The paper considers the receiver of a digital data-transmission system which is fed from a number of different transmitters in an arrangement where separate short messages are transmitted, in turn, from the different transmitters, with arbitrary time gaps between adjacent messages when no signal is transmitted. The signals received from the different transmitters may be subjected to widely different levels of amplitude and phase distortions, so that a training or synchronising signal must be transmitted at the start of each message, during which the receiver must obtain a reasonably accurate estimate of the sampled impulse response of the channel. The latter may then be used for the initial adjustment of an adaptive equaliser, but this or any other application of the channel estimate is not studied here. Since a satisfactory estimate of the sampled impulse response of the channel is needed in the shortest possible time, in order to maximise the transmission rate of useful data over the duration of the message, an optimum, estimation process must be used together with the training signal that enables the best estimate of the channel response to be obtained. The paper derives the maximum-likelihood estimation process for the sampled impulse response of the channel, which, under the assumed conditions, gives the unbiased estimate with the minimum mean-square error. The paper then determines the conditions that must be satisfied by the training signal to minimise the mean-square error in the channel estimate, and lists a number of such optimised training signals using two-, three-and four-level symbols. Finally, the new systems are compared with the more promising of the techniques previously described.