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In Chapter 2 the equalizers operated under the assumption of perfect channel estimation, where the receiver always had perfect knowledge of the CIR. However, the CIR is typically time variant and consequently the receiver has to estimate the CIR or the coefficients of the equalizer, in order to compensate the IS1 induced by the channel. for Algorithms have been developed in order to automatically adapt the coefficients of the equalizer directly [ l 181 or by utilizing the estimated CIR [124,125] | Adaptive Wireless Tranceivers L. Hanzo C.H. Wong M.S. Yee Copyright 2002 John Wiley Sons Ltd ISBNs 0-470-84689-5 Hardback 0-470-84776-X Electronic Chapter Adaptive Equalization In Chapter 2 the equalizers operated under the assumption of perfect channel estimation where the receiver always had perfect knowledge of the CIR. However the CIR is typically time variant and consequently the receiver has to estimate the CIR or the coefficients of the equalizer in order to compensate for the ISI induced by the channel. Algorithms have been developed in order to automatically adapt the coefficients of the equalizer directly 118 or by utilizing the estimated CIR 124 125 . These algorithms can be generally classified into three categories which were depicted in Figure 2.1. The first category involves the steepest descent methods where a considerable amount of the pioneering work was achieved by Lucky 126 127 . The second class of adaptive algorithms incorporates the stochastic gradient method which is more commonly known as the Least Mean Square LMS algorithm that was widely documented by Widrow et al. 119-121 . The third and final category includes the Least Square LS algorithms. In this section we shall concentrate on the LS algorithms in particular on the well known Recursive LS RLS or Kalman algorithm 86 . The Kalman algorithm was first formulated by Kalman 86 in 1961. This was followed by the application of the algorithm for adaptive equalizers 104 128-132 Lee and Cunningham 133 extended the adaptive equalizer to QPSK modems which invoked the complex version of the adaptive Kalman equalizer. Adaptive channel estimators utilizing the recursive Kalman algorithm were also researched by Cheung 104 Godard 123 Messe et al. 134 Harun et al. 124 and Shukla et al. 125 . In order to ensure stability and to reduce the complexity variants of the Kalman algorithm have been developed by amongst others by Hsu 135 which was referred to as the Square Root Kalman algorithm and by Falconer
