The use of Volterra series for simiulating the nonlinear behaviour of musical instruments
Lamberto Tronchin; Andrea Venturi

Measurement and emulation of audio systems (devices, environments and soundboxes) have been walked in these years. The most used methods to obtain information about an audio system are those based on measuring its impulse response (IR). Once the IR has been caught it is possible to recreate, by the use of linear convolution, the output signal that the audio system will generate when it is physically driven by any input signal. This method gives great results if the system is linear and time-invariant (environments behaviour is much linear and therefore its reverberant effect can be faithfully recreated using IRs) but not satisfactory in other cases, such as the emulation of tube preamps (mainly nonlinear) and musical instruments. Since the musical instruments cannot be considered completely linear, their musical performance might be analysed properly considering also their nonlinear behaviour.
By the use of Volterra series it is possible to represent the input-output relationship of nonlinear systems. This mathematical theory uses a set of impulse responses to describe the system and not only one as before. By an enhanced impulse response measurement method it is possible to obtain this set of impulses and then by Volterra series it will be possible to have the output of the audio system driven by any input. A special numerical tool has been developed to recreate the system behaviour by using this method. Satisfactory results have been obtained in comparison with the traditional linear convolution based approach.