A family of Wiener-type methods is discussed in a general context. These methods share the concept of expansion of an unknown transducer as an orthogonal series. The terms of the series are drawn from a hierarchy of subspaces of transducers that are orthogonal with respect to a particular stimulus ensemble. Choices of specific stochastic ensembles lead to previously described analytical methods, including the classical one of Wiener.
It is proposed that a sum of incommensurate or nearly incommensurate sinusoids forms a signal that leads to a useful orthogonal expansion. The family of orthogonal subspaces are presented explicitly. Projection of unknown transducer into an orthogonal subspace amounts to isolation of Fourier components of the output of the unknown transducer at certain harmonics and combination frequencies of the input frequencies. Practical advantages of this technique include i) the ease of computation of the higher-order kernels, and ii) the opportunity for digital filtering of the response, which enhances the signal-to-noise ratio.
Finally, it is shown that the kernels obtained using a sum-of-sinusoids signal approach the Fourier transforms of the Wiener kernels as the number of sinusoids grows without bound. Thus, the sum-of-sinusoids technique retains a major theoretical advantage of the Wiener white-noise method: the kernels of simple model transducers have simple analytic forms.