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Recently, the idea of compressed sensing (CS) has been used in radar system, and the concept of compressed sensing radar (CSR) has been proposed in which the target scene can be sparsely represented in the range-Doppler plane. With sufficiently incoherent transmission waveform, the target scene can be reconstructed by the technique of CS. With the idea that the transmission waveform can adapt in response to the operational information in cognitive radar system, we propose the notion of adaptive compressed sensing radar (ACSR) whose transmission waveform and sensing matrix can be updated by the target scene information fed back by the recovery algorithm. The methods for optimizing the transmission waveform and sensing matrix separately and simultaneously are both presented to decrease the cross correlations between different target responses. The principle for an ACSR system to synthesize the transmission waveform and sensing matrix matched to the target scene is also investigated. This novel ACSR system offers more degrees of freedom than classical radar system and better recovery performance than the CSR system.