تجزیه و تحلیل عملکرد از فیلتر شبکه ای خودتطبیقی

This paper describes new self-orthogonalising adaptive lattice filter (SALF) structures to enhance slow convergence rates caused by eigenvalue spread. Firstly, we propose the variable stepsize self-orthogonalising adaptive lattice filtering (VSALF) structure to speed up the convergence rate of partial correlation (PARCOR) coefficients. Secondly, the partial self-orthogonalising adaptive lattice filtering (PSALF) structure is proposed in order to enhance the tracking ability for nonstationary environments. Moreover, the PSALF structure can reduce computational complexity whilst maintaining a fast convergence rate. A performance analysis based on the convergence model of the lattice predictor is given in terms of mean-squared error and variance of the PARCOR coefficient error. Computer simulations are undertaken to verify the performance and applicability of the proposed filter structures.

Adaptive filtering algorithms based on the stochastic gradient method are widely used in many applications such as system identification, noise cancellation, active noise control and communication channel equalisation. These algorithms have attracted the attention of many researchers because of their low complexity and robustness to implementation error [11]. However, the eigenstructure of the correlation matrix of a transversal filter's tap input signal has a profound impact on the convergence behaviour of the least mean-squares (LMS) algorithm. When the tap input signals are drawn from a white noise process, they are uncorrelated and the eigenvalue spread of the correlation matrix is unity, with the result that the LMS algorithm has a nondirectional convergence. At the other extreme, when the tap input signals are highly correlated, the LMS algorithm takes on a directional nature which results in a slow convergence [6] and [12]. Our approach to resolve the eigenvalue spread problem is explained briefly in Fig. 1. The plant model extends the concept of the transform domain adaptive filter [2], [20] and [25] by combining an adaptive lattice predictor and a linear combiner. Henceforth, this structure is described as the self-orthogonalising adaptive lattice filter (SALF). This filter structure has previously been referred to as joint process estimator, it was first developed by Griffiths [8] and [9] to be used in a multichannel noise canceling application, but, focus was not placed on the theoretical analysis. Full-size image (6 K) Fig. 1. Block diagram of adaptive plant modelling. Figure options Similar filtering structures using a prewhitening scheme have already been developed in [5], [23], [24] and [27]. In [23], Mboup proposed the prewhitening filter structure to speed up convergence rates. This work employed a simple LMS predictor to decorrelate the input signal. In this case, the input signal was modelled as a stationary complex sinusoidal signal in a broad-band noise background to make the theoretical analysis simple. Proudler [27] proposed the preconditioning LMS filtering structure based on the conventional lattice predictor. The performance was judged purely by computer simulation. Other related work was published by Farhang-Boroujeny [5] and Moustakides [24]. This work is based on the fast Newton algorithm and an autoregressive (AR) model in order to reduce the predictor order. We first propose the variable stepsize self-orthogonalising adaptive lattice filter (VSALF) structure utilising the squared output error of linear combiner at the second stage to speed up the slow convergence rate caused by correlated input signals and to reduce gradient noise in steady state. Secondly, we propose the partial self-orthogonalising adaptive lattice filter (PSALF) which is capable of reducing computational complexity whilst maintaining fast convergence rates. The proposed structure is implemented as a partial lattice predictor (reduced order) which is dependent on the level of the correlation of the input signal. The PSALF is particularly efficient in memory usage. The filter transfer function of the SALF depends on both the partial correlation (PARCOR) coefficients and the coefficients of the linear combiner. Thereby, the PSALF enhances the tracking ability for nonstationary environments due to the partial update of the PARCOR coefficients. A performance analysis based on the convergence model of the lattice predictor is given in terms of mean-squared error and variance of the PARCOR coefficient error. Computer simulation results demonstrating the accuracy of the model and the performance of the proposed filter structures are also presented. Acronyms of the algorithms introduced in this paper are summarised below: • SALF: Self-orthogonalising adaptive lattice filter. • FSALF: Fixed stepsize self-orthogonalising adaptive lattice filter. • VSALF: Variable stepsize self-orthogonalising adaptive lattice filter. • NSALF: Normalised self-orthogonalising adaptive lattice filter. • PSALF: Partial self-orthogonalising adaptive lattice filter. • PNSALF: Partial normalised self-orthogonalising adaptive fattice filter. The rest of this paper is organised as follows: In Section 2, the SALF structure is presented. In Section 3, the convergence property of the SALF structure is investigated. In Section 4, we propose the VSALF and the PSALF structures and evaluate their computational complexity. In Section 5, the performance analysis of the proposed structures using the convergence model of the lattice predictor is presented and the variance of the PARCOR coefficient error is derived. Computer simulations of system identification and acoustic echo cancellation applications are evaluated to verify the applicability of the proposed structure in Section 6. Finally, conclusions are presented in the Section 7.

The VSALF structure which incorporates a time-varying stepsize dependent on the overall squared output error of the second stage for the estimate of the PARCOR coefficient and a PSALF structure with a partial lattice predictor order have been presented. A performance analysis based on a convergence model of the adaptive lattice predictor was given in terms of the mean-squared error and the variance of the PARCOR coefficients. Monte Carlo simulation was found to give a good fit to our theoretical analysis predictions. The VSALF structure has a convergence rate that is greater than that of the plain LMS algorithm as well as those of the FSALF and NSALF algorithms. However, it was shown that the VSALF structure may not be appropriate in places with a low signal-to-noise ratio although a fast convergence can be achieved by employing a variable stepsize.