Identification of the dynamic characteristics of nonlinear structures

HIGHER-ORDER FREQUENCY RESPONSE FUNCTIONS
3.3.1 THEORY OF HIGHER-ORDER FREQUENCY
RESPONSE FUNCTIONS
The 
Volterra kernel transform or 
Volterra transfer function is simply
defined as the n-dimensional Fourier transform of the 
Volterra kernel
Since the 
kernel
is real, symmetric (the value of 
does not depend on the order of 
for example, in the case of second-order
kernel, 
and causal, its Fourier transform 
is
symmetric and also possesses complex conjugate symmetry such that
For a linear system, if the frequency response functions (only the first-order) have been
determined, the output x(t) can be calculated for any form of inputs. The same argument
holds if all the Volterra transfer functions 
have been determined and
since 
are unique (independent of input and output of the system), the
Volterra series representation is mathematically very attractive because under this
representation, the identification of a nonlinear system reduces to the measurement of
these unique Volterra kernel transforms. However, it will be shown that due to the
interactions between kernels, these uniquely defined Volterra transfer functions cannot be
uniquely measured in practice and all that can be measured are approximations which, in
general, are input/output dependent.

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3
Identification of Nonlinearity Using Higher-order 
7 4
The 
frequency response function 
is defined as the measured
Volterra kernel transform 
The relationship between
and 
is discussed below. The input-output
relationship of a nonlinear system based on Volterra series representation has been
discussed in detail 
for different forms of input and only the sinusoidal form of input
is considered here. To make the analysis convenient, it is necessary to introduce the