Abstract
Nonlinear systems which have finite memories and are time invariant can be completely described by the Wiener functional expansion, in which a series of multidimensional kernels provide a polynomial approximation to the nonlinear behaviour. The kernels give a best fitting estimation to the total system behaviour in the least mean square sense and can therefore be used to describe systems in which the nonlinearities include discontinuous functions. A modification of the Wiener method described by Lee and Schetzen, which uses kernels defined in terms of cross correlation functions, has been used in most practical attempts to analyse nonlinear systems, but we have previously described how the cross correlations may be replaced with complex multiplications in the frequency domain. The speed of domain translation offered by the fast Fourier transform makes this method more efficient than time domain estimation. In this paper the practical implementation of the technique on a medium sized digital computer is described for nonlinear systems whose outputs are continuous or pulsatile signals. This description should be adequate to allow others to implement the analysis scheme. The technique is well suited to the analysis of nonlinear biological systems, particularly those encountered in neurophysiology, because of its generality, ability to deal with hard nonlinearities and ease of use with systems having pulsatile outputs.
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References
Beastall, H. R.: White noise generator. Wireless World 78, 127 (1972)
Bedrosian, E., Rice, S. O.: The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Proc. IEEE 59, 1688 (1971)
Bendat, J.S., Piersol, A.G.: Measurement and analysis of random data. New York: John Wiley 1966
Bingham, M.D., Godfrey, M. D., Tukey, J. W.: Modern techniques of power spectrum estimation. IEEE Trans. Audio and Electroacoustics 15, 56 (1967)
Bryant, H., Segundo, J. P.: How does the neuronal spike trigger read Gaussian white noise membrane current? In: Proceedings of the First Symposium on Testing and Identification of Nonlinear Systems, Ed.: McCann, G.D., Marmarelis, P.Z. (1975)
Cooley, J. W., Tukey, J. W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297 (1965)
French, A. S., Holden, A. V.: Alias-free sampling of neuronal spike trains. Kybernetik 8, 165 (1971)
French, A. S.: Automated spectral analysis of neurophysiological data using intermediate magnetic tape storage. Comput. Prog. Biomed. 3, 45 (1973)
French, A. S., Butz, E.G.: Measuring the Wiener kernels of a nonlinear system using the fast Fourier transform algorithm. Int. J. Control 17, 529 (1973)
French, A. S., Butz, E. G.: The use of Walsh functions in the Wiener analysis of nonlinear systems. IEEE Trans. on Computers 23, 225 (1974)
French, A.S.: Compiling and interpreting identical programs in a high level scientific language. Submited to the Journal of the Digital Equipment Users Society (1976)
French, A. S., Wong, R. K. S.: The response of trochanternal hair plate sensilla in the cockroach to periodic and random displacements. Biol. Cybernetics 22, 33 (1976)
Harris, G. H., Lapidus, L.: The identification of nonlinear systems. Indust. Engng. Chem. 59, 66 (1967)
Krausz, H.I.: Identification of nonlinear systems using random impulse train inputs. Biol. Cybernetics 19, 217 (1975)
Landau, J.J.: Sampling, data transmission and the Nyquist rate. Proc. IEEE 55, 1701 (1967)
Lee, Y.W., Schetzen, M.: Measurement of the Wiener kernels of a nonlinear system by cross correlation. Int. J. Control 2, 237 (1965)
Lipson, E.D.: White noise analysis of Phycomyces light growth response system. Biophys. J. (in press)
Marmarelis, P.Z., Naka, K.: Nonlinear analysis and synthesis of receptive field responses in the catfish retina. J. Neurophysiol. 36, 605 (1973)
McCann, G.D., Marmarelis, P.Z. (Eds): Proceedings of the First Symposium on Testing and Identification of Nonlinear Systems. California Institute of Technology, Pasadena, California (1975)
Nyquist, H.: Certain topics in telegraph transmission theory. Trans. AIEE 47, 617 (1928)
Palm, G., Poggio, T.: The Volterra representation and the Wiener expansion: validity and pitfalls. Submitted to S.I.A.M. on Applied Mathematics (1976)
Stark, L.: The pupillary control system: Its nonlinear adaptive and stochastic engineering design characteristics. Automatica 5, 655 (1969)
Stein, R. B., French, A. S., Holden, A. V.: The frequency response, coherence and information capacity of two neuronal models. Biophys. J. 12, 295 (1972)
Udwadia, F.: The identification of building structural systems. In: Proceedings of the First Symposium on Testing and Identification of Nonlinear Systems, Ed.: McCann, G.D., Marmarelis, P.Z. (1975)
Wiener, N.: Nonlinear problems in random theory. New York: John Wiley and Sons (1958)
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French, A.S. Practical nonlinear system analysis by Wiener kernel estimation in the frequency domain. Biol. Cybernetics 24, 111–119 (1976). https://doi.org/10.1007/BF00360650
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DOI: https://doi.org/10.1007/BF00360650