ESPRIT-like estimation of real-valued sinusoidal frequencies

Subspace-based estimation of multiple real-valued sine wave frequencies is considered in this paper. A novel data covariance model is proposed. In the proposed model, the dimension of the signal subspace equals the number of frequencies present in the data, which is half of the signal subspace dimension for the conventional model. Consequently, an ESPRIT-like algorithm using the proposed data model is presented. The proposed algorithm is then extended for the case of complex-valued sine waves. Performance analysis of the proposed algorithms are also carried out. The algorithms are tested in numerical simulations. When compared with ESPRIT, the newly proposed algorithm results in a significant reduction in computational burden without any compromise in the accuracy.     

Comparative performance study of SVD-based and QRD-based high-order Yule-Walker methods for frequency estimation

The high-order Yule-Walker (HOYW) method of sinusoidal frequency estimation based on a singular value decomposition (SVD) is known to have excellent statistical performance. Here, we show that the SVD-based step of the HOYW method can be replaced by a computationally more convenient QR decomposition (QRD)-based step, without affecting the asymptotic properties of the frequency estimates.     

Asymptotic bias of the high-order autoregressive estimates of sinusoidal frequencies

Fitting a high-order autoregressive model to data is often used to estimate the frequencies of sinusoids from their noisy measurements. This paper analyzes the asymptotic bias of the frequency estimates obtained by this method. It is shown that the bias decreases significantly with increasing model order.The work of B. Friedlander was supported by the Office of Naval Research under contract No. N000014-84-C-0408.     

State-space and Yule-Walker approaches to pole estimation

The pole estimates provided by the state-space and Yule-Walker approaches are shown to coincide in the cases where the Hankel matrix of sample covariances used by both methods has full column rank.<>     

Damped sinusoidal signals parameter estimation in frequency domain

Parameter estimation of noisy damped sinusoidal signals in the frequency domain is presented in this paper. The advantage of the frequency domain approach is having the spectral energy concentrated in frequency domain samples. However, the least squares criterion for frequency estimation using frequency domain samples is nonlinear. A low complexity three-sample estimation algorithm (TSEA) for solving the nonlinear problem is proposed. Using the TSEA for initialization, a frequency domain nonlinear least squares (FD-NLS) estimation algorithm is then proposed. In the case of white Gaussian noise, it yields maximum likelihood estimates, verified by simulation results. A time domain NLS (TD-NLS) estimation algorithm is also proposed for comparison.The Cramer-Rao lower bound (CRLB) of the frequency domain estimation algorithms is derived. The theoretical analysis shows that the FD-NLS can yield a near-optimal performance with few energy-concentrated samples. On the other hand, the TD-NLS does not have the energy concentration property and requires more time domain samples to perform satisfactory estimation. Simulation results verify that the frequency domain estimation algorithms provide better tradeoff between computational complexity and estimation accuracy than time domain algorithms.     

多正弦窗谱估计的性能分析

多正弦窗谱估计是一种非参数的现代谱估计方法。本文较为全面地分析了它的性能,包括谱的聚集性,偏差,方差,最优窗口数,分辨率,计算复杂度等。以四种信号模型为例,分析了窗口数和小波阈值法去除估计噪声对多正弦窗谱估计的偏差和方差的影响,并与周期谱图进行了比较。理论分析和实验结果表明与周期谱图相比,多正弦窗谱具有小得多的偏差和方差,可将其应用于谱估计的性能要求较高的场合。     

Optimal estimation of multicast membership

This paper addresses optimal online estimation of the size of a multicast group. Three distinct approaches are used. The first one builds on Kalman filter theory to derive the MSE-optimal estimator in a heavy-traffic regime. Under more general assumptions, the second approach uses Wiener filter theory to compute the MSE-optimal linear filter. The third approach develops the best first-order linear filter from which an estimator that holds for any on-time distribution is derived. Our estimators are tested on real video traces and exhibit good performance. The paper also provides guidelines on how to tune the parameters involved in the schemes in order to achieve high-quality estimation while simultaneously avoiding feedback implosion.     

Asymptotic zero distribution of orthogonal polynomials in sinusoidal frequency estimation

It is shown that the zeros of polynomials orthogonal with respect to certain positive measures on the unit circle have an asymptotic distribution which is uniform on a circle of radius less than or equal to one. These results explain some phenomena observed when various linear prediction methods are used to estimate sinusoidal frequencies. They also describe the asymptotic zero distribution of the prediction error filter polynomials for a class of time series including autoregressive moving average (ARMA) models.     

A two-stage autocorrelation method for frequency estimation of sinusoidal signal

A two-stage autocorrelation method is proposed for frequency estimation of a complex sinusoidal signal in complex additive white Gaussian noise. In the first stage, a function about signal frequency is constructed via an autocorrelation procedure. By the Least-Square (LS) principle, the signal frequency is estimated from the function. In the second stage, the estimated error of signal frequency in the first stage is defined. Applying the Taylor series, a function about the estimated error of signal frequency is conducted from the autocorrelation function. Applying the LS estimation once more, the estimate of estimated error of frequency in the first stage is obtained. Then the fine estimate of signal frequency is obtained. Simulation results show that the performance of proposed method can approach the Cramer–Rao lower bound in the whole frequency range.     

Adaptive frequency estimation of sinusoidal signals in colored non-Gaussian noises

This paper studies the problem of sinusoidal frequency estimation in colored non-Gaussian ARMA noises. A new adaptive approach is proposed by using the second-and third-order statistics of the measurements. Because of the simultaneous establishment of the signal and noise models, the new approach is applicable for tracking the frequencies at each time instant for stationary and nonstationary signal and/or noise cases. The effectiveness of the new approach is demonstrated by extensive computer simulations. As expected, the approach proposed in this paper outperforms the correlation-based approaches in suppressing the effects of the colored non-Gaussian ARMA noises.     

Optimal reduced-rank estimation and filtering

This paper provides a unified view of, and a further insight into, a class of optimal reduced-rank estimators and filters. An alternating power (AP) method for computing the optimal reduced-rank estimators and filters is derived and analyzed. The AP method is a generalization of the conventional power method for subspace computation, which is shown to be globally and exponentially convergent under weak conditions. When the rank reduction is relatively large, the AP method is computationally more efficient than the conventional methods. The AP method is useful for adaptive computation of the canonical components of a desired reduced-rank estimate, which in turn facilitates the detection of a time-varying rank. The study shown in this paper is particularly useful for applications that involve a large number of sources and a large number of receivers, where rank reduction is either inherent in the multivariate system or required to reduce the model complexity and/or the computational load     

Analytical formula for three points sinusoidal signals amplitude estimation errors

In this note, we show that the amplitude estimation of sinusoidal signals proposed in Wu and Hong [Wu, S.T., and Hong, J.L. (2010), ‘Five-point Amplitude Estimation of Sinusoidal Signals: With Application to LVDT Signal Conditioning’, IEEE Transactions on Instrumentation and Measurement, 59, 623–630] is a particular case of Vizireanu and Halunga [Vizireanu, D.N, and Halunga, S.V. (2011), ‘Single Sine Wave Parameters Estimation Method Based on Four Equally Spaced Samples’, International Journal of Electronics, 98(7), pp. 941–948]. An analytical formula for amplitude estimation errors as effects of sampling period deviation is obtained.     

On estimation of multiple frequencies in undersampled complexvalued waveforms

In this correspondence, the determination of multiple frequencies in undersampled waveforms is studied using multiple smaller size discrete Fourier transforms (DFT). Given the sizes of multiple DFT, a range for the detectable frequencies in undersampled waveforms is presented     

Statistical estimation of access frequencies in data broadcasting environments

In a data publishing environment, the server periodically broadcasts data to users based on a broadcast program. The program is constructed using knowledge of access frequencies, which is assumed to be available and accurate, on the broadcast data. For example, the program may broadcast frequently accessed data more often in a broadcast cycle. However, it remains an open question as to how to obtain such access frequencies. The difficulty of obtaining such access frequencies is that in such an environment, mobile users are only listening to the channel they are interested in and do not request for the data items from the server. A promising approach in the literature is to make use of broadcast misses to understand the access patterns in a data publishing environment. In this case, mobile users may decide whether to wait for the required item to arrive or to make an explicit request for it even though it will be published. However, estimation of access frequencies based on broadcast misses may not be accurate because the number of broadcast misses to the data depends on how frequently the data is broadcast: if a piece of data is more frequently broadcast than the others, then the broadcast misses to that piece of data will be low because the average waiting time is low. In this paper, we propose a statistical estimation model that is based on maximum likelihood estimation to estimate the access frequencies. Our approach is novel in that it exploits knowledge that is available – broadcast misses and broadcast frequencies – to refine the program to better meet the needs of the user population. We report our simulation study that demonstrates the effectiveness of our approach. This revised version was published online in August 2006 with corrections to the Cover Date.     

Linear prediction based adaptive algorithm for a complex sinusoidal frequency estimation

The complex direct frequency estimation (CDFE) adaptive algorithm is developed and proposed in this paper. The motivation of this work is obtained from the previous real DFE (RDFE) adaptive algorithm. The methodology of the CDFE is based on the linear prediction property of complex sinusoidal signals. The proposed algorithm is unbiased and computationally efficient. Moreover, it is easy to implement and appropriate for real-time applications. In addition, the convergence behavior is analyzed and the steady-state mean square error (MSE) of the frequency estimate is derived in closed form. Computer simulations are treated to corroborate the theoretical analysis.     

Optimal recursive estimation with uncertain observation

In classical estimation theory, the observation is always assumed to contain the signal to be estimated. In practice, certain observations, or sequences of observations, may contain noise alone, only the probability of occurrence of such cases being available to the estimator. An example is trajectory tracking where the signal is first detected and then the estimator is allowed to process it for tracking purposes. However, any detection decision is associated with a false-alarm probability, which is the probability that the detected signal contains only noise. Minimum mean-square estimators are derived for two different forms of this problem; 1) when it is possible that the observation at any sample time contains signal or is noise alone, independent of the situation at any other sample, and 2) when the entire sequence of observations contains signal or is only noise. The estimators derived are of recursive form. A simple example is given for illustration.     

Optimal kernels for nonstationary spectral estimation

Current theories of a time-varying spectrum of a nonstationary process all involve, either by definition or by difficulties in estimation, an assumption that the signal statistics vary slowly over time. This restrictive quasistationarity assumption limits the use of existing estimation techniques to a small class of nonstationary processes. We overcome this limitation by deriving a statistically optimal kernel, within Cohen's (1989) class of time-frequency representations (TFR's), for estimating the Wigner-Ville spectrum of a nonstationary process. We also solve the related problem of minimum mean-squared error estimation of an arbitrary bilinear TFR of a realization of a process from a correlated observation. Both optimal time-frequency invariant and time-frequency varying kernels are derived. It is shown that in the presence of any additive independent noise, optimal performance requires a nontrivial kernel and that optimal estimation may require smoothing filters that are very different from those based on a quasistationarity assumption. Examples confirm that the optimal estimators often yield tremendous improvements in performance over existing methods. In particular, the ability of the optimal kernel to suppress interference is quite remarkable, thus making the proposed framework potentially useful for interference suppression via time-frequency filtering     

Optimal detection and estimation of straight patterns

This correspondence illustrates the optimal detector and slope estimator of straight patterns. In particular, it is recognized that the output of the likelihood processor, constituted by the Radon transform and a whitening filter, provides a sufficient statistic for both problems of signal detection as well as orientation and offset estimation.     

Optimal data-based kernel estimation of evolutionary spectra

Complex demodulation of evolutionary spectra is formulated as a two-dimensional kernel smoother in the time-frequency domain. First, a tapered Fourier transform, y_v(f, t), is calculated. Then the log-spectral estimate, is smoothed. As the characteristic widths of the kernel smoother increase, the bias from the temporal and frequency averaging increases while the variance decreases. The demodulation parameters, such as the order, length, and bandwidth of spectral taper and the kernel smoother, are determined by minimizing the expected error. For well-resolved evolutionary, spectra, the optimal taper length is a small fraction of the optimal kernel halfwidth. The optimal frequency bandwidth, w, for the spectral window scales as w²~λ/τ, where τ is the characteristic time and λ_F is the characteristic frequency scalelength. In contrast, the optimal halfwidths for the second stage kernel smoother scales as h~1/(τλ_F )¹(p+2)/ where p is the order of the kernel smoother. The ratio of the optimal-frequency halfwidth to the optimal-time halfwidth is determined