Neural Network Modelling of Nonlinear Channels with Memory

Several nonlinear adaptive techniques have been proposed for modelling nonlinear channels with memory. These techniques include Volterra series, wavelet networks, neural networks, etc. See [11, 6] for a review. For example, adaptive Volterra series have been applied in [1] for modelling digital satellite channels. These Volterra models do not characterize each element of the channel. They provide a model only for the overall channel input-output transfer function. However, Volterra models provide an estimation of the global system memory and the complexity of the nonlinear transfer function. Non adaptive parametric techniques also have been proposed for modelling nonlinear systems with memory (e.g. [3, 4, 7, 8, 9]). Nikias and Petropulu [7] review higher order statistic-based methods used for detection and characterization of nonlinearities. In particular, they present examples for the characterization of Volterra series coefficients. Block-oriented methods have been largely used for nonlinear system identification. These methods are based on the idea that the system to be identified is composed of several simple subsystems. For example, several authors studied Hammerstein systems which consist of the cascade of systems composed of a nonlinear memoryless element followed by a linear dynamic one. See for instance [4, 8, 9]. The key issue in adaptive system identification is to find the best model structure within which an optimal model has to be found by using an appropriate adaptive algorithm. This paper proposes adaptive neural network approaches for modelling nonlinear channels with memory. It is shown that a good choice of the neural network structure should follow directly from the application and the prior knowledge on the physical system to be modelled. This paper presents only the algorithms and their learning behavior. In [5] two typical problems are addressed: i) identification and characterization of digital channels which are composed of physically separable parts (e.g. linear filters with memory and nonlinear memoryless devices). An example is a digital satellite channel, composed of a linear filter followed by a memoryless nonlinear travelling wave tube amplifier (TWT) and a second linear filter. The neural network approach models the global nonlinear channel inputoutput transfer function and characterizes each component of the channel separately. The learning process uses only the channel input-output signals. ii) modelling nonlinear channels which cannot be simply represented by separable parts. An example is the solid state power amplifier (SSPA) (nonlinear amplifiers with memory) used typically in satellite communications. The analytic analysis of neural network algorithms applied for modelling nonlinear channels can be found in [2] and [5]. The paper is organized as follows. Section 2 gives an example of a nonlinear channel. Sections 3 and 4 present the neural network algorithms. Finally, section 5 is devoted to the algorithm learning behavior.