In this paper stochastic aradient adaptive filters using the Sign or Sign-Sign Algorithm are analyzed based upon general assumptions on the reference signal, additive noise and particularly jointly distributed tap errors. A set of difference equations for calculating the convergence process of the mean and covariance of the tap errors is derived with integrals involving characteristic function and its derivative of the tap error distribution. Examples of echo canceller convergence with jointly Gaussian distributed tap errors show an excellent agreement between the empirical results and the theory.

This letter develops theoretical analysis of the normalized LMS algorithm (NLMSA) for use in complex-domain adaptive filters in the presence of impulse noise at filter input. We propose a new "stochastic" model for such impulse noise, and assume that filter reference input process is a white process, e.g., digital QAM data, White & Gaussian process, etc. In the analysis, we derive a simple difference equation for mean square tap weight misalignment (MSTWM). Experiment is carried out to demonstrate effectiveness of the NLMSA in robust filtering in the presence of the impulse noise at the filter input. Good agreement between simulated and theoretically calculated filter convergence, in a transient phase as well as in a steady-state, proves the validity of the analysis.

The paper presents an adaptive algorithm named adaptive threshold nonlinear algorithm for use in adaptive filters in the complex-number domain (c-ATNA) in applications to digital QAM systems. Although the c-ATNA is very simple to implement, it makes adaptive filters highly robust against impulse noise and at the same time it ensures filter convergence as fast as that of the well-known LMS algorithm. Analysis is developed to derive a set of difference equations for calculating transient behavior as well as steady-state performance. Experiment with simulations and theoretical calculations for some examples of filter convergence in the presence of Contaminated Gaussian Noise demonstrates that the c-ATNA is effective in combating impulse noise. Good agreement between simulated and theoretical convergence proves the validity of the analysis.

Recursive least absolute(RLA) error algorithm is derived which is basically the sign algorithm (SA) combined with recursive estimation of the inverse covariance matrix of the reference input. The name RLA comes from the absolute error criterion. Analysis of the transient behavior and steady-state performance of the RLA algorithm is fully developed. Results of experiment show that the RLA algorithm considerably improves the convergence rate of the SA while preserving the robustness against impulse noise. Good agreement between the simulation and the theoretically calculated convergence validates the analysis.

In this paper, a new set of difference equations is derived for transient analysis of the convergence of adaptive FIR filters using the Sign-Sign Algorithm with Gaussian reference input and additive Gaussian noise. The analysis is based on the assumption that the tap weights are jointly Gaussian distributed. Residual mean squared error after convergence and simpler approximate difference equations are further developed. Results of experiment exhibit good agreement between theoretically calculated convergence and that of simulation for a wide range of parameter values of adaptive filters.

In this letter, we derive a probability density function (PDF) for a modulus of product of two complex-valued Gaussian random variables. The PDF is expressed using Modified Bessel Functions, and the probability distribution is named Gaussian Product Modulus Distribution. Some examples of expectation calculation are provided.

This paper derives a set of orthogonal polynomials for a complex random variable that is uniformly distributed in two dimensions (2D). The polynomials are used in a series expansion to approximate memoryless nonlinearities in digital QAM systems. We also study stochastic identification of nonlinearities using the orthogonal polynomials through analysis and simulations.

In this paper, we propose a transmitter structure in digital QAM systems where pre-compensator compensates for nonlinearity with "memory effects" at the output amplifier. The nonlinearity is modeled as a linear time-invariant filter cascaded by memoryless nonlinearity (Wiener model), whereas the pre-compensator comprises an FIR-type adaptive filter that follows a memoryless predistorter based on a series expansion with orthogonal polynomials for digital QAM data. The predistorter and the adaptive filter of the pre-compensator are stochastically and directly adapted using the error signal. The theoretically optimum parameters of the predistorter are approximately solved whence the steady-state mean square compensation error is calculated. Simulations show that the proposed pre-compensator can be adapted to achieve a sufficiently small compensation error, restoring the original QAM constellation through linearization and equalization of the nonlinearity with memory effects.

This letter develops convergence analysis of normalized sign-sign algorithm (NSSA) for FIR-type adaptive filters, based on an assumption that filter tap weights are Gaussian distributed. We derive a set of difference equations for theoretically calculating transient behavior of filter convergence, when the filter input is a White & Gaussian process. For a colored Gaussian input and a large number of tap weights, approximate difference equations are also proposed. Experiment with simulations and theoretical calculations of filter convergence demonstrates good agreement between simulations and theory, proving the validity of the analysis.