**Identification and Supervised Equalization of a MIMO Non-linear Communication Channel**

**Nabiha SAIDI**, **Anouar BEN AMOR**, **Hassani MESSAOUD**

Unité de recherche ATSI, Ecole Nationale d’Ingénieurs de Monastir

Rue Ibn ELJazzar, 5019 Monastir – Tunisia

**Abstract**: Due to their general non linear structural and their linearity with respect to their parameters, Volterra models are widely used to describe the behaviour of non linear process. In this paper we are concerned by the modelling and the supervised equalization of a Multi Input Multi Output non linear communication channel using Volterra models. To overcome the burden induced by the parameter number increasing, we develop the Volterra kernel on the General Orthogonal Basis GOB to provide a reduced complexity model known as GOB-Volterra model.

**Keywords**: MIMO Volterra model, identification, modelling, supervised equalization, non linear communication channel.

**Saidi Nabiha** was born in 1981. She received her Electrical Engineering diploma in 2005 from the School of Engineers in Monastir, Tunisia. She is currently preparing her doctorate in the School of Engineers in Monastir about Modelling, identification and equalization of numerical communication channel.

**Ben Amor Anouar** was born in 1981. He received his Electrical Engineering diploma in 2005 from the School of Engineers in Gabes, Tunisia. He is currently preparing his doctorate in the School of Engineers in Monastir about Modelling, identification of non linear systems.

**Hassani Messaoud** is a Professor at the School of Engineers in Monastir, Tunisia. He was graduated from the University of Tunis. He is majoring in process identification and control. Actually his main interest is numerical communication channel equalization using reduced Volterra models.

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**CITE THIS PAPER AS**:

Nabiha SAIDI, Anouar BEN AMOR, Hassani MESSAOUD, **Identification and Supervised Equalization of a MIMO Non-linear Communication Channel**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 18 (2), pp. 149-158, 2009.

**1. Introduction**

Till while ago Volterra models still the most usual and popular way to describe non linear system behaviour as it provides a model linear with respect to its parameters [3]. Truncated Volterra filters constitute a class of non recursive polynomial models without output feedback which guarantees their stability. Such models can approximate any time invariant nonlinear system with fading memory [3], [28]. These models have been successfully applied to a wide variety of engineering problems such as modelling a nonlinear communication channels, biological systems and acoustic noise cancellation [15]. In communication systems, Volterra models have been used for modelling communication channels exhibiting nonlinear behaviours [10] and [14] that is the case of those including amplifiers and optical fiber. Indeed, high power amplifiers, currently used in mobile radio and satellite communication channels, have to operate near their nonlinear region for maximizing the utilization of the available power. However, the main drawback of Volterra models is their complexity due to the high parameter number. To eliminate this disadvantage, two ways can be followed to consider simplified models like Hammerstein or Wiener models or to reduce the number of parameters associated with a Volterra series.

During the last decade, the issue of Volterra model complexity reduction has been addressed in two main different ways. The first is based on the use of Singular Value Decomposition (SVD) of the second kernel and PARAFAC tensor decompositions [7] and [17] of the higher kernels as these latter can be described by tensors. The resulting reduced model known as SVD PARAFAC Volterra model [19], [20], [21]. The second is based on expanding the Volterra kernels on orthonormal basis (OB) such as the Laguerre functions basis [4], [5] and [6] or the Generalized Orthonormal Basis (GOB) [16], [22] and [23]. The complexity reduction depends on the choice of the basis parameter structure such as poles and truncating order. Recently, many approaches are given to identify Multiple-Input- Multiple-Output (MIMO) Volterra models to describe wireless communication channel [11][12][13]. The provided model can be used to design equalizers to restitute the transmitted signals. Channel identification and equalization consist in the retrieval of unknown information about the transmission channel and source signals, respectively. In order to reach a desired quality of service, broadband wireless communication systems classically perform channel identification and/or equalization using pilot symbols, i.e. training sequences composed of a priori known signals.

In this paper we are interested to the identification and supervised equalization of a MIMO non linear communication channel described by a reduced MIMO Volterra model. This reduction is ensured by developing Volterra kernels on Generalized Orthogonal Bases (GOB). The resulting model is used to synthesise the supervised equalizer.

**5. Conclusion**

In this paper we have proposed a MIMO supervised equalizer based on GOB MIMO Volterra model for a Multi Input Multi Output non linear communication channel. The equalizer synthesised is tested in simulation on numerical example to resituate inputs of two input two output non linear communication channel and results are satisfactory. Simulations are carried out to evaluate the equalizer performances and the influence of an additive noise on these performances.

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