Wiener kernels

They differ from the kernels it (ti, ..., r ) of the Volterra series only by a faster signal decay with increasing time arguments [Bliil]. For coinciding time arguments the crosscorrelation function is the sum of the n-dimensional impulse-response function h with the impulse-response functions hm of lower orders m < n. The stochastic impulse-response functions h are the kernels of an expansion of the system response y(t) similar to the Volterra series (4.2.4) but with functionals orthogonalized for white-noise excitation x t) [Bliil, Marl, Leel, Schl], This expansion is known by the name Wiener series, and the h are referred to as Wiener kernels. 

Fourier transformation of the Wiener kernels > > a ) over the time delays... 

The Wiener kernels depend on the GWN input power level P (because they correspond to an orthogonal expansion), whereas the Volterra kernels are independent of any input characteristics. This situation can be likened to the coefficients of an orthogonal expansion of an analytic function being dependent on the interval of expansion. It is therefore imperative that Wiener kernel estimates be reported in the literature with reference to the GWN input power level that they were estimated with. When a complete set of Wiener kernels is obtained, then the complete set of Volterra kernels can be evaluated. Approximations of Volterra kernels can be obtained from Wiener kernels of the same order estimated with various input power levels. Complete Wiener or Volterra models can predict the system output to any given input. 

The orthogonality of the Wiener series allows decoupling of the various Wiener functionals and the estimation of the respective Wiener kernels from input-output data through cross-correlation [Lee and Schetzen, 1965]... 

French, A.S. 1976. Practical nonlinear system analysis by Wiener kernel estimation in the frequency domain. Biol. Cybern. 24 111. 

Lee, Y.W. and Schetzen, M. 1965. Measurement of the Wiener kernels of a nonlinear system by cross-... 

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