Single-tone Frequency Estimation 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).

Single-Tone Frequency Estimation by Weighted Least-Squares Interpolation of Fourier Coefficients

Frequency estimation of a single complex exponential signal embedded in additive white Gaussian noise is a major topic of research in many engineering areas. This work presents further investigations on this problem with regards to the fine estimation task, which is accomplished through a suitable interpolation of the discrete Fourier transform (DFT) coefficients of the observation data. The focus is on fast real-time applications, where iterative estimation methods can hardly be applied due to their latency and complexity. After deriving the analytical expression of the Cramér-Rao bound (CRB) for general values of the system parameters, we present a new DFT interpolation scheme based on the weighted least-squares (WLS) rule, where the optimum weights are precomputed through a numerical search and stored in the receiver. In contrast to many existing alternatives, the proposed method can employ an arbitrary number of DFT samples so as to achieve a good trade-off between system performance and complexity. Simulation results and theoretical analysis indicate that, at sufficiently high signal-to-noise ratios, the estimation accuracy is close to the relevant CRB at any value of the frequency error. This provides some advantage with respect to non-iterative competing schemes, without incurring any penalty in processing requirement.

Single-tone frequency estimation based on reformed covariance for half-length autocorrelation

Single-tone frequency estimation based on reformed covariance for half-length autocorrelation

On statistical analysis of least squares matching methods for frequency estimation of a real sinusoid

A statistical analysis of the least squares matching methods for single-tone frequency estimation is carried out in this paper. Based on asymptotic derivations, an accurate frequency variance expression is provided for different least squares matching methods with variable parameters. Two representative least squares matching methods are involved to show the validity of the variance expression, however, their performances have been proved suboptimal with current parameter settings. After wide discussion, we propose new strategies to choose parameters and achieve further improvements both in accuracy and computational efficiency.

A Novel Four-Point Model Based Unit-Norm Constrained Least Squares Method for Single-Tone Frequency Estimation

A Novel Four-Point Model Based Unit-Norm Constrained Least Squares Method for Single-Tone Frequency Estimation

An Improved Fine-Resolution Method for Frequency Estimation of Real-Valued Single-Tone Using Three DFT Samples

With respect to the frequency estimation of a real-valued single-tone, a direct frequency estimation method is proposed to acquire fine-resolution parameter by only utilizing three discrete Fourier transform (DFT) samples around the peak. Inspired by the Quinn, Rife and Candan methods, a novel Candan combined Quinn Rife estimator (CCQRE) is proposed, and the theoretical analysis of the corresponding MSE is also given. It has been verified that the proposed method is more robust to the interbin frequency and has better performance than these existed state-of-the-art methods. Both simulations and real data experiments demonstrate the effectiveness of the proposed method. In addition, the proposed methods are suitable for real-time applications due to the simplicity, efficiency and low computational burden.

Multidimensional Single-Tone Frequency Estimation Based on QR Decomposition

In this paper, the problem of multidimensional single-tone frequency estimation of sinusoids embedded in white Gaussian noise is investigated. By extracting the two-dimensional (2-D) slice matrices from the multidimensional data, we construct a covariance matrix associated with only one dimension, from which the corresponding frequency is estimated with the utilization of a QR decomposition based iterative method. The frequencies of the remaining dimensions are then obtained following similar procedures. Moreover, the mean square error of the estimated frequencies is devised. The computer simulations are also included to evaluate the performance of the proposed method by comparing with the several state-of-the-art algorithms and Cramer–Rao lower bound.

Real-valued single-tone frequency estimation using half-length autocorrelation

In this paper a new autocorrelation method called half-length autocorrelation (HLA) is introduced. Unlike regular (biased or unbiased) autocorrelations, the exact statistical characteristics of the HLA can be calculated, which makes the analytical calculation of the variance of the frequency estimator possible. The HLA is also used in two different frequency estimation methods namely, Modified Covariance (MC) and Modified Pisarenko Harmonic Decomposition (MPHD). It is shown that the application of the HLA in these frequency estimators improves their performance, particularly in lower signal-to-noise ratios (SNR). The analytical calculation of the variance (mean square error) of the estimators is also provided in this paper. Simulation results are in agreement with the theoretical calculations. For the MPHD estimator using the HLA (MPHD-HLA), the estimated frequency is very close to the theoretical calculations for SNRs as low as −2 dB.

Two-Stage Estimator for Frequency Rate and Initial Frequency in LFM Signal Using Linear Prediction Approach

We propose a two-stage estimator to estimate chirp rate and initial frequency of the chirp signals in the presence of additive white Gaussian noise. In the first stage, the chirp rate estimation problem is reformulated as a single tone frequency estimation problem. Then, the frequency of single tone is estimated through a linear prediction approach. Using the chirp rate estimate in the first stage, we can convert the linear frequency modulated signal to a single tone. Similar to the first stage, the initial frequency is estimated via the linear prediction approach. The performance of the present method is assessed by comparison with Cramer–Rao lower bound and other existing methods through computer simulations. The proposed algorithm estimates well for different values of the chirp rate and initial frequency, as well as for different number of samples. In other words, this algorithm has uniform performance for various values of the signal parameters.

A Closed-Form ARMA-Based ML-Estimator of a Single-Tone Frequency

The new synthesis methodology for single-tone frequency estimation is proposed on the basis of the maximum likelihood (ML) method and autoregressive moving average (ARMA) models. The novel frequency estimator for the scenario with a strongly correlated interference and an additive white Gaussian noise is synthesized in a closed form. The assumption about a constant value of that interference allows to simplify the mixture and transform it to ARMA (3,3) model. The synthesized estimator is invariant to the interference power when it is actually constant (has a unitary correlation coefficient). The particular and much more usual scenario without the interference and just with the noise is considered as well. For that case, the ML-estimator synthesized using the proposed methodology is equal to the existing modified least squares estimator. It is shown that the proposed ML-estimator has an advantage over the known one when the interference is present in the mixture.

Efficient Range Estimation in Beat Radio Interferometry

The beat radio interferometry (BRI) is a novel ranging scheme that achieves high accuracy with low-cost devices. A major problem with BRI is the lack of an efficient range estimation method that is valid in general measurement configurations and accommodates large-scale scenarios. In this paper, we present a new model of BRI and reveal that range estimation in it is equivalent to single tone frequency estimation. Such a revelation has two natural outcomes: the first is the Cramer-Rao lower bound (CRLB) of BRI range estimation; the second is an efficient fast Fourier transform (FFT)-based method that works in arbitrary measurement configurations and large-scale scenarios. Simulation results show that the FFT-based method asymptotically attains the CRLB and is orders of magnitude faster than currently available methods.

An Effective Pre-processing Technique for Robust ESPRIT-Based Single-Tone Frequency Estimation against an I/Q Mismatch

This letter proposes an effective pre-processing technique for robust estimation of signal parameters via rotational invariance techniques (ESPRIT)-based single-tone frequency estimation against an I/Q mismatch. For the implementation of a high-accuracy ranging/radar system, parametric-based algorithms such as MUSIC, ESPRIT and their variations have been adopted due to their superior estimation capability for single-tone frequency estimations. Since most ranging/radar system have the direct conversion architecture, I/Q imbalance problem occurs in the form of spurious frequencies of the frequency spectrum. For this reason, the proposed pre-processing technique is designed to make use of the Hilbert transform structure for received signals. The estimation performance of the proposed pre-processing technique in single-tone frequency is derived and compared with the results from a Monte-Carlo simulation. The root-mean-square error (RMSE) of the proposed method is compared with that of an ESPRIT-based method for various parameters. DOI: http://dx.doi.org/10.5755/j01.eee.21.6.13757

An Effective Pre-processing Technique for Robust ESPRIT-Based Single-Tone Frequency Estimation against an I/Q Mismatch

An Effective Pre-processing Technique for Robust ESPRIT-Based Single-Tone Frequency Estimation against an I/Q Mismatch

Accurate Frequency Estimation Based On Three-Parameter Sine-Fitting With Three FFT Samples

Abstract This paper presents a simple DFT-based golden section searching algorithm (DGSSA) for the single tone frequency estimation. Because of truncation and discreteness in signal samples, Fast Fourier Transform (FFT) and Discrete Fourier Transform (DFT) are inevitable to cause the spectrum leakage and fence effect which lead to a low estimation accuracy. This method can improve the estimation accuracy under conditions of a low signal-to-noise ratio (SNR) and a low resolution. This method firstly uses three FFT samples to determine the frequency searching scope, then – besides the frequency – the estimated values of amplitude, phase and dc component are obtained by minimizing the least square (LS) fitting error of three-parameter sine fitting. By setting reasonable stop conditions or the number of iterations, the accurate frequency estimation can be realized. The accuracy of this method, when applied to observed single-tone sinusoid samples corrupted by white Gaussian noise, is investigated by different methods with respect to the unbiased Cramer-Rao Low Bound (CRLB). The simulation results show that the root mean square error (RMSE) of the frequency estimation curve is consistent with the tendency of CRLB as SNR increases, even in the case of a small number of samples. The average RMSE of the frequency estimation is less than 1.5 times the CRLB with SNR = 20 dB and N = 512.

Experimental and Analytical Studies on Improved Feedforward ML Estimation Based on LS-SVR

Maximum likelihood (ML) algorithm is the most common and effective parameter estimation method. However, when dealing with small sample and low signal-to-noise ratio (SNR), threshold effects are resulted and estimation performance degrades greatly. It is proved that support vector machine (SVM) is suitable for small sample. Consequently, we employ the linear relationship between least squares support vector regression (LS-SVR)’s inputs and outputs and regard LS-SVR process as a time-varying linear filter to increase input SNR of received signals and decrease the threshold value of mean square error (MSE) curve. Furthermore, it is verified that by taking single-tone sinusoidal frequency estimation, for example, and integrating data analysis and experimental validation, if LS-SVR’s parameters are set appropriately, not only can the LS-SVR process ensure the single-tone sinusoid and additive white Gaussian noise (AWGN) channel characteristics of original signals well, but it can also improves the frequency estimation performance. During experimental simulations, LS-SVR process is applied to two common and representative single-tone sinusoidal ML frequency estimation algorithms, the DFT-based frequency-domain periodogram (FDP) and phase-based Kay ones. And the threshold values of their MSE curves are decreased by 0.3 dB and 1.2 dB, respectively, which obviously exhibit the advantage of the proposed algorithm.

Correlation-based algorithm for multi-dimensional single-tone frequency estimation

In this paper, parameter estimation for a R-dimensional (R-D) single cisoid with R⩾2 in additive white Gaussian noise is addressed. By exploiting the correlation of the data samples, we construct R single-tone sequences which contain the R-D frequency parameters. Based on linear prediction and weighted linear squares techniques, two proposals are developed for fast and accurate frequency estimation from each constructed sequence. The two devised estimators are proved to be asymptotically unbiased while their variances achieve Cramér–Rao lower bound when the signal-to-noise ratio and/or data length tend to infinity. Computer simulations are also included to compare the proposed approach with conventional R-D harmonic retrieval schemes in terms of mean square error performance and computational complexity.

Efficient frequency estimation of a single real tone based on principal singular value decomposition

The problem of single-tone frequency estimation for a discrete-time real sinusoid in white Gaussian noise is addressed. We first show that the frequency information is embedded in the principal singular vectors of a matrix which stores the observed data with no repeated entry. The technique of weighted least squares is then utilized for finding the frequency from the singular vectors. It is proved that the variance of the frequency estimate approaches Cramer-Rao lower bound when the data observation length tends to infinity. The computational efficiency and estimation accuracy are demonstrated via computer simulations.

An exact analysis of Modified Covariance frequency estimation algorithm based on correlation of single-tone

The Modified Covariance (MC) algorithm, which exploits the time-domain correlation of sinusoidal signals, has been shown to be a simple but efficient method for the frequency estimation of single-tone. Recently MC method based on correlation sequence was investigated. Both simulation and theoretical analyses show that it is more robust to noise than the conventional MC method that is directly based on the recorded signal [6]. However, the analysis in [6] is only valid when lower-order lags of correlations are applied. In this communication we provide new analytical result to tackle this difficult.

Closed-Form Real Single-Tone Frequency Estimator Based on Phase Compensation of Multiple Correlation Lags

A new frequency estimator for a single real-valued sinusoid signal in white noise is proposed. The new estimator uses the Pisarenko Harmonic Decomposer (PHD) estimator to get a coarse frequency estimate and then makes use of multiple correlation lags to obtain an adjustment term. For the limited-length single sinusoid, its correlation has the same frequency as itself but with a non-zero phase. We propose to use Taylor series to expand the correlation at the PHD coarse estimated frequency with amplitude and phase of the correlation into consideration. Simulation results show that this new method improves the estimation performance of the PHD estimator. Moreover, when compared with other existing estimator, the mean square frequency error of the proposed method is closer to the Cramer-Rao Lower Bound (CRLB) for certain SNR range.

A closed-form expanded autocorrelation method for frequency estimation of a sinusoid

A closed-form expanded autocorrelation method for real single-tone frequency estimation is proposed. Firstly, the modified covariance (MC) method based on multiple autocorrelation lags is applied to provide a coarse frequency estimate. Then, a closed-form adjustment term based on a least square cost function is derived to get the fine frequency estimate. Simulation results show that the performance of the proposed algorithm, when compared with several existing closed-form time-domain estimators, is closer to the Cramer–Rao bound (CRB). Moreover, the proposed method has lower computation complexity than other autocorrelation-based approaches, which also use multiple autocorrelation lags.