  In energy harvesting (EH) relay networks, the coherent communication requires
accurate estima- tion/tracking of the instantaneous channel state information
(CSI) which consumes extra power. As a remedy, we propose two noncoherent EH
relaying protocols based on the amplify-and-forward (AF) relaying, namely,
power splitting noncoherent AF (PS-NcAF) and time switching noncoherent AF
(TS-NcAF), which do not require any instantaneous CSI. We develop a noncoherent
framework of simultaneous wireless information and power transfer (SWIPT),
embracing PS-NcAF and TS-NcAF in a unified form. For arbitrary M-ary
noncoherent frequency-shift keying (FSK) and differential phase- shift keying
(DPSK), we derive maximum-likelihood detectors (MLDs) for PS-NcAF and TS-NcAF
in a unified form, which involves integral evaluations yet serves as the
optimum performance benchmark. To avoid expensive integral computations, we
propose a closed-form detector using the Gauss-Legendre approximation, which
achieves almost identical performance as the MLD but at substantially lower
complexity. These EH-based noncoherent detectors achieve full diversity in
Rayleigh fading. Numerical results demonstrate that our proposed PS-NcAF and
TS-NcAF may outperform the conventional grid- powered relay system under the
same total power constraint. Various insights which are useful for the design
of practical SWIPT relaying systems are obtained. Interestingly, PS-NcAF
outperforms TS-NcAF in the single-relay case, whereas TS-NcAF outperforms
PS-NcAF in the multi-relay case.
