The Cramer-Rao Bound and DMT Signal Optimisation for the Identification of a Wiener-Type Model
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The Cramer-Rao Bound and DMT Signal Optimisation for the Identification of a Wiener-Type Model H. Koeppl Christian Doppler Laboratory for Nonlinear Signal Processing, Graz University of Technology, 8010 Graz, Austria Email: [email protected]
A. S. Josan Department of Electronics and Communication Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India Email: [email protected]
G. Paoli System Engineering Group, Infineon Technologies, 9500 Villach, Austria Email: [email protected]
G. Kubin Christian Doppler Laboratory for Nonlinear Signal Processing, Graz University of Technology, 8010 Graz, Austria Email: [email protected] Received 2 September 2003; Revised 8 January 2004 In linear system identification, optimal excitation signals can be determined using the Cramer-Rao bound. This problem has not been thoroughly studied for the nonlinear case. In this work, the Cramer-Rao bound for a factorisable Volterra model is derived. The analytical result is supported with simulation examples. The bound is then used to find the optimal excitation signal out of the class of discrete multitone signals. As the model is nonlinear in the parameters, the bound depends on the model parameters themselves. On this basis, a three-step identification procedure is proposed. To illustrate the procedure, signal optimisation is explicitly performed for a third-order nonlinear model. Methods of nonlinear optimisation are applied for the parameter estimation of the model. As a baseline, the problem of optimal discrete multitone signals for linear FIR filter estimation is reviewed. Keywords and phrases: Wiener model, Cramer-Rao bound, signal design, nonlinear system identification.
1.
INTRODUCTION
In the design of optimal excitation signals for system identification, the Cramer-Rao bound plays a central role. For a given model structure, it gives a lower bound on the variance of the unbiased model parameter estimates for a given perturbation scenario [1]. The problem of signal optimisation for the identification of linear models is considered in [2]. We focus on a nonlinear model structure proposed in [3], which is nonlinear in the parameters and can be considered a generalisation of the classical Wiener model [4, page 143]. For the classical Wiener model, the Cramer-Rao bound was derived in [5]. The goal of this work is to gain further insight into the design of optimal excitation signals for the identification of nonlinear cascade systems. The application that drove our investigations is adaptive nonlinear filtering
for ADSL data transmission systems. The block diagram in Figure 1 shows an application of the nonlinear model as a nonlinear canceler of the hybrid echo for the receive path of an ADSL transceiver system. System distortion analysis revealed that the line-driver circuit is the main source of nonlinearity. In the subsequent simulation experiments, a nonlinear Wiener-type model of this line-driver circuit is used as a reference model. As excitation signal the class of discrete mul
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