Pseudorandom signals to estimate apparent transfer and coherence functions of nonlinear systems: applications to...

There is an increasing need in physiology to estimate nonparametric linear transfer functions from data originating from biological systems which are invariably nonlinear. For pseudorandom (PRN) input stimuli, we derive general expressions for the apparent transfer (Z) and coherence (gamma 2) functions of nonlinear systems that can be represented by a Volterra series. It is shown that in the case of PRN signals in which the frequency components are integer multiples of other components the estimates of Z are seriously biased due to harmonic distortion and crosstalk among frequency components of the input. When the PRN signal includes components that are not integer multiples of other components harmonic distortion is avoided, but not necessarily cross talk. Here the estimates of Z remain poor without a noticeable influence on gamma 2. To avoid the problems associated with harmonic distortions and minimize the influence of crosstalk, a family of pseudorandom signals is proposed which are especially suited for the estimation of Z and gamma 2 in mechanical measurements of physiological systems at low frequencies. The components in the signals cannot be reproduced as linear combinations of two or more frequency components of the input. In a second-order system, this completely eliminates the bias, while in higher-order, but not strongly nonlinear systems, the interactions among the components are reduced to a level that the response can be considered as if it was measured with independent sine waves of an equivalent amplitude. It is also shown that the values of gamma 2 are not appropriate to assess linearity of the system. The theory is supported by simulation results and experimental examples brought from the field of respiratory mechanics by comparing the input impedance of the respiratory system of a dog measured with various PRN signals.