Antonin Novak - homepage

 Postdoctoral Research Fellow Laboratoire d'Acoustique de l'Université du Maine UMR-CNRS 6613 Université du Maine Av. Olivier Messiaen 72085 Le Mans Cedex 9 Email Address: ant.novak[at]gmail.com

Synchronized-Swept-Sine method is a nonlinear system identification method based on "nonlinear convolution" presented by Angelo Farina in AES 108th convention in Paris in 2000. The method can analyze a nonlinear system in the terms of higher order impulse responses using a single swept sine signal with user-defined frequencies and duration. The synchronization of the swept sine (Fig.1), firstly presented by Antonin Novak in AES 124th convention in Amsterdam in 2008 and mathematically supported in an IEEE paper entitled Nonlinear System Identification Using Exponential Swept-Sine Signal (2009) allows the phase synchronization of the higher order impulse responses. The well synchronized higher order impulse responses can be next used, in several ways, to nonlinear modeling and synthesis.

The synchronized swept sine is an exponential swept sine signal, i.e. a signal exhibiting an instantaneous frequency fi(t) which increases exponentially with time (Fig. 1). Such a signal is sometimes called an exponential chirp and is defined as

where L is defined as

The properties of the swept sine signal are defined by start and end frequencies f1 and f2 and by the approximate time support T̂.

Figure 1.: Swept-sine signal xs(t) in time domain (below) with the time length chosen according to instantaneous frequency fi(t) (above).

Next, an inverse filter, defined as

is generated. Then, the ordinary linear convolution (calculated in frequency domain) between the captured output signal and the inverse filter is calculated. The result of the convolution is a set of time shifted impulse responses called higher-order nonlinear impulse responses (Fig. 2). As the impulse responses are separated in time, they can be easily windowed. Afterwards, the Fourier Transform of each separated higher order impulse response can be calculated. The results are called higher order frequency responses Hi(f). The i-th response corresponds to the frequency evolution in amplitude and phase of the i-th higher harmonic when exciting the system with a harmonic signal. Thus, the method can within one measurement of time length T̂ characterise the nonlinear system in amplitude and phase not only for the fundamental harmonic as usual, but also for higher nonlinear harmonics. For further information, please read the IEEE paper entitled Nonlinear System Identification Using Exponential Swept-Sine Signal, where the principles of the method are studied in detail.

Figure 2.: Result of the nonlinear convolution process in the form of set of higher-order nonlinear impulse responses hi(t).