Geometrical properties of optimal Volterra filters for signal detection

Absfrucf -Volterra filters are a particular class of nonlinear filters defined by an extension of the concept of impulse response to the nonlinear case. Linear-quadratic filters are a special example of Volterra filters limited to the second order. In the first part of this paper it is shown that all the results recently published, valid in the linear-quadratic case, can be extended with the appropriate notations to Volterra filters of arbitrary order. In particular, the optimum Volterra filter giving the maximum of the deflection for the detection of a signal in noise is wholly calculated. In the second part several geometrical properties of optimal Volterra filters are investigated by introducing appropriate scalar products. In particular the concept of space orthogonal to the signal and the noise alone reference (NAR) property are introduced allowing a decomposition of the optimal filter that exhibits a relation between detection and estimation. Extensions to the infinite case and relations with the likelihood ratio are also investigated.