Fast Recursive Least Squares Algorithms Research Articles

Equalizer of Amplitude-Frequency Response of Acoustic Channels and its Adaptive Filter Based on Recursive Least Squares Algorithm with Diagonalized Correlation Matrix

The paper presents an equalizer of the amplitude-frequency response of the acoustic wave propagation channels in the rooms. The considered equalizer uses the so-called modified architecture based on the adaptive filter with the input signal filtering, which weights are calculated using the Recursive Least Squares (RLS) algorithm. The equalizer of the acoustic channels amplitude-frequency response usually requires the usage of the adaptive filters with the large number of weighs that means the large computational complexity of the adaptive filter. The adaptive filters based on the fast RLS algorithms are efficient in the terms of their complexity but often unstable if the number of adaptive filter weights is large. Therefore, the algorithms are poorly suited for usage in the amplitude-frequency response equalizers of the acoustic channels. Adaptive filters based on the RLS algorithm, which use the Matrix Inversion Lemma (MIL), are stable, but computationally complex. The equalizer presented in this paper uses a simplified adaptive filter based on the MIL RLS algorithm with a diagonalized correlation matrix of the filter input signal. In addition, in the filter, each of the square submatrices on the simplified correlation matrix diagonal is inverted using a computationally efficient version of the MIL that takes into account the symmetrical structure of the mentioned submatrices relatively their main diagonals. Computer simulation confirms the performance of the proposed equalizer and demonstrates its efficiency comparing to the unsimplified equalizer. The demo includes the graphs of the equalizer steady-state amplitude-frequency response, the acoustic channel amplitude-frequency responses before and after the equalization, and the power spectral density plots of the equalizer input signal and the signals passed through the equalizer and the acoustic medium. The equalizer can be used in the equipment for the high-quality voice and music reproduction in the conditions of the limited computing resources for the implementation of the equalizer.

Data-Reuse Recursive Least-Squares Algorithms

There are different strategies to improve the overall performance of the recursive least-squares (RLS) adaptive filter. In this letter, we focus on the data-reuse approach, aiming to improve the convergence rate/tracking of the algorithm by reusing the same set of data (i.e., the input and reference signals) several times. First, we present a computationally efficient data-reuse RLS algorithm, which is the result of a low complexity implementation of the data-reuse process. Moreover, we extend the idea to the fast RLS algorithm. Simulations performed in the context of echo cancellation support the performance gain.

A unified view of adaptive algorithms for finite impulse response filters using the H∞ framework

The normalized least mean squares (NLMS) and recursive least squares (RLS) algorithms are widely used for adaptive filtering. Interestingly, the NLMS algorithm has been shown to be strictly optimal in the sense of H∞ filtering, whereas the forgetting factor RLS algorithm has not been clearly related to a solution to the H∞ filtering problem. This paper describes a method for further optimizing the solutions to the ordinary H∞ filtering problem over an assumed system model set and a predetermined norm weight set. The extended H∞ filtering problem offers a framework for constructing a unified view of adaptive algorithms for finite impulse response (FIR) filters. The framework enables a discussion of the relationships among the NLMS algorithm, the forgetting factor RLS algorithm, and the H∞ filter over the common parameter space, and facilitates the development of new fast adaptive algorithms that outperform the existing algorithms, such as the NLMS and the fast RLS algorithms. The validity of the discussion based on the H∞ framework is verified using numerical examples.

Removal of Ocular Artifacts from EEG Signals by Fast RLS Algorithm using Wavelet Transform

This paper presents an adaptive filtering method to remove ocular artifacts in the electroencephalogram (EEG) records. The major concern in analyzing EEG signal is the presence of ocular artifacts in EEG records caused due to various factors. It is essential to design specific filters to remove the artifacts in EEG records. Here, we proposed an adaptive filtering method that uses RLS (Recursive Least Square) algorithm and FRLS (Fast Recursive Least Squares) to remove ocular artifacts from EEG recordings through wavelet transform. We compared RLS & FRLS algorithms with wavelet transforms. Elapsed time can be decreased by using the FRLS algorithm compared to other techniques and also we can compare the PSNR and MSE values.

A Stabilized Multichannel Fast RLS Algorithm for Adaptive Transmultiplexer Receivers

The transmultiplexer (TMUX) system has been studied for its application to multicarrier communications. The channel impairments including noise, interference, and distortion draw the need for adaptive reconstruction at the TMUX receiver. Among possible adaptive methods, the recursive least squares (RLS) algorithm is appealing for its good convergence rate and steady state performance. However, higher computational complexity due to the matrix operation is the drawback of utilizing RLS. A fast RLS algorithm used for adaptive signal reconstruction in the TMUX system is developed in this paper. By using the polyphase decomposition method, the adaptive receiver in the TMUX system can be formulated as a multichannel filtering problem, and the fast algorithm is obtained through the block Toeplitz matrix structure of received signals. In addition to the reduction of complexity, simulation results show that the adaptive TMUX receiver has a convergence rate close to that of the standard RLS algorithm and the performance approaches the minimum mean square error solution.

Fast RLS Fourier analyzers capable of accommodating frequency mismatch

Adaptive Fourier analyzers are used to estimate the discrete Fourier coefficients (DFC) of sine and cosine terms of noisy sinusoidal signals whose frequencies are usually assumed known a prior. The recursive least squares (RLS) Fourier analyzer provides excellent performance, but is computationally very intensive. In this paper, we first present four fast RLS (FRLS) algorithms based on the inherent characteristics of the DFC estimation problem. These FRLS algorithms show approximately the same performance and indicate estimation capabilities that are quite similar to those of the RLS, while requiring considerably less computational cost. Second, the performance of the proposed FRLS algorithms is analyzed in detail. Difference equations governing their dynamics as well as closed-form expressions for their steady-state mean square errors (MSE) are derived and compared with those of the LMS Fourier analyzer. Third, the RLS and four FRLS algorithms are modified by incorporating an adaptive scheme, to alleviate the influence of undesirable frequency mismatch (FM) on their performance. Extensive simulations as well as application to real noise signals are provided to demonstrate the relative performance capabilities of the RLS and four FRLS algorithms, the validity of analytical findings, and ability of the modified RLS and FRLS algorithms to mitigate the influence of the FM.

Adaptive noise reduction using numerically stable fast recursive least squares algorithm

AbstractThis paper is concerned with adaptive noise reduction based on the fast recursive least squares (FRLS) algorithm. It is well known that the fast recursive least squares (FRLS) algorithm suffers from numerical instability when operating under the effects of finite precision arithmetic. Several numerical solutions of stabilization were proposed in the case of stationary signals. In this work a new version of a numerically stable FRLS algorithm (NS‐FRLS) is proposed. The stability characteristics of this new stabilized algorithm are analysed. The analysis is based on a linear model for the errors in the states of the adaptive filter. Experimental results confirm the merits of adaptive filtering with the NS‐FRLS algorithm over optimum filtering using the solution provided by Wiener–Hopf equations. Copyright © 2006 John Wiley & Sons, Ltd.

The fast normalized cross-correlation double-talk detector

In this paper, we present a method for handling double-talk. This approach uses a robust fast recursive least-squares algorithm (FRLS) and the normalized cross-correlation double-talk detector (NCC DTD). The NCC DTD is developed into a fast version, called FNCC, by reusing computational results from the FRLS algorithm. The major advantage of this detector is that it is much less dependent on the actual echo path attenuation than, e.g., the Geigel DTD. This makes it much more suitable for the acoustic application.

Multichannel parallelizable sliding window RLS and fast RLS algorithms with linear constraints

This paper presents new sliding window (SW) recursive least squares (RLS) and fast RLS algorithms for adaptive filtering with linear constraints. The algorithms are formulated for the general case of multichannel adaptive filters with complex-valued weights, and are based on Papaodysseus's matrix inversion lemma. The lemma is used for SW correlation matrix inversion and for the inversion of some other matrices that appear in constrained SW RLS algorithms. The algorithms have a form that can be implemented by means of at least two parallel processors. The proposed algorithms can be used for time-domain adaptive filtering of non-stationary signals, whilst restricting the frequency response of the filter to specific values at particular frequencies. In addition, the algorithms can be used in linearly constrained adaptive beamforming, dealing with non-stationary interference.

Efficient fast recursive least-squares adaptive complex filtering using real valued arithmetic

In this paper, a novel implementation of the fast recursive least squares (FRLS) algorithm for complex adaptive filtering, is presented. Complex signals are treated as pairs of real signals, and operations are re-organized on a real arithmetic basis. A new set of signals and filtering parameters is introduced, which are expressed either as the sum, or as the difference between the real and the imaginary part of the original variables. The algorithmic strength reduction technique is subsequently applied, resulting in significant computational savings. In this way, an efficient, FRLS algorithm is derived, where only two real valued filters are propagated, based on novel three-terms recursions for the time updating of the pertinent parameters. The proposed algorithm is implemented using real valued arithmetic only, whilst reducing the number of the required real valued multiplication operations by 23.5%, at the expense of a marginal 2.9% increase in the number of the real valued additions. The performance of the proposed FRLS algorithm is investigated by computer simulations, in the context of adaptive system identification and adaptive channel equalization.

Adaptive linearly constrained inverse QRD-RLS beamforming algorithm for moving jammers suppression

A general, linearly constrained (LC) recursive least squares (RLS) array-beamforming algorithm, based on an inverse QR decomposition, is developed for suppressing moving jammers efficiently. In fact, by using the inverse QR decomposition-recursive least squares (QRD-RLS) algorithm approach, the least-squares (LS) weight vector can be computed without back substitution and is suitable for implementation using a systolic array to achieve fast convergence and good numerical properties. The merits of this new constrained algorithm are verified by evaluating the performance, in terms of the learning curve, to investigate the convergence property and numerical efficiency, and the output signal-to-interference-and-noise ratio. We show that our proposed algorithm outperforms the conventional linearly constrained LMS (LCLMS) algorithm, and the one using the fast linear constrained RLS algorithm and its modified version.

V-vector algebra and its application to Volterra-adaptive filtering

In this paper, we describe a new algebraic structure called V-vector algebra, which is a formal basis for the development of Volterra-adaptive filter algorithms as an extension of linear-adaptive techniques. In this way, fast and numerically stable adaptive Volterra filtering algorithms can be easily derived from the known linear theory. V-vector algebra can also be applied to deal with linear multichannel filters with channels of different memory lengths. A reformulation of the Lee-Mathews fast recursive least squares (RLS) algorithm and a new fast and stable Givens rotation-based square root RLS algorithm, both derived using V-vector algebra, are finally presented.

Stabilizing fast RLS algorithms by leakage

In this paper a new stabilization method for the problem of numerical instability of fast recursive least-squares algorithms is introduced. The new approach arose from implementation aspects of common stabilized fast transversal filters (SFTF) algorithms in the compensation of acoustical echoes. Real-time situations are characterized by non-persistent excitation and time-variant systems. Analysing these cases we found out insufficiencies in the performance of SFTF algorithms that exploit redundancy by feedback of numerical errors. Out of these results arose a new concept to cope with stabilization problems: A leakage in the prediction part of the FTF algorithm turned out to make the algorithm very stable and robust. The new found Leaky FTF (LFTF) algorithm, first derived in Binde, IEEE Signal Process. Lett. 2(6) (June 1995) 114–116, is analysed in this article. It has proven to be stable given an appropriate value for the leakage factor.

Adaptive filter bank reconstruction using fast multichannel RLS algorithms

The recursive least-squares (RLS) algorithm has been used in the adaptive synthesis filter bank for fast convergence. However, because of interpolation operations involved in the synthesis process, fast RLS algorithms cannot be applied. In this Letter, an approach is proposed that can formulate subband signal reconstruction as a multichannel filtering problem. This formulation allows the application of fast multichannel RLS algorithms and substantial reduction in computational complexity.

Adaptive filtering in cascade form using a fast multichannel RLS algorithm

This work proposes an approach to apply least-squares techniques to adaptive FIR filtering in cascade form. A triangular structure is combined with an adaptive multichannel filter, which can exploit a fast recursive least squares (RLS) algorithm. The resulting approximation and the behavior of the system in the learning case is discussed. Simulations results are given for the linear prediction case, for which an additional adaptive procedure is derived. Sinusoidal signals are used at the input, since the inherent cascade structure directly yields the frequencies of the sinusoids from the section coefficients. The proposed method brings the efficiency of least-squares techniques, in terms of accuracy and convergence rate, to the adaptive filter in cascade form.

Structured Matrices and Fast RLS Adaptive Filtering *

Abstract We present a new approach to the discrete-time Chandrasekhar recursions and some generalizations thereof. We extend the recursions to a class of structured time-variant state-space models, and discuss connections with the (generalized) Schur algorithm. We also apply the extended recursions to the adaptive filtering problem and give a transparent derivation of fast recursive least-squares algorithms.

Backward consistency concept and round-off error propagation dynamics in recursive least-squares algorithms

A new approach for the analysis of the propagation of roundoff errors in recursive algorithms is presented. This approach is based on the concept of backward consistency. In general, this concept leads to a decomposition of the state space of the algorithm, and in fact, to a manifold. This manifold is the set of state values that are backward consistent. Perturbations within the manifold can be interpreted as resulting from perturbations on the input data. Hence, the error propagation on the manifold corresponds exactly (without averaging or even linearization) to the propagation of the effect of a perturbation of the input data at some point in time on the state of the algorithm at future times. We apply these ideas to the Kalman filter and its various derivatives. In particular, we consider the conventional Kalman filter, some minor variations of it, and its square-root forms. Next we consider the Chandrasekhar equations, which apply to time-invariant state-space models. Recursive least-squares (RLS) parameter estimation is a special case of Kalman filtering, and hence the previous results also apply to the RLS algorithms. We also consider in detail two groups of fast RLS algorithms: the fast transversal filter algorithms and the fast lattice/fast QR RLS algorithms.

Fast adaptive RLS algorithms: a generalized inverse approach and analysis

A generalized inverse approach is used to derive two fast adaptive recursive least squares (RLS) algorithms and an exact and stable initialization algorithm for the prewindowed signal case. The partitioning techniques are directly applied to the signal matrix rather than the covariance matrix. The simulations show that the method has better numerical properties than existing fast algorithms, especially when the Kalman gain is concerned. Interesting relations among the variables in the fast RLS algorithms are provided. These relations can be used to derive other variations of fast RLS algorithms. They can also be used to develop new rescue schemes. Finally, a numerical analysis illustrates some pitfalls of the fast RLS algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Easy and effective stabilisation measure for fast recursive least squares algorithms for adaptive transversal filters

Fast recursive least squares (FRLS) algorithms are known for their favourable convergence characteristics. They are of special interest for the fast adaptation of transversal filters of high order N. Unfortunately FRLS algorithms have a tendency towards numeric instability. A new and effective stabilisation measure is presented for a 0(7N) FRLS algorithm which needs only one additional multiplication per iteration step.

A numerically stable relay structure for fast RLS adaptive filtering

Long-term numerical instability is a critical problem in fast recursive least squares (RLS) adaptive algorithms. The author presents a numerically stable fast RLS relay structure (FRLS-RS), in which a conventional fast transversal RLS algorithm and a mixed time-and-order updating procedure are combined in a relay form to provide, with O(M) operations, the instantaneous filter-coefficient solution for each time step, where M is the order of the filter. Since continuous propagation of the FRLS-RS is limited to 2M+1 time steps, accumulation of the numerical errors is isolated. Both fixed-point analysis and computer simulations show that in the case of moderate precision, and when the order is not very high, the present structure is long-term numerically stable. Efficient implementation and exact initialization of the FRLS-RS are also considered. >