Frequency Estimation Error Research Articles

Correction to: Development of a GNSS‑IR instrument based on low‑cost positioning chips and its performance evaluation for estimating the reflector height

Global Navigation Satellite System interferometric reflectometry (GNSS-IR) can be used to monitor a series of geophysical parameters in a cost-effective manner with high temporal resolution. The technique makes use of the simultaneous reception of direct and reflected GNSS signals with a single antenna. Based on the low-cost u-blox M8N chips, a GNSS-IR instrument is developed, which could be used to collect and process GNSS signals automatically and save and transmit generated GNSS data. Details about the instrument are described here for the first time. Then, the recorded SNR observation characteristics are analyzed by comparing three in-situ SNR sequences, which are simultaneously collected by the instrument with a low-cost patch antenna and a geodetic antenna and by a geodetic GNSS receiver with a geodetic antenna. Based on the developed function relating the peak power spectral density to peak frequency estimation error of the low-cost instrument, a weighting method is proposed to fuse multiple estimations of the reflector height to improve the estimation accuracy of the GNSS-IR-based reflector height. The performances of the developed low-cost instrument and the proposed reflector height estimation method are evaluated using a data set collected in Xinxiang City, Henan, China, over 6 days. The results show that there exists good agreement between the instrument-based reflector height estimates and the ground-truth estimates, with root-mean-square errors of 1.1 cm and 0.4 cm for the normal average and the proposed weighted average results, respectively, when the antenna height is in the range of 0.65 m to 2.15 m and the reflecting surface is flat, silty loam soil ground.

Exploring Implicit Pilots for Precise Estimation of LEO Satellite Downlink Doppler Frequency

The performance of low earth orbit (LEO) satellite Doppler positioning is fundamentally dominated by the precision with which the Doppler frequency is estimated, generally using an explicit pilot from the received signal. This letter proposes an algorithm for the precise estimation of the LEO satellite downlink Doppler frequency based on the joint use of the explicit pilot signal and an implicit pilot, which is a signal segment opportunistically available in an LEO downlink frame with all zero bits, all one bits, or interleaved zero and one bits. The theoretical performance is derived, and simulation results show that the proposed algorithm reaches the Cramer-Rao lower bound (CRLB). Verifications are conducted on signals from the Iridium mobile satellite communications system. The results show that 58.41% of Iridium frames in the ring alert and paging channels have implicit pilots and that for frames with implicit pilots, the proposed algorithm reduces the Doppler frequency estimation error by 70% compared to algorithms that use only the explicit pilot.

Contribution of an interharmonic component to the sine-wave parameters estimators returned by the interpolated Discrete Fourier transform algorithm

<p class="Abstract">This article investigates the contribution of a small-amplitude interharmonic component to the sine-wave parameter estimators returned by the classical interpolated discrete Fourier transform (IpDFT) algorithm. The analytical expressions for the frequency, amplitude, and phase estimation errors are derived herein by considering the IpDFT algorithm based on the maximum sidelobe decay (MSD) windows and by assuming the interharmonic frequency located at least one bin apart the unknown sine-wave frequency. The derived expressions allow us to analyse the impact of an interharmonic on the accuracies of the IpDFT frequency, amplitude, and phase estimators. The accuracies of the derived expressions are verified by means of both computer simulations and experimental results.</p>

Novel Method for Vibration Sensor-Based Instantaneous Defect Frequency Estimation for Rolling Bearings Under Non-Stationary Conditions.

It is proposed a novel instantaneous frequency estimation technology, multi-generalized demodulation transform, for non-stationary signals, whose true time variations of instantaneous frequencies are unknown and difficult to extract from the time-frequency representation due to essentially noisy environment. Theoretical bases of the novel instantaneous frequency estimation technology are created. The main innovations are summarized as: (a) novel instantaneous frequency estimation technology, multi-generalized demodulation transform, is proposed, (b) novel instantaneous frequency estimation results, obtained by simulation, for four types of amplitude and frequency modulated non-stationary single and multicomponent signals under strong background noise (signal to noise ratio is −5 dB), and (c) novel experimental instantaneous frequency estimation results for defect frequency of rolling bearings for multiple defect frequency harmonics, using the proposed technology in non-stationary conditions and in conditions of different levels of noise interference, including a strong noise interference. Quantitative instantaneous frequency estimation errors are employed to evaluate performance of the proposed IF estimation technology. Simulation and experimental estimation results show high effectiveness of the proposed estimation technology.

A Fine-Tuned Positioning Algorithm for Space-Borne GNSS Timing Receivers.

To maximize the usage of limited transmission power and wireless spectrum, more communication satellites are adopting precise space–ground beam-forming, which poses a rigorous positioning and timing requirement of the satellite. To fulfill this requirement, a space-borne global navigation satellite system (GNSS) timing receiver with a disciplined high-performance clock is preferable. The space-borne GNSS timing receiver moves with the satellite, in contrast to its stationary counterpart on ground, making it tricky in its positioning algorithm design. Despite abundant existing positioning algorithms, there is a lack of dedicated work that systematically describes the delicate aspects of a space-borne GNSS timing receiver. Based on the experimental work of the LING QIAO (NORAD ID:40136) communication satellite’s GNSS receiver, we propose a fine-tuned positioning algorithm for space-borne GNSS timing receivers. Specifically, the proposed algorithm includes: (1) a filtering architecture that separates the estimation of satellite position and velocity from other unknowns, which allows for a first estimation of satellite position and velocity incorporating any variation of orbit dynamics; (2) a two-threshold robust cubature Kalman filter to counteract the adverse influence of measurement outliers on positioning quality; (3) Reynolds averaging inspired clock and frequency error estimation. Hardware emulation test results show that the proposed algorithm has a performance with a 3D positioning RMS error of 1.2 m, 3D velocity RMS error of 0.02 m/s and a pulse per second (PPS) RMS error of 11.8ns. Simulations with MATLAB show that it can effectively detect and dispose outliers, and further on outperforms other algorithms in comparison.

The Effects of Forced Oscillation Frequency Estimation Error on the LS-ARMA+S Mode Meter

The least squares autoregressive moving average plus sinusoids (ls-arma+s) mode meter has been previously shown to successfully estimate power system electromechanical modes without bias in the presence of a forced oscillation (FO). In order to do this, it requires knowledge of the FO frequency, which is estimated as a byproduct of the periodogram-based FO detection scheme. Previous work has not fully addressed the effect that FO frequency estimation error has on the effectiveness of LS-ARMA+S. This letter demonstrates that if a FO is powerful enough, its frequency must be estimated with immense accuracy, something that the existing technique may not be able to achieve. Here it is demonstrated that easy-to-implement alternative frequency estimation algorithms can overcome this, resulting in the recommendation that for any future use of the LS-ARMA+S mode meter, the FO frequency be estimated with an algorithm separate from that which is used for FO detection.

Performance-aware predictive-model-based on-chip body-bias regulation strategy for an ULP multi-core cluster in 28 nm UTBB FD-SOI

The performance and reliability of Ultra-Low-Power (ULP) computing platforms are adversely affected by environmental temperature and process variations. Mitigating the effect of these phenomena becomes crucial when these devices operate near-threshold, due to the magnification of process variations and to the strong temperature inversion effect that affects advanced technology nodes in low-voltage corners, which causes huge overhead due to margining for timing closure. Supporting an extended range of reverse and forward body-bias, UTBB FD-SOI technology provides a powerful knob to compensate for such variations. In this work we propose a methodology to maximize energy efficiency at run-time exploiting body biasing on a ULP platform operating near-threshold. The proposed method relies on on-line performance measurements by means of Process Monitoring Blocks (PMBs) coupled with an on-chip low-power body bias generator. We correlate the measurement performed by the PMBs to the maximum achievable frequency of the system, deriving a predictive model able to estimate it with an error of 9.7% at 0.7 V. To minimize the effect of process variations we propose a calibration procedure that allows to use a PMB model affected by only the temperature-induced error, which reduces the frequency estimation error by 2.4x (from 9.7% to 4%). We finally propose a controller architecture relying on the derived models to automatically regulate at run-time the body bias voltage. We demonstrate that adjusting the body bias voltage against environmental temperature variations leads up to 2X reduction in the leakage power and a 15% improvement on the global energy consumption when the system operates at 0.7 V and 170 MHz.

Research on Data Privacy Protection of Internet of Vehicles Based on Differential Privacy

The private data of intelligent vehicles connected to the Internet are easy to leak through the network. However, all existing solutions require the intervention of trusted third parties. In reality, even the confidentiality commitment provided by vehicle service providers is questioned by users. Aiming at this problem, a local differential privacy algorithm is introduced to design a data privacy protection scheme adapted to the characteristics of Internet of vehicles. The original data are only stored locally, and all the statistical data provided to the collector are disturbed. On the basis of ensuring the necessary data collection of the service center, it is guaranteed by mathematical principle that the collector cannot reverse the original data and the original data provider. The experimental results show that when the number of users is 10 million, the maximum relative error of frequency estimation is 0.0031 and the maximum relative error of mean estimation is 0.00006, which are far less than the tolerance value of 0.01, indicating that the scheme guarantees the application of Internet of vehicle safety situational awareness and other applications that need a lot of data support on the basis of protecting user privacy.

A High-Resolution Algorithm for Supraharmonic Analysis Based on Multiple Measurement Vectors and Bayesian Compressive Sensing

Supraharmonics emitted by electrical equipment have caused a series of electromagnetic interference in power systems. Conventional supraharmonic analysis algorithms, e.g., discrete Fourier transform (DFT), have a relatively low frequency resolution with a given observation time. Our previous work supplied a significant improvement on the frequency resolution based on multiple measurement vectors and orthogonal matching pursuit (MMV-OMP). In this paper, an improved algorithm for supraharmonic analysis, which employs Bayesian compressive sensing (BCS) for further improving the frequency resolution, is proposed. The performance of the proposed algorithm on the simulation signal and experimental data show that the frequency resolution can be improved by about a magnitude compared to that of the MMV-OMP algorithm, and the signal frequency estimation error is about 20 times better. In order to identify the signals in two adjacent frequency grids with one resolution, a normalized inner product criterion is proposed and verified by simulations. The proposed algorithm shows a potential for high-accuracy supraharmonic analysis.

Practical Synchronization Waveform for Massive Machine-Type Communications

This paper proposes a practical synchronization waveform that is resilient to frequency error for machine-type communications with applications in massive Internet of Things (IoT). Mathematical properties of the waveform are derived, which are keys to addressing the practical issues. In particular, it is shown that this type of waveform is asymptotically optimal in the presence of a frequency error, in the sense that its asymptotic performance is the same as the optimal matched-filter detector that is free of frequency error. This asymptotic property enables optimization of the waveform under the constraints imposed by an application. It is also shown that such optimized waveform comes in pairs, which facilitates the formation of a new waveform capable of frequency error estimation and timing refinement at the receiver. Detailed comparisons with the LTE narrowband IoT primary synchronization signal are provided.

An Improved Method Based on Time-Frequency Distribution to Detect Time-Varying Interference for GNSS Receivers With Single Antenna

This paper presents a regroup method based on reassigned smoothed pseudo Wigner-Ville distribution (RSPWVD) to detect time-varying interference for global navigation satellite system (GNSS) receivers with a single antenna. In order to prove the effectiveness and superiority of the introduced method, the theoretical analysis and performance simulation has been performed between the proposed method and the classical time-frequency analysis methods. The performance of the proposed method has been accessed by experiments with the real BDS BII signals corrupted by the linear frequency modulation (chirp) interference. The actual performance of experiments has been used to test theoretical analysis by the quantitative metric root mean square error of the instantaneous frequency estimation. The results of experiments depict that the regroup method based on RSPWVD completely eliminates not only cross-term interferences but also end effects and provide better detection performance of time-varying interference compared with the classical time-frequency analysis methods for GNSS receivers with a single antenna.

Accurate Frequency Estimation of a Real Sinusoid by Three New Interpolators

Interpolators are widely employed in the frequency estimation of sinusoidal signals. However, for a real-valued sinusoidal signal, the existing interpolators do not consider the negative frequency component, which leads to a frequency estimation error floor. To solve this problem and further improve the estimation accuracy, three new interpolators are proposed in this paper. These three new interpolators consist of two initial interpolators and one fine interpolator. In the proposed algorithm, the initial interpolators consider both the positive and negative frequency components and are utilized first to eliminate this error floor and obtain an initial frequency estimate. Then, the fine interpolator for the frequency estimation of a complex sinusoid is exploited to improve the accuracy of the initial frequency estimate. The theoretical analysis demonstrates that the frequency estimation mean square error of the proposed algorithm is almost equal to the Cramer-Rao lower bound. Compared with the existing time-domain analysis algorithms, the proposed algorithm has better estimation performance, especially in the case of a low signal-to-noise ratio. Compared with the existing frequency-domain analysis algorithms, the proposed algorithm has lower computational complexity and a wider valid estimation range.

Voltage Harmonic Suppression by Means of Grid-Connected Converters Using only Local Measurements

A single-phase grid-connected converter is considered in this paper in the presence of harmonic problems introduced non-linear loads. In order to compensate the harmonics caused by the loads, a local voltage support scheme is proposed. This is an added feature because its implementation is in parallel with a conventional current control method. Distinctively, the measurements of the grid or load current are not needed since the scheme is based on only local measurements. On top of a fundamental part for desired power injection, the converter output current comprises a harmonic part for compensation. Thus, the grid current harmonic distortion is minimized and the enhancement of the local voltage quality is achieved. A comprehensive model analysis indicates that the proposed strategy can help to attenuate harmonics of the local voltage without compromising on the quality of the fundamental current injection. Experimental results validate the effectiveness of the proposed control scheme. Moreover, the impact of grid frequency estimation error on the control strategy’s performance is quantified theoretically and experimentally.

Windowed Fourier ridges for demodulation of carrier fringe patterns with nonlinearity: a theoretical analysis.

Accurately extracting phase or phase derivative is the most important requirement in optical metrology. However, in practice, there are many error sources, among which nonlinear distortion in fringe patterns is often encountered. Several techniques have been proposed over time to remove the nonlinearity error. Among these techniques, the windowed Fourier ridges (WFR) algorithm has been shown to be an effective solution insensitive to nonlinearity, but it lacks a theoretical justification. In this paper, we theoretically analyze the local frequency estimation error and phase extraction error, which not only proves the mentioned insensitivity, but also supports the performance prediction and error control, and thus is very important and useful in optical measurement. The theoretical results have been verified by computer simulations. Other error sources such as model error and noise are also compared and discussed.

High accuracy frequency extraction based on sigma-delta ADC applied in resonant dew point sensor

One of the design challenges for the resonant dew point measurement was optimizing the rapidity and accuracy of frequency extraction to meet the requirements. We have developed a 1-bit second-order sigma-delta (ΣΔ) analog-to-digital (ADC) to be applied in it, which was anticipated to have a highest resolution and design flexibility than traditional ADC. Based on the Butterworth filter, the design of ΣΔ modulator was modeled firstly to evaluate the performance of filters scheme before the hardware implementation. Furthermore, the design and implement of the digital decimation filters was presented to extend the design flexibility and application. To improve the accuracy and instantaneous of frequency estimation, we developed a new frequency estimation algorithm, which combined the Fast Fourier transform (FFT) and the chirp-Z transform. We built a complete hardware implementation system. We tested the performance of the designed ΣΔ ADC and found that has 11-bit resolution and the high flexibility. The experiment results showed the root mean square (RMS) error of frequency estimation was 0.0917 Hz with minimum error 0.01 Hz, and the SNR can achieve 123.04 dB. Results showed the proposed frequency estimation error decreases approximately 35 times under the premise of the same computation with FFT. Moreover, every completion of the frequency estimation only takes less than 1 s. Finally, the experiment of dew point measurement was carried out. The results indicated that the proposed measurement system was feasible for the frequency extraction of the resonant dew point sensor.

Frequency estimation of a sinusoidal signal with time-varying amplitude and phase

Abstract This paper is devoted to frequency estimation of a non-stationary sinusoidal signal. The amplitude is supposed to be a known function within a constant factor, the phase should be known. Example of such problem statement is sensorless angular velocity estimation for permanent magnet synchronous motors. On the first step by reparametrization, a third order linear regression model is obtained. On the next step, an estimation algorithm is constructed based on a standard gradient approach. The frequency estimate can be computed from one of the model parameters using inverse trigonometric functions. To improve estimates quality for noisy measurements we propose a new identification method, which can be tuned to attenuate the noise influence. It is shown that the frequency estimation error converges to zero exponentially fast. The described algorithm does not require measuring or calculating derivatives of the input signal. The efficiency of the proposed approach is demonstrated through the set of numerical simulations.

Frequency Estimation of the Plenoptic Function Using the Autocorrelation Theorem

The frequency estimation of the plenoptic function (POF) is an important research topic in spectral analysis for determining the minimum sampling rate of image-based rendering. In this paper, we mathematically derive a frequency estimation function of the POF using the autocorrelation theory. The autocorrelation function (ACF) of the POF is studied along both the spatial and image plane frequency axes. The influence of the scene's complexity and depth on the ACF of the POF is analyzed. Furthermore, we study the frequency estimation error to analyze the performance of the method. Existing techniques typically use Fourier transformation to determine the frequency of the POF. The technique presented herein simply starts from measurements of the light field in a finite number of positions, and the information from the light field is analyzed in the spatial domain. Finally, experimental results are presented to demonstrate that the proposed method can effectively estimate the frequency of the POF. The results are shown to be in good agreement with traditional methods.

An Enhanced Non-Coherent Pre-Filter Design for Tracking Error Estimation in GNSS Receivers.

Tracking error estimation is of great importance in global navigation satellite system (GNSS) receivers. Any inaccurate estimation for tracking error will decrease the signal tracking ability of signal tracking loops and the accuracies of position fixing, velocity determination, and timing. Tracking error estimation can be done by traditional discriminator, or Kalman filter-based pre-filter. The pre-filter can be divided into two categories: coherent and non-coherent. This paper focuses on the performance improvements of non-coherent pre-filter. Firstly, the signal characteristics of coherent and non-coherent integration—which are the basis of tracking error estimation—are analyzed in detail. After that, the probability distribution of estimation noise of four-quadrant arctangent (ATAN2) discriminator is derived according to the mathematical model of coherent integration. Secondly, the statistical property of observation noise of non-coherent pre-filter is studied through Monte Carlo simulation to set the observation noise variance matrix correctly. Thirdly, a simple fault detection and exclusion (FDE) structure is introduced to the non-coherent pre-filter design, and thus its effective working range for carrier phase error estimation extends from (−0.25 cycle, 0.25 cycle) to (−0.5 cycle, 0.5 cycle). Finally, the estimation accuracies of discriminator, coherent pre-filter, and the enhanced non-coherent pre-filter are evaluated comprehensively through the carefully designed experiment scenario. The pre-filter outperforms traditional discriminator in estimation accuracy. In a highly dynamic scenario, the enhanced non-coherent pre-filter provides accuracy improvements of 41.6%, 46.4%, and 50.36% for carrier phase error, carrier frequency error, and code phase error estimation, respectively, when compared with coherent pre-filter. The enhanced non-coherent pre-filter outperforms the coherent pre-filter in code phase error estimation when carrier-to-noise density ratio is less than 28.8 dB-Hz, in carrier frequency error estimation when carrier-to-noise density ratio is less than 20 dB-Hz, and in carrier phase error estimation when carrier-to-noise density belongs to (15, 23) dB-Hz (26, 50) dB-Hz.

Reconstructing Tone Sequences from Functional Magnetic Resonance Imaging Blood-Oxygen Level Dependent Responses within Human Primary Auditory Cortex.

Here we show that, using functional magnetic resonance imaging (fMRI) blood-oxygen level dependent (BOLD) responses in human primary auditory cortex, it is possible to reconstruct the sequence of tones that a person has been listening to over time. First, we characterized the tonotopic organization of each subject’s auditory cortex by measuring auditory responses to randomized pure tone stimuli and modeling the frequency tuning of each fMRI voxel as a Gaussian in log frequency space. Then, we tested our model by examining its ability to work in reverse. Auditory responses were re-collected in the same subjects, except this time they listened to sequences of frequencies taken from simple songs (e.g., “Somewhere Over the Rainbow”). By finding the frequency that minimized the difference between the model’s prediction of BOLD responses and actual BOLD responses, we were able to reconstruct tone sequences, with mean frequency estimation errors of half an octave or less, and little evidence of systematic biases.

How to optimize the precision of allele and haplotype frequency estimates using pooled-sequencing data.

Sequencing pools of individuals rather than individuals separately reduces the costs of estimating allele frequencies at many loci in many populations. Theoretical and empirical studies show that sequencing pools comprising a limited number of individuals (typically fewer than 50) provides reliable allele frequency estimates, provided that the DNA pooling and DNA sequencing steps are carefully controlled. Unequal contributions of different individuals to the DNA pool and the mean and variance in sequencing depth both can affect the standard error of allele frequency estimates. To our knowledge, no study separately investigated the effect of these two factors on allele frequency estimates; so that there is currently no method to a priori estimate the relative importance of unequal individual DNA contributions independently of sequencing depth. We develop a new analytical model for allele frequency estimation that explicitly distinguishes these two effects. Our model shows that the DNA pooling variance in a pooled sequencing experiment depends solely on two factors: the number of individuals within the pool and the coefficient of variation of individual DNA contributions to the pool. We present a new method to experimentally estimate this coefficient of variation when planning a pooled sequencing design where samples are either pooled before or after DNA extraction. Using this analytical and experimental framework, we provide guidelines to optimize the design of pooled sequencing experiments. Finally, we sequence replicated pools of inbred lines of the plant Medicago truncatula and show that the predictions from our model generally hold true when estimating the frequency of known multilocus haplotypes using pooled sequencing.