Radar waveform design is facing severe challenges with the development of electronic reconnaissance technology such as intra-pulse feature recognition and signal deinterleaving. Amplitude modulation can improve the anti-identification ability of the waveform by increasing the signal complexity in time domain, which is a newly developing modulation method. A novel radar waveform with joint amplitude-phase coding modulation is proposed in this paper. It improves the complexity of radar waveform through Amplitude-Phase Joint Modulation(APJM), which is promising in anti-reconnaissance. In addition, regarding the processing of this signal, an echo signal processing method combining matched filtering and Compressed Sensing(CS) is proposed, which uses the sparse sampling characteristics of amplitude. The proposed signal processing method effectively improves the detection probability under the condition of a low Signal-to-Noise Ratio(SNR). In the end, simulation results show the effectiveness of the proposed waveform design and processing method of the joint amplitude-phase modulation radar waveform.