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In this paper, we derive a maximum-likelihood (ML) decoder of the differential data in a decode-and-forward (DF)-based cooperative communication system utilizing uncoded transmissions. This decoder is applicable to complex-valued unitary and nonunitary constellations suitable for differential modulation. The ML decoder helps to improve the diversity of the DF-based differential cooperative system using an erroneous relaying node. We also derive a piecewise linear (PL) decoder of the differential data transmitted in the DF-based cooperative system. The proposed PL decoder significantly reduces the decoding complexity, as compared with the proposed ML decoder, with no significant degradation in the receiver performance. Existing ML and PL decoders of the differentially modulated uncoded data in the DF-based cooperative communication system are only applicable to binary modulated signals like binary phase shift keying and binary frequency shift keying, whereas the proposed decoders are applicable to complex-valued unitary and nonunitary constellations suitable for differential modulation under uncoded transmissions. We derive a closed-form expression of the uncoded average symbol error rate (SER) of the proposed PL decoder with <formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula> phase-shift keying constellation in a cooperative communication system with a single relay and one source&#x2013;destination pair. An approximate average SER by ignoring higher order noise terms is also derived for this setup. It is analytically shown on the basis of the derived approximate SER that the proposed PL decoder provides full diversity of second order. In addition, we also derive the approximate SER of the differential DF system with multiple relays at asymptotically high signal-to-noise ratio (SNR) of the source&#x2013;relay links. It is shown by simulations that the proposed PL decoder in the differential DF cooperative system with more than one relay also achieves the maximum possible diversity.

Decode-and-Forward-Based Differential Modulation for Cooperative Communication System With Unitary and Nonunitary Constellations