The Volterra functional analysis remains the basic mathematical tool for nonlinear systems identification. Volterra functional analysis can be viewed as an extension to nonlinear systems of the notion of impulse response derived for linear system analysis. This is obtained as a Taylor series expansion of the system input-output mapping, an extensive presentation of which can be found in [28, 29]. An introduction to Volterra series analysis has been given in Chapter 2 as applied to system identification with "natural" real signals. In this section we will present the extension of Volterra series analysis to band-pass systems for which the excitation is a complex envelope signal, as previously described.
Let us consider as stated above a causal and stable nonlinear bloc functioning in impedance-matched conditions. In these assumptions, (3.5) describing the bloc dynamics can be seen as a functional ?:[ t - T ?, t] 2 ? ?,which links the excitation 
to the response ?( t) on the time interval [ t - T ?, t]. This is illustrated in Figure 3.9.
Carrying a Taylor series expansion of the functional ? readily gives birth to the Volterra series describing the system, as follows. In fact, sampling the input signal, as sketched, we may intuitively express the sample ?( T) of the system response at time abscissa t as a function of t and the...
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