Derived PDF of maximum likelihood signal estimator which employs anestimated noise covariance

A probability density function (PDF) for the maximum likelihood (ML) signal vector estimator is derived when the estimator relies on a noise sample covariance matrix (SCM) for evaluation. By using a complex Wishart probabilistic model for the distribution of the SCM, it is shown that the PDF of the adaptive ML (AML) signal estimator (alias the SCM based minimum variance distortionless response (MVDR) beamformer output and, more generally, the SCM based linearly constrained minimum variance (LCMV) beamformer output) is, in general, the confluent hypergeometric function of a complex matrix argument known as Kummer's function. The AML signal estimator remains unbiased but only asymptotically efficient; moreover, the AML signal estimator converges in distribution to the ML signal estimator (known noise covariance). When the sample size of the estimated noise covariance matrix is fixed, it is demonstrated that there exists a dynamic tradeoff between signal-to-noise ratio (SNR) and noise adaptivity as the dimensionality of the array data (number of adaptive degrees of freedom) is varied, suggesting the existence of an optimal array data dimension that will yield the best performance     

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.     

Scattering function estimation

The estimation of the scattering function of a random, zero-mean, homogeneous, time-variant, linear filter is considered. The sum of the random filter output and independent noise is the input to an estimator. The estimator structure is equivalent to a bank of linear filters followed by squared-envelope detectors; the envelope detector outputs are the input to a final linear filter. The estimator output is shown to be an unconstrained linear operation on the ambiguity function of the estimator input. Except for a bias term due to the additive noise, the mean of the estimator output is an unconstrained linear operation on the scattering function of the random filter. The integral variance of the output is found for a Gaussian channel. The mean and variance clearly indicate the tradeoff between resolution and variance reduction obtained by varying the estimator structure. For any well-behaved channel it is shown that an effectively unbiased estimate of the scattering function can be obtained if the input signal has both sufficient energy and enough time and frequency spread to resolve the random filter; the random filter is not required to be underspread. The variance of an estimate can be further reduced by increasing the time or frequency spread of the transmitted signal.     

Stationary and nonstationary learning characteristics of the LMS adaptive filter

This paper describes the performance characteristics of the LMS adaptive filter, a digital filter composed of a tapped delay line and adjustable weights, whose impulse response is controlled by an adaptive algorithm. For stationary stochastic inputs, the mean-square error, the difference between the filter output and an externally supplied input called the "desired response," is a quadratic function of the weights, a paraboloid with a single fixed minimum point that can be sought by gradient techniques. The gradient estimation process is shown to introduce noise into the weight vector that is proportional to the speed of adaptation and number of weights. The effect of this noise is expressed in terms of a dimensionless quantity "misadjustment" that is a measure of the deviation from optimal Wiener performance. Analysis of a simple nonstationary case, in which the minimum point of the error surface is moving according to an assumed first-order Markov process, shows that an additional contribution to misadjustment arises from "lag" of the adaptive process in tracking the moving minimum point. This contribution, which is additive, is proportional to the number of weights but inversely proportional to the speed of adaptation. The sum of the misadjustments can be minimized by choosing the speed of adaptation to make equal the two contributions. It is further shown, in Appendix A, that for stationary inputs the LMS adaptive algorithm, based on the method of steepest descent, approaches the theoretical limit of efficiency in terms of misadjustment and speed of adaptation when the eigenvalues of the input correlation matrix are equal or close in value. When the eigenvalues are highly disparate (λmax/λmin> 10), an algorithm similar to LMS but based on Newton's method would approach this theoretical limit very closely.     

Relationships between adaptive minimum variance beamforming andoptimal source localization

For many years, the popular minimum variance (MV) adaptive beamformer has been well known for not having been derived as a maximum likelihood (ML) estimator. This paper demonstrates that by use of a judicious decomposition of the signal and noise, the log-likelihood function of source location is, in fact, directly proportional to the adaptive MV beamformer output power. In the proposed model, the measurement consists of an unknown temporal signal whose spatial wavefront is known as a function of its unknown location, which is embedded in complex Gaussian noise with unknown but positive definite covariance. Further, in cases where the available observation time is insufficient, a constrained ML estimator is derived here that is closely related to MV beamforming with a diagonally loaded data covariance matrix estimate. The performance of the constrained ML estimator compares favorably with robust MV techniques, giving slightly better root-mean-square error (RMSE) angle-of-arrival estimation of a plane-wave signal in interference. More importantly, however, the fact that such optimal ML techniques are closely related to conventional robust MV methods, such as diagonal loading, lends theoretical justification to the use of these practical approaches     

Intersymbol interference and error probability

Using a criterion of minimum average error probability we derive a method for specifying an optimum linear, time invariant receiving filter for a digital data transmission system. The transmitted data are binary and coded into pulses of shapepm s(t). The linear transmission medium introduces intersymbol interference and additive Gaussian noise. Because the intersymbol interference is not Gaussian and can be correlated with the binary digit being detected, our problem is one of deciding which of two waveforms is present in a special type of correlated, non-Gaussian noise. For signal-to-noise ratios in a range of practical interest, the optimum filter is found to be representable as a matched filter followed by a tapped delay line--the same form as that of the least mean square estimator of the pulse amplitude. The performance (error probability vs.S/N) of the optimum filter is compared with that of a matched-filter receiver in an example.     

Parameter estimator based on a minimum discrepancy criterion: aBayesian approach

A new estimation criterion based on the discrepancy between the estimator's error covariance and its information lower bound is proposed. This discrepancy measure criterion tries to take the information content of the observed data into account. A minimum discrepancy estimator (MDE) is then obtained under a linearity assumption. This estimator is shown to be equivalent to the maximum likelihood estimator (MLE), if one assumes that a linear efficient estimator exists and the prior distribution of parameters is uniform. Moreover, it is equivalent to the minimum variance unbiased estimator (MVUE) if the MDE is required to be unbiased. Illustrative examples of MDE and its comparisons with other estimators are given     

On the Joint Synchronization of Clock Offset and Skew in RBS-Protocol

Motivated by the necessity of having a good clock synchronization amongst the nodes of wireless ad-hoc sensor networks, the joint maximum likelihood (JML) estimator for clock phase offset and skew under exponential noise model for reference broadcast synchronization (RBS) protocol is formulated and found via a direct algorithm. The Gibbs sampler is also proposed for joint clock phase offset and skew estimation and shown to provide superior performance relative to JML- estimator. Lower and upper bounds for the mean-square errors (MSE) of JML-estimator and Gibbs Sampler are introduced in terms of the MSE of the uniform minimum variance unbiased (UMVU) estimator and the conventional best linear unbiased estimator (BLUE), respectively.     

On the existence of efficient estimators

The common signal processing problem of estimating some nonrandom parameters of a signal in additive noise is considered. The problem investigated in this paper is under what conditions an efficient estimator exists, i.e., an unbiased estimator with a variance equal to the Cramer-Rao lower bound (CRB). It is well known that if the signal is linear or, more generally, affine in the parameters and the noise Gaussian, an efficient estimator does exist. This paper shows that under some conditions, this is the only case where an efficient estimator exists     

Adaptive recovery of a chirped sinusoid in noise. II. Performanceof the LMS algorithm

For pt.I see ibid., vol.39, no. 3, p.583-94 (1991). The authors present a methodology for evaluating the tracking behavior of the least-mean square (LMS) algorithm for the nontrivial case of recovering a chirped sinusoid in additive noise. A complete closed-form analysis of the LMS tracking properties for a nonstationary inverse system modeling problem is also presented. The mean-square error (MSE) performance of the LMS algorithm is calculated as a function of the various system parameters. The misadjustment or residual of the adaptive filter output is the excess MSE as compared to the optimal filter for the problem. It is caused by three errors in the adaptive weight vector: the mean lag error between the (time-varying mean) weight and the time-varying optimal weight; the fluctuations of the lag error; and the noise misadjustment which is due to the output noise. These results are important because they represent a precise analysis of a nonstationary deterministic inverse modeling system problem with the input being a colored signal. The results are in agreement with the form of the upper bounds for the misadjustment provided by E. Eweda and O. Macchi (1985) for the deterministic nonstationarity     

Second-order parameter estimation

This work provides a general framework for the design of second-order blind estimators without adopting any approximation about the observation statistics or the a priori distribution of the parameters. The proposed solution is obtained minimizing the estimator variance subject to some constraints on the estimator bias. The resulting optimal estimator is found to depend on the observation fourth-order moments that can be calculated analytically from the known signal model. Unfortunately, in most cases, the performance of this estimator is severely limited by the residual bias inherent to nonlinear estimation problems. To overcome this limitation, the second-order minimum variance unbiased estimator is deduced from the general solution by assuming accurate prior information on the vector of parameters. This small-error approximation is adopted to design iterative estimators or trackers. It is shown that the associated variance constitutes the lower bound for the variance of any unbiased estimator based on the sample covariance matrix. The paper formulation is then applied to track the angle-of-arrival (AoA) of multiple digitally-modulated sources by means of a uniform linear array. The optimal second-order tracker is compared with the classical maximum likelihood (ML) blind methods that are shown to be quadratic in the observed data as well. Simulations have confirmed that the discrete nature of the transmitted symbols can be exploited to improve considerably the discrimination of near sources in medium-to-high SNR scenarios.     

MEMS陀螺的抗野值自适应滤波降噪方法

针对微机电系统(MEMS)陀螺随机漂移较大及量测信息中野值对滤波的不利影响,提出了一种抗野值自适应滤波降噪方法。该方法采用Allan方差信息估计量测噪声方差参数,避免了Kalman滤波器与量测噪声估值器之间的相互关联,能有效抑制滤波发散。在此基础上引入新息抗野值算法,通过修正新息去除野值的不利影响,增强对随机漂移的滤波效果。实测数据试验结果表明,采用该文方法滤波后的MEMS陀螺输出信号均方差及角度随机游走都比滤波前明显降低,验证了提出的滤波方法在MEMS陀螺降噪中的有效性。     

Robust adaptive recovery of spread-spectrum signals with short data records

The problem under consideration is the adaptive reception of a multipath direct-sequence spread-spectrum (SS) signal in the presence of unknown correlated SS interference and additive impulsive noise. An SS receiver structure is proposed that consists of a vector of adaptive chip-based Hampel nonlinearities followed by an adaptive auxiliary-vector linear tap-weight filter. The nonlinear receiver front end adapts itself to the unknown prevailing noise environment providing robust performance over a wide range of underlying noise distributions. The adaptive auxiliary-vector linear tap-weight filter allows rapid SS interference suppression with a limited data record. Numerical and simulation studies under finite-data-record system adaptation show significant improvement in bit-error-rate performance over the conventional linear minimum variance-distortionless-response (MVDR) SS receiver or conventional MVDR filtering preceded by vector adaptive chip-based nonlinear processing.     

Reduced-rank multistage receivers for DS-CDMA in frequency-selective fading channels

Multistage (MS) implementation of the minimum mean-square error (MMSE), minimum output energy (MOE), best linear unbiased estimation (BLUE), and maximum-likelihood (ML) filter banks (FBs) is developed based on the concept of the MS Wiener filtering (MSWF) introduced by Goldstein et al. These FBs are shown to share a common MS structure for interference suppression, modulo a distinctive scaling matrix at each filter's output. Based on this finding, a framework is proposed for joint channel estimation and multiuser detection (MUD) in frequency-selective fading channels. Adaptive reduced-rank equal gain combining (EGC) schemes for this family of FBs (MMSE, MOE, BLUE, and ML) are proposed for noncoherent blind MUD of direct-sequence code-division multiple-access systems, and contrasted with the maximal ratio combining counterparts that are also formed with the proposed common structure under the assumption of known channel-state information. The bit-error rate, steady-state output signal-to-interference plus noise ratio (SINR), and convergence of the output SINRs are investigated via computer simulation. Simulation results indicate that the output SINRs attain full-rank performance with much lower rank for a highly loaded system, and that the adaptive reduced-rank EGC BLUE/ML FBs outperform the EGC MMSE/MOE FBs, due to the unbiased nature of the implicit BLUE channel estimators employed in the EGC BLUE/ML schemes.     

BLUE‐Based Channel Estimation Technique for Amplify and Forward Wireless Relay Networks

The best linear unbiased estimator (BLUE) is most suitable for practical application and can be determined with knowledge of only the first and second moments of the probability density function. Although the BLUE is an existing algorithm, it is still largely unexplored and has not yet been applied to channel estimation in amplify and forward (AF)‐based wireless relay networks (WRNs). In this paper, a BLUE‐based algorithm is proposed to estimate the overall channel impulse response between the source and destination of AF strategy‐based WRNs. Theoretical mean square error (MSE) performance for the BLUE is derived to show the accuracy of the proposed channel estimation algorithm. In addition, the Cramér‐Rao lower bound (CRLB) is derived to validate the MSE performance. The proposed BLUE channel estimation algorithm approaches the CRLB as the length of the training sequence and number of relays increases. Further, the BLUE performs better than the linear minimum MSE estimator due to the minimum variance characteristic exhibited by the BLUE, which happens to be a function of signal‐to‐noise ratio.     

Performance of linear reduced-rank multistage receivers for DS-CDMA in frequency-selective fading channels

The performance of a set of linear reduced-rank multistage filter banks is studied in the context of multiuser detection for direct-sequence (DS) code-division multiple-access (CDMA) systems. The set of filter banks under consideration is comprised of the minimum mean-square error (MMSE), the minimum output energy (MOE), the best linear unbiased estimator (BLUE), and the maximum-likelihood (ML) detector. Based on a common framework for the multistage implementations of the aforementioned filter banks, the signal-to-interference plus noise ratios (SINRs) and bit-error rates (BERs) of these reduced-rank filter banks are studied for multipath Rayleigh-fading channels. A generic BER formula is provided for coherent detection and noncoherent differential detection schemes constructed under this common framework. Analysis shows that all of these performance measures are characterized by a kernel matrix K/sub mmse/ whose trace forms the output SINR of the MMSE filter bank. Through investigating the recursive structure of K/sub mmse/, the output SINRs are proven to be monotonically increasing with the number of stages and upper-bounded by a number equal to the paths of the desired user's channel. The condition for asymptotically achieving this upper bound is also provided, which leads to the notion of effective user capacity of linear reduced-rank multiuser detection as well as serves as a test for the existence of a BER floor for coherent detection. In addition, the channel mismatch due to differential detection is also shown to yield a BER floor for noncoherent detection. Based on this analysis, a simple yet effective rule for choosing the number of stages is provided for both coherent and noncoherent linear multistage multiuser detection.     

A stochastic gradient adaptive filter with gradient adaptive stepsize

The step size of this adaptive filter is changed according to a gradient descent algorithm designed to reduce the squared estimation error during each iteration. An approximate analysis of the performance of the adaptive filter when its inputs are zero mean, white, and Gaussian noise and the set of optimal coefficients are time varying according to a random-walk model is presented. The algorithm has very good convergence speed and low steady-state misadjustment. The tracking performance of these algorithms in nonstationary environments is relatively insensitive to the choice of the parameters of the adaptive filter and is very close to the best possible performance of the least mean square (LMS) algorithm for a large range of values of the step size of the adaptation algorithm. Several simulation examples demonstrating the good properties of the adaptive filters as well as verifying the analytical results are also presented     

基于小波-卡尔曼的语音增强方法研究

提出了一种基于小波变换和卡尔曼滤波相结合的语音增强方法,这样既保留了小波变换对自相似过程的去相关作用和多分辨分析的功能,同时又保持了卡尔曼滤波器对未知信号的线性无偏最小方差估计的特点,可以有效地减小非平稳噪声;并引入基于声学模型的感知滤波器,以提高语音信号的可懂度。实验证明该方法对于低信噪比的有色噪声干扰条件下的语音信号的增强效果要优于一般的语音增强系统。     

Adaptive robust spread-spectrum receivers

We consider the problem of robust detection of a spread-spectrum (SS) signal in the presence of unknown correlated SS interference and additive non-Gaussian noise. The proposed general SS receiver structure is comprised by a vector of adaptive chip-based nonlinearities followed by an adaptive linear tap-weight filter and combines the relative merits of both nonlinear and linear signal processing. The novel characteristics of our approach are as follows. First, the nonlinear receiver front-end adapts itself to the unknown prevailing noise environment providing robust performance for a wide range of underlying noise distributions. Second, the adaptive linear tap-weight filter that follows the nonlinearly processed chip samples results in a receiver that is proven to be effective in combating SS interference as well. To determine the receiver parameters, we propose, develop, and study three adaptive schemes under a joint mean-square error (MSE), or a joint bit-error-rate (BER), or a joint MSE-BER optimization criterion. As a side result, we derive the optimum decision fusion filter for receivers that utilize hard-limiting (sign) chip nonlinearities. Numerical and simulation results demonstrate the performance of the proposed schemes and offer comparisons with the conventional matched-filter (MF), the decorrelator, the conventional minimum-variance-distortionless-response (MVDR) filter, and the sign-majority vote receiver     

Adaptive recovery of a chirped sinusoid in noise. I. Performance ofthe RLS algorithm

The authors study the ability of the exponentially weighted recursive least square (RLS) algorithm to track a complex chirped exponential signal buried in additive white Gaussian noise (power P _n). The signal is a sinusoid whose frequency is drifting at a constant rate Ψ. lt is recovered using an M-tap adaptive predictor. Five principal aspects of the study are presented: the methodology of the analysis; proof of the quasi-deterministic nature of the data-covariance estimate R(k); a new analysis of RLS for an inverse system modeling problem; a new analysis of RLS for a deterministic time-varying model for the optimum filter; and an evaluation of the residual output mean-square error (MSE) resulting from the nonoptimality of the adaptive predictor (the misadjustment) in terms of the forgetting rate (β) of the RLS algorithm. It is shown that the misadjustment is dominated by a lag term of order β^-2 and a noise term of order β. Thus, a value β_opt exists which yields a minimum misadjustment. It is proved that β_opt={(M+1)ρΨ²} ^1/3, and the minimum misadjustment is equal to (3/4)P_n(M+1)β_opt, where ρ is the input signal-to-noise ratio (SNR)