## 2 Related Work on Buffer-Aided CRN
In another recent work [17], isolation between the transmitter and receiver was achieved by using a signal splitter and a single antenna  
The concept of using multiple antennas can be utilized to mitigate the effect of self-interference and to isolate the receiver from the transmitter. This can be realized by integrating antennas of physically-separated/distributed nodes, into one system. While each node will be operating in half-duplex mode, but their receivers will not get saturated due to physical distance from the transmitter. We would apply such a virtual full-duplex (VFD) relay scheme formed by two-half duplex relays in our proposed buffer-aided cognitive relaying scheme. As previously mentioned, this can be seen as the distributed version of full-duplex relaying based on multiple antennas. At each time slot, one of the relays will be in receiving mode, accepting data from the source, while another relay will be in transmitting mode, sending decoded data to the destination, in the same time slot. The role of the relays is swapped in each time interval. This relaying operation is known as successive relaying [18; 19]. This way, the source can send a new message to the destination in every time slot as if a full-duplex relay was employed. It is thought-provoking to notice that the network topology so formed seems closer to the well-known diamond relay network, with the addition of one interfering link between the two relays, as shown in Figure 2. Here, one relay will be in transmit mode, while the other will be in receiving. As in full-duplex relaying, one of the main bottleneck is self-interference, thus successive relaying is handicapped by inter-relay interference. For the given successive relaying operation, an upper bound on the achievable rate is easily obtained as follows:  
\[C(SNR)=B\times log_{2}(1+SNR)\, \tag{1}\]  
where \(C\) is the channel capacity and \(B\) is the bandwidth; this will be referred to as the successive upper bound. Figure 3 gives an overview of full-duplex and virtual full-duplex, i.e., the successive relaying also discussed in [9]. Figure 2: Virtual full-duplex relaying concept. Various coding schemes can also be used to improve performance under interference. For non-interfering relaying models, it was deliberated in [20] that the full-duplex (FD) scheme performs better than others. This can be realized by locating each relay far from others or in a fixed relay network using highly-directional antennas [21, 22]. Again, it was argued in [23] that since the source has accurate information of the relays, transmit signal, and inter-relay interference channel \(\gamma_{RR}\), by using the dirty paper coding (DPC) method, it can completely remove the known interference at the desired receiver. Therefore, successive relaying in combination with DPC can mimic the performance of an ideal full-duplex relay in a two-hop network with each relay still operating in half-duplex mode. However in this work, a multi-hop network has not been well studied. In [24], the authors presented a multi-hop decode and forward (DF) scheme and derived a closed-form expression. A multiple-hop DF scheme cannot outperform an ideal full-duplex scheme because of the delays involved in decoding the message first. Thus, the resulting scheme achieves a gap that scales logarithmically with the number of nodes. Furthermore, the proposed quantize-map-and-forward (QMF) scheme is a special case of short-message NNC(SNNC) [25] and has lower decoding delay and complexity through successive decoding instead of joint simultaneous decoding as in [26, 27]. In addition, it was shown that compute and forward (CoF) could achieve the upper bound within a 0.5-bit duration if the inter-relay interference level tends to be an integer. An upper bound was also derived that was independent of the number of relays (K) and coincided with the successive upper bound \(log_{2}(1+SNR)\).