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## Methods for the Volterra Model

There exists many methods for identification of NLS by the Volterra series. One of the first basic methods is the method based on weighted Dirac impulses (Schetzen, M., 1989), where the Volterra kernels are estimated recursively from the first kernel to the higher one. Other methods have also been proposed, such as the maximum length sequences based method (Reed, M.J. & Hawksford, M.O., 1996b), the periodic signal based method (Evans, C. et al., 1996), the least squares method (Westwick, D.T. & Kearney, R.E., 1998), or the third moment based method (Hong-Zhou Tan & Aboulnasr, T., 2006). All these methods are generally not used in practice because of the large number of parameters required to represent the higher-order Volterra kernels.

### Methods for the Wiener and Hammerstein Models

The identification methods for Simple Wiener and Hammerstein systems depend upon whether the model is parametric or not. If the model is parametric, the zero-memory nonlinear part is represented by a polynomial function, or by general orthogonal series, and thus by a finite number of parameters (Billings, S.A. & Fakhouri, S.Y., 1979). The parameters to be estimated are those of the zero-memory part and those of the linear part of the model. The original method uses an iterative scheme to determine both, the linear and zero-memory nonlinear parts separately (Chang, F. & Luus, R., 1971). More recent algorithms are based on estimation of general NARMAX models (Chen, S. & Billings, S, 1989). If the model is nonparametric, the estimation consists in estimation of finished point of the nonlinear function that can further serve for polynomial representation (Greblicki, W. & Pawlak, M., 1989), (Greblicki, W., 1997), (Greblicki, W., 2004).

#### Method for the Multiple-Input Single-Output Model

The identification methods based on MISO model depend upon whether the model is parametric or not. Some parametric methods can be found in literature (Boutayeb, M. et al., 1993), (Kortmann, M. & Unbehauen, M., 1987). The method proposed by (Rice, H.J. & Fitzpatrick, J.A., 1988) for general nonparametric model is based on power- and cross-spectral density estimations.

**1 Introduction**

**2 Nonlinear System Identiﬁcation**

2.1 Introduction

2.2 Nonlinear Models

2.2.1 The Volterra Series Model

2.2.2 Simple Wiener and Hammerstein Models

2.2.3 Polynomial Hammerstein and Wiener Models

2.2.4 Multiple-Input Single-Output Model

2.3 Excitation Signals

2.3.1 White Gaussian Noise

2.3.2 Pseudo-Random Signals

2.3.3 Harmonic Signals

2.3.4 Multitone Signals

2.3.5 Swept Sine Signals

2.4 Methods of Identiﬁcation

2.4.1 Methods for the Volterra Model

2.4.2 Methods for the Wiener and Hammerstein Models

2.4.3 Method for the Multiple-Input Single-Output Model

2.4.4 Nonlinear Convolution Based Method

2.5 Summary

**3 MISO Method based on Power Series**

3.1 Correlation and Power Spectral Density

3.2 Expression of Decorrelated Inputs

3.3 Estimation of Filters of the MISO model

3.4 Zooming Eﬀect

3.5 Simulation of Nonlinear Systems

3.5.1 Memoryless Nonlinear Systems

3.5.2 Nonlinear System with Memory

3.6 Summary

**4 Method Based on Swept Sine Excitation**

4.1 Introduction

4.2 Input Signal

4.2.1 New Redesign of the Input Signal

4.2.2 Time Domain Properties

4.2.3 Nonlinear Convolution Properties

4.2.4 Frequency Domain Properties

4.3 Inverse Filter

4.3.1 Extension of the Inverse Filter

4.4 Modiﬁed Nonlinear Convolution Method

4.4.1 Case of Blind Identiﬁcation

4.4.2 Case of Known Inputs

4.5 Simulation of a NLS with Memory

4.6 Summary

**5 Nonlinear Systems with Hysteresis**

5.1 Introduction

5.2 Hysteresis as a Hammerstein System

5.2.1 Static Hysteresis with Local Memory

5.2.2 Dynamic Hysteresis

5.3 Simulation Experiment

5.3.1 Static Hysteresis Nonlinearity

5.3.2 Hysteresis Nonlinearity with Frequency Dependency

5.4 Summary

**6 Nonlinear Systems in Acoustics**

6.1 Measuring Methodology

6.2 Objective Criteria for Veriﬁcation

6.3 Validation on an Audio Limiter

6.4 Validation on a Acoustic Waveguide

6.5 Application to Electrodynamic Loudspeaker

6.5.1 Context

6.5.2 Experimental Setup

6.5.3 Preliminary Results

6.5.4 Total Harmonic Distortion Measurement

6.6 Summary

**7 Conclusion**

Biblioghraphy

List of Author’s Publications

Vita