation results show that the proposed method has a better anti-noise and unbiased performance compared with some traditional estimator methods. Furthermore, the proposed method is utilized to process the raw data of MEX and Tianwen-1 satellites received by Chinese Deep Space Stations (CDSS). The results show that the Doppler estimation accuracy of MEX and Tianwen-1 are both about 3 millihertz (mHz) in 1-s integration, which is consistent with that of ESA/EVN/CDSN and a little better than that of the Chinese VLBI network (CVN). Generally, this proposed method can be effectively utilized to support Chinese future deep space navigation missions and radio science experiments.

A new frequency estimation method for a real-valued waveform with multiple sub-Nyquist channels based on the ESPRIT algorithm

Real-valued waveform frequency measurement problems are often encountered in signal processing and they can be solved by designing a linear system. However, they require the sampling frequency to reach twice the maximum of the measured frequency, that is, to meet the Nyquist condition, which in many cases cannot be satisfied considering the hardware limitation. Therefore, how to design a nonlinear method to reconstruct the signal frequency is of great significance. To solve the frequency estimation problem for a real-valued waveform with sub-Nyquist channels, we propose a frequency estimator based on multiple sub-Nyquist channels. Our algorithm mainly consists of the following parts: 1) Calculate the folded frequencies which are lying on the first Nyquist region for each channel based on ESPRIT. 2) Reconstruct the signal frequency by a search method according to the set of folded frequencies. Finally, the simulation results verify the good performance of our proposed algorithm.

3D Imaging With Moving Fringe Structured Illumination Microscopy

Structured Light Illumination Microscopy (SIM) has proved itself as a very effective method to improve the resolution of a widefield (WF) fluorescent microscope. In this paper, we demonstrate a new approach to three-dimensional (3D) imaging with the SIM, using a moving fringe (MF) illumination pattern. Instead of the standard three-beam standing wave illumination pattern, our method requires a two-beam one, varying along the optical axis. Each axial layer of the MF illumination pattern contains single-spatial-frequency interference fringes, traversing the space with its own speed (temporal frequency), proportional to the axial offset of such layer from the excitation plane. The different axial layers of a fluorescent object, excited with the MF illumination, will emit a continuous amplitude modulated fluorescent signal with the frequency of modulation proportional to the temporal frequency of the moving fringe pattern. The fine 3D image reconstruction is achieved via extracting the spatial location of the fluorescent object from the temporal frequency of amplitude modulated signal emitted by it. Since in our approach the problem of 3D image reconstruction is reduced to the problem of accurate temporal frequency estimation, any of the well-known spectrum estimation techniques can be applied to the problem, allowing the axial resolution improvement far beyond the limits of the classical 3D SIM. In this research, we suggest using the Minimum-norm method for the proposed MF SIM system, which gives a superior resolving power in spectrum estimation. Simulation results show that such a simple and rapid hardware implementation in combination with a straightforward signal processing method can, however, deliver an improvement in axial resolution far beyond the classical 3D SIM approach.

Frequency estimation under multiparty differential privacy

We study the fundamental problem of frequency estimation under both privacy and communication constraints, where the data is distributed among k parties. We consider two application scenarios: (1) one-shot, where the data is static and the aggregator conducts a one-time computation; and (2) streaming, where each party receives a stream of items over time and the aggregator continuously monitors the frequencies. We adopt the model of multiparty differential privacy (MDP), which is more general than local differential privacy (LDP) and (centralized) differential privacy. Our protocols achieve optimality (up to logarithmic factors) permissible by the more stringent of the two constraints. In particular, when specialized to the ε-LDP model, our protocol achieves an error of √ k /(ε Θ(ε) − 1) using O ( k max{ε, log 1/ε}) bits of communication and O ( k log u ) bits of public randomness, where u is the size of the domain.

Empirical Bayes identification of stationary processes and approximation of Toeplitz spectra

We study Statistical Consistency of an approximate subspace identification procedure for the infinite dimensional a posteriori model of a frequency estimation problem in an Empirical Bayesian framework. By first imposing a natural uniform prior probability density on the unknown frequencies, the estimation of the hyperparameters of the a priori distribution can be accomplished by a sequence of subspace identification techniques. These techniques exploit the special structure of the covariance matrix of the a posteriori process which is discovered by making a connection with classical results on energy concentration in deterministic signal processing. The convergence of the spectrum of the subspace estimates to the (nonrational) spectral density of the a posteriori process has an analytic counterpart in the approximation of symmetric positive definite Toeplitz matrices by submatrices of finite rank. This is proven by a weak-sense convergence theorem for Toeplitz spectra.

Learning Feedback Control Strategies for Quantum Metrology

We consider the problem of frequency estimation for a single bosonic field evolving under a squeezing Hamiltonian and continuously monitored via homodyne detection. In particular, we exploit reinforcement learning techniques to devise feedback control strategies achieving increased estimation precision. We show that the feedback control determined by the neural network greatly surpasses in the long-time limit the performances of both the "no-control" strategy and the standard "open-loop control" strategy, which we considered as benchmarks. We indeed observe how the devised strategy is able to optimize the nontrivial estimation problem by preparing a large fraction of trajectories corresponding to more sensitive quantum conditional states.

An Efficient and Accurate Multi-Sensor IF Estimator Based on DOA Information and Order of Fractional Fourier Transform.

Instantaneous frequency in multi-sensor recordings is an important parameter for estimation of direction of arrival estimation, source separation, and sparse reconstruction. The instantaneous frequency estimation problem becomes challenging when signal components have close or overlapping signatures and the number of sensors is less than the number of sources. In this study, we develop a computationally efficient method that exploits the direction of the IF curve in addition to the angle of arrival as additional features for the accurate tracking of IF curves. Experimental results show that the proposed scheme achieves better accuracy compared to the-state-of-art method in terms of mean square error (MSE) with a slight increase in the computational cost, i.e., the proposed method achieves MSE of −50 dB at the signal to noise ratio of 0 dB whereas the existing method achieves the MSE of −38 dB.

A fast robust numerical continuation solver to a two‐dimensional spectral estimation problem

This paper presents a fast algorithm to solve a spectral estimation problem for two-dimensional random fields. The latter is formulated as a convex optimization problem with the Itakura-Saito pseudodistance as the objective function subject to the constraints of moment equations. The structure of the Hessian of the dual objective function is exploited in order to make possible a fast Newton solver. Then the Newton solver is incorporated to a predictor-corrector numerical continuation method which is able to produce a parametrized family of solutions to the moment equations. Two sets of numerical simulations are performed to test the algorithm and spectral estimator. The simulations on the frequency estimation problem show that our spectral estimator outperforms the classical windowed periodograms in the case of two hidden frequencies and has a higher resolution. The other set of simulations on system identification indicates that the numerical continuation method is more robust than Newton's method alone in ill-conditioned instances.

SpaceSaving ±

In this paper, we propose the first deterministic algorithms to solve the frequency estimation and frequent item problems in the bounded-deletion model. We establish the space lower bound for solving the deterministic frequent items problem in the bounded-deletion model, and propose Lazy SpaceSaving ± and SpaceSaving ± algorithms with optimal space bound. We develop an efficient implementation of the SpaceSaving ± algorithm that minimizes the latency of update operations using novel data structures. The experimental evaluations testify that SpaceSaving ± has accurate frequency estimations and achieves very high recall and precision across different data distributions while using minimal space. Our experiments clearly demonstrate that, if allowed the same space, SpaceSaving± is more accurate than the state-of-the-art protocols with up to logU - 1/ logU of the items deleted, where U is the size of the input universe. Moreover, motivated by prior work, we propose Dyadic SpaceSaving ± , the first deterministic quantile approximation sketch in the bounded-deletion model.

Simultaneous Frequency and Amplitude Estimation for Grid Quality Monitoring: New Partitioning with Memory Based Newton-Schulz Corrections

High penetration level of renewable energy sources and introduction of controllable loads will result in significant harmonic emissions and sag and swell events in the future electricity networks. Development of the methods for high performance simultaneous estimation of the frequency and amplitude events is required for stable operation of the future electricity networks since zero crossing frequency detection method (which is widely used nowadays in industry) is not accurate enough and does not allow estimation of the amplitude events. The multiple model method which is suitable for simultaneous estimation of the frequency and amplitude is extended in this paper with introduction of a new decomposition technique based on stepwise partitioning, which allows simultaneous construction and accurate and computationally efficient inversion of the information matrix. Recursive calculations of the inverse introduce error accumulation and a new general high order memory based Newton-Schulz iteration is proposed in this paper for correction and reduction of the accumulated error. Moreover, parallel Richardson iterations which are based on partitioning method are proposed in this paper for reduction of the computational complexity. The methods are especially efficient for approximation of the signals with large number of harmonics. The approaches were tested for simultaneous estimation of the frequency and sag and swell signatures in the one-phase synchronized voltage waveform measured at the wall outlet. Simulation results show that the multiple model method provides more accurate frequency estimation in comparison to zero crossing method.In addition, the cascade multiple model method which is based on the multi-windowing technique (where the components of the signals are separated via a proper choice of the window sizes) is introduced in this paper for estimation of the significantly separated frequencies of the electrical signals. The approach was tested in the problem of frequency estimation using the measurement record from the electric vehicle with on-board charger connected to the supply voltage in the laboratory.

Automatic Detection and Tracking of Random Frequency Signals Using Magnitude and Phase Information

This paper presents an approach to the problems of detection and frequency estimation of a frequency modulated narrow-band signal in additive complex Gaussian noise. The signal is assumed to have an unknown amplitude, initial phase and frequency trajectory over time, while a <i>priori</i> information regarding random frequency variability is taken to be available. The proposed approach operates in the frequency domain and uses the magnitude and phase of the discrete-time Fourier transforms computed over nonoverlapping signal segments. Robustness at low signal-to-noise ratios is achieved by suppressing the segment-related likelihoods for which the phase estimation error is large. The approach utilises unthresholded transform data and thus works in a track-before-detect manner, and the frequency trajectory is estimated by applying a search over the frequency-time bins. The results of a simulation study involving two signal types with different frequency variability are described.