The system architecture for cognitive radar-communication (CRC) transceiver is proposed. A cognitive waveforms design approach, which is suitable for simultaneously performing both data communication and target detection, is presented. This approach aims at estimating target scattering coefficient (TSC) from the radar scene and facilitating high data rate communications. In order to minimize the mean square error (MSE) of the TSC, a convex cost function is established. The peak to average power ratio-(PAPR-) constrained optimal solution is achieved by applying the Kalman filtering-based strategy to design the set of ultrawideband (UWB) transmission pulses and embed into them the information data with the M-ary position phase shift keying modulation technique. In addition to theoretical considerations, the simulation results show an improvement in target scattering coefficient (TSC) estimation and target detection probability as the number of iterations increases, while still transmitting data rates in the range of several Mbps with low bit error rates between CRC transceivers.