Truncated Volterra expansions model nonlinear systems encountered with satellite communications, magnetic recording channels, and physiological processes. A general approach for blind deconvolution of single-input multiple-output Volterra finite impulse response (FIR) systems is presented. It is shown that such nonlinear systems can be blindly equalized using only linear FIR filters. The approa...

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function...

In this paper, we investigate periodic solutions of linear and nonlinear discrete Volterra equations of convolution or non-convolution type with unbounded memory. For linear discrete Volterra equations of convolution type, we establish Fredholm’s alternative theorem and for equations of non-convolution type, and we prove that a unique periodic solution exists for a particular bounded initial fu...

This article introduces two Tensor Train-based iterative algorithms for the identification of high order discrete-time nonlinear MIMO Volterra systems. The system identification problem is rewritten in terms of a Volterra tensor, which is never explicitly constructed, thus avoiding the curse of dimensionality. It is shown how each iteration of the two identification algorithms involves solving ...

Abstract: A new exact method of measuring the Volterra kernels of finite order discrete nonlinear systems is presented. The kernels are rearranged in terms of multivariate crossproducts in the vector form. The one-, two-, . . . , and `-dimensional kernel vectors are determined using a deterministic multilevel sequence with ` distinct levels at the input of the system. It is shown that the defin...

The robust nonlinear adaptive filtering problem based on Volterra model is researched when the input and output observation data are both corrupted by noise in this paper. On the basis of minimizing Volterra total mean square error (VTMSE), a robust total least mean square adaptive filtering algorithm for nonlinear Volterra filter is proposed. The performance analysis demonstrates that the robu...

Abstract— A new polynomial method to solve Volterra–Fredholm Integral equations is presented in this work. The method is based upon Shifted Legendre Polynomials. The properties of Shifted Legendre Polynomials and together with Gaussian integration formula are presented and are utilized to reduce the computation of Volterra–Fredholm Integral equations to a system of algebraic equations. Some num...

We consider the rate of convergence to equilibrium of Volterra integrodifferential equations with infinite memory. We show that if the kernel of Volterra operator is regularly varying at infinity, and the initial history is regularly varying at minus infinity, then the rate of convergence to the equilibrium is regularly varying at infinity, and the exact pointwise rate of convergence can be det...

We propose a new class of mapping-based nonlinear quadratic Volterra filters for image enhancement. The input signal is mapped prior to filtering by a memoryless one-to-one nonlinear mapping function. The mapping function can be described by any arbitrary function and implemented by a simple look-up table. If necessary, an inverse mapping function is applied after filtering. Using this general ...

Over the last decade, the field of geometric integration has rapidly established itself as one of the core research areas in numerical ordinary differential equations. Geometric integrators are numerical methods which preserve some of the mathematical or physical properties of the system they are approximating. In the case of the Lotka-Volterra equations, which are a Poisson system, a good geom...