Comparison of discrete-time approximations for continuous-time nonlinear systems

Abstract

The work addresses the problem of approximating the sampled input-output (i/o) behavior of continuous-time nonlinear systems using discrete-time Volterra models. For an exactly band-limited nonlinear system for which a Volterra representation exists, the discrete-time Volterra model exactly corresponds to the sampled continuous-time Volterra kernels. Physical systems, as they are causal, are never exactly band-limited. Thus, a modeling error is introduced. By relaxing the causality condition and allowing a small processing delay, it is shown through simulation that more accurate discrete-time Volterra models compared to sampled continuous-time Volterra models can be generated.

Evaluating the potential of Volterra-PARAFAC IIR models

“…for 1 ≤ p ≤ P , where H p is the p-th order frequency domain kernel associated with the PM. Finally, assuming that a sampling frequency Ω s is appropriately chosen, the discrete-time reference p-th order kernel can be obtained by samplingH p uniformly over the domain −Ω s /2 < Ω i ≤ Ω s /2 and then computing its p-dimensional IFFT of length N p [12,14].…”

Efficient Kernel Computation for Volterra Filter Structure Evaluation

“…With the frequencydomain characterization of eq. (4), we can, by using the approach of [9], impose that the (discrete-time) GFRFs associated with the CVF be such that…”

“…From the above discussion we see that, in order to determine the GFRFs of the CVF and, consequently, its kernels, it is necessary to know G 1 , G 2 and the GFRFs of V . It should be noted that this frequency-domain derivation is advantageous over the direct sampling of the time-domain kernels, because it prevents the aliasing of components with frequency greater than Ω N [9].…”

“…In principle, with the GFRFs H p and the filters G 1 and G 2 , the kernels of the CVF can be determined through the inverse transform of (5). In practice, assuming that the p-th kernel contains little energy after some instant T p , an approximation of the anti-transforms of (5) that will suffice is [9]…”

“…and N p = ⌈T p Ω s /2π⌉. As observed in [9], it is important to note that this corresponds to sampling a circular convolution of the kernels with the sinc function in the time-domain, and thus there will be a non-causal portion of the kernels appearing at samples next to N p , which may impair the results. One way of attenuating this effect is to multiply the right hand side of eq.…”