IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO.9, OCTOBER 2000 1227 Performance Analysis of Signed Self-Orthogonalizing Adaptive Lattice Filter Dai I. Kim and P. De Wilde, Fellow, IEEE Abstract—This paper describes the novel signed self-orthogo- nalizing adaptive lattice filter (SSALF) structure to enhance the slow convergence rate caused by an eigenvalue disparity whilst con- straining the level of the convergence rate and the misadjustment required by a specification. The SSALF structure is also imple- mented by the partial lattice predictor in order to reduce a compu- tational complexity. The performance analysis based on the con- vergence model of the lattice predictor is given in terms of the mean-squared error and the variance of the reflection coefficient error. Computer simulations are undertaken to verify the perfor- mance and the applicability of the proposed filter structure. Index Terms—Adaptive lattice filter, lattice predictor, reflection coefficient, self-orthogonalizing. I. INTRODUCTION A DAPTIVE 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 equalization. These algo- rithms have attracted the attention of many researchers because of their low complexity and robustness to the implementation error. However, the eigenstructure of the correlation matrix of a linear combiner’s tap input signal has a profound impact on the convergence behavior of the least mean square (LMS) algorithm. When the tap input signals are highly correlated, the LMS algorithm takes on a directional nature which results in a slow convergence [1], [2]. As one of alternative techniques to circumvent the direction- ality of a convergence, self-orthogonalizing adaptive lattice fil- tering (SALF) structure [1], [2], [4] has been developed. The basic frame of the SALF structure extends the concept of the transform-domain adaptive filter [3] by combining an adaptive lattice predictor and a linear combiner, which is referred to here as the SALF structure. This filter structure has been called the joint process estimator, which was first developed by L. J. Grif- fiths [4] to be used in multichannel noise-canceling application. Similar filtering structures by the prewhitening scheme have al- ready been developed in [5]–[7]. In [5], Mboup et al. proposed the prewhitening filter structure to speed up the convergence rate. This work employed the simple LMS predictor to decorre- late an input signal. An input signal was modeled by only a sta- tionary complex sinusoidal signal in a broad-band background noise to make the theoretical analysis simple. Other related work Manuscript received November 1999; revised June 2000. This paper was rec- ommended by Associate Editor A. Skodras. The authors are with the Department of Electrical and Electronic Engi- neering, Imperial College of Science, Technology and Medicine, the University of London, London SW7 2BT, U.K. Publisher Item Identifier S 1057-7130(00)07755-7. was published by B. Farhang-Boroujeny [6], and G. V. Mous- takides et al. [7]. These work is fast Newton algorithm based on the autoregressive (AR) model in order to reduce the predictor order. In this framework, we propose the signed self-orthogonal- izing adaptive lattice filtering (SSALF) structure which utilizes the variable stepsize algorithm to speed up the slow convergence rate caused by a correlated input signal. Partial self-orthogonal- izing adaptive lattice filtering (PSSALF) are also implemented by a partial lattice predictor stage. The performance analysis based on the convergence model of the lattice predictor is given in terms of the mean-squared error and the variance of the reflection coefficient error. Computer simulation results demonstrating the accuracy of the model and the performance of the proposed filter structure are also pre- sented. The rest of this paper is organized as follows: in Section II, the SSALF structure is presented. In Section III, the convergence properties of the SSALF structure are investigated. In Section IV, the performance analysis of the proposed structure using the convergence model of the lattice predictor is presented and the variance of the reflection coefficient error is also derived. In Sec- tion V, the computer simulations through the system identifica- tion and the acoustic echo cancellation are undertaken to verify the applicability of the proposed structure. Finally, conclusions are presented in the Section VI. II. SALF STRUCTURE A. SALF Algorithm The SALF structure is a general digital filtering implementa- tion [1] which exploits the attractive orthogonal property of the lattice predictor. This structure has two distinct stages: a lattice predictor and a linear combiner. In the first stage, a lattice pre- dictor preprocesses an input signal. This stage can be viewed as the preprocessor or decorrelator. The update equations of back- ward and forward prediction errors are as follows: (1) (2) where forward prediction error; backward prediction error; reflection coefficient; for a filter of order . Since may be considered as the zeroth-order forward and backward prediction errors, the initialization for above recur- sions is given by . 1057–7130/00$10.00 © 2000 IEEE