SPTM-15.1

Statistical Analysis of the LMS Algorithm with a Zero-Memory Nonlinearity After the Adaptive Filter
Marcio H Costa (Universidade Catolica de Pelotas), Jose C. M Bermudez (Universidade Federal de Santa Catarina), Neil J Bershad (Univerity of California Irvine)

This paper presents a statistical analysis of the Least Mean Square (LMS) algorithm when a zero-memory nonlinearity appears at the adaptive filter output. The nonlinearity is modelled by a scaled error function. Deterministic nonlinear recursions are derived for the mean weight and mean square error (MSE) behavior for white gaussian inputs and slow adaptation. Monte Carlo simulations show excellent agreement with the behavior predicted by the theoretical models. The analytical results show that a small nonlinear effect has a significant impact on the converged MSE.

SPTM-15.2

Convergence Properties of the Block Orthogonal Projection Algorithm
Kazushi Ikeda, Hideaki Sakai (Grad Sch Informatics, Kyoto Univ)

The normalized LMS (N-LMS) algorithm has a disadvantage that the convergence rate is much worse when the input signal is colored. To overcome this, the affine projection algorithm and the block orthogonal projection (BOP) algorithm which are applied the block signal processing technique to the N-LMS algorithm are proposed although the reason why they are tough against the coloredness is not given yet. This paper gives the convergence rate of the BOP algorithm for colored input signals, which shows the superiority of the BOP algorithm. To put it concretely, we derive the expression of the convergence rate, propose an approximation method to calculate it, and confirm the result by computer simulations. We also consider the relation between the block size and the convergence rate formally and geometrically.

SPTM-15.3

Exact Convergence Analysis of Affine Projection Algorithm: The Finite Alphabet Inputs Case
Besbes Hichem, Ben Jemaa Yousra, Jaidane Meriem (L.S.Telecoms , ENIT/ESPTT Tunisia)

The affine projection algorithm (APA)is a very promising algorithm that has good convergence properties when the input signal is correlated. In particular, it's used to perform communications systems: echo cancellation, equalization...However, due to its complexity, there is no available transient and steady state analysis. In this paper, we present an exact analysis approach tailored for digital transmission context. In such context, the input signal remains in a finite alphabet set. With a discrete Markov chain model of the inputs, we can describe accurately the APA's behavior without any unrealistic assumption. In particular we can calculate the exact value of the critical and optimum step size. Moreover, we provide the exact Mean Square Deviation for all step size and input correlation. The influence of high order statistics can be enhanced.

SPTM-15.4

On the Stability of the Inverse Time-Varying Prediction Error Filter Obtained with the RWLS Algorithm
Roberto Lopez-Valcarce, Soura Dasgupta (Dept. of Electrical and Computer Engineering, University of Iowa)

This work provides conditions on the input sequence that ensure the exponential asymptotic stability of the inverse of the forward prediction error filter obtained by means of the Recursive Weighted Least Squares algorithm. Note that this filter is in general time varying. Thus this result is a natural extension to the well-known minimum phase property of forward prediction error filters obtained by the autocorrelation method.

SPTM-15.5

Stability Bounds on Step-Size for the Partial Update LMS Algorithm
Mahesh Godavarti (Student, Dept. of EECS, The University of Michigan), Alfred O Hero III (Professor, Dept. of EECS, The University of Michigan)

Partial Updating of LMS filter coefficients is an effective method for reducing the computational load and the power consumption in adaptive filter implementations. Only in the recent past has any work been done on deriving conditions for filter stability, convergence rate, and steady state error for the Partial Update LMS algorithm. In [5] approximate bounds were derived on the step size parameter mu which ensure stability in-the-mean of the alternating even/odd index coefficient updating strategy. Unfortunately, due to the restrictiveness of the assumptions, these bounds are unreliable when fast convergence (large mu) is desired. In this paper, tighter bounds on mu are derived which guarantee convergence in-the-mean of the coefficient sequence for the case of wide sense stationary signals.

SPTM-15.6

Adaptive Parameter Esitmation Using Interior Point Optimization Techniques:Convergence Analysis
Kaywan H Afkhamie (Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada L8S 4K1), Zhi-Quan Luo (Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada L8S 4k1)

Interior Point Optimization techniques have recently emerged as a new tool for developing parameter estimation algorithms. These algorithms aim to take advantage of the fast convergence properties of interior point methods, to yield, in particular, fast transient performance. In this paper we develop a simple "analytic center" based algorithm, which updates estimates with a constant number of computation (independent of number of samples). The convergence analysis shows that the asymptotic performance of this algorithm matches that of the well-known least squares filter (provided some mild conditions are satisfied). Some numerical simulations are provided to demonstrate the fast transient performance of the interior point algorithm.

SPTM-15.7

Affine Projection Methods in Fault Tolerant Adaptive Filtering
Robert A. Soni (Lucent Technologies), Kyle A. Gallivan (Florida State University), W. K. Jenkins (University of Illinois at Urbana-Champaign)

Reliable performance is very important for high speed channel equalizers and echo cancellers used in high speed communications channels. A common type of hardware fault occurs when the coefficients get ``stuck'' at an uncontrollable value. Such faults lead to larger overall mean square errors, and generally poor performance. Redundancy can provide the ability to compensate for these types of faults if the proper design is introduced into the adaptive filter structure. Unfortunately, this form of redundancy can lead to poor convergence performance for the adaptive filter after the fault occurrence. This paper examines the use of affine projection and {\em row projection} techniques to improve the convergence performance of the fault tolerant adaptive filtering structure. Algorithms are developed for two cases: fault knowledge and no fault knowledge incorporated in the adaptive filtering update. These algorithms are introduced in this paper and simulations are presented to illustrate the effectiveness of these approaches.

SPTM-15.8

Performance Analysis of Third-Order Nonlinear Wiener Adaptive Systems
Shue-Lee Chang, Tokunbo Ogunfunmi (Santa Clara University)

This paper presents a detailed performance analysis of third-order nonlinear adaptive systems based on the Wiener model. In earlier work, we proposed the discrete Wiener model for adaptive filtering applications for any order. However, we had focused mainly on first and second-order nonlinear systems in our previous analysis. Now, we present new results on the analysis of third and higher-order systems. This results can be extended to higher-oder systems. The Wiener model has many advantages over other models such as the Volterra model. These advantages include less number of coefficients and faster convergence. The Wiener model performs a complete orthogonalization procedure to the truncated Volterra series and this allows us to use linear adaptive filtering algorithms like the LMS to calculate all the coefficients efficiently. Unlike the Gram-Schmidt procedure, this orthogonalization method is based on the nonlinear discrete Wiener model. It contains three sections. It contains three sections: a single-input multi-output linear with memory section, a multi-input, multi-output nonlinear no-memory section and a multi-input, single-output amplification and summary section. Computer simulation results are also presented to verify the theoretical performance analysis results.

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Last Update: February 4, 1999 Ingo Höntsch