Unbiased Frequency Estimator Research Articles

Low complexity subspace approach for unbiased frequency estimation of a complex single-tone

We propose a single-tone frequency estimator of a one-dimensional complex signal in complex white Gaussian noise. The estimator is based on the subspace approach and the unitary transformation. Due to its low space and time-complexity, we name the estimator as Low complexity Unitary Principal-singular-vector Utilization for Model Analysis (LUPUMA). Regardless of the observation length, LUPUMA provides a uniform estimation variance over the whole frequency range, while achieving the lowest time-complexity among subspace methods. The proposed estimator asymptotically reaches the Cramér-Rao Lower Bound. For short observations, the signal-to-noise ratio threshold of LUPUMA corresponds to the threshold of the maximum likelihood estimator. The low space and time-complexity along with the stable and state-of-the-art estimation performance for short observations make LUPUMA an ideal candidate for applications with a limited number of signal samples, limited computational power, limited memory, and for applications that require rapid processing time (low latency).

Frequency of the loss of CAA interruption in the HTT CAG tract and implications for Huntington disease in the reduced penetrance range

PurposeIn some Huntington disease (HD) patients, the “loss of interruption” (LOI) variant eliminates an interrupting codon in the HTT CAG-repeat tract, which causes earlier age of onset (AOO). The magnitude of this effect is uncertain, since previous studies included few LOI carriers, and the variant also causes CAG size misestimation. We developed a rapid LOI detection screen, enabling unbiased frequency estimation among manifest HD patients. Additionally, we combined published data with clinical data from newly identified patients to accurately characterize the LOI’s effect on AOO. MethodsWe developed a LOI detection polymerase chain reaction (PCR) assay, and screened patients to estimate the frequency of the LOI variant and its effect on AOO. ResultsMean onset for LOI carriers (n=49) is 20.4 years earlier than expected based on diagnosed CAG size. After correcting for CAG size underestimation, the variant is still associated with onset 9.5 years earlier. The LOI is present in 1.02% of symptomatic HD patients, and in 32.2% of symptomatic reduced penetrance (RP) range patients (36–39 CAGs). ConclusionThe LOI causes significantly earlier onset, greater than expected by CAG length, particularly in persons with 36–39 CAG repeats. Detection of this variant has implications for HD families, especially for those in the RP range.

Non-iterative and accurate frequency estimation of a single cisoid using two DFT coefficients

An accurate and unbiased frequency estimator is devised in this paper, which utilizes the magnitudes of two mid-bin points around the highest magnitude discrete Fourier transform coefficient to perform sinc function fitting. The computational complexity of our algorithm is O(Nlog⁡(N)) with N being the data length. Its standard deviation is also analyzed, and is shown to achieve 1.0073 times of the square root of Cramér-Rao lower bound. Simulation results are presented to evaluate the performance of the proposed method via comparison with several benchmarking interpolation based frequency estimation schemes.

Highly accurate frequency estimation of brief duration signals in noise

A highly accurate frequency estimation providing suppression of windowing effects, denoising performances and frequency resolutions in excess of Gabor–Heisenberg limit, is proposed for brief duration signals. It is shown that unbiased frequency estimation with vanishing frequency variances is achieved far below Cramer–Rao lower bound when signal-to-noise ratio reaches vicinity of threshold values. Observed performances provide novel and valuable perspectives for efficient and accurate frequency estimation for brief duration signals in noise.

Complex-Valued Least Squares Frequency Estimation for Unbalanced Power Systems

A class of fundamental frequency estimation algorithms for unbalanced three-phase power systems is proposed. This is achieved in the complex domain by establishing and exploiting three time-series relationships among equidistantly spaced Clarke’s transformed system voltage samples. To enable meaningful frequency estimation with high immunity to noise and higher order harmonic pollution, complex-valued least squares (CLS) framework is employed in conjunction with those relationships. Theoretical bias and variance analysis are further conducted to yield the optimal distance (phase interval) between the voltage samples within the models considered. It is shown that although all the CLS approaches, with their own optimal phase intervals, ideally have identical minimum mean square estimation errors, the CLS approaches based on four-point relationships are favorable in practical applications, since the optimal phase interval attains statistically unbiased frequency estimation in the presence of both noise and harmonic pollution, which is impossible when using their three-point counterpart. Simulations over a range of unbalanced system conditions and in the presence of noise and higher order harmonics, as well as for real-world measurements, support the analysis.

The use of topology in fracture network characterization

In two-dimensions, a fracture network consist of a system of branches and nodes that can be used to define both geometrical features, such as length and orientation, and the relationship between elements of the network – topology. Branch lengths are preferred to trace lengths as they can be uniquely defined, have less censoring and are more clustered around a mean value. Many important properties of networks are more related to topology than geometry.The proportions of isolated (I), abutting (Y) and crossing (X) nodes provide a basis for describing the topology that can be easily applied, even with limited access to the network as a whole. Node counting also provides an unbiased estimate of frequency and can be used in conjunction with fracture intensity to estimate the characteristic length and dimensionless intensity of the fractures. The nodes can be used to classify branches into three types – those with two I-nodes, one I-node and no I-nodes (or two connected nodes). The average number of connections per branch provides a measure of connectivity that is almost completely independent of the topology. We briefly discuss the extension of topological concepts to 3-dimensions.

Frequency estimation of single tone signals with bit transition

Frequency estimation of a single tone signal is important in many applications. In this study, a frequency estimation algorithm is proposed for single tone signals modulated with unknown data. The proposed unbiased frequency estimator (UFE) uses discrete Fourier transform (DFT) coefficients to calculate the accurate frequency offset. To achieve better performance in a noisy environment, a biased frequency estimator (BFE) is obtained by applying trigonometric approximation to UFE. The Taylor expansion of BFE is also proposed to meet the computational complexity requirement in some real-time applications. The performance of the proposed algorithm is evaluated through Monte Carlo simulations, compared with conventional interpolated DFT-based algorithm and typical non-data-aided frequency estimation algorithm.

A novel phase difference frequency estimator based on forward and backward sub-segmenting

In engineering applications, many measurements of physical quantities can be converted into the problems of frequency estimation. The current frequency estimators are mainly divided into two categories: iterative approaches and direct approaches. However, iterative approaches are not suitable for rapid physical measurement occasions due to its complicated process. But most of the direct approaches are the biased estimators, which are incapable of providing quantitative estimates of variance expression. To enhance the accuracy of the direct frequency estimator and derive the closed-form theoretic expression of the estimated error variance, this paper proposes a novel phase difference frequency estimator based on the forward and backward sub-segmenting. This estimator implements forward and backward fast Fourier transform (FFT) on the given samples separately, and then extracts the phase difference from the peak FFT bins of these two sub-segments to estimate the frequency. And it is emphasized that the proposed method is an unbiased frequency estimator, whose closed-form theoretic expression of the frequency estimated error variance is also derived. Moreover, simulation not only verifies the correctness of this closed-form expression but also proves that the proposed frequency estimator's mean square error is closer to the Cramer-Rao lower bound than that of apFFT/FFT phase difference estimator and Candan estimator. In conclusion, the proposed estimator has higher accuracy in measuring frequencies and has a wide application prospect.

Widely Linear Modeling for Frequency Estimation in Unbalanced Three-Phase Power Systems

Real-time frequency estimation in three-phase power systems is revisited from the state space point of view, in order to provide a unified framework for frequency tracking in both balanced and unbalanced system conditions. This is achieved by using a novel class of widely linear complex valued Kalman filters, which provide unbiased frequency estimation and are faster converging and more robust to noise and harmonic artifacts than the existing methods. It is shown that the Clarke's transformed three-phase voltage is circular for balanced systems and noncircular for unbalanced ones, making the proposed widely linear estimation perfectly suited both to identify the fault and to provide accurate estimation in unbalanced conditions, critical issues where standard models typically fail. The analysis and simulations show that the proposed model outperforms the recently introduced widely linear stochastic gradient-based frequency estimators, based on the augmented complex least mean square. Comprehensive simulations on synthetic and real-world power system data, in both balanced and unbalanced conditions, support the approach.

Widely Linear Adaptive Frequency Estimation of Unbalanced Three-Phase Power Systems

A novel technique for online estimation of the fundamental frequency of unbalanced three-phase power systems is proposed. Based on Clarke's transformation and widely linear complex domain modeling, the proposed method makes use of the full second-order information within three-phase signals, thus promising enhanced and robust frequency estimation. The structure, mathematical formulation, and theoretical stability and statistical performance analysis of the proposed technique illustrate that, in contrast to conventional linear adaptive estimators, the proposed method is well matched to unbalanced system conditions and also provides unbiased frequency estimation. The proposed method is also less sensitive to the variations of the three-phase voltage amplitudes over time and in the presence of higher order harmonics. Simulations on both synthetic and real-world unbalanced power systems support the analysis.

A Complex Adaptive Notch Filter

A complex adaptive notch filter is developed, for tracking single-sided (a.k.a. analytic or complex) tones immersed in background noise. A complex all-pass based realization is pursued which inherits useful properties from its real counterpart: independent tuning of the notch frequency and attenuation bandwidth, easy realization of the complementary band-pass filter, unbiased frequency estimation, and faster convergence and tracking than a gradient descent algorithm.

Closed-form real single-tone frequency estimator based on a normalized IIR notch filter

A novel unbiased frequency estimator for a single real-valued sinusoid in white noise is proposed, which is based on a normalized second-order infinite impulse response (IIR) notch filter, and can be seen as an extension of the Pisarenko harmonic decomposer (PHD) estimator. The new estimator inherits the simplicity of the PHD estimator and the high accuracy and selectivity achieved by an IIR filter. As the coefficient determining the pole angle of the IIR notch filter depends on the unknown frequency value, an iterative scheme is proposed in which this coefficient is computed from the frequency estimate from the previous iteration. An analytic expression for the mean square error at each iteration is derived. Simulation results show that this iterative method can approach the Cramer–Rao lower bound in a few iterations under different signal to noise ratio levels, provided that the data record length is sufficiently large.

Computationally Efficient Joint Frequency Synchronization for OFDM-Based DVB-T

This letter proposes a computationally efficient way of jointly estimating the residual frequency offset (RFO) and sampling frequency offset (SFO) by using a continual pilot (CP) defined in OFDM-based DVB-T system. In order to devise an unbiased joint frequency estimator in the current DVB-T system, a CP subset is selected to offset the effects of RFO and SFO simultaneously.

Pitch estimation of speech signal based on adaptive lattice notch filter

A novel approach to pitch estimation of speech signals using an adaptive lattice notch filter (ALNF) is proposed. Based on a good convergence property and an unbiased frequency estimate with respect to the noise variance, the pitch period is estimated using the ALNF. Simulation results have verified that the ALNF is very effective in estimating the pitch period of the given speech signal under background noise environments.

Reformulation of Pisarenko Harmonic Decomposition Method for Single-Tone Frequency Estimation

Based on the linear prediction (LP) property of sinusoidal signals, a closed-form unbiased frequency estimator for a real sinusoid in white noise is proposed. The frequency estimator, which is derived by minimizing a constrained least squares cost function, can be considered as a reformulation of the well known Pisarenko harmonic decomposer (PHD). Online computation of the frequency estimate can be achieved in a very simple manner, and its variance is derived. Computer simulations are included to corroborate the theoretical development and to contrast the estimator performance with the PHD, maximum likelihood, and LP-based methods as well as Crame/spl acute/r-Rao lower bound.

Closed-form frequency estimation using second-order notch filters

In this paper the problem of the frequency estimation of a sinusoid embedded in white noise is considered. The approach used herein is the minimization of the sample variance of the output of constrained notch filters fed by the noisysinusoid. In particular, this paper focuses on closed-form expressions of the frequency estimate, which can be obtained using notch filters having an all-zeros FIR structure. In this paper it is shown that the FIR notch filters obtained from standard 2nd-order IIR filters are inadequate, and an alternate 2nd-order IIR notch filter is proposed, which provides an unbiased estimate of the frequency. The FIR filter obtained from the new IIR filter provides a closed-form unbiased frequency estimate.

Closed-form unbiased frequency estimation of a noisy sinusoid using notch filters

In this note, the problem of the frequency estimation of a sinusoid embedded in white noise is considered. The approach used herein is the minimization of the sample variance of the output of constrained notch filters fed by the noisy sinusoid. In particular, this note focuses on closed-form expressions of the frequency estimate, which can be obtained using notch filters having an all-zeros finite-impulse response (FIR) structure. The results presented in this note are as follows: 1) it is shown that the FIR notch filters obtained from standard second-order infinite-impulse response (IIR) filters are inadequate; 2) a new second-order IIR notch filter is proposed, which provides an unbiased estimate of the frequency; 3) the FIR filter obtained from the new IIR filter provides a closed-form unbiased frequency estimate; and 4) the closed-form frequency estimate obtained using the new FIR notch filter asymptotically converges toward the Pisarenko harmonic decomposition estimator and the Yule-Walker estimator.

FSK detection using frequency estimator

A receiver for FSK in noise is proposed based on the use of an unbiased minimum-variance frequency estimator. It allows a complete digital implementation, and is robust in that knowledge of the frequency deviation and carrier phase is not required. The BER of the receiver is obtained using computer simulations.

Eine verbesserte Methode zur Frequenzmessung auf der Basis der gewichteten Burg-Technik

The measurement of physical quantities based on estimating of single-sinusoid frequency is the starting point of this paper. The described method is particular interesting for robust estimating of frequency from shot data records. Potential applications of such a frequency estimator include the estimation of time-varying Doppler frequency shift in radar or Laser-Doppler-Anemometer. The starting point is a computationally efficient method based on coefficient-constrained lattice filterstructures (CLF) that proposed by Hu and Phan [1]. This method yields an unbiased frequency estimate independent of the initial phase, but the variance of the frequency estimation depends strongly on signal-to-noise ratio (SNR). The objective of this paper is to devise a modification based on weighted Burg technique to reduce the variance of the frequency estimation. The superiority of this modification will be proved.

The Cramer-Rao bound on frequency estimates of signals closely spaced in frequency (unconditional case)

The paper analyzes the Cramer-Rao (CR) bound on frequency estimation covariance for the unconditional (or stochastic) signal model. It addresses the problem of n signals closely spaced in (temporal or spatial) frequency. The main result is that for this regime, the CR bound decomposes into a product of simple scalar factors that individually reflect frequency separation, signal powers and covariances, data sampling grid, and sample size. The factored expression provides useful insight into the behavior of the bound for closely spaced frequencies. The result also leads to a new formula for the signal-to-noise (SNR) threshold at which an unbiased frequency estimator can resolve signals closely spaced in frequency. Interestingly, with a simple modification, the formulae are identical to those recently obtained for the conditional (or deterministic) signal model. >