The paper describes a number of novel and promising techniques for use in the estimation of a flat fading channel with a rapid fading rate, such as occurs in many land mobile radio systems. The algorithms of the various techniques are first presented, followed by the results of a series of computer simulation tests, carried out on these algorithms when operating with an idealised model of the channel. The quantities measured in the tests include the mean-square error in the channel estimate, with the correct detection of the data symbols, and the error rate in the detected data symbols, with perfect channel estimation and also when operating with a channel estimator. The channel estimate considered here is the one- or two-step prediction of the channel that is fed to the detector. A further aim of this investigation is to study the feasibility of the simultaneous transmission of two bandlimited 4-level QAM (quaternary PSK) signals simultaneously over two independent Rayleigh fading channels to a single receiver, where both signals occupy the same frequency band and no coding or diversity techniques are employed to improve the performance, other than differential coding needed to avoid prolonged error bursts. The results obtained suggest that it should be possible to achieve the satisfactory coherent detection of a single Rayleigh-fading 4-level QAM signal, under conditions of even quite severe and rapid fading, but for the simultaneous transmission of two independently Rayleigh-fading signals over the same frequency band, the appropriate diversity techniques will be required to reduce the resultant depth of the fades. The techniques proposed here should make it possible to achieve a significantly more efficient use of bandwidth in cellular mobile radio systems than is possible by conventional methods.