Estimating the frequency of a noisy sinusoid by linear regression (Corresp.)

The estimation of the parameters of a sinusoid from observations of signal samples corrupted by additive noise is investigated. At high signal-to-noise ratios the additive noise is viewed as an equivalent phase noise, suggesting frequency and phase estimation by linear regression on the signal phase. The variances of the regression estimates are shown to achieve the Cramer-Rao bounds. A formula for the variance of the regression frequency estimator is derived in terms of the noise power spectrum. A simple formula for the variance with1/f^{2}phase noise is presented.     

Optimal sub-Nyquist nonuniform sampling and reconstruction formultiband signals

We study the problem of optimal sub-Nyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support ℱ that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau's (1967) lower bound equal to the measure of ℱ. For signals with sparse ℱ, this rate can be much smaller than the Nyquist rate. Unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a previous study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and designing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with ℱ that tiles under translation. For signals with nontiling ℱ, which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with sub-Nyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate     

Effects of sampling and quantization on single-tone frequencyestimation

The effects of sampling and quantization on frequency estimation for a single sinusoid are investigated. The Cramer-Rao bound for 1-bit quantization is derived and compared with the limit of infinite quantization. It is found that 1-bit quantization gives a slightly worse performance, however, with a dramatic increase of variance at certain frequencies. This can be avoided by using four times oversampling. The effect of sampling when using nonideal antialiasing lowpass filters is therefore investigated through derivation of the Cramer-Rao lower bounds. Finally, fast estimators for 1-bit quantization, in particular, correlation-based estimators, are derived, and their performance is investigated. The paper is concluded with simulation results for four times oversampled 1-bit quantization     

Sequence estimation and synchronization from nonsynchronizedsamples

The authors study the problem of maximum-likelihood sequence estimation and synchronization from samples of the output of a matched filter taken at integer multiples of the symbol rate, which is assumed perfectly known by the receiver. A general analysis is presented of sampled receivers that handle arbitrary baseband pulse shapes and arbitrary sampling rates. It is observed that the optimal processing of the matched-filter samples consists of digital interpolation, followed by symbol-by-symbol decoding when sampling is at (or above) the Nyquist rate or Viterbi decoding when sampling is below the Nyquist rate. Performance is studied through the Cramer-Rao bound on mean-square estimation error and a lower-bound on error-probability     

A generalization of nonuniform bandpass sampling

Nth-order nonuniform sampling is described for generalized bandpass signal frequency position, bandwidth, sampling rate, frequency-shift and phase-shift. A bandpass extension to the Nyquist criterion is derived, showing that restrictions on bandpass frequency position for odd orders of nonuniform sampling tend to zero as N tends to infinity. Bandpass interpolants based on the sinc function are derived for the generalized Nth-order sampled bandpass signals. It is shown that, for minimum (Nyquist) rate sampling, these interpolants are comprised of N bandpass filters, each with independent phase. The number of bandpass filters comprising the interpolant is found to decrease as the sample rate increases. The advantage of describing Nth-order sampling as the Nth replication and uniform sampling of a signal is demonstrated. Finally, digital implementation of the Nth-order bandpass sampling interpolants is discussed. It is established that it is not practicable to attempt to perform nonuniform bandpass sampling at the theoretical minimum rate, where the interpolation is to be performed digitally     

Doppler frequency estimation and the Cramer-Rao bound

Addresses the problem of Doppler frequency estimation in the presence of speckle and receiver noise. An ultimate accuracy bound for Doppler frequency estimation is derived from the Cramer-Rao inequality. It is shown that estimates based on the correlation of the signal power spectra with an arbitrary weighting function are approximately Gaussian-distributed. Their variance is derived in terms of the weighting function. It is shown that a special case of a correlation-based estimator is a maximum-likelihood estimator that reaches the Cramer-Rao bound. These general results are applied to the problem of Doppler centroid estimation from SAR (synthetic aperture radar) data     

A subspace method for direction of arrival estimation ofuncorrelated emitter signals

A novel eigenstructure-based method for direction estimation is presented. The method assumes that the emitter signals are uncorrelated. Ideas from subspace and covariance matching methods are combined to yield a noniterative estimation algorithm when a uniform linear array is employed. The large sample performance of the estimator is analyzed. It is shown that the asymptotic variance of the direction estimates coincides with the relevant Cramer-Rao lower bound (CRB). A compact expression for the CRB is derived for the ease when it is known that the signals are uncorrelated, and it is lower than the CRB that is usually used in the array processing literature (assuming no particular structure for the signal covariance matrix). The difference between the two CRBs can be large in difficult scenarios. This implies that in such scenarios, the proposed methods has significantly better performance than existing subspace methods such as, for example, WSF, MUSIC, and ESPRIT. Numerical examples are provided to illustrate the obtained results     

Minimum variance in biased estimation: bounds and asymptotically optimal estimators

We develop a uniform Cramer-Rao lower bound (UCRLB) on the total variance of any estimator of an unknown vector of parameters, with bias gradient matrix whose norm is bounded by a constant. We consider both the Frobenius norm and the spectral norm of the bias gradient matrix, leading to two corresponding lower bounds. We then develop optimal estimators that achieve these lower bounds. In the case in which the measurements are related to the unknown parameters through a linear Gaussian model, Tikhonov regularization is shown to achieve the UCRLB when the Frobenius norm is considered, and the shrunken estimator is shown to achieve the UCRLB when the spectral norm is considered. For more general models, the penalized maximum likelihood (PML) estimator with a suitable penalizing function is shown to asymptotically achieve the UCRLB. To establish the asymptotic optimality of the PML estimator, we first develop the asymptotic mean and variance of the PML estimator for any choice of penalizing function satisfying certain regularity constraints and then derive a general condition on the penalizing function under which the resulting PML estimator asymptotically achieves the UCRLB. This then implies that from all linear and nonlinear estimators with bias gradient whose norm is bounded by a constant, the proposed PML estimator asymptotically results in the smallest possible variance.     

Some results on the stochastic signal parameter estimation problem

The problem of finding maximum-likelihood estimates of the partially or completely unknown autocorrelation function of a zero-mean Gaussian stochastic signal corrupted by additive, white Gaussian noise is analyzed. It is shown that these estimates can be found by maximizing the output of a Wiener estimator-correlator receiver biased by a smoothed version of the output noise-to-signal ratio of the Wiener estimator over the class of admissible autocorrelation functions. For the case where the autocorrelation function is known except for an amplitude scale parameter, an illuminating expression for the Cramer-Rao minimum estimation variance is derived. Detailed study of this expression yields, among other results, an interpretation of the maximum-likelihood estimator as an adaptive processor.     

DOA estimation of wideband sources using a harmonic source modeland uniform linear array

We consider the problem of estimation of the DOAs of multiple wideband sources incident on a uniform linear array (ULA) in the presence of spatially and temporally white Gaussian noise (WGN). The approach presented builds up on the IQML algorithm suggested by Bresler and Macovski (1986) for the case of narrowband DOA estimation. It is shown that the concept of an ARMA model for the observed data vector for the narrowband case can be generalized to model an appropriately stacked, space-time data vector obtained by combining the space-time samples. The coefficients of the corresponding 2-D predictor polynomial can be used to represent the null subspace of the wideband array steering matrix, and rooting of the polynomial at each frequency, separately, gives the DOA estimates. These separate estimates at multiple frequencies are combined into a single DOA estimate in a least squares sense. This leads to the formulation of an IQML like procedure for the spatial parameter estimation of wideband sources. Like its narrowband counterpart, the proposed approach is applicable to both noncoherent and coherent sources. The performance of the proposed method is studied via extensive computer simulations and by comparison with the Cramer-Rao bounds     

Cramer-Rao bounds of SNR estimates for BPSK and QPSK modulatedsignals

We derive Cramer-Rao lower bounds (CRLBs) for the estimation of signal-to-noise ratio (SNR) of binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulated signals. The received signal is corrupted by additive white Gaussian noise (AWGN). The lower bounds are derived for non-data-aided estimation where the transmitted symbols are unknown at the receiver. The bounds are compared to those for data-aided estimations (known symbols at the receiver). It is shown that at low SNR there is a significant difference between the bounds for non-data-aided and data-aided estimations     

On the estimation of variance for autoregressive and moving average processes (Corresp.)

The sample variance is commonly used to estimate the variance of stationary time series. When the second-order statistics of the process are known up to a scaling factor, this estimator is generally inefficient. In the case of an autoregressive (AR) process with unknown parameters, the sample variance is shown to be asymptotically efficient. However, the sample variance of a moving-average (MA) process with unknown parameters is generally an inefficient estimator. Closed-form expressions are derived for the Cramer-Rao hound associated with the variance estimation problem and for the variance of the sample-variance estimator, for both AR and MA processes.     

Time-domain analysis of cross correlation for time delay estimationwith an autocorrelated signal

The estimation accuracy of the time difference of arrival (TDOA) of an exponentially autocorrelated signal at two sensors in white noise is analyzed. The estimate is obtained by cross correlating samples taken as short-term integrals of the noisy signals from the two sensors. This technique avoids ambiguities in the cross correlation, and it is shown that the best sampling rate is double the Nyquist rate, for which the Cramer-Rao lower bound (CRLB) is met in practice     

A general class of lower bounds in parameter estimation

A general class of Bayesian lower bounds on moments of the error in parameter estimation is formulated, and it is shown that the Cramer-Rao, the Bhattacharyya, the Bobrovsky-Zakai, and the Weiss-Weinstein lower bounds are special cases in the class. The bounds can be applied to the estimation of vector parameters and any given function of the parameters. The extension of these bounds to multiple parameter is discussed     

Interpolation of electromagnetic radiated fields over a plane bynonuniform samples

An iterative algorithm is developed which allows the interpolation of two-dimensional band-limited functions from nonuniformly distributed samples. Such a scheme is based on optimal sampling expansions of central type and is applied to the interpolation of electromagnetic fields over a plane. The algorithm can be applied whenever the “average” sample's density is higher than the Nyquist one and there is a one-to-one correspondence between the nonuniform samples and a lattice of the uniform ones, which associates at each uniform sampling point the nearest nonuniform sampling point. As compared with the optimal linear estimation algorithm, the procedure is computationally much more efficient. Many numerical simulations have assessed the rapid convergence of the method as well as its stability with respect to both absolute and relative errors in the data     

Interpolation and extrapolation in nonuniform sampling sequences with average sampling rates below the Nyquist rate

Coefficients for interpolation and extrapolation, including average sampling rates below the Nyquist rate, have been derived. The method is applicable for nonuniform patterns obtained through omission of samples in a uniform pattern with a sampling rate slightly higher than the Nyquist rate. The interpolations and extrapolations are valid for band-limited signals, except those theoretical signals that are zero valued at all sampling points.     

Gradient estimation for stochastic optimization of opticalcode-division multiple-access systems .I. Generalized sensitivityanalysis

For optimizing the performance of optical code-division multiple-access (CDMA) systems, there is a need for determining the sensitivity of the bit-error rate (BER) of the system to various system parameters. Asymptotic approximations and bounds, used for system bit-error probabilities, seldom capture the sensitivities of the system performance. We develop single-run gradient estimation methods for such optical CDMA systems using a discrete-event dynamic systems (DEDS) approach. Specifically, computer-aided techniques such as infinitesimal perturbation analysis (IPA) and likelihood ratio (LR) methods are used for analyzing the sensitivity of the average BER to a wide class of system parameters. It is shown that the above formulation is equally applicable to time-encoded and frequency-encoded systems. Further, the estimates derived are unbiased, and also optimality of the variance of these estimates is shown via the theory of common random variates and importance sampling techniques     

Bounds on the accuracy of estimating the parameters of discretehomogeneous random fields with mixed spectral distributions

This paper considers the achievable accuracy in jointly estimating the parameters of a real-valued two-dimensional (2-D) homogeneous random field with mixed spectral distribution, from a single observed realization of it. On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and evanescent. An exact form of the Cramer-Rao lower bound on the error variance in jointly estimating the parameters of the different components is derived. It is shown that the estimation of the harmonic component is decoupled from that of the purely indeterministic and the evanescent components. Moreover, the bound on the parameters of the purely indeterministic and the evanescent components is independent of the harmonic component. Numerical evaluation of the bounds provides some insight into the effects of various parameters on the achievable estimation accuracy     

The bispectrum of sampled data. II. Monte-Carlo simulations ofdetection and estimation of the sampling jitter

For pt.I see IEEE Trans. Acoust. Speech, Signal Process., vol.41, no.1, p.296-312 (1993). The effect of jitter in sampling on the spectrum and bispectrum of the sampled data has been considered previously. Methods of detecting the presence of jitter in a uniform sampling process and of estimating its variance based on a test statistic calculated from the bispectrum estimates have been proposed. The present authors demonstrate, by means of Monte-Carlo simulations, how these results can be applied in an actual case. For this purpose, samples taken from a stationary band-limited process in sampling times given by a random jitter process are generated by computer. The authors then apply the jitter detection and estimation methods that have been developed in previous work and study how their performance depends on signal duration and on jitter variance. They examine the actual simulation results concerning detection probability, estimation bias, and estimation variance in comparison with the theoretical results. This comparison indicates that the bispectrum is a domain where jitter detection and estimation with high performance can be achieved, provided that a signal with sufficiently long duration and high skewness is available     

Maximum-likelihood direction-of-arrival estimation in the presenceof unknown nonuniform noise

We consider the problem of estimating directions of arrival (DOAs) of multiple sources observed on the background of nonuniform white noise with an arbitrary diagonal covariance matrix. A new deterministic maximum likelihood (ML) DOA estimator is derived. Its implementation is based on an iterative procedure which includes a stepwise concentration of the log-likelihood (LL) function with respect to the signal and noise nuisance parameters and requires only a few iterations to converge. New closed-form expressions for the deterministic and stochastic direction estimation Cramer-Rao bounds (CRBs) are derived for the considered nonuniform model. Our expressions can be viewed as an extension of the well-known results by Stoica and Nehorai (1989, 1990) and Weiss and Friedlander (1993) to a more general noise model than the commonly used uniform one. In addition, these expressions extend the results obtained by Matveyev et al. (see Circuits, Syst., Signal Process., vol.18, p.479-87, 1999) to the multiple source case. Comparisons with the above-mentioned earlier results help to discover several interesting properties of DOA estimation in the nonuniform noise case. To compare the estimation performance of the proposed ML technique with the results of our CRB analysis and with the performance of conventional “uniform” ML, simulation results are presented. Additionally, we test our technique using experimental seismic array data. Our simulations and experimental results both validate essential performance improvements achieved by means of the approach proposed