Higher-order Statistical Signal Processing With Volterra

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Abstract
This paper presents a review of some applications of Volterra filters (VFS) in statistical signal processing. VFS are a particular class of nonlinear filters defined by an extension of the idea of impulse response to the nonlinear case. Linear and linear-quadratic filters are special examples of VFS limited to the first or second order respectively. The pnrpose of this paper is to extend to VFS the main ideas used in applications of linear filters for signal processing problems. To do this the first task is to simplify the notation of the input-outpnt relationship of VFS which appears very tedions. Noting that this relation is nonlinear in terms of the input but linear in terms of the parameters defining the VF, it is possible to dednce that the ontput can be written as a scalar product quite similar to that nsed in the linear case. Using this form of scalar product, which is very easy to manipulate, and also the higher order statistics of the signals, we study some fundamental problems snch as detection, estimation, array processing, etc. In doing this we avoid analytic calculations which are very complex with VFS and focus our attention on geometrical methods.

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