Linear multichannel blind equalizers of nonlinear FIR Volterra channels

Abstract

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 approach requires that the Volterra kernels satisfy a certain coprimeness condition and that the input possesses a minimal persistence-of-excitation order. No other special conditions are imposed on the kernel transfer functions or on the input signal, which may be deterministic or random with unknown statistics. The proposed algorithms are corroborated with simulation examples.

Minimum Mean-Square Error Equalization for Second-Order Volterra Systems

“…The big channel matrix containing all products up to fourth order is given as Now there is no correlation between the useful term and the noise terms and , since we've used the reduced Kronecker notations and zero-mean properties for and , respectively. To make things more clear, the final equation for is rewritten in a matrix form (18) The formula for the MMSE expression for is now easily derived. Let us define the correlation matrices as (19) (20) and (21) One can show that the correlation between the data vector and the noise vector is zero since the assumption that data and noise are uncorrelated even holds for this modified data and noise vector.…”

“…Note that initial work on these approaches has been reported in [17], but the equalizers developed there are only approximations of the true MMSE equalizers. In [18], linear equalizers are proposed to achieve equalization of a nonlinear Volterra system. Such an equalizer requires oversampling at the receiver front end increasing the complexity significantly.…”

Blind equalization and identification of nonlinear and IIR systems-a least squares approach

“…Blind linear identification and equalization problems have been studied by many researchers; however, relatively little work has been done in the field of blind nonlinear system identification and equalization. Blind approaches have been proposed for restricted classes of nonlinearities such as linear-zero memory nonlinearity-linear systems [7] or strict Volterra models of nonlinearity [3]. The work in [3] describes a very elegant approach for finding linear FIR equalizers for nonlinear channels.…”

“…Blind approaches have been proposed for restricted classes of nonlinearities such as linear-zero memory nonlinearity-linear systems [7] or strict Volterra models of nonlinearity [3]. The work in [3] describes a very elegant approach for finding linear FIR equalizers for nonlinear channels. However, the results in [3] only apply to nonlinear channels that can be exactly described by Volterra filters.…”

“…The work in [3] describes a very elegant approach for finding linear FIR equalizers for nonlinear channels. However, the results in [3] only apply to nonlinear channels that can be exactly described by Volterra filters. Another class of approaches are based on assuming a finite alphabet [11]; these suffer from ambiguity in the solution.…”