Quantum mechanical linear filtering of vector signal processes

The problem of estimating a member of a discrete-time vector process from past and present quantum mechanical measurements is considered; specifically, the minimum-variance linear estimator based on optimal present measurements is studied. Necessary and sufficient conditions that characterize the optimal processing matrix coefficients and the optimal measurements are discussed and interpreted. The optimal linear filter is compared to the optimal quantum estimator without postprocessing of past data. When the signal sequence is pairwise Gaussian and the optimal quantum measurement without postproccssing has the properties that it is linear in a specific sense and that its outcome and the corresponding element of the signal sequence are jointly Gaussian, then the optimal linear filter separates. That is, the optimal measurement can be taken to be the same as thc optimal measurement without regard to past data, and the past and present data are processed classically. Thc results are illustrated by considering the filtering of the in-phase and quadrature amplitudes of a laser field received in a single-mode cavity along with thermal noise. In this case, when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively in a very efficient manner.     

Information fusion Wiener filter for the multisensor multichannel ARMA signals with time-delayed measurements

For the multisensor multichannel autoregressive moving average (ARMA) signals with time-delayed measurements, a measurement transformation approach is presented, which transforms the equivalent state space model with measurement delays into the state space model without measurement delays, and then using the Kalman filtering method, under the linear minimum variance optimal weighted fusion rules, three distributed optimal fusion Wiener filters weighted by matrices, diagonal matrices and scalars are presented, respectively, which can handle the fused filtering, prediction and smoothing problems. They are locally optimal and globally suboptimal. The accuracy of the fuser is higher than that of each local signal estimator. In order to compute the optimal weights, the formulae of computing the cross-covariances among local signal estimation errors are given. A Monte Carlo simulation example for the three-sensor target tracking system with time-delayed measurements shows their effectiveness.     

A denoising approach to multisensor signal estimation

Multisensor array processing of noisy measurements has received considerable attention in many areas of signal processing. The optimal processing techniques developed so far usually assume that the signal and noise processes are at least wide sense stationary, yet a need exists for efficient, effective methods for processing nonstationary signals. Although wavelets have proven to be useful tools in dealing with certain nonstationary signals, the way in which wavelets are to be used in the multisensor setting is still an open question. Based on the structure for optimal linear estimation of nonstationary multisensor data and statistical models of spatial signal coherence, we propose a multisensor denoising algorithm that fully exploits, in a statistically optimal fashion, the additional information afforded by multisensor measurements. Under certain conditions, we show that the proposed estimator can be realized efficiently and robustly in a completely blind fashion, employing only wavelet and discrete Fourier transforms while entailing only a small loss in performance     

Restricted structure optimal linear estimators

The restricted structure optimal deconvolution filtering, smoothing and prediction problem for multivariable, discrete-time linear signal processing problems is considered. A new class of discrete-time optimal linear estimators is introduced that directly minimises a minimum variance criterion but where the structure is prespecified to have a relatively simple form. The resulting estimator can be of much lower order than a Kalman or Wiener estimator and it minimises the estimation error variance, subject to the constraint referred to above. The numerical optimisation algorithm is simple to implement and the full-order optimal solutions are available as a by-product of the analysis. Moreover, the restricted structure solution may be used to compute both IIR and FIR estimators. A weighted H/sub 2/ cost-function is minimised, where the dynamic weighting function can be chosen for robustness improvement. The signal and noise sources can be correlated and the signal channel dynamics can be included in the system model. The algorithm enables low-order optimal estimators to be computed that directly minimise the cost index. The main technical advance is in the pre-processing, which enables the expanded cost expression to be simplified considerably before the numerical solution is obtained. The optimisation provides a direct minimisation over the unknown parameters for the particular estimator structure chosen. This should provide advantages over the simple approximation of a high-order optimal estimator. The results are demonstrated in the estimation of a signal heavily contaminated by both coloured and white noise.     

Fast estimators of time delay and Doppler stretch based ondiscrete-time methods

Estimation of the time delay and the Doppler stretch of a signal is required in several signal processing applications, This paper is focused on the joint fine estimation of these two parameters by a fast interpolation of the estimated ambiguity function. Four discrete-time methods (viz. multirate, piecewise scaling, linear scaling, and indirect estimators), based on an orthogonal model, are introduced. Their mean square error is mathematically derived for random signals corrupted by additive random noises. The obtained expressions have been evaluated for some typical parameter sets in the reference case of Gaussian signals and noises. The numerical results, compared with the accuracy of a continuous-time estimator, show the near efficiency of the multirate estimator for a wide range of SNRs. In fact, the adjustable multirate estimator can operate under near optimal conditions, unlike the approximate (i.e., piecewise and linear) scaling and the indirect methods based on constant sampling rates     

Information fusion steady-state white noise deconvolution estimators with time-delayed measurements and colored measurement noises

White noise deconvolution or input white noise estimation problem has important application backgrounds in oil seismic exploration, communication and signal processing. By the modern time series analysis method, based on the Auto-Regressive Moving Average (ARMA) innovation model, under the linear minimum variance optimal fusion rules, three optimal weighted fusion white noise deconvolution estimators are presented for the multisensor systems with time-delayed measurements and colored measurement noises. They can handle the input white noise fused filtering, prediction and smoothing problems. The accuracy of the fusers is higher than that of each local white noise estimator. In order to compute the optimal weights, the formula of computing the local estimation error cross-covariances is given. A Monte Carlo simulation example for the system with 3 sensors and the Bernoulli-Gaussian input white noise shows their effectiveness and performances.     

The structure of least-mean-square linear estimators for synchronousM-ary signals (Corresp.)

Several authors have shown that the structure of the least-mean-square linear estimator of the sequence of random amplitudes in a synchronous pulse-amplitude-modulated signal that suffers intersymbol interference and additive noise is a matched filter whose output is periodically sampled and passed through a transversal filter (tapped delay line). It is our purpose in this paper to generalize this result to synchronousm-ary signals (e.g., FSK, PSK, PPM signals). We prove that the structure of the least-mean-square linear estimator of the sequence of random parameters in a synchronousm-ary signal, which suffers intersymbol interference and additive noise, is a parallel connection ofmmatched filters followed by tapped delay lines. A similar structure is derived for the continuous waveform estimator of a synchronousm-ary signal. Finally, we present a structure for estimation-decision detection of synchronousm-ary signals, which is based on least-mean-suare linear estimates of aposterioriprobabilities.     

A Recursive Frequency Estimator Using Linear Prediction and a Kalman-Filter-Based Iterative Algorithm

This paper proposes a new Kalman-filter-based recursive frequency estimator for discrete-time multicomponent sinusoidal signals whose frequencies may be time-varying. The frequency estimator is based on the linear prediction approach and it employs the Kalman filter to track the linear prediction coefficients (LPCs) recursively. Frequencies of the sinusoids can then be computed using the estimated LPCs. Due to the coloredness of the linear prediction error, an iterative algorithm is employed to estimate the covariance matrix of the prediction error and the LPCs alternately in the Kalman filter in order to improve the tracking performance. Simulation results show that the proposed Kalman-filter-based iterative frequency estimator can achieve better tracking results than the conventional recursive least-squares-based estimators.     

A competitive minimax approach to robust estimation of random parameters

We consider the problem of estimating, in the presence of model uncertainties, a random vector x that is observed through a linear transformation H and corrupted by additive noise. We first assume that both the covariance matrix of x and the transformation H are not completely specified and develop the linear estimator that minimizes the worst-case mean-squared error (MSE) across all possible covariance matrices and transformations H in the region of uncertainty. Although the minimax approach has enjoyed widespread use in the design of robust methods, we show that its performance is often unsatisfactory. To improve the performance over the minimax MSE estimator, we develop a competitive minimax approach for the case where H is known but the covariance of x is subject to uncertainties and seek the linear estimator that minimizes the worst-case regret, namely, the worst-case difference between the MSE attainable using a linear estimator, ignorant of the signal covariance, and the optimal MSE attained using a linear estimator that knows the signal covariance. The linear minimax regret estimator is shown to be equal to a minimum MSE (MMSE) estimator corresponding to a certain choice of signal covariance that depends explicitly on the uncertainty region. We demonstrate, through examples, that the minimax regret approach can improve the performance over both the minimax MSE approach and a "plug in" approach, in which the estimator is chosen to be equal to the MMSE estimator with an estimated covariance matrix replacing the true unknown covariance. We then show that although the optimal minimax regret estimator in the case in which the signal and noise are jointly Gaussian is nonlinear, we often do not lose much by restricting attention to linear estimators.     

Minimax estimation of unknown deterministic signals in colorednoise

The estimation of a deterministic signal corrupted by random noise is considered. The strategy is to find a linear noncausal estimator which minimizes the maximum mean square error over an a priori set of signals. This signal set is specified in terms of frequency/energy constraints via the discrete Fourier transform. Exact filter expressions are given for the case of additive white noise. For the case of additive colored noise possessing a continuous power spectral density, a suboptimal filter is derived whose asymptotic performance is optimal. Asymptotic expressions for the minimax estimator error are developed for both cases. The minimax filter is applied to random data and is shown to solve asymptotically a certain worst-case Wiener filter problem     

Minimax MSE-ratio estimation with signal covariance uncertainties

In continuation to an earlier work, we further consider the problem of robust estimation of a random vector (or signal), with an uncertain covariance matrix, that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix. While, in the earlier work, we developed and proposed a competitive minimax approach of minimizing the worst-case mean-squared error (MSE) difference regret criterion, here, we study, in the same spirit, the minimum worst-case MSE ratio regret criterion, namely, the worst-case ratio (rather than difference) between the MSE attainable using a linear estimator, ignorant of the exact signal covariance, and the minimum MSE (MMSE) attainable by optimum linear estimation with a known signal covariance. We present the optimal linear estimator, under this criterion, in two ways: The first is as a solution to a certain semidefinite programming (SDP) problem, and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure. We then show that the linear minimax ratio regret estimator can also be interpreted as the MMSE estimator that minimizes the MSE for a certain choice of signal covariance that depends on the uncertainty region. We demonstrate that in applications, the proposed minimax MSE ratio regret approach may outperform the well-known minimax MSE approach, the minimax MSE difference regret approach, and the "plug-in" approach, where in the latter, one uses the MMSE estimator with an estimated covariance matrix replacing the true unknown covariance.     

The Gaussian test channel with an intelligent jammer

The problem of transmitting a sequence of identically distributed independent Gaussian random variables through a Gaussian memoryless channel with a given input power constraint, in the presence of an intelligent jammer, is considered. The jammer taps the channel and feeds back a signal, at a given energy level, for the purpose of jamming the transmitting sequence. Under a square-difference distortion measure which is to be maximized by the jammer and to be minimized by the transmitter and the receiver, this correspondence obtains the complete set of optimal (saddle-point) policies. The solution is essentially unique, and it is structurally different in three different regions in the parameter space, which are determined by the signal-to-noise ratios and relative magnitudes of the noise variances. The best (maximin) policy of the jammer is either to choose a linear function of the measurement he receives through channel-tapping, or to choose, in addition (and additively), an independent Gaussian noise sequence, depending on the region where the parameters lie. The optimal (minimax) policy of the transmitter is to amplify the input sequence to the given power level by a linear transformation, and that of the receiver is to use a Bayes estimator.     

Fast wavelet estimation of weak biosignals

Wavelet-based signal processing has become commonplace in the signal processing community over the past decade and wavelet-based software tools and integrated circuits are now commercially available. One of the most important applications of wavelets is in removal of noise from signals, called denoising, accomplished by thresholding wavelet coefficients in order to separate signal from noise. Substantial work in this area was summarized by Donoho and colleagues at Stanford University, who developed a variety of algorithms for conventional denoising. However, conventional denoising fails for signals with low signal-to-noise ratio (SNR). Electrical signals acquired from the human body, called biosignals, commonly have below 0 dB SNR. Synchronous linear averaging of a large number of acquired data frames is universally used to increase the SNR of weak biosignals. A novel wavelet-based estimator is presented for fast estimation of such signals. The new estimation algorithm provides a faster rate of convergence to the underlying signal than linear averaging. The algorithm is implemented for processing of auditory brainstem response (ABR) and of auditory middle latency response (AMLR) signals. Experimental results with both simulated data and human subjects demonstrate that the novel wavelet estimator achieves superior performance to that of linear averaging.     

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.     

Restricted-structure linear estimators for multiple-model systems

A new class of discrete-time optimal linear estimators is introduced for multiple-model systems that minimises a minimum-variance criterion but where the structure is prespecified to have a simple low-order form. The restricted-structure estimator can be of much lower order than a Kalman (1961) or Wiener (1949) estimator and it minimises the estimation-error variance, subject to the constraint referred to. The numerical optimisation algorithm is simple to implement and full-order optimal solutions are available as a by-product of the analysis. The algorithm enables low-order optimal estimators to be computed that directly minimise the cost index across a set of possible linear signal or noise source models. The main technical advances lie in the theoretical analysis that enables the expanded cost expression to be simplified before the numerical solution is obtained, and the extension of the restricted-structure optimisation technique to multiple-model systems     

A new recursive filter for systems with multiplicative noise

An optimal linear recursive minimum mean-square-error estimator was previously developed by the authors (see IEEE Trans. Autom. Control, vol.34, no.5, p.568-74, May 1989) for a zero-mean signal corrupted by multiplicative noise in its measurement model. This recursive filter cannot be obtained by the recursive structure of a conventional Kalman filter where the new estimate is a linear combination of the previous estimate and the new data. Instead, the recursive structure was achieved by combining the previous estimate with recursive innovation, a linear combination of the most recent two data samples and the previous estimate. In this work the signal is extended to be nonzero-mean. In the conventional Kalman filter, the superposition principle can be applied to both the signal and the measurement models for this nonzero-mean extension. However, when multiplicative noise exists, the measurement model becomes nonlinear. Therefore, a new recursive structure for the innovation process needs to be developed to achieve a recursive filter     

Probabilistic data association techniques for target tracking in clutter

In tracking targets with less-than-unity probability of detection in the presence of false alarms (FAs), data association-deciding which of the received multiple measurements to use to update each track-is crucial. Most algorithms that make a hard decision on the origin of the true measurement begin to fail as the FA rate increases or with low observable (low probability of target detection) maneuvering targets. Instead of using only one measurement among the received ones and discarding the others, an alternative approach is to use all of the validated measurements with different weights (probabilities), known as probabilistic data association (PDA). This paper presents an overview of the PDA technique and its application for different target tracking scenarios. First, it describes the use of the PDA technique for tracking low observable targets with passive sonar measurements. This target motion analysis is an application of the PDA technique, in conjunction with the maximum-likelihood approach, for target motion parameter estimation via a batch procedure. Then, the PDA technique for tracking highly maneuvering targets and for radar resource management is illustrated with recursive state estimation using the interacting multiple model estimator combined with PDA. Finally, a sliding window (which can also expand and contract) parameter estimator using the PDA approach for tracking the state of a maneuvering target using measurements from an electrooptical sensor is presented.     

Performance of iterative data detection and channel estimation for single-antenna and multiple-antennas wireless communications

In iterative data-detection and channel-estimation algorithms, the channel estimator and the data detector recursively exchange information in order to improve the system performance. While a vast bulk of the available literature demonstrates the merits of iterative schemes through computer simulations, in this paper analytical results on the performance of an iterative detection/estimation scheme are presented. In particular, this paper focus is on uncoded systems and both the situations that the receiver and the transmitter are equipped with either a single antenna or multiple antennas are considered. With regard to the channel estimator, the analysis considers both the minimum mean square error and the maximum likelihood channel estimate, while, with regard to the data detector, linear receiver interfaces are considered. Closed-form formulas are given for the channel-estimation mean-square error and for its Crame/spl acute/r-Rao bound, as well as for the error probability of the data detector. Moreover, the problem of the optimal choice of the length of the training sequence is also addressed. Overall, results show that the considered iterative strategy achieves excellent performance and permits, at the price of some complexity increase, the use of very short training sequences without incurring any performance loss. Finally, computer simulations reveal that the experimental results are in perfect agreement with those predicted by the theoretical analysis.     

Combined Coding and Training for Unknown ISI Channels

Combined Coding and Training for Unknown ISI Channels The traditional method of sending a training signal to identify a channel, followed by data, may be viewed as a simple code for the unknown channel. Results in blind sequence detection suggest that performance similar to this traditional approach can be obtained without training. However, for short packets and/or time-recursive algorithms, significant error floors exist due to the presence of sequences that are indistinguishable without knowledge of the channel. In this paper, we reconsider training signal design in light of recent results in blind sequence detection. Specifically, we consider the tradeoff between the complexity of receiver processing and the amount of training overhead required. More generally, we design training codes which combine modulation and training. In order to design these codes, we find an expression for the pairwise error probability of the joint maximum-likelihood (JML) channel and sequence estimator. This expression motivates a pairwise distance for the JML receiver based on principal angles between the range spaces of data matrices. The general code-design problem (generalized sphere packing) is formulated as the clique problem associated with an unweighted, undirected graph. We provide optimal and heuristic algorithms for this clique problem. For both long and short packets, we demonstrate that significant improvements are possible by jointly considering the design of the training, modulation, and receiver processing.