Fast Affine Projection Algorithm Research Articles

A comparative survey of fast affine projection algorithms

The affine projection (AP) algorithm is one of the most celebrated adaptive filtering algorithms, and it achieves a good tradeoff between the convergence rate and computational complexity. However, the complexity of the AP algorithm increases with the projection order. A wealth of fast AP algorithms have been proposed to reduce the complexity in the last two decades. However, those low-complexity methods have not been well analyzed and compared. To fill this gap, this paper reviews the fast AP algorithms, including both fast filtering approaches and efficient solutions of the linear system of equations. The advantages and disadvantages of each fast implementation version are clarified based on an extensive performance evaluation, which could be useful to engineers when selecting a suitable algorithm for their specific applications and could also be a starting point for experts in this field to develop better solutions.

A two-sensor Gauss–Seidel fast affine projection algorithm for speech enhancement and acoustic noise reduction

This paper deals with the problem of noise reduction and speech enhancement by adaptive filtering using two-sensor configuration. The well-known Gauss–Seidel fast affine projection (GSFAP) algorithm is used in various applications such as acoustic echo cancelation (AEC), and adaptive noise control. In this paper, we use the GSFAP algorithm in noise reduction and speech enhancement application. Moreover, we derive a new two-sensor GSFAP (TS-GSFAP) algorithm for speech enhancement and acoustic noise reduction application. The application of the proposed TS-GSFAP algorithm to the forward blind source separation (FBSS) structure leads to a significant improvement of the output speech signals quality even in noisy conditions. A fair comparison of the proposed TS-GSFAP algorithm with other standard two-sensor type algorithms is presented. This comparison is based on the evaluation of several objective and subjective criteria such as: the segmental signal-to-noise ratio (SegSNR), the Cepstral distance (CD) and the system mismatch (SM), the perceptual evaluation of speech quality (PESQ), and the mean opinion score (MOS) criteria. The obtained results show the good performances of the proposed TS-GSFAP algorithm.

Efficient Implementation of the Affine Projection Sign Algorithm

-norm optimization-based sign algorithms (SAs) are more robust against impulsive interference than -norm optimization-based adaptive filtering algorithms. However, most SAs suffer from slow convergence rate, especially for highly correlated input signals. In order to overcome this problem, recently, an affine projection SA (APSA) has been proposed , which exhibits fast convergence rate. In this letter, we first analyze the computational complexity of the APSA in detail and then apply a recursive approach proposed for the affine projection algorithm (APA) to the APSA to reduce its computational complexity. Analysis results show that the computational complexity of the APSA with the efficient implementation method is even lower than that of the classical fast affine projection (FAP) algorithm.

A Family of Adaptive Filter Algorithms in Noise Cancellation for Speech Enhancement

In many application of noise cancellation, the changes in signal characteristics could be quite fast. This requires the utilization of adaptive algorithms, which converge rapidly. Least Mean Squares (LMS) and Normalized Least Mean Squares (NLMS) adaptive filters have been used in a wide range of signal processing application because of its simplicity in computation and implementation. The Recursive Least Squares (RLS) algorithm has established itself as the ultimate adaptive filtering algorithm in the sense that it is the adaptive filter exhibiting the best convergence behavior. Unfortunately, practical implementations of the algorithm are often associated with high computational complexity and/or poor numerical properties. Recently adaptive filtering was presented, have a nice tradeoff between complexity and the convergence speed. This paper describes a new approach for noise cancellation in speech enhancement using the two new adaptive filtering algorithms named fast affine projection algorithm and fast Euclidean direction search algorithms for attenuating noise in speech signals. The simulation results demonstrate the good performance of the two new algorithms in attenuating the noise.

Fast affine projection algorithm for adaptive noise canceling and its application on the fetal electrocardiogram extraction

Aimed at the problem of adaptive noise canceling (ANC), three implementary algorithms, which are least mean square (LMS) algorithm, recursive least square (RLS) algorithm and fast affine projection (FAP) algorithm, have been researched. The simulations were made for the performance of these algorithms. The extraction of fetal electrocardiogram (FECG) is applied to compare the application effect of the above algorithms. The proposed FAP algorithm has obvious advantages in computational complexity, convergence speed and steadystate error.

Nonlinear ANC using a third-order Volterra filter with an LDLT-FAP algorithm

Active noise control (ANC) systems employing the conventional fast affine projection (FAP) algorithms may lead to low ANC performance when the non-unity step size is chosen. To solve the problem, an LDL T - fast affine projection algorithm was proposed recently for the linear ANC. In this paper, the LDL T -FAP algorithm is further utilized, along with a third-order Volterra filter, for nonlinear ANC. Simulation results show that the proposed approach yields the good nonlinear ANC performance even in wide range of step sizes.

Fast Affine Projection Adaptation Algorithms With Stable and Robust Symmetric Linear System Slovers

This paper proposes two noniterative approaches to solve a symmetric linear system associated with the fast affine projection adaptation algorithm. The first approach, using matrix LDL <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> factorization, can provide an exact solution at a moderate complexity, in contrast with the fact that existing stable approaches are all approximate. Based on a reciprocating recursion scheme, the second approach has a very low complexity and gives a good approximate solution. Steady-state and transient properties of the proposed and certain previous FAP algorithms are studied in detail. Being stable and optimal under all step size conditions, fast affine projection algorithms incorporating the proposed approaches are promising in telecom and other applications of adaptive filtering

Pseudo-Affine Projection Algorithms for Multichannel Active Noise Control

For feedforward multichannel active noise control (ANC) systems, the use of adaptive finite-impulse response (FIR) filters is a popular solution, and the multichannel filtered-x least-mean-square (FX-LMS) algorithm is the most commonly used algorithm. The drawback of the FX-LMS is the slow convergence speed, especially for broadband multichannel systems. Recently, some fast affine projection algorithms have been introduced for multichannel ANC, as an interesting alternative to the FX-LMS algorithm. They can provide a significantly improved convergence speed at a reasonable additional computational cost. Yet, the additional computational cost or the potential numerical instability in some of the recently proposed algorithms can prevent the use of those algorithms for some applications. In this paper, we propose two pseudo-affine projection algorithms for multichannel ANC: one based on the Gauss-Seidel method and one based on dichotomous coordinate descent (DCD) iterations. It is shown that the proposed algorithms typically have a lower complexity than the previously published fast affine projection algorithms for ANC, with very similar good convergence properties and good numerical stability. Thus, the proposed algorithms are an interesting alternative to the standard FX-LMS algorithm for ANC, providing an improved performance for a computational load of the same order

Acoustic Echo Cancellation Using Frequency-Domain Spline Identification

Low-complexity delayless acoustic echo cancellation techniques based on frequency-domain spline-identification are proposed and investigated. Two methods of approximation of the acoustic frequency response, both using B-splines, are considered: the optimal-spline method and the local-spline method. The optimal-spline method seeks the solution of a least squares problem. The most computationally demanding part of the method, solution of the normal equations, is implemented by using the low-complexity dichotomous coordinate descent algorithm. The local-spline method avoids solving the normal equations, enabling further simplification; this is at the expense of a slight degradation in the cancellation performance. A novel efficient double-talk detector is also proposed, being an inherent feature of the frequency-domain identification. Open-loop and closed-loop identification schemes with cubic splines are studied by simulation and compared with the fast affine projection (FAP) algorithm. The proposed techniques provide cancellation performance better than that of the FAP algorithm, especially in double-talk and noisy environments, with a lower complexity

Stability improvement of relaxed FAP algorithm

The fast affine projection (FAP) algorithms are widely used in many applications for their advantages. However, the propagation and accumulation of errors always degrade the algorithm performance. Proposed is a modification for the FAP algorithm to improve the stability with limited computational burden increase.

Detection-Guided Fast Affine Projection Channel Estimator for Speech Applications

In various adaptive estimation applications, such as acoustic echo cancellation within teleconferencing systems, the input signal is a highly correlated speech. This, in general, leads to extremely slow convergence of the NLMS adaptive FIR estimator. As a result, for such applications, the affine projection algorithm (APA) or the low-complexity version, the fast affine projection (FAP) algorithm, is commonly employed instead of the NLMS algorithm. In such applications, the signal propagation channel may have a relatively low-dimensional impulse response structure, that is, the number m of active or significant taps within the (discrete-time modelled) channel impulse response is much less than the overall tap length n of the channel impulse response. For such cases, we investigate the inclusion of an active-parameter detection-guided concept within the fast affine projection FIR channel estimator. Simulation results indicate that the proposed detection-guided fast affine projection channel estimator has improved convergence speed and has lead to better steady-state performance than the standard fast affine projection channel estimator, especially in the important case of highly correlated speech input signals.

An efficient affine projection algorithm for 2-D FIR adaptive filtering and linear prediction

In this paper, a fast affine projection algorithm (FAPA), for the two-dimensional (2-D) adaptive linear filtering and prediction, is presented. The derivation of the proposed algorithm is based on the spatial shift-invariant properties of the 2-D discrete time signals. The proposed algorithm has low computational complexity, comparable to that of the 2-D LMS algorithm. The performance of the proposed scheme is comparable to that of the higher complexity 2-D RLS algorithms. The convergence speed and the tracking ability of the proposed 2-D FAPA algorithm are illustrated by computer simulation.

Coordinate descent iterations in fast affine projection algorithm

We propose a new approach for real-time implementation of the fast affine projection (FAP) algorithm. This is based on exploiting the recently introduced dichotomous coordinate descent (DCD) algorithm, which is especially efficient for solving systems of linear equations on real-time hardware and software platforms since it is free of multiplication and division. The numerical stability of the DCD algorithm allows the new combined DCD-FAP algorithm also to be stable. The convergence and complexity of the DCD-FAP algorithm is compared with that of the FAP, Gauss-Seidel FAP (GS-FAP), and modified GS-FAP algorithms in the application to acoustic echo cancellation. The DCD-FAP algorithm demonstrates a performance close to that of the FAP algorithm with ideal matrix inversion and the complexity smaller than that of the Gauss-Seidel FAP algorithms.

A fast exact weighted subband adaptive algorithm and its application to mono and stereo acoustic echo cancellation

The fast affine projection (FAP) algorithm (Gay and Tavathia, Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing, 1995, 3023) is known to outperform the NLMS with a slight increase in complexity, but it involves the fast calculation of the inverse of a covariance matrix of the input data that could undermine the performance of the algorithm. The block subband adaptive algorithm in (Courville and Duhamel, IEEE Trans. Signal Processing 46(9) (1998) 2359) has also illustrated significant improvement in performance over the NLMS and other frequency domain adaptive algorithms. However, it is known that block processing algorithms have lower tracking capabilities than the their sample-by-sample counterparts. In this paper, we present a sample-by-sample version of the algorithm in (Courville and Duhamel, IEEE Trans. Signal Processing 46(9) (1998) 2359) and develop a low complexity implementation of this algorithm. As a sample-by-sample algorithm, it avoids the reduced tracking capability of block algorithms. Because it does not use matrix inversion, it avoids the numerical problems of FAP algorithms. We will show that the new sample-by-sample algorithm approximates the affine projection algorithm and possesses a similar property in reducing coefficient bias that appears in monophonic and stereophonic teleconferencing when the receiving room impulse responses are undermodeled. The new fast sample-by-sample algorithm is extended for stereo acoustic echo cancellation. Simulations of echo cancellations in actual rooms are presented to verify our findings.

The Gauss‐Seidel fast affine projection algorithm for multichannel active noise control and sound reproduction systems

AbstractIn the field of adaptive filtering, the fast implementations of affine projection algorithms are known to provide a good tradeoff between convergence speed and computational complexity. Such algorithms have recently been published for multichannel active noise control systems. Previous work reported that these algorithms can outperform more complex recursive least‐squares algorithms when noisy plant models are used in active noise control systems. This paper proposes a new fast affine projection algorithm for multichannel active noise control or sound reproduction systems, based on the Gauss–Seidel solving scheme. The proposed algorithm has a lower complexity than the previously published algorithms, with the same convergence speed and the same good performance with noisy plant models, and a potential for better numerical stability. It provides the best performance/cost ratio. Details of the algorithm and its complexity are presented in the paper, with simulation results to validate its performance. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Multichannel affine and fast affine projection algorithms for active noise control and acoustic equalization systems

In the field of adaptive signal processing, it is well known that affine projection algorithms or their low-computational implementations fast affine projection algorithms can produce a good tradeoff between convergence speed and computational complexity. Although these algorithms typically do not provide the same convergence speed as recursive-least-squares algorithms, they can provide a much improved convergence speed compared to stochastic gradient descent algorithms, without the high increase of the computational load or the instability often found in recursive-least-squares algorithms. In this paper, multichannel affine and fast affine projection algorithms are introduced for active noise control or acoustic equalization. Multichannel fast affine projection algorithms have been previously published for acoustic echo cancellation, but the problem of active noise control or acoustic equalization is a very different one, leading to different structures, as explained in the paper. The computational complexity of the new algorithms is evaluated, and it is shown through simulations that not only can the new algorithms provide the expected tradeoff between convergence performance and computational complexity, they can also provide the best convergence performance (even over recursive-least-squares algorithms) when nonideal noisy acoustic plant models are used in the adaptive systems.

Multichannel active noise control and acoustic equalization using fast affine projection algorithms

In the field of adaptive signal processing, it is well known that affine projection algorithms or their low-computational implementations, fast affine projection algorithms, can produce a good trade-off between convergence speed and computational complexity. Although these algorithms typically do not provide the same convergence speed as recursive-least-squares algorithms, they can provide a much improved convergence speed compared to stochastic gradient descent algorithms, without the high increase of the computational load or the instability often found in recursive-least-squares algorithms. In this presentation, multichannel fast affine projection algorithms are introduced for active noise control or acoustic equalization. Multichannel fast affine projection algorithms have been previously published for acoustic echo cancellation, but the problem of active noise control or acoustic equalization is a very different one, leading to different structures. The computational complexity of the new proposed algorithm is evaluated, and it is shown through simulations that not only can the new algorithm provide the expected trade-off between convergence performance and computational complexity, it can also provide the best convergence performance (even over recursive-least-squares algorithms) when non-ideal noisy acoustic plant models are used in the adaptive systems.

A block exact fast affine projection algorithm

This paper describes a block (affine) projection algorithm that has exactly the same convergence rate as the original sample-by-sample algorithm and smaller computational complexity than the fast affine projection algorithm. This is achieved by (1) introducing a correction term that compensates for the filter output difference between the sample-by-sample projection algorithm and the straightforward block projection algorithm, and (2) applying a fast finite impulse response (FIR) filtering technique to compute the filter outputs and to update the filter. We describe how to choose a pair of block lengths that gives the longest filter length under a constraint on the total computational complexity and processing delay. An example shows that the filter length can be doubled if a delay of a few hundred samples is permissible.