Fast start-up equalization with periodic training sequences

Prior knowledge of the typical signal spectrum at the equalizer input can be used to speed up the convergence of a least mean-square error nonrecursivc equalizer by placing a weighting matrix in the path of the coefficient (tap gain) corrections. When the training sequence is periodic with period equal to the time-spread of the nonrecursive equalizer, the weighting matrix is not only symmetric and Toeplitz, but also circulant. Thus the fast update algorithm can be implemented by inserting a single nonrecursive filter in the path of the periodic input sequence before it is used for computing coefficient correction terms. The stability and convergence of the fast start-up algorithm with "stochastic" update in the presence of noise are examined. In the absence of prior information about the channel characteristics, this fast converging training procedure may be easily extended to a one-shot method of equalizer setup. The one-shot method, which involves estimation of the inverse filter from one period of the received signal, proves to be robust in the presence of noise. Simulation resuits are given for class I partial-response and Nyquist (20 percent roll-off raised cosine) data transmission systems with quadrature amplitude modulation over a dial-up telephone connection.