equivalent to a phase rotation u03c6n e is a phase value of the same predetermined sequence as for the transmitting modem which is provided by sequence block 24 but at baud time n the signal vn which is processed corresponds in fact to the symbol which has been processed in the transmitting modem at baud time n 1 where 1 is a time delay induced by the transmitting modem processing the telephone line and the receiving modem processing the phase rotated signal issued from multiplier 22 is then decoded in data decoding device 26 which determines the point of the constellation which is estimated to correspond to the point of the constellation which has been originally transmitted the estimate a and the phase rotated signal are used as inputs in the error circuit 28 which computes the mean square error in order to minimize the expression equ1 over a number of baud times if the value found from the above expression is sufficiently important proving that the receiving modem is out of synchronization because the time delay has changed the phase sequence provided by sequence block 24 is advanced or delayed in order to subsequently minimize the error there is a minimum error due to noise it should be noted that in addition correct equalizer updating requires multiplication of the decision error by e j u03c6 n e before applying it in the update algorithm ideally the phase sequence should be designed such that signal decoding is incorrect in all other shift positions of the sequence this generates a large mean squared.