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We consider a multicarrier chirp-based waveform for joint radar and communication (JRC) systems and derive its time discrete periodic ambiguity function (AF). A set of waveform parameters (e.g., chirp rate) can together with the transmit sequence be selected to shape the AF to be thumbtack-like, or to be ridge-like, either along the delay axis or the Doppler axis. We demonstrate how these shapes are applicable for different use cases, e.g., radar target detection or time- and frequency synchronization. The results show that better signal detection performance than OFDM and DFT-s-OFDM can be achieved on channels with large Doppler shift. Furthermore, it is shown how transmit sequences can be selected in order to achieve low peak-to-average-power-ratio (PAPR) of the waveform.