Yongbin Wei

The stability of variable step-size LMS algorithms

Variable step-site LMS (VSLMS) algorithms are a popular approach to adaptive filtering, which can provide improved performance while maintaining the simplicity and robustness of conventional fixed step-size LMS. Here, we examine the stability of VSLMS with uncorrelated stationary Gaussian data. Most VSLMS described in the literature use a data-dependent step-size, where the step-size either depends on the data before the current time (prior step-size rule) or through the current time (posterior step-size rule). It has often been assumed that VSLMS algorithms are stable (in the sense of mean-square bounded weights), provided that the step-size is constrained to lie within the corresponding stability region for the LMS algorithm. For a single tap fitter, we find exact expressions for the stability region of VSLMS over the classes of prior and posterior step-sizes and show that the stability region for prior step size coincides with that of fixed step-size, but the region for posterior step-size is strictly smaller than for fixed step-size. For the multiple tap case, we obtain bounds on the stability regions with similar properties. The approach taken here is a generalization of the classical method of analyzing, the exponential stability of the weight covariance equation for LMS. Although it is not possible to derive a weight covariance equation for general data-dependent VSLMS, the weight variances can be upper bounded by the solution of a linear time-invariant difference equation, after appropriately dealing with certain nonlinear terms. For prior step-size (like fixed step-size), the state matrix is symmetric, whereas for posterior step-size, the symmetry is lost, requiring a more detailed analysis. The results are verified by computer simulations.

Noise-constrained least mean squares algorithm

We consider the design of an adaptive algorithm for finite impulse response channel estimation, which incorporates partial knowledge of the channel, specifically, the additive noise variance. Although the noise variance is not required for the offline Wiener solution, there are potential benefits (and limitations) for the learning behavior of an adaptive solution. In our approach, a Robbins-Monro algorithm is used to minimize the conventional mean square error criterion subject to a noise variance constraint and a penalty term necessary to guarantee uniqueness of the combined weight/multiplier solution. The resulting noise-constrained LMS (NCLMS) algorithm is a type of variable step-size LMS algorithm where the step-size rule arises naturally from the constraints. A convergence and performance analysis is carried out, and extensive simulations are conducted that compare NCLMS with several adaptive algorithms. This work also provides an appropriate framework for the derivation and analysis of other adaptive algorithms that incorporate partial knowledge of the channel.

A noise constrained LMS algorithm for fading channel communications

In many identification and tracking algorithms, an accurate estimate of the measurement noise variance is available. A partially adaptive LMS-type algorithm is developed which can exploit this information while maintaining the simplicity and robustness of LMS. This noise constrained LMS (NCLMS) algorithm is a type of variable stepsize LMS algorithm, which is derived by adding constraints to the mean-square error optimization. The convergence and steady state performance are analyzed. Both the theoretical results and simulations show that NCLMS can dramatically outperform LMS and other variable step size LMS algorithms in a sufficiently noisy environment.

Uniform observability and exponential convergence rate of the Kalman filter for the FIR deconvolution problem

In this paper we carry out a detailed stability analysis of the Kalman filter for the time-varying finite impulse response (FIR) deconvolution problem, with white input and white noise. Although the exponential stability of the Kalman filter is known from general results (since the system matrix is stable), the exponential convergence rate is unknown. The problem of determining the convergence rate is complicated by the singularity of the system matrix corresponding to the FIR response. A new definition of uniform observability is given which allows for singular system matrices, and a simple characterization of this property for the FIR system/channel is found. Under this observability condition, bounds on the exponential convergence rate of the Kalman filter are obtained which show qualitatively the worst-case dependence of the rate on the channel.

Reliable uplink code-timing synchronization for cellular DS-CDMA

For the uplink of a CDMA system, some knowledge about the active users is known to the base station and can be used to improve code-timing acquisition of a new arrival. Based on this idea, two maximum-likelihood (ML) code-timing estimation algorithms are proposed. It is shown that both estimators can effectively cancel the multiple-access interference giving robustness to near-far ratio and system load. This fact is verified by simulation results, which also show their superiority to known algorithms.

Noise constrained LMS algorithm

In many identification and tracking problems, an accurate estimate of the measurement noise variance is available. A partially adaptive LMS-type algorithm is developed which can exploit this information while maintaining the simplicity and robustness of LMS. This noise constrained LMS (NCLMS) algorithm is a type of variable step-size LMS algorithm, which is derived by adding constraints to the mean-square error optimization. The convergence and steady-state performance are analyzed. Both the theoretical results and simulations show that NCLMS can dramatically outperform LMS, RLS and other variable step-size LMS algorithms in a sufficiently noisy environment.

Maximum-likelihood code-timing acquisition of DS-CDMA signals for multipath channels

Future CDMA systems may be required to operate with a low processing gain in order to accommodate high rate users. The resulting increase in channel dispersion will have a detrimental impact on code-timing acquisition. Two maximum-likelihood code-timing acquisition algorithms are proposed for multipath channels: a multiuser estimator and a single-user estimator. Multipath diversity is exploited in the estimators via maximum ratio combining to increase the signal-to-interference ratio. In addition, correlations between signals from different paths are explicitly incorporated in the test statistics of both estimators. Both help to improve the estimation accuracy and decrease mean acquisition time. Extensive simulations indicate that the estimators are robust to near-far power ratio and channel dispersion. Moreover, the gain achieved by incorporating channel dispersion in estimator design is significant.

Analysis of direct and indirect adaptive linear equalization for slowly varying channels

The coefficients of an equalizer can be adjusted to track the underlying channel variations either directly, or indirectly where the channel is first estimated, and the equalizer is then computed based on the channel estimate. In this paper, the performance of direct and indirect linear adaptive equalizers for slowly-varying channels with autoregressive (AR) coefficients is investigated. Explicit expressions are derived which characterize the performance of the equalizers. The results indicate that the indirect equalizer outperforms the direct equalizer. Simulations are conducted to verify the analytic results.

Multiple user maximum likelihood code-timing acquisition for uplink DS-CDMA systems

Future CDMA systems will sometimes be required to operate with low processing gain in order to accommodate high rate users. The resulting increase in channel dispersion can have a detrimental impact on code-timing acquisition. Two maximum likelihood code-timing acquisition algorithms are proposed for multipath channels: a multiuser estimator and a single-user estimator. Multipath diversity is exploited in the estimators via maximum ratio combining to increase the signal-to-interference ratio. In addition, correlations between signals from different paths are explicitly incorporated in the test statistics of both estimators. The Cramer-Rao bounds are found and the performances of the estimators are examined via simulations. The results indicate that the estimators are robust to near-far power ratio and channel dispersion. Moreover, the gain achieved by incorporating channel dispersion in estimator design is significant.

Code-timing acquisition for future DS-CDMA systems

Future CDMA systems will sometimes be required to operate with low processing gain in order to accommodate high rate users. The resulting increase in channel dispersion can have a detrimental impact on code-timing acquisition. In this paper, two maximum likelihood code-timing acquisition algorithms are proposed for multipath channels: a multiuser estimator and a single-user estimator. Multipath diversity is exploited in the estimators via maximum ratio combining to increase the signal-to-interference ratio. In addition, correlations between signals from different paths are explicitly incorporated in the test statistics of both estimators. The computational complexity of each estimator is found and the performance of the estimators are examined via simulations. The results indicate that the estimators and robust to near-far power ratio and channel dispersion. Moreover, the gain achieved by incorporating channel dispersion in estimator design is significant.