Approximate Conditional Mean Research Articles

Sequentially Estimating the Approximate Conditional Mean Using Extreme Learning Machines.

This study examined the extreme learning machine (ELM) applied to the Wald test statistic for the model specification of the conditional mean, which we call the WELM testing procedure. The omnibus test statistics available in the literature weakly converge to a Gaussian stochastic process under the null that the model is correct, and this makes their application inconvenient. By contrast, the WELM testing procedure is straightforwardly applicable when detecting model misspecification. We applied the WELM testing procedure to the sequential testing procedure formed by a set of polynomial models and estimate an approximate conditional expectation. We then conducted extensive Monte Carlo experiments to evaluate the performance of the sequential WELM testing procedure and verify that it consistently estimates the most parsimonious conditional mean when the set of polynomial models contains a correctly specified model. Otherwise, it consistently rejects all the models in the set.

A New Method for Continuous Wave Interference Mitigation in Single-Frequency GPS Receivers

This paper proposes a new method for jamming suppression in the global positioning system (GPS) receivers. The proposed mitigation technique is based on cascading the adaptive finite impulse response (FIR) filter and the approximate conditional mean (ACM) filter. Adaptive FIR filter puts a notch at the interference frequency, but it causes self-noise in presence of high power jammers. ACM filter is an effective filter for estimating the jamming with known autoregressive process model parameters. However, its performance is degraded in the cases of high power interferences. Cascading these two filters has two advantages: first, collaboration of these two filters can help removing high power jammers. Second, the depth of notch can be limited by a threshold in presence of high power jammers to prevent self-noise effect. In such a case, the ACM filter can help to remove the remaining jamming effects on GPS signal. The performance of proposed method is analyzed in presence of single and multiple continuous wave interferences. Experimental results show the benefits of proposed method on terms of signal to noise power ratio improvement and mean square error compared to previous methods.

Adaptive Rake receiver based on the nonlinear ACM technique

AbstractUltra‐wideband (UWB) system is one of the possible solutions to future short‐range indoor data communications with large frequency bandwidth. However, it must coexist with other narrowband wireless systems that may cause interference to each other, and furthermore a large bandwidth will inevitably result in multi‐path fading. The Rake receiver is applicable to combat multi‐path fading but its performance degrades greatly when the narrowband interference (NBI) is present. Although some optimized Rake receivers were proposed to suppress the NBI, such as the minimum mean square error (MMSE) one, their computational complexities are usually too high to be practically implemented. In this paper, we present a new adaptive Rake receiver which can effectively suppress the NBI, based on the nonlinear Masreliez‐type approximate conditional mean (ACM) technique. Simulation results show that it outperforms the previous schemes and even it achieves almost the same performance as that of a MMSE Rake receiver but with much lower complexity. Copyright © 2010 John Wiley & Sons, Ltd.

Optimum Estimation via Gradients of Partition Functions and Information Measures: A Statistical-Mechanical Perspective

In continuation to a recent work on the statistical--mechanical analysis of minimum mean square error (MMSE) estimation in Gaussian noise via its relation to the mutual information (the I-MMSE relation), here we propose a simple and more direct relationship between optimum estimation and certain information measures (e.g., the information density and the Fisher information), which can be viewed as partition functions and hence are amenable to analysis using statistical--mechanical techniques. The proposed approach has several advantages, most notably, its applicability to general sources and channels, as opposed to the I-MMSE relation and its variants which hold only for certain classes of channels (e.g., additive white Gaussian noise channels). We then demonstrate the derivation of the conditional mean estimator and the MMSE in a few examples. Two of these examples turn out to be generalizable to a fairly wide class of sources and channels. For this class, the proposed approach is shown to yield an approximate conditional mean estimator and an MMSE formula that has the flavor of a single-letter expression. We also show how our approach can easily be generalized to situations of mismatched estimation.

Approximate Conditional Mean Particle Filtering for Linear/Nonlinear Dynamic State Space Models

We consider linear systems whose state parameters are separable into linear and nonlinear sets, and evolve according to some known transition distribution, and whose measurement noise is distributed according to a mixture of Gaussians. In doing so, we propose a novel particle filter that addresses the optimal state estimation problem for the aforementioned class of systems. The proposed filter, referred to as the approximate conditional mean particle filter (ACM-PF), is a combination of the approximate conditional mean filter and the sequential importance sampling particle filter. The algorithm development depends on approximating a mixture of Gaussians distribution with a moment-matched Gaussian in the weight update recursion. A condition indicating when this approximation is valid is given. In order to evaluate the performance of the proposed algorithm, we address the blind signal detection problem for an impulsive flat fading channel and the tracking of a maneuvering target in the presence of glint noise. Extensive computer simulations were carried out. For computationally intensive implementations (large number of particles), the proposed algorithm offers performance that is comparable to other state-of-the-art particle filtering algorithms. In the scenario where computational horsepower is heavily constrained, it is shown that the proposed algorithm offers the best performance amongst the considered algorithms for these specific examples.

New approach for suppression of FM jamming in GPS receivers

A new approach is proposed for the suppression of FM jammers in C/A code GPS receivers. This approach is based on the cascading of the augmented-state approximate conditional mean (ASACM) filter and the discrete wavelet transform (DWT) filter. An ASACM filter for single interference suppression in spread-spectrum systems has been reported in the recent literature. However, the formulation of the ASACM filter and its performance analysis is lacking in the case of suppression of multiple FM interferences. The ASACM filter is formulated here for the suppression of multiple FM jammers. Further, a DWT filter using biorthogonal wavelet is suggested for the suppression of FM jammers. Finally, these two filters are cascaded to get an optimum performance for higher jammer to signal ratio, which is of importance in the GPS, for the case of suppression of multiple FM jammers. The performance of the proposed filters is analyzed through simulation examples for the suppression of single and multiple FM jammers in GPS receivers

Narrow-band interference rejection in ds/cdma systems using adaptive (qrd-lsl)-based nonlinear acm interpolators

An Mth order adaptive lattice filter automatically generates all M of the outputs that would be provided by M separate transversal filters. This feature may effectively suppress narrow-band interference (NBI) of either unknown or time-varying bandwidth (or number of frequency bands) in direct-sequence code-division multiple access systems for which the order of the interference rejection filter that achieves the optimal performance is unknown or constantly changing. Moreover, a lattice filter may significantly outperform its transversal counterpart in complex jamming environments in which the adaptive lattice filter must suppress multiple jammers, since each stage of a lattice filter adapts to suppress an orthogonal component of the NBI. The paper develops a computationally efficient and numerically stable adaptive QR-decomposition-based least squares lattice (QRD-LSL)-based nonlinear approximate conditional mean interpolator to suppress NBI effectively. Simulation results demonstrate that both the signal-to-noise ratio improvement and the convergence rate achieved by the proposed interpolators outperform those of other existing prediction-based techniques.

Narrow-band interference rejection in DS/CDMA systems using adaptive (QRD-LSL)-based nonlinear ACM interpolators

An Mth order adaptive lattice filter automatically generates all M of the outputs that would be provided by M separate transversal filters. This feature may effectively suppress narrow-band interference (NBI) of either unknown or time-varying bandwidth (or number of frequency bands) in direct-sequence code-division multiple access systems for which the order of the interference rejection filter that achieves the optimal performance is unknown or constantly changing. Moreover, a lattice filter may significantly outperform its transversal counterpart in complex jamming environments in which the adaptive lattice filter must suppress multiple jammers, since each stage of a lattice filter adapts to suppress an orthogonal component of the NBI. The paper develops a computationally efficient and numerically stable adaptive QR-decomposition-based least squares lattice (QRD-LSL)-based nonlinear approximate conditional mean interpolator to suppress NBI effectively. Simulation results demonstrate that both the signal-to-noise ratio improvement and the convergence rate achieved by the proposed interpolators outperform those of other existing prediction-based techniques.

A new narrowband interference suppression scheme for spread-spectrum CDMA communications

This paper presents a new nonlinear approach for narrowband interference (NBI) suppression in code-division multiple-access (CDMA) systems. The proposed scheme is an adaptive nonlinear predictor that consists of an (N+1)-level quantizer, four adders, and two adaptive linear filters, where N is the number of users in the CDMA system. Both adaptive filters have the same coefficients at each iteration: one for feedforward estimation of NBI and the other for feedback compensation for the estimated result. It could be regarded as an improved version of the nonlinear predictor with offset outputs presented recently by Wang et al. (1996). Computer simulation results support that the improved offset predictor performs much better than the original one under the same complexity. As compared with the nearly optimal approximate conditional mean filter, it achieves almost the same performance even at very low signal-to-noise ratio but involves much less complexity.

Penalized likelihood smoothing in robust state space models

In likelihood-based approaches to robustify state space models, Gaussian error distributions are replaced by non-normal alternatives with heavier tails. Robustified observation models are appropriate for time series with additive outliers, while state or transition equations with heavy-tailed error distributions lead to filters and smoothers that can cope with structural changes in trend or slope caused by innovations outliers. As a consequence, however, conditional filtering and smoothing densities become analytically intractable. Various attempts have been made to deal with this problem, reaching from approximate conditional mean type estimation to fully Bayesian analysis using MCMC simulation. In this article we consider penalized likelihood smoothers, this means estimators which maximize penalized likelihoods or, equivalently, posterior densities. Filtering and smoothing for additive and innovations outlier models can be carried out by computationally efficient Fisher scoring steps or iterative Kalman-type filters. Special emphasis is on the Student family, for which EM-type algorithms to estimate unknown hyperparameters are developed. Operational behaviour is illustrated by simulation experiments and by real data applications.

Adaptive nonlinear filters for narrow-band interference suppression in spread-spectrum CDMA systems

This paper presents a novel nonlinear filter and parameter estimator for narrow band interference suppression in code division multiple access spread-spectrum systems. As in the article by Rusch and Poor (1994), the received sampled signal is modeled as the sum of the spread-spectrum signal (modeled as a finite state independently identically distributed (i.i.d.) process-here we generalize to a finite state Markov chain), narrow-band interference (modeled as a Gaussian autoregressive process), and observation noise (modeled as a zero-mean white Gaussian process). The proposed algorithm combines a recursive hidden Markov model (HMM) estimator, Kalman filter (KF), and the recursive expectation maximization algorithm. The nonlinear filtering techniques for narrow-band interference suppression presented in Rusch and Poor and our proposed HMM-KF algorithm have the same computational cost. Detailed simulation studies show that the HMM-KF algorithm outperforms the filtering techniques in Rusch and Poor. In particular, significant improvements in the bit error rate and signal-to-noise ratio (SNR) enhancement are obtained in low to medium SNR. Furthermore, in simulation studies we investigate the effect on the performance of the HMM-KF and the approximate conditional mean (ACM) filter in the paper by Rusch and Poor, when the observation noise variance is increased. As expected, the performance of the HMM-KF and ACM algorithms worsen with increasing observation noise and number of users. However, HMM-KF significantly outperforms ACM in medium to high observation noise.

Narrow-band interference suppression in spread-spectrum CDMA communications using pipelined recurrent neural networks

This paper investigates the application of pipelined recurrent neural networks (PRNN's) to the narrow-band interference (NBI) suppression over spread-spectrum (SS) code-division multiple-access (CDMA) channels in the presence of additive white Gaussian noise (AWGN) plus non-Gaussian observation noise. Optimal detectors and receivers for such channels are no longer linear. A PRNN that consists of a number of simpler small-scale recurrent neural network (RNN) modules with less computational complexity is conducted to introduce best nonlinear approximation capability into the minimum mean-squared error nonlinear predictor model in order to accurately predict the NBI signal based on adaptive learning for each module from previous non-Gaussian observations. Once the prediction of the NBI signal is obtained, a resulting signal is computed by subtracting the estimate from the received signal. Thus, the effect of the NBI can be reduced. Moreover, since those modules of a PRNN can be performed simultaneously in a pipelined parallelism fashion, this would lead to a significant improvement in its total computational efficiency. Simulation results show that PRNN-based NBI rejection provides a superior signal-to-noise ratio (SNR) improvement relative to the conventional adaptive nonlinear approximate conditional mean (ACM) filters, especially when the channel statistics and exact number of CDMA users are not known to those receivers.

New nonlinear algorithms for narrowband interference suppression in CDMA spread-spectrum systems

The narrow-band interference suppression capability of spread-spectrum systems can be further enhanced by employing interference suppression filters. This paper proposes a number of new nonlinear algorithms for narrow-band interference suppression in code division multiple access spread-spectrum systems. Our research consists of two parts. (1) We propose a multiuser decision-directed Kalman (MDK) filter, which has the same performance as the nonlinear approximate conditional mean (ACM) filter but a much simpler algorithm. (2) We use the nonlinear function in the ACM and the MDK filters to develop nonlinear adaptive least mean square filters with significantly improved performance. Simulation results indicate that our nonlinear algorithms outperform conventional linear ones.

New nonlinear algorithms for estimating and suppressing narrowband interference in DS spread spectrum systems

It has been shown that the narrowband (NB) interference suppression capability of a direct-sequence (DS) spread spectrum system can be enhanced considerably by processing the received signal via a prediction error filter. The conventional approach to this problem makes use of a linear filter. However, the binary DS signal, that acts as noise in the prediction process, is highly non-Gaussian. Thus, linear filtering is not optimal. Vijayan and Poor (1990) first proposed using a nonlinear approximate conditional mean (ACM) filter of the Masreliez (1975) type and obtained significant results. This paper proposes a number of new nonlinear algorithms. Our work consists of three parts. (1) We develop a decision-directed Kalman (DDK) filter, that has the same performance as the ACM filter but a simpler structure. (2) Using the nonlinear function in the ACM and the DDK filters, we develop other nonlinear least mean square (LMS) filters with improved performance. (3) We further use the nonlinear functions to develop nonlinear recursive least squares (RLS) filters that can be used independently as predictors or as interference identifiers so that the ACM or the DDK filter can be applied. Simulations show that our nonlinear algorithms outperform conventional ones.

Nonlinear techniques for interference suppression in spread-spectrum systems

The binary nature of direct-sequence signals is exploited to obtain nonlinear filters that outperform the linear filters hitherto used for this purpose. The case of a Gaussian interferer with known autoregressive parameters is considered. Using simulations, it is shown that an approximate conditional mean (ACM) filter of the Masreliez type performs significantly better than the optimum linear (Kalman-Bucy) filter. For the case of interferers with unknown parameters, the nature of the nonlinearity in the ACM filter is used to obtain an adaptive filtering algorithm that is identical to the linear transversal filter except that the previous prediction errors are transformed nonlinearly before being incorporated into the linear prediction. Two versions of this filter are considered: one in which the filter coefficients are updated using the Widrow LMS algorithm, and another in which the coefficients are updated using an approximate gradient algorithm. Simulations indicate that the nonlinear filter with LMS updates performs substantially better than the linear filter for both narrowband Gaussian and single-tone interferers, whereas the gradient algorithm gives slightly better performance for Gaussian interferers but is rather ineffective in suppressing a sinusoidal interferer. >