V.J. Mathews

Adaptive polynomial filters

Adaptive nonlinear filters equipped with polynomial models of nonlinearity are explained. The polynomial systems considered are those nonlinear systems whose output signals can be related to the input signals through a truncated Volterra series expansion or a recursive nonlinear difference equation. The Volterra series expansion can model a large class of nonlinear systems and is attractive in adaptive filtering applications because the expansion is a linear combination of nonlinear functions of the input signal. The basic ideas behind the development of gradient and recursive least-squares adaptive Volterra filters are first discussed. Adaptive algorithms using system models involving recursive nonlinear difference equations are then treated. Such systems may be able to approximate many nonlinear systems with great parsimony in the use of coefficients. Also discussed are current research trends and new results and problem areas associated with these nonlinear filters. A lattice structure for polynomial models is described.<<ETX>>

Image enhancement via adaptive unsharp masking

This paper presents a new method for unsharp masking for contrast enhancement of images. The approach employs an adaptive filter that controls the contribution of the sharpening path in such a way that contrast enhancement occurs in high detail areas and little or no image sharpening occurs in smooth areas.

A stochastic gradient adaptive filter with gradient adaptive step size

The step size of this adaptive filter is changed according to a gradient descent algorithm designed to reduce the squared estimation error during each iteration. An approximate analysis of the performance of the adaptive filter when its inputs are zero mean, white, and Gaussian noise and the set of optimal coefficients are time varying according to a random-walk model is presented. The algorithm has very good convergence speed and low steady-state misadjustment. The tracking performance of these algorithms in nonstationary environments is relatively insensitive to the choice of the parameters of the adaptive filter and is very close to the best possible performance of the least mean square (LMS) algorithm for a large range of values of the step size of the adaptation algorithm. Several simulation examples demonstrating the good properties of the adaptive filters as well as verifying the analytical results are also presented. >

Improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm

Convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper. The methods of analysis currently available in literature assume that the input signals to the filter are white. This restriction is removed for Gaussian signals in our analysis. Expressions for the second moment of the coefficient vector and the steady-state error power are also derived. Simulation results are presented, and the theoretical and empirical curves show a very good match.

A fast recursive least squares adaptive second order Volterra filter and its performance analysis

A fast, recursive least squares (RLS) adaptive nonlinear filter modeled using a second-order Volterra series expansion is presented. The structure uses the ideas of fast RLS multichannel filters, and has a computational complexity of O(N/sup 3/) multiplications, where N-1 represents the memory span in number of samples of the nonlinear system model. A theoretical performance analysis of its steady-state behaviour in both stationary and nonstationary environments is presented. The analysis shows that, when the input is zero mean and Gaussian distributed, and the adaptive filter is operating in a stationary environment, the steady-state excess mean-squared error due to the coefficient noise vector is independent of the statistics of the input signal. The results of several simulation experiments show that the filter performs well in a variety of situations. The steady-state behaviour predicted by the analysis is in very good agreement with the experimental results. >

Tracking analysis of the sign algorithm in nonstationary environments

A tracking analysis of the adaptive filters equipped with the sign algorithm and operating in nonstationary environments is presented. Under the assumption that the nonstationary can be modeled using a random disturbance, it is shown that the long-term time average of the mean-absolute error is bounded, and that there exists an optimal choice of the convergence constant mu which minimizes this quality. Applying the commonly used independence assumption, and under the assumption that the nonstationarity is solely due to the time-varying behavior of the optimal coefficients, it is shown that the distributions of the successive coefficient misalignment vectors converge to a limiting distribution when the adaptive filter is used in the system identification mode. Finally, under the additional assumption that the signals involved are zero mean and Gaussian, a set of nonlinear difference equations is derived that characterizes the mean and mean-squared behavior of the filter coefficients and the mean-squared estimation error during adaptation and tracking. Results of several experiments that show very good correlation with the theoretical analyses are presented. >

Adaptive unsharp masking for contrast enhancement

A new scheme of unsharp masking for image contrast enhancement is presented. An adaptive algorithm is introduced so that a sharpening action is performed only in locations where the image exhibits significant dynamics. Hence, the amplification of noise in smooth areas is reduced. An adaptive directional filtering is also performed so as to provide suitable emphasis to the different directional characteristics of the detail. Because it is capable of treating high-detail and medium-detail areas differently, this algorithm also avoids unpleasant overshoot artifacts in regions of sharp transitions. Experimental results demonstrating the usefulness of the adaptive operator in an application involving preprocessing of images for enhancement prior to zooming are also included.

A stable adaptive Hammerstein filter employing partial orthogonalization of the input signals

This paper presents an algorithm that adapts the parameters of a Hammerstein system model. Hammerstein systems are nonlinear systems that contain a static nonlinearity cascaded with a linear system. In this paper, the static nonlinearity is modeled using a polynomial system, and the linear filter that follows the nonlinearity is an infinite-impulse response (IIR) system. The adaptation of the nonlinear components is improved by orthogonalizing the inputs to the coefficients of the polynomial system. The step sizes associated with the recursive components are constrained in such a way as to guarantee bounded-input bounded-output (BIBO) stability of the overall system. This paper also presents experimental results that show that the algorithm performs well in a variety of operating environments, exhibiting stability and global convergence of the algorithm.

Stochastic gradient adaptive filters with gradient adaptive step sizes

Two adaptive step-size gradient adaptive filters are presented. The step sizes are changed using a gradient descent algorithm designed to minimize the squared estimation error. The first algorithm uses the same step-size sequence for all the filter coefficients, whereas the second algorithm uses different step-size sequences for different adaptive filter coefficients. An analytical performance analysis of the first algorithm is also presented. Analyses and experiments indicate that (1) the algorithms have fast convergence rates and small midadjustment errors and (2) in nonstationary environments, the algorithms tend to adjust the step sizes so as to give close to the best possible performance. Several simulation examples demonstrating the good properties of the adaptive filters are also presented.<<ETX>>

A fast recursive least-squares second order Volterra filter

A fast, recursive least-squares (RLS) adaptive nonlinear filter is presented. The nonlinearity is modeled using a second-order Volterra-series expansion. The structure makes use of the ideas of fast RLS multichannel filters and has a computational complexity of O(N/sup 3/) multiplications. This compares with O(N/sup 6/) multiplications required for direct implementation. Simulation examples in which the filter is employed to identify nonlinear systems using noisy output observations are also presented. Further simplification to the structure through a simplified model is discussed.<<ETX>>