In this paper, we derive a maximum likelihood (ML) decoder of thedifferential data in a decode-and-forward (DF) based cooperative communicationsystem utilizing uncoded transmissions. This decoder is applicable tocomplex-valued unitary and non-unitary constellations suitable for differentialmodulation. The ML decoder helps in improving the diversity of the DF baseddifferential cooperative system using an erroneous relaying node. We alsoderive a piecewise linear (PL) decoder of the differential data transmitted inthe DF based cooperative system. The proposed PL decoder significantly reducesthe decoding complexity as compared to the proposed ML decoder without anysignificant degradation in the receiver performance. Existing ML and PLdecoders of the differentially modulated uncoded data in the DF basedcooperative communication system are only applicable to binary modulatedsignals like binary phase shift keying (BPSK) and binary frequency shift keying(BFSK), whereas, the proposed decoders are applicable to complex-valued unitaryand non-unitary constellations suitable for differential modulation underuncoded transmissions. We derive a closedform expression of the uncoded averagesymbol error rate (SER) of the proposed PL decoder with M-PSK constellation ina cooperative communication system with a single relay and onesource-destination pair. An approximate average SER by ignoring higher ordernoise terms is also derived for this set-up. It is analytically shown on thebasis of the derived approximate SER that the proposed PL decoder provides fulldiversity of second order. In addition, we also derive approximate SER of thedifferential DF system with multiple relays at asymptotically highsignal-to-noise ratio of the source-relay links.
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