We present a decoupled parameter estimation (DPE) algorithm for estimating sinusoidal parameters from both 1-D and 2-D data sequences corrupted by autoregressive (AR) noise. In the first step of the DPE algorithm, we use a relaxation (RELAX) algorithm that requires simple fast Fourier transforms (FFTs) to obtain the estimates of the sinusoidal parameters. We describe how the RELAX algorithm may be used to extract radar target features from both 1-D and 2-D data sequences. In the second step of the DPE algorithm, a linear least-squares approach is used to estimate the AR noise parameters. The DPE algorithm is both conceptually and computationally simple. The algorithm not only provides excellent estimation performance under the model assumptions, in which case the estimates obtained with the DPE algorithm are asymptotically statistically efficient, but is also robust to mismodeling errors.

Efficient mixed-spectrum estimation with applications to target feature extraction

A frequently encountered problem in signal processing is that of estimating the frequencies and amplitudes of harmonics observed in additive colored Gaussian noise. In practice, the observed signals are contaminated with spatially and temporally colored noise of unknown power spectral density. A cumulant-based approach to these problems is proposed. The cumulants of complex processes are defined, and it is shown that specific 1-D slices of the fourth-order cumulant of the noisy signal for the direction of arrival (DOA) and retrieval of harmonics in noise (RHN) problems are identical to the autocorrelation of a related noiseless signal. Hence correlation-based high-resolution methods may be used with fourth-order cumulants as well. The effectiveness of the proposed methods is demonstrated through standard simulation examples. >

Cumulant-based approach to harmonic retrieval and related problems

The problem of estimating the parameters of complex-valued sinusoidal signals (cisoids, for short) from data corrupted by colored noise occurs in many signal processing applications. We present a simple formula for the asymptotic (large-sample) Cramer-Rao bound (CRB) matrix associated with this problem. The maximum likelihood method (MLM), which estimates both the signal and noise parameters, attains the performance corresponding to the asymptotic CRB, as the sample length increases. More interestingly, we show that a computationally much simpler nonlinear least-squares method (NLSM), which estimates the signal parameters only, achieves the same performance in large samples.

Cisoid parameter estimation in the colored noise case: asymptotic Cramer-Rao bound, maximum likelihood, and nonlinear least-squares

We present the use of continuous-time autoregressive moving average (CARMA) models as a method for estimating the variability features of a light curve, and in particular its power spectral density (PSD). CARMA models fully account for irregular sampling and measurement errors, making them valuable for quantifying variability, forecasting and interpolating light curves, and for variability-based classification. We show that the PSD of a CARMA model can be expressed as a sum of Lorentzian functions, which makes them extremely flexible and able to model a broad range of PSDs. We present the likelihood function for light curves sampled from CARMA processes, placing them on a statistically rigorous foundation, and we present a Bayesian method to infer the probability distribution of the PSD given the measured lightcurve. Because calculation of the likelihood function scales linearly with the number of data points, CARMA modeling scales to current and future massive time-domain data sets. We conclude by applying our CARMA modeling approach to light curves for an X-ray binary, two AGN, a long-period variable star, and an RR-Lyrae star, in order to illustrate their use, applicability, and interpretation.

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Flexible and Scalable Methods for Quantifying Stochastic Variability in the Era of Massive Time-Domain Astronomical Data Sets

A new approach for the estimation of the parameters of exponentially damped sinusoids is introduced based on third- or fourth-order statistics of the observation signal. The method may be seen as an extension of the minimum norm principal eigenvectors method to higher order statistics domains. The strong points and limitations of the method are discussed as well as sufficient conditions for the existence of the solution. The utilization of the method in the case of finite length signals in the presence of additive Gaussian noise (white or colored) is addressed. Monte Carlo simulations demonstrate the effectiveness of the new method when the additive noise is colored Gaussian with unknown autocorrelation sequence for different signal-to-noise ratios and a single data record. The case of an ensemble of data records is studied when the exponentially damped sinusoids are assumed to have random phase. >

Parameter estimation of exponentially damped sinusoids using higher order statistics

This paper considers the estimation of the two close dominant frequencies in the signal when it is known a priori that the observation is a sum of two close sinusoids and an additive colored noise whose spectral density is unknown. Earlier attempts have assumed the additive noise to be independent. Next we develop decision rules for checking whether the observed signal has only one sinusoid or two close sinusoids. All of the earlier studies assumed that there are two close sinusoids in the signal. Another model discrimination problem considered is the determination of the causal structure of the observed periodicity. A rule is given to test whether the observation comes from a sinusoid plus additive noise, possible colored, or the observation comes from a stationary autoregressive model. We also present a numerical study showing the efficacy of the robust estimation procedure for estimating the two frequencies and the decision rules for checking the number of dominant frequencies and the causal mechanism. Finally, we compare the proposed method to well-known methods used for two close sinusoids in white noise.

G. Boray

Papers

Estimation of close sinusoids in colored noise and model discrimination