## 2 Related Work  
For multi-hop transmissions, there are two typical relaying schemes: AF and DF relaying. In the AF relaying scheme, a relay amplifies a signal received from a previous node (or a source) and then forwards the amplified signal to a next node (or a destination). Because of the simple AF relaying operation, low complexity is required at the relay, but there is a noise propagation problem, which can induce a certain performance degradation. On the other hand, in the DF relaying scheme, a relay decodes and re-encodes a received signal from a previous node and then forwards the encoded signal to a next node. Because of the decoding operation in the DF relaying scheme, the noise propagation does not occur in multi-hop transmissions. However, DF relaying needs a relay with the higher complexity than AF relaying for its decoding and re-encoding operations. To achieve spatial diversity in multi-hop relaying systems, the multi-hop cooperation techniques have been widely studied, where multiple cooperative nodes between a source and a destination are used as virtual antennas of the source or the destination. A simple but efficient multi-hop cooperation scheme is the BPS. In [11], the BPS schemes were proposed for dual-hop cooperative systems with AF relaying as well as DF relaying in order to achieve full spatial diversity. Also, their outage probability and diversity-multiplexing trade-off were analyzed for Rayleigh and Ricean fading channels. In [23; 24], average bit error rate of the BPS was evaluated for multi-hop DF cooperative systems in Rayleigh and Nakagami-\(m\) fading channels, respectively, where it was noted that more than two hops for each of every path were considered. For green wireless communication, energy-efficient radio technologies have been presented in [12], where the issues on channel feedback information and CSI feedback overhead have been addressed to improve the energy efficiency. In [13], it was shown that the spectral efficiency and energy efficiency can be affected by channel feedback information in wireless multiple-input multiple-output transmission. In [14], the reduction in the CSI feedback overhead was studied to enhance the energy efficiency in downlink orthogonal frequency division multiple access. In [15], the relay and user selection scheme with one-bit CSI feedback was proposed for energy-efficient multiple-relay and multiple-user systems, and also exact expressions for outage probability and feedback requirement were provided to quantify the trade-off between complexity and performance. As mentioned previously, CSI feedback has been widely studied to achieve high energy efficiency in wireless communication. The operation of the BPS is done by global CSI, and thus full CSI feedback is required. Hence, the CSI feedback for BPS can be limited from the viewpoint of energy efficiency for wireless communication systems. Therefore, in [16], the PPS was introduced in dual-hop AF relaying systems to utilize local CSI for path selection, and the statistical analysis of its end-to-end SNR was presented for Rayleigh fading channels. The PPS simply uses the CSI for one of two hops on each path. Thus, its end-to-end performance can be very limited. To improve the performance of the PPS, in [17], the efficient PPS was hence proposed for dual-hop AF relaying systems, and its outage probability was investigated in Rayleigh fading channels. In [17], it was also shown that the efficient PPS can provide the similar outage performance to the BPS under non-identical channel conditions in two-hopnetworks. In [25; 26], average bit error rate and outage probability of the efficient PPS were analyzed for dual-hop DF relaying systems in Rayleigh and Nakagami-\(m\) fading channels. The above-mentioned works assumed ideal CSI feedback, i.e., no feedback delay and no channel estimation error. However, in real wireless communication environments, there are the time delay between the CSI feedback and the data transmission as well as the channel estimation error at a receiver.