What is the Volterra series analysis of a weakly nonlinear amplifier?

Volterra series analysis extends linear system theory to weakly nonlinear systems (like an amplifier operating near but not at its compression point) by representing the output as a sum of multi-dimensional convolutions (Volterra kernels) of the input signal. The output is: y(t) = H1{x(t)} + H2{x(t)} + H3{x(t)} + ..., where H1 is the linear transfer function (equivalent to S21), H2 is the second-order Volterra kernel (responsible for second harmonic and IMD2), and H3 is the third-order kernel (responsible for third harmonic and IMD3). In the frequency domain, these kernels become the nonlinear transfer functions: H1(f) is the small-signal gain, H2(f1,f2) describes how two input frequencies at f1 and f2 interact to create outputs at f1+f2 and f1-f2, and H3(f1,f2,f3) describes three-frequency interactions creating outputs at f1+f2+f3, 2f1-f2, etc. The key advantage of Volterra analysis over simple power series analysis is that Volterra captures frequency-dependent nonlinearity: the distortion at a given frequency depends not only on the signal amplitude but also on the frequency content of the signal and the amplifier's impedance at the mixing products' frequencies. This makes Volterra analysis essential for predicting intermodulation in circuits where the gain and impedance vary with frequency (i.e., all practical amplifiers with matching networks).