Input Autocorrelation Matrix Research Articles

GNSS array receiver faced with overloaded interferences: anti-jamming performance and the incident directions of interferences

Anti-jamming solutions based on antenna arrays enhance the anti-jamming ability and the robustness of global navigation satellite system (GNSS) receiver remarkably. However, the performance of the receiver will deteriorate significantly in the overloaded interferences scenario. We define the overloaded interferences scenario as where the number of interferences is more than or equal to the number of antenna arrays elements. In this paper, the effect mechanism of interferences with different incident directions is found by studying the anti-jamming performance of the adaptive space filter. The theoretical analysis and conclusions, which are first validated through numerical examples, reveal the relationships between the optimal weight vector and the eigenvectors of the input signal autocorrelation matrix, the relationships between the interference cancellation ratio (ICR), the signal to interference and noise power ratio (SINR) of the adaptive space filter output and the number of interferences, the eigenvalues of the input signal autocorrelation matrix. In addition, two simulation experiments are utilized to further corroborate the theoretical findings through soft anti-jamming receiver. The simulation results match well with the theoretical analysis results, thus validating the effect mechanism of overloaded interferences. The simulation results show that, for a four elements circular array, the performance difference is up to 19 dB with different incident directions of interferences. Anti-jamming performance evaluation and jamming deployment optimization can obtain more accurate and efficient results by using the conclusions.

A Fast Quasi-Newton Adaptive Algorithm Based on Approximate Inversion of the Autocorrelation Matrix

The Newton adaptive filtering algorithm in its original form is computationally very complex as it requires inversion of the input-signal autocorrelation matrix at every time step. Also, it may suffer from stability problems due to the inversion of the input-signal autocorrelation matrix. In this paper, we propose to replace the inverse of the input-signal autocorrelation matrix by an approximate one, assuming that the input-signal autocorrelation matrix is Toeplitz. This assumption would help us in replacing the update of the inverse of the autocorrelation matrix by the update of the autocorrelation matrix itself, and performing the multiplication of R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> x in the update equation by using the Fourier transform. This would increase the stability of the algorithm, in one hand, and decrease its computational complexity, on the other hand. Since the objective of the paper is to enhance the stability of the Newton algorithm, the performance of the proposed algorithm is compared to those of the Newton and the improved quasi-Newton (QN) algorithms in noise cancellation and system identification settings.

Optimal preconditioned regularization of least mean squares algorithm for robust online learning1

Despite its low computational cost, and steady state behavior, some well known drawbacks of the least means squares (LMS) algorithm are: slow rate of convergence and unstable behaviour for ill conditioned autocorrelation matrices of input signals. Several modified algorithms have been presented with better convergence speed, however most of these algorithms are expensive in terms of computational cost and time, and sometimes deviate from optimal Wiener solution that results in a biased solution of online estimation problem. In this paper, the inverse Cholesky factor of the input autocorrelation matrix is optimized to pre-whiten input signals and improve the robustness of the LMS algorithm. Furthermore, in order to have an unbiased solution, mean squares deviation (MSD) is minimized by improving convergence in misalignment. This is done by regularizing step-size adaptively in each iteration that helps in developing a highly efficient optimal preconditioned regularized LMS (OPRLMS) algorithm with adaptive step-size. Comparison of OPRLMS algorithm with other LMS based algorithms is given for unknown system identification and noise cancelation from ECG signal, that results in preference of the proposed algorithm over the other variants of LMS algorithm.

Frequency domain quaternion adaptive filters: Algorithms and convergence performance

Recently, adaptive filtering was extended to quaternion-valued systems. Quaternion-valued algorithms exhibit improved geometrical properties compared with real- and complex-valued algorithms. Moreover, working in the frequency domain allows a fast execution along with a good convergence performance. In this work, we propose three different quaternion-valued adaptive algorithms operating in the frequency domain. Convergence properties are also analyzed: in particular, the step size stability range is obtained in relation to the eigenvalues of the input autocorrelation matrix and the Excess Mean Square Error (EMSE) is expressed in relation to the algorithm parameters. Finally, simulations support the proposal.

Adaptive Polynomial Filtering using Generalized Variable Step-Size Least Mean pth Power (LMP) Algorithm

This correspondence presents the adaptive polynomial filtering using the generalized variable step-size least mean $$p$$ p th power (GVSS-LMP) algorithm for the nonlinear Volterra system identification, under the $$\alpha $$ ? -stable impulsive noise environment. Due to the lack of finite second-order statistics of the impulse noise, we espouse the minimum error dispersion criterion as an appropriate metric for the estimation error, instead of the conventional minimum mean square error criterion. For the convergence of LMP algorithm, the adaptive weights are updated by adjusting $$p\ge 1$$ p ? 1 in the presence of impulsive noise characterized by $$1<\alpha <2$$ 1 < ? < 2 . In many practical applications, the autocorrelation matrix of input signal has the larger eigenvalue spread in the case of nonlinear Volterra filter than in the case of linear finite impulse response filter. In such cases, the time-varying step-size is an appropriate option to mitigate the adverse effects of eigenvalue spread on the convergence of LMP adaptive algorithm. In this paper, the GVSS updating criterion is proposed in combination with the LMP algorithm, to identify the slowly time-varying Volterra kernels, under the non-Gaussian $$\alpha $$ ? -stable impulsive noise scenario. The simulation results are presented to demonstrate that the proposed GVSS-LMP algorithm is more robust to the impulsive noise in comparison to the conventional techniques, when the input signal is correlated or uncorrelated Gaussian sequence, while keeping $$1<p<\alpha <2$$ 1 < p < ? < 2 . It also exhibits flexible design to tackle the slowly time-varying nonlinear system identification problem.

Enhancing the tracking capability of recursive least p-norm algorithm via adaptive gain factor

The main drawback of the recursive least p-norm (RLpN) adaptive-filtering algorithm is a poor tracking performance in the presence of abrupt changes in the model. In this paper, a new method to enhance tracking capability of the RLpN (ET-RLpN) algorithm is proposed, which uses the adaptive gain factor in the cross-correlation vector and the input-signal autocorrelation matrix to enhance tracking capability. Simulation results in system identification and echo cancellation applications are presented, which demonstrate that the ET-RLpN achieves improved tracking capability compared to the conventional RLpN and controlled adaptive combination of two RLpN filters (CAC-RLpN).

Small-Sample-Support Suppression of Interference to PN-Masked Data

In the context of secure wireless communications, pseudo-noise (PN) masking of transferred data has proven to be an effective technique against eavesdropping (notable examples are military-grade communication and global-positioning systems). At the same time, PN-masked transmissions are thought to be vulnerable to interference/jamming due to lack of a minimum-mean-square-error (MMSE) disturbance suppressing solution. In this paper, for the first time we establish the MMSE operation for masked data in the form of a time (mask) varying linear filter, suggest an implementation that avoids repeated input autocorrelation matrix inversion, and develop an auxiliary-vector (AV) MMSE filter estimator with state-of-the-art short-data-record estimation performance. Simulation examples included herein illustrate the theoretical developments.

Robust Recursive Least-Squares Adaptive-Filtering Algorithm for Impulsive-Noise Environments

A new robust recursive least-squares (RLS) adaptive filtering algorithm that uses a priori error-dependent weights is proposed. Robustness against impulsive noise is achieved by choosing the weights on the basis of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norms of the crosscorrelation vector and the input-signal autocorrelation matrix. The proposed algorithm also uses a variable forgetting factor that leads to fast tracking. Simulation results show that the proposed algorithm offers improved robustness as well as better tracking compared to the conventional RLS and recursive least-M estimate adaptation algorithms.

Improved Quasi-Newton Adaptive-Filtering Algorithm

An improved quasi-Newton (QN) algorithm that performs data-selective adaptation is proposed whereby the weight vector and the inverse of the input-signal autocorrelation matrix are updated only when the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> error exceeds a prespecified error bound. The proposed algorithm also incorporates an improved estimator of the inverse of the autocorrelation matrix. With these modifications, the proposed QN algorithm takes significantly fewer updates to converge and yields a reduced steady-state misalignment relative to a known QN algorithm proposed recently. These features of the proposed QN algorithm are demonstrated through extensive simulations. Simulations also show that the proposed QN algorithm, like the known QN algorithm, is quite robust with respect to roundoff errors introduced in fixed-point implementations.

On the convergence of the LMS algorithm with a rank-deficient input autocorrelation matrix

In all books and papers on adaptive filtering, the input autocorrelation matrix R xx is always considered positive definite and hence the theoretical Wiener–Hopf normal equations ( R xx h= r xd ) have a unique solution h= h opt (“ there is only a single global optimum”, [B. Widrow, S. Stearns, Adaptive Signal Processing, Prentice-Hall, 1985, p. 21]) due to the invertibility of R xx (i.e., it is full-rank). But what if R xx is positive semi-definite and not full-rank? In this case the Wiener–Hopf normal equations are still consistent but with an infinite number of possible solutions. Now it is well known that the filter coefficients of the least mean square (LMS), stochastic gradient algorithm, converge (in the mean) to the unique Wiener–Hopf solution ( h opt) when R xx is full-rank. In this paper, we will show that even when R xx is not full-rank it is still possible to predict the (convergence) behaviour of the LMS algorithm based upon knowledge of R xx , r xd and the initial conditions of the filter coefficients.

Cholesky Factorization-Based Adaptive BLAST DFE for Wideband MIMO Channels

Adaptive equalization of wireless systems operating over time-varying and frequency-selective multiple-input multiple-output (MIMO) channels is considered. A novel equalization structure is proposed, which comprises a cascade of decision feedback equalizer (DFE) stages, each one detecting a single stream. The equalizer filters, as well as the ordering by which the streams are extracted, are updated based on the minimization of a set of least squares (LS) cost functions in a BLAST-like fashion. To ensure numerically robust performance of the proposed algorithm, Cholesky factorization of the equalizer input autocorrelation matrix is applied. Moreover, after showing that the equalization problem possesses an order recursive structure, a computationally efficient scheme is developed. A variation of the method is also described, which is appropriate for slow time-varying conditions. Theoretical analysis of the equalization problem reveals an inherent numerical deficiency, thus justifying our choice of employing a numerically robust algebraic transformation. The performance of the proposed method in terms of convergence, tracking, and bit eror rate (BER) is evaluated through extensive computer simulations for time-varying and wideband channels.

Wavelet-Based MPNLMS Adaptive Algorithm for Network Echo Cancellation

The µ-law proportionate normalized least mean square (MPNLMS) algorithm has been proposed recently to solve the slow convergence problem of the proportionate normalized least mean square (PNLMS) algorithm after its initial fast converging period. But for the color input, it may become slow in the case of the big eigenvalue spread of the input signal's autocorrelation matrix. In this paper, we use the wavelet transform to whiten the input signal. Due to the good time-frequency localization property of the wavelet transform, a sparse impulse response in the time domain is also sparse in the wavelet domain. By applying the MPNLMS technique in the wavelet domain, fast convergence for the color input is observed. Furthermore, we show that some nonsparse impulse responses may become sparse in the wavelet domain. This motivates the usage of the wavelet-based MPNLMS algorithm. Advantages of this approach are documented.

Subspace Direction Finding With an Auxiliary-Vector Basis

We develop a new subspace direction-of-arrival (DOA) estimation procedure that utilizes a noneigenvector basis. Computation of the basis is carried out by a modified version of the orthogonal auxiliary-vector (AV) filtering algorithm. The procedure starts with the linear transformation of the array response scanning vector by the input autocorrelation matrix. Then, successive orthogonal maximum cross-correlation auxiliary vectors are calculated to form a basis for the scanner-extended signal subspace. As a performance evaluation example, our studies for uncorrelated sources demonstrate a gain in the order of 15 dB over MUSIC, 7 dB over ESPRIT, and 3 dB over the grid-search maximum likelihood DOA estimator at probability of resolution 0.9 with a ten-element array and reasonably small observation data records. Results for correlated sources are reported as well

Adaptive Volterra filtering using M-band wavelet transform and Gram-Schmidt orthogonalisation

A new adaptive Volterra filter with fast convergence is proposed, where the M-band discrete wavelet transform and Gram-Schmidt orthogonalisation are utilised. In particular, the cascade combination of two such procedures reduces the eigenvalue spread of the Volterra input auto-correlation matrix, thus improving the convergence speed of the adaptive nonlinear filtering.

Short Data Record Adaptive Filtering: The Auxiliary-Vector Algorithm

Karystinos, G. N., Qian, H., Medley, M. J., and Batalama, S. N., Short Data Record Adaptive Filtering: The Auxiliary-Vector Algorithm, Digital Signal Processing 12 (2002) 193–222 Based on statistical conditional optimization criteria, we developed an iterative algorithm that starts from the matched filter (or constraint vector) and generates a sequence of filters that converges to the minimum variance distortionless response (MVDR) solution for any positive definite input autocorrelation matrix. Computationally, the algorithm is a simple recursive procedure that avoids explicit matrix inversion, decomposition, or diagonalization operations. When the input autocorrelation matrix is replaced by a conventional sample-average (positive definite) estimate, the algorithm effectively generates a sequence of MVDR filter estimators: The bias converges rapidly to zero and the covariance trace rises slowly and asymptotically to the covariance trace of the familiar sample matrix inversion (SMI) estimator. For short data records, the early, nonasymptotic, elements of the generated sequence of estimators offer favorable bias–covariance balance and are seen to outperform in mean-square estimation error constraint-LMS, RLS-type, and orthogonal multistage decomposition estimates (also called nested Wiener filters) as well as plain and diagonally loaded SMI estimates. The problem of selecting the most successful (in some appropriate sense) filter estimator in the sequence for a given data record is addressed and two data-driven selection criteria are proposed. The first criterion minimizes the cross-validated sample average variance of the filter estimator output. The second criterion maximizes the estimated J-divergence of the filter estimator output conditional distributions. Illustrative interference suppression examples drawn from the communications literature are followed throughout this presentation.

An iterative algorithm for the computation of the MVDR filter

Statistical conditional optimization criteria lead to the development of an iterative algorithm that starts from the matched filter (or constraint vector) and generates a sequence of filters that converges to the minimum-variance-distortionless-response (MVDR) solution for any positive definite input autocorrelation matrix. Computationally, the algorithm is a simple, noninvasive, recursive procedure that avoids any form of explicit autocorrelation matrix inversion, decomposition, or diagonalization. Theoretical analysis reveals basic properties of the algorithm and establishes formal convergence. When the input autocorrelation matrix is replaced by a conventional sample-average (positive definite) estimate, the algorithm effectively generates a sequence of MVDR filter estimators; the bias converges rapidly to zero and the covariance trace rises slowly and asymptotically to the covariance trace of the familiar sample-matrix-inversion (SMI) estimator. In fact, formal convergence of the estimator sequence to the SMI estimate is established. However, for short data records, it is the early, nonasymptotic elements of the generated sequence of estimators that offer favorable bias covariance balance and are seen to outperform in mean-square estimation error, constraint-LMS, RLS-type, orthogonal multistage decomposition, as well as plain and diagonally loaded SMI estimates. An illustrative interference suppression example is followed throughout this presentation.

Time-Skew Hebb rule in a nonisopotential neuron.

In an isopotential neuron with rapid response, it has been shown that the receptive fields formed by Hebbian synaptic modulation depend on the principal eigenspace of Q(0), the input autocorrelation matrix, where Qij(tau) = <xi i(t) xi j(t-tau)> and xi i(t) is the input to synapse i at time t (Oja 1982). We relax the assumption of isopotentiality, introduce a time-skewed Hebb rule, and find that the dynamics of synaptic evolution are determined by the principal eigenspace of Q. This matrix is defined by Qij = integral of 0 infinity (Qij * psi i) (tau) Kij (tau) d tau, where Kij (tau) is the neuron's voltage response to a unit current injection at synapse j as measured tau seconds later at synapse i, and psi(tau) is the time course of the opportunity for modulation of synapse i following the arrival of a presynaptic action potential.

Analysis of overshoot phenomena in initialisation stage of RLS algorithm

The well-known recursive least-squares (RLS) algorithm experiences overshoot in the norm of the parameter error vector temporarily during the initialisation stage when the initial value of the input autocorrelation matrix is chosen very small. We have had no guide to choose a suitable initial value of the input autocorrelation matrix in order to cause the RLS algorithm not to fluctuate during the initialisation phase. In this paper, deriving the bound of parameter error norm square, we give a useful guide to choose the initial value in order to avoid the initial overshoot in the RLS algorithm. Besides, we show that what are the crucial factors of the overshoot phenomena. The ratio of the initial value to the input power should be chosen according to the signal-to-noise ratio of the system and the order of the filter. We present various simulation results to validate our analysis.

On the effect of input signal correlation on weight misadjustment in the RLS algorithm

New expressions are derived for the mean weight misadjustment in the recursive least squares (RLS) algorithm for first-order Markov channel estimation. The expressions derived are general in that they take into account the correlation in the input. It is shown that the additive system noise is amplified by a correlation amplification factor that is defined as a function of the input autocorrelation matrix eigenvalues. However, input correlation has almost no effect on the misadjustment due to time-varying system weights. These results are checked by simulations demonstrating excellent agreement with the theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Active Noise Control for Automobiles Based on Least Squares Algorithm.

The purpose of this paper is to propose an algorithm for active noise control (ANC) using the recursive least squares (RLS) algorithm. As a main algorithm for ANC, the multiple error filtered-x LMS (Mefx-LMS) algorithm has been used for its simplicity. However, the Mefx-LMS algorithm has a Problem of the convergence speed in the case of the broadband noise as the road noise which is one of the passenger compartment noises in a car running on rough roads. This problem is caused by the ill-conditioning of the input autocorrelation matrix. In the ANC for the road noise, many accelerometers used as the input signals are necessary for a high multiple coherence. Using many input signals also causes a cross-correlation among the input signals. The ill-conditioning is mainly brouht about by the cross-correlation. In order to solve this problem, the Mefx-RLS algorithm is considered. By computer simulations, the RLS algorithm has proved to be more effective than the LMS algorithm in the convergence speed for cross-correlated input signals.