Cross-correlation Vector Research Articles

A generalized data windowing scheme for adaptive conjugate gradient algorithms

The performance of the modified adaptive conjugate gradient (CG) algorithms based on the iterative CG method for adaptive filtering is highly related to the ways of estimating the correlation matrix and the cross-correlation vector. The existing approaches of implementing the CG algorithms using the data windows of exponential form or sliding form result in either loss of convergence or increase in misadjustment. This paper presents and analyzes a new approach to the implementation of the CG algorithms for adaptive filtering by using a generalized data windowing scheme. For the new modified CG algorithms, we show that the convergence speed is accelerated, the misadjustment and tracking capability comparable to those of the recursive least squares (RLS) algorithm are achieved. Computer simulations demonstrated in the framework of linear system modeling problem show the improvements of the new modifications.

Unified approach to adaptive filters and their performance

A streamlined theory is presented for adaptive filters within which the major adaptive filter algorithms can be seen as special cases. The algorithm development part of the theory involves three ingredients: a preconditioned Wiener Hopf equation, its simplest possible iterative solution through the Richardson iteration, and an estimation strategy for the autocorrelation matrix, the cross-correlation vector and a preconditioning matrix. This results in a generalised adaptive filter in which intuitively plausible parameter selections give the major adaptive filter algorithms as special cases. This provides a setting where the similarities and differences between the many different adaptive filter algorithms are clearly and explicitly exposed. Based on the authors' generalised adaptive filter, expressions for the learning curve, the excess mean square error and the mean square coefficient deviation are developed. These are general performance results that are directly applicable to the major families of adaptive filter algorithms through the selection of a few parameters. Finally, the authors demonstrate through simulations that these results are useful in predicting adaptive filter performance.

Robust extended multidelay filter and double-talk detector for acoustic echo cancellation

We propose an integrated acoustic echo cancellation solution based on a novel class of efficient and robust adaptive algorithms in the frequency domain, the extended multidelay filter (EMDF). The approach is tailored to very long adaptive filters and highly auto-correlated input signals as they arise in wideband full-duplex audio applications. The EMDF algorithm allows an attractive tradeoff between the well-known multidelay filter and the recursive least-squares algorithm. It exhibits fast convergence, superior tracking capabilities of the signal statistics, and very low delay. The low computational complexity of the conventional frequency-domain adaptive algorithms can be maintained thanks to efficient fast realizations. We also show how this approach can be combined efficiently with a suitable double-talk detector (DTD). We consider a corresponding extension of a recently proposed DTD based on a normalized cross-correlation vector whose performance was shown to be superior compared to other DTDs based on the cross-correlation coefficient. Since the resulting DTD also has an EMDF structure it is easy to implement, and the fast realization also carries over to the DTD scheme. Moreover, as the robustness issue during double talk is particularly crucial for fast-converging algorithms, we apply the concept of robust statistics into our extended frequency-domain approach. Due to the robust generalization of the cost function leading to a so-called M-estimator, the algorithms become inherently less sensitive to outliers, i.e., short bursts that may be caused by inevitable detection failures of a DTD. The proposed structure is also well suited for an efficient generalization to the multichannel case

Adaptive filtering algorithms designed using control Liapunov functions

The standard conjugate gradient (CG) method uses orthogonality of the residues to simplify the formulas for the parameters necessary for convergence. In adaptive filtering, the sample-by-sample update of the correlation matrix and the cross-correlation vector causes a loss of the residue orthogonality in a modified online algorithm, which, in turn, results in loss of convergence and an increase of the filter quadratic mean error. This letter extends a recently proposed control Liapunov function analysis of the CG method viewed as a dynamic system in the standard feedback configuration to the case of adaptive filtering.

Blind adaptive channel equalization using multichannel linear prediction-based cross-correlation vector estimation

Blind equalization of transmission channel is important in communication areas and signal processing applications because it does not need training sequences, nor does it require a priori channel information. In this paper, an adaptive blind MMSE channel equalization technique based on second-order statistics is investigated. We present an adaptive blind MMSE channel equalization using multichannel linear prediction error method for estimating cross-correlation vector. They can be implemented as RLS or LMS algorithms to recursively update the cross-correlation vector. Once cross-correlation vector is available, it can be used for MMSE channel equalization. Unlike many known subspace methods, our proposed algorithms do not require channel order estimation. Therefore, our algorithms are robust to channel order mismatch. Performance of our algorithms and comparison with existing algorithms are shown for real measured digital microwave channel.

A Multidelay Double-Talk Detector Combined with the MDF Adaptive Filter

The multidelay block frequency-domain (MDF) adaptive filter is an excellent candidate for both acoustic and network echo cancellation. There is a need for a very good double-talk detector (DTD) to be combined efficiently with the MDF algorithm. Recently, a DTD based on a normalized cross-correlation vector was proposed and it was shown that this DTD performs much better than the Geigel algorithm and other DTDs based on the cross-correlation coefficient. In this paper, we show how to extend the definition of a normalized cross-correlation vector in the frequency domain for the general case where the block size of the Fourier transform is smaller than the length of the adaptive filter. The resulting DTD has an MDF structure, which makes it easy to implement, and a good fit with an echo canceler based on the MDF algorithm. We also analyze resource requirements (computational complexity and memory requirement) and compare the MDF algorithm with the normalized least mean square algorithm (NLMS) from this point of view.

Tracking performance of the RLS algorithm applied to an antenna array in a realistic fading environment

In this paper, frequency domain techniques are used to derive the tracking properties of the recursive least squares (RLS) algorithm applied to an adaptive antenna array in a mobile fading environment, expanding the use of such frequency domain approaches for nonstationary RLS tracking to the interference canceling problem that characterizes the use of antenna arrays in mobile wireless communications. The analysis focuses on the effect of the exponential weighting of the correlation estimation filter and its effect on the estimations of the time variant autocorrelation matrix and cross-correlation vector. Specifically, the case of a flat Rayleigh fading desired signal applied to an array in the presence of static interferers is considered with an AR2 fading process approximating the Jakes' fading model. The result is a mean square error (MSE) performance metric parameterized by the fading bandwidth and the RLS exponential weighting factor, allowing optimal parameter selection. The analytic results are verified and demonstrated with a simulation example.

A multichannel acoustic echo canceler double‐talk detector based on a normalized cross‐correlation matrix

AbstractMultichannel acoustic echo cancellation is basically composed of two parts. One part is a multichannel system identification problem which is nontrivial to solve. The other part, which is also nontrivial, is the so‐called double‐talk detection problem. Near‐end speech detection is based on a test statistic. Recently, a new double‐talk test statistic based on the normalized cross‐correlation vector was proposed For the single‐channel case. Obviously, in the multichannel case, there are several solutions, but what should be the optimal one is not yet known. Then, a fundamental question arises: how do we deal optimally with multiple channels? In this paper, we generalize the idea of normalized cross‐correlation vector (single‐channel) to the matrix case (multichannel), derive a frequency‐domain version, and show how to combine both the multichannel frequency‐domain adaptive filter and the multichannel double‐talk detector.

A frequency-domain double-talk detector based on a normalized cross-correlation vector

A key component for hands-free, full-duplex, communication technology is the echo canceler. An echo canceler consists primarily of an adaptive filter and a control device called the double-talk detector. We derive a test statistic based on a normalized cross-correlation vector for the multichannel frequency-domain adaptive algorithm. The advantages of this approach are: low computational complexity, simplicity of implementation, and better performance than the classical Geigel algorithm.

Direct blind MMSE channel equalization based on second-order statistics

A family of new MMSE blind channel equalization algorithms based on second-order statistics are proposed. Instead of estimating the channel impulse response, we directly estimate the cross-correlation function needed in Wiener-Hopf filters. We develop several different schemes to estimate the cross-correlation vector, with which different Wiener filters are derived according to minimum mean square error (MMSE). Unlike many known sub-space methods, these equalization algorithms do not rely on signal and noise subspace separation and are consequently more robust to channel order estimation errors. Their implementation requires no adjustment for either single- or multiple-user systems. They can effectively equalize single-input multiple-output (SIMO) systems and can reduce the multiple-input multiple-output (MIMO) systems into a memoryless signal mixing system for source separation. The implementations of these algorithms on SIMO system are given, and simulation examples are provided to demonstrate their superior performance over some existing algorithms.

A Modular Analog NLMS Structure for Adaptive Filtering

This paper proposes a modular Analog Adaptive filter (AAF) algorithm, in which the coefficient adaptation is carried out by using a time varying step size analog normalized LMS (NLMS) algorithm, which is implemented as an external analog structure. The proposed time varying step size is estimated by using the first element of the crosscorrelation vector between the output error and reference signal, and the first element of the crosscorrelation vector between the output error and the adaptive filter output signal, respectively. Proposed algorithm reduces distortion when additive noise power increases or DC offsets are present, without significatively decreasing the convergence rate nor increasing the complexity of the conventional NLMS algorithms. Simulation results show that proposed algorithm improves the performance of AAF when DC offsets are present. The proposed VLSI structure for the time varying step size normalized NLMS algorithm has, potentially, a very small size and faster convergence rates than its digital counterparts. It is suitable for general purpose applications or oriented filtering solution such as echo cancellation and equalization in cellular telephony in which high performance, low power consumption, fast convergence rates and small size adaptive digital filters (ADF) are required. The convergence performance of analog adaptive filters using integrators like first order low pass filter is analyzed.

EDQNM model of a passive scalar with a uniform mean gradient

Dynamic equations for the scalar autocorrelation and scalar-velocity cross correlation spectra have been derived for a passive scalar with a uniform mean gradient using the Eddy Damped Quasi Normal Markovian (EDQNM) theory. The presence of a mean gradient in the scalar field makes all correlations involving the scalar axisymmetric with respect to the axis pointing in the direction of the mean gradient. Equivalently, all scalar spectra will be functions of the wave number k and the cosine of the azimuthal angle designated as μ. In spite of this complication, it is shown that the cross correlation vector can be completely characterized by a single scalar function Q(k). The scalar autocorrelation spectrum, in contrast, has an unknown dependence on μ. However, this dependency can be expressed as an infinite sum of Legendre polynomials of μ, as first suggested by Herring [Phys. Fluids 17, 859 (1974)]. Furthermore, since the scalar field is initially zero, terms beyond the second order of the Legendre expansion are shown to be exactly zero. The energy, scalar autocorrelation, and scalar-velocity cross correlation were solved numerically from the EDQNM equations and compared to results from direct numerical simulations. The results show that the EDQNM theory is effective in describing single-point and spectral statistics of a passive scalar in the presence of a mean gradient.

Fast transversal versions of the RIV algorithm

The system of linear equations A(n)w(n)=b(n) where A(n) is the sample cross-correlation matrix between an observed process and an instrumental variable process and b(n) is the cross-correlation vector between some desired process and the instrumental variable process, is frequently encountered. For example, the ‘normal’ equations for the AR parameters of ARMA processes based on cumulants can be interpreted as cross-correlation matrices. For a p × p matrix A(n) the recursive instrumental variable (RIV) algorithm requires computations of order p2. We develop exact fast versions which require computations of order p. This is a generalization of the fast transversal filter algorithms of Cioffi et al., who assume the matrix A to be Hermitian. Additionally, we analyse the tracking behaviour and misadjustment aspects of RIV when λ, the forgetting factor, is less than unity.

Analyzing pulmonary capillary pressure

A new mutual wavelet packets scheme is proposed for subband adaptive filtering. Its relation to the existing literature on the subject is outlined. The proposed scheme performs a decomposition of the two signals used in the adaptation process onto the same optimal basis. That basis is chosen so that it favors the adaptation performance. A new criterion, based on a functional of the two signals, is introduced. This proposed criterion consists in the maximization of the magnitude of the cross-correlation samples of the two signals considered. This maximizes the cross-correlation vector of the well known normal equations. At the expense of a slightly higher computational complexity, the mutual wavelet packets scheme reduces the aliasing drawback of regular subband adaptive filtering, preserving its main advantages. Simulations have been performed on pulmonary capillary pressure transients for respiratory interference cancelling. The method proved to be more efficient than classical methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of the adaptive correlation enhancer

Adaptive filtering has become a major component in the field of digital signal processing, presumably because of the self-adjusting property of the filters. Because noise statistics are often unknown or are time-varying, designing a fixed filter that performs well in all cases may not be possible. The adaptive filter is able to adjust the impulse response according to the input data and normally provide a filter suitable for processing the current input data. The majority of adaptive algorithms require the use of feedback and the minimization of an error criterion, such as mean-squared error (MSE). To minimize the error, the coefficients of the filter are recursively updated in a manner that forces error reduction. The most well-known technique for adjusting the filter weights is the method of steepest descent, which searches for the minimum point on a performance surface by finding the gradient and adjusting the filter weights in a direction opposite the gradient [l, 21. Computation of the gradient requires knowledge of the correlation matrix of the input data and a cross-correlation vector between the input data and data which represent the desired filter output. These statistical quantities are normally not known in advance, making the method of steepest descent extremely difficult to implement. Because the necessary statistical parameters were not available, instantaneous estimates were used, which led to the popular least-mean-square (LMS) adaptive filter update algorithm [ 1,2 1. The LMS algorithm, while easy to implement, is capable of providing excellent performance, even though it has relatively slow convergence characteristics and its performance is governed by the ratio of the maximum and

Multichannel recursive least squares adaptive filtering without a desired signal

The author presents a pair of adaptive QR decomposition-based algorithms for the adaptive mixed filter in which no desired signal is available, but the signal-to-data cross-correlation vector is known. The algorithms are derived by formulating the recursive mixed filter as a least-squares problem and then applying orthogonal QR-based techniques in its solution. This leads to algorithms with the performance, numerical, and structural advantages of the RLS/QR algorithm, but without the requirement of a desired signal. Both Givens and square-root-free Givens rotations are used in implementing the recursive QR decomposition. Because of their structural regularity, the algorithms are easily implemented by triangular systolic array structures. Simulations show that these algorithms require fewer computations and less precision than recursive sample matrix inversion approaches.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The performance of the SMI method in the constrained LMS array and the Griffiths array

The sample matrix inversion (SMI) method is applied in the constrained least mean square (LMS) array of the Griffiths (1959) array. The mean square error (MSE) performance measure of the SMI method in both arrays is presented. The total output power (TOP) performance of the SMI method in the constrained LMS array is also presented. In using the SMI method to obtain the MSE and the TOP of the constrained LMS array, the adaptive weight vector is computed based on the estimate of the received signal correlation matrix and the knowledge of the arrival angle of the desired signal. The weight vector of the Griffiths array is also computed, based on the estimate of the correlation matrix and the knowledge of the cross correlation vector between the reference signal and the received signal.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fuzzy Notions in Nonlinear System Classification

Abstract Pattern Recognition methods like the A-nearest neighbor technique and clustering have been successfully used to classify nonlinear systems to a predetermined set of well separated classes, demonstrating similarities in their input-output behavior. The cross-correlation function has been used as a feature vector which yielded a classification with a relatively small misclassification error. Further categorization of the nonlinearity within a class is not possible through the above approach. However, a membership factor in a fuzzy sense can be assigned to each nonlinear system within a certain class. This factor is dependent on the coefficients of the polynomial expansion of the nonlinearity involved and may be obtained through the cross-correlation vector. This way, every class of nonlinear systems may be defined as a fuzzy set. The investigation of the properties of such a set as well as its contribution to the further classification of nonlinear systems is discussed in this paper.