Optimal blind nonlinear least-squares carrier phase and frequency offset estimation for general QAM modulations

This paper introduces a family of blind feedforward nonlinear least-squares (NLS) estimators for joint estimation of the carrier phase and frequency offset of general quadrature amplitude modulated (QAM) transmissions. As an extension of the Viterbi and Viterbi (1983) estimator, a constellation-dependent optimal matched nonlinear estimator is derived such that its asymptotic (large sample) variance is minimized. A class of conventional monomial estimators is also proposed. The asymptotic performance of these estimators is established in closed-form expression and compared with the Cramer-Rao lower bound. A practical implementation of the optimal matched estimator, which is a computationally efficient approximation of the latter and exhibits negligible performance loss, is also derived. Finally, computer simulations are presented to corroborate the theoretical performance analysis and indicate that the proposed optimal matched nonlinear estimator improves significantly the performance of the classic fourth-power estimator.     

Blind feedforward cyclostationarity-based timing estimation for linear modulations

By exploiting a general cyclostationary (CS) statistics-based framework, this letter develops a rigorous and unified asymptotic (large sample) performance analysis setup for a class of blind feedforward timing epoch estimators for linear modulations transmitted through time nonselective flat-fading channels. Within the proposed CS framework, it is shown that several estimators proposed in the literature can be asymptotically interpreted as maximum likelihood (ML) estimators applied on a (sub)set of the second- (and/or higher) order statistics of the received signal. The asymptotic variance of these ML estimators is established in closed-form expression and compared with the modified Crame/spl acute/r-Rao bound. It is shown that the timing estimator proposed by Oerder and Meyr achieves asymptotically the best performance in the class of estimators which exploit all the second-order statistics of the received signal, and its performance is insensitive to oversampling rates P as long as P/spl ges/3. Further, an asymptotically best consistent estimator, which achieves the lowest asymptotic variance among all the possible estimators that can be derived by exploiting jointly the second- and fourth-order statistics of the received signal, is also proposed.     

一种数据辅助的前向位定时估计算法

以平方法为代表的非数据辅助(NDA)类前向位定时估计算法在低信噪比和小成性系数场合估计性能严重恶化,而采用数据辅助(DA)类估计是值得考虑的选择。但传统的数据辅助类算法需要大量的搜索以获取位定时的精确估计,运算量大实现困难。本文将三角内插技术应用于位定时估计,提出了一种新的数据辅助前向位定时估计算法,该算法简单、便于实现。仿真结果表明,本文算法估计性能逼近修正卡美罗界(MCRB),且对成形滚降系数不敏感,适合于低信噪比突发通信。最后,本文针对突发信号给出了前导码捕获与位定时估计一体化的实现结构。      

Low-complexity feedforward symbol timing estimator using conditional maximum-likelihood principle

A low complexity feedforward symbol-timing estimator based on the conditional maximum-likelihood principle is proposed. An approximation is applied to the Fourier series expansion of the conditional maximum-likelihood function such that implementation complexity is greatly reduced. It is shown that the proposed estimator can be viewed as a generalization of the well-known square nonlinearity estimator proposed by Oerder and Meyr in 1988. Simulation results show that the performance of the proposed estimator is very close to the conditional Cramer-Rao bound and is better than that of the square nonlinearity estimator.     

Optimal blind carrier recovery for MPSK burst transmissions

The paper introduces and analyzes the asymptotic (large sample) performance of a family of blind feedforward nonlinear least-squares (NLS) estimators for joint estimation of carrier phase, frequency offset, and Doppler rate for burst-mode phase-shift keying transmissions. An optimal or "matched" nonlinear estimator that exhibits the smallest asymptotic variance within the family of envisaged blind NLS estimators is developed. The asymptotic variance of these estimators is established in closed-form expression and shown to approach the Cramer-Rao lower bound of an unmodulated carrier at medium and high signal-to-noise ratios (SNR). Monomial nonlinear estimators that do not depend on the SNR are also introduced and shown to perform similarly to the SNR-dependent matched nonlinear estimator. Computer simulations are presented to corroborate the theoretical performance analysis.     

Blind code-timing estimation for CDMA systems with bandlimited chip waveforms in multipath fading channels

In this paper, we present a blind code-timing estimator for asynchronous code-division multiple-access (CDMA) systems that use bandlimited chip waveforms. The proposed estimator first converts the received signal to the frequency domain, followed by a frequency deconvolution to remove the convolving chip waveform, and then calculates the code-timing estimate from the output of a narrowband filter with a sweeping center frequency, which is designed to suppress the overall interference in the frequency domain. The proposed estimator is near-far resistant, and can deal with time- and frequency-selective channel fading. It uses only the spreading code of the desired user, and can be adaptively implemented for both code acquisition and tracking. We also derive an unconditional Crame/spl acute/r-Rao bound (CRB) that is not conditioned on the fading coefficients or the information symbols. It is a more suitable lower bound than a conditional CRB for blind code-timing estimators which do not assume knowledge of the channel or information symbols. We present numerical examples to evaluate and compare the proposed and several other code-timing estimators for bandlimited CDMA systems.     

A new non-data-aided feedforward symbol timing estimator using twosamples per symbol

A new non-data-aided feedforward symbol timing estimator is proposed, which can apply to two samples per symbol. Numerical results show the proposed estimator is approximately unbiased. Computational complexity of the new estimator is even simpler than conventional estimators with two samples per symbol. Estimation accuracy of the new estimator is similar to that of the conventional estimator with four samples per symbol presented by Oerder and Meyr in 1988, which is the simplest among the estimators using four samples per symbol. While some conventional estimators have better estimation accuracy than the proposed, this estimator has an attractive advantage over the formers from an implementation point of view     

Adaptive context trees and text clustering

In the finite-alphabet context we propose four alternatives to fixed-order Markov models to estimate a conditional distribution. They consist in working with a large class of variable-length Markov models represented by context trees, and building an estimator of the conditional distribution with a risk of the same order as the risk of the best estimator for every model simultaneously, in a conditional Kullback-Leibler sense. Such estimators can be used to model complex objects like texts written in natural language and define a notion of similarity between them. This idea is illustrated by experimental results of unsupervised text clustering     

Phase Estimators with Predetection Feedback for Optical Communication

Causal, minimum mean-square error (MMSE) estimators of a Gauss-Markov process observed through a conditional Poisson process whose rate parameter is a linear function of the estimation error are presented. Although the conditional estimation performance is data dependent, precomputable upper bounds on the average estimation performance are obtained. Approximate expressions are also presented for the Cramer-Rao lower bound, and it is shown that the estimator performance achieves that bound with equality when the estimator is operating considerably "above threshold." The estimator structure is applied to the problem of phase-tracking receivers for optical communication. Two receiver structures that use predetector phase feedback are considered: one uses local reference fields to allow the detector to observe phase error (homodyne/heterodyne), while the other is a novel direct detection receiver that depends explicitly on the closed-loop nature of the phase estimator. It is concluded that a large local oscillator amplitude is desirable to improve the phase-tracking performance in the homodyne/heterodyne case, and that as few as 4/8 detected signal photons per phase coherence time are required to keep the estimator above threshold. The direct detection scheme achieves the same performance as the homodyne system only in the limit of no dark current-background noise counts, and in general may require considerably more signal photons to keep the estimator "locked."     

An alternative blind feedforward symbol timing estimator using two samples per symbol

Recently, S.J. Lee proposed a blind feedforward symbol timing estimator that exhibits low computational complexity and requires only two samples per symbol (see IEEE Commun. Lett., vol.6, p.205-7, 2002). We analyze Lee's estimator rigorously by exploiting efficiently the cyclostationary statistics present in the received oversampled signal; its asymptotic (large sample) bias and mean-square error (MSE) are derived in closed-form expression. A new blind feedforward timing estimator that requires only two samples per symbol and presents the same computational complexity as Lee's estimator is proposed. It is shown that the proposed new estimator is asymptotically unbiased and exhibits smaller MSE than Lee's estimator. Computer simulations are presented to illustrate the performance of the proposed new estimator with respect to Lee's estimator and existing conventional estimators.     

The statistical performance of some instantaneous frequencyestimators

The authors examine the class of smoothed central finite difference (SCFD) instantaneous frequency (IF) estimators which are based on finite differencing of the phase of the analytic signal. These estimators are closely related to IF estimation via the (periodic) first moment, with respect to frequency of discrete time-frequency representations (TFRs) in L. Cohen's (1966) class. The authors determine the distribution of this class of estimators and establish a framework which allows the comparison of several other estimators such as the zero-crossing estimator and one based on linear regression on the signal phase. It is found that the regression IF estimator is biased and exhibits a large threshold for much of the frequency range. By replacing the linear convolution operation in the regression estimator with the appropriate convolution operation for circular data the authors obtain the parabolic SCFD (PSCFD) estimator, which is unbiased and has a frequency-independent variance, yet retains the optimal performance and simplicity of the original estimator     

Cramer–Rao bound and minimum variance unbiased estimator for joint sampling clock offset and channel taps in MC‐CDMA systems

Exact closed‐form expressions of the Cramer–Rao bound (CRB) for joint sampling clock offset and channel taps are obtained in multi‐carrier code division multiple access systems. CRB is undoubtedly the most well known variance's bound to determine. It provides a benchmark against which we can compare the performance of any unbiased estimator. Furthermore, minimum variance unbiased (MVU) estimator for these parameters is proposed. Moreover, maximum likelihood (ML) and least‐squares estimators for joint sampling clock offset and channel taps are presented. Best linear unbiased estimator is also introduced just for channel taps. The performances of the estimators are compared through simulation results with the proposed CRB. Our results show the better performances of MVU and ML estimators with more computational complexity compared with the others. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal error exponents in hidden Markov models order estimation

We consider the estimation of the number of hidden states (the order) of a discrete-time finite-alphabet hidden Markov model (HMM). The estimators we investigate are related to code-based order estimators: penalized maximum-likelihood (ML) estimators and penalized versions of the mixture estimator introduced by Liu and Narayan (1994). We prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works. We prove a version of Stein's lemma for HMM order estimation and derive an upper bound on underestimation exponents. Then we prove that this upper bound can be achieved by the penalized ML estimator and by the penalized mixture estimator. The proof of the latter result gets around the elusive nature of the ML in HMM by resorting to large-deviation techniques for empirical processes. Finally, we prove that for any consistent HMM order estimator, for most HMM, the overestimation exponent is .     

Estimators of the Nakagami-m parameter and performance analysis

The Nakagami-m parameter is known to capture the envelope distribution of various fading channel conditions in wireless communications. The value of m is indicative of the severity of fading, and is a measure of channel quality, making its estimation necessary in many applications. In this paper, we summarize the existing estimators for the Nakagami-m parameter and propose a new class of estimators, whose performance is analyzed by deriving the asymptotic variance and comparing with the Cramer-Rao bound (CRB). Moreover, we develop a novel estimator robust to the presence of additive white Gaussian noise (AWGN) or any other additive noise with a symmetrical distribution. We also discuss practical issues not previously addressed in the literature including adaptation and computational complexity of these estimators. We conclude that our novel integer-moment based estimators are the best choice from a computational complexity and performance point of view. Simulation results corroborate our analysis.     

On Least Squares Weighted Recursive Estimation of Clock Skew and Offset in Wireless Sensor Networks

Three clock synchronization algorithms for Wireless sensor networks (WSNs) in Pairwise broadcast synchronization (PBS) mechanism are derived.They include the joint Least squares estimator (LS),joint Least squares weighted estimator (LSW) and joint Least squares weighted Recursive estimator (R-LSW).For these estimators,the corresponding algorithms are derived and described by assuming a Gaussian random delay model.Unlike PBS,these estimators can achieve the Cramer-Rao lower bound (CPLB) for both listening node and active node without knowledge of the deterministic delay.The purpose of considering R-LSW is to reduce the use of storage space with the method of estimating while observing.Simulation and analytical results verify that the estimators are efficient.     

Parametric signal restoration using artificial neural networks

The problem of parametric signal restoration given its blurred/nonlinearly distorted version contaminated by additive noise is discussed. It is postulated that feedforward artificial neural networks can be used to find a solution to this problem. The proposed estimator does not require iterative calculations that are normally performed using numerical methods for signal parameter estimation. Thus high speed is the main advantage of this approach. A two-stage neural network-based estimator architecture is considered in which the vector of measurements is projected on the signal subspace and the resulting features form the input to a feedforward neural network. The effect of noise on the estimator performance is analyzed and compared to the least-squares technique. It is shown, for low and moderate noise levels, that the two estimators are similar to each other in terms of their noise performance, provided the neural network approximates the inverse mapping from the measurement space to the parameter space with a negligible error. However, if the neural network is trained on noisy signal observations, the proposed technique is superior to the least-squares estimate (LSE) model fitting. Numerical examples are presented to support the analytical results. Problems for future research are addressed     

Lower bound on training-based channel estimation error for frequency-selective block-fading Rayleigh MIMO channels

A lower bound on the error correlation matrix of training-based channel estimators is derived for multiple-input multiple-output (MIMO) systems over block-fading frequency-selective channels with symbol-spaced receivers. The bound is obtained in a constructive way by evaluating the asymptotic performance of an estimator that fully exploits the algebraic structure of the multipath channel. In particular, the estimator is assumed to be able to estimate the long-term features of the channel consistently (e.g., second order statistics of fading, delays, angles) while tracking the fast-varying fading fluctuations by Wiener filtering. Known estimators that are able to attain the bound under simplified settings are referred to, and general guidelines for designing novel estimators are discussed. Based on the simple analytical expression of the bound, the impact of channel estimation error on the link capacity is investigated for different system parameters and channel characteristics (e.g., Doppler shift, spatial correlation of fading). Numerical results are provided to corroborate the analysis.     

A class of shrinkage estimators for the variance of a normal population

The crux of this paper is to propose a class of shrinkage estimators for the variance of a normal population and study its properties. Some estimators are generated from a proposed class and compared with the usual unbiased estimator, minimum mean squared error (MMSE) estimator and Pandey and Singh, South African Statistical Journal (1976) and J. Indian Statistical Assoc. 15, 141–150 (1977) estimator.     

Nonparametric density estimation for classes of positive randomvariables

A kernel-based density estimator for positive random variables is proposed and analyzed. In particular, a nonparametric estimator is developed which takes advantage of the fact that positive random variables can be represented as the norms of random vectors. By appropriately choosing the dimension of the assumed vector space, the estimator can be structured to exploit a priori knowledge about the density to be estimated. The asymptotic properties (e.g., pointwise and L₁-consistency) of this density estimator are investigated and found to be similar to the desirable features of the standard kernel estimator. An upper bound on the expected value of the L₁ error is also derived which provides insight into the behavior of the estimator. Upon using this upper bound, the optimal form for the estimator (i.e., the kernel function, the smoothing factor, etc.) is selected via a minimax strategy. In addition, this upper bound is used to compare the asymptotic performance of the proposed estimator to that of the standard kernel estimator and to boundary-corrected kernel estimators. Numerical examples illustrate that the proposed scheme outperforms the standard and boundary-corrected estimators for a variety of density types     

Estimation of Dependability Measures and Parameter Sensitivities of a Consecutive-$k$-out-of-$n$: F Repairable System With $(k-1)$-Step Markov Dependence by Simulation

In this paper, we present two methods, direct simulation (DS), and conditional expectation estimation (CEE), for estimating the unreliability, transient unavailability, steady unavailability, mean time to failure (MTTF), mean time between failure (MTBF), and their parameter sensitivities of consecutive-k-out-of-n: F repairable systems with (k - 1)-step Markov dependence. After an expression of the likelihood ratio estimator of parameter sensitivity is introduced, the direct estimators, and conditional expectation estimators are derived. The analytical results in a study by Lam & Ng were used to verify the algorithms presented in this paper. Numerical examples of highly reliable linear, and circular C (4, 50: F), and C (4,100: F) systems were illustrated to compare the algorithm efficiencies of DS, and CEE and CEE was found to be remarkably more efficient than DS.