Enhanced Wideband Frequency Estimation via FFT: Leveraging Polynomial Interpolation and Array Indexing

Accurate frequency estimation in a wideband digital receiver using the FFT algorithm encounters challenges, such as spectral leakage resulting from the FFT’s assumption of signal periodicity. High-resolution FFTs pose computational demands, and estimating non-integer multiples of frequency resolution proves exceptionally challenging. This paper introduces two novel methods for enhanced frequency precision: polynomial interpolation and array indexing, comparing their results with super-resolution and scalloping loss. Simulation results demonstrate the effectiveness of the proposed methods in contemporary radar systems, with array indexing providing the best frequency estimation despite utilizing maximum hardware resources. The paper demonstrates a trade-off between accurate frequency estimation and hardware resources when comparing polynomial interpolation and array indexing.

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Here the estimating problem of a single sinusoidal signal in the additive symmetricα-stable Gaussian(ASαSG)noise is investigated.The ASαSG noise here is expressed as the additive of a Gaussian noise and a symmetricα-stable distributed variable.As the probability density function(PDF)of the ASαSG is complicated,traditional estimators cannot provide optimum estimates.Based on the Metropolis-Hastings(M-H)sampling scheme,a robust frequency estimator is proposed for ASαSG noise.Moreover,to accelerate the convergence rate of the developed algorithm,a new criterion of reconstructing the proposal covar-iance is derived,whose main idea is updating the proposal variance using several previous samples drawn in each iteration.The approximation PDF of the ASαSG noise,which is referred to the weighted sum of a Voigt function and a Gaussian PDF,is also employed to reduce the computational complexity.The computer simulations show that the performance of our method is better than the maximum likelihood and the lp-norm estimators.

Architecture design of GPS software receiver and implementation of its acquisition algorithm with fine frequency estimation

The design of a global positioning system （GPS） software receiver is introduced. This design uses the concept of software radio, and it consists of the following parts： front-end, acquisition, tracking, synchronization, navigation solution and some assisting modules. In the acquisition module, the acquisition algorithm based on circular correlation is utilized. The input data and the local code are converted into the frequency domain by means of the fast Fourier transform （FFT）. After performing circular correlation, the initial phase of the C/A code can be obtained and the cartier frequency can be found in 1 kHz frequency resolution, which is too coarse to use for the tracking loop. In order to improve the frequency resolution, the fine frequency estimation through a phase relationship is then achieved, by which, the frequency resolution is improved dramatically. Experiments show that the inaccuracy of the carrier frequency can be estimated within a few hertz by the fine frequency estimation method, and the fine frequency attained can be directly used for the tracking loop.

Application of frequency estimation algorithm based on a single vector sensor in underwater acoustic communication

A method of high resolution frequency estimation based on a single vector sensor using ESPRIT （Estimating Signal Parameters via Rotational Invariance Techniques） algorithm is proposed and applied to the underwater acoustic （UWA） communication system of frequency modulation. Higher resolution frequency estimation can be obtained by this algorithm using fewer snapshots comparing with the sound intensity frequency estimation. Results of simulation and lake experiment show that the proposed algorithm can improve the communication data rate and reduce the bandwidth of the system. Because higher signal-to-noise ratio （SNR） is demanded, range UWA communication at oresent. this algorithm can be used in high speed short

An algorithm for frequency estimation of signals composed of multiple single-tones

The high-accuracy, wide-range frequency estimation algorithm for multi-component signals presented in this paper, is based on a numerical differentiation and central Lagrange interpolation. With the sample sequences, which need at most 7 points and are sampled at a sample frequency of 25600 Hz, and computation sequences, using employed a formulation proposed in this paper, the frequencies of each component of the signal are all estimated at an accuracy of 0.001% over 1 Hz to 800 kHz with the amplitudes of each component of the signal varying from 1 V to 200 V and the phase angle of each component of the signal varying from 0° to 360°. The proposed algorithm needs at most a half cycle for the frequencies of each component of the signal under noisy or non-noisy conditions. A testing example is given to illustrate the proposed algorithm in Matlab environment.

Instantaneous frequency estimation of multi-component Chirp signals in noisy environments

A classical time-varying signal, the multi-component Chirp signal has been widely used and the ability to estimate its instantaneous frequency （IF） is very useful. But in noisy environments, it is hard to estimate the 1F of a multi-component Chirp signal accurately. Wigner distribution maxima （WDM） are usually utilized for this estimation. But in practice, estimation bias increases when some points deviate from the true IF in high noise environments. This paper presents a new method of multi-component Chirp signal 1F estimation named Wigner Viterbi fit （WVF）, based on Wigner-Ville distribution （WVD） and the Viterbi algorithm. First, we transform the WVD of the Chirp signal into digital image, and apply the Viterbi algorithm to separate the components and estimate their IF. At last, we establish a linear model to fit the estimation results. Theoretical analysis and simulation results prove that this new method has high precision and better performance than WDM in high noise environments, and better suppression of interference and the edge effect. Compared with WDM, WVF can reduce the mean square error （MSE） by 50% when the signal to noise ration （SNR） is in the range of-15dB to -11dB. WVF is an effective and promising 1F estimation method.

Robust Frequency Estimation Under Additive Mixture Noise

In many applications such as multiuser radar communications and astrophysical imaging processing,the encountered noise is usually described by the finite sum ofα-stable(1≤α<2)variables.In this paper,a new parameter estimator is developed,in the presence of this new heavy-tailed noise.Since the closed-formPDF of theα-stable variable does not exist exceptα=1 andα=2,we take the sum of the Cauchy(α=1)and Gaussian(α=2)noise as an example,namely,additive Cauchy-Gaussian(ACG)noise.The probability density function(PDF)of the mixed random variable,can be calculated by the convolution of the Cauchy’s PDF and Gaussian’s PDF.Because of the complicated integral in the PDF expression of the ACG noise,traditional estimators,e.g.,maximum likelihood,are analytically not tractable.To obtain the optimal estimates,a new robust frequency estimator is devised by employing the Metropolis-Hastings(M-H)algorithm.Meanwhile,to guarantee the fast convergence of the M-H chain,a new proposal covariance criterion is also devised,where the batch of previous samples are utilized to iteratively update the proposal covariance in each sampling process.Computer simulations are carried out to indicate the superiority of the developed scheme,when compared with several conventional estimators and the Cramér-Rao lower bound.

STUDY ON FREQUENCY ESTIMATION BASED ON WEIGHTED LEAST SQUARE METHOD WITH THREE FOURIER COEFFICIENTS

In this paper,a sinusoidal signal frequency estimation algorithm is proposed by weighted least square method.Based on the idea of Provencher,three biggest Fourier coefficients in the maximum periodogram are considered,the Fourier coefficients can be written as three equations about the amplitude,phase,and frequency,and the frequency is estimated by solving equations.Because of the error of measurement,weighted least square method is used to solve the frequency equation and get the signal frequency.It is shown that the proposed estimator can approach the Cramer-Rao Bound(CRB)with a low Signal-to-Noise Ratio(SNR)threshold and has a higher accuracy.

Polynomial Root Finding on Frequency Estimation with Sub-Nyquist Temporal Sampling

This paper addresses an algebraic approach for wideband frequency estimation with sub-Nyquist temporal sampling. Firstly, an algorithm based on double polynomial root finding procedure to estimate aliasing frequencies and joint aliasing frequencies-time delay phases in multi-signal situation is presentcd. Since the sum of time delay phases determined from the least squares estimation shows the characteristics of the corre- sponding parameters pairs, then the pairmatching method is conducted by combining it with estimated parameters mentioned above. Although the proposed method is computationally simpler than the conventional schemes, simulation results show that it can approach optimum estimation performance.

l_(1)-norm Based GWLP for Robust Frequency Estimation

In this work,we address the frequency estimation problem of a complex single-tone embedded in the heavy-tailed noise.With the use of the linear prediction(LP)property and l_(1)-norm minimization,a robust frequency estimator is developed.Since the proposed method employs the weighted l_(1)-norm on the LP errors,it can be regarded as an extension of the l_(1)-generalized weighted linear predictor.Computer simulations are conducted in the environment of α-stable noise,indicating the superiority of the proposed algorithm,in terms of its robust to outliers and nearly optimal estimation performance.

Improving the utility of locally differentially private protocols for longitudinal and multidimensional frequency estimates

This paper investigates the problem of collecting multidimensional data throughout time(i.e.,longitudinal studies)for the fundamental task of frequency estimation under Local Differential Privacy(LDP)guarantees.Contrary to frequency estimation of a single attribute,the multidimensional aspect demands particular attention to the privacy budget.Besides,when collecting user statistics longitudinally,privacy progressively degrades.Indeed,the“multiple”settings in combination(i.e.,many attributes and several collections throughout time)impose several challenges,for which this paper proposes the first solution for frequency estimates under LDP.To tackle these issues,we extend the analysis of three state-of-the-art LDP protocols(Generalized Randomized Response–GRR,Optimized Unary Encoding–OUE,and Symmetric Unary Encoding–SUE)for both longitudinal and multidimensional data collections.While the known literature uses OUE and SUE for two rounds of sanitization(a.k.a.memoization),i.e.,L-OUE and L-SUE,respectively,we analytically and experimentally show that starting with OUE and then with SUE provides higher data utility(i.e.,L-OSUE).Also,for attributes with small domain sizes,we propose Longitudinal GRR(L-GRR),which provides higher utility than the other protocols based on unary encoding.Last,we also propose a new solution named Adaptive LDP for LOngitudinal and Multidimensional FREquency Estimates(ALLOMFREE),which randomly samples a single attribute to be sent with the whole privacy budget and adaptively selects the optimal protocol,i.e.,either L-GRR or L-OSUE.As shown in the results,ALLOMFREE consistently and considerably outperforms the state-of-the-art L-SUE and L-OUE protocols in the quality of the frequency estimates.

Polynomial rooting based frequency offset estimation for MIMO OFDM systems

Based on the frequency domain training sequences, the polynomial-based carrier frequency offset （CFO） estimation in multiple-input multiple-output （ MIMO ） orthogonal frequency division multiplexing （ OFDM ） systems is extensively investigated. By designing the training sequences to meet certain conditions and exploiting the Hermitian and real symmetric properties of the corresponding matrices, it is found that the roots of the polynomials corresponding to the cost functions are pairwise and that both meger CFO and fractional CFO can be estimated by the direct polynomial rooting approach. By analyzing the polynomials corresponding to the cost functions and their derivatives, it is shown that they have a common polynomial factor and the former can be expressed in a quadratic form of the common polynomial factor. Analytical results further reveal that the derivative polynomial rooting approach is equivalent to the direct one in estimation at the same signal-to-noise ratio（SNR） value and that the latter is superior to the former in complexity. Simulation results agree well with analytical results.

Sarve:synthetic data and local differential privacy for private frequency estimation

The collection of user attributes by service providers is a double-edged sword.They are instrumental in driving statistical analysis to train more accurate predictive models like recommenders.The analysis of the collected user data includes frequency estimation for categorical attributes.Nonetheless,the users deserve privacy guarantees against inadvertent identity disclosures.Therefore algorithms called frequency oracles were developed to randomize or perturb user attributes and estimate the frequencies of their values.We propose Sarve,a frequency oracle that used Randomized Aggregatable Privacy-Preserving Ordinal Response(RAPPOR)and Hadamard Response(HR)for randomization in combination with fake data.The design of a service-oriented architecture must consider two types of complexities,namely computational and communication.The functions of such systems aim to minimize the two complexities and therefore,the choice of privacy-enhancing methods must be a calculated decision.The variant of RAPPOR we had used was realized through bloom flters.A bloom filter is a memory-efficient data structure that offers time complexity of O(1).On the other hand,HR has been proven to give the best communication costs of the order of log(b)for b-bits communication.Therefore,Sarve is a step towards frequency oracles that exhibit how privacy provisions of existing methods can be combined with those of fake data to achieve statistical results comparable to the original data.Sarve also implemented an adaptive solution enhanced from the work of Arcolezi et al.The use of RAPPOR was found to provide better privacy-utility tradeoffs for specific privacy budgets in both high and general privacyregimes.

Further result in the fast and accurate estimation of single frequency

A new fast and accurate method for estimating the frequency of a complex sinusoid in complex white Gaussian environments is proposed. The new estimator comprises of applications of low-pass filtering, decimation, and frequency estimation by linear prediction. It is computationally efficient yet obtains the Crazner-Rao bound at moderate signal-to-noise ratios. And it is well suited for real time applications requiring precise frequency estimation. Simulation results are included to demonstrate the performance of the proposed method.

Clock frequency estimation under spontaneous emission

We investigate the quantum dynamics of a driven two-level system under spontaneous emission and its application in clock frequency estimation. By using the Lindblad equation to describe the system, we analytically obtain its exact solutions, which show three different regimes： Rabi oscillation, damped oscillation, and overdamped decay. From the analytical solutions, we explore how the spontaneous emission affects the clock frequency estimation. We find that under a moderate spontaneous emission rate, the transition frequency can still be inferred from the Rabi oscillation. Our results enable potential practical applications in frequency measurement and quantum control under decoherence.

Estimating mean trabecular bone spacing based on the combination of Hilbert transform and fundamental frequency estimation

Ultrasonic backscatter signals from cancellous bone are sensitive to the microstructure of trabecular bone,and thus enable the feasibility to extract microstructural information of trabecular bone.The mean trabecular bone spacing(MTBS)is an important parameter for characterizing bone microstructure.This paper proposes an MTBS estimation method based on the combination of Hilbert transform and fundamental frequency estimation(CHF). The CHF was verified with ultrasonic backscatter signals from simulations and in vitro measurements at a central frequency of 5MHz.The CHF method was compared with the simplified inverse filter tracking(SIFT)method,Simons' Quadratic Transformation(QT)method,Singular Spectrum Analysis(SSA)method,and Spectral Autocorrelation(SAC)method.Monte-Carlo simulations were performed by varying the MTBS,signal-to-noise ratio(SNR),standard deviation of regular spacing(SDRS),amplitude ratio of diffuse scattering to regular scattering(Ad)and frequency dependent attenuation(nBUA).The simulation results showed that the CHF method had a better performance in MTBS estimation under almost all the examination conditions except for SNR.The estimation percentage correct(EPC)was greater than 90% when the MTBS was in the range of 0.4to 1.4mm.In the in vitro measurements,the estimated EPC by the CHF method was91.25±7.81%(mean±standard deviation).A significant correlation was observed for the CHF-estimated MTBS and micro-computed tomography(μ-CT)-measured values(R^2=0.75,p<0.01).These results demonstrate that the CHF method is anti-interference for MTBS estimation and can be used to estimate trabecular bone spacing.

The frequency estimation of harmonic signals embedded in multiplicative and additive noise

A method to separate a harmonic signal from multiplicative and additive noises is proposed. The method is to square the signal x(t), which consists of a harmonic signal embedded in multiplicative and additive noises, to form another signal y(t) = x2(t)-E[x2(t)]. After y(t) having been gotten, the Fourier transform is imposed on it. Because the information of x(t) (especially about frequency) is included in y(t), the frequency of x(t) can be estimated from the power spectrum of y(t). According to the simulation, under the condition where frequencies divided by resolution dω are integer, the maximum relative error of estimated frequencies is less than 0.4% when the signal-to-noise ratio (SNR) is greater than -23 dB. If frequencies divided by resolution dω are not integer, the maximum relative error will be less than 2.9%. But it is still small in terms of engineering.

ESTIMATION OF CARRIER FREQUENCY OFFSETS FOR MIMO SYSTEMS WITH DISTRIBUTED TRANSMIT ANTENNAS

The problem of estimating the carrier frequency offsets in Multiple-Input Multiple-Output (MIMO) systems with distributed transmit antennas is addressed. It is supposed that the transmit antennas are distributed while the receive antennas are still centralized, and the general case where both the time delays and the frequency offsets are possibly different for each transmit antenna is considered. The channel is supposed to be frequency flat, and the macroscopic fading is also taken into consideration. A carrier frequency offset estimator based on Maximum Likelihood (ML) is proposed, which can separately estimate the frequency offset for each transmit antenna and exploit the spatial diversity. The Cramer-Rao Bound (CRB) for synchronous MIMO (i.e., the time delays for each transmit antenna are all equal) is also derived. Simulation results are given to illustrate the per- formance of the estimator and compare it with the CRB. It is shown that the estimator can provide satisfactory frequency offset estimates and its performance is close to the CRB for the Signal-to-Noise Ratio (SNR) below 20dB.

Two-step compressed acquisition method for Doppler frequency and Doppler rate estimation in high-dynamic and weak signal environments

To acquire global navigation satellite system(GNSS)signals means four-dimension acquisition of bit transition,Doppler frequency,Doppler rate,and code phase in high-dynamic and weak signal environments,which needs a high computational cost.To reduce the computations,this paper proposes a twostep compressed acquisition method(TCAM)for the post-correlation signal parameters estimation.Compared with the fast Fourier transform(FFT)based methods,TCAM uses fewer frequency search points.In this way,the proposed method reduces complex multiplications,and uses real multiplications instead of improving the accuracy of the Doppler frequency and the Doppler rate.Furthermore,the differential process between two adjacent milliseconds is used for avoiding the impact of bit transition and the Doppler frequency on the integration peak.The results demonstrate that due to the reduction of complex multiplications,the computational cost of TCAM is lower than that of the FFT based method under the same signal to noise ratio(SNR).

Estimation of DOA and Doppler Frequency on Nonideal UCA

The problem of estimating direction of arrivals (DOA) and Doppler frequency for many sources is considered in the presence of general array errors (such as amplitude and phase error of sensors, setting position error of sensors). Adopting direct array manifold in a uniform circular array (UCA), the estimation of Doppler frequency can be obtained by DOA matrix. Based on analyzing the statistic characters of general array errors, the estimation of DOA can be obtained by Weight Total Least Squares. Numerical results illustrate that the estimator is robust to general array errors and show the capabilities of the estimator.