Complex Extended Kalman Filters for Training Recurrent Neural Network Channel Equalizers

The Kalman filter was named after Rudolph E. Kalman published in 1960 his famous paper (Kalman, 1960) describing a recursive solution to the discrete-data linear filtering problem. There are several tutorial papers and books dealing with the subject for a great variety of applications in many areas from engineering to finance (Grewal & Andrews, 2001; Sorenson, 1970; Haykin, 2001; Bar-Shalom & Li, 1993). All applications involve, in some way, stochastic estimation from noisy sensor measurements. This book chapter deals with applications of Complex Valued Extended Kalman Filters for training Recurrent Neural Networks particularly RTRL (Real Time Recurrent Learning) neural networks. Gradientbased learning techniques are usually used in back-propagation and Real-Time Recurrent Learning algorithms for training feed forward Neural Networks and Recurrent Neural Network Equalizers. Known disadvantages of gradient-based methods are slow convergence rates and long training symbols necessary for suitable performance of equalizers. In order to overcome such problems Kalman filter trained neural networks has been considered in the literature. The applications are related to mobile channel equalizers using realistic channel responses based on WSSUS (Wide-Sense Stationary Uncorrelated Scattering) models. The chapter begins with a detailed description showing the application of Extended Kalman Filters to RTRL (Real Time Recurrent Learning) neural networks. The main equations are derived in a state space framework in connection to RTRL training. Then applications are envisioned for mobile channel equalizers where WSSUS models are adequate for handling equalization in presence of time-varying channels. This chapter proposes a fully recurrent neural network trained by an extended Kalman filtering including covariance matrices adjusted for better filter tuning in training the recurrent neural network equalizer. Several structures for the Extended Kalman Filter trained equalizer are described in detail, and simulation results are shown comparing the proposed equalizers with traditional equalizers and other recurrent neural networks structures. Conclusions are drawn in the end of the chapter and future work is also discussed.