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In the cooperative communication system, due to the distributed nature of the relay system, multiple carrier frequency offsets (CFOs) at different relays and multipath channels between relays cause the whole cooperative communication channel to be a doubly time-frequency selective channel. Zero-Padding Orthogonal Frequency Division Multiplexing (ZP-OFDM), which has been adopted for some wideband communication standards and applications, possesses a unique linear structure. In this paper, we use a cooperative tall Toeplitz scheme to achieve the full cooperative and multipath diversity, and combat the CFOs simultaneously. Importantly, this full diversity scheme only requires linear equalizers (LEs), such as zero-forcing (ZF) and minimum mean square error (MMSE) receivers, an issue which reduces the system complexity. Theoretical analysis of the proposed cooperative tall Toeplitz scheme is provided based on the analytical upper bound of the channel orthogonality deficiency derived in this paper. The above mentioned theoretical issues are verified by the simulation results as well.