Sinusoids In White Gaussian Noise Research Articles

Unitary PUMA Algorithm for Estimating the Frequency of a Complex Sinusoid

One-dimensional (1-D) and two-dimensional (2-D) frequency estimation for a single complex sinusoid in white Gaussian noise is a classic signal processing problem with numerous applications. It is revisited here through a new unitary principal-singular-vector utilization modal analysis (PUMA) approach, which is realized in terms of real-valued computations. The 2-D unitary PUMA is first formulated as an iteratively weighted least squares optimization problem. Recognizing that only one iteration is sufficient when 2-D unitary PUMA is initialized using least squares, a computationally attractive closed-form solution is then obtained. A variant of 2-D unitary PUMA is also developed for the 1-D case. Due to the real-valued computations and closed-form expression for the frequency estimate, the unitary PUMA is more computationally efficient than a number of state-of-the-art methods. Furthermore, the asymptotic variances of 1-D and 2-D unitary PUMA estimators are theoretically derived, and numerical results are included to demonstrate the effectiveness of the proposed methods.

A Contraction Mapping Approach for Robust Estimation of Lagged Autocorrelation

We consider the zero-crossing rate (ZCR) of a Gaussian process and establish a property relating the lagged ZCR (LZCR) to the corresponding normalized autocorrelation function. This is a generalization of Kedem’s result for the lag-one case. For the specific case of a sinusoid in white Gaussian noise, we use the higher-order property between lagged ZCR and higher-lag autocorrelation to develop an iterative higher-order autoregressive filtering scheme, which stabilizes the ZCR and consequently provide robust estimates of the lagged autocorrelation. Simulation results show that the autocorrelation estimates converge in about 20 to 40 iterations even for low signal-to-noise ratio.

Relabeling and Summarizing Posterior Distributions in Signal Decomposition Problems When the Number of Components is Unknown

This paper addresses the problems of relabeling and summarizing posterior distributions that typically arise, in a Bayesian framework, when dealing with signal decomposition problems with an unknown number of components. Such posterior distributions are defined over union of subspaces of differing dimensionality and can be sampled from using modern Monte Carlo techniques, for instance the increasingly popular RJ-MCMC method. No generic approach is available, however, to summarize the resulting variable-dimensional samples and extract from them component-specific parameters. We propose a novel approach, named Variable-dimensional Approximate Posterior for Relabeling and Summarizing (VAPoRS), to this problem, which consists in approximating the posterior distribution of interest by a simple---but still variable-dimensional---parametric distribution. The distance between the two distributions is measured using the Kullback-Leibler divergence, and a Stochastic EM-type algorithm, driven by the RJ-MCMC sampler, is proposed to estimate the parameters. Two signal decomposition problems are considered, to show the capability of VAPoRS both for relabeling and for summarizing variable dimensional posterior distributions: the classical problem of detecting and estimating sinusoids in white Gaussian noise on the one hand, and a particle counting problem motivated by the Pierre Auger project in astrophysics on the other hand.

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.

Joint Model Selection and Parameter Estimation by Population Monte Carlo Simulation

In this paper, we study the problem of joint model selection and parameter estimation under the Bayesian framework. We propose to use the Population Monte Carlo (PMC) methodology in carrying out Bayesian computations. The PMC methodology has recently been proposed as an efficient sampling technique and an alternative to Markov Chain Monte Carlo (MCMC) sampling. Its flexibility in constructing transition kernels allows for joint sampling of parameter spaces that belong to different models. The proposed method is able to estimate the desired a posteriori distributions accurately. In comparison to the Reversible Jump MCMC (RJMCMC) algorithm, which is popular in solving the same problem, the PMC algorithm does not require burn-in period, it produces approximately uncorrelated samples, and it can be implemented in a parallel fashion. We demonstrate our approach on two examples: sinusoids in white Gaussian noise and direction of arrival (DOA) estimation in colored Gaussian noise, where in both cases the number of signals in the data is a priori unknown. Both simulations show the effectiveness of our proposed algorithm.

Approximate Maximum-Likelihood Algorithms for Two-Dimensional Frequency Estimation of a Complex Sinusoid

Starting with the maximum-likelihood (ML) formulation, three iterative algorithms for approximate ML frequency estimation of a two-dimensional (2-D) complex sinusoid in white Gaussian noise are developed. Mean and variance analyses of the proposed methods are provided, which show that they are approximately unbiased and their performance achieves Cramer-Rao lower bound (CRLB) at sufficiently high signal-to-noise ratio (SNR) conditions. Computer simulation results are included to corroborate the theoretical development as well as to contrast the performance of the proposed algorithms with Kay's estimators and the CRLB

An eigenvalue residuum-based criterion for detection of the number of sinusoids in white Gaussian noise

It is understood that the eigenvalues of a covariance matrix, deriving from the data of multiple sinusoidal signals in white Gaussian noise, can be divided into two groups when the dimensions of the covariance matrix are larger than the number of sinusoidal signals. One is composed of the sinusoidal signals plus the noise. Another is called ‘small eigenvalues’ which is produced by the noise only. It is also known that the small eigenvalues are an asymptotic Gaussian random process. Based on these facts, an eigenvalue-based residuum criterion for the detection of the number of sinusoids in white Gaussian noise is presented in this paper. Using this criterion, the small eigenvalues of the covariance matrix can gradually be ruled out. While the eigenvalues which belong to the signal and noise are separated, the number of sinusoids is detected successfully. Simulation results exhibit that the proposed method gives a superior performance over the Akaike information criterion (AIC) and the minimum description length principle (MDL), especially for situations where the analyzed signals exhibit a low signal-to-noise ratio (SNR), short data records, and a high number of sinusoids. In addition, the implementation of the new criterion is simpler and faster.

Subspace-based frequency estimation of sinusoidal signals in alpha-stable noise

In the frequency estimation of sinusoidal signals observed in impulsive noise environments, techniques based on Gaussian noise assumption are unsuccessful. One possible way to find better estimates is to model the noise as an alpha-stable process and to use the fractional lower order statistics (FLOS) of the data to estimate the signal parameters. In this work, we propose a FLOS-based statistical average, the generalized covariation coefficient (GCC). The GCCs of multiple sinusoids for unity moment order in S αS noise attain the same form as the covariance expressions of multiple sinusoids in white Gaussian noise. The subspace-based frequency estimators FLOS–multiple signal classification (MUSIC) and FLOS–Bartlett are applied to the GCC matrix of the data. On the other hand, we show that the multiple sinusoids in S αS noise can also be modeled as a stable autoregressive moving average process approximated by a higher order stable autoregressive (AR) process. Using the GCCs of the data, we obtain FLOS versions of Tufts–Kumaresan (TK) and minimum norm (MN) estimators, which are based on the AR model. The simulation results show that techniques employing lower order statistics are superior to their second-order statistics (SOS)-based counterparts, especially when the noise exhibits a strong impulsive attitude. Among the estimators, FLOS–MUSIC shows a robust performance. It behaves comparably to MUSIC in non-impulsive noise environments, and both in impulsive and non-impulsive high-resolution scenarios. Furthermore, it offers a significant advantage at relatively high levels of impulsive noise contamination for distantly located sinusoidal frequencies.

Joint Bayesian model selection and estimation of noisy sinusoids via reversible jump MCMC

In this paper, the problem of joint Bayesian model selection and parameter estimation for sinusoids in white Gaussian noise is addressed. An original Bayesian model is proposed that allows us to define a posterior distribution on the parameter space. All Bayesian inference is then based on this distribution. Unfortunately, a direct evaluation of this distribution and of its features, including posterior model probabilities, requires evaluation of some complicated high-dimensional integrals. We develop an efficient stochastic algorithm based on reversible jump Markov chain Monte Carlo methods to perform the Bayesian computation. A convergence result for this algorithm is established. In simulation, it appears that the performance of detection based on posterior model probabilities outperforms conventional detection schemes.

The ’’Harmonic‘‘ Rejecting Correlation Function

A classical problem in signal processing is accurate estimation of fundamental frequency/periodicity of periodic signals at low SNR. Typically, researchers address the estimation problem, assuming that the signal environment is a sum of sinusoids in white Gaussian noise. If the signals and noise are pulsed, the situation is much more complex since normal FFT based methods result in spectra which are sums of harmonic structures. Sorting radar signals can be especially difficult since there may be many pulsed signals present in a low SNR impulsive noise environment. In this paper, a method equivalent to integration along a hyperbola on the Wigner distribution is presented. This transform, which is closely related to both the Fourier transform and the correlation function, has the property that a periodic signal produces an expected non-zero complex-valued bulge at only the fundamental. The phase, magnitude and position of the correlation bulge are sufficient to characterize the time-domain pulse train. Finally, a simple super-resolution method is presented which may be used to refine the fundamental frequency/period estimate.

Nonmatrix Cramer-Rao bound expressions for high-resolution frequency estimators

Analytical expressions are derived for the Cramer-Rao (CR) lower bound on the variance of frequency estimates for the two-signal time-series data models consisting of either one real sinusoid or two complex sinusoids in white Gaussian noise. The expressions give the bound in terms of the signal-to-noise ratio (SNR), the number N of data samples, and a function dependent on the frequency separation and the initial phase difference between the two signal components of each model. The bounds are examined as the phase difference is varied, and the largest and smallest bound expressions and the corresponding critical values of the phase difference are obtained. The exact expressions are analyzed for the case of small frequency separations /spl delta//spl omega/. It is found that the largest bound is proportional to (N/spl middot//spl delta//spl omega/)/sup -4//N/sup 3//spl middot/SNR and that the smallest bound is proportional to (N/spl middot//spl delta//spl omega/)/sup -2/N/sup 3//spl middot/SNR for small /spl delta//spl omega/. Examples indicate that the small /spl delta//spl omega/ results closely approximate the exact ones whenever the frequency separation is smaller than the Fourier resolution limit. Based on the asymptotic results, it is found that the threshold SNR at which an unbiased estimator can resolve the two signal frequencies is at least proportional to (N/spl middot//spl delta//spl omega/)/sup -6//N for the worst phase difference case and to (N/spl middot//spl delta//spl omega/)/sup -4//N for the best phase difference case for small /spl delta//spl omega/. The results are applicable to the general case of sampling where the samples are taken at arbitrary instants.

Two-Dimensional Spectral Estimation by Free Energy Minimization

Kiernan, P., Two-Dimensional Spectral Estimation by Free Energy Minimization,Digital Signal Processing6 (1996), 160–168.We present a high-resolution 2-D minimum free energy (MFE) spectral estimation technique which is a 2-D extension of the MFE spectral estimation method. We demonstrate the performance of the technique for spectral estimation of closely spaced 2-D sinusoids in white Gaussian noise. Results from our tests on the effect of signal processing temperature illustrate that our method provides accurate low model order autoregressive spectral estimation. The method provides superior spectral estimation with similar computational burden to that achieved with the Levinson algorithm for a number of cases involving sinusoids in white noise at various signal-to-noise ratio (SNR) levels. An example is provided. The method is faster than the modified covariance method (MCV). An example is given from a number of cases involving two closely spaced sinusoids at low SNR, where MFE estimates are as well resolved as MCV estimates. MFE models may also be used for correlation extension and for field modeling and synthesis. We indicate possible extensions to the MFE method for computational efficiency improvement, a priori temperature determination, and the use of higher-order statistics.

A model selection rule for sinusoids in white Gaussian noise

The model selection problem for sinusoidal signals has often been addressed by employing the Akaike (1974) information criterion (AIC) and the minimum description length principle (MDL). The popularity of these criteria partly stems from the intrinsically simple means by which they can be implemented. They can, however, produce misleading results if they are not carefully used. The AIC and MDL have a common form in that they comprise two terms, a data term and a penalty term. The data term quantifies the residuals of the model, and the penalty term reflects the desideratum of parsimony. While the data terms of the AIC and MDL are identical, the penalty terms are different. In most of the literature, the AIC and MDL penalties are, however, both obtained by apportioning an equal weight to each additional unknown parameter, be it phase, amplitude, or frequency. By contrast, we demonstrate that the penalties associated with the amplitude and phase parameters should be weighted differently than the penalty attached to the frequencies. Following the Bayesian methodology, we derive a model selection criterion for sinusoidal signals in Gaussian noise which also contains the log-likelihood and the penalty terms. The simulation results disclose remarkable improvement in our selection rule over the commonly used MDL and AIC.

Two-dimensional AR spectral estimation using a two-dimensional minimum free energy method

We present a 2-D extension of the minimum free energy (MFE) autoregressive parameter and spectral estimation technique. MFE can outperform the multidimensional Levinson algorithm producing superior spectral estimation of 2-D sinusoids in white Gaussian noise. MFE spectral estimates are found to be comparable to modified covariance method estimates. MFE also outperforms the conventional Fourier method for closely spaced sinusoids at low signal-to-noise ratio and low data set size.

Multiscale identification of real sinusoids in noise

We consider the problem of the identification of the real sinusoids in white Gaussian noise. Since the maximum likelihood function is ill-conditioned and possesses numerous local maxima, we propose a different cost function that preserves statistical efficiency. It allows to reparametrise the problem and to consider a whole set of different parametrizations that correspond to different time scales. To each such parametrization one associates in a trivial fashion an initial point. By change of scale, this point is transformed into an initial estimate of the parameters of interest. One then selects the parametrization with lowest initial cost. This procedure already attains the standard Cramer-Rao bound when the difficulty is to resolve two closely spaced sinusoids. Any further processing, such as fusion of the different scales or iterative optimization, is thus useless. We consider situations with SNRs below 10 dB and frequency separations not resolvable by the basic periodogram.

Estimation of Parameters of Real Sinusoids in Noise

We consider real sinusoids in white gaussian noise. Since the maximum likelihood function is ill conditioned, we propose a different cost function which preserves statistical efficiency. It allows to introduce and to compare a whole set of different parametrisations of the problem. To each parametrisation one associates in a nautral fashion a initial point. One then selects the parametrisation with lowest initial cost. As a matter of fact, this procedure already attains the standard Cramer-Rao bound when the difficulty is to resolve two closely spaced sinusoids. We consider situations with SNR's below 10dB and frequency separations not resolvable by the basic periodogram.

Maximum likelihood estimation of superimposed complex sinusoids in white Gaussian noise by reduced effort coarse search (RECS)

An efficient implementation for the maximum likelihood estimator for the frequencies of superimposed complex sinusoids in white Gaussian noise is presented. The algorithm uses one-dimensional searches and classical optimization techniques and produces data that can be used for model order selection. Its convergence and statistical properties are verified through simulation. The method does not require polynomial rooting, frequent manipulation of large matrices, or the use of approximate constraints.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Effects of beamformer output signal characteristics on the detection performance of passive sonar receivers

When one builds a mathematical model of the passive sonar equation, it is neither sufficient nor appropriate to represent the sonar receiver (processor-display-operator) by a single, constant value of the recognition differential. The detectability index d′ is presented as the most appropriate performance metric for realistic sonar receivers, and it is shown to be a primary function of the working environment and the character of real beamformer output signal and noise. It is shown that the characteristics that are of significance include the probability density distributions and the autocorrelation functions of the signal and the noise. Having motivated the need to consider these characteristics, two sets of operational beamformer output data were analyzed, This paper summarizes the analyses of the three basic constituents of the data, namely, signal, broadband noise, and signal-like noise (due to surface shipping). It was found that both the signal and the noise were subject to large fluctuations over time periods that were comparable to typical processor integration times. In addition, the signal-like noise components were found to have rather unique probability distributions. They did not in fact exhibit the probability structure of stable sinusoids in white Gaussian noise. The probability density distributions of the data at the input and the output of the receiver are fundamentally important. Consequently, the paper closes with the description of a methodology for determining the parameters of the receiver operator characteristic (ROC) curve for a particular receiver in situ. [Work supported by the Naval Undersea Center.]