A fast algorithm for general Volterra filtering

Volterra filters are a classical instrument for nonlinear channels and systems modeling, noise and echo cancellation, signal estimation and detection, and various other applications. As is well known, the computational weight of Volterra filters exponentially grows with the nonlinearity degree. This work presents a contribution to the efficient computation of Volterra filters with generic order nonlinearity found in many telecommunication applications. Our technique rests on the interpretation of the Mth-order one-dimensional Volterra filters in terms of M-dimensional linear convolution, and it adopts a multidimensional fast convolution scheme. This makes the method applicable to any M. Interestingly enough, fast convolution based on the standard multidimensional fast Fourier transform (MD FFT) in the case of Volterra filters is outperformed by direct computation. Our method is efficient due to the use of a special MD FFT which can exploit the symmetries of the signals entering the computation of Volterra filters and which makes it superior to direct computation. The points of interests of the results presented are both the generality and the fact that they show that the well-known nonlinearity/multidimensionality tradeoff of Volterra filters can have computational implications.